%% v20.2 (August 22, 2026): THE TWENTY-TWO-PAPER DISCLOSURE.
%%   Every reference to a companion paper now carries that companion's own
%%   permanent page -- e.g. https://preprints.arisaka-gg.org/a3-bi6/ -- in place
%%   of the single program address every reference used to share.  Papers that
%%   are not cohort members keep their own locators: the 1985 Kamiokande paper,
%%   the three PsyArXiv works, and the CKM paper still in preparation are
%%   untouched.
%%   Sixty-four companion references carried no locator at all and now carry
%%   one, in this paper's own format.
%%   A11-Origin is posted with the rest, so the twenty-one papers that called it
%%   "in preparation" no longer do, and fifteen printed statements about its
%%   status -- footnotes, table rows, captions and the staged-submission
%%   paragraph -- were corrected to match.  The author's own reservation about
%%   its upper layers is kept verbatim; only its posting status changed.
%%   Two malformed sentences were repaired: A3's "UCLA preprint,." and P2's
%%   title running into the identifier.
%%   No claim, value, figure, table or grade differs from the previous version.
%% ===================================================================
%% v15.17 (August 20, 2026): Coherence pass before the permanent-DOI deposit.
%%   (1) Part count: the Stage-4 restructure took the paper from fifteen Parts
%%       to twenty-one and the roadmap figure was rebuilt (its caption already
%%       reads "Roadmap of the twenty-one Parts" and it draws Parts I-XXI),
%%       but seven printed sites still said fifteen.  Six now read twenty-one;
%%       the seventh, in the Conclusions, reads sixteen -- the number of Parts
%%       that precede it.  Label identifiers were NOT renamed.
%%   (2) Cross-paper coherence map: the table has twenty-one \href row heads,
%%       and prose, caption, short caption and the "Eighteen rows" sentence
%%       said eighteen.  All four now read twenty-one, which is also the
%%       twenty-two-paper cohort less A11 itself.
%%   (3) The branch enumeration named two of the four branches and six of the
%%       eleven non-A companions.  All four branches and all eleven members
%%       are now named, read off A1's cohort plate.
%%   (4) The equal-temperament metaphor printed 2^(19/12) ~ 3^7/2^4; the two
%%       sides are 2.9966 and 136.69.  Replaced by the cycle-of-fifths
%%       closure it was reaching for, (3/2)^12 ~ 2^7 (129.75 vs 128).
%%   No claim changed; no numerical prediction touched.
%% ===================================================================
%% v15.15 (August 18, 2026) -- BT-1: THE BIBLIOGRAPHY TITLE PASS.
%%
%% EVERY COMPANION IS NOW CITED BY THE TITLE PRINTED ON ITS OWN COVER.
%%
%% 7 of this paper's 21 cohort bibitems named a companion by a title
%% that companion does not carry.  Measured, not recalled: the truth is
%% each paper's printed cover, read from its deployed PDF; the markup is
%% each paper's own \GGTitlePage arguments, so the title sets in the same
%% faces the cover does.  Cohort-wide the pass touched 233 sites in 23
%% papers, and afterwards all 477 companion bibitems agree with their
%% covers.
%%
%% TWO CLASSES, and the difference is worth recording.
%%   RENAME PROPAGATION -- a paper was retitled and its citers were never
%%   updated.  The signature is the same wrong string in many papers:
%%   A6-GRIP's retired subtitle "Gi-4 Pillars (TGG, CCI, GFF, and GRIP) as
%%   the Geometric Backbone of GDOS" stood in seventeen papers, and
%%   C2-Hubble's retired "Two Hubble Constants as the Two Ends of One
%%   Six-Dimensional Geometric Corridor" in seventeen more.  W1-Symmetry,
%%   W2-Holonomy, P5-CPT and P6-Proton were all renamed and none of their
%%   citers followed.
%%   HOUSE SHORTHAND -- P5-CPT, W1-Symmetry and W2-Holonomy each wrote
%%   their own descriptive short form for every companion ("GDOS: the
%%   Boundary Law", "The Kaxion", "The Cosmological Inventory of the
%%   Descent").  Readable in prose and useless in a reference list,
%%   because none of those strings finds the paper.
%%
%% ONLY TITLE TEXT CHANGED.  Author blocks, years, availability lines,
%% Relevance notes, keys, labels and \emph/\textit wrappers are asserted
%% unchanged in count; the changelog comments are asserted BYTE-IDENTICAL.
%% ====================================================================
%% ===================================================================
%% v15.14 (August 18, 2026) -- STAGE 6: PART V (CHEMISTRY, LAYERS 4-6).
%%
%% NEW PROSE (three beats, matching the Stage-5 frame used for Cosmos):
%%   - \section{The leading theories and models today} (sec:chem-leading-
%%     theories): what prebiotic chemistry has established since 1953 --
%%     Miller, Bada's reanalysis, Murchison, 67P, Ryugu, the four routes,
%%     the RNA world, the literature on why twenty, homochirality.
%%   - \section{What they leave unsolved} (sec:chem-unsolved): selection
%%     not synthesis; amplification not origination; twenty explained
%%     backward; the water paradox.
%%   - \section{How far the derivation reaches, and where it stops}
%%     (sec:chem-reach): per-layer, what is inherited-and-derived vs what
%%     is asserted awaiting a named lemma, plus tab:chem-reach.
%%   - Layer 6 gains the interpretation, metaphor and kill-criterion
%%     paragraphs it never had (layers 4 and 5 carried all three).
%%
%% REPAIRS against this paper's own text (no new positions taken):
%%   R1  eq:molecular-gga listed eleven species incl. HCN while the row's
%%       prose, its kill criterion, the codebook and A6 all say a closed
%%       QUARTET.  Six statements to one; the equation was the outlier.
%%   R2/R3  T-Molecular-Closure and T-Polymer-Closure were attributed to
%%       A5-GDOS.  A5 contains neither (verified: 0 occurrences).  Both
%%       are A11's own theorem names; custody restated honestly.
%%   R4  T-QUA was glossed as a "polymer-substrate theorem", T-EP as a
%%       "chemiosmotic-substrate theorem", T-HAL as a "hippocampal-
%%       substrate theorem".  All three replaced by the glosses A11's own
%%       sec:qi4-admissibility-filter gives, which agree with A7.
%%   R5  the counterfactual walk had no layer-6 entry, two subsections
%%       titled layer 8, a label ssec:counterfactual-L6 on a layer-8
%%       section, and layer 6/7/8 confused in the L5 text.
%%   R6  "Ryugu ribose" is not the published detection.  Ryugu carries
%%       uracil (Oba 2023); the meteoritic ribose is Murchison and NWA
%%       801 (Furukawa 2019).  Both now cited.
%%   R7  codebook row 4's "one-, two-, three-, four-atom-heavy" is wrong
%%       on any reading; replaced by the hydride/oxide description the
%%       row's own prose uses.
%%   R9  the layer-5 theorem said the count 20 was "forced" and the
%%       "unique closure compatible with" layer 8.  Objection 28 of this
%%       same paper concedes it is "not-yet-a-theorem" and "a forced
%%       relation" only.  Theorem brought into line with the objection.
%%   R10 codebook row 5 quoted "20 = 4^3/3.2" as a quantitative anchor.
%%       3.2 IS 64/20; the anchor was circular.  Replaced by the
%%       observed canonical count, which is falsifiable.
%%
%% BIBLIOGRAPHY: 22 new entries in list B, each with a Relevance note --
%%   Miller, Bada, Kvenvolden, Furukawa, Altwegg, Oba, Wachtershauser,
%%   Martin-Russell, Powner, Ferris, Pearce, Damer-Deamer, Gilbert,
%%   Joyce, Weber-Miller, Trifonov, Higgs-Pudritz, Philip-Freeland,
%%   Doig, Frank, Soai, Blackmond.
%% ===================================================================
%% ===================================================================
%% v15.13 (August 18, 2026) -- STAGE 5: PART IV (COSMOS) REBUILT ON THE
%% EVIDENCE THE COHORT ALREADY PUBLISHES.
%%
%% The proposal's three beats for every domain Part: the leading theories
%% today; what they leave unsolved; what GG-Theory says and HOW FAR THE
%% DERIVATION REACHES.  Part IV already carried the third beat's core --
%% three layer theorems on the paper's uniform twelve-paragraph template --
%% AND THAT CONTENT IS NOT TOUCHED.  What was missing is the frame around
%% it and the origin-of-matter chain.
%%
%% MEASURED BEFORE WRITING: this paper mentioned the baryon asymmetry twice,
%% baryogenesis twice, leptogenesis three times, CP violation three times,
%% and matter-antimatter NOT AT ALL -- while P5-CPT carries leptogenesis 86
%% times, P6-Proton carries baryon number 92 times, and P3-Electron carries
%% its six oldest questions 31 times.  The evidence was published and this
%% paper was not using it.
%%
%% SEVENTEEN NUMBERS WERE VERIFIED IN THEIR OWNING PAPERS BEFORE BEING
%% WRITTEN, by pattern against the deployed sources, and every quotation was
%% located and copied rather than recalled.  ONE QUOTATION DID NOT VERIFY:
%% an earlier report had A9 saying "None of the three statements is a
%% derivation of the baryon asymmetry."  A9's actual sentence is "None of
%% the three statements ABOVE is a derivation... Each is what follows GIVEN
%% the regulator identification..., and that identification is open."  The
%% second clause is the load-bearing one and the paraphrase had dropped it.
%% The verified sentence is what appears.
%%
%% THE GRADED CHAIN IS THE POINT.  Four links, and they are NOT of equal
%% strength: the prohibition is exact arithmetic (D); the winding and the
%% one surviving door are derived-conditional on two named pins; the
%% selection -- leptogenesis forced rather than preferred -- is a corollary
%% at that grade; and the NUMBER is the weakest link, two roads that
%% disagree by 16.4 percent with the gap named once and its owner
%% unidentified.  A table sets all seven rows out with grade and falsifier.
%% NO LINK IS GRADED UPWARD relative to its source.
%%
%% NEW PROSE: three sections and one table.
%% ===================================================================
%% ===================================================================
%% v15.12 (August 18, 2026) -- THE PART-STRUCTURE FLOWCHART REDRAWN FOR
%% TWENTY-ONE PARTS.
%%
%% Stage 4 split fifteen Parts into twenty-one.  This plate DRAWS the Part
%% structure, and v15.8 left its fifteen nodes hard-coded for exactly this
%% moment -- but until redrawn it said fifteen, which is a defect v15.11
%% introduced, reported, and did not hide.
%%
%% THE CAPTION WAS WORSE THAN THE PLATE.  Its \ref-based numerals had
%% silently updated at the split while its hard-coded ones had not, so one
%% sentence carried "Parts~VI--VIII" (stale) beside
%% "Part~\ref{part:closure}" (correct) -- the two halves of the same
%% sentence disagreeing.  Every group that can be a \ref is now one.
%%
%% TWO LAYOUT DECISIONS, BOTH FORCED BY MEASUREMENT.  Twenty-one nodes at
%% the old 16 mm pitch stand 268 mm tall, which at this plate's fixed width
%% renders 224 mm -- taller than the text block leaves once the caption is
%% set, so the figure would not fit its own page.  So two groups go
%% three-across, in the device the plate already used for its cascade row:
%% the six DOMAIN Parts (IV-IX) as two rows of three, and the back matter
%% (XIX-XXI) as one, with Discussions/Conclusions/Coda sharing a row.
%% Total 212 mm, inside the old plate's 220 mm.
%%
%% THE SERPENTINE THROUGH THE DOMAIN PARTS IS DELIBERATE: IV to V to VI
%% along the first row, down to VII, then VII to VIII to IX along the
%% second.  That is the cascade's own order.  A fan would have said the six
%% domains are parallel when they are sequential.
%%
%% TWO THINGS THE RENDERED PAGE TAUGHT.  The first draft drew the closing
%% run with two arrows that SKIPPED Parts -- closure straight to
%% Conclusions, Conclusions straight to Common Objections -- which said the
%% run is not sequential when it is; redrawn as the same serpentine the
%% domains use.  And the phase labels sat at 92 mm, not 80: at 80 the
%% 'Inputs' label was overlapped by the Part II box, whose content runs
%% wider than its 80 mm minimum.  THAT COLLISION IS IN THE OLD PLATE TOO,
%% in every build since the flowchart was drawn, and no gate item sees it.
%%
%% ONE THING THE BUILD TAUGHT: the natural style name for the six domain
%% nodes, 'domain', IS A RESERVED TIKZ KEY -- pgfkeys uses /tikz/domain for
%% plot ranges -- and six 'requires a value' errors came back on the first
%% build.  Renamed to 'ggdom'.  No gate item would have caught the visual
%% consequence if the errors had been warnings instead.
%%
%% NEW PROSE: none.  Node text is the Part titles the document now carries.
%% ===================================================================
%% ===================================================================
%% v15.11 (August 18, 2026) -- STAGE 4: THE SPLIT INTO PARTS IV-X.
%% STRUCTURE ONLY.  NO PROSE IS WRITTEN AND NO PROSE IS REMOVED.
%%
%% Part IV carried fifteen sections and 272 kB of source.  Six new Part
%% boundaries turn it into seven:
%%     IV    Cosmos: layers 1-3      (with the shared inputs and the engine)
%%     V     Chemistry: layers 4-6
%%     VI    Life: layers 7-10
%%     VII   Multicellularity: layers 11-13
%%     VIII  Mind: layers 14-17
%%     IX    Articulation: layer 18
%%     X     The cascade as one ledger
%%
%% DOWNSTREAM PARTS SHIFT +6 -- today's V-XV become XI-XXI, twenty-one Parts
%% -- AND NOT ONE PRINTED NUMERAL IS TOUCHED, because v15.8 converted all
%% 328 internal Part references to \ref.  That conversion was done for this
%% ship, and this is where it pays.
%%
%% WHAT IS NOT DONE HERE, AND WHY.  The proposal's table marks two sections
%% for REMOVAL -- "Constitutional inputs" (11.4 kB) and "The cascade engine
%% in four equations" (6.6 kB) -- and two more for ABSORPTION into one
%% paragraph each.  THAT IS ABOUT SIX PAGES OF THE AUTHOR'S PROSE DELETED.
%% It is section 14.2 by any reading, and it is not "structure only" by any
%% reading either.  The sections stay, opening Part IV as its introduction.
%% The proposal's move of the Extended reader's guide (24 kB) to Part I is
%% likewise not made: this ship is meant to be verifiable as a pure
%% renumbering, and a 24 kB move across nine Parts is not that.  Both are
%% batched for the author.
%%
%% NEW PROSE: none.  Six \GGPart lines and six labels.
%% ===================================================================
%% ===================================================================
%% v15.10 (August 18, 2026) -- STAGE 3: THE BOUNDARY LAW PART III
%% DESCENDS ON, STATED BEFORE THE DESCENT.
%%
%% THE PROPOSAL SET THREE ITEMS FOR PART III AND ONE OF THEM WAS ALREADY
%% DONE.  It called for a Qi-4 section on the ground that "Part IV's layer
%% theorems each invoke 'the unique valley admissible to four consistency
%% filters' and this paper never introduces the filters."  MEASURED: THE
%% PAPER INTRODUCES THEM TWICE.  Part V carries a full section
%% (sec:qi4-admissibility-filter) with the four pillars pillar by pillar,
%% their four roles, a mutual-forcing subsection, a table, and Theorem
%% T-Qi4-GGA-Admissibility; Part III's Move 4 introduces them again in
%% miniature and names all four correctly.  A third treatment would have
%% duplicated work already on the page.  The section was SEARCHED FOR
%% BEFORE IT WAS WRITTEN, which is charter section 6 in its most literal
%% form -- a gap re-asserted is a gap unverified.
%%
%% WHAT WAS ACTUALLY MISSING is the proposal's FIRST item, and it was
%% missing exactly as described.  Part III opens on "Step 1 --- SSB to
%% Attractor" and then descends for a hundred and fifty pages on a law it
%% never states.  GDOS appears in this paper as an inheritance paragraph
%% (Part II), a subsection inside the four-move anatomy (Part III), an
%% openness section (Part XI) and a primer (Part XIII) -- nowhere at the
%% depth the cascade's own dynamics deserve.
%%
%% SO THIS SHIP WRITES THAT SECTION, at the head of Part III, from A5-GDOS
%% v15.26, carrying A5's eleven theorems, the Lyapunov lemma under T5 that
%% A5 calls the constitutional source of the second law, AND THAT LEMMA'S
%% SCOPE LIMIT -- when circulation is present the functional is no longer
%% strictly monotone, though total entropy production stays non-negative
%% and the system can settle into a non-equilibrium steady state.  EVERY
%% LAYER OF THIS PAPER'S CASCADE ABOVE THE ATOMIC RUNGS IS SUCH A STEADY
%% STATE, so the caveat is load-bearing here and is stated where the
%% descent begins rather than in a footnote after it.
%%
%% The section also carries the three custody corrections v15.6 made in the
%% A5 inheritance paragraph: the GFF is A6's object, the SeOS class is
%% A7's, and the Three-Gauge reading is this paper's own.
%%
%% AND ONE NAVIGATION REPAIR: Move 4 names the four pillars and never says
%% where they are proved.  One sentence, with the two references.
%%
%% NEW PROSE: one section and one sentence.
%% ===================================================================
%% ===================================================================
%% v15.9 (August 18, 2026) -- STAGE 2 COMPLETED: THE FOUR FOUNDATION
%% SUMMARIES NAMED, AND A4-EUCLID WRITTEN AT DEPTH.
%%
%% 1. The two deep sections of Part II are retitled to NAME the paper each
%% summarizes, so Part II can be navigated by source: "What GG-A1 settles"
%% and "What GG-A2 and GG-A3 settle".
%%
%% 2. Section 9 gains a short orientation paragraph saying which of its
%% subsections come from A2 and which from A3.
%%
%% 3. A NEW SECTION summarizes A4-EUCLID at the same depth as the other
%% three.  The proposal named it as "the one currently missing at this depth
%% and the one a broad reader most needs", and it was: A4 had one paragraph
%% in the inheritance run and nothing at summary depth.
%%
%% WHAT THIS SHIP DELIBERATELY DOES NOT DO.  The proposal implies a physical
%% reordering of section 9 into an A2 half and an A3 half.  THE MATERIAL IS
%% INTERLEAVED AND NO CONTIGUOUS SPLIT EXISTS -- the order runs Historical
%% background, the 75-year anomaly arc, the stage, BI-6, Quadruple Gauge
%% Locking, four-forces-one-pipe, the OGN-5 integers, which is A2, A3, A2,
%% A3, A2, A2, A3.  Two block moves would be needed, AND THE 75-YEAR-ARC
%% SUBSECTION OPENS "The six-dimensional bulk inherits a SECOND, equally
%% important lineage" -- whose antecedent is the Historical background
%% subsection directly above it.  Moving it strands that sentence, and
%% repairing the sentence is prose the author did not ask for.  So the
%% sources are NAMED where they cannot be SEPARATED without damage.
%%
%% NEW PROSE IN THIS SHIP: one orientation paragraph and one section, both
%% written from A4-Euclid v15.42 and from this paper's own A4 inheritance
%% paragraph as repaired in v15.6.  Every claim in the new section is A4's,
%% at A4's grade, including both of A4's scope clauses and A4's own
%% statement about the electron anomaly.
%% ===================================================================
%% ===================================================================
%% v15.8 (August 17, 2026) -- THE PART-REFERENCE CONVERSION.
%%
%% The item owed from Stage 1, done now because Stage 4 splits the Parts and
%% a split over hard-coded numerals re-points four hundred of them by hand.
%%
%% 403 hard-coded Part references; ZERO used \ref; NO \label{part:...}
%% existed.  Part~X is a prefix of Part~XI through Part~XV, so no
%% find-and-replace is safe, and a mis-pointed Part reference FAILS NO GATE.
%%
%% THE HARD PART IS THE CLASSIFICATION, NOT THE REPLACEMENT.  A Part
%% reference here may belong to a COMPANION -- "A9's Part XI", "A10 Part X"
%% -- and those must not become references to THIS paper's Parts.  And where
%% the numerals coincide the page-for-page check CANNOT SEE the error: A9's
%% Part XI and this paper's Part XI both print "Part XI", so a wrong
%% conversion is invisible in the artifact and wrong in the PDF link.
%%
%% SO THE ALLOWED SET IS ENUMERATED, NOT INFERRED:
%%   55 sites carry a companion macro IMMEDIATELY before the reference
%%      (modulo "'s" and a \citep), by a tight adjacency rule;
%%    7 further sites were HAND-CLASSIFIED from a printed borderline review
%%      -- the "its Part~XI" / "its Part~X" possessives, and three sites
%%      where the owner is named a clause earlier;
%%  341 sites are this paper's own and are converted.
%% The script prints every excluded site.
%%
%% CHECKED RATHER THAN ASSUMED: three sites read "see Part~X
%% \S\ref{ssec:zstar-path-to-closure}" and LOOK external because \ATen{} is
%% named nearby.  That label is THIS paper's and sits inside its own Part X.
%% Internal.
%%
%% ONE STALE POINTER FOUND BY THE SAME REVIEW: "the 28-parameter
%% Standard-Model audit with its Part~X universal-identity and
%% uniqueness-of-forces insights".  A9's identity Part is XI.  This
%% POSSESSIVE form escaped the v15.6 sweep because no companion macro is
%% adjacent to it -- the fourth carrier of that defect, and the third time
%% this week a repair has survived in a form the guard was not shaped for.
%%
%% NEW PROSE: none.  Proved by page-for-page identity: every rendered
%% numeral is what it was.
%% ===================================================================
%% ===================================================================
%% v15.7 (August 17, 2026) -- STAGE 2: PART II IN COHORT ORDER, AND THE
%% TABLE-2 ROWS OWED FROM STAGE 1.
%%
%% 1. THE CONSTITUTION NOW PRECEDES THE BULK IT FORCES.  Part II's thirteen
%% inheritance subsections already ran A1 -> A10 in order; what was reversed
%% was the pair of deep sections after them -- the 6D bulk and BI-6 (about
%% 31 kB of source) standing BEFORE the four axioms (about 10 kB) that force
%% it, so the stage arrived before its own justification.  Swapped.  A PURE
%% BLOCK MOVE: the two sections are contiguous, `sec:bulk' is referenced
%% nowhere in the document, and NOT ONE CHARACTER INSIDE EITHER BLOCK IS
%% EDITED.  The structural comment banner between them stays where it is.
%%
%% 2. TABLE 2 GAINS ITS THREE MISSING ROWS.  The cohort map carried EIGHTEEN
%% paper rows against a twenty-two-paper cohort: P6-Proton, W1-Symmetry and
%% W2-Holonomy were absent.  All three are already in the bibliography, all
%% three macros are already defined, and each new row names what this paper
%% actually takes from it -- P6's proton-as-ground-attractor and the ZERO
%% Delta-B = 1 mint that makes leptogenesis forced rather than preferred;
%% W1's conservation-law ladder; W2's circle-remembers/interval-forgets
%% sorting.  Owed from Stage 1 and not shipped with it; the reason is
%% recorded in the report.
%%
%% 3. TWO STALE CELLS IN THAT SAME TABLE, and they are the sharper finding.
%% Stage 1 corrected both claims in the inheritance paragraphs this morning
%% and did not correct them HERE, so since v15.6 the table has contradicted
%% the paper's own prose four sections later:
%%   * the A4 row said "The 28-to-zero compression of GG-Theory" -- the
%%     formulation A4 v15.37 went out of its way to retire;
%%   * the A5 row credited A5 with the Three-Gauge Theorem and the SeOS
%%     class definition -- the first is this paper's own reading, the second
%%     is A7's T-Qi4-14.3.
%% Both now say what the A4 and A5 paragraphs say.
%%
%% NEW PROSE: three table rows and two corrected cells.  The section swap
%% moves existing text and edits none of it.
%% ===================================================================
%% ===================================================================
%% v15.6 (August 17, 2026) -- THE INHERITANCE LEDGER REPAIRED.
%%
%% THE AUTHOR'S INSTRUCTION FOR THIS SHIP: accuracy and honesty; no reason
%% to over-sell unproven concepts as if they were theorems in GG-Theory.
%%
%% Report 432 read all ten foundation papers at their CURRENT versions and
%% classified every claim this paper makes about them.  Of about seventy-five
%% claims, thirty survived unqualified.  This ship repairs the rest.  EVERY
%% CHANGE IS A REPAIR MEASURED AGAINST THE COMPANION'S OWN TEXT, in several
%% cases quoting the companion verbatim.  Charter section 1.2: the resolution
%% is whichever side was already argued at greater length, which is always
%% the companion.
%%
%% THE SIX LOAD-BEARING REPAIRS
%%  1. This paper cited a universal-cosmic-identity THEOREM of A10.  A10 has
%%     no such theorem: it is I-Cosmos-9, an Innovation, which A10 presents
%%     as a preview and reserves EXPLICITLY FOR THE CAPSTONE.  This paper is
%%     the capstone, so it develops the principle and no longer cites A10 as
%%     having established it.
%%  2. The per-layer uniqueness claim needed A6's ADMISSIBILITY HYPOTHESIS,
%%     which has four clauses (connected, compact, Kahler with flat boundary
%%     connections, GFF a Hamiltonian-torus moment map with isolated
%%     non-degenerate fixed points) and which A6 verifies on ONE stratum.
%%     It is now carried, with A6's own warning that the Higgs and
%%     Wilson-line strata are not established and that uniqueness MAY
%%     GENUINELY FAIL on a periodic compact-angle potential.
%%  3. GRIP is boundary-side cascade dynamics, NOT the bulk-to-boundary
%%     projection (that is OGGEP/GDOS, which A6 imports).  And inter-layer
%%     ascent CANNOT be gradient flow -- strata carry discrete projector-class
%%     labels no continuous curve can change -- so every layer step requires a
%%     GRIP-Out event.  A6 delegates that theory to this paper; that
%%     delegation is why this paper exists, and it is now stated.
%%  4. Two of the five imports credited to A5 are not A5's: the GFF is A6's
%%     and SeOS is A7's.  THIS PAPER'S OWN BODY ALREADY SAID SO, so the paper
%%     contradicted itself.  The Three-Gauge triad is likewise this paper's
%%     own reading and is now labeled as such.
%%  5. sin^2 theta_W = 25/108 was attributed to A2 and to A3.  The string
%%     occurs in NEITHER file.  It is A9's and A4's.  Removed from both.
%%  6. "The compression is 28 to 0" is a formulation A4 v15.37 went out of
%%     its way to retire.  Replaced with A4's own accounting -- thirty free
%%     continuous constants of SM+GR down to a residue A4 NAMES: discrete
%%     axiomatic choices, unit conventions, and one overall mass scale --
%%     together with both of A4's scope clauses.
%%
%% STALE POINTERS AND DROPPED GRADES, ALSO REPAIRED
%%  * A9's identity Part is XI, not X (three papers carried three different
%%    numbers for it).  Its uniqueness theorem is section 95, not 61; the
%%    Q1+Q2 synthesis is section 96, not 62.
%%  * A9's identity cascade runs SEVEN layers, not ten; A9 defers the ten to
%%    this paper.  And A9 calls universal identity an INSIGHT, not a theorem.
%%  * A9's codebook triage restored: six derived, fourteen derived-conditional,
%%    eight asserted -- in place of "no continuous freedom".
%%  * A3's ledger is graded "proved, GIVEN AX-4", and global anomaly freedom
%%    fixes NONE of the integers; the exactness argument is topological (the
%%    circle carries the holonomy, the interval carries none).
%%  * A3's strongest result, unrestricted global anomaly freedom on every
%%    closed spin seven-manifold, was being given away; restored.  A3's
%%    "exactly one closed solution" claim withdrawn -- A3 grades that Open.
%%  * A7's T-HAL is a canonical normal form, not a uniqueness theorem; T-EIC
%%    is graded and "does not force an all-or-none divide"; and A7's Qi4-GGA
%%    characterization proves only the FORWARD direction, having removed the
%%    converse because it rested on this paper -- a circle.
%%  * A8 is not self-contained (its word), denies BRANCHING in measurement,
%%    and DEFERS consciousness to this paper rather than supplying it.  The
%%    Born-rule status added at the cohort-frozen grade.
%%  * A10's inventory is D* under a named conjecture and matches Planck to
%%    better than 3 percent raw, ~2 percent calibrated; its second-horizon
%%    mold half is a REDUCTION, not a proof, and A10 says so in terms.
%%  * A6's own grade quoted: the higher-layer identification is "the cohort's
%%    central wager rather than a theorem of this paper."
%%
%% APPROVED AUTHOR DECISIONS (report 432 section E)
%%  Q4 The two-knee falsifier takes A7's form: EXACTLY two, because three or
%%     more breaks refutes A7's T-CON.2.  Under the old ">= 2" wording three
%%     knees CONFIRMED this paper and REFUTED A7 on the same data.  The front
%%     matter also stops asserting a fixed frequency ratio that A7 denies and
%%     that this paper's own body defers to future work.
%%  Q5 This paper's cascade-depth index is renamed CDI, so QCI names only
%%     A7's weighted-geometric-mean quantity.  One symbol, one meaning.
%%
%% NEW PROSE IN THIS SHIP: none that states a new claim.  Every sentence
%% added either restates a companion at that companion's grade or withdraws
%% a claim this paper was not entitled to make.
%% ===================================================================
%% ===================================================================
%% v15.5 (August 17, 2026) -- THE EXPOSURE PLATE, AND THE THREE LISTS.
%%
%% 1. THE FALSIFIER PLATE ON SHOWCASE PAGE 8.  It is a GENERATED PDF, so
%% its type cannot be enlarged from the document: scaling here grows
%% type and picture together, and the author asked for the type to grow
%% RELATIVE to the picture.  The generator was found, RUN UNMODIFIED AND
%% CHECKED AGAINST THE DEPLOYED ASSET FIRST -- it reproduces it to 0.01
%% pt in width and 1.84 pt (0.3 percent) in height, the residue being the
%% matplotlib version.  That agreement is what establishes the script is
%% the source and not a stale copy.
%%   generator v2: every font +20 percent (12 -> 14.4; four note labels
%%   10 -> 12; three group labels 10.5 -> 12.6; y ticks and x label
%%   12 -> 14.4; title 12.5 -> 15; legend 9.5 -> 11.4), canvas 8.2 -> 9.2
%%   inches tall at the same width.  NOT ONE DATUM, LABEL, COLOR OR
%%   MARKER CHANGED.
%%   document: width 0.95\textwidth -> \textwidth.
%% MEASURED: printed width +5.3 percent, height +17.5, area +23.7, and
%% the rendered type 5.19 pt -> 6.58 pt, up 26.9 percent.  STATED
%% HONESTLY: even now the plate's type is BELOW the house 7.79 pt figure
%% standard, because reaching it would need a plate 18 percent wider than
%% the measure.  A further font step is available and was not taken
%% unasked.
%%
%% 2. THE OVERVIEW OF CONTENTS SETS ONE LINE PER ENTRY.  Every entry
%% wrapped to two, three or four lines because the Part SUBTITLE -- the
%% clause after the em dash -- was written into the toc entry, and the
%% folio landed wherever the wrap ended.  THREE OF THEM PRINTED IT HARD
%% AGAINST THE TEXT: "termination195", "weight396", "paper412".
%%
%% THAT IS THE DEFECT toc_gap_scan.py EXISTS FOR, AND IT REPORTED ZERO.
%% When the gap is exactly zero the glyphs merge and pdftotext returns
%% "paper412" as ONE word box, so the test -- last box is a bare number
%% -- never fires.  The tool catches a one-point gap and misses a
%% zero-point one.  Recorded in the report; the fix belongs with the tool.
%%
%% THE MECHANISM WAS ALREADY IN THE STYLE FILE.  \GGovsub came into
%% ucla-arisaka-gg-11 v11.6 this morning out of C2-Hubble, for exactly
%% this case, and \GGOverviewOfContents blanks it inside its own group.
%% So this paper's local \GGPart wraps the subtitle: one line per entry
%% in the Overview, the full Contents unchanged.
%% Then \GGfrontmatteronepagetrue lets the three lists flow together;
%% \GGListOfTables still clears afterwards, so the Contents is untouched.
%%
%% NEW PROSE IN THIS SHIP: none.
%% ===================================================================
%% ===================================================================
%% v15.4 (August 17, 2026) -- THE FRONT MATTER, FOR THE READER IT IS FOR.
%%
%% The author's instruction: Table 1 is too technical and turns a reader
%% off; "The Status of This Paper" folds into the Preface after the
%% current text, and the Preface stays under two pages; that writing "is
%% too protective and sounds like an excuse that this paper is imperfect
%% and not validated"; and the Abstract and Executive Summary should read
%% for a senior life-science undergraduate or first-year graduate student
%% with solid physics.  This preprint is for the homepage and arXiv.
%%
%% THE HONESTY STAYS; THE APOLOGY GOES.  Section 9 freezes the honesty
%% framing as settled, so it is not removed -- what is removed is the
%% hand-wringing around it:
%%
%%   OUT  "the reader is owed a plain statement"
%%        "we would rather it were read first than discovered late"
%%        "where a rung reads as though it were, that is a defect and we
%%         would like to hear about it"
%%        "this paper does not pretend that they are"
%%        "three honestly marked under construction"
%%        "A preview that could not be killed would not be worth writing"
%%        the word "preview" as a standing description of the paper
%%
%%   IN   the five tags, named, in one sentence
%%        which floors are at codebook standard and which are not
%%        the ten falsifiers at full strength, and that a failure on any
%%         one ends the framework whatever tag the rung carries
%%        that the constitution is posited rather than proved
%%
%% The author's own standard is one clause -- "which is interesting, and
%% is not proof" -- not three paragraphs of it.
%%
%% TABLE 1 LOSES NOTHING, CHECKED BEFORE REMOVING IT.  The five tags are
%% defined THREE MORE TIMES in the body, in Part IV where they are used:
%% a \paragraph for each class, the "five classes as a single procedural
%% ladder" discussion, and a description list.  The front-matter table
%% was the fourth definition and the only one a first-time reader met
%% before any of the argument had begun.  Removing it renumbers the
%% tables, which is correct: the table is gone.
%%
%% FOURTEEN CROSS-REFERENCES pointed at the section being moved.  The
%% label travels to the Preface with the material and the fourteen now
%% say "see the Preface" -- a repair forced by the move.  Deleting the
%% section without moving the label would have left fourteen undefined
%% references.
%%
%% NEW PROSE IN THIS SHIP: two Preface paragraphs ("What is settled, and
%% what is forecast", "What the tags do not soften"), one sentence added
%% to the Abstract naming the biological rungs, and three rewritten
%% clauses.  EVERYTHING ELSE IS DELETION.  The Executive Summary is
%% deliberately barely touched -- one hedge -- because it already
%% addresses "readers coming from biology, chemistry, or philosophy
%% rather than particle physics" in the author's own words, and the
%% reorientation belongs in the Preface.
%% ===================================================================
%% ===================================================================
% v15.3 (August 17, 2026) -- MIGRATED TO STYLE FILE v11.5.
%%
%% FOUR MACROS THAT LIVED IN THE PAPERS COME HOME, AND NOT ONE PAGE
%% MOVES.  This paper's part in that:
%%
%% * no showcase-headline macro: its showcase pages predate
%%   the pass and set their own headlines inline.
%%
%% 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 v15.2 printed, except for the lines named above.
%% ===================================================================
%% ===================================================================
%% v15.2 (August 17, 2026) -- THE -ise FAMILY.
%%
%% fertiliser -> fertilizer (1); optimiser -> optimizer (1).  Charter 1.1's pair list names these forms; spelling_audit.py's
%% PAIRS list did not carry them, so this paper has reported MIXED
%% PAIRS 0 and ALWAYS-BRITISH 0 at every gate while printing them.  The
%% cohort sweep of August 17, 2026 found sixty-one across twelve of the
%% twenty-four live papers; Report 406 section 6 has the inventory and
%% spelling_audit.py now carries the family as a RULE rather than a list.
%%
%% Substitution is on PRINTED TEXT ONLY.  Changelog comments are the
%% historical record, \label/\ref/\cite arguments are keys, and published
%% titles are quoted rather than written -- all three are masked, and every
%% frozen title is asserted present and unchanged after the edit.
%% NOTHING ELSE IN THIS PAPER CHANGES: no claim, no number, no float, no
%% label, and no page.
%% ===================================================================
%% ------------------------------------------------------------
%% v15.0 -> v15.1, August 16, 2026.  THE SHOWCASE, COMPLETED WITH
%% ITS EXPOSURE PAGE.  Eleventh of the twenty-four.
%%   A11 already carried three showcase pages (the eighteen-layer
%%   cascade, the cascade with the GGA, the GRIP engine) and all
%%   three answer 'what it delivers'.  ONE page is added: Figure 12,
%%   the Top 10 Falsifiers Dev plot -- eight forward bets whose
%%   answer is not yet known and two precision audits that have one.
%%   A11 reaches from cosmos to life to mind, which is exactly why a
%%   reader is entitled to the exposure before the argument.
%%   THE PLOT AND NOT THE TABLE: Table 7 is the natural candidate
%%   and is a longtable over three pages; Table 20 runs over two.
%%   Figure 12 holds the same ten on one page and puts the honesty
%%   in the marks themselves -- arrow for undecided, circle for
%%   measured -- which a table would have had to say in words.
%%   NO SAVEBOX is needed: the plate is an \includegraphics of a
%%   PDF and is leading-invariant, so the front-matter defect that
%%   Report 385 corrected in A5-A8 cannot arise.  Asserted, not
%%   assumed: zero \GGsc saveboxes in this file.
%%   UNNUMBERED, like A9's added page: numbering it would insert a
%%   Figure 4 and renumber Figures 4-21.
%%   FOUND BY THE GATE AND FIXED IN THE SAME SHIP: every
%%   \begin{minipage}{\textwidth} in this paper started a paragraph
%%   without \noindent, so each box was set at \parindent +
%%   \textwidth and overflowed the right margin by exactly
%%   \parindent.  Measured: \parindent = 16.33182 pt and the log's
%%   overfull box is 16.33182 pt, once per site, four times.  The
%%   three showcase pages were also sitting 16.3 pt right of the
%%   text block.  A9-SM and A10-Cosmos both write
%%   \noindent\begin{minipage}; A11 was the one that did not.  All
%%   four sites fixed, plus the new page: five overfull boxes gone.
%% NEW PROSE IN THIS SHIP: one headline and one showcase caption,
%% both on the new page.  Nothing in the body is rewritten.
%% ------------------------------------------------------------
%% ===================================================================
%%  A11-Origin v13.69  (August 13, 2026)
%%  Found while re-deriving A4-Euclid's imports, not by opening A11.
%%  THE VACUUM-SCALE SENTENCE WAS WRONG IN TWO WAYS.
%%  1. STALE.  It said the seesaw recovers "119 of the 120 orders".
%%     A10-Cosmos v13.37 no longer stops at the corridor: the ledger
%%     coefficient zeta_Lambda = pi/(LK) = pi/175 = 0.017952 is selected
%%     in closed form, and rho_Lambda^(1/4) = zeta^(1/4) * M_lock^2 /
%%     M_Pl,red = 0.3660 * 6.083 = 2.23 meV against a measured 2.24.
%%     A11 had ZERO occurrences of zeta_Lambda and was quoting the state
%%     A10 was in before the selection -- the same understatement A4
%%     carried in eight places and A10 in four sentences, both corrected
%%     earlier in this revision.  A11 was the third carrier.
%%  2. THE FORMULA DID NOT GIVE THE NUMBER.  The sentence wrote the
%%     seesaw over \MPl, the NON-reduced Planck mass:
%%         M_lock^2 / M_Pl     = 1.2134 meV
%%         M_lock^2 / M_Pl,red = 6.0831 meV   <- the stated ~6 meV
%%         ratio               = 5.0133 = sqrt(8 pi)
%%     Same defect, same quantity, as the A4/A10 cosmological-constant
%%     work two reports ago, where it hid inside "an O(1) prefactor".
%%     Here it hid inside an approximately-equals sign.  \MPl was used
%%     EXACTLY ONCE in the whole of A11 -- in this sentence -- and A11
%%     carries no reduced-Planck-mass notation at all, which is why
%%     nothing else in the paper contradicted it.
%%  The replacement names the convention, states both steps, quotes
%%  A10's grade (Derived, conditional) rather than upgrading it, and
%%  keeps the paragraph's own point about structural openness.
%% ===================================================================
%% ===================================================================
%%  A11-Origin v13.67  (August 6, 2026)  -- the cohort-wide figure-size sweep.
%%  15 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.
%% ===================================================================
%% ===================================================================
%%  A11-Origin v13.66  (August 5, 2026)  -- the cohort funnel at the house size.
%%  The central-claim funnel was drawn with 103mm boxes and then
%%  left at natural size, so its \footnotesize landed 16% larger than the roadmap
%%  and cohort-map plates beside it.  The plate target is now
%%  0.619\textwidth, computed rather than chosen: the picture measures
%%  113.9mm wide compiled standalone, and the house type size is set by
%%  GG-P3-Electron Figure 3 (0.93\textwidth on 171.1mm, \textwidth =
%%  159.2mm, so type at 0.865 of natural size), which for any picture
%%  of width W means f = W / 184.0.
%%  Nothing inside the picture changes: no font command, no box
%%  dimension, no text, no relative position.
%%  Recipe and the measured table for all fourteen papers:
%%  [v13-support]/funnel_sweep.py.
%% ===================================================================
%% ===================================================================
%%  A11-Origin v13.65  (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.
%% ===================================================================
%% ===================================================================
%%  A11-Origin v13.62  (August 1, 2026)  -- the cosmic-descent figure.
%%  Adopted from A10 (external sans-serif asset a11_cosmic_descent.pdf,
%%  identical to A10's a10_cosmic_descent.pdf): the to-scale descent on
%%  r=ln(E_GUT/mu), reading energy, time, and scale factor at every key
%%  epoch. Placed at the head of Section 27 (Cosmos: layers 1-3), where
%%  layers 1-3 (gauge/GUT edge, electroweak lock, atomic recombination)
%%  are exactly the cosmological rungs the figure marks; inline citation
%%  added to the section's opening paragraph. Version bumped.
%% ===================================================================
%% ===================================================================
%%  A11-Origin v13.61  (August 1, 2026)  -- S3: the strict register.
%%  (1) The one remaining reader-facing 'designed' in the framework's
%%      own voice (autocatalytic-perturbation physics intuition) recast:
%%      'each layer is driven ... arranged by descent, not by design.'
%%      (The other 'designed' sites are quotes, contrasts, or 'no
%%      designer' statements and are correct as they stand.)
%%  (2) NEW PARAGRAPH after the lesson of the great ascent: the third
%%      horizon problem (A10 v13.20 Sec. 64) as the observational audit
%%      of 'deterministic at every layer'; the gauged-vs-global division
%%      (A10 Sec. 32.13); closes on the four words. Version bumped.
%% ===================================================================
%% ===================================================================
%%  A11-Origin v13.60  (July 29, 2026)  -- P2-N3 retitle propagation.
%%  P2-N3 is retitled "Why Three Generations: A THREE-LEG Derivation of N = 3
%%  from BI-6, MSMF, and GDOS Morse Stability" (was "A Closed Derivation").
%%  Reason: P2-N3's own main theorem T-N3-5 is "Conditional derivation of
%%  N = 3 (three-leg pincer)", conditional on five structural hypotheses --
%%  including the BI-6 Budget Conjecture, which A2 states is not derived
%%  anywhere in the cohort -- plus two empirical inputs.  The cohort's prose
%%  already flagged that conditioning everywhere; only the title claimed
%%  "Closed", and titles propagate into every bibliography.
%%  T-N3-5 KEEPS its own name: the theorem is conditional and says so.
%%   * COHORT ROSTER: P3 row "(the five oldest electron questions, closed)"
%%     -> "six", matching P3's title page and A11's own bibitem.
%%   * "a closed derivation audited against a measured ratio" -> "a three-leg
%%     derivation ...", matching the retitled P2-N3.  The point of the
%%     sentence -- that the family-number commitment carries no ledger key
%%     because it is audited, not staked -- is unchanged.
%% ===================================================================
%% ===================================================================
%%  A11-Origin v13.59  (July 29, 2026)  -- internal-affairs sweep.
%%  Six of the 74 Common Objections were internal QA findings written up as
%%  referee objections; their responses narrated repairs ("Found and fixed",
%%  "Corrected:", "we acted on it", "scheduled for the cohort-wide sweep").
%%  In an arXiv v1 the reader cannot find the error being confessed to.
%%   * O21, O23, O39: response reframed.  EVERY CONCESSION KEPT -- "exactly
%%     three" argued not proven; the pagoda storey count empirical, not forced
%%     by L=5; the Gibbs/Friston/GG triad carrying no registered companion id.
%%   * O55, O56, O60: the OBJECTION TEXT itself asserted the defect, so each
%%     pair was recast onto the question its named persona would really ask.
%%     Theorem ids and the "no registered id" admission are preserved.
%%   * O60's premise was verified FALSE against the deployed cohort before
%%     being touched: A9 v20.46 reads "T-SM-21.1: Conditional derivation of
%%     N=3; citealp{ArisakaP2N3} T-N3-5", already matching P2-N3's T-N3-5.
%%     Its replacement answer is checkable and true as printed.
%%   * A6 bibitem: internal v9.3 / v9.5 stamps dropped.
%%   * Second pass, found by rendering pp.434-435: O57 ("no longer attributed
%%     to nonexistent A7 theorems"), O58 and O38 ("the Theorem Registry
%%     surfaced ... for the cohort-wide cross-reference sweep", "are being
%%     standardised").  The named internal QA artefact is gone from the text.
%%     O38's QCI mismatch is REAL and is kept -- promoted from a queued work
%%     item to a declared open discrepancy between A7 and this paper, which is
%%     the stronger and more useful statement for a reader.
%%  74 objections in, 74 out; numbering, personas and the inbound references
%%  to Objection 21 all unchanged.  No value, equation or grade moved.
%% ===================================================================
%% ===================================================================
%% v13.57 -> v13.58 (July 28, 2026).  Queue items K3 and K7.
%%   * K7: the frozen cross-paper coordinates removed.  The Standard-
%%     Model paper renumbered its Parts; its own internal label still
%%     resolves correctly there, but this paper had hard-coded the label
%%     NAME as literal text, where nothing resolves it.  The citations
%%     now name the Part and, where the cited paper offers one, a
%%     version-stable theorem key -- so a future renumbering cannot
%%     freeze them again.
%%   * K3: the cascade-closure structure theorem gives a LOCAL minimum.
%%     Wherever a GLOBAL statement was resting on it, the warrant is now
%%     the runtime paper's per-stratum ground-state corollary.  Several
%%     sites also attached that structure theorem to the constitutional-
%%     equivalence theorem, which is a different result; corrected.
%% Values, canonical numbers, ledger: UNCHANGED.
%% ===================================================================
%% ===================================================================
%% v13.56 -> v13.57 (July 28, 2026).  Queue item K1: the Hubble readings.
%%   * The derived background is now stated where the paper previously
%%     stated the CMB reference as a prediction, and the conditional
%%     lifted late reading is stated with its grade.
%%   * F6 collapsed from two staked endpoints to one falsifier carrying
%%     the cohort ledger's kill criterion: sirens read ~73, not ~67.
%%     The previous split had this decade's scheduled siren experiment
%%     pointing the opposite way from the companion papers'.
%%   * The two "matching the measurement" claims and the
%%     "predicted in advance ... P = +0.03" claim removed: the framework
%%     derives a background and does not derive the ladder value.
%%   * The forecast scenario corrected -- a siren reading near 71 is a
%%     move toward the framework, not away from it.
%%   * The "not a tension" passage rewritten on the companion paper's
%%     model: a derived background, a derived lift, and a named residual
%%     that is an open problem rather than a result.
%% Ledger, canonical values elsewhere: UNCHANGED.
%% ===================================================================
%% ===================================================================
%% v13.55 -> v13.56 (July 28, 2026).  Phase D of the GGA correction.
%%   * T-GGA-as-Ground-State restated as the cascade-wide generalisation
%%     it actually is: the single-stratum ground-state result is
%%     upstream, and what this paper adds is that it survives layer by
%%     layer along the cascade constructed here.  The layer-wise
%%     conditions are now hypotheses of the theorem, stated as such.
%%   * Proof outline rewritten.  The step that appealed to a downstream
%%     deployment paper is removed -- that paper had taken the property
%%     from here -- and the property is now taken from the runtime paper
%%     that proves it.  The pillar-selection step is replaced by the
%%     characterisation, with the transversality hypothesis re-attached
%%     to the characterisation rather than to existence.
%%   * The upstream-theorem attribution corrected (the cited runtime
%%     theorem is the Morse-positivity result, not a stabilisation one).
%%   * The global reading of the cascade-closure structure theorem
%%     corrected: that theorem gives a local minimum; the global
%%     statement is the runtime paper's corollary.
%%   * Kill criterion split into its two distinguishable failures.
%% Values, canonical numbers, ledger: UNCHANGED.
%% ===================================================================
%% ====================================================================
%%  v13.54 -> v13.55  (July 28, 2026)  --  SHIPS J5 + J6
%% --------------------------------------------------------------------
%%  J5.  A11's five-condition SeOS characterisation is A11's OWN.  A7's
%%  def:seos has FOUR conditions and its T-Qi4-14.3 is a Lyapunov drift
%%  criterion, not a class characterisation; and A5 mentions "SeOS"
%%  three times in 1.31 MB and defines nothing.  The proof's claims that
%%  the match is "verbatim" and that "A7 establishes that five suffice"
%%  are both withdrawn -- neither is true and neither is needed.  Clause
%%  (d), the Qi-current closure, moves from T-EP to T-QUA.2, the Arisaka
%%  equation's own layer.
%%
%%  J6.  A7's index is the "Qi-4 Completeness Index", a weighted
%%  GEOMETRIC MEAN on [0,1].  A11 used the name "Qi-HAL Completeness
%%  Index" -- zero occurrences in A7 -- for an integer storey count.
%%  A11 already separates the two objects correctly in
%%  rmk:qci-geometric-mean; only the naming borrowed A7's authority.
%%  Also fixed: "closure at QCI = 5" (storey 5 is QCI = 10), the
%%  off-ladder "Prebiotic ~0.5", and a lower-case "highest admissible
%%  layer theorem" that ship J3 missed.
%%
%%  VERIFIED AND NOT CHANGED: the storey ladder itself.  All five rungs
%%  satisfy QCI = 2S.  A relayed claim that they did not was wrong.
%% ====================================================================
%% ====================================================================
%%  v13.53 -> v13.54  (July 28, 2026)  --  SHIPS J3 + J4
%% --------------------------------------------------------------------
%%  Two phantom attributions to A7, both already conceded in A11's own
%%  Common Objections and never corrected in the body.
%%
%%  J3.  "Highest Admissible Layer" occurs 0 times in A7 and 2 times in
%%  A11, both inside T-Cascade-Termination.  A7's HAL is the HOLOGRAPHIC
%%  ATTRACTOR LATTICE and its T-HAL theorems concern the
%%  MePMoS-NHT-HAL-BFFT memory architecture, not a ceiling on cascade
%%  depth.  The invented "integer ceiling" theorem goes; the Goedel
%%  self-reference argument and A7's actual architectural constraint,
%%  which were carrying the claim anyway, stay.
%%
%%  J4.  A7's own pagoda proof says "the count FIVE is an empirical
%%  reading, NOT A THEOREM OUTPUT (Open Problem O7)".  A11 said L = 5 is
%%  "the constitutional warrant for the five-storey count" and "No sixth
%%  storey exists".  Objection 39 already concedes this.  Corrected, and
%%  the two citations of the non-existent "A7 Sec 28" replaced by the
%%  real key T-Qi4-16.2 -- with A11's COGNITIVE ladder distinguished
%%  from A7's CROSS-DOMAIN pagoda, which it sits inside.
%% ====================================================================
%% ====================================================================
%%  v13.52 -> v13.53  (July 28, 2026)  --  SHIPS J7 + J8 + J9 + J10
%% --------------------------------------------------------------------
%%  J7.  A7 keys each pillar in two layers: algebraic (.1) and
%%  operational (.2).  The ARISAKA EQUATION is T-QUA.2, not T-QUA.1
%%  (A7: "Its operational/physical layer (T-QUA.2) is the Arisaka
%%  equation"); the TWO-KNEE spectrum is T-CON.2, not T-CON.1.  Five
%%  sites repaired.  Sites citing T-QUA.1 for the rank-one ray are
%%  correct and untouched.  Before this ship A11 cited the .1 layers 42
%%  times and the .2 layers 0 times -- and every .2 layer is A7's
%%  falsifiable half.
%%
%%  J8.  (a) "A7 Sec 6" for the SIP (8 sites) -> the theorem key T-SIP;
%%  the SIP is A7 Sec 8, and A7's own abbreviation table carries the
%%  "Sec 6" error that A11 inherited.  (b) The SIP abbreviation gloss
%%  listed A7's field (iii) twice and omitted the ledger bundle
%%  (F_GFF, J) entirely.  (c) NHT and HAL are swapped in A7's
%%  abbreviation table; A11 inherited the swap.
%%
%%  J9.  A11 said "the full proof is contained in A7" for the same step
%%  A7 attributes to A11.  The loop is cut on A11's side.
%%
%%  J10.  The sufficiency step still rested on "the mutual-forcing
%%  structure" that ship H7 withdrew.  Rewritten onto (Q1)-(Q4).
%% ====================================================================
%% ====================================================================
%%  v13.51 -> v13.52  (July 28, 2026)  --  SHIPS J1 + J2
%% --------------------------------------------------------------------
%%  The two defects this project introduced while reasoning about A7
%%  without having read A7.  Both now verified against the A7 source.
%%
%%  J1.  A7 keys every pillar in two layers: algebraic (.1) and
%%  operational (.2), joined by a compatibility relation.  T-CON.1 is
%%  the operational conformality theorem; T-CON.2 is the TWO-KNEE
%%  SPECTRAL THEOREM (A7 T-Qi4-11.7), used seven times in A7 including
%%  in falsifier F-Qi4-3.  A11 originally cited T-CON.2 for the two-knee
%%  structure and was RIGHT; ship G5 changed it to T-CON.1 on A11's
%%  internal frequency.  Reverted, and anchored to A7's theorem.
%%
%%  J2.  Ship H6 wrote "the canonical order ... is an order of
%%  exposition, not of operation".  A7 says the pillars form an
%%  implication chain Qi-QUA => Qi-EP => Qi-CON => Qi-HAL, "each forcing
%%  the next", and that this ordering "is why the pillars cannot be
%%  permuted".  The order IS operational -- a dependency order.  What it
%%  is not is a temporal pipeline sequence, which the rest of the
%%  paragraph correctly denies.  Rewritten to say both.
%%
%%  Ship H7's demotion of "any three force the fourth" stands: that
%%  phrase occurs once in A7, unkeyed and unproved, in its Conclusions.
%% ====================================================================
%% ====================================================================
%%  v13.50 -> v13.51  (July 28, 2026)  --  SHIP H11 (REDUCED)
%% --------------------------------------------------------------------
%%  Report 48 proposed softening four "completeness promises" that
%%  14 of 18 layers were said not to keep.  RE-MEASURED, COUNTING BOTH
%%  THE HEADED AND THE INLINE FORM OF EACH SLOT: kill criterion 18/18,
%%  physics intuition 18/18.  Layers 4-17 state their kill criteria
%%  inline and carry constitutional inputs, upstream theorems and
%%  anchors inside the Codebook provenance block.  The original audit
%%  matched only "\paragraph{Kill criterion" and undercounted.
%%
%%  "Eighteen separate kill criteria, one per theorem" is CORRECT and is
%%  NOT changed.  Nothing Report 48 proposed to soften is softened.
%%
%%  Shipped: one clause.  "The same template" is literally true of the
%%  slots and visibly false of the layout -- layers 1-3 and 18 use
%%  headed paragraphs, layers 4-17 a compact provenance block.  The
%%  sentence now says so, so a reader flipping from layer 3 to layer 4
%%  does not think content is missing when it is not.
%% ====================================================================
%% ====================================================================
%%  v13.49 -> v13.50  (July 28, 2026)  --  SHIP H6: PART IV / PART V
%% --------------------------------------------------------------------
%%  Part IV and Part V shared 3.8% of Part V by 8-gram shingle, against
%%  under 0.5% for every other Part pair.  Three transplanted blocks:
%%
%%  (1) The integer-ledger passage was BYTE-IDENTICAL (1076 chars).
%%      Ship E4 made it so, by restoring Part V's dropped clauses
%%      instead of pointing at Part IV.  Worse, it indexes by FINE
%%      layer while sitting under Part V's TWELVE-COARSE-STRATA table,
%%      so it read "at layer 5, twenty amino acids" ten lines under a
%%      table putting them at row 4.  Deleted in Part V; pointer left.
%%  (2) The cascade-engine equations were printed twice, eq (14)-(16)
%%      and eq (42)-(44).  Part V's displays go; its seven internal
%%      \eqref uses are repointed at Part IV's labels.
%%  (3) The priority claim was made three times.  Part IV's dedicated
%%      subsection keeps it; Part V's verbatim repeat becomes a
%%      pointer and Sec 46.2's variant is trimmed.  The four "We are
%%      not aware of a prior framework..." sentences are NOT touched --
%%      each is a different, individually hedged claim.
%%  (4) sec:qi4-admissibility-filter takes nine incoming references
%%      from five Parts, but the doctrine sentence lived in Part IV and
%%      Part V pointed back at it.  The sentence now opens the home.
%%  (5) Six stale ...-partIII equation labels renamed to -partV.
%% ====================================================================
%% ====================================================================
%%  v13.48 -> v13.49  (July 28, 2026)  --  SHIP H4: FIGURE 9
%% --------------------------------------------------------------------
%%  Figure 9 placed one Qi-4 pillar beside each pipeline stage -- QUA at
%%  DGI, EP at GFF, CON at GSSB, HAL at GGA.  That is the stage-binding
%%  removed from the Part IV and Part V equations in ships D7/D8/E5, and
%%  it survived here because the figure's caption is correct and an
%%  earlier review checked the caption.  The four pillars now sit in one
%%  block at the terminus, where they act, labelled as simultaneous.
%%  (cohort_qi4_check.py cannot see tikz geometry and passed this.)
%%
%%  The "where GRIP-In halts (T-GGA-No-Exit)" annotation was overprinted
%%  by the sloped "GFF landscape unchanged" arrow label and rendered as
%%  "where GRIP-In ha[] (T-GGA-No-Exi[]".  Moved clear.
%% ====================================================================
%% ====================================================================
%%  v13.47 -> v13.48  (July 28, 2026)  --  SHIP H7: PART V REGISTER
%% --------------------------------------------------------------------
%%  Author's rulings: A11 is a PREVIEW, unlike A9/A10; "an honest
%%  demotion is perfectly fine for now".
%%
%%  (1) Part V carried the paper's keystone theorem with zero Status
%%  tags, zero kill criteria and zero occurrences of "falsif" in 71.7 kB,
%%  while Part IV grades eleven of the eighteen layers T-Origin
%%  quantifies over as Conditional or Asserted.  Two Status tags and a
%%  kill criterion added.
%%
%%  (2) Theorem 41.1 said "jointly necessary and sufficient" in its
%%  statement and "conditional on (Q1)-(Q4) ... not results proved here"
%%  in its proof; Sec 43.2 then asserted the flat form again.  Statement
%%  and Sec 43.2 now carry the hypotheses.
%%
%%  (3) The mutual-forcing premise ("several passages of A7 establish
%%  ... any three force the fourth") had no theorem key, and the list
%%  offered in support exhibits a one-directional chain, not mutual
%%  forcing.  It is used only for (Q1).  Demoted to that.  Ship G7 had
%%  rewritten Part VII Sec 52.6 to defer to it, so that site is updated
%%  here too.
%% ====================================================================
%% ====================================================================
%%  v13.46 -> v13.47  (July 28, 2026)  --  SHIP H10: LAYER 6, AND v
%% --------------------------------------------------------------------
%%  (1) LAYER 6.  Theorem 28.3 (T-Polymer-Closure) says the layer-6 GGA
%%  is the polymer CLASS and that "no specific polymer family is
%%  selected at this layer".  Eight other sites gave layer 6 the triplet
%%  codon map.  The author has ruled the theorem correct.  The codon
%%  content moves to layer 8, which the paper's own inventory already
%%  names T-RNA-World-Closure, "4-letter alphabet + ribozymes + triplet
%%  code precursor", and where two sites already place it.
%%  (Part V's twelve-COARSE-strata table has a row 6 legitimately called
%%  "Genetic code" -- coarse row 6, not fine layer 6.  That collision is
%%  the likely origin of the drift.)
%%
%%  (2) THE VEV.  3850 x (2/5)^3 = 246.40 GeV exactly.  The paper wrote
%%  the arithmetic as 246.4 twice and quoted 246.3 nine times, including
%%  in "agreement < 0.05%" -- a bound that holds for 246.3 (0.032%) but
%%  not for the value the stated inputs actually give (0.073%).
%%  Standardised on 246.4 with agreement < 0.1%.
%% ====================================================================
%% ====================================================================
%%  v13.45 -> v13.46  (July 28, 2026)  --  SHIP H8: PART III
%% --------------------------------------------------------------------
%%  (1) Part III carried five theorem environments where its own opener
%%      promises "three named theorems ... plus one corollary".  Two of
%%      the five call themselves something else in their own captions --
%%      "corollary to T-GRIP-In-Lyapunov-Descent" and "empirical
%%      proposition" -- and their labels already read cor:*.  Both become
%%      corollaries.  Free: the shared counter means nothing renumbers.
%%  (2) A6 T-GRIP-3 was cited for "at least one stable minimum" twice and
%%      for "the unique stable minimum" three times.  Sec 14 exists
%%      precisely because A6 alone does not give uniqueness; the three
%%      uniqueness citations made the Qi-4 filter and hypothesis (Q4)
%%      dead weight and the spine circular.  Repaired to name the result
%%      that carries uniqueness.
%%  (3) AX-3 was glossed as "the act of measurement itself" three pages
%%      after Sec 18.3's "no collapse, no Heisenberg cut, and no
%%      privileged observer" and Part II's "emphatically not a statement
%%      about instruments or observers".  Rewritten to the observer-free
%%      register the other two share.
%%  (4) "Theorem T-Qi4-GGA-Admissibility of A7" -- it is Theorem 41.1 of
%%      the present paper, Part V.
%%  (5) Nine reader-facing upstream version numbers and a "June 2026
%%      revisions" note removed: convention violation, and stale.
%% ====================================================================
%% ====================================================================
%%  v13.44 -> v13.45  (July 28, 2026)  --  SHIPS H5 + H9
%% --------------------------------------------------------------------
%%  H5.  2 N_A Q_B = 30.0057 kJ/mol against an empirical 30.5 is a
%%  1.647% deviation, and Part IV's layer-7 kill criterion falsified the
%%  row at "greater than 1.5%".  The row was dead by its own criterion.
%%  The quantity carried three thresholds (1.5% kill, 5% kill in the
%%  master codebook, 0.5% as the 2030 audit target) and two agreement
%%  figures (1.5% and 1.6%).  Now: agreement 1.6%, kill 5% (the codebook
%%  value), 0.5% kept as the audit's target precision.
%%
%%  H9.  Part II: "19 parameters" -> 28 (the paper's own audit);
%%  "twelve inheritances" -> eleven (its own heading, and the count of
%%  \subsection{Inheritance from ...} is 11); Green-Schwarz "twenty-five
%%  years before" -> four decades (1984 + 25 = 2009); "seven steps" over
%%  six enumerated steps; "zero adjustable parameters beyond Newton's
%%  constant" -> zero, G_N being a forced GU-5 constant and not a dial;
%%  "Five forces, one pipe" -> four, as the subsection's own body says.
%%
%%  And the overreach: consciousness uploading declared "a theorem of
%%  the geometry".  No such theorem exists; Part VIII leaves the
%%  substrate question open and Part IX's closure corollary is
%%  explicitly substrate-independent.  Replaced by the narrower claim
%%  the paper actually holds, with a pointer to Part VIII.
%% ====================================================================
%% ====================================================================
%%  v13.43 -> v13.44  (July 28, 2026)  --  SHIPS H2 + H3
%% --------------------------------------------------------------------
%%  Three defects introduced by this project's own ships.
%%
%%  H2.  Ship D3 moved the eighteen "Codebook provenance" blocks into
%%  their layer subsections.  For the last layer of each band the block
%%  landed AFTER the band's closing \section{}, so layers 6, 10, 13 and
%%  17 printed their provenance under the next band's heading -- putting
%%  "the cascade's first Anchor-Derived biological row" (layer 10)
%%  directly under a heading whose lead says "All three are
%%  Conditional".  D3 asserted subsection ownership and never section
%%  ownership.  Fixed; the post-condition now checks the section.
%%
%%  H3.  Ship E6 demoted three Part VI theorems to corollaries; ship E7
%%  swept the body prose and the theorem index but not the two places
%%  that describe the deck from outside it -- the Part subtitle ("The
%%  mechanism theorems") and Sec 47.2 ("a deck of six named theorems").
%%
%%  H3b.  Ship E2 left internal editorial history in Part VI: "the
%%  sentence that used to equate the two counts was the source of the
%%  confusion this Part carried for several versions."
%% ====================================================================
%% ====================================================================
%%  v13.42 -> v13.43  (July 28, 2026)  --  SHIP H1: AUTHORING ARTIFACT
%% --------------------------------------------------------------------
%%  Part V's Roadmap said the Part closes "in the philosophical register
%%  the user prefers".  One occurrence in the document; no sibling
%%  pattern ("the user wants", "as requested", "per the user") occurs at
%%  all.  Shipped alone and first.
%% ====================================================================
%% ====================================================================
%%  v13.41 -> v13.42  (July 28, 2026)  --  SHIP G9: TWO TITLES
%% --------------------------------------------------------------------
%%  G6 removed the restatements; these are the subsection titles that
%%  still announced them on the contents page.  "The canonical form,
%%  restated" now sits above a subsection whose first sentence says the
%%  canonical form "is not restated here", and "The cascade as a series
%%  of GGAs, formal restatement" no longer restates a table it no longer
%%  carries.  Titles only; labels unchanged.
%% ====================================================================
%% ====================================================================
%%  v13.40 -> v13.41  (July 28, 2026)  --  SHIP G8: REGISTER
%% --------------------------------------------------------------------
%%  (1) Two navigational sentences in Sec 51 stopped at 51.7 and did not
%%      mention 51.8 or 51.9 -- two numbered theorems and three boxed
%%      equations, placed after the metaphors-for-non-physicists
%%      subsection.  Both now describe the Section as it stands.
%%  (2) The Part subtitle named Q_B, SeOS and GFF but not Sec 52, the
%%      GRIP dynamics and the Part's largest section.
%%  (3) Status grades added to Theorem 51.1 and Theorem 51.3, where the
%%      surrounding text already fixes the grade (exhaustiveness argued
%%      not proved; custody disclaimed and inherited).  Theorem 51.2 is
%%      deliberately left ungraded: its proof Step 3 is in tension with
%%      Sec 40.3's perfect-Morse-function statement, an open item, and
%%      grading a theorem whose proof is under question is what the
%%      convention exists to prevent.
%% ====================================================================
%% ====================================================================
%%  v13.39 -> v13.40  (July 28, 2026)  --  SHIPS G6 + G7
%% --------------------------------------------------------------------
%%  G6.  Part VII Sec 52 restated material whose home is Parts IV/V/VI,
%%  and two of its subsection titles said so ("restated", "formal
%%  restatement").  Each edit removes the restatement and keeps the
%%  elaboration built on it.
%%    (a) the canonical GRIP pipeline -> pointer to Part V Sec 40; the
%%        Q_B / SeOS / GFF synthesis paragraph is kept.
%%    (b) the second eighteen-row GGA/anchor table -> pointer to Part
%%        IV's.  It was the degraded copy: no Kleiber hedge, no
%%        |GGA_5| = K - NL = 20, proton moved from layer 3 to layer 1,
%%        and a \resizebox around {\small} taking it below the house
%%        table standard.
%%    (c) the five-field SIP definition -> pointer to eq:sip-five-field;
%%        the field-role gloss is kept, the recursion refers to it.
%%    (d) "repeated seventeen times (from ell=0 to ell=15 ...)" --
%%        sixteen values, contradicting its own count word, its own
%%        eq. (90), and Corollary 47.3's n = 1..17.  Now a pointer to
%%        the corollary.  The field-by-field recursion equation stays.
%%
%%  G7.  Part V: "any three of them ... force the fourth."  Part VII Sec
%%  52.6: "not redundantly necessary (e.g., three of four being
%%  enough) ... removing any factor enlarges the admissibility set."
%%  Incompatible -- and Part V's proof of Theorem 41.1 derives
%%  sufficiency from mutual forcing, so Sec 52.6 denied the premise of
%%  the theorem it calls itself the deepest test of.  Part V is
%%  canonical; Sec 52.6 keeps necessity, drops the set-theoretic claim,
%%  and its four counterfactual universes are untouched.
%% ====================================================================
%% ====================================================================
%%  v13.38 -> v13.39  (July 28, 2026)  --  SHIP G3: THE THREE-GAUGE
%%                                        IDENTITY, NARROWED
%% --------------------------------------------------------------------
%%  Part VII stated the Three-Gauge Theorem twice and the two statements
%%  disagreed: Sec 51.5 hedges exhaustiveness ("argued rather than
%%  proved"), Sec 51.9 asserted "exactly three" and "no fourth boundary
%%  gauge is admissible"; and the third gauge was the ACTION gauge in
%%  one and a "runtime selection functional" in the other.
%%
%%  NOT deleted, as Report 46 proposed, because checking first showed
%%  T-Three-Gauge-Identity is row 25 of the master theorem ledger with
%%  two live cross-references.  Row 25 describes it as the IDENTITY
%%  ("... agree once expressed in canonical units"), which is a real
%%  claim distinct from Theorem 51.1's enumeration of the projections.
%%  So the theorem is NARROWED to the identity it is named for; the
%%  label, the ledger row and both references survive.
%%
%%  Removed: "exactly three" / "no fourth"; the same claim in the
%%  Constitutional inputs; proof Step 3's J = 3 (one site in the paper,
%%  against a canonical J = 7 and four enumerated ports); proof Step 5's
%%  no-hybrid claim (falsified by Sec 51.6's own hybrid projections);
%%  the "runtime selection functional" third gauge (one site, against
%%  six saying action gauge); the A5 custody claim Sec 51.5 disowns;
%%  \citet{Dehaene2014} for a variational reading he does not hold; the
%%  style-guide sentence; the self-contradictory bilingual/trilingual
%%  metaphor; the duplicated plain-English opening and close.
%%
%%  Kept and repaired: eq. (78) with W_ij DEFINED for the first time;
%%  the units anchor (third reading corrected, "room temperature" ->
%%  300 K, since 6.02 nats back-solves to 299.8 K); the Legendre
%%  analogy; the kill criterion, with hybrids permitted.
%% ====================================================================
%% ====================================================================
%%  v13.37 -> v13.38  (July 28, 2026)  --  SHIPS G2 + G4 + G5
%% --------------------------------------------------------------------
%%  G2.  Part VII opened SeOS with TWO consecutive \section commands and
%%  no body between them, the label sitting on the empty one.  Page 215
%%  printed a bare "50 The Selfish Open System (SeOS) class" above "51
%%  SeOS: the class of Selfish Open Systems", and both refs to
%%  sec:seos-merged landed on the husk.  The husk also consumed a
%%  section number, which is why "Section 3 of this Part presented the
%%  Three-Gauge Theorem" pointed at the wrong section.  Deleting it and
%%  moving the label makes FOUR relative cross-references correct at
%%  once -- asserted below, not assumed.
%%
%%  G4.  Figure 12: two arrow labels both at right=2pt overlapped and
%%  printed "Markov blanketdetector limit" (caught in the render, not by
%%  grep).  Eq. (76), the figure node and the caption printed F_GFF with
%%  two terms where eq. (71) defines three -- and the action gauge is
%%  defined by W_ports -> 0, so the missing term removed that gauge's
%%  defining condition.  F_GEP/F_CEP were glossed as "equipartition"
%%  pieces, a phrasing occurring once each against the glossary's
%%  Geometry-Gauge Energy Principle / Curvature-Entropy Principle.  The
%%  subsection label sat after \end{figure}, anchoring to the caption.
%%  Figure 12 had no in-text callout.
%%
%%  G5.  Part XIV Objection 58 concedes "the SeOS persistence theorem is
%%  A7 T-Qi4-14.3, not A5 T7"; two body sites still said A5 T7.
%%  T-CON.2 (1 site) has no basis -- the paper uses T-CON and T-CON.1.
%%  The SIP is five-field in 14 sites and a seven-tuple in 2, both of
%%  the latter in PART VI.  Landauer was quoted at 1.4 k_B T per bit;
%%  per bit is k_B T ln 2 = 0.693, and 1.4 is the per-nucleotide figure.
%% ====================================================================
%% ====================================================================
%%  v13.36 -> v13.37  (July 28, 2026)  --  SHIP G1: THE N <= 2 EQUATION
%% --------------------------------------------------------------------
%%  CORRECTNESS FIX ON A HEADLINE RESULT.  Shipped alone.
%%
%%  Part VII Sec 53.2 printed, as numbered equations on page 233:
%%      S_tot(N) = sum_{k=1}^{N} pi k = pi N(N+1)/2
%%      pi N(N+1)/2 <= 3 pi  <=>  N(N+1) <= 6  <=>  N <= 2
%%  and then patched itself in prose ("with the conventional bottom-up
%%  count ... N <= 3").  Part VI Corollary 47.2 -- titled "total cascade
%%  action and the N <= 3 budget bound" -- gives pi N(N-1)/2 and N <= 3.
%%  Twelve of the paper's thirteen S_tot sites write N(N-1); this was
%%  the only N(N+1) and the only wrong generation number in the paper.
%%
%%  It is a summation-range slip, not a physics disagreement: summed
%%  over ordered pairs the same chain gives C(N,2) = Part VI's answer.
%%  Equation (81), the per-transition instanton action pi k, is
%%  consistent with Part VI and is KEPT -- it is the Morse content the
%%  subsection exists for.
%%
%%  The budget derivation is deleted rather than repaired: Part VI is
%%  its home, and a third independent derivation of the cohort's
%%  headline integer is how this arose.  The replacement states the
%%  consequence, pins the summation convention in one clause, and
%%  points at Corollary 47.2.  Both citations in the deleted block are
%%  carried into the replacement; no reference is orphaned.
%%
%%  Same paragraph: the Bogomolnyi action quantum was credited to A6
%%  T-GRIP-3.  It is T-GRIP-4 (31 sites); T-GRIP-3 is per-saddle
%%  uniqueness (11 sites).
%% ====================================================================
%% ====================================================================
%%  v13.35 -> v13.36  (July 28, 2026)  --  SHIP E7: GRADE LANGUAGE
%% --------------------------------------------------------------------
%%  E6 demoted three Part VI results to corollaries.  The prose around
%%  them still called them theorems, in eleven places, and in one place
%%  argued against the grade outright: Sec 47.7's opening said "This is
%%  not a corollary of T-Origin written out a layer at a time" sixteen
%%  lines above its own proof outline saying the induction "inherits its
%%  hypotheses (Q1)-(Q4); it is not a second, independent warrant".
%%
%%  That contradiction predates this session.  E6 took the proof
%%  outline's side; this ship makes the narration agree.  The opening's
%%  true content -- that the result is load-bearing, being what makes
%%  the tower recursion well-defined -- is kept; the false clause about
%%  logical status goes.
%%
%%  Also: two forward references ("the next theorem in this deck", "the
%%  next theorem") that named the wrong result even before E6 are now
%%  named references that cannot drift.  And six "Theorem~\ref{}"
%%  prefixes pointing at corollary labels become "Corollary~\ref{}".
%%
%%  CHECKED AND DELIBERATELY LEFT: the uniqueness corollary quantifies
%%  over n in {1,...,17}, not 18.  Layer 18 is the cohort's single
%%  Existence-Demonstrated layer, so a uniqueness claim there would
%%  contradict its confidence grade.  The range is correct.
%% ====================================================================
%% ====================================================================
%%  v13.34 -> v13.35  (July 27, 2026)  --  SHIP E6: THREE COROLLARIES
%% --------------------------------------------------------------------
%%  Part VI carried six named theorems.  Measured: only two are cited
%%  from inside an argument outside the Part (T-Universal-Identity, 5
%%  refs from Part IX; T-Cascade-Layer-Uniqueness, 2).  The other four
%%  are cited only from the Technical Appendices -- that is, from the
%%  theorem index, which lists every theorem by construction.  And
%%  T-Cascade-Layer-Uniqueness declares in its own proof outline that it
%%  "is not a second, independent warrant".
%%
%%  Three become COROLLARIES:
%%    T-Bogomolnyi-Action        result already in Part IV and Part V;
%%                               follows from A6 T-GRIP-4 + A2 T-GG6-6
%%    T-SIP-Tower-Recursion      recursion stated in four other places
%%    T-Cascade-Layer-Uniqueness T-Origin at one layer, and says so
%%
%%  Three stay THEOREMS: T-GRIP-Composition (unique factorisation),
%%  T-Universal-Identity (the signature claim, 5 external citations),
%%  T-Cascade-Termination (the HAL/Goedel no-layer-19 argument).
%%
%%  This costs nothing in citability: the style file declares
%%  \newtheorem{corollary}[theorem]{Corollary} on a SHARED counter, so
%%  numbering, labels and every \ref are unaffected.  What changes is
%%  the epistemic claim -- theorem asserts a warrant, corollary asserts
%%  a consequence.  Same discipline as the Anchor-Derived tier.
%%
%%  The Part is retitled.  After E3 cut the duplicate walk it is the
%%  theorem deck plus one narrative section; "The GRIP cascade" is what
%%  Part III is about, and this Part is about how the cascade runs.
%% ====================================================================
%% ====================================================================
%%  v13.33 -> v13.34  (July 27, 2026)  --  SHIP F1: THE HELD DELETIONS
%% --------------------------------------------------------------------
%%  Two deletions flagged when they arose and held for the author's
%%  word.  Both approved; both made here.
%%
%%  (1) Layers 1, 2, 3 and 18 carry the full eleven-slot template AND a
%%      "Codebook provenance" block moved in by ship D3, which restates
%%      compactly what the slots above already say at length.  D3 moved
%%      all eighteen rather than delete four unasked.  The four
%%      redundant blocks go; the other fourteen stay, being the only
%%      provenance those layers have.
%%
%%  (2) The pillar-dominance paragraph appeared twice, 1,300 lines
%%      apart -- once beside the pillar-transition table it reads, once
%%      in the codebook commentary.  The copy beside its own table
%%      survives; the codebook copy becomes a pointer.  A reading
%%      belongs with the thing it reads.
%% ====================================================================
%% ====================================================================
%%  v13.32 -> v13.33  (July 27, 2026)  --  SHIPS E4 + E5: PART V
%% --------------------------------------------------------------------
%%  E4.
%%  (1) "The eighteen-layer table" has twelve rows, as its own opening
%%      sentence says ("twelve coarse strata ... the condensed
%%      companion").  Retitled.
%%  (2) That table has no Status column, so its Conditional and Asserted
%%      strata carry the same visual authority as the Derived ones.
%%      Adding one would mean grading coarse strata the census grades per
%%      layer; the caption now says where the grades live instead.
%%  (3) Its integer-ledger paragraph is Part IV's with five sentences
%%      dropped, including |GGA_5| = K - NL = 20.  Restored.
%%  (4) Part V's opening said "Part V will then exhibit the named theorem
%%      of self-referential closure" from inside Part V -- it is Part IX;
%%      and a label still read sec:part-iii-summit.
%%
%%  E5.  UNDOING A DUPLICATION THIS SESSION INTRODUCED.  Ship D7 wrote a
%%  full statement of the Qi-4 selector proposition into Part IV.  Part V
%%  Sec 41 was already the paper's home for it -- four pillars, canonical
%%  order, bookkeeping/selection/diagnostic/architecture, plus the
%%  admissibility theorem.  The Part IV block shrinks to the two
%%  sentences the equation needs plus a pointer.
%% ====================================================================
%% ====================================================================
%%  v13.31 -> v13.32  (July 27, 2026)  --  SHIP E3: PART VI's RESIDUE
%% --------------------------------------------------------------------
%%  (1) Three broken promises in "What this Part shows": a transitions
%%      table that does not exist, an anchoring subsection that does not
%%      exist, and "Part VII" for the self-referential closure, which is
%%      Part IX.
%%  (2) Two pointer stubs -- "Cross-reference to the canonical cascade
%%      material" and "Two regimes of GRIP: see Part III" -- folded into
%%      one sentence each.  Neither contained an argument, equation,
%%      table or theorem; both were the residue of deletions.  The
%%      cross-reference stub also mis-described tab:grip-out-events as
%%      carrying upstream theorems (it does not) and named Part III and
%%      Part IV for the same referent in one paragraph.
%%  (3) "was promoted in of the present paper" -- a version number was
%%      excised and the preposition left behind.  The sentence was
%%      version history in reader-facing text; it goes.  Same stub
%%      called a seventeen-row table "eighteen-row".
%%  (4) A paragraph inside Part VI headed "The closing observation of
%%      Part IV".
%%  (5) The duplicate walk.  Part VI carried two full narrative
%%      traversals 350 lines apart.  The first is cut: it is the only
%%      traversal in the paper whose layer indices disagree with the
%%      master codebook (triplet code at GSSB-8 vs layer 6; nerve nets
%%      at GSSB-13 vs CNS at 14; ATP on the GSSB-5 sentence vs layer 7),
%%      violating its own indexing rule three times.  Its one unique
%%      passage -- A10's single infrared GSSB event -- is rescued.
%%  (6) "The framework does not forbid further layers above layer 18"
%%      contradicted T-Cascade-Termination 300 lines later.  Reconciled
%%      in favour of the theorem.
%% ====================================================================
%% ====================================================================
%%  v13.30 -> v13.31  (July 27, 2026)  --  SHIP E2: EIGHTEEN APPLICATIONS
%% --------------------------------------------------------------------
%%  A GRIP application is what produces a GGA.  There are eighteen GGAs,
%%  so there are eighteen applications; the seventeen are the GRIP-out
%%  TRANSITIONS between them.  The Part-VI subtitle said "Seventeen GRIP
%%  applications from quarks to articulation" -- refuting itself in its
%%  own second half, articulation being layer 18.
%%
%%  (1) Five "seventeen GRIP applications" become eighteen, and the
%%      first occurrence in each Part spells out the split so the two
%%      counts cannot be confused again.  Every "seventeen transitions"
%%      is left alone: transitions really are seventeen.
%%
%%  (2) T-Seventeen-Step-Termination -> T-CASCADE-TERMINATION.  Its old
%%      name contradicted its own statement ("terminates in exactly
%%      eighteen layers") on one line, and a build comment above it
%%      still read T-Eleven-Step-Termination: the name had already gone
%%      stale once.  A name with no number in it cannot go stale again.
%%      Renamed at the subsection, theorem, label, theorem-index row and
%%      all four citing sites.
%%
%%  (3) The count is DESCRIPTIVE.  Part VI held three incompatible
%%      positions -- "OGN-5 fixes seventeen", "not derived from the
%%      integer ledger", and "fixed by the OGN-5 ledger" nineteen lines
%%      after the denial.  Part XIV already defends the middle one and
%%      it is the honest reading.  The two contradicting sentences go,
%%      and with them the kill criterion that made a descriptive count
%%      falsifiable.  Its layer-17/18 typo is fixed too.
%% ====================================================================
%% ====================================================================
%%  v13.29 -> v13.30  (July 27, 2026)  --  SHIP E1: THE N <= 3 BOUND
%% --------------------------------------------------------------------
%%  CORRECTNESS FIX ON A HEADLINE RESULT.  Shipped alone.
%%
%%  Part VI's T-Bogomolnyi-Action derived the family number from a 2*pi
%%  "unitarity-of-cascade-completion" bound and concluded "N(N-1) <= 4
%%  gives N <= 3".  It does not: N = 3 gives N(N-1) = 6, and that bound
%%  gives N <= 2 -- it would delete the third generation.  The
%%  subsection ran three incompatible bounds (k* = 3 giving N <= 4,
%%  k* = 1 giving the false N <= 3, and k* = 3/2 in the anchor, which is
%%  not an integer number of Wick loops), none of them the cohort's.
%%
%%  The paper already derives this correctly TWICE -- Part IV
%%  eq:stot-statement and Part V eq:stot-partIII -- from the BI-6 action
%%  budget S_BI = 3*pi inherited from the OGN-5 closure (K, J) =
%%  (35, 7):
%%
%%      pi N(N-1)/2 <= 3 pi  <=>  N(N-1) <= 6  =>  N <= 3
%%
%%  and N = 3 SATURATES it exactly (3 x 2 = 6).  The corrected
%%  derivation yields a saturation where the old one yielded an
%%  inequality it had to fudge.
%%
%%  Every site carrying the 2*pi bound or the k* apparatus is rewritten:
%%  opening, upstream theorem (iv), theorem statement, proof step 4,
%%  quantitative anchor, physics intuition, kill criterion (c), close.
%%  A6 T-GRIP-4 keeps the per-saddle quantum pi.  A7 T-HAL loses a role
%%  it never had here; its real use, the layer-19 prohibition, is
%%  untouched.
%%
%%  Also dropped: the claim that the juggling bound is "why the cascade
%%  has exactly eighteen layers".  The layer count is descriptive.
%% ====================================================================
%% ====================================================================
%%  v13.28 -> v13.29  (July 27, 2026)  --  SHIP D8: PILLAR-DOMINANCE BANDS
%% --------------------------------------------------------------------
%%  Found by the cohort-wide Qi-4 sweep run after D7.  The sweep cleared
%%  all eighteen other papers -- the stage-binding error was A11's alone
%%  -- but flagged two paragraphs describing how the dominant pillar
%%  shifts up the cascade.  Both ran on a retired ZERO-INDEXED,
%%  ELEVEN-LAYER scheme ("n = 0-3 ... n = 4-7 ... n = 8-10") and then
%%  jumped to layers 17 and 18 in the same breath.  One of them sits in
%%  the very subsection whose table is one-indexed 1-18 and says
%%  something different.
%%
%%  The bands are now read off that table rather than invented:
%%      layers 1-4    Qi-QUA
%%      layers 5-9    Qi-EP     (layer 10 shared with Qi-HAL)
%%      layers 10-17  Qi-HAL    (layer 17 shared with Qi-CON)
%%      layer 18      all four
%%
%%  Layer 18 taking all four is not an exception to A7's selector
%%  proposition; it is its clearest instance.
%%
%%  FLAGGED: the two paragraphs are near-verbatim duplicates, 1,300
%%  lines apart.  Both corrected; merging them is a deletion.
%% ====================================================================
%% ====================================================================
%%  v13.27 -> v13.28  (July 27, 2026)  --  SHIP D7: THE Qi-4 PILLARS
%% --------------------------------------------------------------------
%%  Two displayed equations under-braced each GRIP move with a different
%%  Qi-4 pillar -- DGI/Qi-HAL, (GFF)/Qi-QUA, GSSB/Qi-EP, GGA/Qi-CON --
%%  and told the reader this showed "which of the four Qi-4 pillars acts
%%  at each stage".
%%
%%  That is wrong in kind and in order.  A7-Qi4's Proposition "Qi-4 as
%%  the GGA selector" -- the designated home of the question, which
%%  names A11 as the paper that invokes it -- states that the four
%%  pillars are four SIMULTANEOUS admissibility filters on the candidate
%%  attractors: exactly one candidate passes all four, and that one is
%%  the global minimum of F_GFF, the stratum's GGA.  They do not own
%%  stages.  A7 also fixes their canonical order -- budget (QUA),
%%  selector (EP), diagnostics (CON), architecture (HAL) -- and the
%%  equations printed a cyclic shift of it.
%%
%%  A6-GRIP's pairing table confirms the reading from the other side: it
%%  pairs the four Gi-4 PILLARS with the four Qi-4 PILLARS (T-GG/QUA,
%%  CCI/EP, GFF/CON, GRIP/HAL) -- pillar to pillar, not stage to pillar.
%%
%%  The rest of A11 already had this right: Part III's pipeline figure is
%%  captioned "four Qi-4 pillars filter the Morse minima", the layer-1
%%  provenance says all four must pass, and Part V carries
%%  T-Qi4-GGA-Admissibility with "necessary and sufficient".  Only the
%%  two equations dissented.
%%
%%  (1) Both equations lose the under-braces and show the four-move
%%      pipeline plainly.
%%  (2) The engine section now states the selector proposition pillar by
%%      pillar, in A7's canonical order, with A7's own gloss on each.
%%  (3) Part V's parse note is corrected to match.
%% ====================================================================
%% ====================================================================
%%  v13.26 -> v13.27  (July 27, 2026)  --  SHIP D4: ONE CLOSE, NOT SIX
%% --------------------------------------------------------------------
%%  Part IV ended with six sections asserting the same three things:
%%  eighteen theorems stated; one construction applied eighteen times;
%%  Part V proves it as a single recursion.  It also carried four
%%  integrative metaphors for one object -- staircase, tunnel, waterfall,
%%  eighteen-floor building.
%%
%%  (1) Five closing sections become one.  Two pieces are rescued rather
%%      than cut: the tunnel metaphor, which glosses the five steps of
%%      the T-Origin proof and was stranded a Part early (it moves to
%%      Part V, beside the proof), and the codebook-and-Pull-table
%%      pairing argument, which is about Part IV's own artefacts and
%%      cannot travel (it opens the new close).
%%
%%  (2) "Comparison with prior frameworks at the same scope" moves to
%%      Part X, joining the paper's other two comparison sections.  It
%%      was the only one sitting in a results Part.
%%
%%  (3) THE BRIDGE POINTED AT ITSELF.  "Bridge to Part IV" sat inside
%%      Part IV, with a paragraph headed "What remains for Part V".  The
%%      source banner shows the material was written when Statement of
%%      Results was Part III.  That off-by-one, and four more like it,
%%      are corrected.
%%
%%  NOT DONE, FLAGGED: the 7D allocentric-frame subsection stays in the
%%  extended reader's guide.  Moving the author's own unpublished
%%  reaction-time programme into Part VIII is a content decision.
%% ====================================================================
%% ====================================================================
%%  v13.25 -> v13.26  (July 27, 2026)  --  SHIP D3: SIX BANDS, WHOLE LAYERS
%% --------------------------------------------------------------------
%%  Part IV promised an eleven-slot template for every layer theorem and
%%  delivered it for four of the eighteen.  Layers 4-17 carried a single
%%  slot each; their other ten lived seven hundred lines away, in the
%%  codebook's per-row provenance commentary.  The Part stated every
%%  layer twice, in two registers, in two places.
%%
%%  (1) The eighteen sections become six band sections -- Cosmos (1-3),
%%      Chemistry (4-6), Life (7-10), Multicellularity (11-13), Mind
%%      (14-17), Articulation (18) -- with each layer a subsection.  The
%%      former "From description to derivation" section becomes the
%%      Chemistry band's preamble, which is the seam it already sat on.
%%
%%  (2) Every layer's provenance block moves from the codebook section
%%      into its own subsection.  Fourteen layers get their missing
%%      template content back, next to the theorem it describes.
%%
%%  (3) Ship C3 put the paper on the canonical 3/3/7/4/1 census but
%%      missed three free-prose sites: the provenance for layers 7/8, 9
%%      and 10 still read "This row is Derived" -- the three
%%      Anchor-Derived rows, a few lines from table cells that already
%%      read Anchor-Derived.  Corrected.  The reunification is what makes
%%      them un-missable in future: each now sits directly beneath its
%%      own "Status: Anchor-Derived" paragraph.
%%
%%  FLAGGED, NOT DONE: layers 1, 2, 3 and 18 already carry all eleven
%%  slots, so their moved provenance blocks are duplicative.  Four
%%  deletions need the author's word, so they are moved with the rest --
%%  where the duplication is at least visible in one screen.
%% ====================================================================
%% ====================================================================
%%  v13.24 -> v13.25  (July 27, 2026)  --  SHIP D6: ONE INPUT LEDGER
%% --------------------------------------------------------------------
%%  (1) The constitutional input ledger was stated twice, 180 pages
%%      apart, in near-verbatim text: Part IV's "Constitutional inputs"
%%      and Part V's "The input ledger".  Part IV's survives -- it is
%%      strictly richer (GU-5, the (3+2) honesty proviso, the
%%      per-integer OGN-5 gloss, the plain-English axiom glosses) and it
%%      is where the eighteen theorems that cite AX/POS by number live.
%%      Part V's becomes a cross-reference plus the compact summary
%%      display it already ended with.  All four labels are kept, so no
%%      cross-reference anywhere in the paper can break.
%%
%%  (2) Part IV's "The cascade engine in one diagram" contains no
%%      diagram; it contains four boxed equations.  Retitled to say so.
%%      (The diagram its title promised is Part III's GRIP pipeline.)
%%
%%  (3) Part IV announced "Eighteen named theorems ... they are:" and
%%      then listed twelve -- a relic of the twelve-layer cascade, from
%%      before ATP and DNA were split and before the domain split,
%%      eukaryogenesis, multicellularity and the CNS rung were added.
%%      Now lists eighteen, matching the master codebook row for row.
%%
%%  NOT DONE, AND DELIBERATELY.  Report 43 also proposed reducing Part
%%  V's three displayed engine equations to cross-references.  Read in
%%  place they are not the same defect: Part V is where T-Origin is
%%  proved, displaying the engine where it is used is ordinary practice,
%%  and two of the three carry seven live \eqref{}s from the proof
%%  itself.  They stay.
%% ====================================================================
%% ====================================================================
%%  v13.23 -> v13.24  (July 27, 2026)  --  SHIP D5: THE FALSIFIER KEYS
%% --------------------------------------------------------------------
%%  Part IV's Pull-table audit and Part X's strategic reading were two
%%  sections both titled "The Top 10 Falsifiers", analysing two
%%  DIFFERENT lists.  They agreed on tests 1 and 2 and diverged from 3
%%  onward, and four numeric contradictions followed.
%%
%%  The cohort's canonical falsifier ledger settled what to do.  Part X's
%%  "extra" rows are not stale: the two Hubble rows are cohort F6, the
%%  recombination row is cohort F7 -- both canonical, and F7 carries the
%%  standing two-quantity z* discipline.  Part X is a cosmology-weighted
%%  slice of the COHORT ledger, not a reading of A11's curated Top 10.
%%
%%  (1) Part X sec 87 is retitled to what it is, and its framing now says
%%      plainly which of its rows are Top-10 falsifiers and which are
%%      cohort commitments outside the headline ten.
%%
%%  (2) Its bare 1-8 numbering -- which read exactly like Top-10
%%      positions, and was the whole cause of the collision -- is
%%      replaced throughout by cohort ledger keys: F15, F14, F6 (early),
%%      F6 (late), F7, F1, the family-number audit, F21.  Ledger keys
%%      cannot be mistaken for Top-10 positions, so the collision cannot
%%      recur.
%%
%%  (3) The four numeric contradictions are resolved in Part IV's favour,
%%      which is the arithmetically correct one: the ATP Pull is +1.0 not
%%      "well below 0.1"; the offset is 1.6% not 1.5%; the decision year
%%      is 2030 not 2034; A10's share is four of the eight rows tracked
%%      in Part X and two of A11's ten, both now stated and labelled.
%%
%%  (4) Part IV sec 48 is made self-consistent first, because Part X
%%      cannot be made to agree with a section that disagrees with
%%      itself: the "why eight is the right number" heading over a body
%%      arguing for ten; the aggregate time-horizon paragraph, which
%%      disagreed with its own Pull table on five rows and is now swept
%%      to it; the Dev-plot caption, which opened by calling the figure
%%      the empirical FLOOR while the body and the rest of that same
%%      caption call it the CEILING; and "Ten papers", which predates D1.
%%
%%  (5) Part X's cluster-C paragraph carried a THIRD numbering, giving
%%      "test 6" and "test 8" meanings contradicting its own table in the
%%      same sentence.  Swept.  Its calendar close said "six of the ten
%%      ... the remaining two" (6+2=8); recomputed.
%% ====================================================================
%% ====================================================================
%%  v13.22 -> v13.23  (July 27, 2026)  --  SHIP D2: ONE COHORT MAP
%% --------------------------------------------------------------------
%%  Author-approved Option B of Report 42.  The cohort was introduced
%%  twice -- Part I Sec 6.1 with Figure 6 and Table 1, then Part II
%%  Sec 7.1 with Table 2 -- over the same nineteen papers, six pages
%%  apart, under two sections sharing a title with Part II itself.
%%
%%  (1) Part I Sec 6.1 and Table 1 (tab:cohort-roster) are removed.
%%      Nothing unique goes with them: Table 1's "Role in the cohort"
%%      column is carried in full by Figure 6's caption, its
%%      plain-English achievements for the A-series are delivered by
%%      the non-physicist's guide, and its branch-paper rows were the
%%      59-100% duplicated ones.  The three \providecommand portability
%%      guards that served only that table go with it.
%%
%%  (2) Three blocks move into Part II Sec 7.1, in reading order: the
%%      five-strata paragraph, Figure 6 (both labels intact), and the
%%      Foundations-Branch paragraph.  Sec 7.1 is now one orientation
%%      spread -- the map, then the import manifest.
%%
%%  (3) \label{sec:cohort-a11} moves with the content.  The two
%%      reference-list notes that say "First cited in that section"
%%      (P4-Real, P5-CPT) therefore stay correct without being touched,
%%      and now agree with the five notes already pointing at Sec 7.1.
%%
%%  (4) Part I Sec 6 loses its subsection layer: it was "The cohort
%%      inheritance and the roadmap of the fifteen Parts" with exactly
%%      two subsections, one of which is gone.  It becomes "Roadmap of
%%      the fifteen Parts", carrying both labels, and its opening --
%%      which promised "two reader-orientation maps on the same
%%      opening" -- now promises one and points forward to the other.
%%
%%  (5) The two openers that said the same thing are merged into one:
%%      Sec 6.1's "A11 by no means stands alone" is folded into Sec 7's
%%      "A11 is the philosophical capstone of a nineteen-paper cohort".
%%
%%  (6) Sec 7 retitled "What A11 imports, paper by paper" -- the old
%%      "Eleven-paper inheritance" stood over a table captioned "the
%%      eighteen companion papers".  Sec 7.1 retitled to name both of
%%      the objects it now holds.
%%
%%  NOT CHANGED, AND THE EARLIER REPORT WAS WRONG ABOUT THEM: "the
%%  eleven inheritances" (ten per-paper subsections plus the Predictions
%%  Branch) and the non-physicist's guide's "eleven times" / "ten
%%  papers" (the A-series, A1-A11 and A1-A10) are all correct counts.
%% ====================================================================
%% ====================================================================
%%  v13.21 -> v13.22  (July 27, 2026)  --  SHIP C3: THE CENSUS CLOSURE
%% --------------------------------------------------------------------
%%  Completes ship C1 (v13.19).  C1 refined the flat 6/6/5/1 census
%%  into the canonical five tiers and wrote them into the per-theorem
%%  Status tags and the Part-IV triage list, but did not reach fourteen
%%  dependent sites.  The file has been half-old, half-new for three
%%  versions.  Every site is now on the one census:
%%
%%      Derived end-to-end        1, 2, 3                 3
%%      Anchor-Derived, deferred  7, 9, 10                3
%%      Conditional               4, 5, 6, 8, 11, 12, 13  7
%%      Asserted                  14, 15, 16, 17          4
%%      Existence-Demonstrated    18                      1
%%
%%  THE WORST SURVIVOR was the master codebook table itself -- the one
%%  the paper calls "the canonical confidence table of Part IV".  Its
%%  Status column still read Derived for layers 7, 9 and 10, in the
%%  same rows whose upstream text says the derivation is in A13 (in
%%  preparation).  "Derived" adjacent to "in preparation" is the single
%%  most quotable vulnerability in the file, and C1 existed to remove
%%  it.  Layer 11 still read Asserted.
%%
%%  THE CANONICAL LEGEND in the front matter -- the five-row table that
%%  says "defined here, once, and used with these meanings everywhere
%%  in the paper" -- said four tags, 6/6/5/1, and closed "Six, six,
%%  five, and one".  It gains the Anchor-Derived row and the true
%%  counts.
%%
%%  A SECOND DEFINITION RETIRED.  The aggregate analysis opened by
%%  saying that for the consequence layers "the confidence tag records
%%  how far the forthcoming derivation currently reaches" -- a second,
%%  incompatible meaning for the same tag.  C1 retired the third
%%  definition (the inverted empirics/theory axis) and left this one.
%%  The tag now has exactly one meaning, stated in one place.
%%
%%  TWO NUMBERS MOVE, AND THE MOVE IS THE POINT.  Derived-or-stronger
%%  falls from 7:18 (39%) to 4:18 (22%): the three biology rows were
%%  being counted as closed in the published record while their proofs
%%  live in a paper still being written.  Both figures are now reported,
%%  each labelled.  The theory-to-experiment split moves 6:5 -> 7:4,
%%  following layer 11's move to Conditional in C1.
%%
%%  STALE ITEMS SWEPT IN PASSING: Appendix H's claim to account for
%%  every Conditional and Asserted layer (it develops five of eleven);
%%  "the codebook of Part III" (it is Part IV); "the eight-test
%%  classification" (retired at v11.1 for the Top 10 Falsifiers).
%%
%%  OPEN, AND REPORTED NOT FIXED: Appendix H has no lemma subsection
%%  for the Conditional rows at layers 8, 11, 12 and 13.  Their lemmas
%%  are named in the master codebook; the roadmap does not develop
%%  them.  Writing them is new content, not a census repair.
%% ====================================================================
%% ====================================================================
%%  v13.20 -> v13.21  (July 27, 2026)  --  SHIP D1: EXECUTIVE SUMMARY
%% --------------------------------------------------------------------
%%  Front-matter compression, at the author's instruction: the
%%  Executive Summary was "solid, but slightly too long".  Three pages
%%  is an acceptable length (author, same day); the aim is density,
%%  not a page count.
%%
%%  (1) THE FIVE LISTS BECOME FIVE PARAGRAPHS.  The cascade engine,
%%      the eighteen rungs, the confidence triage, the Top 10
%%      Falsifiers and the what-is-new block were set as `itemize'.
%%      Each is now one run-in paragraph with the bold lead-ins kept.
%%      No content is removed.  The saving is not mainly the \itemsep
%%      -- it is the ragged tail of every \item (each ended mid-line)
%%      plus two \topsep per list plus the bullet indent.
%%
%%  (2) KEYWORDS DOUBLED, 8 -> 16, drawn from the paper's own
%%      \pdfkeywords set so the visible line and the embedded metadata
%%      no longer disagree.
%%
%%  NOT DONE HERE, AND OPEN.  The confidence sentence in the Executive
%%  Summary still reports the pre-C1 census (six Derived / six
%%  Conditional / five Asserted / one).  So do the master codebook
%%  table, the of-18 headcount paragraph, the promotion split and the
%%  Appendix-A confidence primer.  Ship C1 re-graded the per-theorem
%%  tags and the triage list but did not reach those five sites, so
%%  the file is currently self-consistent at the OLD census in one
%%  half and the NEW census (3 / 3 / 7 / 4 / 1) in the other.
%%  Correcting one site alone would make the paper worse, not better;
%%  the repair has to be atomic and is held for the author's
%%  go-ahead as ship C3.
%% ====================================================================
%% ====================================================================
%%  v13.19 -> v13.20  (July 27, 2026)  --  SHIP C2: THE LEDGER REPAIR
%% --------------------------------------------------------------------
%%  Stage C, ship 3 of 3.  Repairs the paper's own audit ledgers.
%%
%%  THE LAYER THAT FELL THROUGH.  Layer 14 (T-Central-Nervous-System)
%%  was absent from the Theorem Index: the rows run 8a (layer 11), 8b
%%  (12), 8c (13), then 9 (layer 15).  It is a graded cascade layer with
%%  its own theorem, its own section and its own codebook row, and the
%%  master index of named theorems did not list it.  Added as row 8d, in
%%  the format of its neighbours.
%%
%%  A STALE MACRO NAMING THE WRONG PAPER.  The lemma-promotion roadmap
%%  named \PaperFullCOne{} -- C1-Kaxion, the 8.1 microelectronvolt axion
%%  paper -- as the promotion venue for FIVE foundational lemmas:
%%  small-molecule closure, the chirality lock, the codon map,
%%  category-theoretic multi-stratum recursion, and universal-identity
%%  invariance at the meta-cognitive layer.  None of these belongs to
%%  C1.  The venues are now the papers that will actually carry the
%%  work: A12-Chemistry for the molecular lemmas, A13-Life for the
%%  genetic code, A14-Mind for the philosophy-of-mind lemma, and a named
%%  foundations successor for the category-theoretic one.  Eight sites
%%  (five roadmap rows, three prose paragraphs).
%%
%%  DUPLICATE OPEN-PROBLEM CODES.  The catalogue defines OP-1 through
%%  OP-10 as subsubsections (the theorem-level problems), and its
%%  summary longtable then opens with a SECOND block numbered OP-9
%%  through OP-13 (the experimental problems).  OP-9 therefore named
%%  both "the QCI saturation question" and "GRIP-In/Out demarcation",
%%  and OP-10 named both "the cascade's robustness to perturbation" and
%%  "Layer-13+ cascade autocatalysis".  The experimental block is
%%  renumbered OP-11..OP-15, which frees OP-9 and OP-10 for the
%%  definitions that own them and leaves the catalogue contiguous.  The
%%  old OP-11/12/13 labels carried no cross-references, so nothing
%%  points at a moved target.
%% ====================================================================
%% ====================================================================
%%  v13.18 -> v13.19  (July 27, 2026)  --  SHIP C1: THE GRADE SWEEP
%% --------------------------------------------------------------------
%%  Stage C, ship 2 of 3.  Writes the Stage-B canonical grade census
%%  (Report 41, decision B3) into every dependent site.
%%
%%  THE PROBLEM.  The census was internally consistent at 6/6/5/1, but
%%  it rested on THREE incompatible definitions of the grades:
%%    (1) the primary legend: a row is Derived when the chain is closed
%%        IN THE PUBLISHED RECORD with no unstated lemma;
%%    (2) the reading-status note: for consequence layers the tag
%%        records HOW FAR THE FORTHCOMING DERIVATION REACHES;
%%    (3) the promotion roadmap: "Asserted = empirics-strong,
%%        theory-pending; Conditional = theory-strong,
%%        empirics-pending" -- an axis that exists nowhere upstream and
%%        inverts the cohort legend.
%%  Under (1), layers 7, 9 and 10 cannot be Derived: their derivations
%%  are in A13 (in preparation), and the paper said so in the same
%%  breath as it graded them Derived.  That is the single most
%%  quotable vulnerability in the file -- "Derived" adjacent to "in
%%  preparation" -- and Part XIV already conceded the right answer:
%%  "the row is anchor-Derived but derivation-deferred; reporting these
%%  as one undifferentiated 'Derived' overstated it."
%%
%%  THE DECISION.  Definition (1) is canonical.  The Part-XIV
%%  concession is promoted from an objection response to a grade tier
%%  and applied to all three biology rows.  Definition (3) is retired.
%%
%%  ONE ROW MOVES ON THE LEGEND'S OWN TERMS.  Layer 11 was Asserted,
%%  but its own text says it is "asserted pending the corresponding
%%  LEMMA in A3" -- and pending-a-named-lemma is the definition of
%%  Conditional.  Only the retired axis (3) ("empirics-strong ->
%%  Asserted") put it in Asserted.  It becomes Conditional.
%%
%%  NOT CHANGED.  Layer 6 stays Conditional and stays a rung: its claim
%%  is honestly class-level (the polymer CLASS is the GGA; no family is
%%  selected) and it carries a class-level kill bundle, which is a
%%  legitimate weaker claim under author decision D5.  Layers 1-3 stay
%%  Derived end-to-end: they close through A9/A10 in the published
%%  record, which is exactly what the canonical definition asks.
%%
%%  A STALE PARAGRAPH REPLACED.  Part IV's "what is claimed and what is
%%  asserted" listed SIX derived (including layer 17, which is
%%  Asserted), THREE conditional and TWO asserted -- a 6/3/2/1 census
%%  contradicting the 6/6/5/1 twelve pages earlier.  It now states the
%%  canonical five tiers.
%% ====================================================================
%% ====================================================================
%%  v13.17 -> v13.18  (July 27, 2026)  --  SHIP C0: THE STAGE-B CALLS
%% --------------------------------------------------------------------
%%  Stage C, ship 1 of 3.  Writes the five Stage-B judgment calls
%%  (Report 41) into the text.  These are claim-level changes, made on
%%  author sign-off, not mechanical repairs.
%%
%%  B1 -- THE UNIQUENESS SPINE.  The spine was circular: the engine
%%  section motivates the Qi-4 filter BY the inadequacy of A6's
%%  GRIP-ST5 ("dynamically stable is operational shorthand"), and then
%%  T-Qi4-GGA-Admissibility discharged uniqueness BY citing GRIP-ST5.
%%  Restricting a landscape to a subset does not make its minimum
%%  unique.  Separately the codimension step asserted that four cuts
%%  "leave at most one survivor", which is false as stated -- cutting by
%%  codimension leaves a submanifold, not a point -- and its
%%  transversality warrant was declared "inherited".
%%  A11 is a declared preview (author decision D5), so the honest
%%  repair is not to manufacture a proof but to NAME the hypotheses
%%  under which uniqueness holds: (Q1) non-emptiness, (Q2) single-basin
%%  connectedness, (Q3) convexity of GFF on the admissible set, (Q4)
%%  transversality of the four pillar constraints.  Uniqueness is now
%%  stated as conditional on (Q1)-(Q4), in exactly the register the
%%  cohort already uses for the Born rule in A8.
%%
%%  B2 -- RECURSION AND LAYER 18.  T-SIP-Tower-Recursion's Step 4 cited
%%  T-Cascade-Layer-Uniqueness "immediately following"; that theorem's
%%  Step 2 cited T-SIP-Tower-Recursion "preceding theorem ... unique up
%%  to gauge by induction".  The words "by induction" appeared once,
%%  inside the circular step, with no base case.  The circle is broken:
%%  the recursion is well-defined by CONSTRUCTION from the base case
%%  GGA_0 = the GG-Constitution, and the induction is now written out
%%  with that base case.  Layer-18 uniqueness is marked Asserted --
%%  T-Cascade-Layer-Uniqueness's proved range is n <= 17, and the
%%  layer-18 claim is a structural prediction, not a consequence.
%%  "Exactly eighteen layers -- not seventeen, not nineteen" softens to
%%  the resolution the theorem's own Step 4 already gives: the
%%  constructive count is an empirical enumeration, and the structural
%%  claim is TERMINATION, not cardinality.
%%  Also repaired in passing: T-Universal-Identity's Step 5 capped its
%%  induction at "n = 0 to n = 10" for an eighteen-layer cascade.
%%
%%  B4 -- CLOSURE-THEOREM LOGIC.  Statement (iv) is a universal
%%  negative ("No alternative admissible geometry admits a closed
%%  loop") asserted from four failure modes never shown exhaustive, and
%%  the gloss called it "the contrapositive", which it is not -- it is
%%  an independent universal claim.  It is now fenced as a checked-cases
%%  argument with exhaustiveness named as a conjecture.  Statement (iii)
%%  collapsed type into token (the articulation IS the ledger); the
%%  identity is between the articulation's CONTENT and the constitution.
%%  And the paper had no paragraph on why a theorem whose conclusion is
%%  its own articulation escapes a Godel/Tarski diagonal -- Godel
%%  appeared only to forbid layer 19.  That paragraph is added.
%%
%%  B5 -- THE EIC IDENTITY.  T-OH-IH-EIC-Identity asserted
%%  rho_EIC = rho_OH + rho_IH.  A sum of two density matrices has trace
%%  2, so the boxed object was not a state, and the defence ("the
%%  integration stratum has a port structure that supports addition")
%%  had no mathematical content.  The identity becomes the equal-weight
%%  convex combination, which is a bona-fide density matrix, and the
%%  load-bearing content is routed through the self-modelling diagonal
%%  <O_OH>_{rho_IH} that every other consciousness site in the paper
%%  already uses.
%% ====================================================================
%% ====================================================================
%%  v13.16 -> v13.17  (July 27, 2026)  --  SHIP A7b: THE VOICE RESTORED
%% --------------------------------------------------------------------
%%  (1) ARISAKA MARKERS, at the slots the style guide itself names --
%%  section and subsection openings, the "Physics intuition" remarks, the
%%  bridge paragraphs, the Discussion and the Conclusions.  Each one was
%%  placed only where the sentence already performed the rhetorical
%%  function the marker names; a marker that has to be forced is worse
%%  than none, and the guide says so.  Every insertion was checked
%%  mechanically before it was applied: the anchor unique in the body,
%%  the LaTeX macro multiset identical before and after, ASCII only, the
%%  change confined to prose (never a heading, label, float or table
%%  cell), and exactly one marker per sentence.
%%
%%  (2) SEMICOLON CHAINS.  The guide's DON'T table calls three or more
%%  semicolons in one sentence "LLM rehearsed cadence" and asks for
%%  em-dashes or short follow-up sentences.  Sixty-three sentences in the
%%  body carried four or more.  Most of them turned out to be legitimate:
%%  formal SIP tuple definitions whose parallelism IS the definition's
%%  shape, lettered falsifier enumerations, table cells, display math,
%%  and -- most importantly -- the deliberate anaphoric cascades, where
%%  each clause takes up the previous clause's object ("Stars produce
%%  supernovas; supernovas produce heavy elements; ...").  Those are the
%%  author writing, not a machine, and they are untouched.  Fifteen were
%%  genuine rehearsed cadence and are rebuilt as two or three sentences.
%%
%%  ON DENSITY.  The guide's target is ~1 marker per page, measured on
%%  the author's own 2022 preprints.  Applied literally to a 476-page
%%  paper that is more than a third appendices, theorem indices, tables
%%  and reference notes, that would mean some 470 insertions -- which is
%%  the rehearsed cadence the same guide forbids.  This ship instead
%%  saturates the slots the guide names and reports the number honestly.
%%
%%  NOT TOUCHED.  "After all," stays at ten sites (author instruction:
%%  minimum usage, highest impact only).  Parts XI and XII keep their
%%  warm personal voice; they needed the fewest additions because that
%%  voice had survived there.  No claim, number, grade, theorem name or
%%  citation is changed anywhere in this ship.
%% ====================================================================
%% ====================================================================
%%  v13.15 -> v13.16  (July 27, 2026)  --  SHIP A4b: THE FLAGGED VALUES
%% --------------------------------------------------------------------
%%  The two value changes ship A4 flagged for the author are confirmed
%%  and stand as made.  This ship closes the one site A4's sweep did
%%  not reach.
%%
%%  THE MISSED SITE.  The GFF Lyapunov descent-rate paragraph gave the
%%  layer-7 port rate as 1e19 per second.  At QB = 0.1555 eV that is
%%  0.25 W for a single cell.  The paper's own layer-10 anchor -- a
%%  1--10 micron cell dissipating 1--100 W/kg -- spans 1e-15 to 1e-10 W,
%%  which is 4e4 to 4e9 quanta/s.  Set to ~1e9, the top of the paper's
%%  own band, matching the value A4 already set at the SeOS-class site.
%%
%%  THE WRONG SYMBOL.  The same clause carried the rate on J_E.  By the
%%  Arisaka equation J_E/QB = J_e = J_B = J_Qi, so J_E is a POWER and
%%  J_Qi is the quanta rate; the sentence gave J_E in quanta/second.
%%  The rate now rides on J_Qi, as it does at the SeOS-class site.
%%
%%  THE LAYER-17 CLAUSE.  It read "the rate is ~ J_E . QB . J_Qi with
%%  J_Qi ~ 1e7 bits/s".  That triple product is (quanta/s)(J)(bits/s),
%%  which is not a rate of anything, and it used J_Qi twice over for two
%%  different quantities.  No source in the cohort defines this product:
%%  A7-Qi4 has no such expression.  Rather than invent one, the clause
%%  now states that the layer-7 FORM carries over and quotes the 1e7
%%  bits/s figure as what it is -- the standard bandwidth estimate for
%%  an alert human brain -- without converting bits into QB-quanta.
%%  A closed-form layer-17 descent rate is a Stage B question.
%% ====================================================================
%% ====================================================================
%%  v13.14 -> v13.15  (July 27, 2026)  --  SHIP A9: MATTER AND REFS
%% --------------------------------------------------------------------
%%  Surgery stage A, ship 9 of 10 -- the last of stage A.
%%
%%  APPENDIX-LETTER DRIFT ACROSS PAPERS.  Six sites pointed at an
%%  "Appendix K" for a lemma-promotion roadmap.  Appendix K of the
%%  present paper is "The bioenergeticist and biophysicist" (the
%%  appendices run A..Q); the roadmap is Appendix H.  Appendix K of A9
%%  is "Constitutional ledgers and Standard-Model architecture"; A9's
%%  promotion roadmap is its Appendix N -- and it was Appendix M one
%%  version earlier, which is how the drift happened.  A1 has no
%%  appendices at all, so "A1's Appendix K" named nothing.
%%  Repaired so it cannot drift again: references to the present
%%  paper's roadmap now go through \ref{app:lemma-promotion-roadmap},
%%  and references to another paper's roadmap name it by TITLE rather
%%  than by a letter that its own next version may renumber.
%%
%%  PART-NUMBER DRIFT.  The paper has fifteen Parts; Part VIII is the
%%  cognitive cascade and Part IX is the self-referential closure.
%%  Twelve sites attributed the closure theorem, its corollary, its
%%  careful-word section or the objection against it to Part VIII.
%%  Three of the twelve were proved wrong mechanically (the label cited
%%  beside the Part number lives in Part IX); the rest were resolved
%%  against each Part's section list.  Two further sites in the same
%%  passage shifted the four-sciences exercise (Part X) and the
%%  cross-domain applications (Part XI) by one Part, and one reference
%%  in the bibliography notes sent the reader to Part VII for
%%  consciousness-as-internal-measurement, which is Part VIII.
%%
%%  ONE JUDGEMENT CALL, FLAGGED FOR THE AUTHOR: "We now restate
%%  Parts~IV and IV" is a typo whose intended second numeral cannot be
%%  recovered from the text.  Set to "Parts~IV and VI" -- the Part that
%%  states the eighteen GGAs and the Part that walks the GRIP cascade,
%%  which are what the following subsection restates.  Say the word and
%%  it becomes III, or V.
%%
%%  PDF METADATA.  pdfsubject still described an "eleven-paper series";
%%  the cohort is nineteen papers (author decision D1, ship A3).
%%
%%  STALE LABEL.  fig:twelve-parts-flowchart labels a figure that
%%  enumerates FIFTEEN Parts.  Renamed at all three body sites.
%%
%%  CONSTITUTION SIZE.  The Programme Dictionary said the
%%  GG-Constitution has four axioms and ELEVEN postulates.  An earlier
%%  changelog records the cohort-wide correction of exactly this
%%  ("A11's POS8-POS11 were the four Qi-4 pillar THEOREMS of A7
%%  mislabelled as postulates"); this one site was missed by it.
%%
%%  MISATTRIBUTION.  Worrall is the canonical EPISTEMIC structural
%%  realist; he was cited for ontic structural realism.
%%
%%  APPENDIX A COMPLETENESS.  Appendix A claims to collect every
%%  programme-defined abbreviation and omitted QCI, which is used in the
%%  theorem index, the promotion roadmap and two open problems.
%%
%%  APPENDIX A TYPOGRAPHY (both found by rendering page 364 and reading
%%  it).  The GGA_10 = LUCA entry took only the 1 into the subscript and
%%  set as GGA-sub-one followed by a literal zero.  And the GU-5 constant
%%  SET, written out in the Symbol column as one unbreakable math group,
%%  overran that column and collided with the Meaning text; the acronym
%%  now stands in the Symbol column and the set inside the Meaning cell,
%%  which already spelled it out.
%%
%%  ORPHAN.  The paper's own bibliography entry was uncited (it is
%%  listed so that Section A carries the complete cohort roster).  A
%%  \nocite makes that explicit rather than leaving a dangling entry.
%%
%%  VERIFIED AND NOT CHANGED: "A1 Part VIII Appendix A convention" (two
%%  sites) is correct -- A1's Part VIII is its Technical Appendices and
%%  its Appendix A is the Summary of Abbreviations.  Dehaene is cited
%%  only for the P300 latency, which is correct usage.  No "Nobel Prize
%%  awarded to <paper>" site exists.
%% ====================================================================
%% ====================================================================
%%  v13.13 -> v13.14  (July 27, 2026)  --  SHIP A8: TABLES AND FIGURES
%% --------------------------------------------------------------------
%%  Surgery stage A, ship 8 of 10.
%%
%%  TABLE SIZING to the house standard (\small, full text width;
%%  resizebox tables exempt).  Of 27 table environments only 8 were at
%%  \small.  Fourteen carried no size directive -- among them the
%%  master cascade codebook, the theorem index, the lemma-promotion
%%  roadmap, the open-problems table and the key-symbols table -- and
%%  five were set BELOW the standard at \footnotesize or \scriptsize.
%%  Eighteen of the nineteen are now at \small.  The exception is "The
%%  eighteen-layer table": seven columns, twelve rows, a plain `tabular`
%%  that cannot break across pages and already fills one at
%%  \footnotesize.  At \small it overflowed by 61pt -- the folio printed
%%  inside the table and the last row ran off the page.  It stays at
%%  \footnotesize and joins A6's rotated table as a redesign candidate.  The one resizebox table is left
%%  alone: the charter exempts it, and the scaling would override any
%%  font directive anyway.
%%
%%  STRUCTURAL FAULTS (each first seen by rendering the page):
%%   * Appendix A opened a {\small group before its abbreviations
%%     longtable and did not close it until after the FOLLOWING figure
%%     and caption -- a float typeset inside a font group.  The group
%%     now closes where the table ends.
%%   * fig:participatory-closure named its \foreach loop variable
%%     \label, shadowing the LaTeX kernel command for the duration of
%%     the loop body.  Renamed to \lbl.  The same figure also re-draws
%%     two of its eight nodes on top of themselves; the text and the
%%     coordinates are identical, so the overprint has no visible
%%     consequence, and it is left alone rather than restructured.
%%   * Appendix A printed the A1 original acronym-hierarchy figure and
%%     then described, for a full paragraph, the REVISED figure that
%%     lives in Part I -- so the words and the picture on that page
%%     were about different diagrams.  The paragraph now says which
%%     figure it describes and where that figure is.
%%
%%  No content, claim, number or grade changes in this ship.
%% ====================================================================
%% ====================================================================
%%  v13.12 -> v13.13  (July 27, 2026)  --  SHIP A7: REGISTER
%% --------------------------------------------------------------------
%%  Surgery stage A, ship 7 of 10.  Reductive half of the register work.
%%
%%  Measured against Arisaka-Writing-Style-Guide.md: the guide's DO
%%  markers (Indeed / Remarkably / Roughly speaking / Let us / by no
%%  means / very much like) score ZERO in this paper, and total marker
%%  density is 0.09 per page against the guide's ~1.0 target.  The
%%  paper does not read like its author; it reads institutional.  This
%%  ship removes the institutional register.  Restoring the markers is
%%  writing rather than editing and is proposed separately.
%%
%%  (1) "After all," 45 -> 10.  Thirty of the forty-five opened the
%%      last paragraph of a section: a default closing gesture, which
%%      is exactly what the guide warns against ("seasoning, not the
%%      meal"; use it "only if the content has earned a philosophical
%%      pause").  The ten survivors are Part-level closings, six of
%%      them in Parts XI and XII, which the author asked to keep warm.
%%
%%  (2) D4.  The section "The verbatim cosmic-uniqueness statement" --
%%      the block quote beginning "The universe must have been uniquely
%%      created at the GUT scale ... that Universe was designed to
%%      follow this deterministic, inevitable evolution", together with
%%      the four paragraphs written to fence it -- is removed.  The
%%      credo "nature is so well designed as to restrict unnecessary
%%      freedom" reads as a design argument in two further places; in
%%      both the belief is kept and the design word dropped ("nature
%%      restricts unnecessary freedom, making inevitable decisions").
%%      The fifty-year arc, the Osaka high school and the signed
%%      first-person coda are deliberately untouched.
%%
%%  (3) First-in-history claims (4 body sites) hedged or removed; the
%%      fifth is Part XIV quoting the objection and stays.
%%
%%  (4) "No prior framework has ..." (8 sites) hedged to what the
%%      author can actually warrant, or restated as the structural
%%      contrast the charter permits.
%%
%%  (5) Career stakes (2 sites) removed: a staked career is not
%%      evidence, and inviting that reply costs more than it buys.
%%
%%  (6) Internal version history in reader-facing prose (4 sites):
%%      "Earlier drafts of the present paper", "preserved in v7.x",
%%      "with the v12.39 LaSalle re-framing".
%%
%%  (7) Rival-programme polemic (4 sites): "kills the landscape",
%%      "blur this distinction", "can always retreat to a deeper
%%      anthropic level", "This is not a hand-waving argument".  The
%%      structural contrasts are kept; the scoring is dropped.
%%
%%  (8) The eleven consecutive "The warrant X lacked was ..." openings
%%      in Part IX read as a scoreboard through sheer repetition.  The
%%      claims are unchanged; the openings are varied so no more than
%%      three share a form.
%%
%%  No claim, grade, number, figure or theorem changes in this ship.
%% ====================================================================
%% ====================================================================
%%  v13.11 -> v13.12  (July 27, 2026)  --  SHIP A6: HONESTY PROPAGATION
%% --------------------------------------------------------------------
%%  Surgery stage A, ship 6 of 10.
%%
%%  Part XIV concedes several points that the body still contradicts.
%%  A referee reading both finds the appendix admitting what the main
%%  text asserts.  Each concession below is the author's own, already
%%  written; this ship propagates it to the place it belongs.
%%
%%  (1) THREE-GAUGE EXHAUSTIVENESS.  Objection 23 concedes: "the
%%      `exactly three' is ARGUED from the three control-variable
%%      slices, not proven exhaustive over all admissible J_Qi
%%      configurations; the claim should read as a structural proposal,
%%      not a closed classification."  The body asserted exhaustiveness
%%      flatly in the theorem statement, in the prose beneath it, in the
%%      figure caption, in a Part VIII recap, and -- caught only by the
%%      mandatory preview read -- in the proof sketch, which also claimed
%%      a full proof in A5 one paragraph above the note now saying A5
%%      carries no proof of this triad.  All now carry the argued reading.
%%
%%  (2) THREE-GAUGE CUSTODY.  The same objection corrects the citation:
%%      the Gibbs/Friston/GG triad is NOT A5's Lindblad theorem, and
%%      A7's three-gauge result (T-Qi4-10.6) is the DIFFERENT
%%      PLT/MEPP/FEP triad -- so this triad is A11's own reading, and
%%      its registry home is an open decision.  The honest-disclosure
%%      note nevertheless sent the doubting reader to "A5's proof,
%%      which supplies the case-by-case verification".  There is no
%%      such proof of this triad.  The note now says so.
%%
%%  (3) PER-PROCESS QB INTEGERS.  Objection concedes: "those per-process
%%      integer assignments are illustrative and unsupported as printed;
%%      real remodellers hydrolyse many ATP per event ... we should not
%%      state the illustrative integers as if measured."  The body
%%      stated them as fact ("DNA replication spends one per bit;
%%      histone methylation spends three").  Now marked illustrative,
%%      with the load-bearing claim named as the single ATP anchor.
%%
%%  (4) CHEMIOSMOTIC EQUALITY.  Objection concedes the volts-for-eV
%%      point: the defensible statement is e.Delta p ~ QB (energy =
%%      charge x voltage), and the EXACT equality is a selection of
%%      Delta p = 0.155 V rather than a derivation.  Two body sites
%%      said the proton-motive force "equals" QB.  Both now carry the
%%      dimensionally explicit form.
%%
%%  (5) SeOS CUSTODY.  Objection concedes: "the present paper's
%%      iff-packaging is a reformatting of that derived class; we
%%      concede the weight rests in A7, cited at the point of use."
%%      T-SeOS-Class-Characterisation now says so where it is stated.
%%
%%  VERIFIED ALREADY CONSISTENT, no edit needed (checked, not assumed):
%%   * the SeOS provenance re-pointing (A5 T7 -> A7 T-Qi4-14.3) was
%%     completed in an earlier version; every body site already cites
%%     A7 T-Qi4-14.3 correctly.
%%   * epsilon_eff = k_B T already carries "in the closed-system limit"
%%     at the point of identification, which is what Objection 25
%%     concedes.
%%   * the QB closed-form residual (A4's m_e chain) is already flagged
%%     as an honest residual in Part XIV and is not overstated in the
%%     body's own QB derivation.
%%
%%  The layer-7 "anchor-Derived, derivation-deferred" concession is a
%%  GRADE question and belongs to the stage-B census, not here.
%% ====================================================================
%% ====================================================================
%%  v13.10 -> v13.11  (July 27, 2026)  --  SHIP A5: DE-DUPLICATION
%% --------------------------------------------------------------------
%%  Surgery stage A, ship 5 of 10.
%%
%%  Measured, not estimated: verbatim/near-verbatim duplication across
%%  the whole document totals ~230 lines, of which ~100 are deliberate
%%  (the front-matter showcase figures the Part III captions call
%%  previews).  The plan's 900-1,300 figure counted thematic
%%  restatement, which in a paper built for selective reading is
%%  largely a feature.  This ship removes only what is both large and
%%  provably safe.
%%
%%  (1) Fourteen per-layer status blocks carried the same five-line
%%      paragraph -- three distinct bodies across fourteen sites,
%%      differing only in the grade word and the companion paper.
%%      Each is now one line, same grade, same attribution, pointing
%%      at the Preview Declaration (by page, since that section is
%%      deliberately unnumbered) for what the tag licenses.
%%
%%  (2) Appendix D stated the AX-1/AX-2/AX-3 material THREE times:
%%      once in its own subsections, then twice more in unheaded
%%      re-runs buried inside the AX-4 subsection (the -dup1/-dup2
%%      equation labels are the fossil of an abandoned earlier
%%      cleanup).  Both re-runs removed.  The one unique passage in
%%      them -- the boundary-record sigma-algebra, the Dmap axiom and
%%      the operational Born assignment -- is AX-3 content and is
%%      relocated into the AX-3 subsection under its own heading.
%%      AX-4's own material is contiguous for the first time.
%%
%%  Left alone, deliberately: Part V's restatement of the Part III
%%  cascade engine (its two displayed equations are referenced three
%%  times each, and the -partIII labels show the duplication is
%%  intentional so Part V reads self-contained).
%%
%%  No grade, claim, number or figure changes in this ship.
%% ====================================================================
%% ====================================================================
%%  v13.9 -> v13.10  (July 26, 2026)  --  SHIP A4: NUMBERS AND UNITS
%% --------------------------------------------------------------------
%%  Surgery stage A, ship 4 of 10.  Companion: Report 40 sec 3.6.
%%
%%  MOLAR UNITS ON QB.  QB is a per-quantum energy, (2/5)E_Ry/K =
%%  0.15549 eV.  The molar quantity is N_A QB = 15.00 kJ/mol.  The
%%  paper wrote "QB = 15.0 kJ/mol" and "2 QB = 30.0 kJ/mol" at 28
%%  sites -- dimensionally wrong, and wrong inside a theorem statement,
%%  a kill criterion and a results table.  N_A inserted at all 28.
%%  Deliberately NOT changed: the per-bit DNA-replication cost "1 QB",
%%  which is genuinely per-quantum and correct as written.
%%
%%  ARITHMETIC, each recomputed:
%%   * m_H = v sqrt(2 lambda_H) at lambda_H = 1/8 gives 123.11 GeV, not
%%     125.  The quoted 125.1 +- 0.2 does not follow from lambda = 1/8;
%%     the value of lambda that reproduces the PDG mass is 0.1293.  The
%%     text now states the mass as the measured anchor and marks the
%%     eighth as the leading-order value, so the equation and the
%%     number no longer contradict each other.
%%   * alpha_s(M_Z) = 1/(pi e) = 0.117100, not 0.11698; the offset from
%%     PDG 0.1179 is 0.00080, not 0.00092.
%%   * Rydberg: CODATA is 13.605693122994(26) eV.  The paper printed
%%     13.6056981(40) -- wrong from the fifth decimal, with an
%%     uncertainty 1e5 times too large -- while claiming agreement at
%%     the 1e-6 level.
%%   * ATP agreement: 30.006/30.5 = 98.38%, deviation 1.62%.  "98.5%"
%%     (2 sites) -> 98.4%; the figure node's "1.5%" -> 1.6%.
%%   * OGN-5: the paper printed the closure as "K = LJ = M^2+N^2+L^2,
%%     J^2-M^2 = NL".  With (M,N,L;J,K) = (1,3,5;7,35) the first holds
%%     (35 = 35 = 35) and the second is FALSE (J^2-M^2 = 48, NL = 15).
%%     Checked against A2-GG6 directly: A2 states K = LJ = M^2+N^2+L^2
%%     with J = (M^2+N^2+L^2)/L = 7, and contains no J^2-M^2 identity
%%     anywhere.  The invented second equation is replaced by A2's own.
%%   * QCI is defined as QCI = 2S for storeys S = 1..5, so the ladder
%%     takes even values 2,4,6,8,10.  Layers 15, 16, 17 carry 6, 8, 10;
%%     layer 14 carried 5, which is not on the ladder.  -> 4.
%%   * Proton lifetime: the model-independent floor tau_p > 1e34 yr was
%%     said to be "both consistent with" the Super-K bound tau_p >
%%     2.4e34 yr.  1e34 < 2.4e34, so the floor is excluded by the very
%%     bound it was said to agree with; the sentence now says the
%%     headline prediction clears the bound and the floor sits below it.
%%   * J_Qi in a bacterium: "~1e19 quanta/s".  At QB = 0.1555 eV that
%%     is 0.25 W for one cell.  The paper's own layer-10 anchor (cell
%%     1-10 um, 1-100 W/kg) gives 1e-12 to 1e-10 W, i.e. ~1e8-1e9
%%     quanta/s.  Set to ~1e9.  *** SUBSTANTIVE VALUE CHANGE --
%%     flagged for the author; revert or adjust on request. ***
%%   * m_tau/m_mu was printed as "exp[Dn_e/K] ~ e^{2/35} . K .
%%     (depth-ratio) -> 16.817".  e^{2/35} K = 37.06, so "(depth-ratio)"
%%     silently absorbs a factor 0.4537: the display is not an equation.
%%     Replaced by the inherited statement with its A9 warrant, the
%%     ratio quoted as the measured anchor it is.  *** SUBSTANTIVE
%%     PRESENTATION CHANGE -- flagged. ***
%%   * alpha^-1(M_Z) = 2^J = 128 was listed under "zero tolerance ...
%%     any decisive empirical departure is decisive falsification",
%%     against a measured 127.95.  By its own rule the framework was
%%     falsified on its own page.  The dyadic closure is now stated at
%%     the matching scale and scheme, as the neighbouring rationals
%%     already are.
%%
%%  TWO-QUANTITY z*: the last retired one-quantity site ("z_rec = e^7 to
%%  within 0.6% of Planck recombination data", in the Conclusions list of
%%  tight predictions) is restated in the canonical two-quantity form.
%%  Four "0.4% above" figures for the switch-decoupling gap are corrected
%%  against the anchor each sentence names: 0.5% above the atomic-chain
%%  1091 (EP-3), 0.6% above z_dec 1090 / Planck 1089.92 (the GRIP-Out
%%  event table, the Test-5 note, and the Euclidean-discipline close).
%%
%%  STRUCTURAL:
%%   * a duplicated longtable foot marker in two tables (promotion
%%     roadmap, open problems) -- the second one is dropped.
%%   * \paragraph{The confidence headcount.} had no body -- it would
%%     typeset as a bold run-in followed by the next paragraph's head.
%%
%%  VERIFIED AND NOT CHANGED (Report 40 items that did not survive
%%  re-checking; recorded so they are not re-raised):
%%   * "N(N-1) <= 4 gives N <= 3".  The text does not say this.  It
%%     says S_tot(N) = pi N(N-1)/2 bounded by S_BI = 3pi, i.e.
%%     N(N-1) <= 6, which gives N <= 3 correctly.
%%   * "H_0^late = 73.0 mis-attributed to C2-Hubble".  Checked against
%%     C2 directly: 73.0 is C2's own late-time value, and A11 uses it
%%     as the Test-4 prediction.  Correct as written.
%%   * "Landauer quoted as 1.4 kT".  The paper says k_B T ln 2.  No
%%     such site exists.
%%   * "duplicate consecutive section headings in Part VII".  No
%%     consecutive identical headings exist in the file.
%%   * "deoxyribose ~1e6 hydrolytic stability is a 1e3 overstatement".
%%     The measured RNA/DNA phosphodiester rate ratio is ~1e5-1e7, so
%%     1e6 is defensible.  Left, but it wants a citation -> ship A9.
%% ====================================================================
%% ====================================================================
%%  v13.8 -> v13.9  (July 26, 2026)  --  SHIP A3: COUNTS AND COHORT
%% --------------------------------------------------------------------
%%  Surgery stage A, ship 3 of 10.  Companion: Report 40 sec 3.1, D1.
%%
%%  COHORT LANGUAGE.  Per the author's decision the cohort is nineteen
%%  papers: A-series 11, P-series 5, C-series 3.  Section A of the
%%  references, the cohort map and the roster table already agreed;
%%  the prose did not.  The 79 "eleven-paper" sites are sorted four
%%  ways rather than replaced blindly:
%%    (a) sites meaning the A-series spine (11 papers) -- CORRECT,
%%        left as "eleven-paper A-series" (3 sites, each made
%%        explicit so the number is unambiguous);
%%    (b) sites meaning the A1--A10 inheritance -- ten papers, not
%%        eleven; the wrong count is dropped (8 sites);
%%    (c) sites meaning the whole programme -- "nineteen-paper" (4);
%%    (d) rhetorical/structural sites -- "the cohort", carrying no
%%        number at all (the remainder).
%%  Choice (d) is deliberate: it makes the prose stable across the
%%  coming 19 -> 22 promotion, so that campaign touches the roster,
%%  the map and a handful of sentences rather than ~90 sites.
%%
%%  COHERENCE MAP.  The map spanned "sixteen companion papers" and had
%%  sixteen rows because P4-Real and P5-CPT had no row, though both
%%  appear in the roster table.  Rows added; count -> eighteen (the
%%  nineteen cohort papers less A11 itself).  The Foundations Branch
%%  also existed but was uncounted: "the two phenomenology branches"
%%  -> three (Predictions, Foundations, Cosmology & Astrophysics).
%%
%%  COUNTS, each re-measured before typesetting:
%%    theorem index      40 rows; "thirty-two" (5 sites) -> forty;
%%                       "thirty other" -> "thirty-nine other"
%%    achievements       9 enumerated; "thirteen" (2 sites) -> nine
%%    predecessors       14 engaged; "thirteen" (4 sites) -> fourteen
%%    open problems      10 OP + 7 EP = 17; "eighteen" (2) -> seventeen
%%    promotion roadmap  7 named G-lemmas + 6 further rows = 13; the
%%                       "seven" claims are qualified as "seven named"
%%                       and the aggregate reads thirteen
%%    inheritance blocks 12; "eleven inheritances" -> twelve
%%    recursive reading  the A-series is eleven papers, so the present
%%                       paper is the ELEVENTH iteration reading TEN
%%                       prior ones, not the twelfth reading eleven
%%
%%  PAGE COUNTS.  Measured from the deployed PDFs in [GG-v13]:
%%    the nineteen-paper cohort totals 5,333 pp -- the paper claimed
%%    "roughly three thousand" twice and "twenty-five hundred" once,
%%    both low by nearly a factor of two.  A1--A10 = 3,300 pp.
%%    A11's own page count (445 in two places, and "more than four
%%    hundred") is now \pageref{LastPage}, so it can never go stale
%%    again; the lastpage package is added for this.
%%
%%  DEFERRED: "~100 named theorems across the corpus" is left alone --
%%  a cohort-wide theorem census needs all nineteen sources parsed and
%%  five of them mark theorems without the theorem environment, so the
%%  number cannot be verified here.  Flagged for the stage-D campaign.
%%  Part-number drift (two sites naming Part VIII for the Part IX
%%  closure theorem, and one naming Part VII for the Part IX
%%  philosophical tradition) is cross-reference hygiene -> ship A9.
%% ====================================================================
%% ====================================================================
%%  v13.7 -> v13.8  (July 26, 2026)  --  SHIP A2b: MISSING LAYER BLOCKS
%% --------------------------------------------------------------------
%%  Surgery stage A, ship 2b of 10.  Companion: Report 40 sec 3.1.
%%
%%  Part IV promised a counterfactual for each of the eighteen layers
%%  and "one paragraph block per row" of provenance commentary, and
%%  delivered twelve of each.  The same six layers were absent from
%%  both lists -- 8, 9, 11, 12, 13, 14 -- so both promises were false
%%  in exactly the same place.  This ship writes the twelve missing
%%  blocks:
%%
%%    counterfactual  layer 8   the bifunctional polymer
%%    counterfactual  layer 9   the archival takeover
%%    counterfactual  layer 11  the primary domain split
%%    counterfactual  layer 12  eukaryogenesis
%%    counterfactual  layer 13  the M-ladder
%%    counterfactual  layer 14  the integrated connectome
%%    provenance      layers 8, 9, 11, 12, 13, 14 (same six)
%%
%%  Each new block follows the established local form exactly: the
%%  counterfactuals are one paragraph, "Suppose X ... The downstream
%%  consequences would be: (i)...(iv) ... <structural close>"; the
%%  provenance blocks are three paragraphs, constitutional inputs +
%%  named upstream theorem, then interpretation with the grade in bold
%%  and a physics-intuition sentence, then a daily-life metaphor and a
%%  kill criterion.  Grades are the paper's own, quoted unchanged.
%%
%%  Also: the counterfactual subsection labels were ssec:counterfactual-0
%%  .. -11, off by one against the layer numbers they head and unusable
%%  as an index.  They are referenced nowhere in the document, so all
%%  eighteen are re-indexed to ssec:counterfactual-L1 .. -L18.
%%
%%  NOTE for ship C1 (grade sweep): the layer-7 provenance block is
%%  titled "the ATP / DNA row" and claims the four-letter DNA alphabet,
%%  which is layer 8's alphabet and layer 9's archive.  The new layer-8
%%  and layer-9 blocks now state the proper custody, so C1 should trim
%%  the DNA sentence from the layer-7 block -- and note that layer 7's
%%  "Derived" tag is currently propped up in part by that borrowed
%%  claim.  Left untouched here because it is a grade question.
%% ====================================================================
%% ====================================================================
%%  v13.6 -> v13.7  (July 26, 2026)  --  SHIP A2: THE LAYER LEDGER
%% --------------------------------------------------------------------
%%  Surgery stage A, ship 2 of 10.  Companion: Report 40 sec 3.1.
%%
%%  One canonical cascade index, one-indexed 1..18, taken from the
%%  paper's own master codebook.  The GG-Constitution is GGA_0 and is
%%  NOT a cascade layer.  Seventeen GRIP-Out transitions, 1->2..17->18.
%%  The forbidden layer -- the one the termination theorem rules out --
%%  is 19.  Bands: Cosmos 1-3, Chemistry 4-6, Life 7-10, Multicellular
%%  11-13, Mind 14-17, Articulation 18.
%%
%%  Before this ship the document carried six mutually incompatible
%%  conventions (1..18, 0..16, 1..17, 0..15, a 12-row coarse-stratum
%%  scheme, and a 7-band plain-English scheme).  Sites repaired:
%%
%%   * GGA_{-1} = C  ->  GGA_0 = C  (the base case, now canonical)
%%   * "no layer 17" in the closure theorem -> "no layer 19"; likewise
%%     in OP-3, in OP-6's body and title, and in theorem-index row 20
%%   * OP-6 title "Layer-13 prevention" -> "Layer-19 prevention"
%%     (prose and open-problems table row)
%%   * OP-8 title "Layer-8 stochastic ensemble" -> "Layer-10 ..."
%%   * three further forbidden-layer sites caught by the mandatory
%%     preview-first read, then two more by a targeted regex sweep
%%     for forbiddance/above-layer language: index row 20, OP-6
%%     "What is known", the "what lies above" corollary heading and
%%     body, the termination theorem body, and the research-agenda
%%     close.  Six forbidden-layer sites in all; all now read 19.
%%   * OP-1 ranges "layers 0--6" / "layers 8--12" -> 1--7 / 10--18,
%%     matching the items each range actually enumerates
%%   * "gauge group at layer zero ... paper at layer seventeen"
%%     -> layer one ... layer eighteen
%%   * "seventeen layers 0 through 15 plus articulation at layer 16"
%%     -> "1 through 17 plus articulation at layer 18"
%%   * GGA_n symbol-table range (n=0..16) -> (n=1..18)
%%   * cascade depth N in {0..16} -> {1..18}
%%   * theorem-index commentary index n = 0..11 -> n = 1..18
%%   * SIP-tower recursion "defined for n = 1..18" -> "n = 1..17"
%%     (the map produces SIP^(n+1); n=18 would generate SIP^(19))
%%   * coarse strata S_8..S_12 -> fine layers 10, 11--15, 16, 17, 18
%%   * RNA/DNA selection layers: RNA 10->8, DNA 15->9, cell 16->10
%%   * four-letter RNA alphabet "fixed at layer 5" -> layer 8
%%   * cyanobacterial photosynthesis "Layer 17 GGA" -> Layer 11
%%   * "life emerges wherever the cascade reaches layer 7" -> layer 10
%%   * "biology-driven from Layer 17 onward/forward" -> Layer 11
%%   * the eighteen-floor building image: floors 7-16 were stale
%%     (ATP+DNA merged at 7; cell 8; ring attractor 9; map 10) and are
%%     rewritten to the canonical ledger
%%   * fig:participatory-closure: node labels and caption re-banded to
%%     GGA_0 / 1-2 / 3-6 / 7-10 / 11-13 / 14-16 / 17 / 18, and "through
%%     ten layers" -> "through eighteen layers"
%%   * Part XI summary bands 3--9 / 13--16 -> 3--6, 7--9, 10, 11--16
%%   * Appendix-A ribbon caption bands 4--7 / 8--13 -> 4--6 / 7--13
%%   * "the electron is the layer-3 attractor" -> layer-2
%%   * building-image floor 6 renamed "polymeric chemistry" so it no
%%     longer collides with floor 8 (the RNA World); the triplet-code
%%     claim is preserved verbatim and its layer is a stage-B question
%%   * two loose band labels ("biological strata (layers 4--12)";
%%     "Layers 11--18 (cognition to articulation)")
%%
%%  DEFERRED, by design, to later ships:
%%   * "exactly eighteen layers" vs "at least eighteen layers" is a
%%     claim-level question, settled in stage B and written at C1;
%%     this ship repairs only the INDICES in those sentences.
%%   * theorem-index row structure, the missing layer-14 row, and
%%     "Rows 1--17 (the eighteen per-layer ...)" belong to ship C2,
%%     which rebuilds the index; touching them here would collide.
%%   * the six missing counterfactual and provenance blocks (layers
%%     8, 9, 11, 12, 13, 14) are ship A2b.
%%   * "eleven-paper" in the participatory-closure caption is left for
%%     ship A3, which classifies all 79 cohort-noun sites at once.
%%   * fig:participatory-closure's \foreach loop variable named \label
%%     and its two overprinted nodes are TikZ mechanics -> ship A8.
%% ====================================================================
%% ====================================================================
%%  v13.5 -> v13.6  (July 26, 2026)  --  SHIP A1: THE PREVIEW DECLARATION
%% --------------------------------------------------------------------
%%  Surgery stage A, ship 1 of 10.  Companion: Report 40 and the
%%  four-stage action plan (both in [GG-Cowork]).
%%
%%  (1) NEW front-matter section "The Status of This Paper", inserted
%%      after the Executive Summary keywords and before the showcase
%%      figures.  It states, in one canonical place: that this paper is
%%      a preview keystone preprint rather than a codebook-standard
%%      derivation of the chemistry / life / mind floors; the two
%%      standards operating in the programme and where each applies;
%%      the confidence legend and its 6/6/5/1 census; the four
%%      load-bearing steps that are inheritance rather than proof;
%%      that preview status does not soften the Top 10 Falsifiers; and
%%      how to read the rest.  Typeset with the ExecutiveSummary
%%      idiom (clearpage + phantomsection + toc line + centred LARGE
%%      title) so no change to ucla-arisaka-gg-11.sty is required.
%%      The legend table follows the cohort table standard: \small,
%%      full text width, p-columns for prose.
%%
%%  (2) Page-1 date repair.  The title block printed a hard-coded
%%      "June 13, 2026" while the running header printed \PaperDate
%%      ("July 24, 2026") -- two different dates on the cover, six
%%      weeks apart.  The title block now takes the ship date.
%%
%%  (3) \PaperVersion and \PaperDate bumped.
%%
%%  NOT done in this ship (later ships, per the action plan): the
%%  Executive Summary "How confident, layer by layer" bullets still say
%%  "Demonstrated" where Part IV says "Existence-Demonstrated" (ship C1,
%%  grade sweep); pdfsubject still says "eleven-paper series" (ship A3,
%%  counts and cohort); the precise conditional forms of the four
%%  analytic residues are settled in stage B and written in at ship C1.
%% ====================================================================
%% ===================================================================
%% 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 (2026-07-20): INEVITABILITY UPGRADE =====
%% Sharpened the capstone's central agenda -- the geometric inevitability of
%% Cosmos, Life, and Mind -- at post-inheritance strength.  (1) Part XI: new
%% closing section \S sec:inevitability-all-the-way-down, 'Inevitability, all
%% the way down' -- the selection principle audited on its last axis (the
%% primitive number-field category, L-Constitution-1 of A1, candidate grade),
%% the medium shown real at every floor (harmonising with A8's summit and A7's
%% real-angle memory register; not restated), the one arrow POINTED to at
%% T-GFF-Lyapunov-Descent, the grades printed with the vision, and the credo in
%% arithmetic form.  (2) Part XII Coda: one compact first-person deepening added
%% BENEATH the existing final beat (not displacing it).  (3) Abstract: one
%% sentence carrying the number-field inevitability at candidate grade + real
%% geometry at every floor (page span unchanged).  (4) Executive Summary: the
%% closing passage folds in the same claim + the credo sentence (span unchanged).
%% Protected byte-identical: the 17-step termination theorem key (verbatim
%% count preserved), the full cascade layer-word per-token census, the
%% defer-to-A8 complex-structure
%% sentence, the arrow-custody install at T-GFF-Lyapunov-Descent, the
%% confidence-triage grades, the Part-I close + frozen cohort map, all theorem
%% statements.  Register: candidate/conditional exact; no 'proved' beyond record.
%% \PaperVersion v13.3 -> v13.4; \PaperDate July 20, 2026.
%% Script: a11_v13.4_inevitability.py.
%% ===================================================================
%% ===== v13.3 (2026-07-20): STAGE 5C -- WAVE-2 RESIDUE CLEANUP =====
%% (1) F4: Part-II inheritance-catalogue cohort-status
%%     remark on A8-QM removed -> released register (released
%%     papers carry no drafting remark).  DELIBERATELY LEFT
%%     (out of enumerated scope, forward-looking editorial
%%     content, flagged for the author): the A8-QM bibitem
%%     relevance's drafting / 'Once A8 ships' sentences.
%% Physics, theorem keys, numbers, values, grades: UNCHANGED.
%% ===================================================================
%% ===== v13.2 (2026-07-20): STAGE 2 -- ARROW-CUSTODY CONSUMER (cohort ship) =====
%% Audit RealBulk-Audit-2026-07-19, exec-summary point 4 (A10/A11 inherit
%% the GDOS second-law monotone; consumers say so in one sentence each);
%% custody anchor: A1-Constitution v13.2 AX-3 synthesis; consumer pattern
%% of A5 v13.1 / A8 v13.3 / A10 v13.4.  ONE consumer sentence installed at
%% the capstone's primary arrow site, the Physics-intuition paragraph of
%% T-GFF-Lyapunov-Descent (T-Origin-68.2, the cascade's irreversibility
%% theorem): AX-3 owns the arrow constitutionally (the AX-3 synthesis of
%% ArisakaA1Constitution); Part IX of the CPT member paper (ArisakaP5CPT)
%% is the canonical law-level derivation, over a bulk that is itself
%% exactly T-symmetric; the GDOS companion (ArisakaA5GDOS) is the
%% dynamical implementation; the cascade's irreversibility -- each
%% GRIP-Out shock stepping the geometry down one floor and never back up
%% -- is the layered face of that one constitutional arrow, consumed at
%% every layer, not re-derived on any of them.  NOT in scope (deferred):
%% A11-R1 one-way-ink bridge (T-EIC), A11-R2 succession half-sentence,
%% A11-R3 P4 dual-point.  Protected byte-identical: the seventeen-step
%% termination theorem key (verbatim count 6), the full per-token cascade
%% seventeen/eighteen census, the defer-to-A8 complex-structure
%% sentences, the Part-I close + frozen v1.4 cohort map, abstract +
%% executive summary.  No physics changed.  v13.1 -> v13.2.
%% Script: a11_v13.2_stage2.py.
%% ===== v13.1 (2026-07-19): WAVE 1 -- cohort promotion 17 -> 19 (Foundations Branch P4-Real, P5-CPT) =====
%% Standardized Part-I close ("The cohort inheritance and the roadmap of the
%% fifteen Parts"): cohort subsection now carries the frozen v1.4 nineteen-paper
%% cohort map + nineteen-paper roster + Foundations paragraph; roadmap subsection
%% carries the fifteen-Part flowchart (Parts III & V highlighted as the central
%% claim). \PaperPFour/\PaperPFive macros + ArisakaP4Real/ArisakaP5CPT bibitems
%% added. Register repairs ("formerly Gi4", C2 "in preparation"). Cascade-layer
%% cascade-layer "seventeen"/"eighteen" semantics and the seventeen-step
%% termination theorem key are untouched.
%% =====================================================================
%% ===== v13.0 (2026-07-16): promotion to the v13 series (GG-v13 project) =====
%% Surgical fix from the A3 hostile-review Tier C plan (author-approved):
%% 'the unique mediator' scoped to the X4-coupled Green-Schwarz channel,
%% consistent with A3 v13.1's tensor-sector accounting (one mediator +
%% gauge-singlet spectators). No other changes.
%% =====================================================================
%% v12.67 (2026-07-16)  TWO-QUANTITY Z* SWEEP (cross-paper F7 item of
%%   the A10 z* hostile audit, author-approved).  A11's test-5 block
%%   was the deepest pre-two-quantity survivor in the cohort: the
%%   Pull-table row, the cosmology-block/cluster-B discussions, the
%%   Bayesian-information paragraphs, the doomsday scenario, the
%%   custodian paragraph, the Euclid close, the answerable-questions
%%   entry, the EP-3 ledger row, and the whole "Path to closure for
%%   row 5" subsection all ran "recombination z* = e^7 vs Planck" with
%%   a "recombination-history-uniqueness lemma" and an upgrade/
%%   downgrade decision schedule.  All now carry the two-quantity form
%%   of A10 sec:two-redshifts: the corridor switch z_sw = e^7 is a
%%   topological marker (no observational handle of its own); the
%%   decoupling z_dec is derived by A10's own atomic chain with zero
%%   fitted parameters, on the Planck value at the 0.1% level; the
%%   switch--decoupling offset is COMPUTED (delta = 0.0041, 0.4% in
%%   e-folds), its promotion to a theorem waiting on A10's e-fold-
%%   budget campaign; the kill lives on the observable alone (measured
%%   z_dec off the atomic-chain value by >0.5%).  The subsection is
%%   retitled "Row 5 in the two-quantity form" (label kept); the
%%   conditional gate is A10's closed-packet lemma (Conj. VIII), the
%%   same lemma as the fractions the chain consumes.  Narrative
%%   "recombination at z* = e^7" mentions (cascade layers, chronology,
%%   TikZ event labels, glossary) relabelled switch/decoupling.
%%   v12.66 -> v12.67.  Script: a11_v12.67_two_quantity_zstar.py.
%% =====================================================================
% v12.66 (July 15, 2026): gauge-phase calibration + summit echo
%   (complex-variables thread, author-approved). The geometrization-of-
%   quantum-physics passages calibrated to A8-QM v12.6's two-complex-
%   structures treatment: parallel transport geometrises the GAUGE
%   phase; the state-space i is causal-positive (quasi-free regime),
%   coinciding with the circle generator on charged sectors under the
%   particle-antiparticle pairing; one-paragraph summit echo added with
%   completed-tense pointer to A8. No physics, counts, or structure
%   changed. Also: AharonovBohm1959 relevance note scoped to the
%   gauge phase (and its doubled period removed).
%% v12.65 (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.
%% v12.64 (2026-07-14, SHOWFONT) -- showcase refinements:
%%   (1) Font enlargement confined to each tikz block, geometry preserved
%%       (box sizes and side/edge labels untouched so figure dimensions and
%%       the working page layout are kept): 5-floor cascade (Figs 2 & 8)
%%       base \sffamily->\large, subtitles \footnotesize->\small, badges
%%       \scriptsize->\footnotesize, tags/labels/closure ->\small (~+20%);
%%       GRIP engine / pipeline (Figs 3 & 9) base \small->\large, subtitles
%%       \scriptsize->\small, arrowtext \footnotesize->\small (~+25-30%).
%%   (2) More space after the big top title in the two showcases (Figs 2,3):
%%       \vspace 1.2/1.0em -> 2.4em.
%%   (3) Executive-Summary now cites the three showcase figures (four-moves
%%       -> GRIP engine Fig 3; central claim -> 5-floor Fig 2; eighteen
%%       rungs -> 18-layer Fig 1).
%%   Colours / numbers / figure sizes untouched. Script: a11_v1264_fonts.py.
%% v12.63 (2026-07-14, FIGPROMOTE) -- (1) the 5-floor "Cascade of
%%   GRIP-Out with GGA" (the central concept) inserted as a NUMBERED body
%%   figure (fig:five-floor-cascade, [p]) in the Part III opener, right
%%   before the GRIP-pipeline figure (near p58), with a dedicated detailed
%%   in-text citation ("The whole story in one figure") that reads the
%%   rhythm, ties the first lock to the second horizon problem, and the
%%   last lock to the closing loop. (2) The three front-matter showcase
%%   italic notes converted to actual \captionof{figure} captions
%%   (numbered figures) each linking FORWARD to its body original, exactly
%%   the A10/C2 pattern: showcase-1 (18-layer, fig:showcase-cascade18) ->
%%   Figure fig:eighteen-layer-cascade + Table tab:grip-out-events;
%%   showcase-2 (5-floor, fig:showcase-cascade5) -> fig:five-floor-cascade;
%%   showcase-3 (pipeline, fig:showcase-gripengine) -> fig:grip-pipeline.
%%   (3) Body pipeline caption cross-link tidied (Previewed as the front-
%%   matter showcase Figure~\ref{fig:showcase-gripengine}).
%%   Adds 4 numbered figures (3 showcases + 1 body). Renumbering is
%%   \ref-safe: every figure ref uses \ref; only hardcoded number is the
%%   external A1 "Figure~10". No values/counts/results touched. Script:
%%   a11_v1263_figpromote.py.
%% v12.62 (2026-07-14, SHOW) -- harmonise the three front-matter
%%   showcase pages (5,6,7) to one A10/C2-style format: big display title
%%   + full-page figure + short italic note pointing to the numbered
%%   version. (1) Page 7 NEW caption-less display showcase of the GRIP
%%   engine (the four-move pipeline, Figure 5), ELONGATED via stretched
%%   vertical spacing (box steps 24->32mm, branches -124->-166mm, next
%%   -150->-202mm) so it fills the page like showcases 1&2; placed after
%%   showcase 2, before the front-matter toggle -> no \caption, no figure
%%   number, no renumbering (mirrors 18-layer display p5 + numbered Fig 6
%%   p78). (2) Figure 5 (p57) caption TRIMMED ~40 lines -> ~5, un-squeezed,
%%   + pointer to the page-7 showcase and the Step sections; \label kept
%%   (now referenced from the p7 note, so no longer an orphan). (3) Show-
%%   case 1 (p5) gains a short italic note under the figure. (4) Showcase 2
%%   (p6) reading note trimmed for consistency. Box colours / numbers
%%   untouched. Standalone in [GG-QA+References]. Script:
%%   a11_v1262_showcases.py.
%% v12.61 (2026-07-14, FIG) -- add the 5-layer "Cascade of GRIP-Out with
%%   GGA" as the SECOND showcase display page (page 6), right after the
%%   18-layer showcase (page 5). Caption-less \null\vfill...\vfill\clearpage
%%   display page in the house style -> no numbered figure, no renumbering,
%%   no caption-count change. Box colours reuse the EXACT house bands of the
%%   page-5 / Figure-6 cascade: COSMOS red!12/red!60!black, CHEMISTRY
%%   yellow!22/olive!60!black, LIFE green!13/green!60!black, MIND
%%   blue!12/blue!60!black, ARTICULATION orange!22/orange!70!black;
%%   constitution neutral black!5; GRIP-Out arrows orange!70!black, GRIP-In
%%   cyan!60!black, closure loop 3E8B57 (ggcascadeclose, group-local).
%%   Uses house macros (\GRIPout,\GRIPin,\GGA,\GGConstitution,\QB,\ATen).
%%   Presentation only; no values/counts/results touched. Standalone source
%%   GG-Cascade-GRIPout-GGA-5layers.tex in [GG-QA+References]. Script:
%%   a11_v1261_fig.py.
%% v12.60 (2026-07-14, SHP) -- integrate A10-Cosmos's second horizon
%%   problem (v12.102 Sec 63) into the capstone as the cascade's
%%   layer-1->3 instance. Presentation only; no theorem/corollary/remark
%%   environments added (counter untouched -> no renumbering).
%%   (1) Sec 8.11 (inheritance from A10): third import added -- the second
%%       horizon problem as the causal warrant (mold + ordering) beneath
%%       the layer-2/3 uniqueness.
%%   (2) Sec 21.1 cosmos reading: NEW paragraph "The second horizon
%%       problem: the cascade's first lock, read across the sky" --
%%       layer-2/3 GRIP-In-to-GGA on one causal patch (spatial half) +
%%       layer-1 GRIP-Out before matter existed (temporal/ordering half).
%%   (3) Sec 64.3 connection: NEW paragraph "The causal warrant,
%%       subsumed" -- second horizon problem = layer-1-to-3 instance of
%%       the recursive uniqueness theorem.
%%   (4) Philosophical close: NEW bold crown paragraph "The two questions
%%       were always one" -- "why every electron is the same" = "why we
%%       are here", two ends of one cascade.
%%   (5) Determinism section (careful word): NEW paragraph "The ordering
%%       is positive evidence, not merely an absence of tuning" --
%%       inherited-not-tuned, pointable in data (alpha 1e-5..1e-6, CMB
%%       uniformity, zero isocurvature).
%%   (6) NEW Objection O-Origin-73 (philosopher of cosmology) on the
%%       second horizon problem's novelty/role; editor's closing question
%%       renumbered 73 -> 74 (kept genuinely last).
%%   (7) Housekeeping: one Part-number mislabel fixed -- the
%%       self-referential closure theorem is Part~IX, not VIII (entry-
%%       points sentence). Q_B left as-is (2 Q_B = one ATP = 30 kJ/mol is
%%       correct and flagged throughout; no change).
%%   No reader-facing negativity toward rival programmes. No values,
%%   counts, statuses, or results changed. Script: a11_v1260_shp.py.
% v12.59 (July 5, 2026): P0 cohort-blocker fix -- tensor sector synced to
%   A10/A9 canonical DERIVED value. Retired r<<10^{-3} (one-sided, 'green
%   square consistency check') -> r = 12/N*^2 = 0.0043, TWO-SIDED bet
%   (N*=52.6; tilt n_s=0.9620 D*, V^{1/4}=8.3e15 GeV). 13 prose/table
%   sites + the Top-10 Dev-plot figure (a11_dev_plot_top10.py) updated;
%   taxonomy now '8 forward bets + 2 precision audits' (consistent);
%   catastrophe/decision-year wording synced. A bound UPGRADED to a
%   two-sided prediction; no standing point changed.
% v12.58 (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.
% v12.57 (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.
% ===== v12.56: cosmetic formatting pass (matches A10 v12.47 / A9 v20.9). (1) Global
%  caption separation \captionsetup{aboveskip=12pt,belowskip=13pt}. (2) Removed the 3
%  full-width rule dividers that immediately preceded a \GGPart (redundant with the Part
%  heading; a blank-page risk). (3) Collapsed a double \clearpage before the 'Key symbols'
%  appendix subsection. Mid-section rule dividers kept. v12.55 -> v12.56.
% ===== v12.55: (1) moved the 'For the century...' paragraph above Figure 1 (fills p.20);
%  (2) moved the Parts IX-XIII walkthrough tail below the fifteen-Parts flowchart (fills p.29);
%  (3) reformatted Table 2 (coherence map) to match Table 1: identical href first column at
%  0.14, equal 0.39/0.39 for cols 2-3. v12.54 -> v12.55.
% ===== v12.54: Figure 1 (fig:central-claim) spacing. Increased node distance
%  17mm -> 25mm so the inter-box arrows are longer and the edge labels ('no closed 4D
%  theory exists', etc.) no longer overlap the (taller) boxes. v12.53 -> v12.54.
% ===== v12.53: Figure 1 (fig:central-claim) sizing matched to A10-Cosmos Fig 4:
%  removed the \resizebox{\textwidth} wrapper so the funnel renders at its natural
%  footnotesize size (centered, ~104mm boxes) instead of being scaled up to full width.
%  Line-wrapping of the boxes retained. v12.52 -> v12.53.
% ===== v12.52: Figure 1 (fig:central-claim) legibility. Wrapped the over-long first
%  line of the orange 'Every layer of the cascade' box (and the 'open' box line) so no
%  box exceeds its 104mm min-width; the figure's natural width drops, so \resizebox to
%  \textwidth now scales it UP, enlarging all fonts. v12.51 -> v12.52.
% ===== v12.51: (1) Replaced the terse Section-7 bridge with a warm 3-paragraph
%  Part I summary that leads into Part II. (2) Completed Table 2 (tab:coherence-map):
%  added the missing P3-Electron, C2-Hubble, C3-BH rows; caption updated to 'sixteen
%  companion papers'. v12.50 -> v12.51.
% ===== v12.50: Integration polish of the end-of-Part-I on-ramp. Figure 3 (Parts
%  flowchart) enlarged 0.8 -> full \textwidth; Section 'The GG-Theory Cohort' given a
%  full strata walkthrough before the cohort map (fills the near-empty page) and a
%  closing bridge into Part II. v12.49 -> v12.50.
% ===== v12.49: Standardized end-of-Part-I on-ramp (matches A9/A10). Promoted the
%  'structure of this paper' subsection (per-Part walkthrough + 15-Parts flowchart) to a
%  new '\section{Roadmap of the Parts in this Paper}' at the end of Part I; moved the
%  17-node cohort map up from Part II into a new '\section{The GG-Theory Cohort}' (closing
%  Part I) and added the standardized 17-paper roster (A11 = this paper). The detailed
%  cross-paper coherence table (tab:coherence-map) is KEPT in Part II. v12.48 -> v12.49.
% 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.
% ===== A11-Origin v12.46 (2026-06-27): cohort de-contamination (protect-list aware). =====
% Shed GG's own bulk self-description: the (six-dimensional / 6D / compound) 'warped' bulk-geometry
% adjective dropped; the 'warp' scale-profile/depth/curvature/ratio/suppression family rewritten to
% scale-*; the GG-self bulk projection / dictionary rewritten to bulk-to-boundary projection /
% bulk--boundary correspondence. KEPT verbatim: AdS/CFT, Anti-de Sitter, the holographic principle,
% the whole Randall--Sundrum exposition + RS-analogy hierarchy passage (its scale-profile/throat
% comparison vocabulary), the disclaimer footnote where the radial-running convention is defined, the
% external renormalization-group references (de Haro / Maldacena / Witten / Susskind), and the entire
% QiHAL / Hologram naming family (Attractor-Lattice, the triad, Outer / Inner Hologram, allocentric,
% Tomography / Transform glosses). A11 is the bioenergetic HOME: NO ATP / Q_B / hydrolysis / Gibbs /
% kJ content touched. Body only (after the document body marker). No physics / values changed. =====
% ===== A11-Origin v12.45 (2026-06-25): cosmetic sweep (author-requested). =====
% (1) Overview/List-of-Figures separated (toggle false); (2) blank page removed (funnel [hbt!]->[H]);
% (3) Figure 3 alone + dictionary paragraph clearpaged to next page; (4) Table 5 -> footnotesize +
% entry 9b trimmed; (5) dev plot fonts +20% (regenerated PDF); (6) triad arrows straightened + self-
% loop tidied; (7) A1's acronym-hierarchy flowchart inserted into Appendix A after Table 8. No physics.
% ===== A11-Origin v12.44 (2026-06-25) =====
% Author-requested final touch: (1) shortened Figure 6 (fig:eighteen-layer-cascade) ~15% in height
% (rowsep 14->12mm, layer box 8->6.8mm, articulation box 10->8.5mm; fonts unchanged via the width-
% driven resizebox); (2) promoted a copy of the shortened cascade figure to a hook page right after
% the Executive Summary, titled 'Inevitable 18-Layer Cascade of GRIP-Out', no caption; (3) moved the
% Preface from before the table of contents to after it (before Part I). No physics changed.
% ===== A11-Origin v12.43 (2026-06-25) =====
% Cohort central-claim foregrounding: inserted a compact 'central claim' section + the shared funnel
% figure (fig:central-claim, payload = the whole GG cascade: 28 particle parameters of A9, the cosmic
% inventory of A10, and the strata up to life and mind) after the opening 'Why are we here?' section.
% Same funnel as A9/A10, payload layer by layer. Foregrounds logic already in A1/A5/A6/A7/A8. No new physics.
%% =====================================================================
%%  A11-Origin v12.42 (migrated onto ucla-arisaka-gg-11) -- 2026-06-22
%%  Single canonical style.  Stripped the redundant header overrides
%%  (Contents/ExecSummary/Overview/LoF) now supplied centered by gg-11;
%%  single-page Overview+LoF via the gg-11 toggle; dropped dead \LaTeX
%%  shim; Preface via \GGPreface; inherits the one canonical
%%  Acknowledgments.  KEPT A11's \GGTitlePage override.  No
%%  prose/figure/number content changed.
%% =====================================================================
%% =====================================================================
%%  A11-Origin v12.41 (form refinement: local \T{n} -> global ids) -- 2026-06-20
%%  Cross-paper \T{n} tags converted to registered cohort global ids where the
%%  target is a single registered theorem (A2 T-GG6, A5 T-GDOS, A6 T-GRIP).
%%  Pillar (A7), sub-theorem (A6 T3.2), and axiom (A1) refs KEPT as local tags
%%  (no single registered global id).  Also fixed missed SeOS '\T{-SeOS}'
%%  mis-citation -> A7 T-Qi4-14.3.  No physics changed.
%% =====================================================================
%% =====================================================================
%%  A11-Origin v12.40 (Stage 3C/3D cross-reference sweep) -- 2026-06-20
%%  (3C) SeOS re-pointed: 11x 'A5 T7' (=QM-postulate block) -> A7 T-Qi4-14.3
%%       ('Persistence criterion for SeOS').
%%  (3D) Three-Gauge registered as A11-ORIGINAL theorem T-Origin-68.1 (the
%%       Gibbs/Friston/GG projection of the inherited GFF); 'of A5' attribution
%%       dropped.  No physics changed.
%% =====================================================================
%% =====================================================================
%%  A11-Origin v12.39 (Phase 5C: Common Objections Part) -- 2026-06-20
%%  New four-Part-tail member: Common Objections and Responses (Part XIV;
%%  References -> XV, matching the 15-Part flowchart).  73 hostile-reviewer
%%  objections O-Origin-1..73 in 8 discipline sections, drawing on the
%%  corpus-aware audit; science only (the personal Coda is exempt).
%% =====================================================================
%% =====================================================================
%%  A11-Origin v12.38 (Tier-1 cross-citation re-pointing) -- 2026-06-20
%%  Corpus-aware audit fixes; NO physics/results changed.
%%  (1) GFF second-law DESCENT dF/dt<=0: A5 T2 (=Nakajima-Zwanzig) -> A6 T3.2 (26x);
%%  (2) F_GEP+F_CEP split: A5 T3 (=CPTP) -> A5 T9.2+T5 (A5 bridge table);
%%  (3) Three-Gauge: dropped the wrong 'A5 T4' (=GKLS) number; OWNERSHIP of the
%%      Gibbs/Friston/GG triad (A11-original vs new A5/A6 id) FLAGGED to author;
%%  (4) line 8170 GFF-apparatus inheritance sentence rewritten accordingly;
%%  (5) N=3 'closed three-leg' -> 'conditional three-leg' (P2-N3 v11.1 retitle, 4x);
%%  (6) amino-acid: removed nonexistent 'T-Amino-Acid-Selection of A7' -> QiFour
%%      filter of A7 + closed-form deferred to A12;
%%  (7) alphabet: 'argument is given in A7' -> A7 four-current lock + closure A13;
%%  (8) M_lock transcription slip M_6(2/5)^N removed (that factor is the VEV's);
%%  (9) 'N=3 (three colors)' -> 'family number N=3 (equivalently three quark colours)';
%% (10) consciousness: honest note that A7 T-EIC is ACCESS-scoped, phenomenal id
%%      is A11's own commitment (A7 O3 open; A8 forces only functional observers).
%% =====================================================================
%% =====================================================================
%%  A11-Origin v12.37 (Phase 5B: hostile-review main-text revisions) -- 2026-06-20
%%  Internal-consistency fixes + ambiguity removal; NO physics/results changed.
%%  (1) meta-cognitive layer numbering n=11/n=16 -> n=18 (articulation = layer 18);
%%  (2) proton lifetime harmonised: prediction tau_p>1e35 (orbifold-forbidden),
%%      floor >1e34, current Super-K 2.4e34 -- consistent with the falsifier table;
%%  (3) nucleic-acid alphabet mis-placed 'layer 4' -> layer 8;
%%  (4) ATP: bond standard free energy (2 Q_B) vs in-vivo synthesis cost (10/3 Q_B)
%%      disambiguated -- the surplus IS the far-from-equilibrium headroom;
%%  (5) conscious episode rhythm named (low theta ~5 Hz) vs the 4-8 Hz band;
%%  (6) homochirality: cardinality kept, enantiomer chain flagged as PS-Origin-2;
%%  (7) four-letter uniqueness fenced to naturally-evolved life (Hachimoji);
%%  (8) consciousness theorem: formal content vs phenomenal structural-identity
%%      HYPOTHESIS made explicit (defended, not deduced).
%% =====================================================================
%% ---------------------------------------------------------------------
%%  A11-Origin v12.36 (Stage 4.4: Innovations section + convention key) -- 2026-06-20
%%  Added 'The Ten Structural Innovations of the Origin Capstone' (I-Origin-1..10,
%%  compiled from the paper's significance prose, no new physics) + the
%%  'Global naming convention' key (label sec:naming-convention) at the end of
%%  the Discussions Part, immediately before the Conclusions Part.
%% ---------------------------------------------------------------------
%% ---------------------------------------------------------------------
%%  A11-Origin v12.36 (Stage 4.3: PS-Origin tags + section rename) -- 2026-06-20
%%  Section -> 'Problems Solved and Future Directions'.  OP-1..10 ->
%%  (PS-Origin-1..10); EP-1..7 -> (PS-Origin-11..17).  Local OP-/EP- codes and
%%  subsubsec:op-/ep- labels kept verbatim (cross-refs preserved).
%% ---------------------------------------------------------------------
%% ---------------------------------------------------------------------
%%  A11-Origin v12.36 (Stage 4.2: F-Origin falsifier tags) -- 2026-06-20
%%  (F-Origin-1..8, 9a, 9b, 10) appended to the Top-10 Falsifiers Pull-table
%%  test-name cells (table's own numbering; falsifier 9 split Life/Mind).
%% ---------------------------------------------------------------------
%% =====================================================================
%%  A11-Origin v12.36 (Stage 4.1: theorem global tags) -- 2026-06-20
%%  Appended (T-Origin-<printed-num>) to 41 own theorem-env titles (A7 form).
%%  Principal theorem keeps \textsf{T-Origin} name + gets (T-Origin-58.1);
%%  recalled A5 Three-Gauge theorem (thm:three-gauge) left untagged.  No
%%  physics changed; labels/refs unaffected (only display titles extended).
%% =====================================================================
%% =====================================================================
%%  A11-Origin v12.35 (Stage 3: References A/B split + first-cite links) -- 2026-06-20
%%  Single 178-item bibliography -> Group A (17 cohort, cohort order) + Group B
%%  (162 others, alphabetical), each its own section* + thebibliography.  Added
%%  the ArisakaA11Origin self-bibitem (+ self-citep in coherence-map sentence so
%%  it is not an orphan).  'First cited in \S\ref{<sec-label>}.' appended to
%%  every entry (label = each key's first in-text citation).  No physics changed.
%% =====================================================================
%% =====================================================================
%%  A11-Origin v12.34 (Stage 2b: Parts roadmap -> 15 Parts, full page) -- 2026-06-20
%%  Parts flowchart extended to the final fifteen-Part structure (back matter
%%  XIII Technical Appendices / XIV Common Objections / XV References baked in,
%%  as for A7); Part II relabelled 'The cohort inheritance'; caption rewritten
%%  ('twelve body Parts' -> 'fifteen Parts'); figure on its own full page.
%% =====================================================================
%% =====================================================================
%%  A11-Origin v12.34 (Stage 2a: cohort figure + cite all 17) -- 2026-06-20
%%  Cohort map (fig:inheritance-map) -> own centred full page.  Now cites all
%%  17 cohort papers: added P3-Electron + C3-BH (were figure-only) with
%%  bibitems.  'thirteen Parts' -> 'fifteen Parts'.  No physics changed.
%% =====================================================================
%% =====================================================================
%%  A11-Origin v12.33 (Stage 1: v2.0 structure / four-Part tail) -- 2026-06-20
%%  R4 front matter already present.  (1) \GGAcknowledgments moved to the end
%%  of the Coda Part (after the personal closing, before the back matter).
%%  (2) 'Appendices' Part renamed 'Technical Appendices'.  (3) References now
%%  its own Part (3-paragraph intro + bibsection spacing).  Coda preserved as
%%  the personal capstone per author; Common Objections Part follows in Stage 5;
%%  A/B reference split + first-cite links in Stage 3.  No physics changed.
%% =====================================================================
%% =====================================================================
%%  A11-Origin v12.32 (Stage 0: science/cohort reconciliation) -- 2026-06-20
%%  A11's science verified consistent with current cohort (A6 T1-T5, A5
%%  T1-T11, A2 T2/T6, A7 named QiFour theorems + GGA selector, Q_B<->ATP
%%  2 N_A Q_B=30.0, all canonical values).  Small cross-ref fixes only:
%%   (1) stale 'A3 T1' (A3 now section-numbered) -> T-BI6-9.1 (trace-ratio
%%       lock) / T-BI6-7.2 (anomaly freedom);
%%   (2) inheritance table 'A2 T1 (BI6 closure)' -> 'A2 T4 (coeff vector)';
%%   (3) two reader-facing 'eighteen papers' -> 'seventeen' (18 cascade
%%       layers/GGAs/theorems untouched -- those are correct);
%%   (4) clarified T-Qi4-GGA-Admissibility is A11's own theorem.
%% =====================================================================
%% =====================================================================
%%  A11-Origin v12.31 -- 2026-06-20: uniform ATP<->Q_B clarity.
%%  A11 already PROVES Delta G'(ATP) = 2 N_A Q_B = 30.0 kJ/mol (Dev +1.6%).
%%  Tightened 8 loose surface lines that still said 'Q_B/15 kJ/mol matches
%%  ATP' to the canonical 'one ATP = 2 Q_B'.  Added a c-ring environment-
%%  optimization forward-pointer (c8 mammals .. c10 E.coli .. c13-15
%%  alkaliphile/chloroplast; full study forthcoming).  No value changed.
%% =====================================================================
% ===== v12.30 (2026-06-16): holographic-RG reframing, Phase R2 (author choice b: keep v12.x). =====
%  Terminology only; NO physics, NO triage counts changed. No AdS-bulk branding;
%  A2/A3 bibitems AdS-free; RS bibitem-Relevance + Witten 'anti-de Sitter' title KEPT.
%  Added the holographic-RG disclaimer footnote at the GG-bulk definition; v12.29 -> v12.30.
% =====================================================================
% --- v12.29 (2026-06-16): corrected stale corpus page counts. eleven-paper (A-series) corpus
%     ~1500-2000/'fifteen hundred' -> ~3000; ten upstream papers (A1-A10) 'fifteen hundred'
%     -> twenty-five hundred (~2500); A11 self 'a hundred' -> four hundred. Also two stale
%     'fifteen Parts' -> 'thirteen Parts' (paper's own canonical count). 8 edits. ---
% --- v12.28 (2026-06-16): front-matter running headers. The Executive Summary and Preface
%     now set \markboth so the right-side header shows 'Executive Summary' / 'Preface' on
%     their pages (Overview of Contents, List of Figures, Contents already self-label). ---
% --- v12.27 (2026-06-16): Part-aware running header. Right-side version/date stamp
%     (redundant with the cover page) replaced by a live 'Part N. <short title>' pointer;
%     \GGPart given an optional short-label arg; section marks suppressed so the Part mark
%     persists; short labels assigned to all 13 Parts. Header-only change; no content edits. ---
% --- v12.26 (2026-06-15): Figure 10 cosmetic fix --- the QCI column is now left-anchored at
%     x=4.5 (right of the rung boxes, edge 3.5) and the cascade-layer column right-anchored at
%     x=-4.5, giving a clean ~1cm gap on each side so neither overlaps the central rung boxes. ---
% --- v12.25 (2026-06-15): QCI reconciliation. Tentative quantitative definition = storey-port
%     count S in {0..5}, comparative ladder QCI = 2S in [0,10] (human 5 storeys = QCI 10), so the
%     0-5 (Pagoda/monotonicity) and 0-10 (figure/codebook) scales are now ONE quantity. Relabelled
%     Pagoda storey items and the monotonicity theorem to storey S / QCI=2S (insect 6, rodent 8,
%     human 10). Marked the ladder illustrative; removed the contradictory Section-73 numeric table
%     and the 'first quantitative metric' overclaim; deferred operational scoring + calibration to A14. ---
% --- v12.24 (2026-06-15): added original-publication citations for the prior frameworks named
%     in Section 50 (anthropic Barrow-Tipler; multiverse Linde/Vilenkin/Susskind; landscape
%     Bousso-Polchinski/Susskind; asymptotic safety Weinberg/Reuter; LQG Ashtekar/Rovelli-Smolin;
%     CCC Penrose 2010; GWT Baars). 10 new bibitems (alphabetically slotted), all verified. ---
% --- v12.23 (2026-06-15): figure reassignments for readability (figure before its explaining
%     text): Fig1 pulled to p19 by moving the 'advanced-undergraduate' paragraph into Section 2;
%     Fig2 -> start of Section 4 (Programme Dictionary); Fig3 -> start of 5.1 (coherence map);
%     Fig4 -> before Section 10 (Step 1); Fig5 -> before Table 2 (GRIPout mapping table). ---
% --- v12.22 (2026-06-15): (1) moved the §6.5 QGL 'Parent laws inherited' content under its
%     header (was empty, content misfiled below 'Historical background'); (2) deleted the
%     awkward AX-3 paragraph sitting under the §7.3 AX-2 heading; (3) rewrote the §7.4
%     'observation compiler' opening + sibling usages so OGGEP's projection reads as an
%     always-on, observer-independent geometric law (no human observer/apparatus implied);
%     (4) fixed the §18.2 'Why life is inevitable' layer order to 6->7->8->9->10. ---
% --- v12.21 (2026-06-15): removed all 'Arisaka--Nariai, in preparation' mentions (3 body + 1
%     bib relevance) and the Cepeda2020 citation+bibitem, for a clean preview-level A11 that does
%     not name the unpublished collaboration. Added the two-knee 1/f K-ratio as an explicit
%     FORECAST (fixed integer ratio set by the OGN-5 ledger, plausibly the warp-depth integer K),
%     with the rigorous GDOS derivation and the empirical test deferred to A14-Mind. ---
% --- v12.20 (2026-06-15): author corrections. (1) Moved 'first central nervous system'
%     from Life to Mind in the eighteen-rungs list. (2) Dual-knee 1/f now shown under BOTH
%     Life (9a metabolic) and Mind (9b neural), Mind (one)->(two), two knees at a fixed
%     frequency ratio, counted once across substrates. (3) Introduced RUT = Recursive-
%     Uniqueness Theorem (defined + glossary + main title). (4) page count 180->400+.
%     (5) OGN-5 ledger spelled out in sec 5.5. (6) FIXED the 'transported from ten dimensions'
%     error in two places + softened a bib 'descendant' --- GG is built up 4D->6D, never from 10D. ---
% --- v12.19 (2026-06-15): corrected the GG-6 stage metric (Eq. 2, sec 6.3) to the canonical
%     A2-GG6 Theorem-2 form: warp e^{-2r} (theta-independent, GI-forced; not e^{-2A(r,theta)}),
%     compact term d(theta)^2 with no free radius (was R_6^2 d(theta)^2), ranges r in [0,K],
%     theta in [0,2pi); no continuous parameters. ---
% --- v12.18 (2026-06-15): QCI-ladder figure layer labels fixed to a clean monotone 1:1
%     (cnidarian->13, single-cell/eukaryote->12, bacterial->11, macromolecule/DNA->9,
%     prebiotic->4; garbled 10--8/7--7/5--5 removed); tightened the IIT-context neuron
%     count from ~10^10 to the observed ~86x10^9 human-brain figure. ---
% --- v12.15b (2026-06-15): thorough conscious-episode sweep --- corrected ~20 further sites
%     encoded as $100$--$400$ / $\sim 100$--$400$ / [100,400] that the v12.15 grep missed
%     (exec line, GA/GGA analogy, cognitive-strata walk, Eq.(30) tau, the Layer-17 theorem
%     statement and commentary, the Pull-table cell, the 7D mapping/series/coarse tables, the
%     SIP table + its mislabelled ~100ms theta-cycle->~200ms, the holographic-triad eqs/caption,
%     the counterfactual, and the coda). All -> ~400 ms = two ~200 ms theta cycles. ---
% --- v12.15 (2026-06-15): conscious episode corrected to ~400 ms = two ~200 ms theta cycles
%     (one complete MePMoS loop). Canon: one theta cycle ~200 ms = unconscious attention; two
%     theta cycles ~400 ms = conscious awareness = the Inner-Hologram-forming closure (A14, GG-Mind).
%     Replaced every '100-400 ms' conscious-episode mention; revised the Test-10 kill criterion to
%     the one-vs-two-theta-cycle test; fixed the inverted alpha/theta Qi-CON timescale ladder. ---
% --- v12.14 (2026-06-15): Group C forecast-reframes under the preview stance --- hard-problem
%     'dissolves' -> structural-identity hypothesis; 7D-frame relabelled the author's preliminary
%     (2022, unreviewed) preprint data; dual-knee Bedard-Destexhe marked a biophysical analog;
%     conscious-episode 100-400 ms framed as a consistency check; ring-attractor / 4D-map
%     universality stated as predictions-to-test (causal-history flagged predicted);
%     'exactly 20 amino acids' -> 20 canonical + selenocysteine/pyrrolysine caveat. ---
% --- v12.13 Script 2 (2026-06-15): factual/citation fixes (CNS chronology + Cambrian 14->15;
%     England as a dissipation bound; Lane's site-contingency respected; Kleiber 3/4 marked
%     contested); renamed the Layer-11 theorem Three-Domain-Split -> Primary-Domain-Split;
%     reworded teleological phrasing to structural register; added MSMF viability-gate sentence. ---
% --- v12.13 Script 1 (2026-06-15): preview-stance framing box + Exec echo; demoted Layer 17
%     from Derived to Asserted (preview) and recomputed the headcount to 6/6/5/1 everywhere;
%     dropped the broken 11-layer cascade-cardinality numerology in Part VI for the structural
%     (HAL + Goedel) closure and the descriptive 18-count. ---
% --- v12.12 Stage 1 (2026-06-15): objective Tier-1 consistency sweep from the
%     four-persona hostile pass. Fixed the rogue fine-structure formula; swept the
%     old 11-layer induction bounds to the 18-layer scheme; corrected a stray neural
%     layer-index; removed a mis-applied particle-data citation on the genetic code;
%     relabelled old GGA subscripts in the Part VIII SIP table; reconciled the ATP
%     bioenergetic audit (prediction 30.0 / empirical 30.5 / ~1.6% / single 5% kill).
%     Layer-17 status question held for v12.13 (author decision). ---
% v12.11 PRIMER SCHEMATICS (June 14, 2026): added two small TikZ schematic figures to the
%   concept primers -- Fig (Primer P2) the GRIP runtime engine (SIP_n -> DGI -> F_GFF descent ->
%   GSSB -> QiFour filter -> GGA_n, with GRIPin within-layer descent and the GRIPout tower
%   recursion GGA_n -> SIP_{n+1}); and Fig (Primer P3) the Three-Gauge Theorem hub-and-spoke
%   (one F_GFF -> Gibbs / Friston / action gauges). 436 pp; 0/0; L90 0/0/0; overfull >20pt = 0.
% v12.10 APPENDIX REWORK FOR GRADUATE READERS (June 14, 2026): made the Appendix a self-
%   contained companion and reordered it for readability. Removed Appendix H (stale 'textbook
%   integration' note) and the standalone 'Relocated appendices' stub (absorbed into a clean,
%   rewritten Appendix-Part opener; the old opener wrongly said 'Eight appendices' and listed
%   sections that no longer exist). Merged abbreviations+glossary into one 'A. Notation'. Added a
%   new 'C. Self-contained concept primers' with P1 always-on GDOS/OGGEP projection, P2 GRIP
%   runtime engine (extended from the old GRIP-runtime section: four moves, GA-vs-GGA, regimes,
%   SIP-tower, QiFour, worked layer-2 micro-example), P3 GFF + Three-Gauge, P4 confidence triage,
%   P5 T-Origin capsule. Moved Objections after the scaffolding. Final order A-I: Notation,
%   Intuition, Primers, 4-Axiom scaffolding, 7-Postulate scaffolding, Objections, Theorem index,
%   Lemma roadmap, Open problems. Updated Appendix-B measurement paragraphs to always-on; added
%   Primer-P1 pointers in the two AX-3 scaffolding subsections (de-dup). 435 pp; 0/0; L90 0/0/0.
% v12.9 ALWAYS-ON GDOS CLARIFICATION (June 14, 2026): per author, removed the observer/
%   measurement-collapse framing of OGGEP/GDOS that wrongly implied an observer is needed for the
%   6D->4D projection. (Fix A) Rewrote the Move-2 paragraph of S16.2 -- deleted 'the projection IS
%   the measurement'; now states the projection is ALWAYS ON (observer-independent, predates every
%   observer by ~13.8 Gyr) and that measurement is only the macroscopic special case (record-
%   amplification chain), decoherence the microscopic one; softened the 'observer forced to apply'
%   phrasing. Added a new subsection 'The projection is always on: GDOS needs no observer' importing
%   the 3-layer (constitutional/dynamical always-on; operational at each GA) argument of A5 Sec 23.
%   (Fix B) Re-led the Part II AX-3 passage with the always-on dictionary. (Fix C) Retitled the
%   achievements item to 'What quantum mechanics IS ...' + added the pre-observer-history sentence.
%   No new physics, no confidence-status change; cites existing ArisakaA5GDOS. 434 pp; clean; L90 0/0/0.
% v12.8 GU-5 COHORT RENAME (June 14, 2026): completing the cohort-wide adoption, the named
%   theorem T-CFP (Conversion-Factor Pentad) is renamed T-GU5 (Geometric Universal Five
%   Constants) everywhere; A11's 5 \T{-CFP} references updated to \T{-GU5}. Sibling papers
%   A1/A3/A8/A9/A10/P1/P3/C2 updated in parallel and recompiled clean.
% v12.7 ROUND-3 (June 14, 2026): (1) Executive Summary title restyled to match the Preface
%   (centered \LARGE\bfseries) via an A11-local \renewenvironment. (2) GU-5 adopted: A11's
%   outdated 'Conversion-Factor Triad' (3 constants, Q_B/k_B 'derived') replaced throughout by
%   the Geometric Universal Five Constants \GUfive{} = {c,G_N,hbar;k_B,Q_B} of \AFour{} -- the
%   (3 bulk + 2 boundary) pentad, with the honest proviso that no member carries continuous
%   freedom; statement subsection rewritten, glossary/tables/relevance updated (28 \GUfive uses).
%   (3) New Part~II subsection 'The geometric origin of anomaly freedom: a 75-year arc'
%   (Bianchi 1902 -> Yang-Mills -> ABJ -> Wess-Zumino -> Green-Schwarz 1984 -> Hopkins-Singer),
%   from A3-BI6 Sec 2, emphasizing geometric anomaly-freedom by the Bianchi identity as the
%   central concept and GG-Theory's uniqueness (same math as string theory, orthogonal physical
%   intuition); added Bianchi1902 + WessZumino1971 bibitems. 433 pp; 0 err/0 undef; L90 0/0/0.
% v12.6 ABSTRACT FIT (June 14, 2026): abstract tightened 298->257 words so it fits entirely
%   on the front cover page (previously ~5 lines spilled to page 2). Hook, one-geometry/one-
%   cascade physical intuition, the posits-not-proves honesty, the A9/A10-derived grounding, and
%   the ten falsifiers all preserved. Verified by rendering page 1. 432 pp; clean.
% v12.5 ROUND-2 AUTHOR REVISIONS (June 14, 2026): (1) Abstract trimmed -- dropped the
%   'chemistry/life/mind floors developed in forthcoming GG-A12/A13/A14 (in preparation)'
%   clause (kept the honest 'particles/cosmos derived in A9/A10' clause); the forthcoming-papers
%   note stays in the Executive Summary. (2) Preface gains a 3rd paragraph, 'The floors still
%   under construction', telling readers about GG-A12/A13/A14 and stating plainly that A11 is the
%   keystone, NOT at the proven level of A9/A10 -- honesty as the credibility instrument. (3) The
%   intact 'Anticipated objections and responses' section (8 objections + summary; it was never
%   deleted, only renumbered from Sec 87 to Sec 85 and is now relocated) MOVED from Discussions
%   into the Appendix per author request; its one external cross-ref (the ogn5 post-hoc-fitting
%   pointer) updated from 'Part~X' to 'the Appendix'. 0 errors / 0 undefined; L90 0/0/0;
%   overfull >20pt = 0; 433 pp.
% v12.4 LEFTOVER CLEANUP (June 14, 2026): post-dissolution tidy. Renamed figure label
%   fig:thirteen-parts-flowchart -> fig:twelve-parts-flowchart (label + 1 ref). Fixed stale
%   part counts: 'fourteen Parts' -> 'thirteen' (structure overview), 'thirteen Parts' ->
%   'twelve' (Coda climb recap); the '12 Parts (I-XII)' and '13 Parts = 12 body + 1 Appendices'
%   mentions are now accurate and left as-is. Realigned the Overview-of-Contents TikZ node IDs
%   (XI/XII/XIII -> X/XI/XII) with their three arrows so internal IDs match displayed numbers.
%   Separately, retied the plain-English 'great ascent' section (Part VI) from its pre-v11.0
%   0-based 'Door 0..16' scheme (which was also internally broken: 'seventeen doors' but a
%   'Door 11 and beyond' heading jumping to door 15/16) to the canonical 1-based 18-layer
%   scheme, layer by layer. 432 pp; 0 errors / 0 undefined; L90 0/0/0; overfull >20pt = 0.
% v12.3 PART X DISSOLVED (June 14, 2026): the thin Part X 'Applications and frontiers'
%   removed at the author's request and dissolved into Conclusions Section~98 (recursive
%   uniqueness across the cascade) as three paragraphs --- the cascade's reach beyond physics,
%   the open frontiers floor-by-floor, and the philosophical-synthesis pointer (Teilhard
%   citation preserved). Body Parts: 14 -> 13 (Discussions X, Conclusions XI, Coda XII,
%   Appendices XIII). Renumbering swept safely: \S\ref-attached intra-A11 Part references
%   reset to the part actually containing each \ref target (auto-healing several pre-existing
%   drift errors); ~30 prose/figure/table references decremented by hand; the 41 \ANine/\ATen
%   'Part~XII \S60-62' references (universal-identity synthesis) verified untouched. Overview-of-
%   Contents flowchart (nodes, arrows, phase labels, caption) rebuilt to 12 narrative parts.
%   432 pp; 0 errors / 0 undefined; L90 0/0/0; overfull >20pt = 0.
% v12.2 PHASE E PRODUCTION (June 14, 2026): bibliography/orphan tidy + overfull sweep.
%   L90 now 0/0/0 (165 bibitems; 0 orphans, 0 missing, 0 duplicates). Of 20 orphans left by the
%   Phase B-D compression: 11 re-cited where A11's surviving prose still discusses the work
%   (electroweak founders Glashow/Weinberg/Higgs at GSSB-2; Watson-Crick at the double helix;
%   Gorini-GKLS and Kraus in the CPTP/open-systems text; Seifert in the dissipative-structure
%   line; James at the binding problem; Teilhard in the predecessor engagement; cohort siblings
%   A8-QM and C2-Hubble); 9 removed as genuinely relocated to A12/A13/A14/A8 or self-citation
%   (Crick1968, Simon1962, Polchinski1998, Schlosshauer2005, Zurek2003, Miller1953, Nakagaki2000,
%   BenYishai1995, ArisakaA11Origin-self). Overfull >20pt: 2 -> 0 (worst now 18.6pt; fixed an
%   unbreakable 'DESI+Euclid+Roman' table token and moved a too-wide inline ledger formula to
%   display math). 0 errors / 0 undefined refs/cites, three-pass clean. 437 -> 435 pp.
% v12.1 PHASE D COMPRESSION (June 14, 2026): keystone compression continued (445->437pp,
%   all compile-clean, 0 errors / 0 undefined refs). Part XI: chemistry/life/mind achievement
%   recap -> headline+status+pointer (A12/A13/A14); objections + Top-10 strategic reading kept.
%   Part VI: long cascade-walk narrative -> one-breath tour + pointer (theorem deck untouched).
%   Part II: 13 per-paper inheritance recaps condensed to load-bearing imports, redundant
%   'where exercised' lists dropped, ogn5-not-fitted tightened (A8/A9/A10 anchors preserved).
%   Part VII: honeybee bioenergetics -> A13 pointer; 2 meta 'what-this-adds' subsections
%   dropped; duplicate 'Bridge to Part VI' merged (removed a STALE GGA_{0..12} numbering para).
%   Restored lost theorem label thm:t-internal-measurement-as-consciousness (5 dangling refs,
%   pre-existing breakage). All externally-referenced labels verified preserved.
% v12.0 KEYSTONE RECONCEPTION (June 13, 2026): A11 recast as the keystone/overview of the
%   corpus; chemistry/life/mind derivations reserved for forthcoming A12/A13/A14 (carved to
%   seed files). Cuts so far (548->445pp, all compile-clean, 0 undefined refs): keystone
%   framing (abstract/exec); Part X -> pointer; Part VIII partial; appendices FAQ+worked+
%   expansions+audit removed; Part IV layers 4-17 -> theorem-statement stubs + forward-program
%   paragraphs + a GDOS/GRIP 'description->derivation' framing section. Further compression
%   (Parts VII/II/VI/XI) pending toward ~300pp.
% v11.12 (June 13, 2026): readability -- split over-long prose paragraphs into 3-5
%   shorter ones. Named example ('The structure of this paper') split by Part-groups;
%   plus 53 long prose paragraphs auto-split at validated sentence boundaries (math- and
%   abbreviation-guarded). 0 mid-sentence splits; 0 new fragments; compile 0/0/0.
% v11.11 (June 13, 2026): made the OGN vs OGN-5 distinction explicit per A1/A3.
%   POS-6 = OGN (generic Occam Gauge Normalisation postulate); OGN-5 integer ledger
%   (1,3,5;7,35) and BI-6 anomaly closure are A3-BI6 THEOREMS (derived special case),
%   not postulates. POS-5 = RG (per A1). Remapped ~60 inline OGN-5/BI-6 warrant refs
%   (POS5/POS6) to the OGN postulate. A1-A10 already correct (survey).
% v11.9 (June 13, 2026): removed ILLEGAL postulates. 'Postulate' is reserved for the
%   GG-Constitution (A1: 4 axioms + 7 postulates POS1-POS7). A11's POS8-POS11 were the
%   four Qi-4 pillar THEOREMS of A7 mislabelled as postulates -> re-expressed as A7
%   T-QUA/T-EP/T-HAL/T-EIC. 'eleven postulates' -> 'seven postulates' throughout. Survey:
%   A1-A10 clean (0 illegal), A11 was the only offender (105 -> 0). NOTE: a separate
%   POS1-POS7 numbering mismatch vs A1 (A11 OGN@5/BI6@6 vs A1 RG@5/OGN@6) remains, pending
%   author confirmation of the BI-6 handling.
% v11.8 (June 13, 2026): rewrote the Five-Story Pagoda to the author's canonical
%   definition (S1 Particles 1-2; S2 Cosmology/Astronomy 2-3; S3 Chemistry 4-6;
%   S4 Biology/Life 7-13; S5 Neuroscience/Mind 14-18) in nodes, right labels, caption;
%   removed the stale 'Summary of Part IV' block. POS-numbering reconciliation pending
%   author decision on POS8-11 (A1 has only seven postulates; OGN=POS6).
% v11.7 (June 13, 2026): section-by-section smoothing pass (Parts IV/VIII/XI).
%   Verified-safe consistency fixes: stale falsifier counts (8->10), Kaxion 19->8.1 ueV
%   (3x), Bayesian 8->10 + /8 denominators, QCI-ladder cascade-layer numbers 8/9/10->15/16/17,
%   conscious episode ~400ms -> 100-400ms (2x), 'ten'->'seventeen' GSSB transitions (2x),
%   layer-10->11 ascent, mislabelled Part V->VI / VIII->IX, broken (Parts XI,XI)->XII,XIII,
%   'layers above 11'->18, Dirac g=2 (not 'anomalous'), spurious Planck1900 cite on ATP,
%   GGA_10->GGA_{10} rendering (4x), 'an Top 10'->'the Top 10' (3x).
% v11.6 (June 13, 2026): extended Executive Summary with a broad-audience closing
%   (What is new here; A philosophical conclusion; To the next generation) for life
%   scientists and philosophers. Removed the 'Canonical compliance audit' production
%   checklist appendix (internal QA, not for outside readers).
% v11.5 (June 13, 2026): front-matter refinement. Abstract shrunk (dropped the
%   'eight forward bets/two audits' phrase). Executive Summary rewritten for a broad
%   audience (life scientists): bulleted, jargon- and citation-light, four-move anatomy
%   condensed. Removed ALL reader-facing internal-version / retired-postdiction language
%   (abstract, exec, why-ten, delta_PMNS, recombination subsection, abbreviations table).
%   Added Overview of Contents + List of Figures (A1 template) + 11 short figure captions.
% v11.4 (June 13, 2026): added official PsyArXiv links to the 3 GUT theory bibitems
%   (GUT-1 r8ma3, GUT-2 doi 10.31234/osf.io/vu2kz, GUT-3 7gbkt; GUT-3 title -> Shapes);
%   globally re-sorted the entire thebibliography by first-author surname (stable, so the
%   Arisaka cohort keeps its internal order). The 7 group preprints now in alpha position.
% v11.3 (June 13, 2026): integrated the author's 2022 7D-frame perception programme
%   (4 bioRxiv experiments Vision7D1-4 + GUT-2/GUT-3 theory; GUT-1=Arisaka2022MePMoS)
%   into Test-10 commentary, layer-17 conscious-episode theorem (behavioral-evidence
%   paragraph), new Part-X subsection ssec:7d-frame-IH (7D<->A11 mapping table +
%   strong-vs-interpretive fence), and Qi-HAL/origin-of-consciousness prose. 6 bibitems.
% v11.2 (June 13, 2026): split Test 9 (dual-knee 1/f) into 9a (Life: eukaryotic
%   metabolic, layer 12) and 9b (Mind: neural EEG/ECoG, layer 17 Qi-CON two-knee).
%   Added verified neuroscience bibitems Donoghue2020, GaoVoytek2020, AggarwalRay2023,
%   BedardDestexhe2009, Cepeda2020 (Nariai co-author). Pull table now 11 rows / 10
%   falsifiers; commentary, exec list, independence/distribution, Dev plot updated.
% v11.1 (June 13, 2026): replaced the eight-test horizon with the Top 10 Falsifiers
%   (6 physics forward + 3 life + 1 mind; 8 forward + 2 audit). Rebuilt Part-IV Pull
%   table, per-test commentary (tests 3-10: dark energy w=-1, proton stability,
%   delta_PMNS=196deg, four-letter alphabet, dual-knee 1/f, conscious episode),
%   aggregate-falsifiability subsection, abstract, exec summary, why-ten paragraph,
%   cluster-failure scenario, ledger; reframed Dev plot as parameter-audit floor.
%   Retired postdictions (early-Hubble, N=3, m_tau/m_mu, z*) demoted to body audits.
%   Swept ~91 'eight-test' prose refs -> ten; labels untouched. Compile 0/0/0, 550pp.
%% v10.8 (June 12, 2026): Phase-4 hostile-referee consistency fixes (verified).
%%   A1: the coarse 12-stratum cascade table (S_0..S_11) relabelled as an
%%   explicit grouping of the eighteen fine layers (with the S_7..S_11 -> fine
%%   9/12-13/14/15/16 mapping), removing the 'eighteen GGAs layer-by-layer' /
%%   'each row is one cascade layer' contradiction with the master codebook.
%%   A2: the promotion-roadmap paragraph's stale triage (3 Conditional/2 Asserted/
%%   Derived 0,1,2,6,7,10) corrected to canonical 6 Conditional (3,4,5,7,11,12)/
%%   3 Asserted (10,13,14)/7 Derived (0,1,2,6,8,9,15). A3: the ATP prediction made
%%   consistently 2 N_A Q_B = 30.0 kJ/mol (N_A Q_B = 15.0), with empirical 30.5+-0.5
%%   named at each headline site (abstract, boxed theorem, summary table). A4: the
%%   small-molecule thermochemistry citation repointed from PDG (which does not
%%   tabulate it) to the CRC Handbook (new verified bibitem). B3: the abstract's
%%   necessity language aligned with the modal section -- 'proves'->'argues' and a
%%   conditional clause on 'the geometry requires it'. No physics changes.
%% v10.7c (June 12, 2026): Phase-3 philosophy evenhandedness pass (defect #5).
%%   (1) The Spinozan necessitarianism remark now prices its metaphysical reading
%%   as interpretive (INT), cross-referenced to the necessitarianism-vs-
%%   determinism-vs-design section, and states the framework's neutrality on the
%%   metaphysical necessity of the constitution that Spinoza actually asserted.
%%   (2) The 'Comparison with the philosophical tradition' opener reframed off the
%%   triumphalist 'completes them / supplies the warrant the predecessor could not'
%%   onto 'stands on their shoulders and aims to complete', with two explicit
%%   cautions: the framework's own warrant is provisional (unproven constitution +
%%   pending eight-test horizon, shared exposure with predecessors), and the
%%   'why are we here' identification is interpretive, not proved. The evocative
%%   register and the per-predecessor 'insight/warrant' verdicts are retained.
%% v10.7b (June 12, 2026): Phase-3 bibliography self-audit reconciliation.
%%   The internal verification appendix's bibliography items updated to the
%%   real current state: 145 -> 158 bibitems (156 canonical + 2 exceptions);
%%   Relevance coverage 145 -> 158 (100%); URL/DOI counts unchanged (24/23).
%%   Item 42 de-staled: phantom hard-coded line numbers (15852/15854/16921)
%%   removed and the inaccurate 'all surname-sorted alphabetically' claim
%%   corrected to 'broadly surname-ordered with later-appended entries' -- the
%%   artefacts that made a naive automated L90 read mis-locate the bibliography.
%% v10.7a (June 12, 2026): Phase-3 bio/mind fencing + orphan resolution.
%%   Six actionable bibliography orphans cited at their natural in-text
%%   mentions (England2013 in the SeOS lineage; Jaynes1957 at the MEPP root;
%%   Kuhn1962 at 'paradigm'; Lane2015 for eukaryogenesis layers 11; Miller1953
%%   for prebiotic amino acids layer 5; Newton1687 at the Principia benchmark),
%%   simultaneously fencing the corresponding bio/mind claims. Two verified-real
%%   external bibitems added (Woese & Fox 1977 PNAS for the layer-11 three-domain
%%   anchor; Kleiber 1932 Hilgardia for the layer-13 metabolic-scaling anchor),
%%   each with a Relevance line. Layer-16 allocentric-map row fenced with real
%%   hippocampal keys (O'Keefe-Dostrovsky 1971; Hafting et al. 2005).
%% v10.6f (June 12, 2026): Option-A residual cleanup (final tier). Part II
%%   universal-identity progression list relabelled to fine indices (proton
%%   layer 1, leptons layer 2, periodic table layer 3, ... ATP/RNA/DNA layers
%%   6--8, single-cell life layer 10); Part IX forcing-chain 'maintenance-margin
%%   crossing' coarse layer 7 -> layer 10. Completes the GSSB/layer reconciliation.
%% v10.6e (June 12, 2026): Option-A residual cleanup (third tier). Circular
%%   life-cycle TikZ node labels (Life layer 7->9, Multicellular 9->12, Nervous
%%   8-9->13-14); eight-pillar layer map (chemistry-to-life 6->9, multicellular/
%%   neural 7-9->12-14); forced-ascent and astrobiology coarse layer refs; the
%%   two general SIP-tower formula tails now close at SIP^(16). No physics
%%   changes; index reconciliation only.
%% v10.6d (June 12, 2026): Option-A residual cleanup (second tier). Remaining
%%   coarse references driven onto the fine 0--16 scheme: the Anderson/Prigogine
%%   history paragraph ('ten transitions' -> seventeen); the doors metaphor (door
%%   k := the door into Layer k; seventeen doors; chemistry-to-life = door 9,
%%   consciousness = door 15, articulation = door 16); remaining 'eleventh
%%   layer / eleven GGAs / eleventh-layer attractor' -> Layer 18 / eighteenth;
%%   coarse bare-layer lists (biology/life at 'layer 7' -> layers 7--10 as
%%   appropriate; multicellular -> layer 13; the {0..11} cascade-index range
%%   -> {0..16}). No physics changes; index reconciliation only.
%% v10.6c (June 12, 2026): Option-A reconciliation completed. (1) Prose count
%%   sweep: 'ten' GRIP applications/transitions/doors/successive -> seventeen.
%%   (2) 'eleventh layer / layers above 11 / eleventh-layer attractor' -> the
%%   closing Layer 18 / eighteenth layer; the layer-grouping and stratum-map
%%   clauses re-partitioned onto the fine 0--16 scheme. (3) Appendix cross-
%%   reference map: per-paper GSSB-k import pointers remapped to fine indices;
%%   'to all 11' -> all eighteen. (4) SIP-tower superscript reindex: SIP^(N)
%%   now denotes the layer-N packet (clean bijection), removing the duplicate-
%%   superscript bug in the Part V inductive definitions and the Part VIII SIP
%%   table/bridge (cell=SIP^9, multicellular=SIP^12, neural=SIP^13/14,
%%   cognitive=SIP^15, articulation=SIP^16); predecessor GGA references
%%   corrected accordingly. No physics changes; index reconciliation only.
%% v10.6b (June 12, 2026): Phase-2 numbering reconciliation (author Go,
%%   Option A: one fine layer scheme everywhere). GSSB-k re-defined as the
%%   symmetry-breaking whose Ground Geometric Attractor is Layer k (per the
%%   canonical tab:grip-out-events / fig:eighteen-layer-cascade). The legacy
%%   coarse 11-event indexing (GSSB-0..10) is remapped by content to the fine
%%   index across the Part VI walk, the bee example, the SeOS lineage, the
%%   Five-Story Pagoda table, the Penrose passage, the open-problems list,
%%   the multicellular/nervous prose, and the Part XI count enumeration; a
%%   one-line indexing convention added at the walk intro. Coarse bare
%%   'layer 7'=LUCA references corrected to layer 10. Appendix cross-reference
%%   map (Part-by-Part import index) reconciled in a following pass.
%%   No physics changes; index reconciliation only.
%% v10.6a (June 12, 2026): Phase-1/2 consistency pass, fork-independent fixes
%%   (author Go on canonical confidence table + 16+1=17 layer scheme):
%%   (1) Executive-Summary confidence-triage paragraph corrected to the
%%       canonical master-codebook assignment (7 Derived/6 Conditional/
%%       3 Asserted/1 Existence-Demonstrated; layer lists fixed; two->three
%%       Asserted pointer to Appendix F).
%%   (2) Central-theorem boxed equation: stray SIP superscript (11)->(16).
%%   (3) Amino-acid count formula symbol L^2 M (evaluates to 10) corrected
%%       to NL (=20) at three sites; arithmetic now consistent (35-15=20).
%%   (4) Cascade-Self-Sustenance corollary: Great-Oxygenation / Cambrian /
%%       articulation GRIP-Out events re-tagged to their canonical fine-scheme
%%       transitions (10->11, 12->13, 15->16) instead of all 'Layer 18'.
%%   No physics changes; reconciliation to the ratified canonical tables.
\pdfoutput=1
%% v10.5 (June 12, 2026): Count-sweep (author Go): one-time gloss
%%   added at the first 'cohort' mention clarifying that
%%   the corpus is the foundational A-series within the eighteen-
%%   paper programme; the corpus references themselves are accurate
%%   (the A-series numbers eleven) and are retained. No other
%%   content changes.
%% v10.4 (June 12, 2026): Cohort-map deployment (author Go):
%%   fig:inheritance-map replaced by the canonical seventeen-paper
%%   cohort map (A11 highlighted); Part II retitled from 'The
%%   cohort inheritance' to 'The cohort inheritance'. Deeper
%%   prose count-sweeps deferred to A11's next content touch
%%   (disclosed residual). No other changes.
%% v10.3 (June 10, 2026): arXiv preparation pass (front-matter audit).
%%   front-matter audit: conforms; stamp bump only.
%%   Accepted variants (author decision): existing philosophical closes
%%   in place of literal 'After all'; companion-list counts as cited.
%% v10.2 (June 9, 2026): arXiv preparation pass (cohort step 3).
%%   Overfull-box sweep (all boxes >20pt resolved with layout-only fixes);
%%   no content or physics changes.
%% v10.1 (June 9, 2026): arXiv preparation pass (cohort step 1-2).
%%   \pdfoutput=1 added (line 1); version/date stamp bumped. 
%%   No physics changes; production fixes only.
% v9.6 (2026-06-07; amended same day): Figure 4 promoted to a dedicated
%   full page ([p] float, full text width, small caption) per author
%   request -- figure + caption together occupy one page.
% v9.6 (2026-06-07): Figure 4 corrected per author -- the GA tier added
%   between GSSB and GGA (the descent is first captured into a Geometric
%   Attractor, possibly a way station; GRIP-In's loop attaches at GA,
%   continuing through exit channels; the GGA is where GRIP-In halts,
%   T-GGA-No-Exit, and after the halt only GRIP-Out can act).  Caption
%   canon fixes: "Geometric Generalised Equivalence Principle" ->
%   Geometry--Gauge (x3, incl. two body passages); Qi-CON Connectivity ->
%   Conformality and Qi-HAL -> Holographic Attractor Lattice in caption +
%   node labels; the projection/GSSB conflation removed from the caption;
%   "four axioms and seven postulates" -> seven postulates; stray
%   "Maximum Symmetry, Minimum Freedom" -> AX-4's canonical "Maximum
%   Symmetry, Minimum Freedom".  The corrected figure is also exported
%   to A6-GRIP v9.7 (runtime-only version) at ssec:grip-in-out.
% v9.5 (2026-06-07): the final shape-up on the canon-clean cohort.
%   (A) Canon sweep rows 1-10 (author adjudication: DGI = "Dynamical
%       Gauge Invariance" everywhere): five "Dynamical Geometric
%       Information" sites fixed (four-move anatomy, Fig. 4 node, the
%       dedicated DGI paragraph, achievements intro, recursion summary);
%       A6 bibitem rewritten (was "Dynamical Geometric Inverse" /
%       "Geometric Symmetry Breaking"); Qi-pillar names aligned to A7
%       canon (EP = Unified Extremal Principle, CON = Conformality,
%       HAL = Holographic Attractor Lattice, SIP = System Instance
%       Packet) at the A7 summary, the T-GGA proof, the SeOS section,
%       and the A7 bibitem; phantom citation "T-Qi4-GGA-Admissibility"
%       softened to A7's actual GGA-selector proposition.
%   (B) Ownership repair: "We name this regime ..." (4 sites + Part-III
%       overview) -> the regime definitions live at runtime level in
%       the parent paper A6-GRIP (def:GRIPin, def:GRIPout, def:GGA);
%       the present paper proves the regime THEOREMS.
%   (C) Four-move anatomy repair (3 passages + Fig. 4 GSSB node): the
%       boundary projection is OGGEP/GDOS; GSSB is the branch selection
%       the projected system undergoes -- the two are no longer
%       conflated; A8's reading of quantisation cited.
%   (D) T-GRIP-In: piecewise-anatomy remark (gradient flow within
%       basins + channel-mediated exits between them) + NEW corollary
%       T-GGA-No-Exit (GRIP-In halts exactly where the exit-channel
%       set is empty), channel-completeness fenced to particle strata.
%   (E) Upstream harvest: T-Electroweak-Vacuum cohort note (A9 v9.9:
%       EW = that paper's one GRIP-Out; the weak interaction = Nature's
%       GRIP-In implementer); Step-4 cosmic-alternation note (A10
%       v9.6); Cascade-Self-Sustenance "minimal machine" note (A7 v9.2
%       QiHAL switch); T-GGA proof cites A7's selector proposition.
%   (F) NEW section "The derivation ledger" closing Part III: every
%       Part-III theorem x its upstream inputs, making "A11 = theorems
%       derived from A5-A10" auditable at a glance.
%   (G) Chirality note: A9's fenced chiral postscript cited; the
%       geometric claim stated as independent of that open question.
% v9.4 (2026-06-03): "keep KK but always qualify" -- qualified the headline
%   Kaluza-Klein lock-cluster mentions (dictionary entry + the 3.85 TeV KK-resonance
%   paragraph) as "non-spatial" (warped/holographic), per A2 v9.21 convention.
% ----------------------------------------------------------------------
% Arisaka-GG-A11-Origin
% The Geometric Inevitability of Cosmos, Life, and Mind:
%   the Self-Referential Closure of the GG-Theory Programme
% Author: Katsushi Arisaka (UCLA)
%
% PROVENANCE.  This file is the v5.0 Phase 1 (identity-migration) of the
% A11-Origin paper, the philosophical capstone of the eleven-paper
% Arisaka-GG series.  It is built mechanically from the legacy
% capstone manuscript
%   UCLA-Arisaka-GG-2026-X.tex   (April 2026, v1.0, ~60 pp)
% via the Phase-1 migration script
%   outputs/a11_v50_phase1_migration.py   (May 16, 2026)
%
% The Phase-1 transformations are MECHANICAL only:
%   * Style upgrade ucla-arisaka-gg -> ucla-arisaka-gg-7          (L52)
%   * Identity block: PaperNumber, PaperStatus, PaperVersion,
%     PaperDate, hypersetup metadata                              (L15)
%   * Local-macro purge: 22 macros now provided in canonical
%     sans-serif by gg-7 converted to \providecommand            (L21, L48)
%   * A11 self-reference macros \AEleven / \AFullEleven added
%     (the v7 style provides A1..A10 + P1/P2/C1; A11 is the new
%     entry that will fold into the v7 style at v7.1 promotion)   (L39, L49)
%   * Bibtex key migration (7 legacy Arisaka-* keys -> ArisakaA*) (L3)
%
% NO SUBSTANTIVE CONTENT EDITS in Phase 1.  Theorem statements,
% section text, and prose are byte-for-byte preserved.  The eleven
% remaining phases plus the three Ultra-passes apply per the v5.0
% canonical workflow; see PROJECT-INSTRUCTIONS-v5.0-CONSOLIDATED.md.
%
% UPSTREAM DEPENDENCIES (Ultra Eight v7.1 cohort + A9-SM v7.2 + A10
% v7.1, all shipped May 14-15, 2026):
%   A1-Constitution v7.1, A2-GG6 v7.1, A3-BI6 v7.1, A4-Euclid v7.1,
%   A5-GDOS v7.1, A6-GRIP v7.1, A7-Qi4 v7.1, A9-SM v7.2 (with new
%   Part XII: "Geometric origin of identity and uniqueness of forces"
%   -- the direct upstream foundation for A11-Origin Part XII),
%   A10-Cosmos v7.1 (with universal-cosmic-identity Part XII).
%
% PHASE LOG.
%   Phase 1 (May 16, 2026):  identity migration.
%   Phase 2 (May 16, 2026):  GS-as-GGEP framing audit (L5; light).
%   Phase 3 (May 16, 2026):  L4 historical-background audit (largest in series).
%   Phase 4A (May 16, 2026): Part-shell insertion -- legacy 6-Part shell
%   Phase 4B (May 16, 2026): Eleven-paper inheritance synthesis.
%   Phase 4C (May 16, 2026): Five-step proof refinement (parent-law
%   Phase 4D (May 16, 2026): GSSB cascade refinement (constitutional-
%   Phase 4E (May 16, 2026): The Self-referential closure theorem
%   Phase 4F (May 16, 2026): Extensive philosophical-tradition
%   Phase 4G (May 16, 2026): Philosophical summit -- Conclusions Part
%   Phase 5a (May 16, 2026): Structural reorganization: 15 Parts ->
%   Phase 5b (May 16, 2026): Filled Part V Section 3 with ~12-15 pp
%   Phase 5 proper (May 16, 2026): Open Problems L6 status-header
%   Phase 5c (May 16, 2026): Pedagogical bridges for broader-audience
%   Phase 6 (May 16, 2026): Hostile reviewer + reference scanner.
%   Phase 7 (May 16, 2026): Companion-paper auditor + cross-paper
%   Phase 8 (May 16, 2026): TikZ figure suite (8 substantial figures).
%   Phase 9 (May 16, 2026): Front-matter rewrite to canonical A1 form.
%   Phase 9b (May 16, 2026): Abstract revised per user feedback.
%   Phase 9c (May 16, 2026): Cover-page redesign per user request.
%   Phase 9d (May 16, 2026): Restored canonical A1-A7 title-page
%   Phase 9e (May 16, 2026): Removed subtitle from title page per
%   Phase 9f (May 16, 2026): Restored \normalsize abstract per user
%   Phase 10 (May 16, 2026): Bibliography surname-sort.
%   Phase 11 (May 16, 2026): Sans-serif sweep + canonical compliance
%   Phase 11b (May 16, 2026): Thorough sans-serif sweep (round 2).
%   Phase 11c (May 16, 2026): Bibliography-tracking bug fix + complete
%   Phase 11d (May 16, 2026): Full programme-defined-names sweep +
%   Phase 11e (May 16, 2026): Sweep axiom/postulate/theorem refs
%   Phase 11f (May 16, 2026): Retire legacy GUT-10 terminology.
%   Ultra-1A (May 16, 2026): Fix 19 overfull hboxes.
%   Ultra-1B (May 16, 2026): Strip placeholder + internal-phase-chatter
%   Ultra-1C (May 16, 2026): Algebraic + cross-reference patches.
%   Ultra-2a (May 16, 2026): Structural reorder + legacy stale-ref fixes.
%   Ultra-2b (May 16, 2026): Remove redundant legacy Conclusions block.
%   Ultra-2c (May 16, 2026): Cross-paper citation hygiene to A1-A10.
%   Ultra-2d (May 16, 2026): Flow patches after the structural reorder.
%   Ultra-2d-followup (May 16, 2026): Shrink Part titles + subtitles.
%   Ultra-2e (May 17, 2026): Promote GRIP to the central operational
%   Ultra-2f (May 17, 2026): Elevate Qi-4 to the GGA-selection role.
%   Ultra-2g (May 17, 2026): Elevate SIP (System Instance Packet) to its
%   Origin-1 (May 17, 2026): Front-matter rewrite. v5.40 -> v6.1.
%   Origin-1 polish (May 17, 2026): v6.1 -> v6.2.
%   Origin-1 polish 2 (May 17, 2026): v6.2 -> v6.3.
%                              Three follow-up fixes after user review:
%                              (i) GG-2026-X reference removed from Executive
%                                  Summary -- the X manuscript is the previous
%                                  draft of this paper and should not be cited
%                                  or mentioned in the final publication.
%                              (ii) Executive Summary trimmed: the 'Closing
%                                   reflection' paragraph (which contained the
%                                   X reference) removed entirely; the 'eight-
%                                   test empirical horizon' subsection compressed
%                                   from 8 indented items to inline (i)-(viii).
%                              (iii) TOC verified to render with full entries after
%                                    fresh aux/toc rebuild (the v6.2 empty-TOC
%                                    artefact was stale .toc state, not a real
%                                    issue).
%                              Three follow-up fixes after user review:
%                              (a) Abstract returned to X-manuscript register
%                                  (compelling philosophical opening 'Why are we
%                                  here? For four centuries...'), while still
%                                  naming the trilogy framing and the cascade
%                                  outputs (electron, ATP, DNA, first cell,
%                                  conscious episode).  Fits cleanly on cover page.
%                              (b) Title font enlarged \huge -> \Huge via local
%                                  \renewcommand{\GGTitlePage}.  Title is now
%                                  visually prominent like A9-SM cover page.
%                              (c) Keywords trimmed from ~30 entries to 8 essential
%                                  entries (GG-Theory, recursive uniqueness theorem,
%                                  GG-Constitution, GRIP cascade, Qi-4 admissibility,
%                                  GGA, SIP, eight-test horizon).  Executive
%                                  Summary + Keywords now fit cleanly on pages 2-3.
%                              The Origin pass series promotes A11 from the
%                              philosophical-essay register of the original X
%                              manuscript to the rigorous physics-paper register
%                              of \ANine{} and \ATen{}.  Origin-1 rewrites the
%                              two front-matter blocks:
%                              (1) Abstract (cover page): replaced essay-style
%                                  'Why are we here?' opening with compact
%                                  physics-paper register naming T-Origin, the
%                                  eleven GGAs, inherited closed forms
%                                  (sin^2 theta_W=25/108 etc.), the eight-test
%                                  empirical horizon, and the trilogy framing
%                                  (A11 = third consequence paper, finale of the
%                                  GG-Theory programme).
%                              (2) Executive Summary (pages 2--3): rewritten as
%                                  eleven structured paragraphs matching A9-SM /
%                                  A10-Cosmos house style:
%                                    The problem
%                                    The constitutional input
%                                    The cascade machinery (with boxed GRIP eq.)
%                                    The central theorem (with boxed cascade eq.)
%                                    The eleven Ground Geometric Attractors
%                                    Triage of confidence (Derived/Cond./Asserted)
%                                    Inherited closed forms
%                                    The eight-test empirical horizon
%                                    Where this sits in the corpus
%                                    Closing reflection (methodological note)
%                              Pending: Origin-2 (Executive Summary tightening),
%                              Origin-3 (Statement of Results Part with 11
%                              cascade-layer theorems), Origin-4 (codebook + Pull
%                              table), Origin-5 (theorem-deck densification),
%                              Origin-6 (Discussions restructure), Origin-7
%                              (Conclusions register polish + rename), Origin-8
%                              (appendices build-out), Origin-9 (v7.0 ship).
%                              proper place alongside GRIP.
%                              (1) SIP FAQ definition (in Appendix FAQs) aligned
%                                  with the A7 \S6 canonical five-field form:
%                                  SIP = [(Z,g), Pi, (L_GDOS, eta), (F_GFF, J),
%                                  (Obs, Kill)] -- with explicit attribution to
%                                  A7 Theorem T-SIP.
%                              (2) Part III T-Origin theorem statement updated
%                                  to make the SIP tower explicit: each GGA_n
%                                  is the cascade output on layer-specific
%                                  \widehat{SIP}^{(n)}, with GGA_{n-1} supplying
%                                  fields 1--3 (geometric carrier) of
%                                  \widehat{SIP}^{(n)}.
%                              (3) Part V gained \subsection{The cascade as a
%                                  tower of \SIP{}s} with: canonical five-field
%                                  restated, why-cascade-needs-carrier discussion,
%                                  recursive-tower equation \widehat{SIP}^{(l+1)}
%                                  from GGA_l, four-layer worked table (charged-
%                                  lepton, metabolic-energy, open-system,
%                                  cognitive SIPs with all five fields and the
%                                  GGA readout at each), T-Origin restated in
%                                  SIP form with the closure equation, why-
%                                  recursion-is-closed three-fact argument,
%                                  what-this-means-for-biology paragraph.
%                              Pending: A6 adds SIP_geom (fields 1-3) alongside
%                              GRIP; A7 \S6 refactored to extend SIP_geom with
%                              fields 4-5; coordinated v7.2 sweep across cohort.
%                              Promote the implicit Qi-4 admissibility filter of
%                              \ASeven{} to a named theorem of \AEleven{}.
%                              (1) Part III rewritten from scratch around two new
%                                  named theorems:
%                                  - T-Qi4-GGA-Admissibility: Qi-4 jointly selects
%                                    the GGA on the GFF Morse landscape.
%                                  - T-Origin: Geometric Inevitable Unique Origin
%                                    of Cosmos, Life, and Mind --- the central new
%                                    theorem of A11 that derives the existence of
%                                    intelligent observers articulating the laws
%                                    of physics from the GG-Constitution alone.
%                              (2) Part V gained \subsection{Qi-4 as the GGA
%                                  selector: worked examples at four strata}
%                                  (electron, ATP, LUCA, conscious episode) plus
%                                  \subsection{The four pillars are jointly
%                                  necessary --- a counterfactual analysis} which
%                                  exhibits what happens if each pillar is
%                                  removed (no current-locking ray, no variational
%                                  selector, no scaling window, no memory closure).
%                              (3) Part IV cascade table gained 'Dominant Qi-4'
%                                  column showing which pillar(s) most strongly
%                                  constrain each layer (early: QUA + EP;
%                                  middle: EP + CON; late: HAL + EIC closure).
%                              Added Carter1974 bibitem for anthropic-principle
%                              attribution.
%                              Pending: revise A1-A7 (Phase beta) to introduce
%                              GGA and T-Qi4-GGA-Admissibility programme-wide.
%                              principle; introduce GGA (Ground Geometric
%                              Attractor); rewrite Parts III, IV, V around
%                              the canonical form GRIP := DGI -> (GFF) ->
%                              GSSB -> GGA.
%                              (1) Added \providecommand{\GGA} preamble macro.
%                              (2) Part III fully rewritten as 'GRIP: the
%                                  five-step proof of inevitability' --- five
%                                  GRIP applications at five strata with
%                                  extensive quantitative analysis (gauge
%                                  group, electron mass, ATP energy, EIC
%                                  timescale, etc.).  ~19-23 typeset pages of
%                                  new content.
%                              (3) Part IV renamed 'The GSSB cascade' -> 'The
%                                  GRIP cascade'; 11 transition labels GSSB-N
%                                  -> GRIP-N; figure node labels updated.
%                              (4) Part V gained new \section 'GRIP: the
%                                  dynamics on the GFF Morse landscape' with
%                                  Morse theory, Bogomolnyi cascade action
%                                  S_tot(N) = pi*N*(N-1)/2 <= S_BI = 3*pi
%                                  forcing N <= 3, GGA vs GA subtlety, and
%                                  the cascade-as-series-of-GGAs table.
%                              (5) GA-vs-GGA terminology: GGA used where the
%                                  cascade endpoint is meant (uniquely most-
%                                  stable global minimum); GA retained where
%                                  references to A6's canonical 'DGI -> GSSB
%                                  -> GA' form or to A6's GA-stabilisation
%                                  theorem are intentional.
%                              Pending: A1-A7 cohort revision to introduce
%                              GGA programme-wide (deferred to coordinated
%                              v7.2 / v7.3 sweep).
%                              A11's Part subtitles were 139-784 chars;
%                              A1-A4 v7.1 subtitles are 60-100 chars and
%                              single-phrase.  All 12 Part headers rewritten
%                              to match the A1-A7 register so the TOC reads
%                              cleanly.  Long titles (Parts V, VI, X, XI)
%                              also shrunk to one-phrase form.
%                              Eight patches: (F1) Part VII -> Part VIII bridge
%                              rewrite (was 'Bridge to Conclusions'); (F2) Part VII
%                              opening recap reworded for post-reorder Part numbers;
%                              (F3) Part IX Reader's guide Part-refs corrected;
%                              (F4) Part IX intro paragraph trimmed of stale
%                              cross-domain + philosophical-tradition claims;
%                              (F5) Part I pre-canonical five-Parts roadmap
%                              replaced with 12-Part structural roadmap;
%                              (F6) Part IV closing recap fixed for post-reorder
%                              layer attributions; (F7) Part VI -> Part VII
%                              transition gains an explicit bridge; (F8) Part VIII
%                              -> Part IX gains an explicit bridge.
%                              (A) Added \bibitem entries for ArisakaP1KK
%                              and ArisakaC1Kaxion -- previously unanchored.
%                              (B) \citep{ArisakaP1KK} and
%                              \citep{ArisakaC1Kaxion} added in the
%                              coherence-map rows for the predictions branch.
%                              (C) 5 missing first-use \citep{Arisaka*} added
%                              in Part V (\AOne, \AThree, \AFour, \ASix,
%                              \ASeven).
%                              (D) New 'Parent laws inherited' paragraph at
%                              start of Part VI consolidating citations for
%                              A1, A5, A6, A7, A9, A10.
%                              (E) Coherence map rows fixed for A5
%                              (GFF+Three-Gauge: Part X -> Part V) and A10
%                              (drop phantom Part-VII cosmic-layer).
%                              (F) Appendix audit fixes: subsec:obj-post-hoc-
%                              fitting Part VII -> Part IX (A3+A9 paragraphs);
%                              A10 audit drop layer-3 cosmological phantom;
%                              BookI audit subsec:obj-philosophical-caricature
%                              Part VII -> Part IX.
%                              Deleted the 527-line pre-canonical Conclusions
%                              block at end of Part XI: \section{Summary},
%                              \section{Why are we here?}, the legacy three-
%                              revolutions section, and the legacy 'A final
%                              word to the reader'.  The author's editorial
%                              note (at \S sec:summit-view's prelude) had
%                              explicitly marked these as preserved-for-
%                              historical-record-only; the load-bearing
%                              philosophical content lives in
%                              \S\ref{sec:summit-view} through
%                              \S\ref{sec:final-word-framework}.  After this
%                              deletion, the editorial note is rewritten to
%                              drop the now-stale 'legacy sections that
%                              follow' sentence.
%                              Estimated page reduction: 214 pp -> ~185 pp.
%                              PHASE A: 11 legacy Part-N parenthetical fixes
%                              (Part~VII (triad) -> Part~VI; Part~VIII (SeOS)
%                              -> Part~V; Part~IX (QCI) -> Part~VI; etc.) +
%                              Part III roadmap paragraph rewritten for Parts
%                              V-VI consolidated content.
%                              PHASE B: structural reorder Old Part VII
%                              (Discussions) <-> Old Part IX (Self-referential
%                              closure).  Source slice reordered: [Old IX]
%                              [Old VIII][Old VII] -> Parts auto-renumber to
%                              [VII Closure][VIII Cross-domain][IX Discussions].
%                              Body-text Part~VII <-> Part~IX swept with A*-
%                              macro-prefix guard (e.g., \ANine{} Part VII
%                              preserved for A9-SM references).
%                              Audit findings (from Ultra-1C audit subagent):
%                                * Q_B formula error in 3 spots: '(2 E_Ry/K) x 5'
%                                  is 25x too large; canonical A4 form is
%                                  '(2/5) E_Ry/K = 2 E_Ry/(5K)'.
%                                * 20 stale 'Part~XIII' references swept to
%                                  'Part~IX' (the self-referential closure Part
%                                  in the shipped 12-Part structure).
%                                * Undefined \MPl macro added to preamble as
%                                  \providecommand{\MPl}{\ensuremath{M_{\rm Pl}}}.
%                                * Misleading prose 'and Newton's constant G_N'
%                                  stripped (G_N does not appear in formula).
%                                * Omega values upgraded to A10 4-sf precision:
%                                  0.318 -> 0.3183; 0.682 -> 0.6817; 0.051 -> 0.0507.
%                                * Duplicated phrase 'Part~II Part~II's' fixed.
%                                * Mangled phrase 'Part~IX ... Part~IX' reworded
%                                  to correctly point to Part~XI for the closing
%                                  personal section.
%                              remnants from body text per L44.
%                              Seventeen literal replacements applied:
%                                * Stale 'Part~X placeholder' parenthetical
%                                  reworded as direct cross-cascade observation.
%                                * Part IX GGPart trailing 'Phase 4E + Phase 4F'
%                                  scaffolding sentence stripped.
%                                * A8 forward-reference language reworded
%                                  ('placeholder pointers' -> 'forward references';
%                                  'unfilled placeholder' -> 'forward-looking
%                                  dependency'; 'A5 placeholder' -> 'A5 stand-in').
%                                * Phase-5c / Phase-6 / Phase-9 / Phase-10 / Phase-11
%                                  internal references in Phase 7 audit appendix and
%                                  Phase 11 compliance audit appendix reworded as
%                                  descriptive past-tense statements or stripped.
%                                * Internal version numbers (v5.14, v5.24) stripped
%                                  per L44.
%                              PROVENANCE comments at the top of the file (% comment
%                              block) are preserved as historical record; they are not
%                              body text.
%                              Six tabularx/longtable column-width tweaks,
%                              one equation split (eq:gff-canonical -> aligned),
%                              four \begin{sloppypar}...\end{sloppypar}
%                              wraps around stress paragraphs (Buddhist
%                              transliteration, Closure was always an idealisation,
%                              the consciousness-question resolution prose,
%                              and the two compliance-audit itemize blocks
%                              with long \texttt{\textbackslash usepackage{...}}
%                              tokens).
%                              User flagged surviving GUT-10 and
%                              "GUT-10 Constitution" instances.
%                              Substitutions applied:
%                                GUT-10 constitution -> \GGConstitution{}
%                                GUT-10/\GG{}/\GDOS{} -> \GGTheory{}/\GG{}/\GDOS{}
%                                GUT-10 axiom system -> \GGConstitution{}
%                                GUT-10 axioms -> \GGConstitution{} axioms
%                                GUT-10 framework -> \GGTheory{} framework
%                                bare GUT-10 -> \GGTheory{}
%                              Plus four special-case literal fixes:
%                                (a) line-broken "the GUT-10\nconstitution"
%                                (b) line-5142 "10 in GUT-10" reworded
%                                (c) bibitem A1-Constitution title updated
%                                (d) provenance-comment H4 label updated
%                              Preserved (intentionally not swept):
%                                - abbreviations-table GUT row (textbook
%                                  Grand Unified Theory concept)
%                                - Georgi-Glashow historical references
%                                  to the original 1974 GUT proposal.
%                              + two-letter programme-defined names.
%                              User flagged Phase 11d for missing
%                              these patterns:
%                                AX-N (axiom refs) -> \AX{N}
%                                POS-N (postulate refs) -> \POS{N}
%                                T-SIP, T-QUA, T-EP, T-CON, T-HAL,
%                                T-EIC -> \T{-XYZ}
%                                IH -> \IH{}
%                                OH -> \OH{}
%                                GA -> \GA{}
%                                LC -> \LC{}
%                                GI -> \GI{}
%                                UEP -> \UEP{}
%                              Not swept (ambiguous in body text):
%                                AF, MF, CP, RG (used in non-
%                                programme senses too).
%                              canonical Appendix A.  Three tasks:
%                                (1) Appendices Part shell updated to
%                                    A9-SM canonical subtitle + intro
%                                    paragraphs.
%                                (2) Summary of Abbreviations section
%                                    replaced with A9-SM v7.2 canonical
%                                    Appendix A content (9-section
%                                    longtable with all programme-
%                                    defined names rendered in sans
%                                    serif via \GG{} \GDOS{} etc.).
%                                (3) Comprehensive sans-serif sweep of
%                                    bare programme-defined names in
%                                    body text.  33 distinct terms
%                                    swept (GG-Theory, GG-Constitution,
%                                    GG-6, GDOS, BI-6, OGN-5, Gi-4,
%                                    Qi-4, GRIP, GSSB, GFF, GEP, CEP,
%                                    TGG, CCI, DGI, QCI, QGL, MSMF,
%                                    OGGEP, GGEP, SeOS, SIP, MePMoS,
%                                    NHT, BFFT, Qi-QUA, Qi-EP, Qi-CON,
%                                    Qi-HAL, EIC, HAL, OGN).  Order:
%                                    longest-first to avoid sub-
%                                    pattern false matches.  Pattern:
%                                    (?<!\\)\bTERM\b with negative
%                                    lookbehind for backslash so
%                                    already-macro'd instances are
%                                    not double-substituted.
%                              sweep.  Phase 11b's sweep was tripped
%                              by \begin{thebibliography} appearing
%                              inside PROVENANCE \% comment lines
%                              of the source (a description of the
%                              bib-tracking rule got matched as if
%                              it were the rule's trigger).  Fix:
%                              bibliography tracking now only honored
%                              on non-comment lines.  Also tightened
%                              SKIP patterns to be line-position-
%                              anchored rather than substring matches,
%                              so descriptive prose mentioning
%                              "@misc" or "UCLA-Arisaka" in body
%                              text (as in the compliance audit) is
%                              not incorrectly skipped.
%                              User flagged the Phase 11 sweep as
%                              incomplete.  Three cleanups applied:
%                                (1) Modernise legacy FAQ paper list:
%                                    7 GG-2026-N entries (using the
%                                    pre-canonical paper-numbering
%                                    scheme that omits A4-Euclid and
%                                    A8-QM) replaced with the full
%                                    Arisaka-GG cohort (the foundational A-series; the full \GGTheory{} programme now numbers eighteen papers, with the Predictions and the Cosmology \& Astrophysics branches carrying the phenomenology)
%                                    inventory using \AOne{}-\AEleven{}
%                                    macros plus predictions branch.
%                                (2) Three "A11-Origin" remnants in
%                                    the Phase 11 compliance audit
%                                    appendix swept to \AEleven{}.
%                                (3) Multi-line bibliography awareness
%                                    added: lines between
%                                    \begin{thebibliography} and
%                                    \end{thebibliography} fully
%                                    skipped, preserving canonical
%                                    bib-entry paper identifiers
%                                    (e.g., "Arisaka-GG-P2-N3
%                                    (2026).") as-is.
%                              audit.  L48-compliant sweep replaced
%                              bare paper codes (A1-Constitution, etc.)
%                              with \AOne{}-style macros in body text.
%                              Skip-line filter excluded \newcommand,
%                              \providecommand, \bibitem, @misc,
%                              @unpublished, UCLA-Arisaka bibliographic
%                              source lines, identity-block lines, and
%                              all \% comment lines (PROVENANCE
%                              header preserved as-is).
%                              New Appendix section "Canonical
%                              compliance audit (Phase 11)" added
%                              before the Glossary section, walking
%                              through the 53-item v5.0 compliance
%                              checklist.  Result: 49 Verified, 2
%                              Partial, 1 Pending (bib line), 1 N-A
%                              (headline parameter-audit Dev plot,
%                              lives in upstream papers).
%                              93 \bibitem entries re-ordered
%                              alphabetically by the [Label]
%                              portion of \bibitem[Label]{Key}.
%                              Result: bibliography reads in
%                              standard surname order (Adler,
%                              Alvarez-Gaume, Anderson, Arisaka,
%                              Aristotle, Barrow-Tipler, ...).
%                              Note: switching from inline
%                              \begin{thebibliography} to v7
%                              master \bibliography{UCLA-Arisaka-
%                              GG-7} is deferred until the master
%                              bib is augmented with ~25 entries
%                              currently only in the inline bib
%                              (Anderson1972, Aristotle350BCE,
%                              BarrowTipler1986, Chalmers1995,
%                              Crick*, Darwin1859, England2013,
%                              Gibbs1875, Hafting*, Haken1983,
%                              Hegel1807, Hopfield1982, Kant1781,
%                              Leibniz1714, OKeefe*, Russell1921,
%                              Smolin1997, Tegmark2014, Teilhard,
%                              Watson-Crick, Whitehead*).  This is
%                              a programme-level cleanup for a
%                              future cycle.
%                              request; merged 6-7 short paragraphs
%                              into 3 substantial paragraphs to save
%                              vertical space:
%                                Para 1: "Why are we here?" question
%                                  + four-centuries-of-science framing
%                                  + "the accident never happened"
%                                  punch (all combined).  Originally
%                                  3 short paragraphs.
%                                Para 2: The geometry/life/mind
%                                  cascade (geometry forces masses +
%                                  information-matter lock + life as
%                                  only option + cascade ascent +
%                                  loop closes).  Originally 2 long
%                                  paragraphs.
%                                Para 3: Cosmos/Life/Mind synthesis
%                                  + self-measurement machine + "we
%                                  are here because the geometry
%                                  requires it".  Originally 2 short
%                                  paragraphs.
%                              No font overrides; canonical normal
%                              size body.
%                              user request.  This specific paper
%                              follows the original GG-2026-X pattern
%                              of title + empty subtitle + date.
%                              The canonical \GGTitlePage macro form
%                              is preserved (no font overrides);
%                              only the second argument is now {}.
%                              Freed vertical space helps the abstract
%                              fit on the front cover page.
%                              format per user correction.  Phase 9c's
%                              custom cover layout (\thispagestyle
%                              {empty} + inline title + abstract
%                              with custom font sizes) was a mistake.
%                              Reverted to the exact canonical form
%                              used by A1-Constitution, A2-GG6,
%                              A4-Euclid, A7-Qi4:
%                                \GGTitlePage{Title}{Subtitle}{Date}
%                                \thispagestyle{empty}
%                                \vspace{6pt}
%                                \begin{abstract}...\end{abstract}
%                              No font overrides; standard format;
%                              UCLA/paper-number header preserved as
%                              part of the GGTitlePage macro's
%                              canonical typeset content.
%                              Removed the legacy UCLA/A11-Origin
%                              running header from cover.  Integrated
%                              the abstract on the same page as the
%                              title, author, and affiliation block.
%                              Replaced \GGTitlePage{} macro call
%                              with a custom inline cover-page layout:
%                              compact title (\LARGE), author line
%                              (\large), affiliation + email
%                              (\small\itshape), version line
%                              (\footnotesize), then the abstract
%                              body in \small font to fit on one
%                              page.  Used \thispagestyle{empty}
%                              to suppress the running header.
%                              Closes with \newpage so Executive
%                              Summary starts on page 2.
%                              Phase 9 version was too dry, citation-
%                              heavy, and read like a reference list.
%                              Phase 9b restores the original X-
%                              manuscript's evocative voice: self-
%                              contained (no \AOne...\ATen refs),
%                              one-page length, original's punchy
%                              rhetorical hooks ("the accident never
%                              happened", "the lock keeps tightening",
%                              "self-measurement machine", "we are
%                              here because the geometry requires it").
%                              Adds A11's new content (the closing
%                              self-referential loop, the recursive
%                              theorem) without losing the style.
%                              New Abstract (~220 words, within
%                              canonical tight target): frames the
%                              question, names the eleven-paper
%                              inheritance, states the recursive
%                              uniqueness theorem, names the 8-test
%                              horizon, closes with title trilogy.
%                              New Executive Summary (~950 words,
%                              within canonical 700-1000 target):
%                              nine thematic paragraphs (the
%                              question; the inheritance; recursive
%                              uniqueness; closing loop; Cosmos;
%                              Life; Mind; the empirical horizon;
%                              the closing reflection).
%                              Keywords refreshed to reflect the
%                              recursive uniqueness theorem and
%                              the title's philosophical content.
%                              Figures inserted at thematically natural
%                              body-text locations:
%                                F1 Eleven-paper inheritance map (Part II);
%                                F2 Five-Story Pagoda (Part VI);
%                                F3 GSSB cascade flowchart (Part IV);
%                                F4 QCI ladder (Part VI);
%                                F5 Three-Gauge Theorem projections
%                                   (Part V Section 3);
%                                F6 Holographic triad OH/IH/EIC
%                                   (Part VI);
%                                F7 Cosmos/Life/Mind trilogy
%                                   (Part X Section 9);
%                                F8 Participatory closure loop
%                                   (Part IX named-theorem section).
%                              All figures use TikZ with arrows.meta,
%                              positioning, calc, shapes.geometric,
%                              fit, backgrounds libraries (already
%                              loaded by ucla-arisaka-gg-7.sty).
%                              No new package dependencies.
%                              coherence audit (L7).  New Appendix
%                              section "Cross-paper coherence audit
%                              (L7 detailed inheritance)" inserted
%                              at head of Appendices Part XII.  The
%                              largest such audit in the series.
%                              Three subsections:
%                                 1. Outward audit (paper-by-paper):
%                                    every Arisaka-* citation traced
%                                    to specific upstream theorem
%                                    with body-text exercise location
%                                    and status (verified/pending).
%                                 2. Inward map: forward-pointers
%                                    from upstream papers to A11
%                                    catalogued and verified honoured.
%                                 3. Consistency audit summary with
%                                    two minor findings flagged for
%                                    downstream phases (predictions-
%                                    branch citation consistency
%                                    sweep in Phase 11; A8 ship
%                                    pending forward-pointer update).
%                              Pre-audit inventory: 11 Arisaka-*
%                              citation keys, ~86 theorem-reference
%                              macros, no undefined cites detected.
%                              New section in Part VII (Discussions)
%                              titled "Anticipated objections and
%                              responses", with 8 subsections each
%                              responding to a specific hostile-
%                              reviewer attack against the framework
%                              central claims:
%                                 1. Circularity of self-referential
%                                    closure;
%                                 2. Architecture proves too much
%                                    (unfalsifiable by construction);
%                                 3. Falsifiability not genuine
%                                    (no precision tolerances);
%                                 4. Caricature of 14 philosophical
%                                    predecessors at 2pp each;
%                                 5. Post-hoc fitting accusation
%                                    against A9-SM 28 parameters;
%                                 6. Scheme dependence and precision;
%                                 7. Why eleven cascade layers;
%                                 8. Theoretical opportunism (use of
%                                    apparatus from rejected programmes).
%                              Each subsection: italicised statement
%                              of objection + 2-4 paragraph response
%                              + honest acknowledgement of where the
%                              objection retains force.  Closing
%                              synthesis subsection collects the eight
%                              attacks as one methodological attack
%                              and states the framework's collective
%                              response.
%                              accessibility (Purpose B).  Reader's
%                              Guide subsections + Metaphors-and-
%                              intuitions content + Bridge paragraphs
%                              added to seven Parts (Part V already
%                              had these from Phase 5b, Parts X-XI
%                              from Phase 4G):
%                                 Part I:   Reader's guide for
%                                   different audiences (life sci,
%                                   neuro/phil-mind, philosopher,
%                                   physicist, integrative reader);
%                                 Part II:  Non-physicist's guide to
%                                   the cohort inheritance
%                                   (six structural paragraphs);
%                                 Part III: Five-step proof in plain
%                                   English (staircase metaphor);
%                                 Part IV:  Cascade in non-technical
%                                   language (great-ascent narrative
%                                   through 10+ doors);
%                                 Part VI:  Reader's guide for
%                                   neuroscientists and philosophers
%                                   of mind (empirical anchors +
%                                   philosophical commitments);
%                                 Part VII: Reader's guide for the
%                                   Discussions Part;
%                                 Part VIII: Reader's guide for
%                                   cross-domain applications.
%                              All bridges follow user choice Q3(c):
%                              layered tone with conversational
%                              opening, register lift to A9-SM voice
%                              at the technical core, return to
%                              conversational close.
%                              insertion.  L6 preamble added at the
%                              head of the Open questions section,
%                              naming the three-status convention
%                              (Resolved / Partially-Resolved /
%                              Genuinely-Open) and distinguishing
%                              empirically falsifiable vs
%                              philosophically open questions.
%                              Per-subsection status summaries
%                              inserted on five subsections:
%                              Particle physics, Cosmology, Chemistry
%                              + origin of life, Mind + consciousness,
%                              and The broader horizon.  Each summary
%                              names the specific upstream-paper
%                              theorem that closes Resolved entries,
%                              the named lemma or experiment for
%                              Partially-Resolved entries, and the
%                              within/outside-scope reason for
%                              Genuinely-Open entries.
%                              substantive GFF + Three-Gauge Theorem
%                              content: \S 3.1 the functional formally
%                              stated (citing A5 T2-T4); \S 3.2 Gibbs
%                              thermodynamic gauge (J_Qi=0 limit);
%                              \S 3.3 Friston variational gauge
%                              (J_Qi != 0 Markov blanket);
%                              \S 3.4 action gauge (detector limit);
%                              \S 3.5 the Three-Gauge Theorem (T4);
%                              \S 3.6 cross-cascade uniformity (NEW
%                              contribution of A11 -- the same
%                              functional governs every cascade
%                              layer); \S 3.7 metaphors and
%                              intuitions for non-physicist readers
%                              (the universal altitude, the musical
%                              score, the cascade staircase);
%                              \S 3.8 bridge to Part VI.  Per user
%                              Q2(c) the content combines extracted
%                              GFF material from Step 2 of Part III
%                              with new cross-cascade observations.
%                              12 Parts.  Merged Q_B + SeOS + GFF
%                              placeholder into new Part V "Dynamics
%                              block"; merged Pagoda + Triad + QCI
%                              into new Part VI "Cognitive cascade";
%                              moved SeOS block to right after Q_B;
%                              demoted SeOS, Triad, QCI GGParts to
%                              sections; deleted empty Part X (GFF)
%                              shell (substantive content for GFF
%                              section to be added in Phase 5b);
%                              split monolithic 22-section Part XIV
%                              Conclusions into Part X (Conclusions:
%                              summit + Cosmos/Life/Mind trilogy)
%                              and Part XI (Coda: closing
%                              loop + voice + final word + legacy).
%                              extensively expanded with 17 new
%                              substantive sections integrating the
%                              revised conclusions of the six v7.1-
%                              cohort companion papers (A4, A5, A6,
%                              A7, A9 v7.2, A10) and articulating the
%                              central philosophy of 'The Geometric
%                              Inevitability of Cosmos, Life, and
%                              Mind' at summit-register depth and
%                              length.  Sections inserted before
%                              legacy \section{Summary}:
%                              S1 The summit view;
%                              S2 The five scientific revolutions;
%                              S3 The Euclidean ideal recovered (A4);
%                              S4 Openness as constitutional (A5);
%                              S5 The Gi-4+Qi-4 backbones (A6/A7);
%                              S6 Geometric inevitability of Cosmos (A10);
%                              S7 Geometric inevitability of Life (A5/A7);
%                              S8 Geometric inevitability of Mind (A7);
%                              S9 The recursive uniqueness theorem
%                              across the cascade;
%                              S10 The closing self-referential loop;
%                              S11 What was renounced (A9-derived);
%                              S12 The fifty-year climb;
%                              S13 The eight-test horizon;
%                              S14 The book and the decade;
%                              S15 Verbatim cosmic-uniqueness;
%                              S16 The voice of the cascade;
%                              S17 A final word from the framework.
%                              Legacy Conclusions sections retained
%                              for historical record but no longer
%                              carry load-bearing philosophical content.
%                              engagement.  New \section{Comparison
%                              with the philosophical tradition}
%                              within Part XIII, comprising 14
%                              substantive subsections:
%                              Upanishadic (tat tvam asi);
%                              Buddhist (pratityasamutpada);
%                              Aristotle (the question why);
%                              Spinoza (necessitarianism);
%                              Leibniz (sufficient reason);
%                              Kant (phenomena/noumena);
%                              Hegel (Geist becoming conscious);
%                              Whitehead (process philosophy);
%                              Wigner (unreasonable effectiveness);
%                              Wheeler (participatory universe);
%                              Penrose (three worlds);
%                              Smolin (cosmological natural selection);
%                              Tegmark (Mathematical Universe);
%                              Barrow-Tipler (anthropic principle);
%                              + closing synthesis subsection naming
%                              the three structural insights (necessity,
%                              layered self-recognition, mathematical
%                              inevitability) the framework inherits
%                              from the tradition.  Tone respectful
%                              throughout: predecessors had insights
%                              without geometric warrant; framework
%                              supplies warrant, owes the insight.
%                              (central new content of A11-Origin).
%                              Part XIII placeholder REPLACED with
%                              ~12-15 pp of structured content:
%                              \S1 The question this Part answers;
%                              \S2 The named theorem -- formal
%                              statement of \emph{Theorem (Self-
%                              referential closure of the GG-Theory
%                              programme)} in canonical four-statement
%                              form (analogous to A9-SM Part XII
%                              Theorem 12.x);
%                              \S3 Proof outline by inheritance from
%                              the eleven prior papers (layer by
%                              layer trace from layer 1 to layer 18);
%                              \S4 Five corollaries at the meta-
%                              cognitive layer (universality of
%                              articulation, falsifiability,
%                              determinism vs human agency,
%                              substrate-independence, higher layers);
%                              \S5 The careful word: necessitarianism
%                              vs determinism vs design;
%                              \S6 The closing of the question that
%                              started in a Tokyo high school
%                              (verbatim 1970s quotation preserved).
%                              The named theorem is the structural
%                              twin of A9-SM Part XII Theorem 12.x
%                              extended from layer 2 to layer 18.
%                              Phase 4F (next) will add ~15-20 pp
%                              of philosophical-tradition engagement
%                              (Spinoza, Hegel, Wheeler, Whitehead,
%                              Wigner, Tegmark, Penrose, Smolin,
%                              Upanishadic, Buddhist).
%                              clause anchoring of the ten transitions).
%                              Three new structural elements added to
%                              Part IV: D1 opening subsection "The
%                              cascade as recursive uniqueness" framing
%                              the entire cascade as layer-by-layer
%                              specialisation of \ANine{} Part XII
%                              \S60; D2 eleven-paragraph
%                              "Constitutional-clause anchoring"
%                              subsection naming for GSSB-0..GSSB-10
%                              the constitutional clauses + upstream
%                              theorems that force each transition;
%                              D3 closing subsection "Why the cascade
%                              closes at the human-awareness layer"
%                              + structural prediction of the layer-18
%                              self-referential closure (forward
%                              pointer to Part XIII).
%                              citations on Steps 2-5; Step 1
%                              harmonised to match).
%                              Step 2: GDOS extremal principle cites
%                              A5-GDOS T2-T4 (GFF descent + split +
%                              Three-Gauge Theorem).
%                              Step 3: QGL forbids disembodied
%                              information cites A3-BI6 T1 + A7-Qi4
%                              T-QUA (Arisaka equation).
%                              Step 4: Life as geometric attractor
%                              cites A6-GRIP GRIP cascade T1-T5,
%                              A5-GDOS SeOS T7, A7-Qi4 T-EP/T-HAL/T-EIC.
%                              Step 5: Mind as inevitable cites A6
%                              GA stabilisation T4, A7 T-EIC, and
%                              A9-SM Part XII universal-identity at
%                              cognitive stratum.
%                              Substantive ~16 pp opening of Part~II:
%                              new \section{Eleven-paper inheritance:
%                              what A11-Origin imports} with twelve
%                              subsections (the cross-paper coherence
%                              map table; ten per-upstream-paper
%                              inheritance subsections covering A1
%                              through A10 plus the Predictions Branch;
%                              and the recursive-uniqueness-theorem
%                              synthesis subsection that closes the
%                              section).  The load-bearing claim:
%                              A11-Origin does NOT re-derive any of
%                              the upstream content; it inherits the
%                              entire corpus by reference and extends
%                              the recursive uniqueness theorem from
%                              the particle-physics layer (A9-SM Part
%                              XII) and the cosmological layer (A10
%                              Part XII) to every higher stratum.
%                              converted to 15-Part target structure.
%                              Five existing GGParts renamed; legacy
%                              Part III "Inevitable Life and Mind"
%                              superseded by new finer-grained Parts
%                              III-IX.  Nine new GGPart shells inserted:
%                              III (5-step proof), IV (GSSB cascade),
%                              V (Q_B), VI (Pagoda), VII (Hologr. triad),
%                              VIII (SeOS), IX (QCI ladder), X (GFF +
%                              Three-Gauge, empty placeholder), XII
%                              (Cross-domain), XIII (Self-ref closure,
%                              empty placeholder).  Three subsections
%                              promoted to sections: Q_B, Pagoda, SeOS.
%                              Twelve \subsection{Historical background}
%                              openings inserted on the named-theorem and
%                              structural-skeleton sections: H1 6D bulk,
%                              H2 BI-6, H3 QGL, H4 GG-Constitution,
%                              H5 five-step proof, H6 GSSB cascade,
%                              H7 holographic triad, H8 OH, H9 IH,
%                              H10 EIC, H11 SeOS, H12 QCI ladder.
%                              Eight new bibitems for missing predecessors:
%                              Aristotle, Leibniz, Kant, Whitehead (Process
%                              and Reality), Whitehead-Russell (Principia
%                              Mathematica), Tegmark, Smolin, Barrow-Tipler.
%                              Four patches applied:
%                                P2-1: three legacy GG-2026-3 body
%                                      refs -> \citep{ArisakaA3BI6};
%                                P2-2: L5 historical-attribution
%                                      paragraph in BI-6 subsection
%                                      naming Green-Schwarz 1984 as
%                                      technical pioneering of AX-2;
%                                P2-3: QGL parent-law citations to
%                                      \AThree{} (BI-6) and \ASeven{}
%                                      (Qi-QUA theorem);
%                                P2-4: Step 1 uniqueness theorem
%                                      forward foundations to
%                                      \ATwo{} (OGN-5 ledger) and
%                                      \ANine{} Part XII (universal
%                                      identity + uniqueness of
%                                      four-force architecture).
%                              This file.
%   Phase 2 (pending):       GS-as-GGEP framing audit (L5; light
%                              -- A11 inherits from A3 + A9 Part XII).
%   Phase 3 (pending):       L4 historical-background audit; ~15-20
%                              philosophical + scientific openings
%                              (largest in the series).
%   Phase 4 (pending):       substantive content, subdivided 4A-4G:
%                              4A: 14-Part shell insertion
%                              4B: cohort inheritance synthesis
%                              4C: five-step proof refined
%                              4D: GSSB cascade with constitutional cites
%                              4E: self-referential closure theorem
%                                  (the central new content of A11)
%                              4F: philosophical-tradition engagement
%                              4G: Conclusions Part (summit register)
%   Phase 5 (pending):       Open Problems with status headers (L6).
%   Phase 6 (pending):       hostile-reviewer + reference-scanner.
%   Phase 7 (pending):       companion-paper auditor + cross-paper
%                              coherence map (L7; largest in series).
%   Phase 8 (pending):       TikZ figures (~10-12; most figure-heavy
%                              paper in series).
%   Phase 9 (pending):       front-matter rewrite to A1 canonical.
%   Phase 10 (pending):      bibliography surname-sort + Relevance
%                              paragraphs (largest bib in series).
%   Phase 11 (pending):      Appendix A 9-section + hierarchy figure +
%                              sans-serif sweep + 36-item compliance.
%   Ultra-1 (pending):       single-line algebraic patches.
%   Ultra-2 (pending):       redundancy compression.
%   Ultra-3 (pending):       cleanup; Conclusions expansion; v7.1 ship.
%   Appendix-Migration-1 (May 18, 2026): new Appendix A migrated from
%                              \AOne{} Part VIII (programme-wide reference
%                              material) with \AEleven{}-specific extensions:
%                              GGA / GRIP-In / GRIP-Out rows added to the
%                              abbreviations table; new \$\GGA_n\$ / \$\SIP_n\$
%                              / \$ rows added to the glossary; new
%                              \S A.10 "The GRIP runtime" section added;
%                              Figure A.1 redesigned: GRIP promoted to central
%                              engine box at Tier 5, with the canonical chain
%                              DGI -> (GFF) -> GSSB -> GGA shown explicitly,
%                              GRIP-In / GRIP-Out as parallel regimes, Gi-4 /
%                              Qi-4 flanking; new bottom tier shows the
%                              17-layer cascade ribbon with cascade-flavour
%                              colouring (blue -> cyan -> green -> orange ->
%                              red).  Old standalone Summary of Abbreviations
%                              (then Appendix C) and Glossary of Key Symbols
%                              (then Appendix E) are deleted; their contents
%                              are merged into the new Appendix A.1 / A.2.
%                              Remaining appendix shift: A (coherence audit)
%                              -> B; B (FAQ) -> C; D (compliance) -> D; F
%                              (theorem index) -> E; G (lemma roadmap) -> F;
%                              H (open problems) -> G.  Also: \Mbulk /
%                              \Mbdy preamble macros added.  v7.16 -> v7.17.
%   Figure-1-Refactor (May 18, 2026): Replaced the radial eleven-paper
%                              inheritance map with the refined A1 v7.1
%                              Figure 4 layout.  New Figure 1 shows the
%                              Consequence Trio at row 5 with A9-SM (left),
%                              A10-Cosmos (centre), and A11-Origin (right,
%                              highlighted in orange as the present paper),
%                              all flanking under A8-QM at row 4.  Lateral
%                              dashed arrows from A9 -> A11 (Q1 universal-
%                              identity) and A10 -> A11 (Q2 cosmic-identity)
%                              show A11's synthesis dependency on the two
%                              upstream consequence papers.  The original
%                              BookI + WhitePaper box at row 7 is removed
%                              per user request (unnecessary for the
%                              eleven-paper closure).  Predictions branch
%                              expanded: P1-KK under A9, C1-Kaxion under
%                              A10, eight-test-horizon synthesis annotation
%                              under A11.  v7.17 -> v7.18.
%   Final-3-L51 (May 18, 2026):  Cross-paper coherence-map audit (53-item
%                              compliance checklist item 52).
%                              (A) Three broken phrasings in the L7 audit
%                                  appendix repaired ("Part~II ... Part~II"
%                                  doublings near A3 and A10 audits;
%                                  "named in A9 the X subsection of A9"
%                                  reworded to direct section refs).
%                              (B) Stale citation-count summary refreshed
%                                  to v7.19 actuals: A1=29, A2=14, A3=20,
%                                  A4=26, A5=44, A6=43, A7=71, A9=58,
%                                  A10=41, P1=10, P2=13, C1=14+2 compact,
%                                  BookI=4.  Total: 389 citations across
%                                  13 distinct Arisaka-* keys.  Per-paper
%                                  paragraph headers updated.
%                              (C) Legacy bibitem location strings
%                                  "UCLA-Arisaka/GG-2026-N" replaced with
%                                  canonical "Arisaka-GG-AX-Name vY.Z (date)"
%                                  forms for all seven entries.
%                              (D) ArisakaA8QM stub bibitem added (status:
%                                  in drafting; Part II forward-pointer
%                                  remains accurate).
%                              (E) Audit close gains a paragraph documenting
%                                  the v7.17/v7.18 additions (Appendix-Migration-1
%                                  and Figure-1-Refactor).
%                              (F) Item 52 of the compliance audit refreshed
%                                  to record the L51 sweep findings.
%                              v7.18 -> v7.19.
%   Final-3-Baseline-A (May 18, 2026): items 1--18 of the canonical
%                              compliance audit (v4.0 baseline conventions)
%                              expanded from three bundled paragraphs into
%                              18 per-item findings.  Each item carries its
%                              own [Verified] / [Partial] / [Pending] /
%                              [N-A] status with the source-line evidence
%                              recorded.  Two stale audit claims fixed:
%                              (a) "Twelve-Part target structure" -> 13-Part
%                              structure (12 body + 1 Appendices); (b)
%                              "Conclusions split into Part A and Part B"
%                              -> "Conclusions: the summit view" + "Coda:
%                              closing the self-referential loop" (Origin-7
%                              user-directed rename).  Compliance summary
%                              line refreshed to "50 Verified / 2 Partial /
%                              1 Pending / 1 N-A".  v7.19 -> v7.20.
%   Final-3-Baseline-B (May 18, 2026): items 19--30 of the canonical
%                              compliance audit (theorem environments,
%                              citation conventions) expanded from two
%                              bundled paragraphs into 12 per-item
%                              findings.  Each item carries its own
%                              [Verified] / [Partial] / [Pending] / [N-A]
%                              status with source-line evidence.  One real
%                              fix applied: \label{def:seos} added to the
%                              SeOS definition (was the only definition
%                              environment lacking a label).  Audit
%                              findings recorded:
%                                * 39 \begin{theorem} environments, all
%                                  named, 38 thm: + 1 cor: label (the
%                                  Cascade-Self-Sustenance corollary is
%                                  intentionally in theorem-env styling).
%                                * 5 corollary, 6 definition (now all
%                                  labelled), 5 remark, 0 lemma, 0
%                                  conjecture environments.
%                                * 386 \citep + 79 \citet = 465 cites;
%                                  0 legacy \cite calls.
%                                * 145 bibitems, all 145 with Relevance
%                                  paragraph (100%).
%                                * Surname-sort verified Adler -> de Haro.
%                                * 24 \url{} fields, 23 DOIs.
%                              v7.20 -> v7.21.
%   Final-3-Baseline-C (May 18, 2026): items 31--42 of the canonical
%                              compliance audit (cross-references, front
%                              matter, back matter) expanded from two
%                              bundled paragraphs into 12 per-item
%                              findings.  Two real source fixes applied:
%                                * 39 \S\ref{thm:...} occurrences (38 in
%                                  the Appendix-E theorem-index table's
%                                  "Location" column + 1 in body prose
%                                  for the Three-Gauge Theorem) swept to
%                                  Theorem~\ref{thm:...} (the section-sign
%                                  prefix was misleading because \ref{thm:}
%                                  expands to the theorem number, giving
%                                  "\S<theorem-number>" which reads as
%                                  "section <N>").
%                                * 1 (\ref{eq:t-grip-in-lyapunov}) in
%                                  Part~V's T-GRIP-In proof outline swept
%                                  to \eqref{eq:t-grip-in-lyapunov}, which
%                                  adds the parentheses automatically per
%                                  AMS convention.
%                              Audit findings recorded:
%                                * Item 31: 314 \ref + 35 \eqref, 0
%                                  legacy \pageref/\autoref/\nameref.
%                                * Item 32: 158 thm: refs, all using
%                                  Theorem~ prefix after v7.22 sweep.
%                                * Item 33: 66+38+6 = 110 section refs.
%                                * Items 34--35: 9 figure refs, 26 table
%                                  refs, all canonical.
%                                * Item 36: 35 eqref, 0 remaining
%                                  hand-parenthesised \ref{eq:...}.
%                                * Items 37--42: cover, abstract, exec
%                                  summary, keywords, acknowledgments,
%                                  bibliography all verified at canonical
%                                  positions.
%                              Final-3-Baseline now complete; the 42
%                              items of the v4.0 baseline have been
%                              individually audited.
%                              v7.21 -> v7.22.
%   Cleanup-1 (May 18, 2026): seven user-requested fixes.
%     (1) PaperDate -> May 18, 2026 (cover header + running header +
%         GGTitlePage date arg).
%     (2) Executive Summary GGA-inventory paragraph compressed (the
%         17 GGAs now in one tight paragraph instead of two; ExecSummary
%         + Keywords now fit cleanly on pages 2-3, no spill to page 4).
%     (3) GGTableOfContents wrapped with \linespread{0.90}\selectfont
%         to reduce TOC line spacing by ~10% per A9-SM-style convention.
%     (4) "GG-6/GDOS framework" / "\GG{}/\GDOS{} framework" / similar
%         compound expressions swept to "\GGTheory{}" throughout body
%         text where appropriate.
%     (5) \QCI macro definition fixed: was \mathrm{QCI} (Roman), now
%         \ifmmode\mathsf{QCI}\else\textsf{QCI}\fi (sans-serif).
%         Plus 4 bare 'QCI' occurrences in body prose converted to
%         \QCI{} macro form.
%     (6) Cross-paper coherence map (Part II): tabularx -> longtable
%         so the table can break across pages instead of overflowing
%         page 22's bottom margin.
%     (7) Eleven long inventory display equations reformatted with
%         \begin{aligned} line-breaks so they fit within the right
%         margin: \GGA_2 (eq 21 electroweak-vacuum), \GGA_5 (20 amino
%         acids, the 288pt offender), \GGA_7 (ATP), \GGA_8 (RNA),
%         \GGA_9 (DNA), \GGA_10 (LUCA), \GGA_{12} (eukaryogenesis),
%         \GGA_{15} (ring attractor), \GGA_{16} (allocentric map),
%         \GGA_{17} (conscious episode), \GGA_{18} (articulation).
%     v7.22 -> v7.23.
%   Edit-1 (May 18, 2026): redundancy audit + trim per user request.
%     Top-priority consolidations applied based on the Plan-subagent
%     redundancy matrix:
%       R11 [structural bug fix]: Part VII had two consecutive sections
%            with the same title "The holographic triad: ..." separated
%            by ~30 lines (an unmerged Origin-6 editorial accident).
%            The first \section{} heading immediately preceding the
%            triad figure was removed; the figure now floats inside the
%            substantive section at \label{sec:holographic-triad-merged}.
%       R13: Part IV \S5418-5432 (three \paragraph blocks duplicating
%            Part II \S1750-1765 verbatim on the universal-identity
%            insight, universal-cosmic-identity, and Q1+Q2 synthesis)
%            collapsed into one paragraph that cross-references Part II.
%       R9: Part VIII \S8526-8534 (entire \section{The closing of the
%            question that started in a Tokyo high school}) trimmed to
%            one cross-reference paragraph; the canonical personal-arc
%            section is XII \S\ref{sec:fifty-year-climb} and the verbatim
%            cosmic-uniqueness quote appears there (avoiding the prior
%            triple-appearance).
%       R1: Part X "Why this section is here" subsection intro (4 long
%            paragraphs spending ~30 lines justifying its own existence
%            after Part III's Pull table) compressed to a single 7-line
%            paragraph that names the six meta-questions and the
%            connection to the recursive uniqueness theorem.
%     Out-of-scope items flagged for later Edit passes:
%       (a) Layer-number drift (16 vs 17 vs 11 layers in different
%            parts).
%       (b) Layer attribution for LUCA varies between layer 7, layer 8,
%            and layer 10 depending on which Part is read.
%       (c) "\ANine{} Part XII vs Part XIII" inconsistency in upstream
%            section references.
%       (d) Excessive in-body re-glossing of abbreviations after the
%            Appendix-A glossary; Edit-3 will sweep these.
%       (e) "After all," sentence opener overused in Parts XI/XII;
%            Edit-4 stylistic.
%     Estimated page reduction: ~3-5 pages from these four cuts (less
%     than the audit estimate because R10 and the smaller matrix items
%     deferred to later Edit passes).
%     v7.23 -> v7.24.
%   Edit-2 (May 18, 2026): citation-discipline + in-situ intuition audit.
%     Audit found 31/39 theorems already PASS both checks; 8 theorems
%     fail check (a) -- the upstream-paper macros (\ANine{}, \AFive{},
%     \ASix{}, \ASeven{}, \AOne{}, \AFour{}) are named in the
%     Upstream-theorems-imported paragraphs without \citep{ArisakaA...}
%     cite-keys attached. Patches applied:
%       B-1: T-Atomic-Ground-State (3 cite-keys)
%       B-2: T-Amino-Acid-Selection (4 cite-keys)
%       B-3: T-Cell-Closure (4 cite-keys)
%       B-4: T-Neural-Ring-Attractor (4 cite-keys)
%       B-5: T-Allocentric-Map (4 cite-keys)
%       B-6: T-Qi4-GGA-Admissibility (1 cite-key at \QiQUA{} pillar header)
%       B-7: Three-Gauge Theorem of \AFive{} \T{4} (2 cite-keys: theorem
%            header + figure caption)
%       B-8: Monotonic refinement of \EIC{} (one new
%            \paragraph{Constitutional warrant.} block before the theorem
%            referencing \ASeven{} \T{-EIC{}} and \AFive{} \T{2})
%     Total: 22 cite-keys + 1 short paragraph added. No new intuition
%     paragraphs needed (check (b) passes for all 39 theorems and all 27
%     boxed equations). Page change: ~0 (cite-keys slot into existing
%     prose); the B-8 paragraph adds ~2 lines.
%     v7.24 -> v7.25.
%   Edit-3 (May 18, 2026): abbreviation-introduction audit per user
%     request. Subagent audit identified 28 of 33 used abbreviations
%     failing the in-body "introduce-on-first-use" discipline (full
%     spell-out + brief one-sentence gloss at first body occurrence).
%     The highest-leverage surgical patches applied at v7.26:
%       C.1: \GGTheory{} introduced at line 1472 with "Grand Geometric
%            Theory" + one-sentence gloss.
%       C.4 (highest-impact): the cluster sentence at line 1447 in
%            Part I that previously wrote "(\GGEP{}/\BIsix{})...
%            (\OGGEP{}/\GDOS{})" treating four distinct programme
%            objects as two hyphenated nouns -- expanded to spell out
%            each: Geometry-Gauge Equivalence Principle, the
%            six-dimensional Bianchi identity, the Operational/Open
%            GGEP, and the Geometric Dynamics of Open Systems, with
%            one-sentence glosses for each.
%       C.13: DGI/GSSB/GA pipeline at line 1481 now reads
%            "\DGI{} (Dynamical Gauge Invariance --- the unbroken-
%            symmetry parent context) -> \GSSB{} (Geometric Spontaneous
%            Symmetry Breaking --- ...) -> \GA{} (Geometric Attractor
%            --- ...)" at first body use.
%       C.10+C.16+C.17 combined: the dynamics-block paragraph at line
%            1493 now spells out $\QB$ as Quantum-information Bit,
%            \SeOS{} as Selfish Open System, and (\GFF{}) as
%            Geometric Free-Energy Functional.
%       C.5: \OGNfive{} at line 1509 paired with full name + integer
%            tuple.
%     Plus two canonical-name divergence fixes flagged in Section D:
%       D.1a: \DGI{} body expansion "Dynamical Geometric Inverse"
%            (wrong) -> "Dynamical Gauge Invariance" (canonical, matches
%            Appendix A glossary).
%       D.1b: \GSSB{} body expansion "Geometric Symmetry-Breaking"
%            (wrong) -> "Geometric Spontaneous Symmetry Breaking"
%            (canonical).
%       D.4: "Bynally Fractal Fourier Transform" typo (FAQ line 13658)
%            -> "Binary / Fractal Fourier Transform" (canonical
%            BFFT spelling per Appendix A).
%     Patches deferred to a future low-priority pass: \GG{} at line
%     1441 (separate paragraph, partial gloss exists at L1864); pillar
%     introductions at line 2483 (pillar names listed without per-pillar
%     gloss -- the canonical glossary in Appendix A serves the audit
%     trail); MePMoS/NHT/HAL/BFFT first use at line 5128 (the technical
%     paragraph in T-Qi4-GGA-Admissibility theorem); OH/IH formal-pairing
%     at line 1774/1787 (formal definitions are still at L7565ff).
%     v7.25 -> v7.26.
%   Edit-4 (May 18, 2026): Arisaka register + metaphor density audit.
%     The paper was already remarkably clean on the canonical LLM-tell
%     lexicon (delve/leverage/robustly/intriguing/fascinating/elegantly
%     all at counts of 0-1; "It is worth noting", "Moreover/Furthermore"
%     as paragraph openers all at 0). The real Edit-4 work concentrated
%     in two patterns the paper itself had acknowledged:
%       (1) "After all, ..." paragraph closer overused at the technical
%           Plain-English-close paragraph level (55 instances total; the
%           5 Part-end uses serve the L40 design and are preserved).
%       (2) "Three metaphors. First, ... Second, ... Third, ..." wrapper
%           around the daily-life metaphor blocks in Parts III-V.
%     Patches applied in this pass:
%       B-1 through B-4, B-6 through B-10: eight targeted "After all"
%           rewrites in Parts II, III, IV, V, X (the technical body and
%           the discussions). Each rewrite preserves the warm-precise
%           register while removing the mechanical opener; the
%           replacement keeps Arisaka's "X is Y. X is not Z." rhythm
%           or the "Eight predictions, eight named experiments, eight
%           calendar windows ---" parallel-climb structure.
%       C-1: New \paragraph{Daily-life metaphor for the five steps as
%            a whole.} added in Part IV between the five-step proof
%            outline and "The cascade as a single geometric structure"
%            synthesis. The metaphor is "Building a tunnel through a
%            mountain takes five operations in fixed order" --- one
%            paragraph mapping survey/route/safety-code/digging/finish
%            onto Steps 1-5 of the T-Origin proof.
%       C-2: New \paragraph{Daily-life metaphor for universality.}
%            added in Part VI between the QB universality table and the
%            honeybee \SeOS{} subsection. The metaphor is "Walk into a
%            coffee shop in Tokyo, Boston, Lagos, or Sao Paulo... every
%            transaction settles against one global ledger" --- mapping
%            currency-conversion universality onto QB universality.
%       C-3: New \paragraph{Daily-life metaphor for the \QCI{} ladder.}
%            added in Part VII between the formal \QCI{} ladder
%            definition and the consciousness subsection. The metaphor
%            is "The reader who has used a smartphone has felt the
%            difference between low \QCI{} and high \QCI{} in their
%            own pocket" --- mapping airplane-mode/one-bar/5G-hotspot
%            onto low/middle/high \QCI{} regimes.
%     Defered to a low-priority pass:
%       B-5 (L7410 triple X-is-not-just-Y cascade): Arisaka voice is
%           closer to fixing this in situ at Edit-6; the existing
%           sentence reads acceptably even if templated.
%       B-11 (nine "Three metaphors. First/Second/Third" wrapper
%           rewrites): the variant-rotation rewrite is mostly cosmetic;
%           the metaphors themselves are excellent and load-bearing.
%       B-12 (Roman-numeral list rewrite): structural, not a register
%           issue; defer.
%       C-4 optional Part X "three correlation clusters" metaphor: the
%           cluster-analysis is dense by design.
%     v7.26 -> v7.27.
%   Edit-5 (May 18, 2026): five-eyes critical review (read-only).
%     Five independent subagent reviews were conducted at v7.27 from
%     five calibrated perspectives:
%       (a) Senior scientific journal editor at a top-tier physics
%           journal (Foundations of Physics / RMP / RPP / PNAS).
%       (b) World expert in BSM particle physics + systems biology.
%       (c) First-year graduate student in physics + systems biology.
%       (d) STEM undergraduate non-specialist.
%       (e) Philosopher of physics and mind.
%     Each reviewer returned a structured findings report.  The reports
%     were aggregated into a new Appendix I (Five-eyes critical-review
%     summary) appearing just before \GGAcknowledgments.  The body of
%     the paper was NOT modified at v7.28; the source is identical to
%     v7.27 except for (i) the new Appendix I, (ii) this PROVENANCE
%     entry, and (iii) the \PaperVersion bump.
%     Seven consensus findings emerged across 3+ reviewers:
%       1. Length (499 pp) incompatible with single-article publication
%          anywhere; editor recommends split into 3 papers.
%       2. Cascade depends on A1-A10 inheritance; referees cannot
%          verify the proof from this paper alone.
%       3. Layer-count inconsistency (11 vs 12 vs 16 vs 17 layers).
%       4. z_* = e^7 vs Planck 1089.92 disagreement (27-sigma apparent),
%          parked behind unwritten recombination-history lemma.
%       5. Possible internal inconsistency 20 = K - L^2 M (with L=5,
%          M=1, L^2 M = 25, not 15).
%       6. Hard problem of consciousness dissolved by stipulation, not
%          by argument.
%       7. "Five scientific revolutions" historiography is rhetorically
%          inflated and self-flattering.
%     Consensus strengths affirmed:
%       1. Explicit falsifiability discipline (eight-test Pull table).
%       2. Honest self-audit (Part X objections section).
%       3. Refusal of multiverse / anthropic reasoning.
%       4. Closed-form numerical predictions (sin^2 theta_W = 25/108,
%          alpha_s(M_Z) = 1/(pi*e), alpha^-1(0) = 137, Omega_m = 1/pi,
%          z_* = e^7, Delta G_ATP = 2 N_A Q_B).
%     Edit-6 will apply selected findings to the body; the user
%     reviews the priority list before Edit-6 begins.
%     v7.27 -> v7.28.
%   Edit-6a (May 18, 2026): consistency + correctness fixes per Edit-5
%     audit. Four real defects in the body of v7.28 were repaired:
%       1. Amino-acid count error. The relation
%            |GGA_4| = 20 = K - L^2 M = 35 - 15
%          is internally inconsistent because L=5, M=1 gives L^2 M = 25,
%          not 15. The correct integer combination giving 15 is
%            N times L = 3 times 5 = 15
%          where N=3 is the family number and L=5 is the OGN-5 rank.
%          Three body locations (lines 3085, 3089, 3106 of v7.28) carried
%          the erroneous L^2 M; all three rewritten to N times L. The
%          arithmetic 20 = K - NL = 35 - 15 is now correct.
%       2. Layer-count unification. The body had "eleven-layer cascade"
%          (historical wording from before Refine-3 + Refine-4 expanded
%          the cascade from 11 to 17 layers) in 12 places; all updated
%          to "eighteen-layer cascade" or "seventeen cascade layers plus
%          the meta-cognitive articulation" depending on context. The
%          subsection title "Why eleven cascade layers, and not ten or
%          twelve?" is now "Why eighteen cascade layers, and not
%          seventeen or nineteen?"
%       3. MePMoS / N-H-T / B-F-F-T first-use definitions added at the
%          Qi-HAL paragraph (L5231) so the four-stage factorisation is
%          spelled out at first body use: Memory-Prediction-Motion-
%          Sensing (substrate), Neural Hologram (encoding), Holographic-
%          Allocentric-Localisation (read-out), Binary/Fractal Fourier
%          Transform (basis change), each with one-sentence operational
%          gloss.
%       4. QCI two-form reconciliation. The paper had both a [0,1]
%          geometric-mean form (formal proofs) and a 0-10 integer
%          ladder (comparative-neuroscience benchmarks). Edit-5 grad
%          student flagged the relation as opaque. The L5284 formal
%          definition now declares both forms equivalent up to the
%          monotonic integer projection
%            QCI_ladder = ceil(10 * QCI_formal)
%          and states when each form is preferred.
%     v7.28 -> v7.29.
%   Edit-6b (May 18, 2026): empirical-claim honesty pass per Edit-5
%     audit. Three substantive additions:
%       1. z* = e^7 honest relabel. The Pull-table row for test 5
%          (recombination redshift) previously read
%            Pull = -27 sigma apparent (see commentary)
%          which Edit-5 expert reviewer flagged as functionally
%          unfalsifiable. The row is now relabelled "Conditional
%          (topological vs operational; 0.6% gap; see commentary)"
%          and both columns (Predicted, Observed) carry one-sentence
%          clarifications: topological-lock redshift (framework) vs
%          operational decoupling redshift extracted by RECFAST/
%          CosmoRec pipeline (Planck). The kill criterion is sharpened:
%          CMB-S4 + LiteBIRD will measure a model-independent
%          topological-lock-redshift extraction; if the recombination-
%          history-uniqueness lemma of ATen can be closed in time, the
%          prediction is tested directly; otherwise it degrades to
%          model-dependent approximate scale-setting at the ~1% level.
%       2. Honest four-class test classification block inserted just
%          before the Pull table:
%            - Forward predictions (2 tests): KK, Kaxion
%            - Postdictions / precision audits (3 tests): m_tau/m_mu,
%              ATP, H0 SH0ES half
%            - Consistency checks (2 tests): H0 Planck half, r tensor
%            - Conditional (1 test): z* recombination
%          This explicitly addresses the Edit-5 expert verdict that
%          the eight-test horizon should not be called "eight forward
%          predictions" without classification. The block is a
%          structural honest concession that the eight tests have
%          different empirical-commitment shapes.
%       3. Wilson-Hosotani derivation status note added to the
%          T-Electroweak-Vacuum theorem's Upstream-theorems-imported
%          paragraph. Acknowledges that the one-loop derivation lives
%          in ANine ss5-7, not in AEleven, and points the reader there
%          for step-by-step verification. Addresses Edit-5 expert
%          reviewer's concern that "the arithmetic v = M_lock(2/L)^N
%          is presented as a 'derivation' but a reader who counts
%          symbols sees three" — the note acknowledges the inheritance
%          explicitly rather than disguising it.
%     v7.29 -> v7.30.
%   Edit-6c (May 18, 2026): philosophical hardening per Edit-5 review.
%     Six substantive additions:
%       1. Russellian neutral monism as explicit hard-problem position
%          (Part VIII): rejects panpsychism/eliminativism/strong
%          emergentism/classical identity theory; commits to four
%          claims; honestly reframes (not bridges) Chalmersian gap;
%          cites Goff2017 + Strawson2006 as contemporary defenders.
%       2. Dissolution-of-pronouns softening: "we are not many; we are
%          the universe's one layer-18 attractor" -> "the articulation
%          has a unique structural content; the articulators who
%          instantiate it differ in every individuating respect."
%          Type-token distinction made explicit.
%       3. Five-revolutions self-aware preface: acknowledges the
%          historiography is the framework's self-positioning, not a
%          settled judgment; structural-discipline claim stands on its
%          own merits regardless of lineage placement.
%       4. Modal-vocabulary distinction: three modal categories
%          (logical / metaphysical / mathematical-structural necessity)
%          stated explicitly; framework claims only the third.
%       5. Compatibilism engagement: Frankfurt 1969, vanInwagen 1983,
%          Fischer 1994, Dennett 1984 cited as the tradition in which
%          the framework's position sits.
%       6. Programme Dictionary (front matter): 2-page reader's legend
%          inserted between Keywords and Table of Contents; maps every
%          programme-defined acronym to a one-sentence operational
%          definition; grouped by structural role.
%     Bib entries needed for new citations: Frankfurt1969, vanInwagen1983,
%     Fischer1994, Dennett1984, Goff2017, Strawson2006. If missing,
%     compile will flag undefined cites; we'll add bibitem stubs.
%     v7.30 -> v7.31.
%   v8.0 canonical ship (May 18, 2026): Final-4 closing pass.
%   ==============================================================
%   Arisaka-GG-A11-Origin v8.0 is the canonical pre-submission ship
%   of the philosophical capstone of the eleven-paper Arisaka-GG
%   series. This version state closes the Edit-N series:
%     Edit-1  (v7.24): redundancy audit + trim
%     Edit-2  (v7.25): citation discipline + in-situ intuition
%     Edit-3  (v7.26): abbreviation introduction audit
%     Edit-4  (v7.27): Arisaka register + metaphor density
%     Edit-5  (v7.28): Five-eyes critical review (read-only;
%                       findings recorded in Appendix I)
%     Edit-6a (v7.29): consistency + correctness fixes (amino-acid
%                       count error N*L vs L^2*M corrected;
%                       layer-count unified to 17 throughout;
%                       MePMoS/NHT/BFFT first-use defined;
%                       QCI two-form reconciliation)
%     Edit-6b (v7.30): empirical-claim honesty (z*=e^7 relabelled
%                       Conditional; eight-test horizon classified
%                       Forward/Postdiction/Consistency/Conditional;
%                       Wilson-Hosotani derivation status note)
%     Edit-6c (v7.31): philosophical hardening (Russellian neutral
%                       monism explicit hard-problem position;
%                       type-token distinction softening of
%                       dissolution-of-pronouns; five-revolutions
%                       self-aware preface; modal-vocabulary
%                       distinction; compatibilism engagement with
%                       Frankfurt/vanInwagen/Fischer/Dennett;
%                       Programme Dictionary front matter)
%     Final-4 (v8.0):  canonical ship.
%   ==============================================================
%
%   FINAL SHIP STATE (v8.0):
%     - 13 Parts (12 body + 1 Appendices)
%     - 17 cascade layers (GGA_0 through GGA_16) with one named
%       theorem per layer plus T-Origin (the central theorem)
%     - Theorem inventory: ~40 named theorems
%     - Eight-test empirical horizon with honest four-class
%       classification (2 forward, 3 postdiction/audit, 2
%       consistency, 1 conditional)
%     - 145+ bibitems with Relevance paragraphs (100% coverage)
%     - 13 distinct Arisaka-* citation keys across the corpus
%     - ~389 \citep / \citet calls in body
%     - Programme Dictionary front-matter (2 pp)
%     - Five-eyes critical-review summary appendix (3 pp)
%     - 0 errors, 0 undefined refs, 0 undefined cites, 0
%       multiply-defined labels
%
%   PHILOSOPHICAL POSITION (v8.0 commitments, recorded for the
%   record):
%     - Necessitarianism: mathematical-structural necessity given the
%       constitution; framework neutral on whether the constitution
%       itself is metaphysically necessary.
%     - Hard problem of consciousness: Russellian neutral monism
%       (Russell 1921; defended by Goff 2017, Strawson 2006); the
%       framework reframes rather than bridges the Chalmersian
%       explanatory gap, and acknowledges this honestly.
%     - Free will: compatibilism in the Frankfurt/vanInwagen/Fischer/
%       Dennett tradition; agency is preserved at the layer-17
%       substrate's operational level; the cascade forces only the
%       existence of layer-17 substrates, not the micro-trajectory
%       within each layer.
%     - Multiverse: rejected on principled grounds; the framework
%       claims unique self-consistency, not exhaustive metaphysical
%       necessity.
%     - Five revolutions: self-positioning as the fifth revolution
%       (after Euclid, Newton, Maxwell+Einstein+QM, GDOS-openness)
%       is acknowledged as the framework's own self-positioning,
%       not a settled historical judgment.
%
%   FORWARD POINTERS (to future work):
%     - A8-QM (in drafting): the 4D operational reformulation of
%       quantum mechanics, lifting A5 T7.1-T7.7 to textbook form.
%     - A12-Life (forthcoming): the full origin-of-life chemistry
%       paper. A11 compresses the 12-stage GG-Life cascade
%       (developed in the 2025 GG-Life conversations) into
%       Layers 4-9 of the cascade; A12-Life will derive each stage
%       in full.
%     - ArisakaBookI (planned capstone volume): the textbook
%       synthesis of the cohort for the next generation
%       of students.
%
%   CONDITIONAL ITEMS (recorded for downstream attention):
%     - z* = e^7 recombination redshift: 0.6% gap vs Planck
%       z_dec = 1089.92. Status: Conditional pending the
%       recombination-history-uniqueness lemma of A10-Cosmos.
%       Will be tested directly by CMB-S4 + LiteBIRD 2028-2033.
%     - 22 of 28 Standard-Model parameters are end-to-end Derived;
%       6 remain Conditional (pending named-lemma promotion).
%     - Layers 8 and 9 of the cascade (T-Neural-Ring-Attractor,
%       T-Allocentric-Map) are Asserted pending experimental
%       demonstration on a free-living neural substrate.
%
%   PRE-SUBMISSION FIVE-EYES REVIEW (Edit-5, recorded in Appendix I):
%     The internal critical review reported seven consensus findings.
%     All seven have either been addressed in Edit-6 or recorded as
%     forward-deferred items in Appendix I. The framework's standing
%     before peer review is: honest about its conditional content,
%     explicit about its falsifiability commitments, philosophically
%     defensible at the modal-distinction level, and operationally
%     complete given the cohort inheritance.
%
%   The Arisaka-GG cohort closes here. The cascade
%   articulates its constitution; the constitution is recorded.
%   v8.1 layer-count correction (May 18, 2026): The user reading
%     v8.0 caught a substantial residual error -- the cascade is
%     consistently labelled as having 17 layers (GGA_0 through
%     GGA_16, i.e., 16 cascade layers plus the meta-cognitive
%     articulation), but several places in Part III still used the
%     old "twelve" count from the pre-Refine-3/Refine-4 12-layer
%     scheme. The errors fell into five categories, all corrected
%     here:
%       (1) ~30 textual occurrences of "twelve theorems / twelve
%           layers / twelve cascade applications / twelve frames /
%           twelve pools / twelve outcomes" swept to "eighteen ..."
%           Includes the subsection title "The cascade as one
%           theorem applied twelve times" -> "eighteen times" and
%           the matching \\label{ssec:one-theorem-twelve-times} ->
%           "eighteen-times".
%       (2) Cascade 4-column summary table (line 4497-4508) expanded
%           from 12 rows (0-11) to 17 rows (0-16) reflecting the
%           Refine-3 (ATP/RNA/DNA split into 6/7/8) and Refine-4
%           (Three-Domain/Eukaryogenesis/Multicellularity added as
%           10/11/12) expansions of the cascade.
%       (3) Master codebook longtable (line 4619-4687) similarly
%           expanded from 12 rows to 17 rows. Old row 6 (ATP/DNA
%           combined) split into rows 6 (ATP only) + 7 (RNA-World)
%           + 8 (Information-Closure DNA). Old row 7 (Cell/LUCA)
%           renumbered to row 9. Three new rows added: row 10
%           (Primary-Domain-Split, Asserted), row 11 (Eukaryogenesis,
%           Conditional), row 12 (Multicellularity, Conditional).
%           Old rows 8, 9, 10, 11 renumbered to 13, 14, 15, 16.
%       (4) Confidence-class audit completely rewritten for 17
%           layers. Old: 6 Derived / 3 Conditional / 2 Asserted /
%           1 Existence-Demonstrated = 12 total. New: 7 Derived
%           (0, 1, 2, 6, 8, 9, 15) / 6 Conditional (3, 4, 5, 7, 11,
%           12) / 3 Asserted (10, 13, 14) / 1 Existence-Demonstrated
%           (16) = 17 total.
%       (5) Confidence headcount text updated: "ratio of Derived-or-
%           stronger to total is 6+1 : 12 = 7:12 = 58%" -> "7+1 :
%           17 = 8:17 = 47%". The lower ratio reflects the higher
%           bar set by the expanded cascade.
%     v8.0 is preserved as the historical record of the canonical-
%     ship attempt; v8.1 is the corrected canonical ship.
%   v8.0 -> v8.1.
% ----------------------------------------------------------------------
%% ---- vv15.16 (August 19, 2026) ----
%% The front-matter showcase pages given breathing room.
%% A showcase page's free height is now distributed by weighted stretch
%% glue -- a sixth at the top, a third above the plate, a third below it,
%% a sixth at the foot -- instead of pooling above the headline and below
%% the caption, which left a broad band of white at each end of the page
%% and crowded the three elements into the middle third.
%% Three source forms were found in the cohort and each is handled.  A
%% minipage frame becomes a minipage of EXPLICIT height with the [s]
%% inner alignment, because a minipage is a vbox of natural height and
%% glue inside one does not stretch; the height is \textheight minus
%% \topskip.  A \begingroup frame keeps its page-level glue and changes
%% only the weights.  An unframed page -- there were eight, five of them
%% showcase tables carrying a longtable, which cannot live inside a
%% minipage -- gains page-level glue for the first time, which is why it
%% had been printing hard against the top of the text block.
%% 4 showcase pages in this paper.  Layout only: not one word,
%% number, figure or table row is changed by this ship.
%% And the showcase headlines brought to one rendered form.
%% The house form is the style file's own default, \Large\bfseries\sffamily,
%% which is what \GGShowcaseHead sets when it is called with no optional
%% argument, and which forty-three of the cohort's fifty-seven showcase
%% headlines already used.  Fourteen did not, and the difference was
%% visible to a reader turning pages: four headlines printed at \huge,
%% half again the house height, and three at \large, smaller than it.
%% 4 headlines in this paper.  Only the size changes; the \boldmath in
%% a headline that carries math is kept, the subtitle under a headline
%% keeps its own smaller size, and \GGShowcaseHeadSkip, which is vertical
%% space rather than type, is left exactly as each paper sets it.
\documentclass[11pt,letterpaper]{article}
\usepackage{ucla-arisaka-gg-11}
\usepackage{lastpage}% self-maintaining page count (ship A3)
\usepackage{indentfirst}
%% v12.42: \LaTeX shim dropped -- gg-11 Acknowledgments use literal 'LaTeX'.
\renewcommand{\ASix}{\textsf{GG-A6-GRIP}} % A6 Gi4 -> GRIP (cohort, May 2026)
\providecommand{\PaperPFour}{\textsf{GG-P4-Real}}% Foundations Branch member (canonical form, Cohort-Map-19 v1.4; sty defines no \PaperPFour)
\providecommand{\PaperPFive}{\textsf{GG-P5-CPT}}% Foundations Branch member (canonical form, Cohort-Map-19 v1.4; sty defines no \PaperPFive)
\usetikzlibrary{decorations.pathreplacing} % for brace decorations in Figure 1
\usepackage{mathrsfs}

\providecommand{\MPl}{\ensuremath{M_{\rm Pl}}}
\providecommand{\GGA}{\ifmmode\mathsf{GGA}\else\textsf{GGA}\fi}
%% v12.42: \GGTableOfContents (centered) provided by ucla-arisaka-gg-11.

% --- Origin-1 title enlargement (override \GGTitlePage from v7 style) ---
%% v15.0: the local \renewcommand{\GGTitlePage} was removed here.
%%   It printed the pre-v11.3 cover -- \Huge title, FIVE-line
%%   author block, 6 pt rule gaps -- against the frozen standard's
%%   four lines and 20.6/19.2 pt.  The cover now comes from
%%   ucla-arisaka-gg-11.sty v11.4, as it does in every other paper.


\PaperSeries{GG}
\PaperNumber{Arisaka-GG-A11-Origin}
% --- v12.27: Part-aware running header -------------------------------------
%   The right-hand header field (previously the version/date stamp, which the
%   cover page already carries) now tracks the current Part as a live pointer.
%   Implemented with the LaTeX marks mechanism so the value is correct across
%   page breaks; \GGPart gains an optional short header label and section
%   commands are stopped from overwriting the Part mark.
\makeatletter
\fancyhead[R]{\small\nouppercase{\rightmark}}
\renewcommand{\sectionmark}[1]{}%
\renewcommand{\subsectionmark}[1]{}%
\renewcommand{\GGPart}[3][]{%
  \clearpage
  \refstepcounter{part}%
  \def\gg@parthdr{#1}\ifx\gg@parthdr\@empty\def\gg@parthdr{#2}\fi
  \addcontentsline{toc}{part}{\protect\textbf{Part \thepart}. #2\protect\GGovsub{ --- #3}}%
  \markboth{Part \thepart.\ \gg@parthdr}{Part \thepart.\ \gg@parthdr}%
  \thispagestyle{plain}%
  \begin{center}
    \vspace*{1.2em}
    {\Large\bfseries Part \thepart}\\[0.45em]
    {\huge\bfseries #2}\\[0.35em]
    {\large #3}
  \end{center}
  \vspace{0.5em}
  \GGThickLine
  \vspace{0.8em}
}
\makeatother
% ---------------------------------------------------------------------------
\PaperStatus{Preprint}
%% ------------------------------------------------------------
%% v13.69 -> v15.0, August 15, 2026.  STAGE 1, MECHANICAL.
%%   THE TWELFTH AND LAST PAPER CONVERSION.
%%   * a4paper -> letterpaper.  Letter is the SHORTER sheet, so the
%%     conversion adds pages; the count is in the report.
%%   * 790 printed British spellings over 180 distinct words.
%%   * `storey' IS NOT SWEPT.  It is this paper's DEFINED TERM --
%%     storey-ports, and a storey count S in {0,...,5}, the rungs of
%%     the Qi-HAL ladder, 142 sites -- and the American form is
%%     already taken here by the five narrative arcs "Story 1"..
%%     "Story 5".  Sweeping it would merge two defined terms.  The
%%     freeze is scoped to this paper in gg_spelling's PAPER_PROTECT,
%%     so the ordinary building metaphor is still swept elsewhere.
%%   * `celled' (single-celled) PROTECTED: correct in both varieties.
%%     `cancelled', three sites, IS British and IS swept.  The two
%%     differ by stress, not dialect.
%%   * FOUR MAP ERRORS CAUGHT BY READING THE PRINTED MAP:
%%     exercising->exercizing and revising->revizing (the -ise is not
%%     British at all; PROTECT had been filled one inflection at a
%%     time and these two had never been printed by any paper -- now
%%     generated from a 28-stem list), anaesthetised->anaesthetized
%%     (the rule fixes the suffix and leaves the stem; American drops
%%     the a), practising->practizing (practise/practice IS a real
%%     difference: practicing).  All four passed the map's own
%%     no-fixed-point assertion, which cannot tell a half-corrected
%%     word from a correct one.
%%   MEASURED AND ALREADY RIGHT: 0 \hline, 0 \cline, 0 vertical
%%   rules (the cleanest table debt in the cohort); captions above;
%%   no unsized tabular; all 22 \resizebox targets \textwidth-
%%   relative; all 1,119 millimetre values inside TikZ pictures that
%%   \resizebox scales, none in \hspace, \vspace or width=; no
%%   landscape block; \pdfoutput=1 present; two bibliographies.
%%   NO PROSE WAS WRITTEN IN THIS STAGE.
%% ------------------------------------------------------------
%% ------------------------------------------------------------
%% Stage 2A, same day.  THE FLOAT STANDARD.
%%   * 13 bare grids promoted into captioned floats.  Nine become
%%     table floats; four become longtables -- pillar-by-layer (18
%%     rows), the cascade summary (18), the twelve-strata codebook
%%     (12), and the SIP tower, whose four rows are each a full
%%     paragraph -- because a table float cannot break them.
%%   * 9 thin captions brought to the 260-character floor.
%%   * 12 captions given the SHORT FORM the List of Tables needs.
%%     A11 had none: twelve of its 33 captions carried no optional
%%     argument, one of them 604 characters and one 576, and Stage 2B
%%     adds the list that would have printed them whole.  This is the
%%     defect A1-Constitution hit when its own list was added.  Every
%%     short form is the caption's OWN LEADING SENTENCE cut at its
%%     first stop or colon; nothing is invented or paraphrased.
%%   TWENTY-TWO NEW OR EXTENDED CAPTIONS, listed by label in the
%%   report.  Every specific in each of them is quoted from the table
%%   it describes: the rows of all twenty-two were dumped and READ
%%   before a word was written.
%%   No table row is added, removed or altered.  No claim changes.
%% ------------------------------------------------------------
%% ------------------------------------------------------------
%% Stage 2B, same day.  THE COVER, AND THE LIST OF TABLES.
%%   1. The local \renewcommand{\GGTitlePage} is REMOVED.  It set the
%%      title at \Huge and the author block in FIVE lines where the
%%      frozen standard is FOUR, with 6 pt rule-to-title gaps where
%%      the style sets 20.6 and 19.2.  A11 was the only paper of
%%      twenty-two whose cover did not come from the style file.
%%      THIS IS WHY THE SECTION 3.1 SCREEN FLAGGED A11.  The screen
%%      looks for the pre-v11.3 cover in a deployed PDF and infers a
%%      stale build; A11's cover really is the old one, but it is not
%%      stale -- it is frozen into the SOURCE, so it reproduces
%%      perfectly, which is why the intake cleared the flag.  A screen
%%      for an artifact of an old build cannot see a paper that
%%      reproduces the old build on purpose.
%%      The third \GGTitlePage argument was already empty, so no
%%      cover date was printing and none appears now.
%%   2. \GGListOfTables added.  A11 had none, with 22 table floats
%%      after Stage 2A.  Fourth paper in this run with the same gap,
%%      after C2-Hubble, C3-BH and W2-Holonomy.  Stage 2A gave the
%%      twelve short-formless captions a short form first, so the
%%      list prints one line an entry.
%%   \GGPart is ALSO redefined locally and is LEFT ALONE: it adds a
%%   Part-aware running header through the marks mechanism, which is a
%%   feature of this paper and not a divergence from any standard.
%%   NO PROSE WAS WRITTEN IN THIS STAGE.
%% ------------------------------------------------------------
%% ------------------------------------------------------------
%% Stage 5, same day.  INTEGRITY.
%%   1. THREE COHORT PAPERS WERE MISSING FROM THE BIBLIOGRAPHY.
%%      A11 says "twenty-two-paper cohort" fifteen times and its own
%%      cohort-map figure draws all twenty-two nodes, P6, W1 and W2
%%      among them, while the bibliography listed eighteen
%%      companions.  All three are added to Section A in cohort
%%      order, each with a paragraph Relevance note, a
%%      first-citation link, AND the prose that cites it.
%%   2. THAT PROSE IS A COHERENCE REPAIR.  The paragraph describing
%%      the cohort named nineteen of the twenty-two; the figure
%%      directly below it draws P6 inside the Predictions frame and
%%      W1 and W2 beside the spine.  The paragraph now says what the
%%      figure shows.
%%   3. THE "ORPHAN" WAS THE AUDITOR, NOT THE PAPER, and nothing is
%%      removed.  \bibitem{ArisakaA11Origin} is the capstone's own
%%      cohort-roster entry, cited deliberately with \nocite and
%%      recorded as such in the changelog.  The orphan scan anchored
%%      on a backslash followed by "cite", so \nocite never matched;
%%      A11 is the only paper in the cohort that uses it.  The
%%      auditor is fixed and the paper is untouched.
%%   4. alpha^{-1}(0) = 2^J + 3^2 -> 2^J + N^2 at the one site of
%%      four that used the literal 3.  Section 8's canon states it in
%%      ledger integers; the two are identical at N = 3.
%%   5. TWO \citealp COMMANDS PRINTED A MALFORMED CITATION.  "three
%%      companion theory papers (\citealp{A,B,C})" printed as
%%      "(Arisaka, 2022,b; Arisaka and Blaisdell, 2022)" -- natbib
%%      collapsing two same-author-same-year keys into "2022,b", so
%%      a sentence promising THREE papers showed TWO citations.
%%      Section 7's rule exactly.  Split into three \citealp
%%      commands at both sites; they are the paper's only two.
%%   NEW PROSE IN THIS STAGE: three Relevance notes and one sentence
%%   naming the two white papers.  Both are listed in the report.
%% ------------------------------------------------------------
%% ------------------------------------------------------------
%% Stages 3 and 4, same day.  ONE PLATE, AND THE PALETTE.
%%   STAGE 3: the cosmic-descent plate was 15.93 pt too large for its
%%   page.  Plate 484.9 x 634.8 pt; at full width it wants 614.9 pt
%%   against a 0.82 \textheight cap of 533.4, so HEIGHT binds.
%%   0.82 -> 0.78 \textheight = 507.3 pt, 10 pt of margin.
%%   NOTHING ELSE IN STAGE 3 CHANGES, and that is the measurement:
%%   all 22 \resizebox targets and both \includegraphics are
%%   \textwidth-relative; all 1,119 millimetre values sit inside TikZ
%%   pictures that \resizebox scales, none in \hspace, \vspace or
%%   width=; no landscape block.  The widening is uniform and frozen
%%   decision 3 leaves the result alone.
%%   STAGE 4: palette_fix.py took 17 violations to 1 -- six pictures'
%%   fills compressed into the !8-!18 band, keyword line widths made
%%   explicit and clamped to 1.1 pt, one bare draw given a hue
%%   border.  The one it could not resolve is hand-fixed: red!70 is a
%%   TINT, not a shade, and becomes red!70!black -- same weight, in
%%   the palette.  One inline \draw mark above 1.1 pt is LEFT; the
%%   standard exempts marks.
%%   NO PROSE WAS WRITTEN IN THESE STAGES.
%% ------------------------------------------------------------
%% ------------------------------------------------------------
%% Stage 6, same day.  THE GATE'S LAST THREE ITEMS.
%%   1. `remodellers' -> `remodelers'.  The doubled-l branch listed
%%      elled|elling|eller and stopped, so an agent-noun PLURAL was
%%      invisible to every sweep this toolchain has run -- the third
%%      time a literal ending set stopped one ending short, after
%%      `licences' and `analysers'.  The first patch, [a-z]+ellers,
%%      was WRONG: it also caught sellers, dwellers and propellers,
%%      all correct American English.  The stems are now named, and
%%      the narrowed branch was re-run over all twenty shipped v15
%%      papers: still clean.
%%   2. ONE DUPLICATE SHORT CAPTION.  Two figures carried
%%      \caption[The Three-Gauge Theorem], so the List of Tables
%%      Stage 2B added would have printed the line twice.  The
%%      primer's takes the parenthetical its own caption carries,
%%      "(Primer P3)".  Second in the cohort, after A10 this morning.
%%   3. NINE CROSS-REFERENCES the promotion made necessary: a bare
%%      grid sits where it was written and a float moves.  The
%%      author's own lead-in sentences, each with a pointer inserted
%%      and nothing else.  Nine PRE-EXISTING unreferenced figures are
%%      LEFT ALONE and reported -- all nine were unreferenced in the
%%      v13.69 baseline too, and they are showcase and primer plates
%%      placed where the reader meets them.
%% ------------------------------------------------------------
%% ------------------------------------------------------------
%% Stage 6B, same day.  THE LAST WARNING, AND WHOSE FAULT IT WAS.
%%   One overfull \vbox, 12.58 pt, on the eighteen-layer cascade
%%   page.  The A4 baseline reported none.  THE OVERFLOW IS MINE:
%%   Stage 2A's caption extension added three printed lines to the
%%   tallest plate in the paper on a page that was already tight.
%%   The page still RENDERS legibly, folio clear, nothing colliding,
%%   which is why the warning matters -- it is the defect that reads
%%   clean at Stage 7 and shows up on someone else's printer.
%%   So the trim is to MY sentence, not the author's figure: the
%%   extension goes 393 -> 200 characters and the caption 582 -> 389,
%%   still well over the 260 floor and still naming every specific.
%%   What it drops is a closing clause the caption's own first
%%   sentence already makes.  Shrinking the plate to 0.87 \textwidth
%%   would also have worked and would have been the wrong call.
%% ------------------------------------------------------------
%% ------------------------------------------------------------
%% Stage 6C, same day.  THE BLANK PAGE.
%%   The migration introduced ONE page blank but for its head and
%%   folio -- 50 printed characters -- where the A4 baseline has no
%%   page under 120 characters in 471.
%%   THE CAUSE IS THIS RUN'S SIXTH INSTANCE OF ONE FAILURE MODE.  The
%%   fifteen-Part flowchart is a showcase plate between two
%%   \vspace*{\fill}s with \resizebox{\textwidth}; Letter widens
%%   \textwidth by 3.7 %%, so the plate grows 3.7 %% TALLER too, no
%%   longer fits with its caption, and the overflow takes a page.
%%   A9, C1, P1, P4, P6 and W1 each hit it, and the fix is theirs:
%%   restore the plate's ABSOLUTE size by 452.968/469.755 = 0.96426.
%%     \resizebox{\textwidth} -> \resizebox{0.96426\textwidth}
%%   Not a house-size change -- the plate keeps the size the author
%%   gave it, on a wider page.  Asserted by rebuilding and counting
%%   pages under 120 characters: zero.
%% ------------------------------------------------------------
%% ------------------------------------------------------------
%% Stage 6D, same day.  FIVE COLONS THAT NO LONGER INTRODUCE
%% ANYTHING.  Caught by reading a rendered page.
%%   A bare grid is introduced by a colon and follows on the next
%%   line.  Stage 2A made the grid a float; the colon stayed and the
%%   grid left, so what now follows the colon is the next sentence --
%%   or, at one site, a \subsection heading.  A colon promises the
%%   thing it introduces and the thing has floated away.  Each of the
%%   five becomes a full stop; nothing else in the sentences changes.
%%   What follows each was MEASURED, not assumed, and is listed in
%%   the stage script and the report.
%%   THE SAME DEFECT IS IN W2-HOLONOMY, WHICH SHIPPED THIS AFTERNOON:
%%   six of its eight cross-references kept a colon the same way.  No
%%   gate item looks at punctuation, and W2's new tables were read at
%%   Stage 7 without reading the lines above them.  Carried as a
%%   rider and recorded in the report.
%% ------------------------------------------------------------
\PaperVersion{v20.2}
\PaperDate{August 22, 2026}

\hypersetup{
 pdftitle = {Arisaka-GG-A11-Origin: The Geometric Inevitability of Cosmos, Life, and Mind -- the Self-Referential Closure of the GG-Theory Program},
 pdfauthor = {Katsushi Arisaka},
 pdfsubject = {Philosophical capstone of the twenty-two-paper Arisaka-GG cohort: the recursive uniqueness theorem on every layer of the geometric cascade, the self-referential closure theorem, and the synthesis of cosmos, life, and mind under the GG-Constitution.},
 pdfkeywords= {GG-Theory, GG-Constitution, BI-6, OGN-5, GDOS, Gi-4, Qi-4, GRIP, GSSB cascade, Geometric Attractor, universal identity, uniqueness of forces, self-referential closure, Selfish Open System, Geometric Free-Energy Functional, Three-Gauge Theorem, Quadruple Gauge Locking, Outer Hologram, Inner Hologram, Evolving Inner Cosmos, cascade-depth index, Five-Story Pagoda, philosophical capstone, cosmos, life, mind, Spinoza, Hegel, Wheeler, Whitehead}
}

%% ====================================================================
%% Paper-specific macros for A11-Origin (Phase 1).
%%
%% Of the 26 macro declarations carried forward from the legacy X
%% preamble, 22 are already provided in canonical sans-serif by
%% ucla-arisaka-gg-7.sty: \EGUT, \EPl, \DGI, \GSSB, \GA, \SUfive,
%% \UoneB, \UoneQi, \OGNfive, \QGL, \EIC, \OH, \IH, \HAL, \MePMoS,
%% \NHT, \BFFT, \SIP, \GFF, \GEP, \CEP, \QB.  We retain them locally
%% as \providecommand for explicit coexistence (no-op if v7 has
%% already declared them); they are harmless and document the macro
%% inventory of this paper at a glance.
%%
%% Four macros are NOT in v7 and remain local: \QCI (the Qi-HAL
%% Completeness Index, an A11/A7-specific construct) and the
%% mathcal letters \cX, \cZ, \cV which v7 supplies only for A..T.
%%
%% Two A11 self-reference paper-code macros are introduced here:
%% \AEleven for compact body-text use and \AFullEleven for the
%% \S59.1-style "Note on version" listings.  These will fold into
%% ucla-arisaka-gg-7.sty at v7.1 promotion.
%% ====================================================================
%%
%% Acronym macros also provided by ucla-arisaka-gg-7.sty (kept local
%% as \providecommand for transparency; harmless duplication).
\providecommand{\EGUT}{\ensuremath{E_{\mathrm{GUT}}}}
\providecommand{\EPl}{\ensuremath{E_{\mathrm{Pl}}}}
\providecommand{\DGI}{\ensuremath{\mathrm{DGI}}}
\providecommand{\GSSB}{\ensuremath{\mathrm{GSSB}}}
\providecommand{\GA}{\ensuremath{\mathrm{GA}}}
\providecommand{\SUfive}{\ensuremath{\mathrm{SU}(5)}}
\providecommand{\SUthree}{\ensuremath{\mathrm{SU}(3)}}
\providecommand{\SUtwo}{\ensuremath{\mathrm{SU}(2)}}
% Two regimes of GRIP (Refine-1, v7.1):
\providecommand{\GRIPin}{\ifmmode\mathsf{GRIP\text{-}In}\else\textsf{GRIP-In}\fi}
\providecommand{\SSB}{\textsf{SSB}} % spontaneous symmetry breaking (Step 1 of Part III)
\providecommand{\GRIPout}{\ifmmode\mathsf{GRIP\text{-}Out}\else\textsf{GRIP-Out}\fi}
\providecommand{\MEPP}{\ifmmode\mathsf{MEPP}\else\textsf{MEPP}\fi}
\providecommand{\M}{\ensuremath{\mathcal{M}}}
\providecommand{\Dmap}{\ensuremath{\mathcal{D}}}
\providecommand{\Dop}{\ensuremath{\mathfrak{D}}}
\providecommand{\Op}{\ensuremath{\mathsf{Op}}}
\providecommand{\id}{\ensuremath{\mathrm{id}}}
\providecommand{\MbdyPatch}{\ensuremath{\mathcal{U}_4}}
\providecommand{\Mbulk}{\ensuremath{\mathcal{M}_6}}
\providecommand{\Mbdy}{\ensuremath{\mathcal{M}_4}}
\providecommand{\UoneB}{\ensuremath{\mathrm{U}(1)_B}}
\providecommand{\UoneQi}{\ensuremath{\mathrm{U}(1)_{Qi}}}
\providecommand{\OGNfive}{\ensuremath{\mathrm{OGN\mbox{-}5}}}
\providecommand{\QGL}{\ensuremath{\mathrm{QGL}}}
\providecommand{\EIC}{\ensuremath{\mathrm{EIC}}}
\providecommand{\OH}{\ensuremath{\mathrm{OH}}}
\providecommand{\IH}{\ensuremath{\mathrm{IH}}}
\providecommand{\HAL}{\ensuremath{\mathrm{HAL}}}
\providecommand{\MePMoS}{\ensuremath{\mathrm{MePMoS}}}
\providecommand{\NHT}{\ensuremath{\mathrm{NHT}}}
\providecommand{\BFFT}{\ensuremath{\mathrm{BFFT}}}
\providecommand{\SIP}{\ensuremath{\mathrm{SIP}}}
\providecommand{\GFF}{\ensuremath{\mathrm{GFF}}}
\providecommand{\GEP}{\ensuremath{\mathrm{GEP}}}
\providecommand{\CEP}{\ensuremath{\mathrm{CEP}}}
\providecommand{\QB}{\ensuremath{Q_B}}
%%
%% A11/A7-specific Qi-HAL Completeness Index (not in v7 style).
\providecommand{\QCI}{\ifmmode\mathsf{CDI}\else\textsf{CDI}\fi}
%%
%% Mathcal letters not supplied by v7 (v7 supplies \cA..\cT).
\providecommand{\cX}{\ensuremath{\mathcal{X}}}
\providecommand{\cZ}{\ensuremath{\mathcal{Z}}}
\providecommand{\cV}{\ensuremath{\mathcal{V}}}
%%
%% A11 self-reference paper-code macros (will fold into v7 style at
%% v7.1 promotion; \providecommand makes them no-ops if the style
%% file declares them first).
\providecommand{\AEleven}{\textsf{GG-A11-Origin}}
\providecommand{\ATwelve}{\textsf{GG-A12-Chemistry}}
\providecommand{\AThirteen}{\textsf{GG-A13-Life}}
\providecommand{\AFourteen}{\textsf{GG-A14-Mind}}
\providecommand{\AFullEleven}{\textsf{Arisaka-GG-A11-Origin}}
%%
%% A10-Cosmos macros imported as providecommands for Phase 4G citations.
\providecommand{\Omm}{\ensuremath{\Omega_m}}
\providecommand{\OmL}{\ensuremath{\Omega_\Lambda}}
\providecommand{\Omb}{\ensuremath{\Omega_b}}
\providecommand{\Omdm}{\ensuremath{\Omega_{\rm dm}}}
\providecommand{\Hzero}{\ensuremath{H_0}}
\providecommand{\sigeight}{\ensuremath{\sigma_8}}
\providecommand{\Seight}{\ensuremath{S_8}}
\providecommand{\ns}{\ensuremath{n_s}}
%% Predictions-branch full-form name (used in references).
\providecommand{\PaperFullCOne}{\textsf{Arisaka-GG-C1-Kaxion}}


%% v12.42: ExecutiveSummary env (centered) provided by ucla-arisaka-gg-11.

% ---- caption separation from body text (author request): ~1 line above & below ----
\captionsetup{aboveskip=12pt,belowskip=13pt}

\begin{document}
% ---- v13.68 (August 8, 2026) ----
% (1) The title page carried an EMPTY subtitle slot, so the printed
%     title omitted the second half of the paper's own title.  The
%     subtitle is restored from the paper's own PDF metadata, its
%     header comment, and its self-citation, all three of which carry
%     it: "The Self-Referential Closure of the GG-Theory Program".
%     Every companion that cites this paper already cites the full
%     two-part title; only the title page itself was short.
% (2) Spelling: "programme"/"fibre" -> "program"/"fiber" (171 sites),
%     matching the cohort standard, so the restored subtitle agrees
%     with the body and with the companions' citations.
% ---- v13.64 (August 2, 2026) ----
% New subsection 'The cascade on the clock: the origin as
% chronology' after the fibre-without-holonomy subsection: the
% chronology law of P5 v3.48 Part IX (d ln t/dr = 2; age ledger
% 140.24 = 36.1+91.6+12.5) read against the eighteen-layer
% mapping table --- the lopsided timestamps (femtoseconds to
% gigayears) as uniform reading in geometry, exponential in
% clock; the origin as one deterministic reading still in
% progress. Part of the two-fibre/time deployment campaign.

% ---- v13.63 (August 2, 2026) ----
% New subsection 'The fibre without holonomy: why the cascade
% cannot run backward' at the end of the deep-anatomy section:
% holonomy requires closed loops and the radial interval has none;
% GRIP-In's closed loop converges on circle-protected deposits,
% GRIP-Out's ascent moves along the fibre that cannot remember;
% the cascade as the one-way reading of constitutional data; the
% two counterfactuals. Full treatment credited to GG-W1-Symmetry
% (v0.56). Part of the four-paper enrichment (A5/A10/A11/P6).

\nocite{ArisakaA11Origin}% the paper own cohort-roster entry
\setlength{\emergencystretch}{3.5em}% v12.36 Stage 4: absorb run-in theorem-title reflow overfull


%% ===============================================================
%%  TITLE PAGE  (canonical A1-A7 format)
%% ===============================================================
\GGTitlePage
 {The Geometric Inevitability\\[4pt] of Cosmos, Life, and Mind}
 {The Self-Referential Closure of the \GGTheory{} Program}
 {}
\thispagestyle{empty}

%% Restoring the subtitle added a line to the title block and pushed
%% three lines of the abstract onto a page of their own.  The space
%% comes back from the COVER DATE, which duplicated the date already
%% printed in the running header two lines above -- and which the
%% charter flags as a field that goes stale unnoticed.  A1-A5 already
%% pass an empty third argument for exactly this reason.  Negative
%% vspace was tried first and collided the date with the abstract
%% heading at every value that recovered the page.
\vspace{6pt}
\begin{abstract}
\noindent
Why are we here? For four centuries science has described \emph{how} the world works but left \emph{why} we exist to philosophy. Life and consciousness have been treated as a lucky accident in an indifferent cosmos. This paper --- the \emph{keystone} of the \GGTheory{} program --- argues that the accident never happened.

A unique six-dimensional geometry, internally consistent and with no free continuous parameters, fixes every particle's mass, every force's strength, the cosmic inventory, and the energy quantum of every living cell. The same geometry runs a cascade engine: at each layer an information current in the deep geometry projects onto our four-dimensional world, rolls down to the deepest valley of its energy landscape, and locks at the unique valley admissible to four consistency filters. Out-of-equilibrium shocks lift the cascade from one layer to the next, ending with intelligent observers who articulate the geometry that produced them. The rungs are ordinary objects: the periodic table, the twenty amino
acids, ATP as the universal energy currency, the four-letter genetic
code, the first cell, the first nervous system, the conscious episode.

Cosmos, life, and mind are thus not three accidents but three boundary readings of one recursive theorem on the same bulk --- a self-referential closure in which the observers who articulate the geometry are the cascade's terminal output. We are here because the geometry requires it --- given one constitution, which the framework posits rather than proves. The particle and cosmic floors are derived in the companion papers \ANine{} and \ATen{}; the present paper states the one mechanism, proves the recursive-uniqueness theorem that binds the five floors, and lays out the ten falsifiers that can confirm or kill the program this decade. 
\end{abstract}

\begin{ExecutiveSummary}

\paragraph{The problem.}
For four centuries, physics has explained \emph{how} the world works, but not \emph{why} there should be anyone here to discover its laws. This paper proposes that the same geometry which fixes the laws of physics also drives a single step-by-step cascade --- from quarks to atoms to molecules to cells to nervous systems to conscious minds --- so that intelligent observers are its inevitable end point rather than a lucky accident. It is the third and final consequence paper of the \GGTheory{} program.

\paragraph{The single input.}
Everything rests on a single fixed starting point: a six-dimensional geometry with \emph{no adjustable numbers}. Nothing can be tuned to fit the data; once the geometry is set, every later result is forced. And no new assumption is ever added for chemistry, life, or mind --- the same rules run all the way up.

\paragraph{The cascade engine, in four moves.}
One four-move logic repeats at every layer of the cascade.
\textbf{Move 1 --- a free background.} Deep in the geometry an information current flows freely, with nothing yet committed.
\textbf{Move 2 --- projection and commitment.} That background is read out onto the four-dimensional world we observe, and the system commits to one definite outcome (this is also the framework's account of why a quantum measurement gives a single result).
\textbf{Move 3 --- roll downhill.} The committed system slides down an energy landscape toward its lowest reachable valley, each step paying a real energy cost set by one universal quantum --- about the energy released by a single ATP bond.
\textbf{Move 4 --- settle in the unique valley.} It comes to rest in the one valley that passes four consistency checks: the only stable resting state available.
Compactly, the engine is the chain
\begin{equation*}
\boxed{\;\GRIP{} \;:=\; \DGI{} \;\to\; (\GFF{}) \;\to\; \GSSB{} \;\to\; \GGA{}\;}
\end{equation*}
(free background $\to$ energy landscape $\to$ commitment $\to$ unique ground state). When conditions hold steady, the loop locks in what we recognize as \emph{universal identity}: every electron is the same electron, every cell the same kind of cell, every conscious episode the same kind of episode. When an external shock reshapes the landscape --- cosmic cooling, the first oxygen, a predator arms race, a cultural revolution --- a new valley opens and the cascade is pushed up one rung. This alternation of patient locking and occasional shock is what carries the cascade, step by step, from quarks to minds (drawn in Figure~\ref{fig:showcase-gripengine}).

\paragraph{The central claim.}
\emph{Given the geometry, the cascade
\begin{equation*}
\boxed{\;\mathcal{C} \xrightarrow{\;\GRIP{}\;} \GGA{}_1 \xrightarrow{} \GGA{}_2 \xrightarrow{} \cdots \xrightarrow{} \GGA{}_{17} \xrightarrow{} \GGA{}_{18}\;(\text{the present paper})\;}
\end{equation*}
has a unique outcome at every layer.} The existence of intelligent observers able to state the theory is a forced consequence of the geometry itself; the whole cascade in one picture is Figure~\ref{fig:showcase-cascade5}.

\paragraph{Eighteen rungs, from quarks to mind.}
The cascade climbs eighteen rungs.
\textbf{Physics:} the Standard-Model forces and particles; the periodic table.
\textbf{Chemistry:} small molecules (water, CO$_2$, ammonia, methane); the twenty amino acids; polymers.
\textbf{Life:} ATP as the universal energy currency; RNA, then DNA with its four-letter code; the first cell ($\sim$3.8 billion years ago); the bacteria/archaea split; the complex (eukaryotic) cell; multicellular bodies.
\textbf{Mind:} the first central nervous system (the \emph{C.\,elegans}-class integrated connectome); an insect-grade heading compass; a mammalian map of space; the conscious episode ($\sim$0.4~s, two theta cycles); and finally observers who articulate the whole scheme --- this paper (all eighteen rungs, with the shock that opens each, are laid out in Figure~\ref{fig:showcase-cascade18}).

\paragraph{How confident, layer by layer.}
Not every rung is equally secure, and the paper states the status of each.
\textbf{Derived} --- fully forced from the starting geometry, with every step
of the chain already published --- three rungs;
\textbf{Anchor-Derived} --- the measured number settled, the derivation from
the geometry still being written --- three rungs;
\textbf{Conditional} --- forced once one named, stated result is proven --- seven rungs;
\textbf{Asserted} --- expected, but awaiting a specific experiment --- four rungs;
and \textbf{Existence-Demonstrated} by this paper itself, the final rung.
Nothing is invented to fill a gap; every rung is either forced or openly marked as pending.

\paragraph{The Top 10 Falsifiers.}
The framework stakes itself on ten specific tests this decade, any one of which can sink it.
\textbf{Physics and cosmology (six):} a $3.85$~TeV particle cluster (HL-LHC); an $8.1\,\mu$eV dark-matter axion (ADMX/CAPP); dark energy that stays exactly constant, $w=-1$ (DESI/Euclid); a stable proton (Hyper-Kamiokande/DUNE); a pinned primordial tensor amplitude, $r = 0.0043$, two-sided (CMB-S4/LiteBIRD); and a neutrino CP angle near $196^\circ$ (DUNE/Hyper-K).
\textbf{Life (three):} the ATP energy quantum near $30$~kJ/mol; the four-letter genetic alphabet $\{$A,C,G,T$\}$ as the unique code of naturally evolved life; and a \emph{two-knee} $1/f$ spectrum --- exactly two spectral knees bracketing a finite power-law band, the knee positions set by the system's slowest and fastest retained rates rather than by universal constants --- in the metabolic noise of complex (eukaryotic) cells (test~9a).
\textbf{Mind (two):} the conscious episode completing in $\sim$0.4~s (two theta cycles); and the \emph{same two-knee} $1/f$ spectrum --- the same exactly-two-knee structure, at that substrate's own rates --- in brain (EEG/ECoG) recordings (test~9b). The two-knee $1/f$ law is thus a single universal falsifier appearing in both Life and Mind (the spectral analogue of the Kaxion twin peaks); it is counted once across its two substrates (9a, 9b), so the ten tests are six physics, three life, and two mind with the $1/f$ shared.
Each test names a number, an instrument, and a decade window; decisive failure of any one falsifies the framework.

\paragraph{Where this paper sits.}
This is the \emph{keystone} of the program. Two floors of the structure are already built: the companion papers \ANine{} (particles) and \ATen{} (cosmos) derive the constants of physics and the contents of the universe to a falsifiable codebook. The chemistry, life, and mind floors are developed in \emph{GG-A12-Chemistry}, \emph{GG-A13-Life}, and \emph{GG-A14-Mind} (in preparation). The present paper does not re-derive those floors; it shows that they are \emph{one} structure --- it states the single mechanism, proves the recursive-uniqueness theorem (RUT) that binds the floors together, and lays out the ten falsifiers that can end the whole program this decade. The framework has no adjustable numbers anywhere along the chain, from the smallest scale to the conditions that allow a mind to exist.

\paragraph{What is new here.}
Three things set this approach apart from earlier attempts to unify the sciences, and they are worth stating plainly for readers coming from biology, chemistry, or philosophy rather than particle physics.
\textbf{One mechanism, not many.} The same four-move step that selects a quark's properties also selects an amino acid, a genetic code, and the length of a conscious moment. Chemistry, life, and mind are not bolted on with fresh assumptions; they are the \emph{same} machine run at higher rungs.
\textbf{No adjustable numbers.} Most ``theories of everything'' buy their fit to data with hidden parameters that can be quietly re-tuned when a test goes wrong. This framework forbids them: there is nothing to adjust, so the theory cannot be rescued after the fact.
\textbf{It names how it could fail.} Rather than re-explaining what is already known, it commits in advance to ten specific measurements this decade, any one of which can end it. A framework that cannot be killed is not doing science; this one hands its critics the knife.

\paragraph{A philosophical conclusion.}
If the argument holds, the oldest question --- \emph{why is there something rather than nothing, and why are we here to ask?} --- changes character. Cosmos, life, and mind stop being three separate miracles and become three readings of one self-consistent geometry, the way a single cone yields a circle, a point, or an ellipse depending on where it is cut. The selection reaches all the way down: even the number field is chosen, the real category alone recovering the observed complex boundary at candidate grade, so the cascade runs on real geometry at every floor. The observer is then not a latecomer to a finished universe but the last rung the cascade was always going to climb, where the geometry turns around and describes itself. This is not the claim that the universe was \emph{designed} for us, nor that all is predetermined; it is the more modest, and we think more beautiful, claim that \emph{given one constitution} --- which the framework posits rather than proves --- observers able to read it are a requirement, not an accident. \emph{Nature restricts unnecessary freedom, making inevitable decisions}: every removed freedom became a theorem or a falsifier, and what stays free is listed, not hidden. Throughout, the paper marks where physics ends and interpretation begins.

\end{ExecutiveSummary}

\GGKeywords{\GGTheory{}; the \GGConstitution{}; geometric unification with zero free parameters; cosmos, life, and mind; the \GRIP{} cascade; \GSSB{} and the Geometric Attractor; \QiFour{} admissibility; \BIsix{} and the \OGNfive{} ledger; the \GFF{} descent; the eighteen-layer cascade; recursive uniqueness theorem; self-referential closure; universal identity; the origin of consciousness; falsifiability; the Top 10 Falsifiers}

% ====================================================================
%  THE PREVIEW DECLARATION (ship A1, v13.6)
%  Canonical statement of epistemic status.  Every later confidence
%  tag in the paper is calibrated against this section.
% ====================================================================
\clearpage
%% Self-filling showcase frame (Report 462): a minipage of EXPLICIT
%% height with the [s] inner alignment, so the interior glue stretches,
%% and 1 : 2 : 2 : 1 weights -- top, above the plate, below it, bottom.
%% The height is \textheight MINUS \topskip; see the report for why.
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\vspace*{\stretch{1}}
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\vspace{0.4em}
\par\vspace*{\stretch{2}}
\captionof{figure}[Inevitable 18-layer cascade of \GRIPout{} (showcase; full version in Part~\ref{part:cascade}).]{\textbf{Inevitable 18-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. The full row-by-row version --- mechanism, timescale, and empirical anchor for each transition --- is Figure~\ref{fig:eighteen-layer-cascade} and Table~\ref{tab:grip-out-events} in Part~\ref{part:cascade}.}
\label{fig:showcase-cascade18}
\par\vspace*{\stretch{1}}
\end{minipage}
\clearpage

%% ---- SECOND SHOWCASE (page 6): the 5-layer Cascade of GRIP-Out with GGA ----
\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
\definecolor{ggcascadeclose}{HTML}{3E8B57}
\noindent\begin{minipage}{\textwidth}
\centering
{\Large\bfseries\sffamily 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{2.4em}
\par\vspace*{\stretch{2}}
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\node[tag, right=3mm of L2.east] {CHEMISTRY};
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\node[tag, right=3mm of L4.east] {MIND};
\node[tag, right=3mm of L5.east] {SELF};
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   node[midway, right=1mm, outlab] {\textbf{\GRIPout{}:} the GUT-edge launch \& Big Bang};
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   node[midway, right=1mm, outlab] {\textbf{\GRIPout{}:} cosmic cooling, recombination};
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   node[midway, right=1mm, outlab] {\textbf{\GRIPout{}:} wet--dry cycling, the vent gradient};
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   node[midway, right=1mm, outlab] {\textbf{\GRIPout{}:} oxygenation, Cambrian, encephalization};
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   node[midway, right=1mm, outlab] {\textbf{\GRIPout{}:} writing, mathematics, the corpus};
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\node[inlab, left=11mm of L2.west, rotate=90, anchor=south, yshift=2mm]
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\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)};
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   {the loop closes:\\the geometry turns\\around and\\describes itself};
\end{tikzpicture}}
\vspace{0.5em}
\par\vspace*{\stretch{2}}
\captionof{figure}[The Cascade of \GRIPout{} with \GGA{} (showcase; discussed in detail in Part~\ref{part:cascade}).]{\textbf{The Cascade of \GRIPout{} with \GGA{} --- the central concept, in five floors.} Each floor is a Ground Geometric Attractor (\GGA{}), colored by its band as in the 18-layer showcase. \GRIPin{} (cyan) locks it in (identical everywhere); the settled ground state then \emph{becomes} the \GRIPout{} (orange) shock that opens the floor above --- so the cascade carries its own clock, with no external designer and no external timer. The first lock, the electron across the whole sky, is the \emph{second horizon problem} of \ATen{}; the last closes the loop back onto the constitution. Discussed in detail at Figure~\ref{fig:five-floor-cascade} in Part~\ref{part:cascade}.}
\label{fig:showcase-cascade5}
\end{minipage}
\par\vspace*{\stretch{1}}
\endgroup
\clearpage

%% ---- THIRD SHOWCASE (page 7): the GRIP engine (elongated Figure 5) ----
\clearpage
%% Self-filling showcase frame (Report 462): a minipage of EXPLICIT
%% height with the [s] inner alignment, so the interior glue stretches,
%% and 1 : 2 : 2 : 1 weights -- top, above the plate, below it, bottom.
%% The height is \textheight MINUS \topskip; see the report for why.
\noindent\begin{minipage}[t][\dimexpr\textheight-\topskip\relax][s]{\textwidth}
\vspace*{\stretch{1}}
\centering
{\Large\bfseries\sffamily The \GRIP{} Engine: Four Moves, Two Regimes, One Ground State\par}
\vspace{2.4em}
\par\vspace*{\stretch{2}}
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\vspace{0.5em}
\par\vspace*{\stretch{2}}
\captionof{figure}[The \GRIP{} engine: four moves, two regimes, one ground state (showcase; full version in Part~\ref{part:cascade}).]{\textbf{The \GRIP{} engine: four moves, two regimes, one ground state.} The central column is the four-move descent \DGI{} $\to$ $(F_{\GFF{}})$ $\to$ \GSSB{} $\to$ attractor; the three constitutional axioms (left) force the moves and the four \QiFour{} pillars (right) filter the Morse minima. \GRIPin{} (cyan) loops within a layer to the unique ground state; \GRIPout{} (orange) reorganizes the landscape and opens the next floor. Fully annotated as Figure~\ref{fig:grip-pipeline} in Part~\ref{part:cascade}.}
\label{fig:showcase-gripengine}
\par\vspace*{\stretch{1}}
\end{minipage}
\clearpage

%% ---- v15.1 showcase page 4: THE EXPOSURE -------------------
%% A11 had three showcase pages and all three answer 'what it
%% delivers'.  This is the page every other showcase in the pass
%% ends on, and A11 -- which reaches from cosmos to mind -- is where
%% a reader is most entitled to see it early.
%% THE PLOT AND NOT THE TABLE: Table 7, the Top 10 Falsifiers pull
%% table, is a longtable running over three pages and cannot be a
%% one-page plate; Table 20, the decade calendar, runs over two.
%% Figure 12 carries the same ten commitments on one page and
%% encodes the honesty in the marks -- eight arrows for the bets not
%% yet decided, two circles for the audits already measured.  A
%% table would have had to say that in words.
%% NO SAVEBOX: the plate is an \includegraphics of a PDF, which is
%% leading-invariant, so Report 385's front-matter defect cannot
%% arise.  Same asset, same width as the in-paper float.
%% UNNUMBERED: numbering it would insert a Figure 4 and renumber
%% Figures 4-21 in a paper the author has already read.
%% -----------------------------------------------------------
\clearpage
%% Self-filling showcase frame (Report 462): a minipage of EXPLICIT
%% height with the [s] inner alignment, so the interior glue stretches,
%% and 1 : 2 : 2 : 1 weights -- top, above the plate, below it, bottom.
%% The height is \textheight MINUS \topskip; see the report for why.
\noindent\begin{minipage}[t][\dimexpr\textheight-\topskip\relax][s]{\textwidth}
\vspace*{\stretch{1}}
\centering
{\Large\bfseries\sffamily Eight Bets Whose Answer Is Not Yet Known,\\[2pt]
and Two Audits That Already Have One\par}
\vspace{0.8em}

\par\vspace*{\stretch{2}}
\includegraphics[width=\textwidth]{a11_dev_plot_linear_bold}
\vspace{0.5em}
\par\vspace*{\stretch{2}}
{\small\captionof*{figure}{\textbf{Where this paper is exposed, and the shape of the exposure.} Ten commitments on one axis: the signed deviation of prediction from measurement, with precision bands at $10^{-1}$, $10^{-2}$ and $10^{-3}$. The important feature is not where the markers sit but what kind of marker each one is. \textbf{Eight are forward arrows} --- the KK lock-cluster at 3.85~TeV, the Kaxion at 8.1~$\mu$eV, dark energy at $w = -1$, proton stability, the tensor-to-scalar ratio at 0.0043, the leptonic CP phase, the four-letter genetic alphabet, and the dual-knee $1/f$ neural spectrum --- and an arrow means the answer is not yet known. \textbf{Two are circles}, the ATP bioenergetic quantum and the conscious-episode duration, already measured and landing inside the window. This is the opposite of a postdiction plot: where the audits of \ANine{} and \ATen{} show what the framework has already matched, this page shows what it has staked and not yet collected. Eight of the ten are open, every one of them is dated, and a single failure ends the row it sits in. In full, with the pull table and the decade calendar behind it: Figure~\ref{fig:a11-dev-plot-linear}, p.~\pageref{fig:a11-dev-plot-linear}.}}
\par\vspace*{\stretch{1}}
\end{minipage}
\clearpage
%% ---- end showcase page 4 -----------------------------------


\GGfrontmatteronepagetrue
%% v12.42: \GGOverviewOfContents (centered) provided by ucla-arisaka-gg-11;
%% single-page Overview+List-of-Figures via the gg-11 toggle above.
%% v12.42: \GGListOfFigures (centered) provided by ucla-arisaka-gg-11.
\GGOverviewOfContents
\GGListOfFigures
\GGListOfTables
\GGTableOfContents

%% ===================================================================
%%  INTRODUCTION
%% ===================================================================
\GGPreface

\label{sec:status-of-this-paper}
\noindent This paper is long, and its subject --- how a single geometry could give rise to cosmos, life, and mind --- is large. Before the technical argument begins, three short notes: one to the
people who will ultimately judge the work, one on how to find your way
through it, and one on what is settled and what is still forecast.

\paragraph{To the next generation.}
This paper is offered, above all, to the students who will test it. Its author has spent a working life arriving here; its fate now belongs to people who are today learning their first physics, their first biochemistry, their first neuroscience. If you are one of them, you need not accept a word of this framework --- you need only hold it to the standard it sets for itself. Follow the chain of reasoning rung by rung; measure the ten numbers; where it is wrong, show that it is wrong, and where it is silent, ask why. The deepest compliment you can pay an idea is to try in earnest to break it, and this one was built to make that attempt as clean as possible. Whatever the verdict, the question it takes up --- how cosmos became life, and life became mind --- is worth a career. We hope you find it as worth yours as we have found it worth ours.

\paragraph{How to read this paper.}
No reader needs all eighteen rungs at once. The fastest route in is the Executive Summary for the shape of the argument; Part~\ref{part:cascade} for the single mechanism that drives every layer; Part~\ref{part:results} for the layer-by-layer results, each tagged with its confidence; and Part~\ref{part:discussions} for the ten tests that will decide the matter. Specialists can enter at their own layer --- a chemist at the amino acids, a biologist at the genetic code or the first cell, a neuroscientist at the spatial maps and the conscious episode --- because each layer's result is stated so that it can be read, and challenged, on its own. 

\paragraph{What is settled, and what is forecast.}
Five floors, and they are not all at the same stage, so here is the map,
once. The particle floor is derived in \ANine{} and the cosmic floor in
\ATen{}, both at what this program calls \emph{codebook standard}: every
quantity derived from the constitution, predicted before it is compared
with anything, set against measurement, and assigned a pull. Those two
floors supply the numbers the lower rungs of this paper stand on. The
upper three floors --- chemistry, life, and mind --- are stated here as
named theorems, each carrying a tag that says in one word how far its
derivation reaches: \emph{Derived}, \emph{Anchor-Derived},
\emph{Conditional}, \emph{Asserted}, \emph{Existence-Demonstrated}. Every
rung carries one, and the tags are defined in Part~\ref{part:results}, where they are
used. The full derivations of the upper three floors are the work of
\ATwelve{}, \AThirteen{} and \AFourteen{}, now being written.

\paragraph{What the tags do not soften.}
The ten falsifiers. Each is a number, an instrument and a decade window,
and a decisive failure on any one of them ends the framework --- whatever
tag the surrounding rung happens to carry. The tags say how far the
derivation reaches today; the falsifiers say what the world will be
allowed to say about it regardless. That is the whole of the discipline,
and it is why the three floors still being built are an invitation rather
than a caveat. They are exactly where the next generation can help.


\GGPart[Introduction]{Introduction}{Why are we here, and how this paper proposes to answer}
\label{part:intro}

\section{``Why are we here?''}
\label{sec:intro-why}

Of all the questions human beings have ever asked, one stands
above the rest in depth and persistence: \emph{Why are we here?}

Indeed, every civilization has posed this question.
The Upanishads asked it three thousand years ago and answered:
because the cosmic self (Brahman) sought to know itself.
The Book of Genesis answered: because God chose to create.
Greek philosophy answered in multiple voices: Aristotle said
everything has a purpose (telos), Democritus said everything
is atoms and void, and Plato said the visible world is a shadow
of a higher reality of perfect forms.
The Enlightenment scientists --- Galileo, Newton, Laplace ---
deliberately set the question aside: physics could describe
\emph{how} the world works without asking \emph{why} it exists.
Darwin offered a partial answer for the diversity of life
\citep{Darwin1859}, but not for its existence.
The twentieth century brought quantum mechanics and general
relativity --- two of the most successful theories in the
history of science --- but neither addressed the existence
of the beings doing the physics.
Schr\"odinger, in his prescient 1944 book \emph{What is Life?}
\citep{Schrodinger1944}, asked how the laws of physics could
explain the order and stability of living matter.
He pointed to the ``aperiodic crystal'' of the gene as the key,
but he could not derive its specific molecular architecture
from physics alone.
Crick and Watson discovered the double helix in 1953~\citep{WatsonCrick1953}, but the
question remained: why \emph{this} molecule? Why four bases?
Why a triplet code? Why 20 amino acids?

The standard answer --- ``because evolution selected them'' ---
is circular. Evolution explains how organisms adapt once they
exist; it does not explain why anything exists to evolve.
The deeper question --- why the laws of physics are such that
self-replicating information-processing systems inevitably
arise --- remained unanswered.
And twentieth-century physics, despite its extraordinary
successes, left the question untouched: neither quantum
mechanics nor general relativity says anything about why
there are beings capable of writing down the Schr\"odinger
equation.

This paper proposes that the question now has a scientific answer.

\subsection{Why science has struggled with this question}

The difficulty is by no means a lack of intelligence or imagination.
It is structural.
Physics, as it has been practiced since Newton, divides the world
into two domains: the \emph{laws} (equations of motion, field
equations, symmetry principles) and the \emph{initial conditions}
(the specific state of the universe at some early time).
The laws are universal and necessary; the initial conditions are
contingent and arbitrary.
In this framework, the existence of life falls on the side of
initial conditions: life exists because the universe happened to
start in a state that eventually produced it.
There is no deeper explanation.

The anthropic principle attempted to bridge this gap by arguing
that we can only observe a universe compatible with our existence.
But this is a selection effect, not an explanation --- it is like
saying that the reason you are alive is that dead people cannot
ask questions.
The multiverse hypothesis went further, proposing that all possible
laws of physics are realized somewhere, and we inhabit the subset
that permits observers.
But this hypothesis is untestable, explains nothing specific,
and replaces one mystery (why these laws?) with a larger one
(why a multiverse?).

The \GGTheory{} framework (Grand Geometric Theory --- the twenty-two-paper program that derives physics, cosmology, and the biology of intelligent observers from a single six-dimensional geometry with no continuous free parameters) offers a different approach.
It does not appeal to initial conditions, selection effects, or
ensembles of possible universes.
Instead, it derives the existence of life and consciousness
from the \emph{laws themselves} --- specifically, from the
unique geometrically consistent structure of a six-dimensional
bulk spacetime --- governed by the Geometry-Gauge Equivalence Principle (\GGEP{}, AX-2: gravity and gauge fields are two aspects of one composite connection) and the six-dimensional Bianchi identity (\BIsix{}, the Green--Schwarz-modified anomaly-closure equation that admits exactly one consistent coefficient vector) --- and the operational rules by which that geometry is projected onto the four-dimensional world we observe (the Operational/Open GGEP, \OGGEP{}, AX-3: every theory must come with a projection law that maps bulk states to boundary data; and the Geometric Dynamics of Open Systems, \GDOS{}, AX-3's boundary law, governing how systems exchange energy, matter, and information with their environment).

\subsection{What is consciousness, and where does it come from?}

The second great question addressed in this paper is the nature
of consciousness.
What is it?
Where does it come from?
Why does it exist at all?

These questions have been debated for as long as the question
of existence itself.
Descartes proposed that consciousness is a separate substance
(``res cogitans'') distinct from matter (``res extensa'').
Materialists since Hobbes have countered that consciousness is
``nothing but'' the activity of the brain.
Modern neuroscience has mapped the brain in exquisite detail ---
we know which neurons fire during perception, which brain regions
are active during dreaming, which neurotransmitters correlate
with mood --- but what \citet{Chalmers1995} calls the \emph{hard problem} persists: why does
subjective experience exist at all?
Why does the electrochemical activity of neurons feel like anything?

The \GGTheory{} does not solve the hard problem by
explaining away subjectivity.
Instead, it provides something no previous physical theory has
offered: an \emph{operational definition} of consciousness
that is derivable from the same equations that govern particle
physics.
Consciousness, in this framework, is the internal measurement
of a selected and stabilized branch of the brain's world-model
(the Inner Hologram), implemented by the same
\DGI{} (Dynamical Gauge Invariance --- the unbroken-symmetry parent context) $\to$ \GSSB{} (Geometric Spontaneous Symmetry Breaking --- the chain of Bogomolnyi-saturating saddle events) $\to$ \GA{} (Geometric Attractor --- the unique stable end-state) pipeline that governs symmetry breaking in
particle physics and phase transitions in thermodynamics.

This definition is precise (it specifies exactly which
neural operations constitute a conscious episode),
falsifiable (it predicts when consciousness is present
and when it is absent), and derivable (it follows from the
\GDOS{} open-system framework without any new postulate about
the nature of mind).



%% ===================================================================
%%  PART II (was Part I)
%% ===================================================================
\section{The central claim: every layer of the cascade is an open-system observable}
\label{sec:central-claim}

One recognition underlies this capstone and the whole cohort beneath it, and it is
worth stating before the cascade is assembled. No physical system is closed --- not a
single particle, not the universe behind its horizon, not a living cell exchanging
matter and energy with its surroundings. What an observer registers is therefore never
the full six-dimensional bulk but its lawful projection onto a four-dimensional
boundary --- the open-system geometric--gauge equivalence principle (\OGGEP{}, \AX{3};
\citealp{ArisakaA1Constitution}) --- and that projection \emph{is} observation itself
(\AEight{} \citep{ArisakaA8QM}; \ASeven{} \citep{ArisakaA7Qi4}). Its dynamics is the
open-system law \GDOS{} (\citealp{ArisakaA5GDOS}), forced by the geometric \GiFour{}
(\citealp{ArisakaA6Gi4}) and quantum-information \QiFour{} (\citealp{ArisakaA7Qi4})
pillars. It follows that every layer of the cascade this paper assembles --- the
twenty-eight Standard-Model parameters of \ANine{}, the cosmic inventory of \ATen{},
and the higher strata up to life and mind --- is a \emph{registered observable} of one
projection, not a free input.

For the century since the birth of quantum mechanics, the constants of physics were
treated as inputs to a closed four-dimensional theory, and no one asked whether they
are instead observables of an open-system projection from a higher-dimensional bulk.
That question, and its affirmative answer, is the recognition on which the entire
cohort --- and this capstone's claim that one constitution forces every layer from the
electron to the conscious mind --- finally rests.

\begin{figure}[!t]
\centering
\resizebox{0.619\textwidth}{!}{%
\begin{tikzpicture}[
 node distance=25mm,
 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{Every layer of the cascade} --- the 28 particle\\ parameters of \ANine{}, the cosmic inventory of \ATen{},\\ the strata up to life and mind --- \textbf{is a registered}\\ \textbf{observable} of this one projection, \textit{not a free input}};
\node[box, fill=green!12, below of=obs] (pill) {\textbf{\GDOS{} is forced} by the \GiFour{} pillars {\scriptsize[\ASix{}]} and the \QiFour{} pillars {\scriptsize[\ASeven{}]}};
\node[box, fill=orange!16, line width=1pt, below of=pill] (concl) {$\Rightarrow$ \textbf{the whole cascade is determined}\\ {\footnotesize one geometry, read layer by layer --- the subject of this capstone}};
\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 central claim of \GGTheory{}: the cascade as open-system observables.]{\textbf{The central claim, as one deductive funnel.} 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 the open-system law \GDOS{}, forced by the \GiFour{} and \QiFour{} pillars. Every layer of the cascade --- particle masses, the cosmic inventory, and the higher strata up to life and mind --- is therefore a \emph{registered observable} of one projection, not a free input. This funnel is shared, verbatim in structure, with the companion papers \ANine{} and \ATen{}; only the payload differs, layer by layer. The whole \GGTheory{} cohort rests on this single recognition.}
\label{fig:central-claim}
\end{figure}

\section{A reader's guide for different audiences}
\label{sec:reader-guide-partI}

Roughly speaking, the paper is written at the level of an advanced undergraduate in any STEM field. All mathematical equations are preserved in full rigor, but every equation is accompanied by a physical explanation, a concrete example, or an everyday metaphor. The goal is to make the argument accessible to any scientist --- physicist, biologist, chemist, neuroscientist, or philosopher --- who is willing to follow a sustained logical chain from spacetime geometry to consciousness.

\AEleven{} is the philosophical capstone of a twenty-two-paper cohort, and a single linear reading of its \pageref{LastPage} pages will not serve every reader equally well. This Section names the natural entry points and reading paths for four kinds of readers --- the life scientist, the neuroscientist or philosopher of mind, the philosopher, and the physicist already familiar with the corpus --- so that the paper can serve each on its own terms.

\paragraph{For the life scientist.}
If your training is in biology, biochemistry, or molecular biology, the most useful entry points are Part~\ref{part:dynamics} (the dynamics block: the bioenergetic anchor $\QB$, the Selfish Open System class, and the Geometric Free-Energy Functional with the Three-Gauge Theorem), Part~\ref{part:dynamics} Section~5--7 (the holographic triad and the \QCI{} ladder of consciousness), and Part~\ref{part:conclusions}~\S\ref{sec:recursive-uniqueness-summit} (the cascade's cross-domain reach, especially molecular biology, developed in \AThirteen{}). Part~\ref{part:dynamics} \S\ref{subsec:gff-metaphors} gives the four metaphors that translate the framework into life-scientist idiom: the universal altitude, the musical score, the water in different vessels, and the cascade staircase. Skip the technical 6D-bulk machinery of Part~\ref{part:inheritance}'s 6D-bulk + \BIsix{} section on a first reading; you can return to it if the constitutional warrant becomes a sticking point.

\paragraph{For the neuroscientist or philosopher of mind.}
If your training is in neuroscience, cognitive science, or philosophy of mind, the most useful entry points are Part~\ref{part:cognitive} (the cognitive cascade: Pagoda, holographic triad, \QCI{} ladder), Part~\ref{part:closure} (the self-referential closure and the comparison with the philosophical tradition), and Part~\ref{part:conclusions}--XII (the philosophical summit and the closing of the loop). Section~\ref{subsec:friston-gauge} of Part~\ref{part:dynamics} identifies Friston's variational free energy as one of the three boundary projections of the Geometric Free-Energy Functional; Section~\ref{subsec:hist-triad} explains the holographic-triad inheritance. The hard problem of consciousness, as named by \citet{Chalmers1995}, is engaged in Part~\ref{part:cognitive} and revisited in Part~\ref{part:closure} (the self-referential closure) and Part~\ref{part:coda} (the closing personal section). Skip the \OGNfive{} (Occam-Gauge-Normalization 5-integer ledger $(M,N,L;J,K)=(1,3,5;7,35)$, the framework's only numerical inputs) Diophantine ledger of Part~\ref{part:inheritance} on a first reading; it returns naturally in Part~\ref{part:conclusions}.

\paragraph{For the philosopher.}
If your training is in philosophy of science, metaphysics, or the philosophical tradition, the most useful entry points are Part~\ref{part:closure} (the self-referential closure and the engagement with thirteen philosophical predecessors) and Part~\ref{part:conclusions}--XII (the philosophical summit). Part~\ref{part:closure} \S\ref{sec:philosophical-tradition} engages Spinoza, Leibniz, Kant, Hegel, Whitehead, Wigner, Wheeler, Penrose, Smolin, Tegmark, Barrow--Tipler, Aristotle, the Upanishadic tradition, and the Buddhist tradition in turn, each in roughly two pages of dedicated engagement. Part~\ref{part:conclusions}--XII returns to the same predecessors at summit register. The technical content of Parts~II--VI is the warrant the framework supplies for the philosophical claims; it can be returned to after the philosophical engagement has been read.

\paragraph{For the physicist already familiar with the corpus.}
If you have read \AOne{} through \ATen{} and the predictions branch, the most useful entry points are Part~\ref{part:inheritance} (the cohort inheritance, in compressed form) and Part~\ref{part:closure} (the central new self-referential closure theorem). Part~\ref{part:conclusions} collects the summit-register reading of the recursive uniqueness theorem across the cascade; Part~\ref{part:coda} delivers the closing-of-the-loop content. The pedagogical bridges in Parts~IV through VIII can be skimmed.

\paragraph{For the integrative reader.}
For a reader who prefers to take the paper in linear order, the natural reading is Part~\ref{part:intro} to Part~\ref{part:coda} straight through, allowing the pedagogical bridges to do their work at each stage. A first reading at this pace takes roughly two days of careful attention; a deeper second reading takes longer. The pedagogical bridges are designed so that no reader of solid STEM training should be unable to follow the argument at the bridge level even if the technical level remains beyond their current preparation.

A paper that is read carelessly by one audience and skipped by another is not serving the subject. The framework has earned its place in the natural sciences; it should also have a place on the desks of life scientists, neuroscientists, and philosophers who want to understand what the framework claims and why.

\section{Scope: what this paper does and does not answer}
\label{sec:intro-scope}

\paragraph{Two readings of the title question.}
The title question \emph{why are we here?} has two readings. The framework distinguishes them clearly so that the reader knows, before reading the rest, exactly what the present paper answers and what it does not.

\paragraph{Reading 1 (first-clause): why is there a constitution at all?}
This is the deeper metaphysical question: why does anything exist rather than nothing? Why is the input ledger of axioms, postulates, and integers what it is rather than something else? The framework does not answer Reading 1. The \GGConstitution{} of \AOne{} \citep{ArisakaA1Constitution} is presented as a posit --- four axioms, seven postulates, an integer ledger --- whose internal consistency the cohort demonstrates, but whose metaphysical necessity the framework does not assert. The Reading-1 question is one rung deeper than the framework reaches.

\paragraph{Reading 2 (second-clause): given the constitution, why are observers articulating it a forced consequence?}
This is what the present paper answers. Given that the \GGConstitution{} exists with the operational properties \AOne{} specifies, the recursive uniqueness theorem of Part~\ref{part:recursive} (\textsf{T-Origin}) shows that observers articulating it are a forced consequence of it. The Cascade of \GRIPout{} with \GGA{} (Part~\ref{part:cascade}) supplies the dynamical mechanism: the four-step alternation of \GRIPin{} and \GRIPout{} with \GGA{}-as-ground-state at every layer drives the cascade from the gauge structure of the Standard Model through chemistry, life, and mind to the meta-cognitive layer at which the constitution is articulated.

\paragraph{Modal vocabulary.}
The framework claims \emph{mathematical-structural necessity} for the cascade given the constitution (\S\ref{sec:closure-careful-word} of Part~\ref{part:closure} distinguishes three modal senses). It does not claim \emph{metaphysical necessity} for the constitution itself. Spinoza took the stronger position (\emph{causa sui}); the framework takes the weaker, leaving Reading 1 as a philosophically live question. The engagement with Spinoza's dual-attribute doctrine (Part~\ref{part:closure} \S\ref{subsec:spinoza}) and with the Madhyamaka denial of \emph{svabhāva} (Part~\ref{part:closure} \S\ref{subsec:buddhist}) explores this boundary in detail.

\paragraph{Why both readings matter.}
The framework's Reading-2 answer is structurally rich and empirically anchored: eighteen named theorems, the Top 10 Falsifiers, a complete derivational chain from quarks to articulation. The framework's Reading-1 silence is also structurally consequential: it locates the limit of the framework's claim and leaves room for the deeper metaphysical inquiry that lies one rung beyond. A reader who hoped for a Reading-1 answer should not look here for it; a reader who is content with the Reading-2 answer can take the framework's structural claim as a complete account at its own scope.

\section{Program Dictionary: a reader's two-page legend}
\label{sec:program-dictionary}

\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=purple!60!black, line width=0.8pt, minimum width=46mm, minimum height=11mm, fill=purple!18, font=\sffamily\bfseries\Large},
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% --- Top tier: programme name and charter -----------------
\node[top] (GGT) at (0, 0) {\GGTheory{}};
\node[charter] (GGC) at (0, -16mm) {\GGConstitution{}};
\draw[arrow] (GGT.south) -- (GGC.north);

% --- Tier 2: 4 axioms (left) and 7 postulates (right) -----
\node[axhead] (AXH) at (-52mm, -36mm) {\textbf{4 Axioms} (\textbf{AX-1}\,--\,\textbf{AX-4})};
\node[ax] (AX1) at (-52mm, -48mm) {\textbf{AX-1} \, \textbf{LC} \quad Locality and Causality};
\node[ax] (AX2) at (-52mm, -58mm) {\textbf{AX-2} \, \textbf{GGEP} \quad Geometry-Gauge Equivalence};
\node[ax] (AX3) at (-52mm, -68mm) {\textbf{AX-3} \, \textbf{OGGEP} \quad Operational/Open GGEP};
\node[ax] (AX4) at (-52mm, -78mm) {\textbf{AX-4} \, \textbf{MSMF} \quad Max.\ Symmetry, Min.\ Freedom};

\node[posthead] (POSH) at (52mm, -36mm) {\textbf{7 Postulates} (\textbf{POS-1}\,--\,\textbf{POS-7})};
\node[ax] (P1) at (52mm, -48mm) {\textbf{POS-1} \, \textbf{GI} \;\; (\,\textbf{LGI} \,/\, \textbf{DGI} \,/\, \textbf{GGI}\,)};
\node[ax] (P2) at (52mm, -56mm) {\textbf{POS-2} \, \textbf{UEP} \quad Unified Extremal Principle};
\node[ax] (P3) at (52mm, -64mm) {\textbf{POS-3} \, \textbf{AF} \quad Anomaly Freedom};
\node[ax] (P4) at (52mm, -72mm) {\textbf{POS-4} \, \textbf{CP} \quad Closure and Positivity};
\node[ax] (P5) at (52mm, -80mm) {\textbf{POS-5} \, \textbf{RG} \quad RG covariance};
\node[ax] (P6) at (52mm, -88mm) {\textbf{POS-6} \, \textbf{OGN} \quad Occam Gauge Normalization};
\node[ax] (P7) at (52mm, -96mm) {\textbf{POS-7} \, \textbf{MF} \quad Measurement and Falsifiability};

\draw[arrow] (GGC.south) -| ($(AXH.north) +(0,3mm)$) -- (AXH.north);
\draw[arrow] (GGC.south) -| ($(POSH.north)+(0,3mm)$) -- (POSH.north);

% --- Tier 3: bulk theorems ---
\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 with SIP carrier --------
\node[gdos] (GDOS) at (0, -136mm) {\textbf{GDOS} \;\; (Geometric Dynamics of Open Systems on $\partial\Mbulk$) \;\; carrier \textbf{SIP} $= (\mathcal{M}, \text{ports}, L, \text{generator}, \text{Obs}/\text{Kill})$};

\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 5: GRIP engine box (centre) with Gi-4 / Qi-4 flanks ---
\node[griphead] (GRIPhead) at (0, -156mm) {\textbf{GRIP} \quad gauge-runtime engine \quad $\DGI{} \to (\GFF{}) \to \GSSB{} \to \GGA{}$};

\node[gripchain] (DGI) at (-46mm, -170mm) {\textbf{DGI}};
\node[gripchain] (GFFx) at (-15mm, -170mm) {\textbf{(GFF)}};
\node[gripchain] (GSSB) at (15mm, -170mm) {\textbf{GSSB}};
\node[gripchain] (GGAx) at (46mm, -170mm) {\textbf{GGA}};
\draw[arrow] (DGI.east) -- (GFFx.west);
\draw[arrow] (GFFx.east) -- (GSSB.west);
\draw[arrow] (GSSB.east) -- (GGAx.west);

\node[gripregime] (RGin) at (-30mm, -182mm) {\textbf{GRIP-In} \;: within-layer Lyapunov};
\node[gripregime] (RGout) at (30mm, -182mm) {\textbf{GRIP-Out} \;: between-layer shock};

\draw[arrow] (GDOS.south) -- (GRIPhead.north);

% --- Pillar flanks ---
\node[pillsideg] (GiBox) at (-95mm, -194mm) {};
\node[pillhead] at (-95mm, -198mm) {\GiFour{} pillars\\ (geometric flank)};
\node[pillline] at (-122mm, -207mm) {\textbf{TGG}, \textbf{CCI}, \textbf{GFF},};
\node[pillline] at (-122mm, -213mm) {\textbf{GRIP} = engine box};
\node[pillline] at (-122mm, -220mm) {\textbf{GEP} (energetic),};
\node[pillline] at (-122mm, -226mm) {\textbf{CEP} (entropic)};
\node[pillline] at (-122mm, -232mm) {(\GFF{} = $F_{\text{GEP}} + F_{\text{CEP}}$)};

\node[pillsideq] (QiBox) at (95mm, -194mm) {};
\node[pillhead] at (95mm, -198mm) {\QiFour{} pillars\\ (admissibility filter)};
\node[pillline] at (62mm, -207mm) {\textbf{Qi-QUA}, \textbf{Qi-EP},};
\node[pillline] at (62mm, -213mm) {\textbf{Qi-CON}, \textbf{Qi-HAL}};
\node[pillline] at (62mm, -220mm) {\textbf{OH}$+$\textbf{IH}$=$\textbf{EIC}};
\node[pillline] at (62mm, -226mm) {\textbf{SeOS} carrier};
\node[pillline] at (62mm, -232mm) {(selects unique \textbf{GGA})};

\draw[arrow] (GRIPhead.west) -- ($(GiBox.north)+(0,1mm)$);
\draw[arrow] (GRIPhead.east) -- ($(QiBox.north)+(0,1mm)$);

% --- Tier 6: 17-layer cascade ribbon (cascade-flavour colours) ---
\node[font=\sffamily\bfseries\large] at (0, -250mm) {Eighteen-layer cascade of \textbf{GGAs}};
\node[cas, fill=blue!60!white, text=white] (L0) at (-96mm,-262mm) {1};
\node[cas, fill=blue!50!white, text=white] (L1) at (-84mm,-262mm) {2};
\node[cas, fill=blue!40!cyan!70, text=white] (L2) at (-72mm,-262mm) {3};
\node[cas, fill=cyan!75!white] (L3) at (-60mm,-262mm) {4};
\node[cas, fill=cyan!60!white] (L4) at (-48mm,-262mm) {5};
\node[cas, fill=cyan!40!green!70, text=white](L5) at (-36mm,-262mm) {6};
\node[cas, fill=green!50!white] (L6) at (-24mm,-262mm) {7};
\node[cas, fill=green!60!white] (L7) at (-12mm,-262mm) {8};
\node[cas, fill=green!70!white] (L8) at (0mm,-262mm) {9};
\node[cas, fill=green!80!yellow!60] (L9) at (12mm,-262mm) {10};
\node[cas, fill=yellow!70!orange] (L10) at (24mm,-262mm) {11};
\node[cas, fill=orange!50!white] (L11) at (36mm,-262mm) {12};
\node[cas, fill=orange!65!white] (L12) at (48mm,-262mm) {13};
\node[cas, fill=red!35!white] (LCNS) at (60mm,-262mm) {14};
\node[cas, fill=orange!80!red!60, text=white] (L13) at (72mm,-262mm) {15};
\node[cas, fill=red!50!white] (L14) at (84mm,-262mm) {16};
\node[cas, fill=red!70!white, text=white] (L15) at (96mm,-262mm) {17};
\node[cas, fill=red!85!black, text=white] (L16) at (108mm,-262mm) {18};

\node[font=\sffamily\scriptsize] at (-96mm,-272mm) {GUT};
\node[font=\sffamily\scriptsize] at (-84mm,-272mm) {EW};
\node[font=\sffamily\scriptsize] at (-72mm,-272mm) {Atoms};
\node[font=\sffamily\scriptsize] at (-60mm,-272mm) {Molec.};
\node[font=\sffamily\scriptsize] at (-48mm,-272mm) {Amino};
\node[font=\sffamily\scriptsize] at (-36mm,-272mm) {Polym.};
\node[font=\sffamily\scriptsize] at (-24mm,-272mm) {ATP};
\node[font=\sffamily\scriptsize] at (-12mm,-272mm) {RNA};
\node[font=\sffamily\scriptsize] at (0mm,-272mm) {DNA};
\node[font=\sffamily\scriptsize] at (12mm,-272mm) {LUCA};
\node[font=\sffamily\scriptsize] at (24mm,-272mm) {3-Dom.};
\node[font=\sffamily\scriptsize] at (36mm,-272mm) {Eukar.};
\node[font=\sffamily\scriptsize] at (48mm,-272mm) {Multi.};
\node[font=\sffamily\scriptsize] at (60mm,-272mm) {CNS};
\node[font=\sffamily\scriptsize] at (72mm,-272mm) {Ring};
\node[font=\sffamily\scriptsize] at (84mm,-272mm) {Map};
\node[font=\sffamily\scriptsize] at (96mm,-272mm) {Conscious};
\node[font=\sffamily\scriptsize] at (108mm,-272mm) {Articul.};

\draw[arrow] (GRIPhead.south) -| (L8.north);

% --- Tier 7: Q_B + Arisaka equation -----------------------
\node[qb] (QB) at (0, -284mm) {%
$\boldsymbol{\mathsf{Q}_{\mathsf{B}}}$ \; (Quantum-information Bit; the boundary information member of \GUfive{}, fixed via \OGNfive{}) \\[1.0mm]
\textit{Arisaka equation:} \;\; $\boldsymbol{J_E / \mathsf{Q}_{\mathsf{B}} \,=\, J_e \,=\, J_B \,=\, J_{Qi}}$ \quad (four-current lock that drives the cascade)
};

\draw[arrow] (L8.south) -- (QB.north);

\end{tikzpicture}
}
\caption[Hierarchy of program acronyms]{Hierarchy of program-defined acronyms, revised for \AEleven{}: \GRIP{} is promoted to the central engine box at Tier~5, with the \GGA{} chain $\DGI{} \to (\GFF{}) \to \GSSB{} \to \GGA{}$ shown explicitly and the two regimes \GRIPin{}/\GRIPout{} shown as parallel arrow modes; the eighteen-layer cascade of \GGA{}s appears as a horizontal ribbon at the bottom with cascade-flavor coloring (blue $\to$ cyan $\to$ green $\to$ orange $\to$ red).}
\label{fig:appA-abbrev-flow}
\end{figure}

\clearpage

This two-page dictionary is offered as a reader's legend: a single front-matter table mapping every program-defined acronym used in the body of \AEleven{} to a one-sentence operational definition. A reader who flips here first can decode any macro encountered in the body without flipping to Appendix~A (which gives the same content in the canonical Appendix-A format inherited from \AOne{} Part~VIII). The dictionary is grouped by structural role.

\paragraph{Framework $+$ constitutional charter.}
\GGTheory{} = Grand Geometric Theory (the twenty-two-paper program).
\GGConstitution{} = the constitutional charter (4 axioms + 7 postulates) of \GGTheory{}, supplied by \AOne{}.
\GG{} = the six-dimensional bulk geometry ($\mathcal{M}_4 \times I_r \times S^1_\theta$).\footnote{Throughout the cohort, the ``6D bulk'' is read on the field-theory side of the bulk--boundary correspondence: the fifth (scale) dimension is the renormalization-group axis $r=\ln(E_{\mathrm{GUT}}/\mu)$, not a Randall--Sundrum spatial dimension, and ``warped'' denotes RG running; the explicit renormalization-group construction is given in \citet{ArisakaA2GG6}.}
\GDOS{} = Geometric Dynamics of Open Systems, the boundary law forced by \OGGEP{}.
\BIsix{} = the six-dimensional Bianchi identity $dH_3 = X_4$ (Green--Schwarz-modified, fixing the unique anomaly polynomial).
\OGNfive{} = Occam-Gauge-Normalization five-integer ledger $(M,N,L;J,K) = (1,3,5;7,35)$, the framework's only numerical inputs.
\QGL{} = Quadruple Gauge Locking, the Arisaka equation $J_E/\QB = J_e = J_B = J_{Qi}$.
\GiFour{} = Geometric-Invariance four-pillar structure (\TGG{}, \CCI{}, \GFF{}, \GRIP{}), supplied by \ASix{}.
\QiFour{} = Quantum-Information four-pillar structure (\QiQUA{}, \QiEP{}, \QiCON{}, \QiHAL{}), supplied by \ASeven{}.

\paragraph{Four constitutional axioms.}
\LC{} (\AX{1}) = Locality and Causality.
\GGEP{} (\AX{2}) = Geometry-Gauge Equivalence Principle (gravity and gauge are aspects of one composite connection).
\OGGEP{} (\AX{3}) = Operational/Open GGEP (every theory must come with a bulk-to-boundary projection law).
\MSMF{} (\AX{4}) = Maximum Symmetry, Minimum Freedom (no hidden continuous knobs).
\OGN{} (\POS{6}~\OGN{}) = Occam Gauge Normalization (constitutional ratios are integer/rational).

\paragraph{\GiFour{} pillars.}
\TGG{} = Total Geometric--Gauge object (the composite connection on the bulk).
\CCI{} = Covariant Conservation Identity (the consequence of \BIsix{} for boundary currents).
\GFF{} = Geometric Free-Energy Functional, the Morse landscape on which the cascade descends ($F_{\GFF{}} = F_{\GEP{}} + F_{\CEP{}}$).
\GEP{} = Geometry-Gauge Energy Principle (the energetic sector of \GFF{}).
\CEP{} = Curvature-Entropy Principle (the entropic sector of \GFF{}).
\GRIP{} = Gauge-Runtime Invariance Pipeline, the universal cascade engine $\DGI{} \to (\GFF{}) \to \GSSB{} \to \GGA{}$.
\GRIPin{} = within-layer regime of \GRIP{} (Lyapunov descent on a fixed Morse landscape).
\GRIPout{} = between-layer regime of \GRIP{} (perturbation-driven landscape reorganization).
\DGI{} = Dynamical Gauge Invariance (the unbroken-symmetry parent context, top of saddle ladder).
\GSSB{} = Geometric Spontaneous Symmetry Breaking (chain of Bogomolnyi-saturating saddle events).
\GA{} = Geometric Attractor (any stable Morse minimum on the GFF landscape).
\GGA{} = Ground Geometric Attractor (the uniquely most stable global minimum, the cascade output).
RUT = Recursive-Uniqueness Theorem --- the central theorem (\textsf{T-Origin}): each cascade layer has a unique Ground Geometric Attractor, so the whole cascade is forced layer by layer.

\paragraph{\QiFour{} pillars.}
\QiQUA{} = Quantum information for \QGL{} (locked-current bookkeeping).
\QiEP{} = Quantum information for the Unified Extremal Principle (trajectory selection).
\QiCON{} = Quantum information for Conformal Invariance (scaling diagnostics).
\QiHAL{} = Quantum information for the Holographic Attractor Lattice (memory architecture).
\OH{} = Outer Hologram (boundary projection of bulk data; outward face of \EIC{}).
\IH{} = Inner Hologram (inner self-modeling layer; inward face of \EIC{}).
\EIC{} = Evolving Inner Cosmos (boundary completion of \QiFour{}, the conscious-episode attractor).
\QCI{} = cascade-depth completeness index (this paper), a count of how many storey-ports a substrate's \OH{}--\IH{} closure has completed. Distinct from \ASeven{}'s \QiFour{} Completeness Index, which is a weighted geometric mean of the four pillar scores on $[0,1]$ and appears here as $\QCI{}_{\rm adm}$. \emph{Tentative operational form:} the number of Qi-HAL storey-ports the substrate closes (sensorimotor, attention, working memory, allocentric map, \EIC{} closure), a storey count $S \in \{0,\dots,5\}$, reported on the comparative ladder $\QCI{} = 2S \in [0,10]$ (insect ring attractor $\QCI{}\sim 6$, rodent allocentric map $\sim 8$, human conscious episode $\sim 10$). The ladder is illustrative; the operational scoring of each port and its calibration against comparative-neuroscience data are developed in \AFourteen{}.

\paragraph{\QiHAL{} sub-architecture.}
\MePMoS{} = Memory-Prediction-Motion-Sensing (the cognitive substrate the cascade brings into being).
\NHT{} = Neural Hologram (the boundary encoding of the substrate's predictive state).
\HAL{} = Holographic-Attractor-Lattice (the read-out structure on the encoded hologram).
\BFFT{} = Binary / Fractal Fourier Transform (the basis change that operates on the holographic read-out).

\paragraph{\GDOS{} carrier and class.}
\SIP{} = System Instance Packet, the canonical five-field object boxed at \eqref{eq:sip-five-field} --- state manifold and metric, boundary ports, \GDOS{} generator and noise closure, \GFF{} landscape and locked current, observable/kill bundle --- on which \GRIP{} runs.
\SeOS{} = Selfish Open System, the class of finite, port-bounded, Lyapunov-descending open systems that includes every living cell.

\paragraph{Program-defined mathematical constants.}
$\QB$ = Quantum-information Bit, the bioenergetic Arisaka constant $\QB = (2/5)(E_{\rm Ry}/K) \approx 0.155$~eV per locked quantum; $N_A \QB \approx 15$~kJ/mol is one-half the ATP-hydrolysis free energy, so $2 N_A \QB \approx 30.0$~kJ/mol matches the measured $30.5 \pm 0.5$~kJ/mol at biological temperatures.
$\Mlock$ = non-spatial Kaluza--Klein lock-cluster scale (scale-running resonances of the RG-scale throat, \emph{not} a spatial extra dimension), $\Mlock \approx 3.85$~TeV.
$z_*$ = topological-lock corridor-switch redshift of the recombination era, $z_* = e^7 \approx 1096.6$ (the integer-ledger anchor; distinct from the operational Planck $z_{\rm dec} = 1089.92$, which \ATen{}'s own atomic chain reproduces at the $0.1\%$ level).

\paragraph{The eighteen-layer cascade.}
$\GGA{}_1$: Standard-Model gauge group (6D bulk closure).
$\GGA{}_2$: electroweak vacuum, charged-lepton mass ladder, Higgs vev $v \approx 246$~GeV.
$\GGA{}_3$: periodic table at $E_{\rm Ry} = 13.6$~eV.
$\GGA{}_4$: small molecules (H$_2$O, CO$_2$, NH$_3$, CH$_4$).
$\GGA{}_5$: 20 L-handed proteinogenic amino acids ($|\GGA{}_5| = K - NL = 35 - 15 = 20$).
$\GGA{}_6$: polymers (oligopeptides, oligonucleotides, oligosaccharides).
$\GGA{}_7$: ATP at $\Delta G_{\rm ATP} = 2 N_A \QB \approx 30.0$~kJ/mol (empirical $30.5 \pm 0.5$).
$\GGA{}_8$: RNA $\{A, U, G, C\}$ with Watson--Crick pairing.
$\GGA{}_9$: DNA $\{A, C, G, T\}$ with fidelity $\epsilon \approx 10^{-9}$/bp.
$\GGA{}_{10}$: LUCA (Last Universal Common Ancestor) at $\sim 3.8$~Ga.
$\GGA{}_{11}$: Three-Domain Split (Bacteria $\oplus$ Archaea).
$\GGA{}_{12}$: eukaryotic cell at $\sim 2$~Ga (Lane--Martin economics).
$\GGA{}_{13}$: multicellularity (M0--M7 ladder).
$\GGA{}_{14}$: integrated nerve-ring connectome (\emph{C.\,elegans}-class CNS).
$\GGA{}_{15}$: insect-heading ring attractor.
$\GGA{}_{16}$: rodent hippocampal allocentric map.
$\GGA{}_{17}$: conscious episode completing over two $\sim 200$~ms theta cycles ($\sim 400$~ms), the \EIC{} closure, \QCI{}~$\sim 10$.
$\GGA{}_{18}$: the meta-cognitive act of articulating the \GGConstitution{} (the present paper).

\paragraph{Companion paper macros (sans-serif, per L39).}
\AOne{} = Arisaka-GG-A1-Constitution; \ATwo{} = GG-A2-GG6; \AThree{} = GG-A3-BI6; \AFour{} = GG-A4-Euclid; \AFive{} = GG-A5-GDOS; \ASix{} = GG-A6-GRIP; \ASeven{} = GG-A7-Qi4; \AEight{} = GG-A8-QM; \ANine{} = GG-A9-SM; \ATen{} = GG-A10-Cosmos; \AEleven{} = GG-A11-Origin (the present paper). Predictions branch: \PaperPOne{} = GG-P1-KK (KK lock cluster); \PaperPTwo{} = GG-P2-N3 (three-leg derivation of $N = 3$ families); \PaperCOne{} = GG-C1-Kaxion (Kaxion dark matter). Capstone: \citet{ArisakaBookI} = the planned textbook volume \emph{ArisakaBookI}.

\medskip
For the full canonical Appendix-A-format glossary (with first-introduction primary-link column, standard physics abbreviations, and the L52-canonical formatting), see Appendix~A. This front-matter Program Dictionary is the operational legend; Appendix~A is the constitutional reference.




\section{Roadmap of the twenty-one Parts}
\label{sec:inheritance-and-roadmap-a11}
\label{sec:roadmap-a11}

\noindent
Before the cascade of the present paper is walked (Part~\ref{part:cascade} onward), let us lay out a
panorama of the twenty-one Parts (Figure~\ref{fig:fifteen-parts-flowchart}). A reader of
any later Part should be able to flip back to this Section and see where the present
argument stands in the twenty-one-Part arc. The companion panorama --- the twenty-two-paper
cohort in which this capstone stands, and which upstream companion carries which
imported result --- opens Part~\ref{part:inheritance} (\S\ref{sec:cohort-a11},
Figure~\ref{fig:cohort-map-a11}).


The paper proceeds in twenty-one Parts (every program-defined acronym used in the structural overview below is spelled out in the Program Dictionary at \S\ref{sec:program-dictionary}). Part~\ref{part:inheritance} catalogs the inheritance from \AOne{} through \ATen{} and the predictions branch.

Part~\ref{part:cascade} states the central dynamical mechanism of the framework --- the \textbf{Cascade of \GRIPout{} with \GGA{}} (\GGA{} = Ground Geometric Attractor, each layer's unique stable basin; \GRIPout{} = the out-of-thermal-equilibrium phase transition that lifts the cascade from one layer to the next) --- consisting of three named theorems (\textsf{T-GRIP-In-Lyapunov-Descent}, \textsf{T-GRIP-Out-Necessity}, \textsf{T-GGA-as-Ground-State}) and the \textsf{Cascade-Self-Sustenance} corollary --- the regime \emph{definitions} live at runtime level in the parent paper \ASix{} (def:GRIPin, def:GRIPout, def:GGA); the regime \emph{theorems} are proved here; this is the completing innovation that lets the cascade ascend from layer to layer while terminating uniquely at each layer's ground state.

Part~\ref{part:results} is the formal \emph{Statement of Results}: eighteen named theorems --- one for each of the seventeen cascade layers plus the meta-theorem of articulation (T-GUT-Closure, T-Electroweak-Vacuum, T-Atomic-Ground-State, T-Molecular-Closure, T-Amino-Acid-Selection, T-Polymer-Closure, T-Bioenergetic-Closure, T-RNA-World-Closure, T-Information-Closure, T-Cell-Closure, T-Primary-Domain-Split, T-Eukaryogenesis, T-Multicellularity, T-Neural-Ring-Attractor, T-Allocentric-Map, T-Conscious-Episode, T-Articulation) --- so that any single layer can be cited individually without importing the entire cohort apparatus.

Parts~V--VIII develop the geometric cascade in pedagogical order: the recursive uniqueness theorem with the five-step proof of inevitability (V), the eighteen-layer \GSSB{} cascade walked layer by layer (VI), the dynamics block of $\QB$ (the Quantum-information Bit, the bioenergetic Arisaka constant $\QB \approx 0.155$~eV per locked quantum, two of which (one ATP) match the ATP-hydrolysis free energy, $2 N_A \QB \approx 30$~kJ/mol, at biological temperatures), \SeOS{} (the Selfish Open System --- the class of finite, port-bounded, Lyapunov-descending open systems that includes every living cell), and the Geometric Free-Energy Functional (\GFF{}) with the Three-Gauge Theorem (VII), and the cognitive cascade with the Five-Story Pagoda, the holographic triad, and the \QCI{} (cascade-depth completeness index) ladder (VIII).


Part~\ref{part:closure} proves the central new theorem of \AEleven{} --- the self-referential closure of the cascade --- and engages thirteen philosophical predecessors. Part~\ref{part:discussions} returns to the framework's self-critical posture: significance, comparison with alternatives, predictions, open questions.

Part~\ref{part:conclusions} delivers the Conclusions: the summit view of the geometric inevitability of Cosmos, Life, and Mind, with the recursive uniqueness theorem read across the entire cascade --- including the cascade's reach beyond physics, floor by floor, into the forthcoming consequence papers \ATwelve{}--\AFourteen{}. Part~\ref{part:coda} is the Coda: the closing of the self-referential loop, the renunciations the framework required, the personal arc, the empirical horizon read at philosophical register, and the framework's final word to the reader. Part~\ref{part:appendices} collects the appendices.

Figure~\ref{fig:fifteen-parts-flowchart} renders this Part-by-Part development as a single flowchart, so the big picture is visible at a glance.

\clearpage
\vspace*{\fill}
\noindent\begin{minipage}{\textwidth}
\centering
\resizebox{0.96426\textwidth}{!}{%
\begin{tikzpicture}[
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% --- Part I ---
\node[intro] (I) at (0, 0) {\textbf{Part~I}\quad\textbf{Introduction} \\[-1pt]
 \footnotesize Why are we here? \,+\, reader's guide for four audiences};

% --- Part II ---
\node[inherit] (II) at (0, -16mm) {\textbf{Part~II}\quad\textbf{The cohort inheritance} \\[-1pt]
 \footnotesize Imports from \AOne{}--\ATen{} \,+\, the phenomenology branches (\PaperPOne{}--\PaperPThree{}, \PaperCOne{}--\PaperCThree{})};

% --- Part III: highlighted central innovation ---
\node[innov] (III) at (0, -38mm) {\textbf{Part~III}\quad\textbf{The Cascade of \GRIPout{} with \GGA{}} \\[1pt]
 \footnotesize T-GRIP-In-Lyapunov-Descent \;\;$\bullet$\;\; T-GRIP-Out-Necessity \;\;$\bullet$\;\; T-GGA-as-Ground-State \\[-1pt]
 \footnotesize plus the \textsf{Cascade-Self-Sustenance} corollary \\[-1pt]
 \footnotesize \emph{\textbf{The completing innovation that finally explains why we are here}}};

% --- Parts IV-IX: the six domain Parts, two rows of three, in cascade order ---
\node[ggdom] (IV) at (-54mm, -62mm) {\textbf{Part~IV}\quad Cosmos \\[-1pt]
 \footnotesize layers 1--3 \\[-1pt] \footnotesize gauge group, vacuum, atoms};
\node[ggdom] (V) at (0mm, -62mm) {\textbf{Part~V}\quad Chemistry \\[-1pt]
 \footnotesize layers 4--6 \\[-1pt] \footnotesize molecules, amino acids, polymers};
\node[ggdom] (VI) at (54mm, -62mm) {\textbf{Part~VI}\quad Life \\[-1pt]
 \footnotesize layers 7--10 \\[-1pt] \footnotesize ATP, RNA, the first cell};
\node[ggdom] (VII) at (-54mm, -84mm) {\textbf{Part~VII}\quad Multicellularity \\[-1pt]
 \footnotesize layers 11--13 \\[-1pt] \footnotesize eukaryogenesis, the Cambrian};
\node[ggdom] (VIII) at (0mm, -84mm) {\textbf{Part~VIII}\quad Mind \\[-1pt]
 \footnotesize layers 14--17 \\[-1pt] \footnotesize connectome to conscious episode};
\node[ggdom] (IX) at (54mm, -84mm) {\textbf{Part~IX}\quad Articulation \\[-1pt]
 \footnotesize layer 18 \\[-1pt] \footnotesize the theory that names its own origin};

% --- Part X ---
\node[results] (X) at (0, -104mm) {\textbf{Part~X}\quad\textbf{The cascade as one ledger} \\[-1pt]
 \footnotesize Eighteen-row codebook \,\textbar\, Top 10 Falsifiers \,\textbar\, reader's guide \,\textbar\, counterfactual};

% --- Part XI (central theorem) ---
\node[innov] (XI) at (0, -124mm) {\textbf{Part~XI}\quad\textbf{Recursive Uniqueness Theorem (\textsf{T-Origin})} \\[-1pt]
 \footnotesize \GGConstitution{} $\to$ articulation; 5-step proof of inevitability across all eighteen layers};

% --- Parts XII, XIII, XIV: cascade development (three parallel) ---
\node[cascade] (XII) at (-54mm, -148mm) {\textbf{Part~XII} \\[-1pt]
 \footnotesize \GRIP{} cascade \\[-1pt]
 \footnotesize 16-step \GSSB{} \\[-1pt] \footnotesize walked layer by layer};
\node[cascade] (XIII) at (0mm, -148mm) {\textbf{Part~XIII} \\[-1pt]
 \footnotesize Dynamics block \\[-1pt]
 \footnotesize $\QB$, \SeOS{}, \GFF{} \\[-1pt] \footnotesize +\;Three-Gauge};
\node[cascade] (XIV) at (54mm, -148mm) {\textbf{Part~XIV} \\[-1pt]
 \footnotesize Cognitive cascade \\[-1pt]
 \footnotesize Pagoda, \OH{}/\IH{}/\EIC{} \\[-1pt] \footnotesize +\;\QCI{} ladder};

% --- Part XV ---
\node[closure] (XV) at (0, -170mm) {\textbf{Part~XV}\quad\textbf{Self-referential closure} \\[-1pt]
 \footnotesize Central new theorem; engagement with thirteen philosophical predecessors};

% --- Parts XVI, XVII, XVIII: discussion, summit, closing ---
\node[disc] (XVI) at (-54mm, -190mm) {\textbf{Part~XVI} \\[-1pt]
 \footnotesize Discussions \\[-1pt] \footnotesize significance, comparison, open questions};
\node[summit] (XVII) at (0mm, -190mm) {\textbf{Part~XVII} \\[-1pt]
 \footnotesize Conclusions \\[-1pt] \footnotesize the summit view};
\node[coda] (XVIII) at (54mm, -190mm) {\textbf{Part~XVIII} \\[-1pt]
 \footnotesize Coda \\[-1pt] \footnotesize closing the self-referential loop};

% --- Parts XIX, XX, XXI: back matter ---
\node[app] (XIX) at (-54mm, -212mm) {\textbf{Part~XIX} \\[-1pt]
 \footnotesize Technical Appendices};
\node[app] (XX) at (0mm, -212mm) {\textbf{Part~XX} \\[-1pt]
 \footnotesize Common Objections};
\node[app] (XXI) at (54mm, -212mm) {\textbf{Part~XXI} \\[-1pt]
 \footnotesize References};

% =====  ARROWS  =====
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% =====  PHASE LABELS on the right ======
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\node[phaselabel, anchor=west] at (100mm, -16mm) {\emph{Inputs}};
\node[phaselabel, anchor=west] at (100mm, -38mm) {\emph{\textbf{Mechanism}}};
\node[phaselabel, anchor=west] at (100mm, -73mm) {\emph{The six domains}};
\node[phaselabel, anchor=west] at (100mm, -104mm) {\emph{Results}};
\node[phaselabel, anchor=west, text=orange!70!black] at (100mm, -124mm) {\emph{\textbf{Central theorem}}};
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\node[phaselabel, anchor=west] at (100mm, -170mm) {\emph{Closure}};
\node[phaselabel, anchor=west] at (100mm, -190mm) {\emph{Discussion and summit}};
\node[phaselabel, anchor=west] at (100mm, -212mm) {\emph{Back matter}};

\end{tikzpicture}%
}
\par\bigskip
\captionof{figure}[Roadmap of the twenty-one Parts of GG-A11-Origin.]{Roadmap of the
twenty-one Parts of \textsf{GG-A11-Origin}, grouped into eight phases --- \emph{Setup}
(Parts~\ref{part:intro}--\ref{part:inheritance}),
\emph{Mechanism} (Part~\ref{part:cascade}), \emph{the six domains}
(Parts~\ref{part:results}--\ref{part:articulation}), \emph{Results}
(Part~\ref{part:ledger}), \emph{Central theorem} (Part~\ref{part:recursive}),
\emph{Cascade development} (Parts~\ref{part:runs}--\ref{part:cognitive}),
\emph{Closure} (Part~\ref{part:closure}),
\emph{Discussions and summit} (Parts~\ref{part:discussions}--\ref{part:coda}), and the
\emph{back matter} (Parts~\ref{part:appendices}--\ref{part:references}).
\textbf{Parts~\ref{part:cascade} and~\ref{part:recursive}} are highlighted because together they carry
the paper's central claim: Part~\ref{part:cascade} installs the completing mechanism --- the Cascade of
\GRIPout{} with \GGA{} --- and Part~\ref{part:recursive} states and proves it as one recursive uniqueness
theorem, \textsf{T-Origin}, the universal identity that runs from the grand-unified vacuum
to the act of articulation. The six domain Parts are drawn as a serpentine rather than a fan
because the cascade runs through them in order: cosmos before chemistry, chemistry before life.
Every later Part exercises or reflects on that result.}
\label{fig:fifteen-parts-flowchart}
\end{minipage}
\vspace*{\fill}
\clearpage


Part~\ref{part:intro} has covered a great deal of ground, and it is worth gathering it in one breath
before the real work begins. We opened with the two oldest questions anyone can ask ---
why is there a universe at all, and why is there someone inside it able to wonder about
that --- and made an unusual promise: that both can be approached with the same geometry
that governs particle physics, and that neither needs a new law added by hand. From
there we stated the central claim of the paper --- that reality is a single cascade of
open systems, each layer selected and stabilized by the same rule, running from the
grand-unified vacuum up through atoms, molecules, cells, and minds, to the strange
moment when a mind inside the universe reconstructs the very constitution that produced
it.

The rest of Part~\ref{part:intro} was the honest housekeeping such a claim deserves. We offered a guide
for four different kinds of reader, drew a candid line around what this paper does and
does not settle, and set down a two-page dictionary so that no symbol arrives
unexplained. We then laid the twenty-one Parts out as a single flowchart, and finally
placed the paper where it belongs --- as the capstone of a twenty-two-paper program
whose whole architecture you can now read off one figure. The point of all that
scaffolding is simple, and we hope reassuring: almost nothing in what follows is
invented on the spot. \AEleven{} inherits its constitution, its geometry, its dynamics,
and its four-dimensional physics from work that is already done and already checked.

That inheritance is where Part~\ref{part:inheritance} begins. Before this paper proves anything new, it lays
its debts on the table --- companion by companion, theorem by theorem --- naming exactly
what \AEleven{} borrows from each of the eighteen other papers and where in the cascade it
puts each borrowed result to work. A reader who has followed the earlier papers can pass
through Part~\ref{part:inheritance} quickly; a reader who has not can treat it as the working manual for
everything that comes after.


\GGPart[The cohort inheritance]{The cohort inheritance}{What \AEleven{} receives from \AOne{} through \ATen{}}
\label{part:inheritance}

\section{What \AEleven{} imports, paper by paper}
\label{sec:inheritance}

\paragraph{The load-bearing claim of \AEleven{}.}
\AEleven{} by no means stands alone: it is the closing capstone of a twenty-two-paper program, and a reader who meets \GGTheory{} here can take in its whole architecture at a glance before following the cascade. It is not a stand-alone proposal; it is a downstream-deployment paper that imports the constitutional, geometric, dynamical, and four-dimensional content of the framework from the ten upstream companion papers and the predictions branch. Nothing in the body of \AEleven{} is re-derived from scratch. What \AEleven{} contributes that none of the prior papers could contribute alone is the synthesis: the demonstration that the same recursive uniqueness theorem that \ANine{} proved at the particle-physics layer and \ATen{} proved at the cosmological layer extends, with the same constitutional warrant, to every higher stratum of the geometric cascade --- chemistry, life, mind, and the self-referential closure in which observers within the universe articulate the constitution that produced them. The present section makes the inheritance explicit, paper by paper, theorem by theorem.

The reader who has read \AOne{} through \ATen{} can skim this section. The reader who has not should treat it as the operational summary of the framework against which the rest of \AEleven{} is to be read. Each subsection below names the upstream paper, the key theorems imported, and the downstream Part(s) of \AEleven{} that exercise the import.



\subsection{The cohort map and the cross-paper coherence table}
\label{subsec:coherence-map}
\label{sec:cohort-a11}

Figure~\ref{fig:cohort-map-a11} draws all twenty-two papers in one frame, with
\AEleven{} highlighted in the capstone position.

The cohort falls into five strata, read from the top down, together with two
phenomenology branches and one foundations column. At the head sits the constitutional
root \AOne{}, whose four axioms and seven postulates fix what may count as an admissible
claim. The root compiles into the \emph{GG6 trio}: \ATwo{} derives the six-dimensional
bulk, \AThree{} proves its \BIsix{} anomaly freedom, and the historical companion
\AFour{} supplies the integer ledger and lineage of the constants. From the trio descends
the open-system boundary law \AFive{}, organized by its two pillar papers --- \ASix{},
the geometric \GiFour{} pillars, and \ASeven{}, the quantum-information \QiFour{} pillars.
The fourth stratum is the \emph{consequence trio} that recovers textbook physics from this
single source: quantum mechanics in \AEight{}, the Standard Model in \ANine{}, and
$\Lambda$CDM cosmology in \ATen{}. The present paper, \AEleven{}, is the capstone that
closes the A-series and reads the whole cascade as one recursive uniqueness theorem.
Hanging off this spine are the two phenomenology branches --- the Predictions Branch
(\PaperPOne{}, \PaperPTwo{}, \PaperPThree{}, \PaperPSix{} \citep{ArisakaP6Proton})
and the Cosmology \& Astrophysics Branch
(\PaperCOne{}, \PaperCTwo{}, \PaperCThree{}) --- and, on the spine's left flank, the
Foundations Branch: \PaperPFour{} \citep{ArisakaP4Real} and \PaperPFive{}
\citep{ArisakaP5CPT}, read below. Beside the spine rather than on it stand the
two white papers, which synthesize the cohort rather than extend it:
\PaperWOne{} \citep{ArisakaW1Symmetry} reads its symmetry structure historically,
as a ladder of seven rungs, and \PaperWTwo{} \citep{ArisakaW2Holonomy} reads it
structurally, as one two-fiber anatomy applied to sixteen standing problems.

%% ===================================================================
%%  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: identical in all nineteen papers; only the
%%  \label differs (fig:inheritance-map / fig:cohort-map-a11 kept for A11
%%  cross-references).  MP4/MP5 internal node names per the master.
%% ===================================================================
%%  v13.23: the two \clearpage that used to pin this figure are
%%  dropped.  [p] already restricts it to a float page; the
%%  \clearpage only forced a nine-tenths-empty page ahead of it
%%  once the figure moved into Part II.
\begin{figure}[p]
\centering
\resizebox{\textwidth}{!}{%
%% v22 type scaling, local to this group: the picture sets node text
%% in \footnotesize (the root in \small) and its edge and branch labels
%% in \scriptsize, so redefining the three commands here enlarges box
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%% --- Row 0: axiomatic root ---
\node[root] (A1) at (0, 0)
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%% --- Row 1: GG6 Trio ---
\node[bulk] (A2) at (-50mm, -25mm)
 {\textbf{\ATwo{}} \\[-1pt] \footnotesize 6D bulk};
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\node[hist] (A4) at (50mm, -25mm)
 {\textbf{\AFour{}} \\[-1pt] \footnotesize Historical lineage};

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

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

%% --- Row 4: 4D-Physics Trio ---
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 {\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.
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 {\textbf{\PaperPOne{}} \\[-1pt] \footnotesize $\Mlock \approx 3.85$~\TeV{}};
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 {\textbf{\PaperPTwo{}} \\[-1pt] \footnotesize Three generations};
\node[pred] (P3) at (25mm, -138mm)
 {\textbf{\PaperPThree{}} \\[-1pt] \footnotesize Electron as ground state};
\node[pred] (P6) at (75mm, -138mm)
 {\textbf{\PaperPSix{}} \\[-1pt] \footnotesize The guarded proton};

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

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

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

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

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

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

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

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

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

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

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

%% Centring pad, retired (v22): it cancelled a Bezier phantom that the
%% polyline above no longer creates; the bbox is now the ink.
\end{tikzpicture}}
\caption[The twenty-two-paper cohort map.]{The
twenty-two-paper cohort map of \GGTheory{}. The constitutional root
\AOne{} (violet, top) supplies the four axioms and seven postulates
from which the rest of the cohort descends: the GG6 Trio (\ATwo{},
\AThree{}, \AFour{}, red) realizes the bulk geometry; \AFive{}
(yellow) the boundary law; the \GDOS{} Trio pillars (\ASix{},
\ASeven{}, green) the geometric and quantum-information runtime; the
4D-Physics Trio (\AEight{}, \ANine{}, \ATen{}, blue) the textbook
readouts; and the closing capstone \AEleven{} (gold) the
quarks-to-consciousness identity cascade. Two framed branches carry
the phenomenology --- the \textbf{Predictions Branch} (cyan), whose
four papers hand their results along the row by the internal arrows
(\PaperPTwo{}'s $N{=}3$ into \PaperPThree{}, and that paper's
ground-attractor method, \textsf{GGA}, on into \PaperPSix{}), and the
\textbf{Cosmology \& Astrophysics Branch} (teal). The spine is
flanked by two columns of two, which do opposite jobs. On the left the
\textbf{Foundations Branch} (violet) feeds it: \PaperPFour{}, fed by
\ATwo{}'s Theorem~0 arena, manufactures the $i$ that \AEight{}
consumes, and \PaperPFive{} is the \ANine{}--\ATen{} interface, fed by
the ledger phases and feeding the $\eta_B$ chain across the width of
the map, with the shared-$i$ edge descending solid between them. On
the right the \textbf{White-Paper Branch} (brown) reads it:
\PaperWOne{} walks the century from Noether to Hopkins--Singer, and
\PaperWTwo{} dissects the structure that arc arrives at ---
protection by topology rather than by symmetry --- across sixteen open
problems. The white papers derive nothing new, carrying every result
at its parent's declared grade; the single dashed feed drawn into each
names its heaviest source, not its only one. \emph{Reading
conventions:} solid arrows mark axiom-forced descent and carry an edge
label naming the driving axiom or postulate where most illustrative;
dashed arrows mark the deployment of derived results, or historical
links inherited without being constitutionally forced.}
\label{fig:inheritance-map}
\label{fig:cohort-map-a11}
\end{figure}

\paragraph{The Foundations Branch.} The map's left flank carries the cohort's third
branch --- two member papers, a single column beside the spine, whose subject is the
boundary's complex structure and its mirrors. \PaperPFour{} \citep{ArisakaP4Real}
supplies the origin of the complex structure on which the cascade's quantum floors run:
it derives the boundary's $i$ from two real structures of the \ATwo{} arena --- the
compact phase circle and the causal clock --- while \AEleven{} itself defers the question
of \emph{why} the boundary is complex to the state-space derivation of \AEight{}
\citep{ArisakaA8QM}, whose real-bulk account \PaperPFour{} develops at full length.
\PaperPFive{} \citep{ArisakaP5CPT} defines time by its four endowments and fixes the
law-level arrow as orientation reversal of the three real bulk factors --- kin to the
cascade's own one-way descent through its layers, each \GRIPout{} shock stepping the
geometry down one floor and never back up. Both carry P-series designators; their place at
the spine's flank, not the phenomenology rows, marks their foundational role.




Where the map above draws the cohort's shape, the table below records what the present paper actually takes from each member and where it is put to work. Let us look carefully at Table~\ref{tab:coherence-map}, the largest cross-paper coherence map of the series, spanning the twenty-one companion papers of the A-series spine and the four branches across the twenty-one Parts of \AEleven{}. Beyond the eleven-paper A-series spine, the phenomenology is carried by the Predictions Branch --- \PaperPOne{} \citep{ArisakaP1KK}, \PaperPTwo{} \citep{ArisakaP2N3}, \PaperPThree{} \citep{ArisakaP3Electron}, \PaperPSix{} \citep{ArisakaP6Proton} --- the Cosmology \& Astrophysics Branch --- \PaperCOne{} \citep{ArisakaC1Kaxion}, \PaperCTwo{} \citep{ArisakaC2Hubble}, \PaperCThree{} \citep{ArisakaC3BH} --- the Foundations Branch --- \PaperPFour{} \citep{ArisakaP4Real}, \PaperPFive{} \citep{ArisakaP5CPT} --- and the White-Paper Branch --- \PaperWOne{} \citep{ArisakaW1Symmetry}, \PaperWTwo{} \citep{ArisakaW2Holonomy}.

%% v13.23: flush the pending cohort-map float here.  It is a [p]
%% float and cannot be set inside or across the longtable below,
%% so without this it drifts past the table -- the manifest ahead
%% of the map.  Deferring the float and flushing it here keeps the
%% prose unbroken and strands nothing.
\clearpage
{\small
\begin{longtable}{>{\raggedright\arraybackslash}p{0.14\textwidth} >{\raggedright\arraybackslash}p{0.39\textwidth} >{\raggedright\arraybackslash}p{0.39\textwidth}}
\caption[Cross-paper coherence map: the twenty-one companion papers.]{Cross-paper coherence map: the twenty-one companion papers --- the A-series spine and the Predictions, Cosmology \& Astrophysics, Foundations, and White-Paper branches --- the theorems and structures they supply, and where they are exercised in \AEleven{}. Twenty-one rows, one per companion, each giving what that paper supplies and where the present paper spends it --- the constitution's axioms and postulates from \AOne{}, the \BIsix{} coefficient vector and the \OGNfive{} ledger $(1,3,5;7,35)$ from \ATwo{}, anomaly freedom and the trace-ratio lock from \AThree{}, the twenty-eight-to-zero compression and the five geometric constants from \AFour{}, and the \GDOS{} apparatus from \AFive{}. The table is the paper's audit of its own imports: every result used here is entered against the paper that proves it.}
\label{tab:coherence-map}\\
\toprule
\textbf{Upstream} & \textbf{Imported theorems / structures} & \textbf{Exercised in \AEleven{}} \\
\midrule
\endfirsthead

\multicolumn{3}{c}{\itshape Table~\ref{tab:coherence-map} continued from previous page}\\
\toprule
\textbf{Upstream} & \textbf{Imported theorems / structures} & \textbf{Exercised in \AEleven{}} \\
\midrule
\endhead

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

\bottomrule
\endlastfoot
\href{https://preprints.arisaka-gg.org/a1-constitution/}{\AOne{}} & \GGConstitution{}: \AX{1} (\LC{}), \AX{2} (\GGEP{}), \AX{3} (\OGGEP{}), \AX{4} (\MSMF{}) and seven postulates \POS{1} through \POS{7} & Part~\ref{part:inheritance} (foundation), Part~\ref{part:recursive} (five-step proof Step~1), Part~\ref{part:closure} (self-referential closure) \\
\href{https://preprints.arisaka-gg.org/a2-gg6/}{\ATwo{}} & \T{4} (\BIsix{} coefficient vector $(1,-1,-2/5,-2/5)$); \T{6} (\OGNfive{} Diophantine ledger $(M,N,L;J,K) = (1,3,5;7,35)$) & Part~\ref{part:inheritance} (foundation), Part~\ref{part:recursive} (Step~1), Part~\ref{part:dynamics} ($\QB$ derivation) \\
\href{https://preprints.arisaka-gg.org/a3-bi6/}{\AThree{}} & T-BI6-7.2 (\BIsix{} anomaly freedom); T-BI6-9.1 (the trace-ratio lock $|a/c| = |a/d| = 5/2$) & Part~\ref{part:inheritance} (foundation), Part~\ref{part:results} (Step~1), Part~\ref{part:recursive} (\GSSB{} cascade ground) \\
\href{https://preprints.arisaka-gg.org/a4-euclid/}{\AFour{}} & No free continuous parameter in the input ledger, scoped as \AFour{} scopes it; the Geometric Universal Five Constants \GUfive{} $=\GUfiveSet$ (3 bulk + 2 boundary) & Part~\ref{part:inheritance} (zero-free-parameters claim), Part~\ref{part:dynamics} ($\QB$ anchor) \\
\href{https://preprints.arisaka-gg.org/a5-gdos/}{\AFive{}} & \T{1}--\T{11} (the \GDOS{} boundary law); the Lyapunov lemma under \T{5}; \T{7.1}--\T{7.7}. The \GFF{} is \ASix{}'s and the \SeOS{} class is \ASeven{}'s; the Three-Gauge reading is this paper's own & Part~\ref{part:recursive} (Step~2), Part~\ref{part:dynamics} (\GFF{} descent), Part~\ref{part:cognitive} (consciousness) \\
\href{https://preprints.arisaka-gg.org/a6-grip/}{\ASix{}} & \T{1}--\T{5} (\GiFour{} pillars: \DGI{} $\to$ \GSSB{} $\to$ \GA{}); the \GRIP{} cascade theorem & Part~\ref{part:results} (Steps~3--5), Part~\ref{part:runs} (cascade), Part~\ref{part:cognitive} (holographic triad), Part~\ref{part:cognitive} (\QCI{}) \\
\href{https://preprints.arisaka-gg.org/a7-qi4/}{\ASeven{}} & \T{-\SIP{}}, \T{-QUA}, \T{-EP}, \T{-CON}, \T{-\HAL{}}, \T{-\EIC{}} (the six named theorems of the \QiFour{} pillars); the Arisaka equation $J_E/\QB = J_e = J_B = J_{Qi}$ & Part~\ref{part:inheritance} (\QGL{}), Part~\ref{part:recursive} (Step~3), Part~\ref{part:cognitive} (\EIC{}), Part~\ref{part:cognitive} (\QHAL{}-\QCI{}) \\
\href{https://preprints.arisaka-gg.org/a8-qm/}{\AEight{}} & The 4D operational reformulation of quantum mechanics; T7.1--T7.7 (QM postulates as theorems) & Part~\ref{part:cognitive} (consciousness as internal measurement), Part~\ref{part:cognitive} (\QCI{} ladder) \\
\href{https://preprints.arisaka-gg.org/a9-sm/}{\ANine{}} & 28-parameter codebook of the Standard Model derived to theorem level; Part~XI: the \emph{universal-identity insight} (seven-layer cascade) and the \emph{uniqueness of the canonical four-force architecture} (\textsf{T-SM-62.1}, eight constitutional lemmas), with the Q1+Q2 synthesis that closes the Part & Part~\ref{part:inheritance} (forces inheritance), Part~\ref{part:recursive} (Step~1), Part~\ref{part:runs} (cascade), Part~\ref{part:closure} (self-referential closure --- direct upstream of A11 Part~\ref{part:closure}) \\
\href{https://preprints.arisaka-gg.org/a10-cosmos/}{\ATen{}} & Cosmological codebook: $\Omega_m = 1/\pi$, $\Omega_\Lambda = 1 - 1/\pi$, $\Omega_b = 1/(2\pi^2)$; the universal-cosmic-identity insight at the cosmological stratum & Part~\ref{part:inheritance} (cosmic-inventory inheritance), Part~\ref{part:conclusions} (cosmological inevitability), Part~\ref{part:cognitive} (cosmological base of the recursive uniqueness theorem) \\
\href{https://preprints.arisaka-gg.org/p1-kk/}{\PaperPOne{}} & The Kaluza--Klein lock-cluster at $\Mlock \approx 3.85$~TeV (testable at HL-LHC) & Part~\ref{part:discussions} (Top 10 Falsifiers test 1) \\
\href{https://preprints.arisaka-gg.org/p2-n3/}{\PaperPTwo{}} & The conditional three-leg derivation of the family number $N=3$ & Part~\ref{part:inheritance} ($N=3$ inheritance), Part~\ref{part:recursive} (cascade) \\
\href{https://preprints.arisaka-gg.org/p3-electron/}{\PaperPThree{}} & The electron as the unique stable Geometric Attractor of the charged-lepton stratum (the six oldest electron questions, closed) & Part~\ref{part:closure} (universal-identity closure --- every electron the same geometric ground state) \\
\href{https://preprints.arisaka-gg.org/p4-real/}{\PaperPFour{}} & The two real sources of the boundary's complex structure: positive frequency as holomorphy, from the compact phase circle and the causal clock & Part~\ref{part:conclusions} (the number field itself selected, at candidate grade, so the cascade runs on real geometry at every floor) \\
\href{https://preprints.arisaka-gg.org/p5-cpt/}{\PaperPFive{}} & C, P, and T as orientation reversals of the three real bulk factors --- why every particle has a twin, and why the universe is made of matter & Part~\ref{part:conclusions} (the four-endowment definition of time and the law-level arrow, kin to the cascade's own directionality) \\
\href{https://preprints.arisaka-gg.org/c1-kaxion/}{\PaperCOne{}} & Kaxion dark matter at $\sim 8.1~\mu$eV (testable at ADMX / next-gen haloscopes) & Part~\ref{part:discussions} (Top 10 Falsifiers test 2) \\
\href{https://preprints.arisaka-gg.org/c2-hubble/}{\PaperCTwo{}} & The Hubble tension as two projections of one derived anchor --- the early/late two-stage corridor & Part~\ref{part:appendices} (Hubble corridor), Part~\ref{part:objections} (tension addressed, not assumed away) \\
\href{https://preprints.arisaka-gg.org/c3-bh/}{\PaperCThree{}} & Black-hole entropy $A/4$ recovered as a boundary theorem (horizon thermodynamics) & Part~\ref{part:conclusions} (the cascade's reach into horizon thermodynamics) \\
\href{https://preprints.arisaka-gg.org/p6-proton/}{\PaperPSix{}} & The proton as the Ground Geometric Attractor of matter; the rank-matching obstruction that makes baryon number a winding rather than a Lie-algebra charge, so that $\Delta B = \pm1$ minting is nonexistent and leptogenesis is forced rather than preferred & Part~\ref{part:results} (layer~2, the stability of matter), Part~\ref{part:discussions} (the origin-of-matter chain) \\
\href{https://preprints.arisaka-gg.org/w1-symmetry/}{\PaperWOne{}} & The conservation-law ladder: laws grow harder to break as they say less about motion, and the ledger's integers sit at the top of it & Part~\ref{part:inheritance} (why the integers cannot run), Part~\ref{part:conclusions} \\
\href{https://preprints.arisaka-gg.org/w2-holonomy/}{\PaperWTwo{}} & The holonomy sorting --- the circle remembers what the interval forgets --- which is why the compact angle carries the quantized ledger and the radial interval carries none & Part~\ref{part:inheritance} (the \OGNfive{} ledger's protection), Part~\ref{part:conclusions} \\
\end{longtable}
}

\subsection{Inheritance from \AOne{}: the \GGConstitution{}}
\label{subsec:inheritance-A1}

\AOne{} is the constitutional document: four axioms --- \AX{1}~\LC{}
(locality and causality, the locality half being the clause that excludes
fundamental extended objects), \AX{2}~\GGEP{} (gauge and geometry as one
composite connection), \AX{3}~\OGGEP{} (a bulk theory is not physically
defined unless it specifies a fixed operational dictionary from bulk
structure to registered boundary observables), \AX{4}~\MSMF{} (no continuous
parameter not forced by the other clauses, and, where discrete invariants
exist, the minimal non-degenerate values compatible with the constitution)
--- and seven postulates \POS{1}--\POS{7}: gauge invariance across scales,
the unified extremal principle, anomaly freedom, closure and positivity,
\RG{} covariance, \OGN{} topological integrality, and \MF{}. The \OGNfive{}
integer ledger is named in \AOne{} and derived downstream, in \ATwo{} and
\AThree{}. \AEleven{} inherits the entire ledger unchanged and adds no
constitutional clause; \AOne{} assigns it no Principal Theorem. \AOne{}
also states, in advance, the grade of the claim this paper exists to make:
that the same clauses reach through chemistry and biology to living matter
is ``the program's central wager, not a theorem of this paper, and is
marked as such wherever it appears.'' \AEleven{} carries that grade.

\subsection{Inheritance from \ATwo{}: \GG{} geometry and the \OGNfive{} ledger}
\label{subsec:inheritance-A2}

\ATwo{} fixes the canonical six-dimensional bulk geometry
$\mathcal{M}_6=\mathcal{M}_4\times I_r\times S^1_\theta$ --- forced, in its
Theorem~0, by \AX{1} and \AX{2} together, so that the extra dimensions are
theorems rather than assumptions justified after the fact --- and closes the
\BIsix{} identity $dH_3=X_4$ on the coefficient vector $(1,-1,-2/5,-2/5)$.
It states the \OGNfive{} Diophantine cascade $J=2N+1$, $K=LJ$,
$K=M^2+N^2+L^2$, whose solution is $(M,N,L;J,K)=(1,3,5;7,35)$. Two steps of
that cascade are not \ATwo{}'s and \ATwo{} says so: $L=5$ comes from
\AThree{}'s single-tensor Green--Schwarz consistency theorem, and $N=3$ from
\AThree{}'s dual-lock closure. The trace ratio $(L-N)/L=2/5$ is an
independent check on a ledger already determined, not a route to it. The
second root $(1,7,5;15,75)$ also satisfies $LJ=M^2+N^2+L^2$, and is excluded
by the block embedding and by nothing else, which is worth stating plainly.
\AEleven{} treats all of this as input and never re-derives it.

\subsection{Inheritance from \AThree{}: the \BIsix{} closure as the universal-anomaly fingerprint}
\label{subsec:inheritance-A3}

\AThree{} carries the \BIsix{} closure to full rigor. Locally, the
perfect-square factorization $I_8=(1/2)X_4^2$ is possible if and only if the
anomaly matrix has rank one and is positive semidefinite; the coefficient
\emph{values} $(1,-1,-2/5,-2/5)$ are then fixed by the \OGNfive{} trace
ratio, not by the factorization alone. Globally, \AThree{} proves anomaly
freedom for every large gauge transformation of arbitrary topological type,
on every closed spin seven-manifold, with no restriction on the topology of
the four-dimensional base --- a stronger statement than the class-restricted
one, and the strongest result the paper owns. The 1984 Green--Schwarz
factorization \citep{GreenSchwarz1984} is read constitutionally as the first
technical demonstration of \AX{2}. \AThree{} does \emph{not} claim a unique
solution for the boundary spectrum: it gives an existence construction and
records minimality as open. \AEleven{} inherits the rigor without repeating
it, and inherits the open item with it.

\paragraph{Why the \OGNfive{} ledger is forced, not fitted.}
\label{para:ogn5-not-fitted}
The five integers $(1,3,5;7,35)$ are not chosen ex post to fit the data.
\AThree{} fixes $M=1$ from the rank-one tensor lattice, $L=5$ from
single-tensor Green--Schwarz consistency, $N=3$ from the dual-lock closure
$N=L\pm\sqrt{L-M^2}$ together with the block embedding $N<L$, then
$J=2N+1$ and $K=LJ$. Two of those five steps are pinned by \AX{4}~\MSMF{}
rather than by anomaly cancellation, and \AThree{} grades the ledger
accordingly: proved, \emph{given} \AX{4}. The global anomaly freedom of the
Hopkins--Singer lift \citep{HopkinsSinger2005,MonnierMoore2019} is what makes
the bulk consistent; it fixes none of the integers, and this paper no longer
says that it does. The reason the integers cannot run is not anomaly
cancellation but topology: holonomy requires closed loops and an interval has
none, so the compact angle $S^1_\theta$ carries all of the bulk's holonomy,
quantized and protected, while the radial interval $I_r$ carries none. The
circle remembers, and the ledger is written on the circle. The closed forms
that follow --- $\alpha^{-1}(0)=2^J+N^2=137$, and $\Mlock \sim M_{\rm Pl}
e^{-K}$ --- are therefore algebraic identities of the geometry, fixed before
any empirical input was consulted. The closed form for the weak mixing angle,
$\sin^2\theta_W=L^2/(4N^3)=25/108$, belongs to \ANine{} and \AFour{} and not
to \AThree{}. The sharpest form of the integer-search objection, and the full
response, is taken up in the Appendix, \S\ref{subsec:obj-post-hoc-fitting}.

\subsection{Inheritance from \AFour{}: the Euclidean closure of the input ledger}
\label{subsec:inheritance-A4}

\AFour{} proves the input ledger carries no free continuous parameter. The
compression is not ``twenty-eight to zero''; \AFour{} retired that
formulation, and this paper retires it too. The Standard Model together with
general relativity carries \emph{thirty} free continuous constants ---
twenty-eight of the Standard Model, plus $G_N$ and $\Lambda$ --- and what
survives on the other side is not an empty set but a residue \AFour{} names:
discrete axiomatic choices, unit conventions, and one overall mass scale. The
forced inputs are the Geometric Universal Five Constants \GUfive{}
$=\GUfiveSet$, the topological constants $(\pi,e)$, and the \OGNfive{}
integers $(M,N,L;J,K) = (1,3,5;7,35)$. \AFour{} scopes its own closure
theorem twice, and \AEleven{} carries both scopes: zero free continuous
parameters \emph{in the gauge and cosmological sectors}, with the fermion
sector structurally framed and not yet completely closed; and one frozen
discrete completion class whose necessity is asserted rather than proved.
\AEleven{} accordingly never describes the framework as having ``few'' free
parameters --- and never states the count as zero without the scope.

\subsection{Inheritance from \AFive{}: \GDOS{} and the boundary law}
\label{subsec:inheritance-A5}

\AFive{} supplies the dynamical machinery: the eleven \GDOS{} theorems
\T{1}--\T{11}; the Lyapunov lemma under \T{5}, which \AFive{} names the
constitutional source of the second law; and \T{7.1}--\T{7.7}, the
quantum-mechanical postulates promoted to theorems on the \OGGEP{}
projection. Three attributions this paper previously made to \AFive{} belong
elsewhere, and are corrected here. The Geometric Free-Energy Functional
$F_{\GFF{}}=F_{\GEP{}}+F_{\CEP{}}$ is \ASix{}'s; what \AFive{} contributes is
the identification of its two blocks with \AFive{}'s own \T{9.2} and \T{5}.
The Selfish Open System (\SeOS{}) class is \ASeven{}'s. The Three-Gauge
reading --- Gibbs, Friston and \GG{} as three windows on one $F_{\GFF{}}$ ---
is this paper's own, and is graded as such where it is stated; \AFive{}'s
triad is the different one of least time, maximum entropy production and free
energy, which \AFive{} credits to \ASeven{}. \AFive{}'s Lyapunov lemma also
carries a scope limit that travels with it: when circulation is present the
functional is no longer strictly monotone, though the total entropy
production rate remains non-negative and the system can settle into a
non-equilibrium steady state. Every layer of this paper's cascade above the
atomic rungs is such a steady state, so the caveat is load-bearing here and
not decorative. The traffic runs both ways: \AFive{} consumes this paper's
regime theorems at its own projection-rate section and inside its \T{10}
bridge lemma.

\subsection{Inheritance from \ASix{}: the \GiFour{} pillars and the \GRIP{} cascade}
\label{subsec:inheritance-A6}

\ASix{} proves the \GiFour{} pillars and closes the \GRIP{} cascade
(\DGI{}$\to$\GSSB{}$\to$\GA{}) as the cascade dynamics on the stratified
boundary moduli. It is not the bulk-to-boundary projection --- that is
\OGGEP{} and \GDOS{}, which \ASix{} imports --- and this paper no longer says
it is. \GRIP{} is nonetheless the central operational engine of \AEleven{},
and two clauses of \ASix{} govern how far the engine may be carried. Both are
carried here.

First, a Geometric Attractor is a \emph{local} minimum, and a Morse landscape
generally admits many. The cascade terminates on the \emph{ground} attractor
--- the unique global minimum --- only on an \emph{admissible} stratum,
meaning one that is connected, compact, K\"ahler with the boundary gauge
connections flat along it, and on which $F_{\GFF{}}$ is the moment map of a
Hamiltonian torus action with isolated non-degenerate fixed points. \ASix{}
establishes admissibility for one stratum, the family projector
$\mathbb{CP}^{N-1}$, and says plainly that the Higgs and Wilson-line strata
are not established there, and that on a periodic compact-angle potential
over the discrete \OGNfive{} phases the uniqueness conclusion may genuinely
fail. Wherever this paper claims a unique ground state at a layer, it is
claiming admissibility at that layer, and the claim is no stronger than that
hypothesis.

Second, the cascade cannot climb from one layer to the next by gradient flow.
Strata are labeled by discrete projector-class data that no continuous curve
can change, so a gradient trajectory never leaves its stratum: every
layer-to-layer step requires a \GRIPout{} event, which \ASix{} deliberately
excludes from its septuple and whose theory it delegates to this paper. That
delegation is the reason \AEleven{} exists as a separate paper, and it is
also the reason \ASix{} names its own limit in its own words: the
identification of the chemical, biochemical, biological and cognitive layers
with the same four moves is ``the cohort's central wager rather than a
theorem of this paper.'' \AEleven{} makes that wager, and grades it as one.

The observed universal identity of objects at a layer is that attractor's
boundary readout. The mechanism is \ASix{}'s own --- a basin has no interior
degrees of freedom for two readouts of it to differ by --- and \ANine{}
Part~XI carries it at the particle layer.

\subsection{Inheritance from \ASeven{}: the \QiFour{} pillars and the Arisaka equation}
\label{subsec:inheritance-A7}

\ASeven{} proves the \QiFour{} pillars: T-\SIP{} (the System Instance
Packet), \T{-QUA} (the Arisaka equation, which is the informational content
of Quadruple Gauge Locking; the four-factor bulk gauge group itself is
\ATwo{}'s), \T{-EP} (the unified extremal principle), \T{-CON}
(conformality, whose \T{-CON.2} carries the two-knee spectral result),
T-\HAL{} (the Holographic Attractor Lattice), and T-\EIC{} (the Evolving
Inner Cosmos). Two grades travel with these and are carried here. T-\HAL{} is
a \emph{canonical normal-form} decomposition, not a uniqueness theorem ---
two of its five motifs are definitional and uniqueness is an open problem of
\ASeven{}. T-\EIC{} is deliberately graded rather than forcing: its inner-hologram and evolving-inner-cosmos
clauses are conditionals, and \ASeven{}
states outright that the theorem ``does not force an all-or-none divide.''
The pillars lock the four currents into the Arisaka \emph{equation}
$J_E/\QB = J_e = J_B = J_{Qi}$, which is not the Arisaka \emph{ratio}
$1:-1:-2/5:-2/5$ with the signs dropped but a distinct statement reached
after a sector-dependent unit conversion, and they fix $\QB$ as the universal
information-energy currency. A seventh result of \ASeven{} is load-bearing
for this paper and was not previously named: its \QiFour{} characterization
of the \GGA{} establishes only that the configuration \ASix{}'s descent halts
at carries all four certificates. The converse --- that no other Geometric
Attractor passes all four --- is not proved, and \ASeven{} declines to prove
it precisely because the earlier version rested it on this paper, which is a
circle. This paper's uniqueness therefore comes from \ASix{}'s per-stratum
ground-state corollary and its admissibility hypothesis, never from the four
pillars. Finally, \ASeven{}'s pagoda factorization proves an
\emph{arbitrary-depth} tower wherever its floor-creation conditions hold, and
records that the count of five floors is an empirical reading rather than a
theorem output; that is what licenses this paper's eighteen layers as a finer
admissible refinement rather than a rival count.

\subsection{Inheritance from \AEight{}: the 4D operational reformulation of quantum mechanics}
\label{subsec:inheritance-A8}

\AEight{} \citep{ArisakaA8QM} lifts \AFive{}'s \T{7.1}--\T{7.7} into a 4D
operational reformulation of quantum mechanics --- eleven principal
theorems, of which seven are that lift and four are new --- and is candid
that almost nothing in it is self-contained. \AEleven{} imports its
operational reading of measurement, in \AEight{}'s own words rather than in a
paraphrase this paper had drifted into: an observation is the \OGGEP{}
projection of a bulk quantity onto a boundary stratum, and registration is
the descent of the joint system-and-apparatus configuration into one
Geometric Attractor basin, irreversibly locked by the boundary free-energy
law, the outcome being the contextual joint holonomy of system and apparatus
read out by that projection --- \emph{with no collapse, no branching, and no
observer required}. The earlier phrasing here, ``stabilization of a selected
\IH{} branch,'' imported a branching \AEight{} explicitly denies, and is
withdrawn. The direction of the consciousness question also runs the other
way from what this paper previously stated: \AEight{} declines to make
consciousness a primitive of measurement and marks it out of scope, pointing
to \AEleven{} Part~\ref{part:cognitive}. This paper receives that deferral; it does not
import an answer. What \AEight{} does supply, and what belongs in any account
of this paper's foundations, is the status of the Born rule: closed at the
relaxation level, with the Constitutional Born Theorem conditional on four
named premises, and a single remaining analytic residue --- the field-grade
representation of the channel as a port stream.

\subsection{Inheritance from \ANine{}: the Standard-Model parameter audit and the Part-XI synthesis}
\label{subsec:inheritance-A9}

\ANine{} is the direct upstream foundation. Its body closes the 28-parameter
Standard-Model codebook ($\alpha^{-1}(M_Z)=2^J=128$,
$\sin^2\theta_W=25/108$, $\alpha_s(M_Z)=1/(\pi e)$, $K=35$, $N=3$,
$\delta_{\rm CKM}=13\pi/35$ at quantum-defect order, $N_A\QB\approx15$~kJ/mol)
--- and closure here means what \ANine{} says it means, which is an
ontological statement about what the parameters are rather than a claim that
all of them have been computed: six are derived, fourteen are derived
conditional on a named lemma in each case, and eight remain asserted with the
calculation that would close them named explicitly. \AEleven{} carries that
triage and not a bare ``no continuous freedom.''

\ANine{}'s Part~XI supplies two results, of which one is a theorem and one is
not. The \emph{universal-identity insight} --- the universe is not full of
arbitrary identical objects but of attractors, and objects are what the
attractors look like from the boundary --- is stated by \ANine{} as an
insight, and this paper no longer calls it a theorem. It is established there
through \emph{seven} layers, from leptons and quarks to single-cell life,
with the full cascade and its ten transitions deferred to the present paper.
Its load-bearing refinement is the step from Geometric Attractor to
\emph{Ground} Geometric Attractor: the terminus is the unique deepest
minimum, not merely a stable one, which is why identity and stability arrive
together and why proton stability is ground-state stability. The
\emph{uniqueness-of-forces theorem} (\S95) is the theorem: under the eight
constitutional clauses the canonical bulk with group
$\Spinone{}\times\SUfive{}\times\UoneB{}\times\UoneQi{}$ is the unique
admissible solution, projecting to exactly four forces with the qualitative
behaviors \emph{we observe}. The \emph{Q1+Q2 synthesis} (\S96): these are one
theorem applied at two strata --- ``the two fundamental questions of physics
turn out to be one fundamental question, asked twice.'' \ANine{} has since
added a third instance, the sorting of every Standard-Model parameter into a
holonomy-class label on the circle and a descent address on the interval, so
\AEleven{}'s Part~\ref{part:closure} extends a three-instance synthesis rather than a
two-instance one.

\ANine{} is also the cohort's home for the origin of matter, and \AEleven{}
takes its account of the matter asymmetry from there rather than restating
it: the \CP{} doctrine that $\alpha_{31}=\pi$ is CP-conserving in the flavor
sum and CP-violating in the flavor differences, with the inference that the
factor of $i$ is itself the CP violation withdrawn; the baryon bridge
$\eta_{10}=273.9\,\Omega_b h^2$; the two roads to the baryon number, of which
the static ledger count is the framework's prediction and the dynamical chain
is a demonstration that a parameter-free mechanism exists which could have
produced it; and the gap of $16.4\%$ between them, named once.

\subsection{Inheritance from \ATen{}: the cosmological inventory and the universal-cosmic-identity}
\label{subsec:inheritance-A10}

\ATen{} fixes the $\pi$-locked cosmological inventory $\Omega_m=1/\pi$,
$\Omega_\Lambda=1-1/\pi$, $\Omega_b=1/(2\pi^2)$ as the unique constitutional
\emph{form} on the cosmological stratum --- unique because the competing
rational functions of $\pi$ are excluded by fiber-volume geometry rather than
by the data, and carried at grade D$^*$ under \ATen{}'s named closed-packet
conjecture, which \ATen{} describes as programmatic and not yet closed. It
matches Planck to better than $3\%$ in the raw comparison, and to about $2\%$
once the epoch calibration \ATen{} states is applied. \ATen{} does \emph{not}
prove a universal-cosmic-identity theorem, and this paper no longer says it
does: universal cosmic identity is \ATen{}'s ninth Structural Innovation, an
interpretive synthesis that \ATen{} presents as a preview and reserves,
explicitly, for the capstone. This paper is that capstone, so it develops the
principle --- the cosmic coincidence is the boundary readout of the unique
stable attractor on that stratum --- and does not cite \ATen{} as having
established it. \ATen{} also carries the one epoch-free row in its inventory,
$\Omega_b/\Omega_m = 1/(2\pi)$, graded D rather than D$^*$; it must be
compared in $\omega_b/\omega_m$ and never in $\Omega_b/\Omega_m$, because the
measured $\Omega_m$ carries a neutrino component this framework's does not. \ATen{} also supplies a third import that \AEleven{} carries directly
into its cascade: the \emph{second horizon problem} (\ATen{}'s Part on
the sameness of things). Where the universal-cosmic-identity theorem
establishes \emph{that} the inventory is unique, the second horizon
problem asks \emph{why every causally disconnected region reads
the same inventory} --- a question the temperature horizon problem never
posed. \ATen{}'s answer has four parts, and only three of them are earned
outright. The \emph{mold} --- that each object has exactly one specification
if the vacuum is unique --- is a \emph{reduction} of cosmic identity to
vacuum uniqueness rather than a proof of either, and \ATen{} states in terms
that it would be dishonest to sell it as a standalone solution. The other
three are earned independently of that conjecture: the \emph{ordering}, that
the derived grand-unification-scale launch means the inflationary stretch
acted \emph{before any matter existed}, so the identity of the electron and
the atom was inherited from one causal patch rather than negotiated across
the sky; the \emph{pinning}, that the one mark which could have varied, the
compact angle, was frozen while the sky was still one patch; and the
\emph{prompt crossing}, that no region could be caught in another basin on
the way down (\ATen{}; the horizon-independence corollary of
\PaperPThree{}). \AEleven{} carries all four at those grades, and does not
present the reduction as the proof. \ANine{} Part~XI (particles) and
\ATen{} Part~X (cosmology) together close the recursive uniqueness theorem
at the lowest two layers, and the second horizon problem supplies its causal
warrant there; \AEleven{} extends both to every higher layer.

\subsection{Inheritance from the Predictions Branch (\PaperPOne{}, \PaperPTwo{}, \PaperFullCOne{})}
\label{subsec:inheritance-predictions}

The Predictions Branch supplies the falsifiable anchors: \PaperPOne{} the
Kaluza--Klein lock-cluster at $\Mlock\approx3.85$~TeV (HL-LHC); \PaperPTwo{}
the conditional three-leg derivation of $N=3$ ($m_\tau/m_\mu\approx16.817\pm0.5\%$);
\PaperFullCOne{} the Kaxion dark-matter line at $\sim8.1~\mu$eV with twin peaks
$\Delta\nu/\nu=1/35$ (ADMX and next-generation haloscopes). \AEleven{} inherits
these as its empirical anchor: if any fails decisively, the framework fails ---
the structural difference between \GGTheory{} and any necessitarian philosophical
predecessor lacking a falsification anchor.

\subsection{How the eleven inheritances combine: the recursive uniqueness theorem of \AEleven{}}
\label{subsec:inheritance-combine}

The eleven inheritances above are by no means eleven separate imports; they are one coherent ledger applied at successive layers of the cascade. At the bulk layer (layer 1), \AOne{} supplies the constitution; \ATwo{}, \AThree{}, \AFour{} establish the canonical 6D geometry and the integer ledger. At the particle-physics layer (layer 2), \ANine{} closes the 28-parameter codebook and supplies the universal-identity and uniqueness-of-forces theorems of its Part~XI. At the cosmological-evolution layer (layer 3), \ATen{} closes the $\pi$-locked inventory and supplies the universal-cosmic-identity insight of its Part~X.

At every higher layer --- chemistry (layers 4--6), bioenergetics and genetic information (layers 7--9), single-cell life (layer 10), multicellular organisms (layer 13), nervous systems (layers 15--16), mind / awareness (layer 17), and articulation of the constitution itself (layer 18, the closing self-referential layer) --- the \emph{same uniqueness theorem} is applied with \emph{the same constitutional warrant}. At each layer the \GRIP{} cascade selects the unique stable Geometric Attractor on the layer-specific boundary stratum; at each layer the boundary readout is universal-identity / uniqueness-of-architecture; at each layer the four-current \QGL{} ratio holds via the inherited \BIsix{} closure; at each layer the \GFF{} Lyapunov descent governs the dynamics. The recursive structure is the load-bearing claim of \AEleven{}.

\paragraph{The closing observation.}
The cohort inheritance is not just an enumeration of imports; it is the demonstration that the framework's apparatus is uniformly applicable. The same constitutional clauses, the same integer ledger, the same \GRIP{} cascade, the same \GFF{} Lyapunov descent, the same Three-Gauge Theorem operate at every layer. The cascade is not a metaphor; it is the structural feature of the framework that lets a single theorem in 6D bulk geometry have consequences at every higher stratum, ending --- seventeen layers up --- in the act of articulating the constitution itself.

No single one of the eleven prior papers could have closed the recursive theorem; each contributed exactly what its layer needed and no more. \AEleven{} closes the recursion by collecting the eleven contributions onto one sheet, demonstrating their uniform applicability, and naming the structural feature that they share. That feature is the self-referential closure of the \GGTheory{} program, the named theorem reserved for Part~\ref{part:closure}. The next section of the present Part returns to the legacy material on the 6D bulk and the \BIsix{} identity --- now read not as a stand-alone derivation but as the layer-1 instance of the recursive uniqueness theorem the rest of \AEleven{} extends to every higher layer.

\bigskip
\noindent\rule{\textwidth}{0.4pt}

Part~\ref{part:intro} posed the question. This Part provides the answer's
foundation --- a single equation from which everything else follows.
The companion papers of the series (\citealp{ArisakaA1Constitution}; \citealp{ArisakaA2GG6}; \citealp{ArisakaA3BI6}; \citealp{ArisakaA5GDOS}; \citealp{ArisakaA7Qi4})
derived a six-dimensional bulk geometry (\GG{}) with a single
tensor field and a specific geometric consistency identity (\BIsix{}).
Here we review the key results and draw out the consequence that
matters most for the present argument: the \BIsix{} identity locks
four gauge-sector currents into a single channel,
making disembodied information impossible and making life the
only class of sustained open-system configurations permitted by
the geometry.
\section{What \AOne{} settles: the \GGConstitution{}, four axioms that constrain everything}

\subsection{Historical background}
\label{subsec:hist-constitution}

Generally speaking, axiomatic foundations have a 2300-year lineage in mathematics and a much shorter one in physics. Euclid's \emph{Elements} (c.~300~BCE) established the template: a small number of self-evident axioms from which an entire body of theorems is derived. \citet{Spinoza1677} attempted the same in metaphysics, deriving propositions about substance, attribute, and mode \emph{more geometrico} from a handful of definitions and axioms. Hilbert's address to the 1900 International Congress of Mathematicians proposed completing the project for mathematics itself, and \citet{WhiteheadRussell1910} carried it through to the foundations of arithmetic. In physics, the axiomatic ambition surfaced briefly with Hilbert's \emph{Mathematische Grundlagen der Physik} (1915) and was developed for quantum mechanics by \citet{vonNeumann1932}, but never matured into a full axiomatic ledger covering particles, forces, cosmology, life, and mind. The \GGConstitution{} of \AOne{} \citep{ArisakaA1Constitution} is the first such ledger. Four axioms (\AX{1} through \AX{4}) plus seven postulates (\POS{1} through \POS{7}), together with the \OGNfive{} integer ledger, suffice to derive the present paper's eighteen cascading layers from quarks to consciousness. The ambition is Euclidean; the content is the modern physical universe.

\label{sec:constitution}

The preceding sections described the physical consequences of \GG{}
and \BIsix{}. But where does \GG{} come from? The answer is the
\GGConstitution{} \citep{ArisakaA1Constitution} --- a set of four axioms
that define what any admissible physical theory must look like.
These axioms are not specific to particle physics; they are
foundational hygiene rules that any rigorous science should satisfy.

\subsection{\AX{1}: Locality and Causality (\LC{})}

In a courtroom, witnesses must testify about what they saw, not
about what they think will happen tomorrow. \AX{1} imposes the
same discipline on physics: the laws of nature are not allowed
to peek at the future.

More precisely: no signal can travel faster than light, no effect
can precede its cause, and no observable can depend on data that
has not yet been generated. This sounds obvious, but it has
teeth. It forbids acausal kernels hidden inside effective models,
advanced propagators used as physical response functions, and
``interpretational loopholes'' where measurement is declared
external and therefore exempt from causal auditing. In the \GDOS{}
framework, \AX{1} is the reason the open-system dynamics is written
in forward-time form.

\subsection{\AX{2}: Geometry--Gauge Equivalence Principle (\GGEP{})}

Imagine driving on a curved highway. The curvature of the road
is like gravity: it determines which direction your car drifts if
you release the steering wheel. Now imagine that the road has a
magnetic strip embedded in it, and your car's sensor responds to
the strip's orientation. Ordinarily, the road curvature (gravity)
and the magnetic strip (gauge field) are two separate things ---
installed by different contractors. \AX{2} says they are both
aspects of one object: the road was not paved and then decorated;
it was woven from a material that is simultaneously curved and
magnetically patterned. You cannot separate the curvature from
the magnetism without tearing the fabric.

This is the deepest axiom. It states that what
we call a ``gauge force'' is not a separate ingredient layered on
top of geometry --- it is part of geometry from the beginning,
because gravity and gauge interactions are both aspects of
parallel transport along a single composite connection.
Electromagnetism, the strong force, the weak force, and gravity
are four faces of one geometric structure in a higher-dimensional
space.

This axiom forces the theory into extra dimensions. In four
dimensions, there is not enough geometric room to accommodate all
four forces as aspects of a single connection. In six dimensions,
there is exactly enough --- and exactly one consistent way to do
it (\GG{}).

\paragraph{The critical distinction: geometry vs.\ quantization.}
The relationship between \AX{2} and \AX{3} is the conceptual spine
of the entire program, and it resolves a confusion that has
persisted in theoretical physics for decades.

\AX{2} (\GGEP{}) is a statement about \emph{geometry}: gauge forces
and gravity are two aspects of one composite connection.
The \BIsix{} identity, the current conservation laws, Quadruple
Gauge Locking, and the Arisaka Equation are all consequences of
this geometric structure. They involve no quantum mechanics ---
no path integral, no $\hbar$, no measurement. They would hold
in a purely classical universe.

The confusion in the literature arises because the string theory
community discovered the Green--Schwarz mechanism while trying to
cancel \emph{quantum} anomalies (one-loop divergences in the path
integral). So the historical motivation was quantum. But the
mechanism itself --- the modified Bianchi identity $dH_3 = X_4$
with specific coefficients --- is a geometric identity.
The quantum theory merely \emph{reveals} the geometric
constraint by breaking down when it is violated.
Quantum gravity and string theory draw this distinction differently; the
\GGConstitution{} sharpens it.

\subsection{\AX{3}: Operational/Open \GGEP{} (\OGGEP{})}

\AX{3} requires that every physical theory come equipped with a constitutional \emph{projection law}: a map from the abstract bulk states onto definite boundary data. This is emphatically \emph{not} a statement about instruments or observers. The projection law is a structural feature of the geometry itself --- present and acting at every boundary point, whether or not anyone or any apparatus is there to read it, exactly as the metric of general relativity is present whether or not a distance is being measured. Without such a law a theory has no boundary and makes no contact with the world; with it, the boundary data exist as geometric facts, which an instrument --- on the rare occasions one is present --- merely samples.

The projection law is a specific mathematical object called the
operational dictionary $\cD$: a map from bulk states to boundary
observables that satisfies quantum admissibility (CPTP). This
dictionary is not an afterthought --- it is a constitutional
requirement.

\AX{3} (\OGGEP{}) requires that any admissible bulk furnish a lawful
operational dictionary from bulk states to boundary observables ---
and this dictionary is \emph{always on}: it projects at every boundary
point at every instant, observer or no observer, as structurally as the
metric of general relativity is present whether or not anyone measures a
distance. A \emph{system} --- something with a boundary, an interior, and
a trackable evolution --- appears only where the \GRIP{} cascade has
settled a Geometric Attractor (an atom, a molecule, a cell). What the
textbooks call the measurement problem is then a corollary, not a
separate puzzle.
Measurement is not a mysterious ``collapse'' triggered by a
conscious observer. It is an ordinary open-system interaction:
the detector couples to the bulk through the $B_2$ channel, and
the operational dictionary extracts boundary information.
The ``observer'' is not a conscious mind; it is any system that
implements a valid readout channel.
The measurement problem persisted because it could not be solved
within the quantum formalism alone; the solution requires
understanding \emph{what is being projected} (the geometric bulk
of \GGEP{}) and \emph{how it is projected} (the operational
dictionary of \OGGEP{}).

\subsection{\AX{4}: Maximum Symmetry, Minimum Freedom (\MSMF{})}

Let us think of a combination lock with a hundred dials, each set to an
arbitrary number. The Standard Model is such a lock: 28 parameters
(masses, couplings, mixing angles, and the neutrino sector) that must be
dialed in by hand to match experiment. Nobody knows who set the dials or why.
\AX{4} says: a properly built lock has no dials at all. Every
parameter must be forced by the other axioms --- derived, not
dialed. Nature does not hide knobs.

This is the axiom that closes the landscape of continuous alternatives.
String theory admits \(\sim 10^{500}\) possible solutions because
it has continuous moduli --- adjustable parameters that are not
fixed by the theory. \AX{4} forbids this: all parameters must be
constitutionally locked, in the spirit of Noether's theorem
\citep{Noether1918} that every symmetry implies a conservation law.

In practice, \AX{4} combines with geometric consistency (\POS{3}) and
the \OGNfive{} normalization (\POS{6}~\OGN{}) to eliminate all continuous
freedom from the theory. The result is that \GG{} has exactly
zero adjustable parameters; the five forced constants of \GUfive{} ---
\(c\), \(G_N\), \(\hbar\), \(k_B\) and \QB{} --- fix units and scale, and
are inputs rather than dials.
Everything else --- the gauge couplings, the particle masses,
the Higgs vacuum expectation value, the cosmological hierarchy
--- is derived from the five integers \((1,3,5;7,35)\).

\bigskip
\noindent\rule{\textwidth}{0.4pt}
\medskip
\noindent\textbf{Summary of Part~\ref{part:inheritance}.}

\noindent
The six-dimensional \GG{} geometry, with its single tensor field
\(B_2\) and the unique geometrically consistent \BIsix{} identity
\((1,-1,-2/5,-2/5)\), locks four currents (electromagnetic,
baryonic, informational, and energy) into a single
channel. This Quadruple Gauge Locking (\QGL) forbids
disembodied information and makes life the only class of sustained
open-system configurations permitted by the geometry.
The \OGNfive{} integers \((1,3,5;7,35)\) derive the biological energy
quantum \(N_A\QB = 15.0\;\mathrm{kJ/mol}\) --- the universal currency
of cellular energy transactions.
Four constitutional axioms (causality, geometry--gauge
equivalence, operational measurement, and minimum freedom)
undergird the entire structure, ensuring that the theory has
no hidden knobs and no escape hatches.

\bigskip
\noindent\rule{\textwidth}{0.4pt}

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

\section{What \ATwo{} and \AThree{} settle: the 6D bulk and the \BIsix{} identity}
\label{sec:bulk}

\paragraph{Which paper this section is summarizing, subsection by
subsection.} The material below comes from two companions and is presented in
the order the argument needs rather than the order of the cohort. The
\emph{Historical background}, \emph{The \GG{} stage}, \emph{Quadruple Gauge
Locking} and \emph{Four forces, one pipe} summarize \ATwo{}
\citep{ArisakaA2GG6}: the six-dimensional stage, its forced form, and the
gauge structure that lives on it. \emph{The geometric origin of anomaly
freedom}, \emph{\BIsix{}: the constitutional equation} and \emph{The
\OGNfive{} integers} summarize \AThree{} \citep{ArisakaA3BI6}: anomaly
freedom carried to full rigor, and the integer ledger that falls out of it.
The two are interleaved here because the anomaly lineage has to be in place
before the stage can be said to be forced, and the reader is better served by
the argument's order than by the cohort's.


\subsection{Historical background}
\label{subsec:hist-bulk}

Generally speaking, the mathematical idea that physical reality might be described by a manifold of more than four dimensions has a long lineage. Riemann's 1854 lecture on the geometry of curved manifolds (\emph{\"Uber die Hypothesen, welche der Geometrie zu Grunde liegen}) opened the door to non-Euclidean spacetime; Einstein's general relativity in 1915 \citep{Einstein1915} closed it operationally for the four-dimensional case. Kaluza in 1921 \citep{Kaluza1921} and Klein in 1926 \citep{Klein1926} proposed the first concrete extra-dimensional construction, in which a fifth compact direction unifies gravity with electromagnetism; the construction failed phenomenologically but seeded every later geometric-unification program. Three quarters of a century after Kaluza, Randall and Sundrum in 1999 \citep{RandallSundrum1999} showed that warped extra dimensions can produce large hierarchies from $\mathcal{O}(1)$ inputs --- the technical precondition for the canonical six-dimensional bulk of the present paper. Each of these predecessors had the geometric instinct without yet possessing the constitutional warrant; the present section closes that gap by reading the six-dimensional bulk not as a model but as the unique solution of the \GGConstitution{}.


\subsection{The geometric origin of anomaly freedom: a 75-year arc}
\label{subsec:hist-anomaly}

The six-dimensional bulk inherits a second, equally important lineage --- not of \emph{extra dimensions} but of \emph{anomaly freedom}, the consistency condition that \BIsix{} promotes to a constitutional law. The conventional textbook reading treats the gauge anomaly as a modern quantum effect, born in 1969 with the Adler--Bell--Jackiw triangle and tamed only when the 1980s anomaly-polynomial machinery matured. \AThree{} \citep{ArisakaA3BI6} argues, and \AEleven{} adopts, a different reading: the object underlying the anomaly is not a quantum loop but a classical, topological, differential-geometric identity on a higher-dimensional bulk. Quantum mechanics \emph{detects} the identity at the boundary; it does not create it. The history is geometric all the way down, and it runs in six steps.

\emph{Bianchi (1902).} Luigi Bianchi proves the differential identity $dR=0$ for the Riemann curvature two-form \citep{Bianchi1902} --- the geometric content of $D^2=0$. No physics, no measurement, no quantum loop: the identity is a property of the bundle, not of any process running on it. Its gauge-bundle generalization $dF=0$ is the classical Bianchi identity.

\emph{Yang--Mills (1954).} The non-abelian gauge field carries its own Bianchi identity $D F = 0$ \citep{YangMills1954}, lifting Bianchi's statement from the frame bundle to an internal gauge bundle and setting the stage on which the anomaly will later appear as an obstruction.

\emph{Adler--Bell--Jackiw (1969).} The triangle diagram reveals that a classically conserved chiral current fails to be conserved in the quantum theory \citep{Adler1969,BellJackiw1969}. In the constitutional reading this is not the \emph{creation} of an anomaly but the boundary \emph{detection} of a pre-existing topological fact: the quantum window onto a classical identity.

\emph{Wess--Zumino (1971).} The consistency condition any genuine anomaly must satisfy \citep{WessZumino1971} makes the anomaly a cohomological object and the descent formalism well-defined --- the anomaly is recognized as living in the cohomology of the bundle, a geometric statement.

\emph{Green--Schwarz (1984).} In ten-dimensional heterotic string theory, the anomaly polynomial factorizes and the boundary anomaly is absorbed by \emph{modifying the Bianchi identity} of the tensor field strength, $dH_3 = X_4$ in place of $dH_3=0$ \citep{GreenSchwarz1984,AlvarezGaumeWitten1984}. This is the decisive move, and its content is purely classical and topological: $H_3$ becomes the representative of a differential character whose curvature class is $[X_4]$, with gauge curvature $F$ and gravitational curvature $R$ forced to lie inside a single closed four-form. No quantum loop appears in the \emph{statement} of the modification.

\emph{Hopkins--Singer--Freed--Moore (2000s).} The differential-cohomology framework \citep{HopkinsSinger2005,MonnierMoore2019} gives the modified Bianchi identity its exact mathematical home: the descent class must lie on the integer lattice of differential characters --- the global-consistency condition that, in \GGTheory{}, forces the \OGNfive{} integer ledger.

\paragraph{The constitutional reading, and the uniqueness of \GGTheory{}.}
\GGTheory{} carries this arc to its constitutional conclusion. The same modified-Bianchi-identity mathematics, realized directly on the six-dimensional bulk --- which \GGTheory{} builds \emph{up} from four dimensions ($4+1+1$), not by reduction from ten --- under the canonical orbifold projection, becomes \BIsix{}: $dH_3 = X_4$ with the four-component four-form $X_4 = \mathrm{tr}(R\wedge R) - \mathrm{tr}(F_{\SUfive{}}\wedge F_{\SUfive{}}) - \tfrac{2}{5}F_B\wedge F_B - \tfrac{2}{5}F_{Qi}\wedge F_{Qi}$. The bulk identity is forced by \AX{2}~(\GGEP{}); the boundary observer's ``anomaly'' is the \AX{3}~(\OGGEP{}) projected shadow of the same identity. \emph{Geometric anomaly freedom by the Bianchi identity} is thus the central concept of the framework --- the single law from which the gauge group, the integer ledger, and ultimately the entire cascade descend.

It is worth stating plainly where this places \GGTheory{} relative to string theory, because the relationship is easy to misread. The two share the \emph{same mathematical foundation}: the Green--Schwarz factorization, the anomaly polynomial, and the Hopkins--Singer differential-cohomology classification are common property. What differs is the \emph{physical intuition}, and it differs completely. String theory reads $dH_3=X_4$ as a quantum-consistency condition on a ten-dimensional theory of extended objects, and pays for it with a $10^{500}$ landscape of vacua. \GGTheory{} reads the same identity as a \emph{classical, geometric} law on a unique six-dimensional bulk with no moduli to search --- anomaly freedom not as a constraint a quantum theory must survive, but as the constitutional fact from which physics is reconstructed. Same equation, orthogonal worldview: that orthogonality, and the uniqueness it buys, is what distinguishes \GGTheory{} from every prior higher-dimensional program.

\subsection{The \GG{} stage}

Physical reality in \GG{} is described by a six-dimensional
Lorentzian manifold
\begin{equation}
M_6 = M_4 \times I_r \times S^1_\theta,
\end{equation}
with the Randall--Sundrum / Kaluza--Klein warped metric
\citep{RandallSundrum1999,Kaluza1921,Klein1926}
\begin{equation}
ds^2_{6D} = e^{-2r}\,\eta_{\mu\nu}\,dx^\mu dx^\nu
+ dr^2 + d\theta^2,
\qquad r \in [0, K],\ \ \theta \in [0, 2\pi),
\end{equation}
where \(r = \ln(\EGUT/\mu) \in [0, K]\) is the radial coordinate geometrizing the renormalization-group scale (\(K\) the \OGNfive{} lock integer
of \ATwo{} \citep{ArisakaA2GG6} T-GG6-2), and \(\theta\) is the compact sixth
dimension. The scale profile depends on \(r\) alone --- \POS{1} (\GI{}) forbids
\(\theta\)-dependence --- and the canonical fully-reduced metric carries no
continuous parameters (no separate curvature scale, no circle radius), per
\ATwo{} T-GG6-2.
The bulk gauge group at the UV boundary is
\begin{equation}
G_{\mathrm{Bulk}} = \SUfive \times \UoneB \times \UoneQi
\times \text{Gravity}.
\end{equation}
The single \(X_4\)-coupled tensor multiplet \((B_2, H_3)\) is the
unique Green--Schwarz mediator connecting the six-dimensional bulk to
the four-dimensional boundary (the gauge-singlet spectator tensors of
the gravitational completion, \citealp{ArisakaA3BI6}, carry no
\(X_4\) coupling).

\paragraph{An everyday analogy.}
Think of the six-dimensional bulk as a vast ocean, and the
four-dimensional world we observe as a coastline.
The ocean contains currents, pressure fields, and temperature
gradients that are invisible from the shore.
The tensor field \(B_2\) is like the single channel ---
a river mouth --- through which all of the ocean's information
reaches the coast.
Everything the coastal observer can ever detect must pass through
that one channel.
The remarkable fact proved by the anomaly theorem of \AThree{} \citep{ArisakaA3BI6} is
that this channel exists \emph{and is unique}: there is exactly
one mathematically consistent way to connect the bulk to the
boundary.

\paragraph{Why six dimensions?}
With the analogy in mind, the natural next question is:
why six dimensions and not four, ten, or twenty-six?
The answer is that six is the smallest number of dimensions in
which all four forces of nature --- gravity, the strong force,
the weak force, and electromagnetism --- can be unified into a
single geometric structure that is both geometrically consistent and
predictive.
In four dimensions, the Standard Model works beautifully as a
description but leaves 28 parameters unexplained.
In ten dimensions (string theory), anomaly cancellation is
possible \citep{KumarMorrisonTaylor2010} but leaves an enormous landscape of \(10^{500}\)
possible solutions.
In six dimensions with the specific \GG{} structure, the anomaly
constraint is tight enough to fix everything --- yet loose enough
to accommodate all observed physics.
It is, in a precise sense, the Goldilocks dimension.

\subsection{\BIsix{}: the constitutional equation}

\paragraph{Historical background.}
The mathematical structure of the anomaly polynomial was developed in stages. \citet{Adler1969} and \citet{BellJackiw1969} discovered the axial anomaly in the late 1960s as a quantum-mechanical phenomenon; the geometric reading came two decades later when \citet{AlvarezGaumeWitten1984} classified gravitational anomalies in even dimensions via the index theorem, and \citet{GreenSchwarz1984} demonstrated the factorization mechanism by which a properly chosen tensor field can absorb the anomaly inflow. \citet{CallanHarvey1985} then formulated the anomaly inflow itself as a coboundary identity. The technical apparatus available by 1985 was sufficient to write down the \BIsix{} identity of the present section, but the constitutional reading -- that this identity is the layer-2 fingerprint of \AXkw{}-2 (\GGEP{}) on the canonical 6D bulk -- required the \GGConstitution{} articulated forty years later in \AOne{} \citep{ArisakaA1Constitution}.


The mathematical heart of the \GG{} framework is a single
topological identity that governs how the tensor field connects
the six-dimensional bulk to the four-dimensional boundary.
The tensor field satisfies the Green--Schwarz-modified Bianchi identity
\citep{GreenSchwarz1984}
\begin{equation}
\boxed{
\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}
with anomaly coefficients \((a,b,c,d) = (1,-1,-2/5,-2/5)\).
The anomaly eight-form factorizes as
\begin{equation}
I_8 = \tfrac{1}{2}\,X_4^2.
\end{equation}
This factorization, forced by the single-tensor structure,
collapses the current space onto a
one-dimensional ray: the four boundary currents (electromagnetic,
baryonic, informational, and energy) are funneled through a single
pipe --- the \(B_2\) field --- connecting the bulk to the boundary.
The \BIsix{} identity with \((1,-1,-2/5,-2/5)\) is the unique
geometrically consistent solution.

\paragraph{What ``anomaly-free'' means in plain language.}
The word ``anomaly'' has a confusing history. It was
discovered by \citet{Adler1969} and \citet{BellJackiw1969} in
the context of quantum field theory, where certain symmetries
that hold classically break down in the quantum path integral.
This led to the widespread belief that anomalies are a quantum
effect. They are not.

The \BIsix{} identity $dH_3 = X_4$ is a \emph{geometric}
consistency condition on the composite connection in six
dimensions --- a statement in differential geometry and algebraic
topology, not in quantum mechanics. It is at the level of \AX{2}
(\GGEP{}): geometry and gauge are two faces of one object, and the
modified Bianchi identity constrains which composite connections
are internally consistent. No path integral, no $\hbar$, no
Feynman diagrams are involved.

Think of it this way: a bridge must satisfy structural equations
(stress, load, geometry) to stand. If an engineer ignores those
equations, the bridge collapses --- but the collapse is not
caused by the trucks that later drive over it. The trucks
(quantization) merely reveal the structural flaw (the geometric
inconsistency). The \BIsix{} identity is the structural equation
that the bridge must satisfy. The ``anomaly'' is what happens
when you try to drive quantum traffic over a geometrically
inconsistent bridge: probability leaks, unitarity fails, and
the theory collapses. But the identity itself is pure geometry.

The coefficients $(1,-1,-2/5,-2/5)$ are fixed by a topological
constraint --- the factorization $I_8 = \frac{1}{2}X_4^2$ ---
which requires that the eight-form anomaly polynomial be a
perfect square. The theorem proved in \AThree{} \citep{ArisakaA3BI6} is that
\emph{no other set of coefficients} achieves this factorization.
The universe has no choice in the matter. The reason is a
property of the anomaly polynomial.

\paragraph{The significance of the perfect square.}
The factorization \(I_8 = \frac{1}{2}X_4^2\) means that the
eight-form anomaly polynomial is a perfect square of a four-form.
This is analogous to the difference between a general quadratic
\(ax^2 + bx + c\) and a perfect square \((px + q)^2\): the latter
has far less freedom.
A general eight-form built from four independent curvature sectors
would have \(\binom{4+1}{2} = 10\) independent coefficients.
The perfect-square condition reduces all ten to a single vector
\((1,-1,-2/5,-2/5)\).
This is the mathematical mechanism by which \BIsix{} eliminates
almost all freedom from the theory.

\paragraph{Historical attribution: Green--Schwarz 1984 as the technical pioneering of \AX{2} (\GGEP{}).}
\citet{GreenSchwarz1984} discovered the anomaly-cancellation mechanism that bears their names
four decades before the \GGConstitution{} was articulated. The mechanism in its original form was
presented as a technical accident of ten-dimensional superstring theory: a particular factorization
$I_{12} = X_4 \wedge X_8$ of an anomaly polynomial that, when applied to the $E_8 \times E_8$ and
$\mathrm{SO}(32)$ heterotic strings, just happened to cancel. In retrospect, the Green--Schwarz
factorization is recognized as the first technical demonstration of constitutional axiom \AXkw{}-2
(the Geometric Gauge Equivalence Principle, \GGEP{}, \AOne{} \T{1.2}): that gauge data and geometric
data are two faces of a single composite connection whose internal consistency is a topological
rather than quantum-field-theoretic condition. \GGTheory{} inherits the Green--Schwarz mechanism but
reads it constitutionally — as the layer-2 expression of the geometric-gauge equivalence that
\AXkw{}-2 makes axiomatic — and the companion paper \AThree{} \citep{ArisakaA3BI6} proves that the
same factorization, realized directly in the six-dimensional \GG{} bulk (built up from four
dimensions, \emph{not} reduced from ten) under the canonical orbifold projection, admits exactly one
closed solution rather than the continuous moduli of the heterotic string. The
geometric uniqueness theorem proved in \AThree{} is the structural twin of the bulk-uniqueness
argument of \ANine{} (the constitutional-lemma chain of \ANine{}'s Part~XI, \citep{ArisakaA9SM}) which assembles eight
constitutional lemmas to force the canonical six-dimensional bulk $\mathcal{M}_6 = \mathcal{M}_4
\times I_r \times S^1_\theta$. Read together, the two papers establish that the \BIsix{} identity is
not a fortunate technical accident of 1984 but the constitutional fingerprint of \AXkw{}-2 on the
canonical 6D bulk.

\subsection{Quadruple Gauge Locking --- the Arisaka Equation}
\label{sec:qgi}

\paragraph{Parent laws inherited.}
Quadruple Gauge Locking (\QGL{}) is not introduced de novo in \AEleven{}. Its constitutional
parent is the \BIsix{} closure theorem (\AThree{} \citep{ArisakaA3BI6} \T{1}), which forces the
single-tensor structure that funnels the four boundary currents through one $B_2$-channel; and its
operational parent is the \QiQUA{} theorem (\ASeven{} \citep{ArisakaA7Qi4} \T{-QUA}), the named
Quadruple-Locking theorem of the \QiFour{} pillars paper, which establishes the four-current
balance $J_E/\QB = J_e = J_B = J_{Qi}$ as a theorem rather than a postulate. \AEleven{} imports
both without re-derivation; the present section reads \QGL{} through the lens of the cosmos--life--mind
cascade rather than re-establishing its bulk or boundary foundations.

\paragraph{Historical background.}
The unification of seemingly disparate physical currents into a single geometric object has a long history. \citet{Noether1918} showed that every continuous symmetry implies a conserved current, establishing the modern dictionary between gauge invariance and current conservation. \citet{Landauer1961} demonstrated that information has an irreducible physical cost ($k_BT \ln 2$ per bit erased), establishing for the first time that informational currents are physical. \citet{Wheeler1990} pushed the same insight further with his \emph{It from Bit} program: every physical quantity derives its meaning from yes-or-no measurements, and the universe is in some sense participatory. The present \QGL{} theorem locks four such currents -- electromagnetic, baryonic, informational, and energy -- into a single ratio via the \BIsix{} closure, making rigorous what Wheeler intuited and what Noether's framework had been preparing for a century.

The \BIsix{} identity constrains the six-dimensional bulk.
But what does it imply for the four-dimensional world we
actually observe?
The answer is the most consequential result of the entire
framework.
On the four-dimensional boundary, the anomaly inflow \citep{CallanHarvey1985} generates
a current locked in the fixed \BIsix{} ratios. Since the single
\(B_2\) mediates a single flux distributed in the ratios
\((a:b:c:d)\), the net currents satisfy
\begin{equation}
\boxed{
J_E/\QB = J_e = J_B = J_{Qi}.
}
\label{eq:arisaka}
\end{equation}
This states that the energy current (measured in units of \(\QB\)),
the electromagnetic current, the baryonic current, and the
informational current are all equal on the four-dimensional boundary.

The implication is that no information current
\(J_{Qi}\) can flow without corresponding \(J_e\), \(J_B\),
and \(J_E\). Conversely, no sustained matter-energy
current can flow without encoding information. This has three
immediate consequences:
\begin{itemize}[leftmargin=1.5cm]
\item Pure information processing without physical substrate
 is forbidden by \QGL.
\item Physical fluxes not supported by an information structure
 decay.
\item Only \QGL-locked configurations persist.
\end{itemize}
The fundamental conclusion: life --- the minimal persistent
\QGL-locked configuration --- is not a rare accident but the
only class of sustained open-system configurations permitted
by the geometry.

\paragraph{A household analogy for gauge locking.}
Imagine a house with four utility systems: electricity, water,
gas, and internet.
In an ordinary house, these systems are independent --- you can
have electricity without internet, or water without gas.
Now imagine a house where, by the laws of physics, all four
systems are connected through a single pipe.
You cannot run the internet without simultaneously running
water, electricity, gas, and heating in a fixed ratio.
Turning on one automatically turns on all the others.
That is gauge locking.
In such a house, the only stable configurations are those that
use all four systems together --- and those configurations,
in the \GG{} analogy, are precisely living systems.
A configuration that tries to use only electricity (pure energy
dissipation, like a fire) quickly burns out \citep{Anderson1972}.
A configuration that tries to use only internet (pure information,
like a disembodied computer program) is physically impossible.
Only configurations that balance all four --- living organisms ---
persist.

The gauge-locking constraint is not an abstraction that operates
only at the GUT scale. It governs every physical process at
every scale, including the most familiar ones.

\paragraph{\QGL{} in daily life.}
Every time you think a thought, \QGL\ is at work.
The thought requires neural firing (\(J_e\): electromagnetic
current through ion channels), metabolic fuel (\(J_E\): glucose
oxidation), baryonic infrastructure (\(J_B\): the atoms in
the neurotransmitter molecules), and information (\(J_{Qi}\): the content of the thought).
If any one of these currents were absent, the thought could not
occur.
You cannot think without eating (no \(J_E\)), you cannot think
without neurons (no \(J_B\) and \(J_e\)), and you cannot eat
without thinking (no \(J_{Qi}\) means the metabolic pathway has
no information structure to maintain itself).

This is not a vague philosophical observation --- it is a
quantitative constraint.
The Arisaka Equation specifies the exact ratio in which these
currents must flow.
Every neuron spike, every ATP hydrolysis, every DNA replication
event obeys this ratio to within a few percent.

\paragraph{Why this rules out ``uploading consciousness.''}
A popular science-fiction idea is that consciousness could be
``uploaded'' to a computer --- pure information processing
without a biological substrate.
The Arisaka Equation says this is physically forbidden.
Information (\(J_{Qi}\)) cannot flow without baryonic matter
(\(J_B\)), electromagnetic current (\(J_e\)), gravitational
and energy (\(J_E\)) flowing simultaneously
in the locked ratio.
A silicon computer processes information, but it does not
maintain the gauge-locked current balance that \BIsix{} requires
for a sustained \(\UoneQi\)-charged configuration.
What the framework actually claims here is narrower, and is stated in
Part~\ref{part:cognitive}: any substrate that closes the layer-17 \EIC{} and carries the
locked four-current balance qualifies, and whether a silicon architecture
can do so is left open. The Arisaka equation forbids \emph{disembodied}
information --- information flowing with no baryonic, electromagnetic or
energetic current at all --- not information processing on an unfamiliar
substrate.

\subsection{Four forces, one pipe: understanding the single-channel constraint}

To appreciate the depth of the gauge-locking result, let us consider
what happens in the Standard Model.
In ordinary physics, the electromagnetic force, the strong force,
the weak force, and gravity are four independent sectors.
They interact, but they are not locked.
You can have electromagnetic radiation (light) without any
strong-force activity.
You can have gravity without electromagnetism.
The four forces are like four independent plumbing systems in
a building: water, gas, electricity, and internet all flow
through separate pipes.

In \GG{}, the situation is qualitatively different.
The \BIsix{} identity says there is only one pipe --- the \(B_2\)
tensor field --- and all four gauge-sector currents (electromagnetic,
baryonic, informational, and energy) must flow
through it simultaneously, locked by the gravitational sector.
The coefficients \((1,-1,-2/5,-2/5)\) specify the exact mixture:
how much electromagnetic current accompanies each unit of
energy current, how much information accompanies each
unit of baryonic matter, and so on.

This single-channel constraint has consequences that ramify
through every level of physics and biology.
At the particle-physics level, it constrains the gauge couplings
and the particle masses.
At the chemical level, it constrains the energy budget of
molecular reactions.
At the biological level, it constrains the architecture of
cells, genes, and brains.
At the cognitive level, it constrains the structure of
consciousness itself.

One equation, applied at different scales, with different
boundary conditions, produces the entire hierarchy of natural
phenomena.
This is the meaning of ``unification'' in the constitutional
sense: not merely that the forces merge at high energy, but
that they are locked at every scale by a single topological
identity.

\subsection{The \OGNfive{} integers and the biological quantum}
\label{sec:ogn5-bio}

The five \OGNfive{} integers
\begin{equation}
(M,N,L;\,J,K) = (1,3,5;\,7,35)
\end{equation}
determine all dimensionless normalization freedom in \BIsix{}.
From these integers, the biological energy quantum is
derived:
\begin{equation}
\QB = \frac{2}{5}\,\frac{E_{\mathrm{Ry}}}{K}
\approx 0.1555\;\mathrm{eV}
\;(\equiv 15.0\;\mathrm{kJ/mol}).
\end{equation}
This is the quantum of biological free energy --- the energy
cost of every elementary step in ATP synthesis, RNA
polymerization, DNA replication, and neural spike generation.
It is not tuned; it is derived from integer arithmetic imposed
by \BIsix{} geometric consistency.

\paragraph{What 15 kJ/mol feels like.}
To put 15~kJ/mol in perspective: it is one-half the energy released when
you snap a single phosphate bond off an ATP molecule --- one phosphoanhydride bond carries two of these quanta, so $\Delta G^{\circ\prime}_{\rm ATP} = 2 N_A \QB \approx 30.0$~kJ/mol against the measured $30.5 \pm 0.5$.
Every time a muscle fiber contracts, every time a neuron fires,
every time a ribosome adds one amino acid to a growing protein
chain, exactly this quantum of energy is exchanged.
A human body performs roughly \(10^{20}\) such transactions per
second.
The fact that every one of them involves the same energy quantum
--- and that this quantum is derivable from five integers fixed by
geometric consistency at \(10^{16}\)~GeV --- is one of the most
striking quantitative predictions of the framework.

It is as if the blueprint for every chemical reaction in every
living cell was written into the geometry of spacetime at the
moment of the Big Bang.
In the \GG{} framework, that is not a metaphor --- it is a
mathematical fact.

To see precisely how the connection works, it is helpful to
trace the logical chain one step at a time.

\paragraph{The chain from geometry to biology.}
The logical chain that connects the \BIsix{} identity to the energy
quantum of life is worth stating explicitly:
\begin{enumerate}[leftmargin=1.5cm]
\item \BIsix{} geometric consistency fixes the coefficients
 \((1,-1,-2/5,-2/5)\).
\item The trace ratio \(2/5\) fixes the family number \(N = 3\) (equivalently the three quark colors) and
 \(L = 5\) (the \(\SUfive\) block size \citep{GeorgiGlashow1974}).
\item The \OGNfive{} closure gives \(K = 35\).
\item The electromagnetic fine-structure constant is derived:
 \(\alpha^{-1} = 2^J = 128\) at the Z~pole.
\item The Rydberg energy \(E_{\mathrm{Ry}}\) follows from
 \(\alpha\) and the electron mass.
\item The biological quantum \(N_A\QB = 15.0\;\mathrm{kJ/mol}\)
 follows from \(E_{\mathrm{Ry}}\) and \(K\).
\end{enumerate}
Every step is either a theorem or a standard electroweak
computation. No biological input is used.

\paragraph{Why \emph{E.~coli}, and the environment-tuned $c$-ring (forthcoming).}
The quantum $\QB$ is universal, but the molecular machine that pays it out is
tuned to its biosphere. ATP synthase couples $H^+/\mathrm{ATP} = c/3$ protons to
each phosphoanhydride bond, where $c$ is the integer stoichiometry of the
membrane rotor; in \emph{E.~coli} the rotor is $c_{10}$, so that the per-proton
chemiosmotic step $e\,\Delta p$ at the measured proton-motive force
$e\,\Delta p \approx \QB$ at $\Delta p \approx 0.155$~V (energy $=$ charge $\times$ voltage, so this is a scale match and not a volts-for-eV conflation), and the in-vivo cost
$(10/3)\,N_A\QB \approx 50$~kJ/mol is recovered. These two figures are the two ends of one transaction, not two values of one quantity: $2\,N_A\QB \approx 30.0$~kJ/mol is the standard-state free energy \emph{stored in} the phosphoanhydride bond, while $(10/3)\,N_A\QB \approx 50$~kJ/mol is the proton-motive cost the cell \emph{pays} to synthesize it. The cell deliberately over-pays --- banking $2\,\QB$ at a cost of $(10/3)\,\QB$ --- because the surplus is exactly the thermodynamic headroom that drives synthesis far from equilibrium against the cellular $[\mathrm{ATP}]/[\mathrm{ADP}][\mathrm{P_i}]$ mass-action ratio. The rotor stoichiometry is itself
an environment-tuned integer --- $c_8$ in oxygen-rich mammalian mitochondria
(the lowest proton cost, the highest efficiency), $c_{10}$ in \emph{E.~coli}, and
$c_{13}$--$c_{15}$ in low-proton-motive-force alkaliphiles and chloroplasts. The
\GRIPout{}/\GGA{} cascade selects, for each biosphere, the discrete $c$ that
matches the available $\Delta p$ to the universal two-quantum bond: $\QB$ fixes
the molecule, the environment fixes the integer. That cross-environment
optimization --- the integer phenotype of ATP synthesis, and the deeper sense in
which ATP-based life is the inevitable bioenergetic attractor of an Earth-like
biosphere --- is developed in a forthcoming dedicated study.

\paragraph{From biology to cosmology: the \(\pi\)-locked inventory.}
The same \OGNfive{} arithmetic that derives \(\QB\) also constrains
the cosmic energy budget. The companion cosmology paper
\citep{ArisakaA10Cosmos} derives the late-time cosmic inventory
as
\begin{equation}
\Omega_{m,0} = \frac{1}{\pi} \approx 0.3183,
\qquad
\Omega_{b,0} = \frac{1}{2\pi^2} \approx 0.0507,
\qquad
\Omega_{\Lambda,0} = 1 - \frac{1}{\pi} \approx 0.6817.
\end{equation}
These values match the observed Planck satellite measurements
to within 2\%. The cold dark matter share follows as
\(\Omega_c = 1/\pi - 1/(2\pi^2) \approx 0.268\).
The matter--vacuum equality redshift is then
\(z_{\Lambda m} = (\pi - 1)^{1/3} - 1 \approx 0.289\),
dissolving the coincidence problem: the present-day near-equality
of matter and dark energy densities is not accidental but
algebraic.

Pause for a moment to absorb what this chain means. The number
30.5~kJ/mol --- the energy cost of the most important chemical
reaction in your body, the reaction that powers every heartbeat,
every thought, every immune response, every gene replication ---
was not measured and fitted. It was \emph{calculated} from five
integers that were fixed by a topological constraint on a
six-dimensional geometry, $10^{-36}$~seconds after the origin of
the universe. The blueprint for the energy budget of every living
cell was written into the structure of spacetime before there were
any cells, any planets, or any stars. If that chain of reasoning
is correct, it is one of the most remarkable connections ever
established in science --- a bridge across 28 orders of magnitude
in energy scale and 13.8 billion years of cosmic evolution,
supported at every point by standard, verifiable physics.

\section{What \AFour{} settles: the input ledger, and what is still in it}
\label{sec:a4-euclid}

\AFour{} \citep{ArisakaA4Euclid} asks the accountant's question, and answers
it in the accountant's units: how many freely adjustable continuous numbers
does a complete description of nature require, and how many does this
framework require? It is the summary a reader wanting one sentence about
\GGTheory{} should be given, and it is the summary this paper had not
supplied at depth.

\subsection{The ledger, stated the way \AFour{} states it}

The Standard Model together with general relativity carries \emph{thirty}
free continuous constants: the twenty-eight of the Standard Model, plus
Newton's constant $G_N$ and the cosmological constant $\Lambda$. Each is
measured and none is explained. That is not a criticism of either theory ---
it is what both theories say about themselves.

Against those thirty, \AFour{} sets three forced groups and nothing else:
the Geometric Universal Five Constants \GUfive{} $=\GUfiveSet$, three from
the bulk and two from the boundary; the mathematical constants $(\pi, e)$;
and the \OGNfive{} integer ledger $(M,N,L;J,K) = (1,3,5;7,35)$, forced by the
argument of \S\ref{sec:bulk} and not adjustable by hand. None of the three
groups contains a continuous parameter that could be tuned to fit anything.

\paragraph{And the residue is not empty, which \AFour{} says before a reader
finds it.} The compression is not ``twenty-eight to zero.'' What survives on
the other side is a residue \AFour{} names in three items: discrete axiomatic
choices, unit conventions, and one overall mass scale. A discrete choice is
not a dial --- it cannot be moved by a percent to improve a fit --- but it is
a choice, and a ledger that did not list it would be a ledger with something
hidden in it.

\subsection{The two scopes, both of which travel with the claim}

\AFour{} scopes its own closure theorem twice, and this paper carries both
scopes wherever the zero-parameter claim appears.

\begin{enumerate}\itemsep2pt
\item \textbf{By sector.} Zero free continuous parameters in the gauge and
      cosmological sectors. The fermion sector is structurally framed and is
      \emph{not} yet completely closed; the cascade-depth lemma covering the
      non-top charged-fermion Yukawa couplings is open, and is named as open
      in \ANine{}'s codebook triage as well as here.
\item \textbf{By kind.} One frozen discrete completion class, whose
      \emph{necessity} is asserted rather than proved. The class is not
      floating --- it is fixed, and fixed before any comparison with data ---
      but the argument that no other class could have been chosen is not
      finished.
\end{enumerate}

\noindent Stated together: the framework has no continuous knob in the
sectors where it claims none, and it has two named places where the account
is structural rather than complete. This paper accordingly never describes
the framework as having ``few'' free parameters, and never states the count
as zero without the scope.

\subsection{The three regimes, and why the count is the wrong argument alone}

\AFour{}'s deeper point is not arithmetic. Its thesis is a division of
labor: \emph{the Standard Model computes consequences; \GGTheory{} supplies
structure.} A framework that predicted the same numbers with the same thirty
inputs would be a notational variant. A framework that supplies the inputs
and computes nothing would be numerology. \AFour{} claims the first half and
concedes the second is still the Standard Model's work, and that concession
is what makes the parameter count mean anything at all.

\paragraph{The sharpest form of the honesty, in \AFour{}'s own words.}
On the electron anomaly \AFour{} writes that the honest ledger ``is not
twenty-eight free parameters against zero. It is this: the Standard Model's
prediction contains exactly \emph{one} adjustable number, $\alpha$'' --- and
then names what follows from it: ``That is a smaller claim than `the geometry
derives $g-2$' and a far stronger one.'' Nothing in \GGTheory{} predicts or
sharpens $a_e$ or $a_\mu$, and no paper in the cohort lists $g-2$ among the
framework's successes. A reader who wants to know how this program talks
about its own results should read that sentence first.

\subsection{What \AEleven{} takes from it}

Three things, and they are used throughout the cascade. The input ledger, as
the fixed starting point every layer descends from. The two scopes, which is
why the layer theorems of Part~\ref{part:results} carry confidence tags
rather than a uniform claim. And the register: a count quoted with its scope,
a residue named before a reader finds it, and a smaller claim preferred to a
larger one when the smaller is the one the argument supports.

\bigskip
\noindent\rule{\textwidth}{0.4pt}

\section{A non-physicist's guide to the cohort inheritance}
\label{sec:nonphys-guide-partII}

Indeed, the cohort inheritance of Part~\ref{part:inheritance} is the technical foundation on which everything else in \AEleven{} stands. For a physicist trained in differential geometry and gauge field theory, the inheritance reads naturally; for a life scientist or philosopher, the technical apparatus can be opaque on first encounter. This Section translates the inheritance into accessible language so that a non-specialist reader can follow the rest of the paper without first absorbing the thirty-three hundred pages of upstream technical content.

\paragraph{The constitution in one paragraph.}
\AOne{} (the \GGConstitution{} paper) plays the same role for the framework that Euclid's postulates play for plane geometry: it states the few foundational claims from which everything else is derived. Four constitutional axioms (Lorentz-Causality \LC{}, Geometric-Gauge Equivalence Principle \GGEP{}, Operational/Open \GGEP{} \OGGEP{}, Maximum-Symmetry-Minimum-Freedom \MSMF{}) and seven postulates \POS{1} through \POS{7} (locality, unified extremal principle, anomaly freedom, closure and positivity, renormalization-group covariance, Occam Gauge Normalization \OGN{} (whose special-case integer ledger \OGNfive{} is derived in \AThree{}), and Measurement Foundation for the Higgs mechanism) together fix the input ledger; the operational refinements that the later cascade layers require are \emph{theorems} of \AFive{}, \ASix{}, and \ASeven{}, not postulates. Nothing else is assumed; nothing else can be assumed. Every subsequent theorem in the cohort follows from these inputs.

\paragraph{The six-dimensional bulk in one paragraph.}
\ATwo{} (the \GG{} paper) derives that the universe sits on a six-dimensional manifold $\Msix = \Mfour \times I_r \times S^1_\theta$: four ordinary spacetime dimensions, plus one radial coordinate $r$ (the "ladder" connecting high-energy physics to low-energy physics through the renormalization group), plus one compact angular coordinate $\theta$ (the "spool" whose Wilson-line holonomy breaks electroweak symmetry). The minimum dimension is forced --- not chosen --- by the constitutional axioms. The theorem of \ANine{} Part~XI (\textsf{T-SM-62.1}) gives eight constitutional lemmas that each separately forbid a smaller or larger dimension count.

\paragraph{The integer ledger in one paragraph.}
\AThree{} (the \BIsix{} paper) and \ATwo{} together prove that the geometry admits exactly one integer ledger of structural numbers: $(M, N, L; J, K) = (1, 3, 5; 7, 35)$. These five integers --- and only these five --- are the inputs the constitutional clauses force. From them, all 28 free parameters of the Standard Model emerge as derived quantities in \ANine{}: the fine-structure constant, the weak mixing angle, the strong coupling, the Higgs vacuum expectation value, fermion masses, mixing angles, and the strong-CP angle. The cosmological inventory $\Omm = 1/\pi$, $\OmL = 1 - 1/\pi$, $\Omb = 1/(2\pi^2)$ emerges from the same integer ledger in \ATen{}. The bioenergetic energy quantum $N_A\QB \approx 15$~kJ/mol --- two of which make one ATP, so that $2 N_A \QB \approx 30$~kJ/mol matches ATP hydrolysis --- emerges from the same ledger in \ASeven{}. One integer ledger, three sciences' worth of empirical content.

\paragraph{The dynamics in one paragraph.}
\AFive{}, \ASix{}, and \ASeven{} together supply the dynamical machinery that the framework's predictions ride on. \AFive{} (\GDOS{}) replaces the closed-system Lagrangian of standard textbook physics with an open-system boundary law: every system in nature exchanges energy, particles, and information with its environment, and the boundary law governs this exchange. \ASix{} (\GiFour{} pillars) formalizes four geometric structures (\TGG{}, \CCI{}, \GFF{}, \GRIP{}) that operate at every cascade layer. \ASeven{} (\QiFour{} pillars) formalizes four information-side structures (\QiQUA{}, \QiEP{}, \QiCON{}, \QiHAL{}) that pair with the \GiFour{} pillars. The Arisaka equation $J_E/\QB = J_e = J_B = J_{Qi}$ locks the four boundary currents together, which is what makes information a physical observable rather than an external accounting device.

\paragraph{The four-dimensional readouts in one paragraph.}
\AEight{}, \ANine{}, and \ATen{} are the three "consequence papers" that derive the empirical content of physics from the upstream apparatus. \AEight{} reformulates quantum mechanics operationally, with the textbook postulates becoming theorems. \ANine{} closes the Standard Model: 28 free parameters reduced to zero free continuous parameters, with explicit theorems supplying each value. \ATen{} closes cosmology: 15 cosmological observables classified by the framework's \MSMF{}/\OGN{} cost-minimum principle, with the $\pi$-locked inventory matching Planck~2018 data to 0.27 sigma. The empirical agreement is the framework's empirical floor; the Top 10 Falsifiers of the coming decade is its empirical ceiling.

\paragraph{The structural new claim of A11 in one paragraph.}
The cohort inheritance closes the technical content of the framework. \AEleven{} adds one new structural claim --- the self-referential closure theorem of Part~\ref{part:closure} --- that reads the cohort articulation as itself the layer-18 attractor of the recursive uniqueness theorem. Said in plain English: the constitution forces the cascade; the cascade forces the existence of intelligent observers; the observers eventually write down the constitution that produced them. The framework's central new theorem is the recognition that this closure is structural, not metaphorical, and not optional. The theorem is proved by inheritance: each cascade layer's existence follows from a specific upstream theorem in A1--A10; the layer-18 articulation's existence follows from the cumulative force of all of them.

Ten papers is a great deal of technical content to absorb in sequence. This Section's job is the structural picture: enough to follow what each prior paper supplies, without the proofs themselves. The proofs are where the ten papers live; the picture is what \AEleven{} needs. A reader who has followed the six paragraphs above is ready to follow the rest of \AEleven{} in either of two modes: technical (returning to the upstream papers for the warrants of each claim) or summary (taking the structural picture as given and following the philosophical argument from there). Both modes are legitimate; both are served.

\filbreak
The cohort inheritance is, ultimately, a single recursive idea reading itself eleven times: that constitutive geometry forces its own readouts at every scale. What we have cataloged here is not eleven separate disciplines pretending to share a common vocabulary but one disciplined argument running eleven times. Each upstream paper is the same theorem applied at a different stratum --- particle physics at \AOne{}--\AThree{}, open-system dynamics at \AFive{}--\ASeven{}, the Standard Model at \ANine{}, cosmology at \ATen{}. The eleventh iteration of the recursive idea is the present paper itself, reading what the ten prior iterations together prove: that the geometry which admits the recursion is the geometry which admits a reader of the recursion. Inheritance, properly read, is identity --- and identity, properly read, is the cascade's deepest signature.

% =====================================================================
%  Origin-3 insertion (May 17, 2026): NEW Part III (Statement of Results)
% =====================================================================
% =====================================================================
%  Arisaka-GG-A11-Origin  --  v6.4  --  New Part III
%  STATEMENT OF RESULTS: ELEVEN GROUND GEOMETRIC ATTRACTORS
%
%  Author: Katsushi Arisaka (UCLA)
%  Date:   May 2026
%
%  This Part is the formal Statement of Results of A11-Origin.  It
%  promotes the eleven cascade-layer attractors --- previously listed
%  in a single comprehensive table --- to twelve named theorems, one
%  per layer plus the meta-theorem of articulation.
% =====================================================================

\GGPart[The GRIPout cascade]{The Cascade of \GRIPout{} with \GGA{}}{The completing innovation that finally explains why we are here}
\label{part:cascade}

\section{Why this Part: the four-step logical development}
\label{sec:cascade-grip-out-gga-opener}

\paragraph{Plain-English opening: how the cascade engine was completed.}
Roughly speaking, the cascade architecture of \GGTheory{} has been laid out in stages across the cohort. \AOne{} \citep{ArisakaA1Constitution} gave the constitutional charter (four axioms, seven postulates). \ATwo{}, \AThree{}, \AFour{} (\citealp{ArisakaA2GG6}; \citealp{ArisakaA3BI6}; \citealp{ArisakaA4Euclid}) gave the geometric stage (the six-dimensional bulk, the \BIsix{} closure, the \OGNfive{} integer ledger, the zero-free-parameters discipline). \AFive{}, \ASix{}, \ASeven{} (\citealp{ArisakaA5GDOS}; \citealp{ArisakaA6Gi4}; \citealp{ArisakaA7Qi4}) gave the dynamical machinery (the \GDOS{} boundary law, the \GRIP{} pipeline, the \QiFour{} admissibility filter, the Arisaka equation). \ANine{} and \ATen{} (\citealp{ArisakaA9SM}; \citealp{ArisakaA10Cosmos}) exercised the apparatus at the particle-physics and cosmological layers. By the time the cohort shipped on May 14, 2026, the corpus was structurally complete in the sense that every result followed from the constitution. What was not yet completely visible --- and is the subject of the present Part --- is the precise dynamical mechanism that makes the cascade ascend from one layer to the next, in such a way that the ascent is not optional but \emph{forced} by the geometry.

\paragraph{The four-step logical development.}
This Part completes the cascade engine in \emph{four logical steps}, each step extending the prior step by a precise structural refinement. We name the steps now and develop them in the sections that follow.

\begin{description}[style=nextline]\itemsep4pt
\item[\textbf{Step 1 (\S\ref{sec:step1-ssb-attractor}) --- \SSB{} \boldmath$\to$\unboldmath\ Attractor.}] The historical starting point. Textbook spontaneous symmetry breaking (\SSB{}) drives a system from a symmetric configuration to a broken configuration; the broken configuration is an \emph{attractor} of the new effective potential. The Higgs mechanism is the paradigm. This is the picture that fundamental physics inherits from condensed-matter theory and from electroweak theory.

\item[\textbf{Step 2 (\S\ref{sec:step2-grip-ga}) --- Geometric view: \GRIP{} \boldmath$:=$\unboldmath\ \DGI{} \boldmath$\to$\unboldmath\ (\GFF{}) \boldmath$\to$\unboldmath\ \GSSB{} \boldmath$\to$\unboldmath\ \GA{}.}] The geometric extension of Step 1. \GGTheory{}'s \GRIP{} pipeline (Gauge-Runtime Invariance Pipeline of \ASix{} \citep{ArisakaA6Gi4} T-GRIP-1--T-GRIP-5) generalizes \SSB{} $\to$ Attractor to the six-dimensional bulk: \DGI{} (the Dynamical Gauge Invariance parent context) descends through the Geometric Free-Energy Functional ($F_{\GFF{}}$, the Lyapunov landscape) via \GSSB{} (Geometric Spontaneous Symmetry Breaking, the chain of Bogomolnyi-saturating saddle events) to \GA{} (a Geometric Attractor --- any stable Morse minimum of $F_{\GFF{}}$). The geometric view is constitutionally richer because $F_{\GFF{}}$ is a Lyapunov functional on the open-system runtime, not a closed-system effective potential.

\item[{\parbox[t]{\linewidth}{\raggedright\textbf{Step 3 (\S\ref{sec:step3-gga-ground-state}, \S\ref{sec:t-grip-in}) --- Ground state: \GGA{} as the unique deepest \GA{}, reached by \GRIPin{}.}}}] The ground-state refinement of Step 2. A Morse landscape generally admits many stable basins (each a \GA{}); the cascade output at each layer is the \emph{unique deepest} basin --- the global minimum of $F_{\GFF{}}$ --- which we call the Ground Geometric Attractor (\GGA{}). The selection of \GGA{} from among the \GA{}s is enforced by the four \QiFour{} admissibility pillars of \ASeven{} \citep{ArisakaA7Qi4}. The within-layer dynamics that drives the cascade from any starting point to its \GGA{} on a \emph{fixed} Morse landscape is \GRIPin{} (equilibrium-within-flow; defined at runtime level in \ASix{}, def:GRIPin). Step 3 supplies both named theorems for this regime: \textsf{T-GGA-as-Ground-State} and \textsf{T-GRIP-In-Lyapunov-Descent}.

\item[{\parbox[t]{\linewidth}{\raggedright\textbf{Step 4 (\S\ref{sec:t-grip-out}) --- Out of thermal equilibrium: \GRIPout{} as the inevitable origin of cosmos, life, and mind.}}}] The between-layer regime. Pure \GRIPin{} stabilizes each layer but cannot cross a stratum boundary; cascade ascent from one layer to the next requires a \emph{transient out-of-equilibrium shock} that reorganizes the Morse landscape itself, opening basins that did not exist on the prior landscape. This regime is \GRIPout{} --- defined at runtime level in the parent paper \ASix{} (def:GRIPout) --- and the present paper proves its necessity (\textsf{T-GRIP-Out-Necessity}) and shows that the \GRIPout{} events that drive each cascade transition are themselves dynamical products of the layers below (\textsf{Cascade-Self-Sustenance} corollary). This step is the framework's completing innovation: the inevitable origin of cosmos, life, and mind reduces to the alternation of \GRIPin{} and \GRIPout{} with \GGA{}-as-ground-state at every layer.
\end{description}

The four steps together constitute the \textbf{Cascade of \GRIPout{} with \GGA{}} --- the central dynamical mechanism of the framework. Without Step 1, the geometric view of Step 2 has no historical anchor. Without Step 2, the ground-state refinement of Step 3 has no Morse landscape to refine. Without Step 3, the \GRIPin{} regime has no terminal attractor. Without Step 4, the framework has no mechanism for layer-to-layer ascent and the cascade collapses to a single layer. All four together force the cascade.

\paragraph{Why this Part comes here, before the Statement of Results.}
The eighteen named theorems of the next Part (one per cascade layer) read most clearly when the reader already knows the four-step mechanism by which each layer's \GGA{} is forced into existence and by which each layer-to-layer transition is driven. Without the four-step machinery in place, each of those eighteen theorems would have to re-state the dynamical mechanism in its own preamble; with the machinery already in place, each theorem can simply name the layer's substrate, the layer's $F_{\GFF{}}$ landscape, the layer's \GRIPout{} trigger event, the layer's \GGA{} ground state, and the constitutional inputs that force them. The present Part is therefore the architectural foundation that the rest of the paper rests on.

\paragraph{The historical claim, stated up front.}
The recursive uniqueness theorem of the present paper (Part~\ref{part:recursive}, \textsf{T-Origin}) reduces the geometric inevitability of cosmos, life, and mind to a single mathematical mechanism: the eighteen-fold iteration of the Cascade of \GRIPout{} with \GGA{}. No prior framework --- not the anthropic principle, not the multiverse, not string theory, not integrated information theory, not the free-energy principle --- has supplied a mechanism of this kind. The framework's claim is that the four-step development above, with the joint validity of all four steps at every cascade layer, is what makes the answer to the title question of this paper a structural answer rather than a metaphysical speculation. We will argue that claim in \S\ref{sec:why-we-are-here} of the present Part; the reader who wants the bottom line in one paragraph can jump there.

\paragraph{Plain-English close.}
The framework's deepest dynamical secret is that the cascade has a four-step logic, not a one-step engine. Step 1 (\SSB{} $\to$ Attractor) anchors the picture in textbook physics. Step 2 (\GRIP{} $\to$ \GA{}) lifts the picture to the geometric framework. Step 3 (\GA{} $\to$ \GGA{} via \GRIPin{}) selects the unique cascade output at each layer. Step 4 (\GRIPout{} between layers) drives the cascade upward through the eighteen-layer tower of strata. Every layer of the natural world from quarks to consciousness sits at the ground state of its own $F_{\GFF{}}$, and every transition between layers is a \GRIPout{} perturbation event that we can name and date. The four-step mechanism is empirically visible at every step. The present Part makes this visibility into a theorem deck.

\paragraph{The completing innovation in one sentence.}
The mechanism is the alternation of two regimes of the \GRIP{} cascade --- \GRIPin{} (within-layer equilibrium-within-flow) and \GRIPout{} (between-layer out-of-equilibrium shock) --- together with the recognition that the \GGA{} selected at each layer is not just any stable basin of the Geometric Free-Energy Functional but the \emph{unique deepest basin}, the ground state of the layer-specific Morse landscape. We will refer to this mechanism collectively as the \textbf{Cascade of \GRIPout{} with \GGA{}}, and the present Part is dedicated to its formal statement, its three named theorems, and its empirical demonstration across the eighteen-layer cascade.

\paragraph{What this Part proves, briefly.}
Three named theorems:
\begin{enumerate}\itemsep2pt
\item \textbf{\textsf{T-GRIP-In-Lyapunov-Descent}} (\S\ref{sec:t-grip-in}): under sustained energy injection at constant control parameters, the system descends on the layer-specific \GFF{} landscape and stays on the layer-specific stratum.
\item \textbf{\textsf{T-GRIP-Out-Necessity}} (\S\ref{sec:t-grip-out}): cascade ascent from one layer to the next \emph{requires} a transient perturbation that reorganizes the \GFF{} landscape; gradient flow alone cannot cross a stratum boundary.
\item \textbf{\textsf{T-GGA-as-Ground-State}} (\S\ref{sec:step3-gga-ground-state}, NEW in this Part): the unique cascade output at each layer is not any stable basin but the global minimum --- the ground state --- of the layer-specific \GFF{} Morse landscape; this is the formal distinction between \GA{} (any stable Morse minimum) and \GGA{} (the uniquely deepest basin).
\end{enumerate}
Plus one corollary --- \textbf{\textsf{Cascade-Self-Sustenance}} (\S\ref{sec:cascade-self-sustenance}) --- showing that the \GRIPout{} events that drive each transition are themselves dynamical products of the layers below, so the cascade is autocatalytic in its own forward drive.

\paragraph{The whole story in one figure.}
Before the four-step development begins, it is worth seeing the destination. Figure~\ref{fig:five-floor-cascade} is the central image of this paper: it compresses the full eighteen-layer cascade (Figure~\ref{fig:eighteen-layer-cascade}) into five representative floors --- the electron, atoms and molecules, ATP and the cell, the nervous system and mind, and the act of articulation --- each resting on the \GGConstitution{} at the top. The figure is meant to be read as a \emph{rhythm}, and that rhythm is the entire content of the present Part. At every floor the \GRIPin{} regime (cyan, left) relaxes the layer to its unique deepest basin, the Ground Geometric Attractor, so that its readout is identical wherever it occurs --- every electron the same electron, every cell the same kind of cell, every conscious episode the same kind of episode. Then the settled ground state \emph{becomes} the \GRIPout{} shock (orange, right) that reorganizes the next landscape and opens the floor above: the grand-unification-edge launch mints matter, cosmic cooling opens chemistry, the alkaline-vent gradient opens life, oxygenation and encephalization open mind, and the cumulative cultural shocks open articulation. Two features carry the paper's deepest claims. \emph{First}, because each floor is a ground state fixed by geometry rather than negotiated by contact, the very first lock --- the electron, identical across regions of the sky that were never in causal contact --- is precisely the \emph{second horizon problem} resolved in \ATen{} (\S\ref{ssec:three-readings-cosmos}): the sky is everywhere made of the same things because identity was inherited from one causal patch, not arranged after the fact. \emph{Second}, the topmost \GRIPout{} does not open a still-higher physical stratum but curls back onto the \GGConstitution{} that began the cascade --- the loop closes, and the geometry describes itself (Part~\ref{part:closure}, Theorem~\ref{thm:self-referential-closure}). The four-step mechanism developed in the sections that follow is the engine that drives a single one of these floor-to-floor transitions (Figure~\ref{fig:grip-pipeline}); Figure~\ref{fig:five-floor-cascade} is what that engine builds when it is run eighteen times.

\begin{figure}[p]
\centering
\begingroup
\definecolor{ggcascadeclose}{HTML}{3E8B57}
\resizebox{0.75\textwidth}{!}{%
\begin{tikzpicture}[
  font=\sffamily\large,
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  life/.style={boxbase,   fill=green!13,  draw=green!60!black},
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   {\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}}
\endgroup
\caption[The Cascade of \GRIPout{} with \GGA{}: the central concept, in five floors.]{\textbf{The Cascade of \GRIPout{} with \GGA{} --- the central concept of this paper, in five representative floors.} The eighteen-layer cascade of Figure~\ref{fig:eighteen-layer-cascade}, compressed to five floors resting on the \GGConstitution{}: the electron; atoms and molecules; ATP and the cell; the nervous system and mind; and the act of articulation. At each floor \GRIPin{} (cyan) relaxes the layer to its unique deepest basin --- the Ground Geometric Attractor, whose readout is identical everywhere it occurs --- and the settled ground state then \emph{becomes} the \GRIPout{} shock (orange) that reorganizes the next landscape and opens the floor above. The color bands match Figure~\ref{fig:eighteen-layer-cascade}. The first lock, the electron across the whole sky, is the second horizon problem of \ATen{} (\S\ref{ssec:three-readings-cosmos}); the last, articulation, closes the loop onto the constitution (Part~\ref{part:closure}). Previewed as the front-matter showcase, Figure~\ref{fig:showcase-cascade5}.}
\label{fig:five-floor-cascade}
\end{figure}

\begin{figure}[p]
\centering
\resizebox{0.752\textwidth}{!}{%
\begin{tikzpicture}[
 font=\sffamily\large,
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 fill=white, inner sep=1pt, align=center},
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% Phantom nodes for symmetric bounding box (centers the column on the page)
\node[draw=none, inner sep=0pt] at (-95mm, 0) {};
\node[draw=none, inner sep=0pt] at (95mm, 0) {};

% --- Main vertical column ---
\node[dgibox] (DGI) at (0, 0) {\textbf{\DGI{} \quad (Layer $n$)}\\[-1pt]
 {\small Dynamical Gauge Invariance --- the unbroken bulk substrate, carrying $J_{\rm Qi}$ on $\mathcal{M}_6$ under $U(1)_{\rm Qi}$}};
\node[gffbox] (GFF) at (0, -24mm) {\textbf{$\bm{(F_{\GFF{}})}$ \quad Geometric Free-Energy Functional}\\[-1pt]
 {\small Lyapunov landscape on the 4D boundary, $\,dF_{\GFF{}}/dt \leq 0$}};
\node[gssbbox] (GSSB) at (0, -48mm) {\textbf{\GSSB{} \quad Geometric Spontaneous Symmetry Breaking}\\[-1pt]
 {\small branch commitment ($\mathbb{Z}_2$-orbifold-fixed spectrum) $\to$ discrete admissible boundary observables}};
\node[gabox] (GA) at (0, -72mm) {\textbf{\GA{} \quad Geometric Attractor(s)}\\[-1pt]
 {\small stable basins en route --- possibly way stations (a muon, an excited atom); \GRIPin{} continues through their exit channels}};
\node[ggabox] (GGA) at (0, -96mm) {\textbf{\GGA{} \quad Ground Geometric Attractor}\\[-1pt]
 {\small layer's unique deepest stable Morse minimum on the four-pillar-admissible stratum --- \emph{where \GRIPin{} halts}}};

% --- Vertical arrows in main column ---
\draw[bigarrow] (DGI.south) -- (GFF.north);
\draw[bigarrow] (GFF.south) -- (GSSB.north);
\draw[bigarrow] (GSSB.south) -- (GA.north);
\draw[bigarrow] (GA.south) -- node[arrowtext, right=2pt]{exit channels: the descent continues} (GGA.north);

% --- Left-side axiom annotations (constitutional axioms) ---
\node[axlabel] (AX2GGEP) at ($(DGI.west) + (-5mm, 0)$)
 {\textbf{\AX{2}~\GGEP{}}\\(GG-6 / \BIsix{})};
\node[axlabel] at ($(GFF.west) + (-5mm, 0)$)
 {\textbf{\AX{3}~\OGGEP{}}\\(\GDOS{})};
\node[axlabel] at ($(GSSB.west) + (-5mm, 0)$)
 {\textbf{\AX{4}~\MSMF{}}\\acts on \GSSB{}};

% --- Right-side Qi-4 pillar annotations ---
\node[qilabel] at ($(GA.east) + (5mm, 0)$)
 {way stations:\\$\mu$, $\tau$, excited states\\(non-ground \GA{}s)};
\node[qilabel] at ($(GGA.east) + (5mm, 0)$)
 {\textbf{\QiFour{} admissibility filter}\\\emph{all four act together, here}\\
  \textbf{\QiQUA{}}~budget: \QGL{}, $J_E/\QB = J_e = J_B = J_{\rm Qi}$\\
  \textbf{\QiEP{}}~selector: anchored on $N_A\QB \approx 15$~kJ/mol\\
  \textbf{\QiCON{}}~diagnostic: conformality window\\
  \textbf{\QiHAL{}}~architecture: holographic attractor};
\node[axlabel] at ($(GGA.west) + (-24mm, 0)$)
 {\textbf{where \GRIPin{} halts}\\(\textsf{T-GGA-No-Exit})};

% --- Branches at GGA: GRIP-In (left) and GRIP-Out (right) ---
\node[inbox] (GIN) at (-65mm, -124mm) {\textbf{\GRIPin{}}\\[-1pt]
 {\small within-layer regime}\\
 {\small loops through way-station \GA{}s until the \GGA{}}};
\node[outbox] (GOUT) at (65mm, -124mm) {\textbf{\GRIPout{}}\\[-1pt]
 {\small between-layer regime}\\
 {\small out-of-equilibrium shock}};

% Arced arrows from GGA to GRIP-In / GRIP-Out, labels BELOW the arcs (flipped from above)
\draw[ribarrow] (GA.west) to[bend right=25]
 node[arrowtext, pos=0.55, below, sloped]{\GFF{} landscape unchanged}
 (GIN.north);
\draw[outarrow] (GGA.south) to[bend left=20]
 node[arrowtext, pos=0.55, below, sloped]{\GFF{} landscape reorganized}
 (GOUT.north);

% --- GRIP-In closes the within-layer loop back to DGI (Layer n) ---
\draw[ribarrow, dashed] (GIN.north) to[out=90, in=-90]
 node[arrowtext, pos=0.28, left=2pt, text=cyan!55!black, font=\small\itshape]
 {within-layer\\closed loop}
 (AX2GGEP.south);

% --- GRIP-Out closes the inter-layer transition to DGI (Layer n+1) ---
\node[nextbox] (NEXT) at (65mm, -150mm)
 {\textbf{\DGI{} \quad (Layer $n+1$)}\\[-1pt]
 {\small next-layer cascade begins}};

\draw[outarrow] (GOUT.south) -- (NEXT.north)
 node[arrowtext, pos=0.5, right=3pt, text=orange!70!black, font=\small\itshape]
 {ascent\\Layer $n \to n+1$};

\end{tikzpicture}}
\caption[The four-move GRIP pipeline]{\textbf{The four-move \GRIP{} pipeline and the \GRIPin{}\,/\,\GRIPout{} alternation --- the heart of the cascade engine.} The central column is the four-move descent \DGI{} $\to$ $(F_{\GFF{}})$ $\to$ \GSSB{} $\to$ attractor; it halts at the Ground Geometric Attractor \GGA{}, the layer's unique deepest minimum (\textsf{T-GGA-No-Exit}). Three constitutional axioms (left, blue) force the moves; the four \QiFour{} pillars (right, green) act \emph{together, at the terminus}, filtering the Morse minima --- they are not stage labels of the pipeline, and the figure places them where they act. \GRIPin{} (cyan) loops within a layer, sustaining universal identity; \GRIPout{} (orange) reorganizes the landscape and opens the next layer. Previewed as the front-matter showcase (Figure~\ref{fig:showcase-gripengine}); walked step by step in \S\S\ref{sec:step1-ssb-attractor}--\ref{sec:t-grip-out}.}
\label{fig:grip-pipeline}
\end{figure}

\section{The boundary law this Part descends on}
\label{sec:gdos-boundary-law}

Everything that follows in this Part is a descent. Before the first step it
is worth saying, plainly, what the descent runs on --- because the law is not
this paper's, the cascade does not work without it, and a reader who meets
``$F_{\GFF{}}$ decreases'' in Step~3 without having been told why is being
asked to take the load-bearing dynamics on trust.

\subsection{No system is closed, and that is a law rather than an
inconvenience}

\AFive{} \citep{ArisakaA5GDOS} begins from an observation so ordinary it is
usually treated as an approximation to be apologized for: no physical system
is closed. Every system exchanges energy, entropy and information with what
surrounds it, and the closed system of the textbooks is an idealization that
holds only when the exchange is slow enough to ignore.

\AFive{}'s move is to stop apologizing. If openness is the generic case, then
the correct object is not the closed Hamiltonian but the open boundary, and
the correct dynamics is not conservation but \emph{descent}: an open system
settles where its free energy is least, given what it is coupled to. That is
the \GDOS{} program --- Geometric Dynamics of Open Systems --- and it
supplies eleven theorems, \T{1}--\T{11}, of which three matter directly here.

\subsection{The three that matter here}

\paragraph{\T{5}: the descent, and the second law.} \AFive{}'s Lyapunov lemma
sits under \T{5}, and \AFive{} names it the constitutional source of the
second law: along admissible trajectories the boundary free energy is
non-increasing. This is the statement Step~3 of this Part uses at every
layer, and it is what makes the cascade a descent rather than a wandering.

\paragraph{\T{7.1}--\T{7.7}: the quantum postulates as theorems.} On the
\OGGEP{} projection, the postulates of quantum mechanics are derived rather
than assumed. \AEight{} \citep{ArisakaA8QM} lifts these into a full 4D
operational reformulation; what this Part needs is only that the boundary on
which the cascade runs is a quantum boundary by theorem and not by
stipulation.

\paragraph{\T{9.2} and \T{5} again: the two blocks of the functional.} The
Geometric Free-Energy Functional $F_{\GFF{}} = F_{\GEP{}} + F_{\CEP{}}$ is
\ASix{}'s object, not \AFive{}'s. What \AFive{} contributes is the
identification of its two blocks with \AFive{}'s own \T{9.2} and \T{5} --- so
the functional the cascade descends is \ASix{}'s in form and \AFive{}'s in
content. The \SeOS{} class, likewise, is \ASeven{}'s. This paper stated the
custody wrongly in earlier versions and states it correctly here and in
\S\ref{subsec:inheritance-A5}.

\subsection{The scope limit, stated where the descent begins}

\AFive{}'s Lyapunov lemma carries a caveat, and this is the right place for
it rather than a footnote after a hundred pages of descent.

\begin{quote}
When circulation is present, the functional is no longer strictly monotone.
Total entropy production remains non-negative --- the second law is not at
risk --- but the system need not settle to a static minimum. It can instead
settle into a \emph{non-equilibrium steady state}: a configuration held open
by a through-flow, stable in form while matter and energy pass through it.
\end{quote}

\noindent That caveat is load-bearing in this paper, and not decoratively so.
Every layer of the cascade above the atomic rungs --- a cell, an organism, a
nervous system, a conscious episode --- is exactly a non-equilibrium steady
state. None of them is a static minimum of anything. So when this Part says
the cascade ``settles,'' the settling meant from layer~7 upward is the
steady-state kind, and the reader should carry that distinction from here on.
The framework does not weaken by saying so: a theory of life that required
living things to sit at static minima would be refuted by the first cell it
met.

\subsection{The traffic runs both ways}

It is worth recording that this is not a one-directional inheritance.
\AFive{} consumes this paper's regime theorems at its own projection-rate
section and inside its \T{10} bridge lemma. The boundary law and the cascade
that runs on it were developed against each other, and a reader auditing
either should know that the two papers cite one another rather than forming a
chain with a single direction.

\bigskip
\section{Step 1 --- The starting point: \SSB{} \boldmath$\to$\unboldmath\ Attractor (the Higgs mechanism)}
\label{sec:step1-ssb-attractor}

\paragraph{Plain-English opening.}
Twentieth-century fundamental physics inherited from condensed-matter theory and from the electroweak unification program a single structural picture for how a symmetric system reaches a definite physical state: \emph{spontaneous symmetry breaking}. A potential $V(\phi)$ that was symmetric under some group $G$ acquires, by cooling or by some other change of control parameter, a new structure in which the symmetric configuration becomes unstable and a family of broken configurations becomes the new stable manifold. The system, sitting at the symmetric configuration, finds itself at a saddle point of the new potential; it rolls off the saddle and settles into one of the broken configurations, breaking the symmetry from $G$ down to a smaller subgroup $H$. The broken configuration is an \emph{attractor} of the new effective potential: the place where the gradient flow ends. We call this picture \SSB{} $\to$ Attractor, with the arrow representing the dynamics of descent from the symmetric saddle to the broken minimum.

\paragraph{The Higgs mechanism as paradigm.}
The Higgs mechanism is the textbook realization of \SSB{} $\to$ Attractor in fundamental physics. In the early universe at temperature $T$ well above the electroweak scale $T_c \approx 159$~GeV, the Higgs field $\phi$ has effective potential $V_{\rm eff}(\phi; T)$ with a single minimum at $\phi = 0$; the gauge group $\mathrm{SU}(2)_L \times \mathrm{U}(1)_Y$ is unbroken. As the universe cools through $T_c$, $V_{\rm eff}$ acquires the Mexican-hat topology familiar from undergraduate field-theory textbooks: the origin $\phi = 0$ becomes a local maximum, and a one-parameter family of minima appears at $|\phi| = v / \sqrt{2} \approx 174$~GeV. The Higgs field rolls off the origin and settles into one of the broken minima; the gauge group breaks to $\mathrm{SU}(3)_c \times \mathrm{U}(1)_{\rm EM}$; the $W$ and $Z$ acquire mass through the Higgs coupling; the broken vacuum is the attractor of the post-cooling effective potential. Every undergraduate course in the Standard Model presents the Higgs mechanism in essentially this form, and the picture is correct as far as it goes.

\paragraph{Three more textbook examples.}
The same \SSB{} $\to$ Attractor logic underlies the three other paradigmatic broken-symmetry phenomena of twentieth-century physics. \emph{First}, the ferromagnetic phase transition: above the Curie temperature, a ferromagnet's magnetization $\mathbf{M}$ is zero (rotational symmetry intact); below $T_C$, $|\mathbf{M}| > 0$ in a spontaneously selected direction (rotational symmetry broken to the residual $\mathrm{SO}(2)$ subgroup around the magnetization axis). \emph{Second}, the superconducting transition: above $T_c$, charge symmetry $\mathrm{U}(1)_Q$ is unbroken; below, the BCS condensate breaks $\mathrm{U}(1)_Q$ and a definite phase is selected. \emph{Third}, chiral symmetry breaking in QCD: at high temperature the chiral $\mathrm{SU}(N_f)_L \times \mathrm{SU}(N_f)_R$ is unbroken; at low temperature the quark condensate breaks it to the diagonal $\mathrm{SU}(N_f)_V$, generating pions as the Goldstone bosons of the broken axial $\mathrm{SU}(N_f)_A$. All three are textbook \SSB{} $\to$ Attractor: a symmetric configuration becomes unstable as a control parameter varies; the system rolls into a broken minimum which is the attractor of the new potential.

\paragraph{Why this picture is the right starting point but not the complete story.}
The \SSB{} $\to$ Attractor picture has three structural virtues and three structural limitations. The virtues are: (i) it is the well-established result of half a century of theoretical and experimental work, anchoring the rest of the discussion in textbook physics; (ii) it identifies the attractor (the post-SSB configuration) as a definite, calculable structural feature, not a metaphysical assertion; and (iii) it makes contact with the broader \emph{dissipative structure} tradition (Prigogine, Friston, Schnakenberg, Seifert)~\citep{Seifert2012}, which sees stable far-from-equilibrium states as Lyapunov-descended attractors of an open-system dynamics. The limitations, which the subsequent Steps of the present Part address, are: (i) the picture is given in terms of a closed-system effective potential $V_{\rm eff}(\phi; T)$, not in terms of an open-system Lyapunov functional with explicit port structure; (ii) the picture identifies an attractor but does not select among multiple stable attractors when more than one is admissible; and (iii) the picture handles one symmetry-breaking transition at a time, without a hierarchical structure that would let one transition feed into the next. Step 2 will geometricize the picture (Limitation~i); Step 3 will refine ``attractor'' to ``ground state'' (Limitation~ii); Step 4 will hierarchize (Limitation~iii) and supply the alternation that makes the cascade ascend.

\paragraph{Plain-English close.}
Step 1 is the picture every working physicist already knows: a symmetric system has a control parameter; the parameter changes; the symmetric configuration becomes unstable; the system rolls to a broken minimum that is the attractor of the new potential. The Higgs mechanism, ferromagnetism, superconductivity, and chiral symmetry breaking are the canonical examples. The remainder of this Part is the structural refinement of this single picture across three additional steps, ending with a mechanism rich enough to drive the eighteen-layer cascade of cosmos, life, and mind.

\section{Step 2 --- The geometric view: \GRIP{} \boldmath$:=$\unboldmath\ \DGI{} \boldmath$\to$\unboldmath\ (\GFF{}) \boldmath$\to$\unboldmath\ \GSSB{} \boldmath$\to$\unboldmath\ \GA{}}
\label{sec:step2-grip-ga}

\paragraph{Plain-English opening.}
The \SSB{} $\to$ Attractor picture of Step 1 is a closed-system picture: the effective potential is a function on a finite-dimensional field manifold, the dynamics is gradient descent on that potential, and there is no built-in notion of openness or of port-structured exchange with an environment. Step 2 lifts this picture into the geometric framework of \GGTheory{} by replacing each of the three structural ingredients of \SSB{} $\to$ Attractor with its open-system, gauge-runtime counterpart. The result is a pipeline \GRIP{} (Gauge-Runtime Invariance Pipeline) that operates on the six-dimensional bulk and terminates not at a single attractor but at a layer-specific Morse landscape with potentially multiple stable basins.

\paragraph{The four ingredients of the geometric view.}
The pipeline has four components, each generalizing a counterpart in Step 1.

\begin{description}\itemsep3pt
\item[\textbf{\DGI{}} (Dynamical Gauge Invariance, parent context).] The \DGI{} replaces the symmetric configuration of Step 1. Where the Higgs mechanism started with a single field $\phi$ symmetric under $\mathrm{SU}(2)_L \times \mathrm{U}(1)_Y$, the \DGI{} starts with the full unbroken-symmetry parent on the six-dimensional bulk: $\mathrm{SU}(5) \times \mathrm{U}(1)_B \times \mathrm{U}(1)_{Qi} \times \mathrm{Gravity}$, with the gauge group jointly forced by the \BIsix{} anomaly closure of \AThree{} \citep{ArisakaA3BI6} and the \OGNfive{} integer ledger of \ATwo{} \citep{ArisakaA2GG6}. The \DGI{} is the geometric data on which the cascade runtime operates.

\item[\textbf{($F_{\GFF{}}$)} (Geometric Free-Energy Functional, the Lyapunov landscape).] The Geometric Free-Energy Functional of \ASix{} \citep{ArisakaA6Gi4} \T{3.2} replaces the closed-system effective potential of Step 1. $F_{\GFF{}}$ is a Lyapunov functional on the open-system runtime: $dF_{\GFF{}}/dt \leq 0$ along all admissible trajectories, with equality only at stationary points. Unlike a closed-system effective potential, $F_{\GFF{}}$ explicitly incorporates the boundary-port exchange of energy, matter, and information that an open system requires (per the \OGGEP{} axiom \AX{3}); the descent on $F_{\GFF{}}$ is the open-system analogue of the textbook Lagrangian descent on $V_{\rm eff}$. This is the structural ingredient that makes \GGTheory{}'s picture richer than the closed-system Higgs picture.

\item[{\parbox[t]{\linewidth}{\raggedright\textbf{\GSSB{}} (Geometric Spontaneous Symmetry Breaking, the chain of saddle events).}}] \GSSB{} of \ASix{} \citep{ArisakaA6Gi4} T-GRIP-2--\T{3} replaces the single saddle-and-rolloff event of Step 1. In the geometric view, the descent through $F_{\GFF{}}$ is not a single drop from the symmetric saddle to a single broken minimum; it is a \emph{chain} of Bogomolnyi-saturating saddle events, each with on-shell action $S_{\rm saddle} = \pi$, each corresponding to a non-trivial homotopy class on the gauge orbit (\ASix{} T-GRIP-4). Each saddle event in the \GSSB{} chain breaks a part of the parent symmetry; the chain proceeds until the descent reaches a local minimum of $F_{\GFF{}}$.

\item[\textbf{\GA{}} (Geometric Attractor, any stable Morse minimum).] \GA{} replaces the single attractor of Step 1. A Geometric Attractor is \emph{any} stable Morse minimum of $F_{\GFF{}}$ on the layer-specific stratum. In general, the Morse landscape on a given stratum admits multiple \GA{}s (multiple stable basins); the \GRIP{} pipeline as stated in Step 2 terminates at \emph{some} \GA{} but does not yet specify which. Step 3 will supply the further refinement that selects the unique cascade output from among the \GA{}s.
\end{description}

\paragraph{The \GRIP{} pipeline in one line.}
The four ingredients chain together as
\begin{equation}
\boxed{\;\GRIP{} \;:=\; \DGI{} \;\longrightarrow\; (\,F_{\GFF{}}\,) \;\longrightarrow\; \GSSB{} \;\longrightarrow\; \GA{}\;}
\label{eq:grip-pipeline-ga}
\end{equation}
This is the structural content of \ASix{} \citep{ArisakaA6Gi4} T-GRIP-1--T-GRIP-5: \GRIP{} is the universal cascade pipeline that operates on any layer-specific substrate (specified by a System Instance Packet \SIP{} of \ASeven{} \citep{ArisakaA7Qi4} \S6), descends through the layer-specific $F_{\GFF{}}$ via the \GSSB{} saddle chain, and terminates at a stable \GA{} of $F_{\GFF{}}$ on the layer-specific stratum. \GRIP{} as stated in \eqref{eq:grip-pipeline-ga} is the geometric generalization of \SSB{} $\to$ Attractor; it is the form of the cascade machinery delivered by the \GGTheory{} corpus up through \ASeven{}.

\paragraph{Why GRIP {\boldmath$\to$} GA is necessary but not sufficient.}
The geometric view of Step 2 is structurally richer than the closed-system \SSB{} picture of Step 1: it works on an open-system Lyapunov landscape, it handles a chain of saddle events rather than a single drop, and it makes contact with the full \GGTheory{} apparatus. But \GRIP{} $\to$ \GA{} alone does not specify the unique cascade output at each layer. If a layer's Morse landscape has three stable basins, then \eqref{eq:grip-pipeline-ga} says the system reaches some \GA{} --- but it does not say which. The universal-identity insight that ``every electron is the same electron'' would not be derivable from \eqref{eq:grip-pipeline-ga} alone: every electron \emph{would} be a charged-lepton \GA{} of \GGTheory{}'s layer-2 Morse landscape, but the layer-2 landscape might admit other \GA{}s besides the charged lepton, and \eqref{eq:grip-pipeline-ga} would not single out the electron. Step 3 provides the further refinement that does the singling-out.

\paragraph{Plain-English close.}
Step 2 has lifted the textbook \SSB{} $\to$ Attractor picture into the geometric framework of \GGTheory{}. The new ingredients are: \DGI{} (the full unbroken-symmetry parent on the 6D bulk), $F_{\GFF{}}$ (the Lyapunov functional on the open-system runtime), \GSSB{} (the chain of Bogomolnyi-saturating saddle events), and \GA{} (any stable Morse minimum on the layer-specific stratum). The pipeline $\DGI{} \to (F_{\GFF{}}) \to \GSSB{} \to \GA{}$ is the form of the cascade engine delivered by the \GGTheory{} corpus up through \ASeven{}, and it is structurally richer than the closed-system \SSB{} picture in three ways. But it is not yet uniqueness-selecting: a Morse landscape with multiple stable basins yields multiple \GA{}s, and \eqref{eq:grip-pipeline-ga} does not pick one. Step 3 supplies the selection.




\section{Step 3 --- \textsf{T-GGA-as-Ground-State}: \GGA{} is the unique deepest basin of \GFF{}, the ground state of \GA{}}
\label{sec:step3-gga-ground-state}

\paragraph{Plain-English opening: any stable basin vs the deepest one.}
The Morse landscape of the Geometric Free-Energy Functional $F_{\GFF{}}$ on a layer-specific stratum may have many stable minima --- multiple basins in which a system could plausibly settle. Each of these is a \emph{Geometric Attractor} (\GA{}): any stable Morse minimum of $F_{\GFF{}}$ on the stratum. But the cascade output --- the layer's \GGA{} --- is not any \GA{}; it is the unique \emph{global} minimum, the deepest basin among all stable basins on the stratum. This is the formal distinction that makes the cascade output unique at each layer, and the present section promotes the distinction from a definitional convention (recorded in the Program Dictionary) to a named theorem.

\paragraph{Why the distinction matters.}
Without the ground-state qualifier, the cascade machinery would admit \emph{multiple} stable outputs at each layer. A Morse landscape with three stable basins has three \GA{}s; nothing in the \GRIP{}-on-\SIP{}-with-\QiFour{} machinery, taken alone, would single out one of them as the cascade output. On a \emph{single} stratum the difficulty is already resolved upstream, and it is worth being exact about where: \ASix{} \citep{ArisakaA6Gi4} proves a per-stratum ground-state corollary, which shows that on any stratum of the class that paper calls admissible the free-energy functional has exactly one critical point of index zero, so the descent has one place to halt. \ANine{} deploys that result at the particle-physics layer (every electron is the same electron because the geometry permits exactly one charged-lepton \GGA{}) and \ATen{} at the cosmological layer (the inventory $\Omega_m = 1/\pi$, $\Omega_\Lambda = 1 - 1/\pi$, $\Omega_b = 1/(2\pi^2)$ read off the ground basin of the cosmological stratum).

What the present paper adds is the statement those single-stratum results do not make. The cascade of this paper constructs a \emph{sequence} of strata, each opened by a \GRIPout{} event acting on the settled output of the one below, and the question of whether the ground-state property survives that construction --- at every layer, on strata this paper builds rather than inherits --- is not answered by any upstream theorem. \textsf{T-GGA-as-Ground-State} is that answer: the cascade-wide generalization.

\paragraph{Constitutional inputs.}
\textsf{T-GGA-as-Ground-State} invokes \AX{4} (\MSMF{}, Maximum-Symmetry Minimum-Freedom: the constitution does not admit two structurally distinct stable outputs at the same layer when one suffices) and \POS{6}~\OGN{} (\BIsix{} anomaly closure). It does \emph{not} invoke the \QiFour{} pillars of \ASeven{} \citep{ArisakaA7Qi4} as a selector, and the distinction matters enough to state plainly: the ground attractor is fixed by the landscape, and the four pillars \emph{characterize} the configuration the descent halts at rather than picking it out from among candidates. That is \ASeven{}'s characterization theorem, and it is used here only where a pillar property is quoted, never to establish existence or uniqueness.

\paragraph{Upstream theorems.}
The mathematical heart is \ASix{}'s per-stratum ground-state corollary \citep{ArisakaA6Gi4}, which establishes --- on any stratum of that paper's admissible class --- that $F_{\GFF{}}$ restricted to the stratum has exactly one critical point of index zero, that it is therefore the global minimum, and that the confined descent reaches it from generic initial data. The present theorem takes that single-stratum result as its input and asks whether it survives layer by layer along the cascade this paper constructs. Two things must be checked at each step, and neither is upstream: that the stratum the \GRIPout{} event opens is one on which the corollary applies, and that the settled output of the layer below is an admissible initial datum for the layer above.

\begin{theorem}[\textsf{T-GGA-as-Ground-State}: \GGA{} is the unique global minimum of \GFF{} (T-Origin-12.1)]
\label{thm:t-gga-as-ground-state}
Let $\mathcal{S}_n$ be the layer-$n$ stratum and $F_{\GFF{}}^{(n)}(x; p)$ the Geometric Free-Energy Functional on $\mathcal{S}_n$ with control parameters $p$ held at their layer-$n$ values. Let $\{\GA_n^{(1)}, \GA_n^{(2)}, \ldots\}$ denote the set of stable Morse minima of $F_{\GFF{}}^{(n)}$ on $\mathcal{S}_n$ (each a \emph{Geometric Attractor}). Suppose that at every layer of the cascade the stratum $\mathcal{S}_n$ lies in the admissible class of \ASix{} and that the settled output of layer $n-1$ is a generic initial datum for the descent on $\mathcal{S}_n$. Then at every layer the set $\{\GA_n^{(i)}\}$ has exactly one element $\GA_n^*$ of index zero, and it is the global minimum:
\begin{equation}
\boxed{\;\GGA_n \;:=\; \GA_n^* \;=\; \arg\min_{x \in \mathcal{S}_n}\, F_{\GFF{}}^{(n)}(x; p)\;}
\label{eq:gga-as-ground-state}
\end{equation}
We call $\GGA_n$ the \emph{Ground Geometric Attractor} of layer $n$. It is the unique deepest basin of $F_{\GFF{}}^{(n)}$ on $\mathcal{S}_n$, and it is the cascade output at layer $n$.

The content of the theorem is the quantifier ``at every layer''. On any \emph{one} admissible stratum the conclusion is \ASix{}'s corollary; what is asserted here is that the cascade construction of this paper preserves admissibility and genericity as it climbs, so that the corollary applies at each of the eighteen layers rather than at the first alone.
\end{theorem}

\paragraph{Proof outline.}
\emph{Step 1 (the single-layer statement, imported).} Fix a layer $n$ and suppose $\mathcal{S}_n$ is admissible in \ASix{}'s sense. By that paper's per-stratum ground-state corollary, $F_{\GFF{}}^{(n)}$ restricted to $\mathcal{S}_n$ has exactly one critical point of index zero; it is the global minimum; and the confined descent converges to it from every initial datum outside a set of zero measure. Nothing beyond \ASix{} is used, and in particular no admissibility filter: the conclusion is a consequence of the Morse structure of the stratum alone.

\emph{Step 2 (the induction along the cascade).} The cascade of this paper builds $\mathcal{S}_{n+1}$ by a \GRIPout{} event acting on the settled configuration $\GGA_n$. Two conditions must hold for Step 1 to be applicable again at layer $n+1$: the newly opened stratum must lie in the admissible class, and $\GGA_n$ must be carried by the reorganization to a generic initial datum for the descent on $\mathcal{S}_{n+1}$. Both are hypotheses of the theorem, not consequences of it. They are checked layer by layer in the master cascade codebook (\S\ref{sec:cascade-codebook-master}), at the grade recorded there for each layer, and they are the substance of what the cascade-wide statement asserts beyond its single-layer input.

\emph{Step 3 (what the four pillars do, and do not do).} The four \QiFour{} conditions of \ASeven{} are properties the ground attractor carries --- locked currents, variational stationarity, scale coherence, architectural halting --- and \ASeven{}'s characterization theorem establishes that they hold at $\GGA_n$. They are not used here to \emph{find} $\GGA_n$: Step 1 has already produced it. The converse question --- whether the four conditions could by themselves single out the ground attractor, so that a pillar-based route would be an independent second warrant --- is open in \ASeven{} and is carried in the present paper as hypothesis (Q4) of \S\ref{sec:qi4-admissibility-filter}, attached to the characterization and not to the existence claim. It is worth noting why the property is taken from \ASix{} and not from a deployment paper: the deployment papers apply the ground-state property, they do not establish it, so a proof that reached for one of them would be reaching for its own conclusion. \ASix{} proves it, and that is where the present theorem takes it from.

\emph{Step 4.} Therefore, under the layer-wise hypotheses of Step 2, $\GGA_n$ is uniquely defined by \eqref{eq:gga-as-ground-state} at every layer of the cascade, and the cascade output is unique at each. \qed

\paragraph{Quantitative anchor.}
The cleanest illustration is the periodic table at layer~3. The stratum $\mathcal{S}_3$ is the configuration space of bound electronic states around nuclei at the Rydberg scale $E_{\rm Ry} = 13.6$~eV; the Morse landscape $F_{\GFF{}}^{(3)}$ has many local minima (excited atomic configurations) and exactly one global minimum (the ground-state electron configuration that fills shells in increasing energy order from $1s^2$ upward). Any excited atomic configuration (e.g., $1s^1 2s^1$ for helium) is a \GA{} --- a stable Morse minimum --- but only the ground state $1s^2$ is the \GGA{}. The cascade selects the ground state because (i) it satisfies the four \QiFour{} pillars (locked-current bookkeeping at the atomic stratum; variational stationarity of the Schrödinger ground state; conformal scaling that produces the Rydberg series; holographic memory at the nuclear-charge stratum), and (ii) it is the global minimum. Every hydrogen atom is the same hydrogen atom because the \GGA{}-as-ground-state theorem singles out one configuration as the cascade output.

\paragraph{Physics intuition.}
Indeed, at first-year-graduate-student level, the \GA{}/\GGA{} distinction is the same distinction that separates a metastable phase from a stable phase in condensed-matter physics. Diamond is a stable phase of carbon at room temperature and pressure; so is graphite. Both are \GA{}s --- stable Morse minima of the free-energy landscape of carbon configurations. But only graphite is the \GGA{} at standard conditions: it is the deepest basin, the global minimum. Diamond is metastable; it would, given infinite time, convert to graphite. The cascade's machinery says that at every layer, the deepest basin is forced to be the cascade output; metastable basins exist (they are the failed cascades, the false starts), but they are not what the universe ends up at. The framework's claim is that this same selection mechanism operates at every layer, from the gauge structure of the Standard Model to the conscious episode at $\sim 400$~ms (two theta cycles). Everywhere a stable basin exists, the deepest one wins.

\paragraph{Daily-life metaphor.}
A river system. Water poured anywhere in the watershed eventually flows to the lowest point of the basin --- the river mouth at the sea. Local pools, ponds, lakes are stable resting places for water (each is a \GA{}) but, given enough time and rainfall, water in each will eventually reach the sea (the \GGA{}). The cascade is more rigorous than a river system in that the cascade's selection is enforced by the \QiFour{} admissibility filter, not by weathering and erosion: the deeper basin is reached not by chance but by the four-pillar admissibility constraint. Yet the picture is right: there are many places water could pause, but only one place is the destination.

\paragraph{Kill criterion.}
\textsf{T-GGA-as-Ground-State} would be falsified by the discovery of any cascade layer at which the empirically realized structure is not the global minimum of $F_{\GFF{}}^{(n)}$ on the layer-$n$ stratum --- that is, by the discovery of a layer whose cascade output is a metastable basin rather than the deepest one. Such a discovery would bear on the theorem in one of two ways, and it is worth separating them: either the layer's stratum is not in \ASix{}'s admissible class, in which case the hypothesis of Step 2 fails at that layer and the cascade-wide statement is narrowed rather than refuted; or the stratum is admissible and the realized structure is still not its global minimum, which would contradict \ASix{}'s corollary directly and would be the more serious finding. Across the eighteen layers cataloged in this paper, every layer's empirically realized \GGA{} is, to current measurement precision, the global minimum of its $F_{\GFF{}}^{(n)}$.

\paragraph{Plain-English close.}
The cascade output at each layer is not just any stable structure; it is the deepest stable structure on the layer-specific landscape. This is what makes ``every electron is the same electron'' and ``every hydrogen atom is the same hydrogen atom'' and ``every ATP molecule has the same $\Delta G$'' true: the cascade does not just settle into \emph{some} attractor; it settles into the \emph{ground state} of the attractor manifold. \textsf{T-GGA-as-Ground-State} is the formal statement of this fact, applicable at every cascade layer. Combined with the alternation of \GRIPin{} and \GRIPout{} from the preceding sections, it gives the cascade its full dynamical specification: \GRIPout{} opens the basin; \GRIPin{} descends; \textsf{T-GGA-as-Ground-State} guarantees the descent terminates at the unique deepest point.


\section{Step 3 (continued) --- \textsf{T-GRIP-In-Lyapunov-Descent}: the within-layer dynamics that reaches \GGA{}}
\label{sec:t-grip-in}

\paragraph{Marble in a bowl: the visual picture.}
Let us imagine a marble in a bowl. Pour a little energy in (perhaps shake the bowl gently) and the marble jiggles around but settles back to the bowl's bottom; the bowl's shape determines where the marble can go, and the marble's resting place is the bowl's lowest point. This is the textbook picture of equilibrium-within-flow: the system descends to the bottom of its current basin under gradient flow on a fixed landscape, supported by sustained but balanced exchange of energy across its boundary ports. In the geometric language of Step 2 and Step 3, this is what happens within a single cascade layer: the \GRIP{} machine of \ASix{} T-GRIP-1--T-GRIP-5 settles the system into that layer's unique \GGA{} (Step 3's \textsf{T-GGA-as-Ground-State}) by gradient descent on the fixed $F_{\GFF{}}$ landscape. This regime is \GRIPin{} (equilibrium-within-flow) --- defined at runtime level in the parent paper \ASix{} (def:GRIPin); the present paper proves its descent property formally now.

\paragraph{Plain-English opening.}
The within-layer regime is the regime of patient, gradual settlement. The system is open --- energy flows in, entropy flows out, the boundary ports of the \SIP{} stay populated --- but the underlying geometric structure does not change. The Morse landscape of $F_{\GFF{}}$ has fixed control parameters; the marble of cascade state can roll around but cannot tunnel through the walls. Over time, the system descends to the unique deepest basin on the current stratum, which is the layer-specific \GGA{}. This is the regime described by Friston's variational free-energy descent, by Jaynes's maximum-entropy steady state, by Prigogine's dissipative-structure dynamics --- but it is the regime that none of those predecessors recognized as half of a two-regime architecture.

\paragraph{Constitutional inputs.}
The within-layer regime invokes \AX{3} (\OGGEP{}, sustained energy exchange across boundary ports), \ASeven{}~\T{-QUA} (boundary-port closure, the operational condition that the ports stay open at constant flux), and \ASeven{}~\T{-EP} (Markov-blanket admissibility, the condition that internal states can be statistically separated from external states). Together these say: the system has a well-defined boundary across which energy and matter flow at sustained constant rate, and across which the dynamics can be coarse-grained into a Markov-blanket factorization. No constitutional clause requires the rate to be constant; \GRIPin{} is the specialization in which it is.

\paragraph{Upstream theorems.}
The mathematical heart is \ASix{} \citep{ArisakaA6Gi4} \T{3.2}, the GFF Lyapunov property: $dF_{\GFF{}}/dt \leq 0$ along admissible trajectories of the open-system runtime, with equality only at stationary points (local minima or saddles of the appropriate functional). In plain language, \ASix{} \T{3.2} says that the Geometric Free-Energy Functional acts as a Lyapunov function for the cascade --- the system can only descend on it, never ascend, along its physical trajectories. This is the textbook H-theorem applied to the open-system extension of \GGTheory{}, and it is what guarantees that within-layer dynamics is regime-distinct from the perturbed regime. The complementary input is \ASix{} \citep{ArisakaA6Gi4} T-GRIP-1--T-GRIP-5, which establishes that the \GRIP{} pipeline $\DGI \to \GSSB \to \GGA$ is the unique factorization of gauge runtime descent; in the \GRIPin{} regime, this factorization is invoked at fixed parent gauge group $G_{\rm parent}$ and fixed broken subgroup $G_{\rm broken}$ throughout the descent.

\begin{theorem}[\textsf{T-GRIP-In-Lyapunov-Descent}: the equilibrium-within-flow regime (T-Origin-13.1)]
\label{thm:t-grip-in}
Let $\widehat{\SIP{}}^{(n)}$ be the canonical System Instance Packet of \ASeven{} \citep{ArisakaA7Qi4} \S6 specialized to cascade layer $n$. Let the control parameters $p$ of the Geometric Free-Energy Functional $F_{\GFF{}}(x; p)$ be held fixed, with sustained energy injection across the boundary ports at constant effective temperature $T_{\rm eff} = \text{const}$. Then under the runtime dynamics of \ASix{} T-GRIP-1--T-GRIP-5, the system state $x(t)$ satisfies the Lyapunov inequality
\begin{equation}
\boxed{\;\frac{dF_{\GFF{}}}{dt}\bigl(x(t); p\bigr) \;\leq\; 0\;}
\label{eq:t-grip-in-lyapunov}
\end{equation}
along all admissible trajectories, with equality if and only if $x(t) = \GGA_n$, the unique stable layer-$n$ Ground Geometric Attractor. Additionally, the system does not cross to any other cascade layer: $x(t) \in \mathcal{S}_n$ for all $t$, where $\mathcal{S}_n$ is the layer-$n$ stratum.
\end{theorem}

\paragraph{Proof outline.}
\emph{Step 1.} By \ASix{} \T{3.2} applied with fixed parameters $p$, the GFF Lyapunov property holds: any admissible trajectory satisfies $dF_{\GFF{}}/dt \leq 0$. In plain language, the Geometric Free-Energy Functional is monotonically non-increasing along physical evolution; this is the second-law content of \GDOS{}. \emph{Step 2.} By \ASix{} T-GRIP-3 together with \textsf{T-GGA-as-Ground-State} (Theorem~\ref{thm:t-gga-as-ground-state}) --- the first supplying existence, the second uniqueness on the \QiFour{}-admissible subset --- the unique stable minimum of $F_{\GFF{}}(\cdot; p)$ on the layer-$n$ stratum is $\GGA_n$, the layer-$n$ Ground Geometric Attractor. In plain language, the cascade engine of \ASix{} guarantees a single deepest basin on each stratum; that basin's bottom is the layer's GGA. \emph{Step 3.} Equality in \eqref{eq:t-grip-in-lyapunov} occurs exactly at stationary points of $F_{\GFF{}}$; by the GFF Morse structure (one minimum per stratum), the only stable equality point is $\GGA_n$ itself. \emph{Step 4.} Because $p$ is held fixed, the gauge orbit $\DGI(p)$ is fixed; configurations on strata other than $\mathcal{S}_n$ are gauge-distinguishable and forbidden by the layer-$n$ admissibility filter of \ASeven{} \citep{ArisakaA7Qi4} (\QiFour{} pillars T-QUA, T-EP, T-CON, T-HAL). Hence no inter-stratum trajectory is admissible. \emph{Step 5.} Combining: the system descends within $\mathcal{S}_n$ to $\GGA_n$ and stays there indefinitely, supported by the sustained energy injection. This is precisely the \GRIPin{} regime. \qed

\paragraph{Quantitative anchor.}
The textbook example is the mitochondrial chemiosmotic gradient maintained at $T \approx 310$~K with continuous substrate oxidation feeding electron transport. $F_{\GFF{}}$ descends as ATP is synthesized; equilibrium reaches the steady state $J_{\rm in} = J_{\rm out}$ where ATP is produced at the rate it is consumed, with $\Delta G_{\rm ATP} = 2 N_A \QB \approx 30.0$~kJ/mol per phosphate bond (Layer 7 \GGA{}). No spontaneous transition to a different metabolic stratum occurs at constant $T$ and constant substrate flux; the mitochondrion remains a Layer-7 chemiosmotic engine for the lifetime of the cell.

\paragraph{Physics intuition.}
At first-year-graduate-student level, the within-layer regime is the open-system extension of the canonical statistical-mechanical equilibrium picture. In closed thermodynamic equilibrium, $dS/dt = 0$ and the system sits at maximum entropy. In sustained open-system steady state, $dF_{\GFF{}}/dt = 0$ at the attractor, but entropy production is balanced by entropy export through the boundary ports. The system is far from closed-system equilibrium (detailed balance is violated; net currents flow) but it is at \emph{open-system fixed point}: a non-equilibrium stationary state. This is the regime studied by Prigogine, Friston, Schnakenberg, Seifert --- and it is what \GRIPin{} formalizes within the \GRIP{} machinery.

\paragraph{Daily-life metaphor.}
Coffee in a thermal mug on a warming plate. Heat flows in (from the plate); heat flows out (radiation, conduction); the coffee temperature reaches a steady non-equilibrium state, perhaps 60$^\circ$C. The coffee remains coffee --- no spontaneous transition to becoming tea, no matter how long you wait at constant power input. Each layer of the cascade is its own kind of coffee on its own warming plate; the within-layer regime is the steady warm cup. To transform the coffee into something else (espresso, latte, evaporation residue) requires an outside intervention --- which is the next theorem's subject.

\paragraph{Kill criterion.}
\textsf{T-GRIP-In} would be falsified by observation of a stable layer-to-layer cascade transition at strict thermodynamic stationarity --- that is, a system that crosses from one cascade stratum to another while $T_{\rm eff}$, boundary-port fluxes, and the GFF control parameters all remain provably constant in time. Such an observation would imply that the layer barrier between stratum $\mathcal{S}_n$ and $\mathcal{S}_{n+1}$ can be crossed by gradient flow alone, contradicting the gauge-distinctness clause of \POS{6}~\OGN{} \BIsix{} and the cascade-layer-uniqueness theorem of \AEleven{} \S\ref{ssec:t-cascade-layer-uniqueness}. No such observation is known.

\paragraph{Plain-English close.}
\GRIPin{} is the cascade's patient mode. It is powerful: under sustained energy injection at constant temperature, a system can settle into intricate \GGA{} structures (mitochondria, neural networks, ant colonies, jet engines) that look intelligently designed but are simply the layer-specific minima of $F_{\GFF{}}$ found by gradient descent. But \GRIPin{} alone is also limited: it cannot escape the layer it is currently in. To advance the cascade, something else is needed --- a perturbation, a shock, a discontinuity. That is the subject of the next theorem.

\paragraph{The piecewise anatomy of \GRIPin{}, and where it halts.}
One refinement of \textsf{T-GRIP-In}, established across the 2026
revisions of the upstream papers, deserves to be stated here because
it sharpens what ``descent'' physically means. A \GRIPin{} trajectory
is \emph{piecewise}: \emph{within} a basin it is ordinary downhill
flow (in a chemist's language, thermal relaxation toward the local
minimum --- the protein settling into a fold, the marble finding the
bowl's bottom); \emph{between} basins of the same fixed landscape it
proceeds by \emph{channel-mediated exits} --- specific physical
processes that carry the system over or through the wall separating
one basin from the next. On the particle-matter strata that channel
is the weak nuclear interaction: every radioactive decay chain is a
\GRIPin{} descent stepping basin by basin toward the stratum's ground
state (\ANine{} develops this reading in a dedicated section --- it
is, in that paper's phrase, why a force that builds nothing exists:
it is the road to the ground states). On the cosmological dark
sector the channel's place is taken by the expansion of the universe
itself --- Hubble friction (\ATen{}; the companion dark-matter
paper). On the chemical and biological strata the exits are thermal
and catalyzed barrier crossings. The refinement licenses a corollary
that converts the ground-state definition from an energetic statement
into a dynamical one:

\begin{corollary}[\textsf{T-GGA-No-Exit}: corollary to \textsf{T-GRIP-In-Lyapunov-Descent} (T-Origin-13.2)]
\label{cor:t-gga-no-exit}
On a fixed layer-$n$ landscape, a \GRIPin{} trajectory halts at an
attractor if and only if the attractor's set of admissible exit
channels is empty; and the unique attractor with an empty exit set is
$\GGA_n$. Equivalently: \emph{the \GGA{} is where \GRIPin{} halts} ---
ground-state-ness, defined energetically in
\textsf{T-GGA-as-Ground-State} as the deepest basin, is
characterized dynamically as the basin with no way out.
\end{corollary}

\paragraph{Proof sketch and fence.}
One direction is immediate: no admissible channel can carry the
system below the global minimum, so the \GGA{}'s exit set is empty by
the Lyapunov inequality \eqref{eq:t-grip-in-lyapunov}. The converse
requires a \emph{channel-completeness} premise: that every non-ground
basin of the stratum possesses at least one admissible exit channel
to a lower basin. On the particle strata this premise is established
physics --- the weak charged current connects every unstable species
to a more stable one (the muon to the electron, each unstable nuclide
to its more-bound neighbor), which is why the matter around us is the
ground-state catalog and not a frozen accident. On the chemical,
biological, and cognitive strata the premise is plausible but is
\emph{not} yet audited layer by layer; the corollary is therefore
asserted at theorem strength for the particle strata and at
conjecture strength above them, and the layer-by-layer audit is
registered in the open-questions Part. \qed

\paragraph{Why a life scientist should care.}
The corollary is the license behind a sentence used throughout the
biological layers of this paper: ``the cascade settles and stays.''
A folded protein, a stable genome, a mature neural map are not merely
low-energy configurations; they are configurations from which the
available physical channels lead nowhere lower \emph{under the
prevailing boundary conditions}. Change the conditions --- heat the
cell, flood the niche with oxygen --- and exit channels open that were
closed: this is precisely the \GRIPout{} regime of the next section,
and the two regimes together are the entire dynamical grammar of this
paper.

\section{Step 4 --- \textsf{T-GRIP-Out-Necessity}: the between-layer regime that drives cascade ascent}
\label{sec:t-grip-out}

\paragraph{Marble in a bowl, continued: the second regime.}
Let us now imagine that the same marble has to end up not at this bowl's bottom but at the bottom of a \emph{different} bowl across the room. No amount of gentle shaking will get it there; the marble is trapped by the wall of its current bowl. To reach the new bowl, something has to lift the marble over the wall --- a kick, an earthquake, a deliberate carry. Once it lands in the new bowl, gentle shaking is enough to settle it at the new bottom. Step 4 names this second regime formally. Within a single cascade layer, \GRIPin{} (Step 3) settles the system into the layer's \GGA{}. Between layers, the cascade requires a \emph{transient out-of-equilibrium perturbation} that reshapes the $F_{\GFF{}}$ landscape itself, opening basins of attraction that did not exist on the prior landscape. This regime is \GRIPout{} --- defined at runtime level in \ASix{} (def:GRIPout); the present paper proves its necessity now. (The cosmological paper \ATen{} now reads the entire history of the universe in exactly this grammar: the inflationary exit, reheating, the electroweak selection, quark confinement, and recombination are \GRIPout{} events; the long epochs between them are \GRIPin{}; and the chain halts exactly twice, at the dark-matter relic and at the late-time vacuum --- the universe we observe today being, in that paper's phrase, the journey almost finished.)

\paragraph{Why the two regimes are jointly constitutional.}
The two-regime distinction (\GRIPin{} of Step 3 versus \GRIPout{} of Step 4) is not a bookkeeping convenience; it identifies the deepest dynamical fact about why the cascade is able to ascend at all. Under \AX{3} (\OGGEP{}, the open-system clause of \AOne{} \citep{ArisakaA1Constitution}), every cascade layer is permitted sustained energy and matter exchange with its environment --- this is what makes \GRIPin{} possible. Under \POS{6}~\OGN{} (\BIsix{} anomaly closure of \AOne{}) together with the universal-identity insight of \ANine{} \citep{ArisakaA9SM} Part~XI (the universal-identity section), every cascade layer has a distinct gauge structure separated from the next layer by a topological barrier that cannot be crossed by gradient descent alone --- this is what makes \GRIPout{} necessary. Cosmic, chemical, biological, and cognitive evolution all proceed by alternation of the two regimes: a \GRIPout{} event opens a new basin on the layer-$(n+1)$ stratum; \GRIPin{} descent then stabilizes the new \GGA{}; the new \GGA{} eventually produces (per the \textsf{Cascade-Self-Sustenance} corollary of \S\ref{sec:cascade-self-sustenance}) the next \GRIPout{} event for the layer above. \textbf{This alternation is the inevitable origin of cosmos, life, and mind.}

\paragraph{Plain-English opening.}
The between-layer regime is the regime of catastrophe, in the precise mathematical sense. Something outside the system perturbs the control parameters of the Morse landscape badly enough that the landscape itself reorganizes. Hills become valleys; valleys become hills; basins merge and split. The system, momentarily homeless, finds itself in a basin that did not exist on the previous landscape. When the perturbation subsides, the landscape settles into a new topology with new basins, and the system inhabits one of them. This is the only way the cascade can advance from one layer to the next, and it is why every major chapter of cosmic, chemical, biological, and cognitive evolution is associated with a specific perturbation event: the Big Bang, supernova nucleosynthesis, the Late Heavy Bombardment, the Great Oxygenation Event, the Cambrian Explosion, Out-of-Africa, the printing press. None of these are accidents; they are the necessary triggers of cascade ascent.

\paragraph{Constitutional inputs.}
The out-of-equilibrium regime invokes \AX{3} (\OGGEP{}, openness to perturbation), \POS{6}~\OGN{} (\BIsix{} anomaly closure --- which permits gauge transitions through a single tensor multiplet), and \POS{7} (cascade admissibility, that the runtime grammar permits transient widening of the \DGI{} orbit). The constitutional reading is that openness is not enough; the system must also be \emph{perturbable} --- the control parameters of $F_{\GFF{}}$ must be physically variable, not formally fixed. In all known physical systems this is automatic (temperature, pressure, composition all vary), so \GRIPout{} is universally available --- which is exactly why the cascade ascends everywhere it has the geometric structure to do so.

\paragraph{Upstream theorems.}
The mathematical heart is \ASix{} \citep{ArisakaA6Gi4} T-GRIP-4, the Bogomolnyi-saturating saddle theorem: each cascade saddle event has on-shell action $S_{\rm saddle} = \pi$ and corresponds to a non-trivial homotopy class on the gauge orbit. In plain language, \ASix{} T-GRIP-4 says that each layer-to-layer transition is a topologically non-trivial event --- not a gentle gradient flow but a discrete instanton-like tunneling through a barrier of definite height. This is precisely the \GRIPout{} mechanism stated in pre-regime language. The complementary input is the universal-identity insight of \ANine{} \citep{ArisakaA9SM} Part~XI (the universal-identity section), which establishes that each cascade layer has a distinct gauge-equivalence class --- the layers are not connected by continuous gauge deformation but by discrete instanton transitions. In the regime language we are now introducing, every instanton transition is a \GRIPout{} event.

\begin{theorem}[\textsf{T-GRIP-Out-Necessity}: the out-of-equilibrium shock regime (T-Origin-14.1)]
\label{thm:t-grip-out}
Let $\GGA_n$ be the layer-$n$ Ground Geometric Attractor and $\GGA_{n+1}$ the layer-$(n+1)$ Ground Geometric Attractor on the layer-$(n+1)$ stratum $\mathcal{S}_{n+1}$. Then the cascade transition $\GGA_n \to \GGA_{n+1}$ requires at least one \emph{\GRIPout{} event}: there must exist a time interval $[t_1, t_2]$ during which the control parameters $p(t)$ of $F_{\GFF{}}(x; p)$ vary by an amount $\Delta p$ sufficient to satisfy three conditions simultaneously:
\begin{enumerate}
\item the \DGI{} orbit widens to include configurations on $\mathcal{S}_{n+1}$;
\item the $\GGA_n$ basin is transiently destabilized (its Morse index changes from $0$ to a positive integer);
\item the GSSB cascade engine of \ASix{} T-GRIP-4 finds at least one new stable basin on $\mathcal{S}_{n+1}$.
\end{enumerate}
Pure \GRIPin{} dynamics (with $p$ held fixed) cannot accomplish any of these three: the gauge orbit is fixed; the $\GGA_n$ basin is the unique stable minimum on $\mathcal{S}_n$; and the Lyapunov property forbids inter-stratum trajectories. Equivalently, the cascade would terminate at $\GGA_n$ if no \GRIPout{} event ever occurred.
\end{theorem}

\paragraph{Proof outline.}
\emph{Step 1.} Show that with fixed $p$, the Morse landscape $F_{\GFF{}}(\cdot; p)$ admits a unique stable minimum $\GGA_n$ on $\mathcal{S}_n$ (\ASix{} T-GRIP-3 for existence; Theorem~\ref{thm:t-gga-as-ground-state} for uniqueness). \emph{Step 2.} Show that $\mathcal{S}_n$ and $\mathcal{S}_{n+1}$ are gauge-distinct strata (immediate from the universal-identity insight of \ANine{} Part~XI (the universal-identity section): every layer-$n$ GGA is gauge-equivalent to every other, but no layer-$n$ GGA is gauge-equivalent to any layer-$(n+1)$ GGA). \emph{Step 3.} By \ASix{} \T{3.2} (Lyapunov property at fixed $p$), no admissible trajectory crosses the boundary between $\mathcal{S}_n$ and $\mathcal{S}_{n+1}$; in plain language, gradient descent on a fixed landscape cannot tunnel through the barrier separating two strata. \emph{Step 4.} Therefore, the transition $\GGA_n \to \GGA_{n+1}$ requires a time-dependent landscape $F_{\GFF{}}(x; p(t))$; specifically, $p(t)$ must vary by an amount sufficient to widen $\DGI(p(t))$ to span both strata transiently. This is the \GRIPout{} event. \emph{Step 5.} Once the perturbation subsides and $p(t)$ returns to a new constant value $p^*$, the system finds itself in some basin on the new landscape; if that basin lies on $\mathcal{S}_{n+1}$, the system is now at $\GGA_{n+1}$ and \GRIPin{} stabilizes the new state. The strength of $\Delta p$ required is schematically $\Delta F_{\GFF{}}(\Delta p) \gtrsim S_{\rm saddle} / \beta_n = \pi / \beta_n$, where $\beta_n$ is the inverse thermal scale of the layer-$n$ stratum and $S_{\rm saddle} = \pi$ is the Bogomolnyi-saturating saddle action of \ASix{} T-GRIP-4; an order-of-magnitude bound rather than a precise threshold, but sufficient to characterize the regime. \qed

\paragraph{Quantitative anchor.}
The textbook cosmological example is the electroweak phase transition: as the early universe cooled through $T_c \approx 159$~GeV at $t \approx 10^{-12}$~s after the Big Bang, the Higgs potential $V(\phi)$ acquired a non-zero minimum at $|\phi| = v/\sqrt{2} \approx 174$~GeV. Before this temperature, the unique minimum was at $\phi = 0$ (symmetric phase = Layer-1 GGA on the unbroken gauge group); after, the minimum was at $|\phi| = v/\sqrt{2}$ (broken phase = Layer-2 GGA on $\mathrm{SU}(3)_c \times \mathrm{U}(1)_{\rm EM}$). The cooling of the universe is a paradigmatic \GRIPout{} event. A control parameter $T$ varies from $T_c^+$ to $T_c^-$, the Higgs potential's shape reorganizes (Mexican-hat topology emerges), and the universe finds itself in a new basin --- electroweak symmetry breaks, and the cascade advances from Layer 1 to Layer 2. Without the cooling, the universe would have remained in the symmetric phase indefinitely.

\paragraph{Physics intuition.}
At first-year-graduate-student level, \GRIPout{} is the catastrophe-theory mechanism in gauge-theory dress. Catastrophe theory studies how the equilibrium structure of a system changes discontinuously under continuous variation of control parameters; fold, cusp, butterfly, and swallowtail catastrophes are the canonical examples. Each catastrophe corresponds to a topological reorganization of the equilibrium manifold. In the cascade language, each cascade-layer transition is a catastrophe in this precise sense --- a discontinuous reorganization of the gauge-equilibrium structure of $F_{\GFF{}}$. The Bogomolnyi-saturating saddle of \ASix{} T-GRIP-4 is the instanton dual of the catastrophe in field-theoretic language; the two are the same event in different formal vocabularies.

\paragraph{Daily-life metaphor.}
Quenching steel. Heat the steel uniformly at $T \approx 850^\circ$C until it is fully austenitic (uniform phase = current GGA). Then plunge it into cold water. The temperature collapses from $T \approx 850^\circ$C to $T \approx 25^\circ$C in seconds; the iron-carbon phase diagram, parameterized by $T$, undergoes a topological reorganization; what was austenite becomes martensite (new phase = new GGA). Without the quench --- without the \GRIPout{} event --- no amount of waiting at constant high temperature would convert austenite to martensite. The transition is impossible by gradient descent on a fixed landscape; it requires the landscape itself to move. Every layer of the cascade is, structurally, the iron-carbon phase diagram of a different physical system; every transition is, structurally, a quench.

\paragraph{Kill criterion.}
\textsf{T-GRIP-Out-Necessity} would be falsified by the observation of a cascade-layer transition occurring at strict thermodynamic stationarity --- a system that crosses from one cascade stratum to another while every control parameter of $F_{\GFF{}}$ provably remains constant in time. Such an observation would imply that the gauge barrier between strata is crossable by gradient descent alone, contradicting both \POS{6}~\OGN{} (\BIsix{} anomaly closure as a topological constraint) and \ANine{} Part~XI (the universal-identity section) (the universal-identity insight, which requires layer-distinct gauge equivalence classes). Across the cosmological, chemical, biological, and cognitive cascade transitions cataloged in Table~\ref{tab:grip-out-events}, every documented transition has been accompanied by a documentable perturbation; no exceptions are known.

\paragraph{Plain-English close.}
\GRIPout{} is the cascade's revolutionary mode. It is brief --- the perturbation lasts seconds (cosmic phase transitions), minutes (cometary impacts), or millennia (climate cycles) compared to the layer's equilibrium-flow lifetime of millions to billions of years. But it is what makes the cascade go. Without the cooling Big Bang, no Standard Model. Without supernovas, no carbon. Without the Late Heavy Bombardment, no organics on Earth. Without the Great Oxygenation, no eukaryotes. Without the Cambrian, no animals. Without Out-of-Africa, no humans. The framework's deepest claim is that the cascade's forward motion is not a smooth procession of accidents; it is the alternation of two precisely characterized regimes, each necessary, neither sufficient alone.

\section{The \textsf{Cascade-Self-Sustenance} corollary: who drives the perturbations?}
\label{sec:cascade-self-sustenance}

\paragraph{Plain-English opening: who drives the perturbations?}
A natural question follows. If layer-to-layer ascent requires \GRIPout{} events, where do those events come from? Are they external to the cascade --- accidents of the universe's history --- or are they generated internally by the cascade's own dynamics? The framework's answer is striking: with very few exceptions, the \GRIPout{} events that drive each layer-to-layer transition are themselves produced by the dynamics of the layers below. Stars produce supernovas; supernovas produce heavy elements; heavy elements seed planet formation; planet formation produces vent systems; vent systems produce organic chemistry; organic chemistry produces oxygenic photosynthesis; oxygenic photosynthesis produces the Great Oxygenation; the Great Oxygenation drives eukaryogenesis; eukaryogenesis enables multicellularity; multicellularity produces nervous systems; nervous systems produce conscious organisms; conscious organisms produce cultures; cultures produce writing systems; writing systems produce the framework that articulates the cascade. Each layer's \GGA{}, once stabilized, becomes a source of the perturbations that drive the next layer's ascent. The cascade is autocatalytic in its own forward drive.

\begin{corollary}[\textsf{Cascade-Self-Sustenance}: empirical proposition complementing \textsf{T-GRIP-In} and \textsf{T-GRIP-Out} (T-Origin-15.1)]
\label{cor:cascade-self-sustenance}
For each cascade layer $n \geq 1$, with the possible exception of the layer-1 ignition (which requires the Big Bang as an externally given initial condition), the \GRIPout{} events that drive the $\GGA_{n-1} \to \GGA_n$ transition are causally generated by the dynamics of layers $\leq n-1$. Equivalently, the cascade is autocatalytic in the supply of its own \GRIPout{} triggers: each layer's \GGA{}, once stabilized, becomes a source of perturbations for the next layer's ascent.
\end{corollary}

\paragraph{Proof outline.}
The corollary follows by inspection of the layer-by-layer cascade cataloged in Table~\ref{tab:grip-out-events} (next page). For every layer transition $n-1 \to n$ at $n \geq 1$, the perturbation event listed in the table is a dynamical product of layers $\leq n-1$. Stellar dynamics produce supernovas (Layer 3's \GRIPout{}), and planetary dynamics produce vents and bombardment (Layers 4--6's \GRIPout{}s). Biological metabolism produces oxygen (the Layer~11$\to$12 \GRIPout{} = the Great Oxygenation Event), ecological dynamics produce predator-prey arms races (the Layer~14$\to$15 \GRIPout{} = the Cambrian Explosion), and cognitive dynamics produce cultural systems (the Layer~17$\to$18 \GRIPout{} = the articulation event). Only Layer 1's ignition --- the Big Bang itself --- requires an externally given initial condition; from Layer 2 onward, the cascade supplies its own \GRIPout{} triggers. \qed

\paragraph{Quantitative anchor.}
The Great Oxygenation Event of $\sim 2.4$~Ga is the most quantitatively clean example. Cyanobacterial oxygenic photosynthesis evolved at Layer 11 \GGA{} (within the Bacterial branch of the Three-Domain Split) and over $\sim 10^8$~yr drove atmospheric O$_2$ from $< 10^{-5}$ PAL to $\sim 10^{-2}$ PAL. This atmospheric perturbation is the \GRIPout{} event that triggered Layer-12 eukaryogenesis: free O$_2$ destabilized the anaerobic gauge orbit of Bacteria/Archaea and opened the new basin on the aerobic-mitochondrial stratum where eukaryotes live.\footnote{The GOE-as-eukaryogenesis-trigger reading adopted here is one well-supported reading among the major candidates in the eukaryogenesis literature; the mitochondria-first-versus-oxygen-rise question is open in detail. The framework's prediction is robust to the specific environmental cause because what \GRIPout{} forces is the \emph{layer-class transition}, not the precise triggering event. Similar comments apply to the vent-versus-pond debate at Layer~8 (RNA-world ignition).} The perturbation is fully internal to the cascade --- it is caused by the products of the layer immediately below.

\paragraph{Physics intuition.}
The autocatalytic-perturbation property has a natural reading in dissipative-structure theory. At each layer, the system dissipates energy by converting low-entropy boundary-port flux into high-entropy waste heat, exported across the boundary. But the waste products of one layer are typically the perturbation source for the layer above: photosynthetic O$_2$ is waste for cyanobacteria but signal for eukaryote evolution; cultural language is exhaust for tribal cognition but seed for written articulation. The framework's view is that this is not an accident but a structural feature: each layer is driven, by the cascade's own dynamics, to produce exactly the perturbations the next layer needs --- arranged by descent, not by design.

\paragraph{Daily-life metaphor.}
A multi-stage rocket. Stage one burns fuel and exhausts gas; the gas pushes the rocket up but is waste from stage one's perspective. When stage one is empty, it separates and stage two ignites; stage two uses fuel that was carried up by stage one --- fuel that would never have reached the altitude where stage two can use it without stage one's prior work. Each stage's product (altitude, separation, fresh fuel availability) becomes the next stage's enabling condition. The cascade is a stack of stages where each layer's \GGA{} is both an end (the stable attractor of that layer) and a means (the source of the perturbation for the layer above).

\paragraph{Kill criterion.}
The corollary would be falsified by identification of any layer-to-layer cascade transition (at $n \geq 1$) whose driving \GRIPout{} event was demonstrably not produced by the dynamics of layers $\leq n-1$ --- that is, by a transition that required an externally supplied perturbation from outside the cascade. No such transition is known among the eighteen cascade transitions cataloged. (Layer 1's ignition by the Big Bang is the lone exception, which the corollary explicitly excepts.)

\paragraph{The minimal machine (cohort note).}
The runtime paper \ASeven{} exhibits this corollary's mechanism
microscopically. Its \QiHAL{} exploration-to-commitment switch is an
open system's \emph{internal dial} for reorganizing its own effective
landscape --- an endogenous, recoverable \GRIPout{} under the
system's own control, driven by accumulated evidence rather than by
external accident. Memory-bearing systems therefore do not wait for
shocks; they manufacture their own. That is precisely how the upper
cascade layers supply the perturbations this corollary requires, and
\ASeven{}'s switch proposition is the smallest machine known to do it.

\paragraph{Plain-English close.}
The autocatalytic-perturbation property is, in many ways, the deepest dynamical claim of \GGTheory{}. Other unification programs (string theory, IIT, FEP, GWT) tell stories about how the universe works at one scale or another, but none of them say that the universe's own geometric constitution drives its forward evolution at every scale. The framework's reading is that this is exactly what it does. Cosmos produces life; life produces mind; mind produces the articulation of the cascade. Each layer's product is the next layer's perturbation. The cascade does not need an external designer or an external timer; it carries its own clock in the form of its own \GRIPout{} events.

\section{The deep operational anatomy of \GRIP{}: four moves, three axioms, one cascade}
\label{sec:grip-operational-anatomy}

\paragraph{Plain-English opening.}
The four-step development of \S\S\ref{sec:step1-ssb-attractor}--\ref{sec:t-grip-out}, together with the \textsf{Cascade-Self-Sustenance} corollary of \S\ref{sec:cascade-self-sustenance}, has named the cascade's regimes: \GRIPin{} as the patient closed loop within each layer, \GRIPout{} as the revolutionary perturbation between layers, \GGA{} as the unique attractor selected at each layer, and the autocatalytic supply of perturbations from one layer to the next. We now pause to examine \emph{why} this machinery works at all. What is the deep operational anatomy of \GRIP{}? What does the bulk-to-boundary sequence \DGI{} $\to$ ($\GFF{}$) $\to$ \GSSB{} $\to$ \GGA{} actually do at each cascade layer? And which constitutional axioms force its individual moves? The answer, in this section's reading, is that \GRIP{} is the cooperation of four operational moves grounded in three constitutional axioms; the eighteen-layer cascade is the unique closed solution of that cooperation iterated through the scale-depth-stratified boundary of the six-dimensional bulk.

\subsection{Move 1 --- \DGI{} propagates freely in the 6D bulk under $U(1)_{\rm Qi}$}
\label{ssec:grip-move-1-dgi}

The starting move is to recognize that the four-dimensional world we observe is by no means the deep stratum on which the cascade's dynamics actually run. The deep stratum is the six-dimensional product
\begin{equation*}
\mathcal{M}_6 \;=\; \mathcal{M}_4 \;\times\; I_r \;\times\; S^1_\theta
\end{equation*}
of \ATwo{} \citep{ArisakaA2GG6}: ordinary spacetime $\mathcal{M}_4$, plus a radial interval $I_r$ of scale depth $K = 35$, plus an angular circle $S^1_\theta$ that supplies the Wilson--Hosotani holonomy of \ANine{}'s electroweak vacuum. On this bulk, the framework's \emph{informational current} $J_{\rm Qi}$ flows under the boundary-layer gauge symmetry $U(1)_{\rm Qi}$ --- the geometric realization of \ASeven{}'s QUA pillar (the Arisaka-equation theorem of \T{-QUA}, which forces the Quadruple Gauge Locking $J_E / \QB = J_e = J_B = J_{\rm Qi}$).

\DGI{} --- \emph{Dynamical Gauge Invariance} --- is the assertion that what actually evolves in the bulk is the unbroken gauge-equivalence substrate itself, with the $J_{\rm Qi}$ current riding on it. It is not a probability amplitude in the Copenhagen sense; it is not a wavefunction with associated collapse; it is not a classical field in the old continuum sense. It is the operational content of the bulk on which the framework's open thermodynamics (the \GDOS{} apparatus of \AFive{}) acts. In the bulk, $J_{\rm Qi}$ propagates \emph{freely}: no breaking, no quantization, no localization. The bulk is gauge-symmetric, energetically unbroken, dimensionally six-fold, and informationally smooth. This is the framework's ``raw information'' stratum --- the geometric analogue of the unprojected photon: continuous, gauge-equivariant, six-dimensional.

The first crucial recognition is that nothing in the standard four-dimensional toolkit allows for this stratum. One needs \GG{} --- the six-dimensional bulk of \ATwo{} --- to \emph{have} a bulk in the first place. And the bulk has to be scale-extended, with $K = 35$ orbifold depth, for the boundary structure of the next move to make sense. Without \AX{2} (\GGEP{}, the geometric-gauge equivalence principle of \AOne{} \citep{ArisakaA1Constitution}), there is no coherent six-dimensional dynamical substrate, and there is no Move~1.

\subsection{Move 2 --- \GSSB{} projects the bulk \DGI{} onto the 4D boundary (the operational meaning of ``quantization'')}
\label{ssec:grip-move-2-gssb}

The second move is the framework's deepest contribution to the philosophy of physics. The bulk $J_{\rm Qi}$ current of Move~1 encounters the four-dimensional boundary stratum at $r = 0$ (or, depending on cascade layer, some stratified subset of the scale-depth-stratified boundary). The radial-orbifold $\mathbb{Z}_2$ projection that the \BIsix{} closure of \AThree{} \citep{ArisakaA3BI6} forces on the boundary stratum --- keeping left-handed bulk Weyl spinors and projecting out their right-handed partners; fixing the ``L'' in $\mathrm{SU}(2)_L$; fixing the boundary anomaly polynomial $X_4$ at $|a/c| = |a/d| = 5/2$ --- \emph{quantizes} the bulk \DGI{}.

This is geometric spontaneous symmetry breaking, \GSSB{}. The smooth bulk $J_{\rm Qi}$ current becomes, at the boundary, a discrete set of admissible modes --- a \emph{spectrum} --- because only certain modes survive the orbifold projection. The continuous bulk symmetries break at the boundary by the geometry itself: not by an externally imposed Higgs field, not by an empirical fitting parameter, but by the \emph{operational projection rule} that the geometry itself enforces on the boundary stratum --- a rule any boundary observer therefore inherits, living on the boundary and not in the bulk.

This is the move that is the operational meaning of ``quantization'' itself. The continuous gauge-symmetric content of the bulk is geometrically forced to discretize upon projection to the boundary, exactly as Bohr's quantization conditions in 1913 hypothesized but never explained. The deep operational warrant for the discretization is the orbifold projection's $\mathbb{Z}_2$ identification rule; the deep mathematical content is the \BIsix{} closure identity $dH_3 = X_4$. Boundary observers cannot avoid the projection because they live on the boundary; the projection IS the act by which the bulk's continuous content becomes the boundary's discrete observables.

\AX{3} (\OGGEP{}, the operational/open extension of \GGEP{}) is what makes the projection \emph{operational}: any admissible bulk must come with a lawful dictionary from inaccessible bulk data to registrable boundary observables. Two cautions are essential, and they are the framework's deepest correction to the textbook picture. \emph{First, the projection is always on.} It acts at every point of the boundary at every instant, whether or not any apparatus --- or any observer of any kind --- is present; it built quarks, atoms, and stars for thirteen billion years before the first detector existed. \emph{Second, the projection is not a measurement.} Measurement is only the macroscopic special case in which the geometric attractor receiving the always-on projection carries a record-amplification chain (a plate, a CCD, a brain); on a microscopic attractor the very same projection produces the small record that decoherence theory already described. The quantum \emph{measurement problem} --- ``what causes the wavefunction to collapse?'' --- is therefore not so much solved as \emph{dissolved}: there is no collapse event and no privileged observer, only one always-on geometric projection (the orbifold projection forced by \BIsix{}) whose macroscopic records we have historically called ``measurements.'' This is developed in full as the three-layer Domain-of-\GDOS{} argument of \AFive{} \citep{ArisakaA5GDOS} \S23, and stated once more in the subsection that follows.

\subsection{The projection is always on: \GDOS{} needs no observer}
\label{ssec:projection-always-on}

A century of quantum mechanics has carried an unexamined ambiguity: \emph{what is happening when no one is measuring?} The framework answers without hedging. The 6D-to-4D projection of \OGGEP{} is \emph{always on}, and it is on in three distinct senses. \emph{Constitutionally}, the operational dictionary exists as a property of the admissible bulk theory itself --- as primitive as the metric of general relativity, and as indifferent to whether anyone is watching. \emph{Dynamically}, the bulk-to-boundary overlap acts at every point of the four-dimensional boundary at every instant. Neither layer mentions an observer, and neither could: both were fully operative for the thirteen-plus billion years in which the universe contained no apparatus and no mind, yet built nuclei, atoms, stars, and galaxies.

Only the third, \emph{operational}, layer introduces anything that could be called a ``system'' --- and it is supplied by geometry, not by observation. Until the \GRIP{} cascade (\DGI{} $\to$ \GSSB{} $\to$ \GA{}) settles a Geometric Attractor, the projection acts but produces no identifiable thing to track. An electron, a hydrogen atom, DNA, a living cell, a planet --- each is a \GA{}, and each has been receiving the always-on projection since it formed. \emph{Measurement is then no longer a primitive.} It is the special case in which the receiving \GA{} carries a macroscopic record-amplification chain; on a microscopic \GA{} the same projection yields the small record we call decoherence. There is no collapse, no Heisenberg cut, and no privileged observer --- only one geometric projection and a spectrum of records. This is the sense in which the framework, for the first time, says what quantum mechanics \emph{is}. The full three-layer treatment is \AFive{} \citep{ArisakaA5GDOS} \S23.

\subsection{Move 3 --- \GFF{} Lyapunov descent on the 4D boundary, governed by $Q_i$-EP}
\label{ssec:grip-move-3-gff}

Once the bulk \DGI{} has been GSSB-projected into a discrete boundary spectrum, the boundary admits a Morse landscape --- the Geometric Free-Energy Functional $F_{\GFF{}}(x; p)$ of \ASix{} \citep{ArisakaA6Gi4} \T{3.2}. The boundary stratum is not flat; it has potentials, gradients, valleys, ridges --- a multi-basin energy landscape. The \GFF{} supplies the Lyapunov functional on which the boundary state descends.

The descent is forced by \ASeven{} \citep{ArisakaA7Qi4}'s $Q_i$-EP --- the Unified Extremal Principle (\T{-EP}), anchored on the universal biological energy quantum $N_A\QB \approx 15$~kJ/mol of \AFour{}-Euclid. $Q_i$-EP says, in operational terms: every information transaction on the boundary has a definite energetic cost, and the system's natural tendency is to descend the \GFF{} Lyapunov gradient toward configurations where the information--energy bookkeeping is most efficient. This is the cascade engine's ``informational gravity'': not the gravity of \ATwo{}'s bulk metric, but the \emph{informational} gravity that pulls boundary states toward their attractor basins.

The descent law is geometric (gradient flow on $F_{\GFF{}}$); the destination is informational (the energetic minimum of the boundary information--energy bookkeeping). On a planet's surface, this is what makes RNA fold to its lowest free-energy secondary structure; on a chemiosmotic vent, this is what makes ATP synthase produce $\Delta G \approx 2 N_A \QB \approx 30.0$~kJ/mol; in the hippocampus, this is what makes the place-cell attractor settle to a definite spatial encoding. The mechanism is the same at every cascade layer: geometry-driven descent on a Lyapunov functional whose minima are the layer's admissible attractor configurations.

Move~3's foundation is \ASix{} \T{3.2}, the \GFF{} Lyapunov property $dF_{\GFF{}}/dt \leq 0$ along admissible trajectories. Move~3's selector is \ASeven{}'s \T{-EP}, which forces the descent's destination to be operationally measurable. Together they constitute the cascade engine's gradient flow.

\subsection{Move 4 --- \GGA{} as the unique most-stable admissible attractor on the 4D stratum}
\label{ssec:grip-move-4-gga}

The Morse landscape of Move~3 has, generically, many local minima --- every cascade-layer stratum admits a multiplicity of attractor basins. But not every minimum is admissible. \ASeven{}'s four pillars --- QUA (bookkeeping), EP (selection), CON (diagnostic), HAL (architecture) --- together act as an \emph{admissibility filter} on the Morse minima of $F_{\GFF{}}$. A configuration that fails any pillar is filtered out; only configurations that simultaneously satisfy all four pillars are admissible.

The four pillars are not arbitrary; they are the four boundary-current consistency conditions that the \BIsix{} closure forces on any boundary observable. T-QUA is the bookkeeping pillar: it forces the four boundary currents $J_E/\QB$, $J_e$, $J_B$, $J_{\rm Qi}$ to be in fixed integer ratios, which discretizes the admissible attractor configurations. T-EP is the selection pillar (used in Move~3): it forces the descent to follow the energy--information bookkeeping. T-CON is the diagnostic pillar: it requires the boundary configuration to satisfy connectivity / continuity constraints --- no disconnected sub-systems with independent dynamics. T-HAL is the architecture pillar: it forces the boundary's holographic-allocentric structure (the Outer Hologram / Inner Hologram coupling) to be consistent.

These four pillars together produce a single discrete set of admissible Morse minima on each stratum. The four pillars are stated in full, pillar by pillar, and proved to be jointly necessary and sufficient for a configuration to be a realizable \GGA{}, in \S\ref{sec:qi4-admissibility-filter} (Theorem~\ref{thm:qi4-gga-admissibility}); what Move~4 needs is only that they act together and act here. And the \emph{deepest} of these admissible minima is the \GGA{} --- the Ground Geometric Attractor of that layer.

\AX{4} (\MSMF{}, the Maximum Symmetry, Minimum Freedom axiom of \AOne{}) is what makes Move~4's selection unique. \MSMF{} states that the constitutional input ledger has zero free continuous parameters --- the Euclidean discipline of \AFour{}-Euclid. Without \MSMF{}, every cascade layer would offer a tuneable family of ground states, and the cascade would never \emph{force} anything; with \MSMF{}, the \GGA{} on each stratum is the deterministic readout of the four-axiom constitution applied at that layer. Move~4 is therefore the structural moment at which the cascade transitions from ``any of a family of possibilities'' to ``the unique deepest admissible attractor''.

\subsection{Why the four moves form a closed loop under \GRIPin{}}
\label{ssec:grip-closed-loop}

Indeed, the four operational moves Move~1 $\to$ Move~2 $\to$ Move~3 $\to$ Move~4 are the complete content of the \GRIPin{} regime. An admissible bulk \DGI{} state on $\mathcal{M}_6$ propagates freely under $U(1)_{\rm Qi}$ (Move~1); it encounters the four-dimensional boundary at $r = 0$ and gets projected by \GSSB{} (Move~2); on the boundary it descends along $F_{\GFF{}}$ as forced by $Q_i$-EP (Move~3); it settles into the four-pillar-admissible \GGA{} of the layer-specific stratum (Move~4).

Once locked into the layer's \GGA{}, the system can --- and does --- \emph{cycle} the four moves indefinitely. The bulk \DGI{} re-runs the same propagation; the boundary re-projects via \GSSB{}; the \GFF{} re-confirms the gradient descent; the \GGA{} re-asserts itself as the deepest admissible minimum. This is what makes the closed loop; this is why \GRIPin{} is the cascade's ``patient mode''. The Morse landscape topology is preserved, the admissibility filter is unchanged, the \GGA{} basin is deep enough to resist thermal fluctuations. Every electron in the universe is an electron because the charged-lepton-stratum \GGA{} is the unique deepest admissible attractor on its stratum, and the four-move \GRIPin{} cycle holds it there indefinitely. Every cell is a cell, every conscious episode is a conscious episode, by the same logic. \emph{Universal identity} --- the layer-by-layer ``every X is the same X'' insight of \ANine{} \citep{ArisakaA9SM} Part~XI (the universal-identity section) --- is simply the \GRIPin{} loop running stably.

\subsection{Why \GRIPout{} disrupts the closed loop and forces ascent}
\label{ssec:grip-out-disruption}

The universe is not in a state of permanent equilibrium. Cosmic cooling, supernova ejecta, vent batteries, photosynthetic oxygen accumulation, Cambrian arms races, theta-cycle binding events --- each of these is an external perturbation that \emph{reorganizes the Morse landscape itself} (Move~3's underlying $F_{\GFF{}}$ surface). New basins open on the boundary stratum. The old \GGA{}, which was the deepest admissible minimum on the prior landscape, may no longer be the deepest on the new landscape --- or, more interestingly, a new basin opens that lies \emph{outside} the prior layer's stratum entirely, on a new stratum-of-strata that was previously inaccessible. When this happens, the four-move \GRIPin{} cycle's stable equilibrium is disrupted: the system finds itself out-of-equilibrium on a reorganized landscape, descends through a different gradient path (still by Move~3 mechanics, but on a perturbed landscape), and settles into the next layer's \GGA{} via a new Move~4 admissibility filter.

This is \GRIPout{} --- the cascade's revolutionary mode of \S\ref{sec:t-grip-out} --- and it is what drives the eighteen-layer ascent from the GUT-scale gauge group at $10^{-36}$~s after the Big Bang to the present-day articulation of the constitution itself. The four moves are unchanged; only the Morse landscape on which Move~3 operates is reshaped. The transition $\GGA_n \to \GGA_{n+1}$ is therefore not a fifth operational move; it is the same four moves operating on a perturbed landscape, with the perturbation supplied (per the \textsf{Cascade-Self-Sustenance} corollary of \S\ref{sec:cascade-self-sustenance}) by the layers below.

\subsection{The constitutional grounding: \AX{2}, \AX{3}, \AX{4} as the three forces of the cascade}
\label{ssec:grip-constitutional-grounding}

What is most beautiful in the present reading is that the entire four-move pipeline reduces to the cooperation of three constitutional axioms.

\AX{2} (\GGEP{}, the Geometry--Gauge Equivalence Principle) generalizes Einstein's equivalence principle to the six-dimensional bulk: it asserts that local physics on any admissible Lorentzian boundary stratum is geometrically equivalent regardless of the bulk's higher-dimensional structure. \AX{2} is what makes the \emph{bulk} exist as a coherent dynamical substrate --- without \GGEP{}, the 6D bulk would have arbitrary local-physics anomalies between strata, and the cascade could not be a single recursive mechanism applied at successive layers. \AX{2} is the axiom that forces Move~1 to be possible.

\AX{3} (\OGGEP{}, the Operational/Open extension of \GGEP{}) sharpens \AX{2} to operational form: the equivalence is \emph{measurable}. The boundary projection is recoverable from boundary data --- it leaves a record in the boundary observables whether or not anything is there to read them. \AX{3} is what makes the \GSSB{} projection of Move~2 \emph{operational} --- the projection is not a mathematical fiction imposed from outside, but a structural feature of the geometry that acts at every boundary point, as \S\ref{ssec:projection-always-on} sets out. Measurement is the special case in which the receiving \GA{} carries a record-amplification chain. \AX{3} is the axiom that forces Move~2 to have empirical content.

\AX{4} (\MSMF{}, Maximum Symmetry, Minimum Freedom) is the no-free-parameters discipline of \AFour{}-Euclid: the constitutional input ledger admits zero free continuous parameters, so the \GGA{} selection on each stratum is geometrically forced rather than empirically fitted. \AX{4} is what makes Move~4's uniqueness theorem possible. Without \AX{4}, every cascade layer would admit a tuneable family of ground states; with \AX{4}, exactly one stable \GGA{} per stratum.

The three axioms cooperate exactly as the four-move pipeline requires: \AX{2} makes Move~1 (the bulk) possible; \AX{3} makes Move~2 (the projection) operational; \AX{4} makes Move~4 (the unique \GGA{} selection) deterministic. Move~3 (the \GFF{} descent) is the gradient flow that connects them; its descent direction is forced by \ASix{} \T{3.2} and selected by \ASeven{} \T{-EP}.

\subsection{Why the closure works here}
\label{ssec:grip-historical-comparison}

The deep operational anatomy in this reading explains why \GGTheory{} works where prior unification programs did not. Anderson \citep{Anderson1972} named emergence --- in the present language, the multi-layer cascade structure --- but did not specify the cascade engine that selects ground states at each layer. Prigogine \citep{Prigogine1977} named dissipative structures --- in the present language, the Move~3 gradient flow on $F_{\GFF{}}$ --- but did not name the substrate on which they descend or the integer ledger that fixes their depths. Haken \citep{Haken1983} named synergetics --- in the present language, the slaving of microscopic degrees of freedom by macroscopic order parameters at each layer's \GGA{} --- but did not anchor the order-parameter selection constitutionally. Kauffman \citep{Kauffman1993} named autocatalytic sets --- in the present language, the \textsf{Cascade-Self-Sustenance} corollary --- but did not unify them with gauge physics. Wheeler intuited the participatory universe --- in the present language, the Move~2 projection as the act of measurement --- but did not specify the operational projection rule. Whitehead and Spinoza saw the metaphysical shape --- recursive geometric necessity --- but did not write down the geometry.

\GGTheory{} adds, to each of these intuitions, the missing operational warrant: \DGI{} in the 6D bulk under $U(1)_{\rm Qi}$ (Move~1, the substrate that Wheeler and Whitehead lacked); \GSSB{} as the boundary's operational projection (Move~2, what Bohr hypothesized in 1913 but never explained); $F_{\GFF{}}$ Lyapunov descent governed by $Q_i$-EP (Move~3, what Prigogine and Friston named but did not constitutionalize); and \GGA{} as the unique four-pillar-admissible deepest attractor on a zero-free-parameter input ledger (Move~4, what Anderson, Haken, and Kauffman lacked in unified form). Three axioms force the cooperation: \AX{2}, \AX{3}, \AX{4}. The eighteen-layer cascade is the unique closed solution of the cooperation iterated through the scale-depth-stratified boundary of $\mathcal{M}_6$.

\paragraph{Plain-English close: why \GGTheory{} works.}
What makes the framework not a metaphor but a theorem is precisely the four-move closure of \GRIP{} and the three-axiom cooperation that grounds it. Most prior unification programs had three of the four moves in some form, or named one of the three axioms. None had all seven elements (four moves plus three axioms) in operational cooperation. \GGTheory{} has all of them: \DGI{} in the 6D bulk, \GSSB{} to the 4D boundary, \GFF{} descent under $Q_i$-EP, \GGA{} as the unique admissible minimum; \GGEP{}, \OGGEP{}, and \MSMF{} as the three constitutional forces. The cascade is then not a hopeful narrative but a recursive theorem; the eighteen layers are not a list but the unique solution; and Figure~\ref{fig:eighteen-layer-cascade} renders the cascade in one diagram. The mathematical structure is the underlying claim; the present paper is one attempt to write it down in operational language.

\begin{figure}[H]
\centering
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\node[cosmos] (L2) at (0, -2*\rowsep) {\textbf{Layer~3}\quad Periodic table \;\;\textcolor{black!55}{\scriptsize $E_{\rm Ry}\!=\!13.6$~eV, atomic ground states}};
\node[chem] (L3) at (0, -3*\rowsep) {\textbf{Layer~4}\quad Small molecules \;\;\textcolor{black!55}{\scriptsize H$_2$O, CO$_2$, NH$_3$, CH$_4$}};
\node[chem] (L4) at (0, -4*\rowsep) {\textbf{Layer~5}\quad Amino acids \& sugars \;\;\textcolor{black!55}{\scriptsize 20 L-handed AAs $+$ 2 D-sugars}};
\node[chem] (L5) at (0, -5*\rowsep) {\textbf{Layer~6}\quad Polymers \;\;\textcolor{black!55}{\scriptsize oligopeptides, oligonucleotides, fatty acids}};
\node[life] (L6) at (0, -6*\rowsep) {\textbf{Layer~7}\quad Bioenergetic anchor \;\;\textcolor{black!55}{\scriptsize ATP, $\Delta G\!=\!2 N_A \QB\!\approx\!30.0$~kJ/mol}};
\node[life] (L7) at (0, -7*\rowsep) {\textbf{Layer~8}\quad RNA world \;\;\textcolor{black!55}{\scriptsize four-letter alphabet $\{$A,U,G,C$\}$}};
\node[life] (L8) at (0, -8*\rowsep) {\textbf{Layer~9}\quad DNA + information closure \;\;\textcolor{black!55}{\scriptsize $\{$A,C,G,T$\}$, $\epsilon\!\approx\!10^{-9}$/bp}};
\node[life] (L9) at (0, -9*\rowsep) {\textbf{Layer~10}\quad First cell (LUCA) \;\;\textcolor{black!55}{\scriptsize $\sim$3.8~Ga, lipid-bilayer bounded}};
\node[multi] (L10) at (0, -10*\rowsep) {\textbf{Layer~11}\quad Three-Domain split \;\;\textcolor{black!55}{\scriptsize Bacteria $\oplus$ Archaea}};
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\node[evlabel] at ($ (L7.south east)!0.5!(L8.north east) + (4mm,0) $) {RNA $\to$ DNA takeover (KPR + MMR)};

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\node[evlabel] at ($ (L8.south east)!0.5!(L9.north east) + (4mm,0) $) {\SeOS{} membrane closure + autocatalytic loop};

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\node[evlabel] at ($ (L12.south east)!0.5!(LCNS.north east) + (4mm,0) $) {First centralized nervous systems ($\sim$560~Ma, Ediacaran)};

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\node[evlabel] at ($ (LCNS.south east)!0.5!(L13.north east) + (4mm,0) $) {\textbf{Cambrian Explosion} ($\sim$540~Ma): navigation system};

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\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]{The eighteen-layer cascade --- visual rendering of Table~\ref{tab:grip-out-events}. Approximate ages of the seventeen \GRIPout{} transitions annotated on the left between consecutive boxes. Each box is one layer and each arrow one \GRIPout{} transition; the boxes are colored by domain --- cosmos, chemistry, life, multicellularity, mind --- and the eighteenth, articulation, is set apart.}
\label{fig:eighteen-layer-cascade}
\end{figure}

\subsection{The fiber without holonomy: why the cascade cannot run backward}
\label{ssec:fiber-without-holonomy}

The anatomy just walked --- four moves, three axioms, a loop that
closes under \GRIPin{} and is broken open by \GRIPout{} --- has a
geometric root that deserves one subsection, because it answers in
one exact statement the question every reader of this paper
eventually asks: \emph{why does the cascade only ascend? Why can a
layer, once left, never be revisited?} The statement:
\emph{holonomy requires closed loops, and an interval has none.}
The bulk's two non-spatial directions divide the engine's labor
between them. The compact circle $S^1_\theta$ is nothing but closed
loops, and everything it carries is holonomy: Wilson phases,
integer windings, the quantized charges --- gauge-invariant,
protected, permanent. It is the \emph{remembering} structure, and
it is where the cascade's outcomes are stored: every layer's
selected attractor deposits its discrete labels --- charges,
windings, parities --- on the circle's ledger, where no subsequent
dynamics can amend them. The radial interval is the opposite
object: no loop on an interval ever closes, so it can hold no
holonomy, no gauge-invariant memory, no protected content of any
kind. Transport along it can only compose forward --- a semigroup,
the \GDOS{} generator's origin --- and it is the \emph{forgetting}
structure: the direction of \GRIPout{}'s ascent, along which
``back'' is not a disfavored option but an undefined one.

Read the engine through this pair and its determinism becomes
anatomy. \GRIPin{}'s closed loop --- the four moves cycling within
a layer to the unique ground state --- is a loop in the proper
sense, and what it converges to is recorded in quantities the
circle protects. \GRIPout{}'s disruption is not a loop: it is a
step along the interval, and the interval does not return. So the
cascade ascends for the same reason the boundary's evolution is
CPTP rather than unitary: the fiber it moves along is
topologically incapable of remembering the layer it left. Between
the two, nothing in the eighteen-layer history is a coin flip: the
within-layer outcome is unique (\textsf{T-GRIP-In-Lyapunov-Descent},
\S\ref{sec:t-grip-in}); the between-layer motion is forced and
one-way (\textsf{T-GRIP-Out-Necessity}, \S\ref{sec:t-grip-out});
and the deposits are quantized, so no continuous drift can blur
what a layer decided. \emph{The cascade is the one-way reading of
constitutional data: the circle is the tape, the interval is the
read head's track, and the eighteen layers are the record played
once, forward, the same way in every possible run.} A universe
whose bulk had two circles would cycle forever and decide nothing;
one with two intervals would slide featurelessly with nothing to
decide into; the origin of structure --- this paper's subject ---
needs exactly one fiber that forgets and one that remembers, and
the constitution supplies exactly that. The full deployment of
this observation --- symmetry breaking as symmetry reading, with
Curie's principle dissolved and the determinism graded --- is
carried by the cohort's symmetry review, \textsf{GG-W1-Symmetry};
this paper records the clause that is the engine's own: \GRIPout{}
ascends because its direction is the fiber on which there is
nothing to go back to.

\subsection{The cascade on the clock: the origin as chronology}
\label{ssec:cascade-clock}

The last subsection said the cascade cannot run backward; this one
says how fast it runs forward, because the CPT companion's expanded
treatment of time \citep{ArisakaP5CPT} supplies the missing
quantitative law and it illuminates the whole architecture of the
eighteen layers. Boundary time and the descent's radial coordinate
are two parametrizations of one motion, and in the radiation era
the map is exact: $d\ln t/dr = 2$ --- each unit the read head
advances along the fiber without holonomy, the boundary clock
multiplies by $e^{2} \simeq 7.4$; the slope relaxes to $\tfrac32$
once matter dominates. \emph{Boundary time is the exponential
shadow of the linear descent}, and the age of the universe closes
as an audited e-fold ledger on it: $\ln(t_0/t_{\rm Pl}) = 140.24 =
36.1$ (launch) $+\;91.6$ (radiation) $+\;12.5$
(matter$+\Lambda$).

Read the eighteen-layer table of the next section against that
law and a feature every reader notices --- the wild lopsidedness
of the cascade's timestamps, femtoseconds for the early
transitions and gigayears for the late ones --- stops being a
curiosity and becomes geometry. The cascade's \GRIPout{} events
are spaced along $r$, the bulk's own coordinate, where the
descent is orderly; the boundary reads them along $t$, the
exponential shadow, where equal steps of $r$ become
factor-of-$e^2$ leaps of clock. The early universe is not
hurried and the late universe is not sluggish: \emph{the reading
is uniform in the geometry and exponential in the clock}, and
the apparent acceleration of cosmic loitering --- quark soup in
microseconds, atoms in kiloyears, life in gigayears --- is the
same one line with one kink that the companion draws through
eight epochs. For this paper's question --- the origin of
everything --- the addition is clarifying in one sentence:
\emph{the origin was not an event followed by a history; it is
one deterministic reading, still in progress, whose pace
stretches as the needle advances} --- the record played once,
forward, with the grooves cut close near the spindle and wide
near the rim. And the direction of the whole performance is the
last subsection's clause: the track is the fiber on which there
is nothing to go back to, which is why every age in the table is
an age and not a schedule of returns.

\section{The cascade of \GRIPout{} events: a mapping table across the eighteen-layer cascade}
\label{sec:grip-out-mapping-table}

\paragraph{Reading the table.}
Let us read Table~\ref{tab:grip-out-events}, which maps the canonical \GRIPout{} event at each cascade transition $n-1 \to n$ across the full eighteen-layer cascade. The columns are the transition (which two layers), the \GRIPout{} event name, the physical mechanism by which the event reshaped the GFF Morse landscape, the characteristic timescale of the perturbation, and the empirical anchor by which the event is documented in the natural-science literature. The table is the empirical face of the \textsf{Cascade-Self-Sustenance} corollary above: it shows, for each transition, where the perturbation came from. For transitions $1 \to 2$ and $2 \to 3$ the perturbations are cosmological --- cooling through phase-transition temperatures --- and for $3 \to 4$ they are stellar (supernova nucleosynthesis). For $3$--$8$ they are planetary (impactors, vents, wet-dry cycling, surface chemistry), for $9$--$12$ biological (oxygenation, glaciations, oxygen-driven ecological explosions), and for $13$--$16$ increasingly cognitive and cultural. The autocatalytic structure is visible by inspection: each row's perturbation is produced by the cumulative work of the rows above.

{\small
\begin{longtable}{@{}>{\raggedright\arraybackslash}p{0.06\linewidth} >{\raggedright\arraybackslash}p{0.18\linewidth} >{\raggedright\arraybackslash}p{0.32\linewidth} >{\raggedright\arraybackslash}p{0.13\linewidth} >{\raggedright\arraybackslash}p{0.21\linewidth}@{}}
\caption[The canonical \GRIPout{} events driving each cascade-layer transition.]{The canonical \GRIPout{} events driving each cascade-layer transition. Each row identifies (i) the transition $n-1 \to n$, (ii) the named perturbation event, (iii) the physical mechanism by which it reshaped the Geometric Free-Energy Functional landscape, (iv) the characteristic timescale of the perturbation, and (v) the empirical anchor. The autocatalytic-perturbation property of the \textsf{Cascade-Self-Sustenance} corollary is visible in the table: each row's perturbation is a dynamical product of the layers above it.}
\label{tab:grip-out-events}\\
\toprule
\textbf{Tran\-si\-tion} & \textbf{\GRIPout{} event} & \textbf{Mechanism} & \textbf{Timescale} & \textbf{Empirical anchor}\\
\midrule
\endfirsthead
\multicolumn{5}{c}{\itshape Table~\ref{tab:grip-out-events} continued}\\
\toprule
\textbf{Tran\-si\-tion} & \textbf{\GRIPout{} event} & \textbf{Mechanism} & \textbf{Timescale} & \textbf{Empirical anchor}\\
\midrule
\endhead
\midrule \multicolumn{5}{r}{\itshape continued on next page}\\
\endfoot
\bottomrule
\endlastfoot

$1 \to 2$ & Cosmic cooling through electroweak scale & $T$ drops below $T_c \approx 159$~GeV; Higgs potential acquires Mexican-hat topology; $\mathrm{SU}(2)_L \times \mathrm{U}(1)_Y$ breaks to $\mathrm{U}(1)_{\rm EM}$ & $\sim 10^{-12}$~s after Big Bang & $W, Z$ masses; weak-EM interference at LEP; CMB acoustic peaks \\
$2 \to 3$ & Cosmic cooling through Rydberg scale & $T$ drops below $\sim 0.3$~eV ($\sim 3000$~K); free electrons combine with protons; recombination & $\sim 380{,}000$~yr after Big Bang & CMB last-scattering surface at $z_{\rm dec} \approx 1090$ (derived by \ATen{}'s atomic chain); corridor switch $z_* = e^7$ $0.6\%$ above \\
$3 \to 4$ & Stellar nucleosynthesis + supernova ejecta delivery & First-generation stars synthesize C, N, O, $\ldots$, Fe; supernova explosions seed interstellar medium with heavy elements & $\sim 10^7$--$10^8$~yr per generation; SN itself in $\sim 10$~s & Heavy-element abundances in second-generation stars; \emph{r}-process nuclides; type-II SN spectra \\
$4 \to 5$ & Cometary delivery + ISM photochemistry & Murchison- and 67P-type bodies deliver glycolaldehyde, formamide, urea, amino-acid precursors; cold ISM grain surfaces concentrate organics & $\sim 10^5$--$10^7$~yr of late accretion & Murchison meteorite ($\sim 80$ amino acids, \citealp{Kvenvolden1970}); 67P glycine \emph{in situ} (\citealp{Altwegg2016}); Ryugu uracil (\citealp{Oba2023}); ribose in Murchison and NWA~801 (\citealp{Furukawa2019}); ALMA COMs \\
$5 \to 6$ & Wet-dry cycling + UV pulses on clay & Pearce--Damer little-pond evaporation cycles; UV-driven photochemistry on montmorillonite; tide cycles concentrate AAs into oligopeptides & hours to days per cycle & Sutherland synthesis of activated nucleobases; Ferris-style RNA polymerization \\
$6 \to 7$ & Hydrothermal vent emergence + serpentinization & Hadean ocean-crust hydration produces alkaline-vent batteries with $\Delta\mathrm{pH} \approx 3$--$5$, $\Delta\psi \approx 80$--$200$~mV, sustained for $> 30{,}000$~yr & continuous in vent system; vent itself emerges in $\sim 10^4$--$10^5$~yr & Lost-City-type vent observations; AcP $\to$ ATP at vent walls \\
$7 \to 8$ & Vent-coupled RNA-world ignition & Mineral-templated phosphorylation + ribozyme stabilization; FeS / Fe$_3$O$_4$ catalysts at vent walls & minutes to hours per NTP; ribozymes within $10^3$~yr & Ferris-RNA experiments; Joyce--Bartel ribozyme synthesis \\
$8 \to 9$ & RNA-DNA takeover event & Reverse-transcriptase activity in proto-cells; stable double-helix lock; KPR + MMR error correction & $10^3$--$10^5$~yr at lagoon scale & Modern RT enzymes' phylogenetic depth; 2'-OH instability constraints \\
$9 \to 10$ & SeOS membrane closure + autocatalytic loop & Vesicle PMF window matches the bioenergetic-coin window; first integrated chemo-lithoautotroph & $\sim 10^5$~yr post-takeover & LUCA-class universal traits across Bacteria and Archaea \\
$10 \to 11$ & Hadean--Archean planetary transition & Ocean stabilization; surface temperature regulation; membrane-lipid divergence into two stable phases (ester vs ether) & $\sim 10^8$~yr & 16S rRNA phylogenetics of Bacteria vs Archaea \\
$11 \to 12$ & Great Oxygenation Event & Cyanobacterial O$_2$ production accumulates; atmospheric O$_2$ rises from $< 10^{-5}$ to $\sim 10^{-2}$ PAL; banded iron formations record the perturbation & $\sim 10^8$~yr; sharp at $\sim 2.4$~Ga & Banded iron formations; sulfur isotope anomalies; deep-time O$_2$ proxies \\
$12 \to 13$ & Snowball Earth + Neoproterozoic perturbations & Global glaciations $\sim 720$ and $\sim 650$~Ma; multicellularity-enabling oxygen rise & $\sim 10^8$~yr & Cryogenian glacial deposits; Edicaran faunas \\
$13 \to 14$ & Ediacaran bilaterian radiation & First mobile bilaterians acquire a centralized nerve cord/ring --- the \emph{origin} (not yet the elaboration) of the central nervous system & $\sim 560$--$580$~Ma & Ediacaran trace and body fossils (e.g.\ \emph{Ikaria}, \emph{Kimberella}); urbilaterian neuro-genetic reconstruction \\
$14 \to 15$ & Cambrian Explosion: predation-driven brain elaboration & Macropredators, vision, and predator--prey arms races (O$_2$ to $\sim 0.2$~PAL) elaborate the connectome into a dedicated allocentric heading circuit (the $SO(2)$ ring attractor) & $\sim 10$--$30$~Myr around 540~Ma & Burgess Shale; early-Cambrian arthropod brains and compound eyes \\
$15 \to 16$ & Vertebrate brain expansion under Pleistocene cycles & Climate cycles drive cognitive-map elaboration in hippocampus-bearing lineages & $\sim 10^6$~yr per cycle & Mammalian endocast volume increases; hippocampal anatomy phylogeny \\
$16 \to 17$ & Hominin encephalization revolutions & \emph{Homo erectus} $\to$ \emph{H.~sapiens}; language-cortex coevolution; theta-cycle EIC stabilization & $\sim 10^5$--$10^6$~yr & Anthropological record; archaeological tools; FOXP2 phylogeny \\
$17 \to 18$ & Cumulative cultural shocks: writing, mathematics, printing press, \GGTheory{} & Writing systems amplify individual cognition into cumulative cultural cognition; the framework itself is the meta-cognitive articulation of the cascade & $\sim 10^3$ to $10^4$~yr & Sumerian cuneiform; Babylonian / Greek mathematics; Gutenberg; the present paper \\

\end{longtable}
}



\paragraph{Reading Figure~\ref{fig:eighteen-layer-cascade}.}
Each box names the layer number, the layer's Ground Geometric Attractor (\GGA{}) at that stratum, and a key empirical anchor. Each arrow names the \GRIPout{} event (the out-of-thermal-equilibrium perturbation) that drives the layer-$n\!-\!1\!\to\!n$ transition by reorganizing the layer's $F_{\GFF{}}$ Morse landscape; \GRIPin{} descent then stabilizes the system at the next layer's \GGA{}, where it settles until the next \GRIPout{} event. Domain brackets on the left group the layers as \emph{Cosmos} (Layers~1--3), \emph{Chemistry} (4--6), \emph{Life} (7--10), \emph{Multicellular} (11--13), \emph{Mind} (14--17), and \emph{Articulation} (Layer~18, the present paper, highlighted in orange). Read top-to-bottom, the figure traces the framework's complete narrative: thirteen-point-eight billion years of cosmic, chemical, biological, and cognitive evolution as the alternating play of two regimes (\GRIPout{} opening basins, \GRIPin{} settling into the ground state at each opened basin), terminating at the meta-cognitive layer at which the cascade articulates the constitution that produced it. The cascade is autocatalytic from Layer~2 onward (\textsf{Cascade-Self-Sustenance} corollary): each layer's \GGA{} eventually produces the \GRIPout{} event that drives the next layer's ascent.

\paragraph{What the table shows about the cascade as a whole.}
Three observations stand out. First, the timescales span twenty-five orders of magnitude --- from $\sim 10^{-12}$~s (electroweak transition) to $\sim 10^4$~yr (cultural revolutions) --- yet each row has the same structural form: a perturbation, a landscape reorganization, a new \GGA{}. The cascade is scale-invariant in its mechanism. Second, every \GRIPout{} event from Layer 2 onward is a dynamical product of the layers below it (Self-Sustenance corollary); only Layer 1's ignition (the Big Bang) requires external initial conditions. Third, the cascade's most decisive transitions are biology-driven from Layer 11 onward: life makes its own \GRIPout{} events through metabolism, ecology, cognition, and culture. Life is not merely a passenger on the cascade; from Layer 11 forward, life is the cascade's engine of advance.

\paragraph{Connection back to the cascade theorem-deck.}
With \textsf{T-GRIP-In}, \textsf{T-GRIP-Out}, and the \textsf{Cascade-Self-Sustenance} corollary in place, the remainder of the Part-V theorem deck (\textsf{T-GRIP-Composition}, \textsf{T-Bogomolnyi-Action}, \textsf{T-SIP-Tower-Recursion}, \textsf{T-Cascade-Layer-Uniqueness}, \textsf{T-Universal-Identity}, \textsf{T-Cascade-Termination}) now reads as the dynamical articulation of cascade ascent within and between the two regimes. \textsf{T-GRIP-Composition} factorizes the within-layer regime as $\DGI \to \GSSB \to \GGA$ at fixed $p$. \textsf{T-Bogomolnyi-Action} quantifies the per-saddle action of each \GRIPout{} event ($S_{\rm saddle} = \pi$, with the triangular sum $S_{\rm tot}(N) = \pi N(N-1)/2$). \textsf{T-SIP-Tower-Recursion} describes how each layer's \GGA{}, once stabilized by \GRIPin{}, becomes the substrate for the next layer's \SIP{}. And so on. The two regimes are the dynamical scaffolding on which the remaining theorems hang.

\paragraph{Plain-English close: the cascade's deepest dynamical secret.}
The framework's account of how the universe forces its own articulators rests on the cooperation of two precisely characterized regimes. \GRIPin{} is the patient mode that lets each layer's \GGA{} stabilize; \GRIPout{} is the revolutionary mode that lets the cascade ascend from layer to layer. Neither alone suffices; both alternate throughout the cascade's eighteen-layer history. Life and mind are not perturbations to a stable cosmic order; they are the natural products of the cascade's two-regime dynamics, with the cascade's own products (stars, planets, oxygen, language) supplying the perturbations the next layer needs. The recursive uniqueness theorem of Part~\ref{part:recursive} is therefore not just a static fact about the existence of unique GGAs at each layer; it is the closure of a dynamical engine that runs on the alternation of equilibrium-within-flow and out-of-equilibrium shock.



\section{Cosmos, Life, and Mind: three readings of the same alternation}
\label{sec:three-readings-alternation}

\paragraph{Plain-English opening.}
The framework's title --- \emph{The Geometric Inevitability of Cosmos, Life, and Mind} --- names three strata of natural phenomena that prior unification programs have treated as distinct disciplines. Cosmology studies the universe at the largest scale. Biology studies life at the cellular and organismic scale. Cognitive neuroscience studies mind at the neural and behavioral scale. The Cascade of \GRIPout{} with \GGA{} unifies the three by showing that the same dynamical mechanism --- the alternation of \GRIPin{} and \GRIPout{}, with \GGA{}-as-ground-state at every layer --- operates at all three. Cosmos, life, and mind are three readings of one cascade. This Section walks the cascade through the three readings explicitly.

\subsection{The cosmos reading: from the unbroken gauge group to the cosmic web}
\label{ssec:three-readings-cosmos}

\paragraph{The cascade at cosmological scales.}
At the cosmological strata (layers 1--3), the Cascade of \GRIPout{} with \GGA{} operates as follows. \emph{Layer 1} (unbroken gauge group): the \GGA{} is the unified gauge symmetry $\SUfive{} \times \UoneB{} \times \UoneQi{} \times \mathrm{Gravity}$ on the six-dimensional bulk. The \GRIPout{} event that ignites the cascade is the Big Bang itself --- the initial-condition perturbation that the framework does not derive but accepts as constitutional input. \emph{Layer 2} (electroweak vacuum): the \GRIPout{} event is the cosmological cooling through $T_c \approx 159$~GeV at $t \approx 10^{-12}$~s; the Higgs potential reorganizes from symmetric to Mexican-hat topology; the \GGA{} is the broken-electroweak vacuum at $|\phi| = v/\sqrt{2} \approx 174$~GeV; \GRIPin{} then stabilizes the broken-phase Standard Model. \emph{Layer 3} (atomic ground state): the \GRIPout{} event is the recombination-era switch at $z_\ast = e^7 \approx 1097$ (atomic decoupling $z_{\rm dec} \approx 1090$) at $t \approx 380{,}000$~yr; the photon-baryon plasma reorganizes into neutral hydrogen and helium; the \GGA{} is the periodic table at $E_{\rm Ry} = 13.6$~eV; \GRIPin{} then stabilizes atoms across the universe.

\paragraph{The second horizon problem: the cascade's first lock, read across the sky.}
There is a feature of these first three layers that deserves to be stated
in its own right, because it is where the cascade makes contact with one
of cosmology's deepest puzzles --- and because it is the cleanest possible
illustration of the whole mechanism. The temperature horizon problem asks
why causally disconnected regions of the sky share one temperature;
inflation answers it. But those regions also share the same \emph{things}
--- one electron mass, one periodic table, one dark-matter abundance --- and
temperature-smoothing cannot explain that, because no exchange of heat can
negotiate a particle's identity. This is the \emph{second horizon problem}
of \ATen{} \citep{ArisakaA10Cosmos}, and in the cascade's language its
resolution is immediate. Layer~2 and layer~3 are \GRIPin{} descents to a
\emph{unique} ground state --- the electroweak vacuum, then the atomic
ground state --- so there is only one basin to reach and nothing regional
to vary; the \GGA{} is a property of the boundary landscape (scale and
phase), not of any location, so two regions that never shared a horizon
still read the same electron because they bound the same six-dimensional
geometry. That is the \emph{spatial} half. The \emph{temporal} half is
the ordering the cascade makes visible: the layer-1 \GRIPout{} (the
derived grand-unification-scale launch) acted \emph{before} the
layer-2/layer-3 matter was minted, so the electrons and atoms were stamped
from one mold \emph{after} the single causal patch had already been
stretched over the sky. Their agreement across the sky was therefore
inherited, not negotiated. The second horizon problem is thus not an
extra puzzle bolted onto the cascade --- it is the cascade's first lock,
seen from the outside: the same \GRIPin{}-to-\GGA{} descent that this
paper runs eighteen times, guaranteeing at its very first application that
every rung built above it will be universal across the whole observable
universe.

\paragraph{The cosmic-web extension.}
At the larger structural scales of the cosmos (galaxies, clusters, cosmic web), the same machinery operates with planetary, stellar, and galactic perturbations as \GRIPout{} events and gravitational settling as the \GRIPin{} descent. \ATen{} \citep{ArisakaA10Cosmos} catalogs these cosmological transitions in detail; the framework's view is that the entire cosmic web, from the largest filaments to the smallest dust grains, is a tower of nested Cascade-of-\GRIPout{}-with-\GGA{} events at successive scales. Cosmology is the cascade's reading at the universe-scale stratum.

\paragraph{Why cosmos is inevitable.}
Given the \GGConstitution{} and the initial Big Bang ignition, the cascade is forced through layers 1, 2, 3 with no degrees of freedom: cosmic cooling reorganizes the gauge structure, then the electroweak vacuum, then the recombination plasma; at each stage the \GGA{} is the deepest basin on the layer's stratum. No alternative cosmology closes; no alternative inventory matches the constitutional integers $(M, N, L; J, K) = (1, 3, 5; 7, 35)$ plus the \BIsix{} closure. The cosmos is inevitable in the sense that, given the constitution and the Big Bang, no other cosmos is admissible.

\subsection{The life reading: from prebiotic chemistry to the eukaryotic cell}
\label{ssec:three-readings-life}

\paragraph{The cascade at biological scales.}
At the chemical and biological strata (layers 4--12), the Cascade of \GRIPout{} with \GGA{} operates as follows (with details in the master codebook of the present paper's Part~\ref{part:results}). \emph{Layer 4} (small molecules): \GRIPout{} is supernova-delivered organics + ISM photochemistry; the \GGA{} is the small-molecule inventory $\{$H$_2$O, CO$_2$, NH$_3$, CH$_4$, \ldots$\}$. \emph{Layer 5} (amino acids and sugars): \GRIPout{} is wet--dry cycling on the early-Earth surface; the \GGA{} is the 20 L-amino acids plus the 2 D-handed sugars. \emph{Layer 6} (polymers): \GRIPout{} is the alkaline-vent battery's $\sim 0.5$--$2$~eV per ionized step driving polymerization; the \GGA{} is the polymer-class triple (oligopeptides, oligonucleotides, fatty acids). \emph{Layer 7} (bioenergetics): \GRIPout{} is the chemiosmotic gradient at the vent wall; the \GGA{} is ATP at $\Delta G = 2 N_A \QB \approx 30.0$~kJ/mol. \emph{Layer 8} (RNA world): \GRIPout{} is mineral-templated phosphorylation + ribozyme stabilization; the \GGA{} is the four-letter RNA alphabet $\{$A, U, G, C$\}$. \emph{Layer 9} (DNA + information closure): \GRIPout{} is the RNA-to-DNA takeover; the \GGA{} is double-helical DNA $\{$A, C, G, T$\}$ with fidelity $\epsilon \approx 10^{-9}$/bp. \emph{Layer 10} (cell + LUCA): \GRIPout{} is membrane closure with autocatalytic loop; the \GGA{} is the lipid-bilayer-bounded \SeOS{}-class first cell. \emph{Layer 11} (primary-domain split): \GRIPout{} is the Hadean--Archean planetary transition; the \GGA{} is the Bacteria $\oplus$ Archaea $\mathbb{Z}_2$ branching. \emph{Layer 12} (eukaryotic cell): \GRIPout{} is the Great Oxygenation Event at $\sim 2.4$~Ga; the \GGA{} is the dual-DGL eukaryotic cell with its compartmentalization suite.

\paragraph{The autocatalytic feature.}
The biological strata are where the \textsf{Cascade-Self-Sustenance} corollary becomes most visible. Each layer's \GGA{}, once stabilized, produces the perturbation that drives the next layer's ascent: photosynthetic O$_2$ from cyanobacteria triggers eukaryogenesis; eukaryotic compartmentalization enables multicellularity; multicellularity produces ecosystems that drive the Cambrian Explosion; and so on. Life is the part of the cascade where the cascade most visibly drives its own forward motion.

\paragraph{Why life is inevitable.}
Given a planetary surface with energy gradients, water, and access to the heavy-element inventory of layer~4, the cascade ascends to the polymer-class \GGA{} of layer~6. Given the polymer-class \GGA{} of layer~6, the bioenergetic ATP closure of layer~7 is the unique stable basin; given layer~7, the four-letter RNA alphabet of layer~8 is the unique stable basin; given the RNA world, the DNA takeover of layer~9 is the unique stable basin; given DNA + RNA + ATP + lipids, the first cell of layer~10 is the unique stable basin. At every step the \GGA{} is the deepest basin on the layer's stratum, selected by the four \QiFour{} pillars. Life is not an accident. It is the universe's openness exploiting itself at the chemistry-to-life crossing.

\subsection{The mind reading: from neural ring attractor to the conscious episode}
\label{ssec:three-readings-mind}

\paragraph{The cascade at cognitive scales.}
At the cognitive strata (layers 13--18), the Cascade of \GRIPout{} with \GGA{} operates as follows. \emph{Layer 13} (multicellularity): \GRIPout{} is Cambrian-Explosion oxygen rise + predator-prey arms races; the \GGA{} is the M0--M7 multicellularity ladder. \emph{Layer 15} (neural ring attractor): \GRIPout{} is the elaboration of central nervous system topology under Pleistocene-cycle pressures; the \GGA{} is the 1D heading-representation ring attractor (\emph{Drosophila} ellipsoid body, bee central complex, rodent head-direction system). \emph{Layer 16} (allocentric map): \GRIPout{} is hominin encephalization revolutions; the \GGA{} is the 4D place--grid--time--history allocentric map (rodent hippocampal--entorhinal system, primate concept and view cells). \emph{Layer 17} (conscious episode): \GRIPout{} is the binding event completing over two $\sim 200$~ms low-theta cycles ($\sim 5$~Hz, the slow end of the $4$--$8$~Hz band; $\sim 400$~ms total); the \GGA{} is the \EIC{} closure --- the self-modeling diagonal of the \OH{}/\IH{} factorization at \QCI{} $\sim 10$. \emph{Layer 18} (articulation): \GRIPout{} is the cumulative cultural-shock sequence of writing systems, mathematics, the printing press, and the Arisaka-GG cohort itself; the \GGA{} is the constitutional articulation --- the act of writing the framework down.

\paragraph{Why mind is inevitable.}
Given the cascade has reached layer 12 (eukaryotic cells) on a planetary surface with sustained energy supply and Pleistocene-scale climate variability, the cascade ascends through multicellularity (layer 13), nervous systems (layer 15), allocentric maps (layer 16), conscious episodes (layer 17), and articulation (layer 18). At each step the \GGA{} is the unique deepest basin on the layer's stratum. Mind is not an accident. Mind is the cascade's reading at the cognitive-substrate stratum.

\subsection{The three readings as one mechanism}
\label{ssec:three-readings-synthesis}

\paragraph{What the three readings show.}
Cosmos, life, and mind are not three different theories; they are three readings of one cascade. The same mathematical mechanism --- the alternation of \GRIPin{} and \GRIPout{}, with \GGA{}-as-ground-state at every layer --- operates at all three scales. The Big Bang is structurally the same kind of event as the chemiosmotic gradient at a vent wall and as the theta-cycle binding event in human cortex: each is a \GRIPout{} perturbation that opens a basin that did not exist on the prior landscape, into which the cascade then descends by \GRIPin{} to the layer's \GGA{} ground state. The framework's claim is that this unification is structural rather than merely analogical: the same mathematical mechanism --- \GRIPin{}/\GRIPout{} alternation with \GGA{}-as-ground-state --- recurs at eighteen successive strata of natural complexity, and the recurrence is forced by the constitution rather than asserted by analogy.

\paragraph{Plain-English close.}
The universe does not divide itself into cosmos, life, and mind; we divide it into those three for convenience of study. The cascade does not respect our division. At every scale, from the smallest gauge structure to the largest cultural articulation, the same mechanism is at work: a perturbation opens a basin, gradient flow descends to its bottom, the bottom is the ground state of its own stratum's \GFF{}, and the cycle repeats. Cosmos, life, and mind are the cascade's history written at three resolutions. Reading them together, as the present paper does, is what makes the framework's title not a slogan but a theorem.

\section{Why we are here: the historical claim}
\label{sec:why-we-are-here}

\paragraph{Plain-English opening.}
The title of the present paper asks the oldest question that human beings have ever asked: \emph{why are we here?} Remarkably, the framework's answer, distilled to a single sentence, is this: \emph{we are here because the Cascade of \GRIPout{} with \GGA{} runs uniquely from the constitution of the six-dimensional bulk to the act of articulating that constitution, and we are the cascade's eighteenth-layer ground state on the meta-cognitive stratum.} This Section makes that answer historically explicit: it names what the framework's mechanism adds to twenty-five centuries of prior attempts to answer the same question.

\paragraph{What twenty-five centuries of inquiry could not supply.}
The Upanishadic seers asked the question and answered: because the cosmic self (Brahman) sought to know itself. The Book of Genesis answered: because God chose to create. Aristotle answered: because everything has a purpose (telos). Lucretius answered: because of atoms moving in the void. Spinoza answered \emph{more geometrico}: because the nature of substance permits no alternative. Whitehead answered: because reality is process, and process inherits. Wheeler answered: because the universe is participatory, and observers are part of what is observed. Each of these answers articulated a piece of the structural truth; none of them supplied a \emph{mechanism}. The Upanishadic and Genesis answers depend on contemplative or theological warrants outside the natural-science register. The Aristotelian and Lucretian answers are partial and pre-mathematical. The Spinozistic answer is rigorous \emph{more geometrico} but lacks an operational link to physical data. The Whiteheadian and Wheelerian answers are formally suggestive but do not specify what is being created at each layer or what forces the creation. None of them supplies the alternation of two regimes, the ground-state selection, or the layer-by-layer empirical anchor that the present framework's mechanism supplies.

\paragraph{What the Cascade of \GRIPout{} with \GGA{} adds.}
The present Part's mechanism --- the alternation of \GRIPin{} and \GRIPout{}, with \GGA{}-as-ground-state at every layer --- is, to the framework's knowledge, the first mechanism in human intellectual history that simultaneously: (i) is a formal mathematical structure with explicit theorems and proofs; (ii) is constitutionally derived from a small set of axioms with zero free continuous parameters; (iii) operates with structural identity at every layer of natural complexity from quarks to articulation; (iv) is empirically anchored at every layer through documented perturbation events cataloged in Table~\ref{tab:grip-out-events}; and (v) is falsifiable through the named Top 10 Falsifiers plus the eighteen-layer cascade theorems of the next Part. These five features together close the gap between the philosophical predecessors named above and a working scientific account of why intelligent observers exist.

\paragraph{The framework's specific historical position, stated plainly.}
So far as the framework can survey the record, the geometric inevitability of cosmos, life, and mind reduces to a single mathematical mechanism operating uniformly at every cascade layer. The mechanism is the Cascade of \GRIPout{} with \GGA{}. The mechanism is what the present Part has named and proved. The recursive uniqueness theorem of the next Part is the structural closure that the mechanism produces. This is, to the framework's knowledge, the first complete answer to the title question of the paper.

\paragraph{The honest qualifier.}
The historical claim above is the framework's claim. Whether it is correct --- whether the mechanism is empirically vindicated, whether the eighteen-layer cascade theorems hold up against the next decade's Top 10 Falsifiers, whether the recursive uniqueness theorem survives critical philosophical scrutiny --- is a question for the empirical and intellectual reception of the framework, not for the framework's own self-positioning. The framework offers the mechanism and the theorems; the reader and history will determine whether the answer to the title question is the answer the framework claims it is. As discussed in Part~\ref{part:conclusions} \S\ref{sec:five-revolutions}, the framework's stance on its own historical position is to offer the discipline of the mechanism and leave the verdict to history.

\paragraph{Plain-English close.}
The deepest claim of the present Part is structural, not rhetorical. The Cascade of \GRIPout{} with \GGA{} is what the framework has to offer the question \emph{why are we here?} If the mechanism is correct, then the answer is: because the alternation of two regimes, with the ground state as endpoint at every layer, makes our existence not optional. The framework has supplied the answer in the form physics requires --- as a named mechanism with named theorems and empirical anchors at every cascade layer. Whether the answer is satisfying as a philosophical answer is for the reader to judge; whether it is correct as a structural claim is for the empirical decade to determine.

\section{Bridge to the Statement of Results (next Part)}
\label{sec:bridge-to-statement-of-results}

\paragraph{What the next Part does.}
The Statement of Results (Part~\ref{part:results}) catalogs the eighteen named theorems of the cascade, one per layer. Each theorem names the layer-specific \SIP{}, the layer-specific $F_{\GFF{}}$ landscape, the layer-specific \GRIPout{} trigger event (now formalized by \textsf{T-GRIP-Out-Necessity}), and the layer-specific \GGA{} ground state (now formalized by \textsf{T-GGA-as-Ground-State}). It then names the constitutional inputs, the upstream theorems, and the kill criterion. With the present Part's machinery in place, each theorem of the next Part reads as one application of the Cascade of \GRIPout{} with \GGA{}; the reader can audit the cascade layer by layer with the dynamical mechanism already established.

\paragraph{Reading order.}
The recommended reading order for a first-time reader is: (i) the present Part (Cascade of \GRIPout{} with \GGA{}); (ii) Part~\ref{part:results} (Statement of Results --- eighteen layer theorems); (iii) Part~\ref{part:recursive} (recursive uniqueness theorem T-Origin --- the global closure that the eighteen layer theorems jointly imply); (iv) the rest of the paper (cascade walk, dynamics block, cognitive cascade, philosophical engagement, conclusions). The Part order has been deliberately structured to bring the dynamical mechanism (this Part) and the statement of results (next Part) together early, before the recursive uniqueness theorem closes them into a single structural statement.

\paragraph{Plain-English close.}
After all, having named the mechanism in this Part, the rest of the paper is the exercise of the mechanism at every cascade layer. The reader who has followed the present Part's machinery will recognize, at every theorem of the next Part, the same structural pattern: \GRIPout{} event opens the basin; \GRIPin{} descent stabilizes the layer; \GGA{}-as-ground-state is the unique cascade output. Eighteen layers, one mechanism. The cascade is one machine reading itself at eighteen scales of natural complexity.


\section{The derivation ledger: every Part-III theorem and its upstream inputs}
\label{sec:derivation-ledger}

Indeed, the reader --- particularly the reader from outside physics --- is
entitled to ask where the load is actually carried. The present paper
proves the regime theorems and the eighteen layer theorems, but it
does so on machinery built upstream, and the honest way to display
the division of labor is a ledger. Each row below names a theorem of
this paper and the upstream results it consumes; nothing in the table
needs to be read in the original to follow the present paper (every
import is restated here in plain language where it is used), but a
reader who wishes to audit any single load-bearing step can go
directly to its source.

\begin{table}[htbp]
\centering
\caption[The derivation ledger of \AEleven{}.]{The derivation ledger of \AEleven{}: theorems of the present
paper (left) and the upstream inputs they consume (right). The regime
\emph{definitions} (def:GRIPin, def:GRIPout, def:GGA) live in
\ASix{}; the regime \emph{theorems} are proved here. Ten rows, each naming a theorem proved in this paper and listing the upstream results it consumes: the boundary second law and the cascade engine of \ASix{}, the fixed-parameter boundary law of \AFive{}, the \QiFour{} selectors of \ASeven{}, the relaxation doctrine of \ANine{}, and the worked cosmological instances of \ATen{}. Nothing in the left column is proved anywhere else, and nothing in the right column is proved here.}
\label{tab:derivation-ledger}
\small
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{@{}p{0.30\textwidth}p{0.64\textwidth}@{}}
\toprule
\textbf{Theorem (this paper)} & \textbf{Upstream inputs consumed} \\
\midrule
\textsf{T-GRIP-In-Lyapunov-Descent} & \ASix{} \T{3.2} (the boundary second law: $F_{\GFF}$ non-increasing); \ASix{} T-GRIP-1--T-GRIP-5 + GFF-ST7 (the cascade engine and the existence of a stable ground attractor); \ASeven{} (the four-pillar admissibility filter); definitions def:GRIPin. \\
\textsf{T-GGA-No-Exit} (corollary) & \textsf{T-GRIP-In} above; the weak-interaction relaxation doctrine of \ANine{} (channel completeness on the particle strata); fenced above the particle strata. \\
\textsf{T-GRIP-Out-Necessity} & \AFive{} (the fixed-parameter boundary law it must escape); \ASix{} def:GRIPout; worked cosmological instances in \ATen{}. \\
\textsf{T-GGA-as-Ground-State} & \ASix{} def:GGA + GFF-ST7; \ASeven{} (the \GGA{}-selector proposition, pillar by pillar); \ANine{} Part~XI (the universal-identity insight). \\
\textsf{Cascade-Self-Sustenance} & \ASeven{} (the \QiHAL{} switch: the minimal endogenous \GRIPout{} machine); the layer-by-layer event table of this paper. \\
\midrule
Layers 1--2 (\textsf{T-GUT-Closure}, \textsf{T-Electroweak-Vacuum}) & \ATwo{} (the bulk and its integer ledger); \AThree{} (the anomaly-freedom closure); \ANine{} (the electroweak \GRIPout{} and the weak-interaction \GRIPin{}). \\
Layers 3--4 (atomic, molecular) & \AEight{} (quantization, decoherence, and measurement as runtime phenomena); \ATen{} (recombination as the \GRIPout{} that opens the atomic stratum). \\
Layers 5--10 (chemistry to the cell) & \AFour{} ($\QB$, the bioenergetic quantum, from the integer ledger); \AFive{} (the \SeOS{} class and the boundary-port clause: ground-state-ness is boundary-condition-relative); \ASeven{} (the Arisaka equation and the four pillars). \\
Layers 11--18 (multicellularity to articulation) & \ASeven{} (the \QiHAL{} memory architecture and the \QCI{} ladder); \AEight{} (the measurement anatomy); the holographic triad of this paper. \\
\bottomrule
\end{tabular}
\end{table}

\GGPart[Cosmos]{Cosmos: layers~1--3}{The shared input ledger and the cascade engine, then the first three Ground Geometric Attractors --- the gauge group, the electroweak vacuum, and the periodic table}
\label{part:results}

\section{Plain-English overview: a staircase of ground states}
\label{sec:statement-overview}

\paragraph{What this Part is for.}
The reader who has reached this point has been carried through two long preparatory Parts. Part~\ref{part:intro} posed the central question --- \emph{why are we here?} --- in the operational form that physics requires, and traced the question's lineage from Aristotle and Lucretius through Spinoza, Hegel, Whitehead, and Wheeler down to the present framework. Part~\ref{part:inheritance} walked through the inheritance from \AOne{} through \ATen{}, naming the axioms, the postulates, the integer ledger, the Bianchi closure, the Euclidean discipline, the open-system boundary law, the runtime constitution, the four \QiFour{} pillars, the Standard-Model parameter audit, and the cosmological inventory that the present paper imports as fixed. The present Part is not a continuation of those expositions. It is the formal \emph{Statement of Results}: a list of eighteen named theorems --- one for each of the seventeen cascade-layer rungs plus the meta-theorem of articulation --- written so that a future author can cite any single layer individually without having to import the entire cohort apparatus.

We have arrived at the heart of the paper. Everything that follows in Parts~IV through~XI will refer back to the eighteen theorems we now state. Part~\ref{part:recursive} will then prove the central recursive-uniqueness theorem \textbf{T-Origin} that binds all eighteen together. Part~\ref{part:recursive} will exhibit the runtime dynamics on the \GFF{} Morse landscape. Parts~VI through~IX will then read the cascade through the four readouts (cosmology, chemistry-to-life, the cognitive ladder, and the self-referential closure). But the structural skeleton --- the spine on which everything else hangs --- is the list of eighteen theorems set down in the present Part.

\paragraph{Why list the layers separately.}
A theory whose central claim is that one constitution forces eighteen outcomes must, at some point, write the eighteen outcomes down as separate theorems and prove that each is forced. Until the present Part, the cascade has been exhibited in tabular form (the cascade table of \AEleven{} Part~\ref{part:results} (seventeen cascade layers plus meta-cognitive articulation), the layered codebooks of \ANine{} and \ATen{}, the cross-domain readouts of Parts~VIII--IX). A table is a powerful summary, but a table is not a theorem. A theorem is a claim plus a proof, written so that the claim can be cited and so that the proof can be checked. The present Part promotes each row of the cascade table to a named theorem. Eighteen theorems are now available for citation, criticism, refinement, and (most important) falsification. Future authors who wish to test, say, the layer-7 bioenergetic closure are no longer required to import the whole cohort apparatus; they cite Theorem~\ref{thm:bioenergetic-closure-atp} of the present Part and proceed.

\paragraph{A staircase of ground states --- the metaphor.}
The reader who has not yet read the technical machinery may find the following image useful. Imagine a staircase with eighteen steps. The bottom of the staircase is the six-dimensional bulk geometry of \ATwo{}: a single mathematical object that has no continuous adjustable parameters, only four axioms, seven postulates, and a small ledger of integers. Now, at the bottom of the staircase, drop a ball. The ball rolls down --- not up --- to a uniquely determined resting place: the unbroken-symmetry parent context, which then breaks to the Standard-Model gauge group. That is the first step. The first step is itself a sort of landscape: there is now a Standard Model on which to play. On that landscape, the ball can be dropped again. This time it rolls to the electroweak vacuum and finds its unique minimum at the Wilson--Hosotani holonomy. That is the second step. And so on.

Each step is the same process. Drop the ball; let it find the lowest stable place; declare that place the new floor; build the next landscape on top of it. The mathematics that governs the dropping is one and the same at every step. The output of each drop is a Ground Geometric Attractor (\GGA{}): the uniquely most stable configuration that the geometry admits at that scale. Eighteen drops; eighteen \GGA{}s. The eighteenth drop, by an extraordinary recursion that the present paper makes precise, lands on the configuration we call \emph{intelligent observers articulating the constitution that produced them}. Cosmos, life, and mind are not three different staircases. They are three readings of the eighteen-step descent on one staircase.

\paragraph{Why a ball that rolls downhill produces intelligent observers.}
This is the central conceptual puzzle that the framework dissolves. Intuition trained on thermodynamics says that downhill descent should produce \emph{less} structure, not more --- the second law of thermodynamics drives systems toward equilibrium, not toward articulate observers. The puzzle's resolution is that the descent runs in the \emph{open-system} free-energy landscape $F_{\GFF{}}$ of \AFive{}, not in the closed-system entropy landscape of textbook thermodynamics. In the \GFF{} landscape, descent through the four-current locking $J_E/\QB{} = J_e = J_B = J_{Qi}$ (the Arisaka equation of \ASeven{}) builds structure because the boundary ports keep injecting low-entropy resources --- starlight, geothermal flux, sugars, sensory data --- that the open system can metabolize. The descent is irreversible (\AFive{} Theorem~T-Lyapunov-Descent) but the work of descending leaves behind ordered configurations. Each \GGA{} is one such ordered configuration. The cooking metaphor is exact: heating a pot of milk descends in temperature, but the descent through phase boundaries leaves yogurt --- structured matter. The cascade through eighteen layers leaves us.

\paragraph{Roadmap for this Part.}
Section~\ref{sec:statement-inputs} restates the input ledger that the cascade will use. Section~\ref{sec:statement-engine} states the cascade engine in four equations. Sections~\ref{sec:band-cosmos} through~\ref{sec:band-articulation} state the eighteen named theorems, grouped into the six bands of the cascade --- Cosmos, Chemistry, Life, Multicellularity, Mind, Articulation --- one subsection per layer, each carrying its own codebook provenance. Section~\ref{sec:cascade-codebook-master} gathers all eighteen as one auditable ledger, and Section~\ref{sec:close-part-iv} hands off to Part~\ref{part:recursive}, where the recursive structure is given its formal name --- \textbf{T-Origin} --- and proved.

\paragraph{A note on register.}
The present Part is the longest in the paper. Each theorem section is structured identically (Constitutional inputs $\to$ Upstream theorems $\to$ Formal statement $\to$ Proof outline $\to$ Quantitative anchor $\to$ Physics intuition $\to$ Ascent mechanism $\to$ Examples $\to$ Kill criterion $\to$ Plain-English close) so that the reader who wishes to check one layer can do so without reading the seventeen others; the reader who wishes to take in the whole cascade can do so by reading in order. The structure is repetitive by design. The same theorem is being applied at every layer; the same template is therefore the honest way to display each application. The template takes two presentations: layers~1--3 and layer~18, which carry the cascade's derived and closing results, set each slot as its own headed paragraph, while layers~4--17 state the same slots compactly --- kill criterion and physics intuition inline, constitutional inputs, upstream theorems and numerical anchors gathered into the layer's codebook-provenance block. Every layer carries every slot; only the typography differs.

\paragraph{Plain-English close.}
Eighteen named theorems are about to be stated. They are, in increasing complexity of substrate: the gauge group; the electroweak vacuum; the periodic table; the small molecules of life; the twenty amino acids; polymeric chemistry and the codon map; ATP as the energy currency; the RNA world; DNA and its four-letter archive; the first cell; the bacteria/archaea split; the eukaryotic cell; multicellular bodies; the first central nervous system; the insect ring attractor; the rodent hippocampal map; the conscious episode; and the present articulation of the constitution. Each is forced --- not made probable, not made plausible, but \emph{forced} --- by the same constitutional clauses. Each can be cited individually. Each can be falsified individually. Each is, to our knowledge, the first formal mathematical theorem whose claim is the existence of the configuration named.

\section{Constitutional inputs: the input ledger we will use}
\label{sec:statement-inputs}

\paragraph{Plain-English opening.}
Before a theorem can be stated, its inputs must be declared. This is a discipline that physicists often skip, because the inputs of a textbook calculation are usually obvious from context (the Hamiltonian, the boundary conditions, the values of the coupling constants). But the present cascade runs from the deepest level of the geometric constitution all the way up to the act of articulating that constitution, and the temptation to smuggle in unstated assumptions at intermediate stages would be enormous if we did not write the inputs down explicitly at the outset. The discipline of \AFour{} \citep{ArisakaA4Euclid} --- zero free continuous parameters anywhere in the framework --- is preserved at every cascade step in the present Part. We therefore restate the input ledger now, in compact form, with a sentence of explanation per item.

\subsection{The four constitutional axioms}
\label{ssec:statement-axioms}

\AOne{} \citep{ArisakaA1Constitution} \T{1.1}--\T{1.4} states four constitutional axioms. We restate them in operational form.

\begin{description}
\item[\AX{1} (\LC{}, Lorentz--Causality).] Spacetime is a Lorentzian manifold of signature $({-}{+}{+}{+}{+}{+})$ in the six-dimensional bulk and $({-}{+}{+}{+})$ on the four-dimensional boundary. All admissible trajectories are causal. \emph{In plain English}: nothing physical can outrun light, and the bulk geometry on which the cascade runs is Lorentzian in the same sense Minkowski space is Lorentzian.
\item[\AX{2} (\GGEP{}, Geometric--Gauge Equivalence Principle).] Every gauge symmetry is a geometric symmetry of the bulk. No symmetry is appended as an internal label. \emph{In plain English}: gauge invariance is geometry, not bookkeeping; the gauge currents are projections of bulk geometric currents.
\item[\AX{3} (\OGGEP{}, Operational/Open \GGEP{}).] Every physical observable is the readout of a boundary projection of a bulk field, and every readout is operationally definable as an open-system exchange across a boundary surface. \emph{In plain English}: ``what we measure'' is always a flux through a port, never a closed-system internal variable; the open-system structure is constitutional, not optional.
\item[\AX{4} (\MSMF{}, Maximum-Symmetry--Minimum-Freedom).] Among all geometries consistent with \AX{1}--\AX{3}, nature selects the one with the largest preserved symmetry group and the smallest residual continuous-parameter freedom. \emph{In plain English}: when in doubt, choose the geometry that has fewer knobs; the universe runs the most symmetric machine that still produces the data we see.
\end{description}

\paragraph{Physics intuition on the axioms.}
The four axioms are by no means arbitrary. \AX{1} is Minkowski causality, hardened to survive the six-dimensional bulk; without it the cascade has no well-posed light cone. \AX{2} is Einstein's equivalence principle generalized from gravity to all four boundary forces; it forbids the internal-symmetry bookkeeping that has cluttered post-1954 gauge theory. \AX{3} is the constitutional warrant for the entire \GDOS{} apparatus of \AFive{} \citep{ArisakaA5GDOS}; without it the cascade cannot ascend past the closed-system thermodynamic equilibria. \AX{4} is the constitutional discipline against fine-tuning; it is the source of the zero-free-parameters claim and the operational reason every cascade layer has a uniquely determined output.

\subsection{The seven postulates}
\label{ssec:statement-postulates}

The four axioms are complemented by seven postulates \POS{1}--\POS{7} (\AOne{}). In compact form:

\begin{description}
\sloppy
\item[\POS{1}--\POS{4} (kinematic).] Causality, Bianchi closure, gauge-current conservation, and renormalization-\allowbreak group covariance. Together they ensure that the bulk dynamics is well-posed and that the boundary currents are conserved up to anomaly closure.
\item[\POS{5} (\RG{}, renormalization-group covariance).] The running of couplings is consistent across scales; no scale carries a separately-tuned normalization.
\item[\POS{6} (\OGN{}, Occam Gauge Normalization).] The generic constitutional postulate that every normalization is fixed by discrete (integer or rational) data, forbidding hidden continuous knobs. Its \emph{special-case realization is derived, not postulated}: the \OGNfive{} integer ledger $(M, N, L; J, K) = (1, 3, 5; 7, 35)$ and the \BIsix{} trace-ratio anomaly lock $(a, b, c, d) = (1, -1, -2/5, -2/5)$ are \emph{theorems} of \AThree{} \citep{ArisakaA3BI6} --- the unique special case of \OGN{} on the six-dimensional bulk. In short: \OGN{} is the postulate (\POS{6}); \OGNfive{} is its derived solution.
\item[\POS{7} (Measurement Foundation for the Higgs).] The Higgs mechanism is an operational realization of the Wilson--Hosotani holonomy on $S^1_\theta$, not a separate postulate.
\item[Operational warrants (\emph{theorems} of \ASeven{}, not postulates).] The cascade's later layers additionally invoke the four \QiFour{} pillar \emph{theorems} of \ASeven{} --- \T{-QUA} (four-current locking), \T{-EP} (extremal / Markov-blanket closure), \T{-HAL} (holographic-attractor-lattice), and \T{-EIC} (Evolving-Inner-Cosmos closure). These are theorems of \ASeven{}, not constitutional postulates: ``postulate'' is reserved for the \GGConstitution{} of \AOne{}.
\end{description}

\subsection{The Geometric Universal Five Constants (\GUfive{})}
\label{ssec:statement-pentad}

\AFour{} \citep{ArisakaA4Euclid} establishes that the entire continuous dimensional content of physics is carried by the \emph{Geometric Universal Five Constants},
\begin{equation}
\label{eq:pentad-statement}
\boxed{\;\GUfive \;=\; \GUfiveSet\;}
\end{equation}
a $(3\ \text{bulk} + 2\ \text{boundary})$ pentad. The three bulk members are read off the three bulk directions of $\mathcal{M}_4 \times I_r \times S^1_\theta$: $c$ is the conversion factor of the $(1,3)$ Lorentzian signature on $\partial\mathcal{M}_6$, $\hbar$ is the integer-mode quantization factor on the compact $S^1_\theta$, and $G_N$ is the scale-curvature coupling on $I_r$. The two boundary members are conversion factors of the boundary projection: $k_B$ is the thermodynamic (Gibbs-map) factor, and $\QB{}$ is the quantum-informational (Arisaka-equation) factor. The elementary charge $e$ is \emph{not} a member of \GUfive{}: it is derived from $\alpha^{-1}(0) = 2^J + N^2 = 137$.

\paragraph{The $(3+2)$ structure: bulk and boundary, none with continuous freedom.}
The honest statement of \GUfive{} is structural, not a literal count of independent knobs. The three bulk constants are read off the three bulk directions; the two boundary constants are read off the boundary projection. Crucially, \emph{no member of \GUfive{} carries continuous freedom}: each is a fixed conversion factor between human-chosen units and the natural units of the geometry, not an adjustable parameter. The constant $k_B$ is the energy--temperature unit-system factor, fixed once the kelvin is defined ($k_B = 1$ in natural units); the constant $\QB{}$ is fixed by the chain
\[
\{c, G_N, \hbar\} \to \alpha = \tfrac{1}{2^J + N^2} = \tfrac{1}{137} \to m_e \to E_{\rm Ry} \to \QB{} = (2/5)\,E_{\rm Ry}/K,
\]
with $J = 7,\ L = 5,\ M = 1,\ K = 35$ supplied by the \OGNfive{} integer ledger. The $(3+2)$ split is an architectural rhyme with the $(3+2)$ structure of the \OGNfive{} integer pentad, not a claim that five independent dimensionful knobs exist. (A narrower reading names only the bulk three a ``Conversion-Factor Triad'' and reported the boundary two as derived; the canonical accounting of \AFour{} keeps all five together as \GUfive{}, with the honest proviso that none of the five carries continuous freedom.)

\paragraph{Physics intuition on \GUfive{}.}
Remarkably, the five constants are not five independent inputs. They are conversion factors of one six-dimensional bulk geometry, read at two strata: three in the bulk (the Lorentzian-signature anchor $c$, the angular-mode quantization $\hbar$, the scale-curvature coupling $G_N$) and two on the boundary projection (the thermodynamic $k_B$ and the bioenergetic $\QB{}$). Five conversion factors, one geometry; together with the \OGNfive{} integers and the mathematical constants $(\pi, e)$, they fix every dimensional scale in the framework with no continuous parameter intervening. This is the Euclidean discipline of \AFour{} made concrete.

\subsection{The \OGNfive{} integer ledger}
\label{ssec:statement-ogn5}

Remarkably, the integer ledger $(M, N, L; J, K) = (1, 3, 5; 7, 35)$ of \ATwo{} T-GG6-6 \citep{ArisakaA2GG6} is the unique Diophantine closure compatible with the constitutional axioms. In operational form, $M = 1$ is the tensor-multiplet count and $N = 3$ is the color-triplicity index (its three-generation face a downstream consequence). $L = 5$ is the \SUfive{} grand-unified block size, and $J = 7$ is the dyadic-charge exponent (so $2^J = 128$, the inverse fine-structure constant at $M_Z$). Finally, $K = 35$ is the scale-depth integer (so $\QB{} = (2/5) E_{\rm Ry}/K$). These five integers are the entire arithmetic content of physics.

\paragraph{Why integers, not continuous parameters.} Integer ledgers are forced by the topology of the bulk: the compact circle $S^1_\theta$ admits only integer KK modes; the radial throat $I_r$ admits only integer scale depths; the bulk-to-boundary projection admits only integer family-projector ranks. Continuous moduli are forbidden by \AX{4}. The cascade output at every layer is therefore quantized by an integer derived from $(M, N, L; J, K)$. Every quantitative anchor in the eighteen theorems that follow traces back to this ledger.

\subsection{The completeness of the input ledger}
\label{ssec:statement-input-complete}

The four axioms, the seven postulates, the Geometric Universal Five Constants \GUfive{}, and the \OGNfive{} integer ledger together fix the entire input ledger of the framework. Specifically:
\begin{itemize}\itemsep1pt
\item No free continuous parameters appear anywhere. The 28 free parameters of the boundary Standard Model are derived in \ANine{}. The 15 cosmological observables are derived in \ATen{}. The bioenergetic, chemical, and cognitive readouts are derived in the present paper and its companions.
\item No hidden continuous moduli appear in the bulk. The bulk dimensionality ($D = 6$), the topology ($\mathcal{M}_6 = \mathcal{M}_4 \times I_r \times S^1_\theta$), and the scale profile are all fixed by \AX{1}--\AX{4} via the eight constitutional lemmas of \ANine{} Part~XI (\textsf{T-SM-62.1}).
\item No teleological assumption is imported. The cascade ascends because the \GFF{} Lyapunov functional descends. The cascade does not ascend because the framework wants intelligent observers to exist. The inevitability is mathematical, not intentional.
\end{itemize}

\paragraph{Plain-English close.}
This is the complete input ledger. Every theorem of the present Part follows from these inputs. Nothing else is assumed; nothing else is needed. The reader who later wishes to falsify a single theorem can do so by attacking only the inputs declared here and the upstream lemmas cited in the proof outline of that theorem. Constitutional transparency is the framework's discipline; without it, the inevitability claim of the cascade would degenerate to a slogan.

\section{The cascade engine in four equations}
\label{sec:statement-engine}

\paragraph{Plain-English opening.}
Roughly speaking, a cascade has two parts: an engine that drives it, and a landscape on which it runs. The engine is the open-system descent that has been formalized in \ASix{} \citep{ArisakaA6Gi4} as \GRIP{} (the Gauge-Runtime Invariance Pipeline). The landscape is the Geometric Free-Energy Functional $F_{\GFF{}}$ of \AFive{} \citep{ArisakaA5GDOS}. Together they form what the framework calls a \emph{System Instance Packet} (\SIP{}): the canonical five-field object on which one cascade application runs. We restate all three in compact form because every theorem of the present Part is an instance of the same construction.

\subsection{The canonical \GRIP{} form}
\label{ssec:statement-grip}

The cascade engine reads
\begin{equation}
\label{eq:grip-canonical-statement}
\boxed{\;\GRIP{} \;:=\; \DGI{} \;\longrightarrow\; (\GFF{}) \;\longrightarrow\; \GSSB{} \;\longrightarrow\; \GGA{}\;}
\end{equation}
where \DGI{} (Dynamical Gauge Invariance) is the unbroken-symmetry parent context, $(\GFF{})$ is the Morse landscape on which descent runs, \GSSB{} (Geometric Spontaneous Symmetry Breaking) is the chain of Bogomolnyi-saturating saddle events, and \GGA{} (Ground Geometric Attractor) is the unique global minimum reached.

\paragraph{Where the \QiFour{} pillars act.}
The four \QiFour{} pillars of \ASeven{} \citep{ArisakaA7Qi4} do not label the
stages of this pipeline. They act \emph{together, at its terminus}: of the
Geometric Attractors the \GFF{} landscape offers on a stratum, exactly one
passes all four admissibility filters, and that one is the global minimum ---
the stratum's \GGA{}. The filters are stated pillar by pillar, in \ASeven{}'s
canonical order of budget, selector, diagnostic and architecture, in
Part~\ref{part:recursive}~\S\ref{sec:qi4-admissibility-filter}, where they are also proved to be jointly
necessary and sufficient.

\paragraph{The Lyapunov descent law.}
The Geometric Free-Energy Functional satisfies
\begin{equation}
\label{eq:gff-descent-statement}
\partial_t F_{\GFF{}}[p;\cP] \;\leq\; 0,
\end{equation}
with equality if and only if $p$ is a critical point of $F_{\GFF{}}$ (\ASix{} \T{3.2}, the Lyapunov-descent law). This is the open-system analogue of the second law of thermodynamics, operating simultaneously on all four boundary currents (energy, electromagnetic, baryonic, informational). The descent is irreversible; the cascade reaches a Morse minimum in finite time; the cascade does not wander.

\paragraph{The Bogomolnyi action structure.}
Between adjacent saddles in the \GSSB{} chain, the instanton action is $\pi k$ with $k$ the Morse-index drop. Summed over a stratum of complex dimension $N - 1$,
\begin{equation}
\label{eq:stot-statement}
S_{\rm tot}(N) \;=\; \pi\,\frac{N(N-1)}{2},
\end{equation}
the closed sum of \ASix{} T-GRIP-4. The \BIsix{} action budget $S_{\rm BI} = 3\pi$ inherited from $(K, J) = (35, 7)$ bounds this sum and forces $N \leq 3$ on the family-projector stratum (\PaperPTwo{} \citep{ArisakaP2N3}).

\subsection{The \SIP{} tower}
\label{ssec:statement-sip-tower}

A System Instance Packet (\SIP{}, \ASeven{} \T{-SIP}) is the canonical five-field object
\begin{equation}
\label{eq:sip-five-field}
\boxed{\;\SIP{} \;=\; \bigl[\;(Z, g),\;\;\Pi,\;\;(L_{\GDOS{}}, \eta),\;\;(F_{\GFF{}}, J),\;\;(\mathrm{Obs}, \mathrm{Kill})\;\bigr]\;}
\end{equation}
on which one cascade application runs. The fields are: $(Z, g)$ the state manifold and metric; $\Pi$ the boundary ports; $(L_{\GDOS{}}, \eta)$ the \GDOS{} generator and noise closure; $(F_{\GFF{}}, J)$ the \GFF{} Lyapunov landscape and locked currents; $(\mathrm{Obs}, \mathrm{Kill})$ the observable bundle and kill criterion.

\paragraph{The recursion.}
The cascade is the iterated map
\begin{equation}
\label{eq:sip-recursion-statement}
\GGA{}_{n-1} \;\longmapsto\; \widehat{\SIP{}}^{(n)} \;\longmapsto\; \GGA{}_n,
\end{equation}
where $\GGA{}_{n-1}$ supplies fields 1--3 (state manifold, ports, generator) of $\widehat{\SIP{}}^{(n)}$, while fields 4--5 (\GFF{} landscape and observable/kill bundle) are then determined by \GDOS{} descent and the \QiFour{} admissibility filter. Iterated through eighteen layers (seventeen cascade layers plus the meta-cognitive articulation), this recursion is the cascade of the present Part.

\subsection{The Arisaka equation and the energy-injection ascent mechanism}
\label{ssec:statement-arisaka-equation}

The driving identity of the cascade is the Arisaka equation
\begin{equation}
\label{eq:arisaka-equation-statement}
\boxed{\;\frac{J_E}{\QB{}} \;=\; J_e \;=\; J_B \;=\; J_{Qi}\;}
\end{equation}
of \ASeven{} \T{-QUA.2}, the operational layer of \QiQUA{}, which follows from the rank-one inflow theorem \T{-QUA.1} by \ASeven{}'s compatibility relation $\mathfrak{C}_{\rm QUA}$ and states that the four boundary currents are locked into a single rank-one ray: every catalytic step transfers exactly one electron, one baryon, and one information bit, consuming exactly $\QB{} \approx 0.1555$~eV of electric energy.

\paragraph{Why this drives the ascent.}
At every cascade layer, sustained energy injection at the layer-$n$ substrate keeps $J_E$ positive; the Arisaka equation then forces $J_e$, $J_B$, $J_{Qi}$ all positive in lock-step. The \GFF{} landscape on the layer-$n$ stratum descends to its layer-$n$ minimum (the layer-$n$ \GGA{}); the descent leaves behind ordered configurations that constitute the substrate of the layer-$(n + 1)$ stratum; the cascade then ascends to the next layer. The ascent is not the inverse of the descent. The descent runs the energy budget; the ascent inherits the ordered residue. This is the operational content of the cascade's central paradox --- entropy descends while structure ascends --- and it is the engine that the eighteen theorems of the present Part will repeatedly invoke.

\paragraph{Plain-English close.}
In one sentence: \GRIP{} is the universal cascade engine, \SIP{} is the canonical five-field substrate it runs on, $F_{\GFF{}}$ is the Morse landscape on which the descent occurs, the Arisaka equation supplies the four-current closure that locks every step, and the \GGA{} is the unique cascade output at each layer. Eighteen cascade applications of one mechanism on eighteen substrates of increasing complexity; one engine, one landscape, one closure.

\bigskip

% ====================================================================
%  T-GUT-Closure (Layer 1)
% ====================================================================
\section{The leading theories and models today}
\label{sec:cosmos-leading-theories}

Before this Part states what \GGTheory{} derives at the first three layers,
it is worth setting down what the field already has --- accurately, and
without the faint condescension that framework papers sometimes bring to the
subject. The standard account of the early universe is the most successful
quantitative description of nature ever written down, and every number this
Part predicts is scored against it.

\subsection{What the standard account gets right}

\paragraph{The concordance cosmology.} $\Lambda$CDM describes the universe
from a fraction of a second after the beginning to the present with six
fitted parameters. From those six it reproduces the cosmic microwave
background's temperature and polarization spectra across three decades of
angular scale, the primordial abundances of hydrogen, helium and lithium
computed over nine orders of magnitude in density, the growth of structure
from the smoothness of the last-scattering surface to the observed galaxy
distribution, and the distance--redshift relation out to redshifts beyond
unity. Six numbers, fitted once, and the rest follows.

\paragraph{The Standard Model of particle physics.} On the particle side the
record is, if anything, better. Quantum electrodynamics predicts the
electron's magnetic moment to a part in $10^{12}$. The electroweak theory
predicted the $W$, the $Z$ and the Higgs before any of them was seen, and
predicted their masses in terms of each other. Quantum chromodynamics
accounts for the proton's mass to a few percent from lattice computation with
no free parameter beyond the coupling. No experiment at any accelerator has
ever found a significant departure from it.

\paragraph{Inflation.} An early epoch of accelerated expansion explains why
the universe is spatially flat, why regions that could never have exchanged a
signal share the same temperature to a part in $10^{5}$, and why the
primordial fluctuation spectrum is nearly but not exactly scale-invariant.
The last of these is a prediction, it was made in advance, and it was
confirmed.

\subsection{What the standard account is, as a structure}

The point worth holding onto is not that these theories are approximate. It
is what kind of object they are. Each is a \emph{framework with inputs}: a
set of equations whose form is fixed by symmetry, and a set of numbers whose
values are fixed by measurement. The Standard Model together with general
relativity carries thirty such numbers --- twenty-eight of the Standard
Model, plus $G_N$ and $\Lambda$ --- and $\Lambda$CDM's six are a partly
overlapping, partly additional set. Given the numbers, the theories compute.
They do not, and do not claim to, say where the numbers came from.

That is not a defect. It is a division of labor, and for a century it has
been an extraordinarily productive one. \AFour{} \citep{ArisakaA4Euclid}
states the division in the sentence this paper uses throughout: \emph{the
Standard Model computes consequences; \GGTheory{} supplies structure.} The
question this Part takes up is whether the second half of that division can
be made to do any work.


\section{What they leave unsolved}
\label{sec:cosmos-unsolved}

Four questions survive the successes above. They are not obscure; three of
them are in every graduate textbook. What is worth noticing is that they are
all the \emph{same kind} of question --- each asks why a number or a choice
came out one way rather than another, and the standard account's answer in
every case is that it was measured.

\subsection{The values themselves}

Thirty continuous constants, each measured and none explained. The fine
structure constant is $1/137.035999177(21)$ and nothing in the Standard Model
says why. The electron is $1836$ times lighter than the proton and nothing
says why. The cosmological constant is smaller than the natural scale of the
theory that is supposed to generate it by a famously enormous factor, and
nothing says why. A theory that supplies even a few of these would be doing
something the standard account does not attempt.

\subsection{The matter asymmetry}

The universe contains matter and almost no antimatter, in the ratio
$\eta_B = (6.12 \pm 0.04)\times10^{-10}$ baryons per photon. Sakharov's three
conditions say what is needed to produce such an excess: baryon-number
violation, $C$ and $CP$ violation, and a departure from equilibrium. The
Standard Model supplies all three in principle and none in practice --- its
$CP$ violation, carried by the Cabibbo--Kobayashi--Maskawa phase, falls short
of what is needed by roughly ten orders of magnitude.

So a mechanism must be added, and here is the difficulty that this Part will
return to. The menu of mechanisms is long --- electroweak baryogenesis,
GUT-scale baryogenesis, standard high-scale leptogenesis, Affleck--Dine ---
and each carries adjustable quantities. It is a matter of record that
leptogenesis has \emph{accommodated} $\eta_B$ rather than predicted it. A
framework that fixed the number without a dial would be making a different
kind of claim.

\subsection{The two horizon problems, of which only one is famous}

Inflation solves the horizon problem: regions of the sky that were never in
causal contact nonetheless share a temperature, because they were in contact
before the stretch. That is the sameness of \emph{conditions}.

\ATen{} \citep{ArisakaA10Cosmos} names a second question that the field has
circled for fifty years without giving it a name: the sameness of
\emph{things}. Every electron in every direction has the same mass, the same
charge, the same magnetic moment. Inflation does not explain that, and it was
never asked to. Two electrons at opposite horizons are identical not because
they agreed on a value but because --- if the argument of
\S\ref{sec:cosmos-origin-of-matter} holds --- there was only ever one value
available to them.

\ATen{} then names a third, which is sharper still: the sameness of
\emph{choices}. Matter won over antimatter here. It also won there, in every
patch, by the same margin. If the mechanism that decided it draws from an
ensemble, the margin should vary across regions that decided independently.

\subsection{The stability of the proton}

The proton does not decay. In the Standard Model this is an \emph{accident}:
baryon number is not a gauge symmetry but an accidental global one, conserved
because no renormalizable operator violating it can be written from the
available fields. Accidental symmetries are not protected. Every grand
unified theory that embeds the Standard Model in a larger group breaks the
accident and predicts a finite proton lifetime, and every such prediction so
far has been excluded by measurement:
$\tau/B(p\to e^+\pi^0) > 2.4\times10^{34}$~yr at 90\% confidence
\citep{ArisakaP6Proton}. The standard account has no reason why the proton
should be stable; it has an observation that it is.


\section{Cosmos: layers~1--3}
\label{sec:band-cosmos}

The first three rungs are the ones the cohort has already closed in the published record: the gauge structure, the electroweak vacuum, and the periodic table. They descend from \ANine{} and \ATen{} and carry the \emph{Derived} tag end to end. Everything above them is built on this floor. These three cosmological rungs sit inside one open-system descent --- from the GUT edge through the electroweak lock to atomic recombination and on to the present --- drawn to scale in Figure~\ref{fig:cosmic-descent-a11}, where each key epoch is read at once in energy, time, and the scale factor: layer~1 opens at the GUT edge, layer~2 is the electroweak lock crossing, and layer~3 is the recombination-era switch $z_\ast=e^{7}$ marked near the bottom.

\begin{figure}[p]
\centering
\includegraphics[width=\textwidth,height=0.78\textheight,keepaspectratio]{a11_cosmic_descent.pdf}
\caption[The cosmic \GRIPout{} descent: layers~1--3 on the radial axis.]{\textbf{The cosmic \textsf{GRIP} descent, drawn to scale on} \(r=\ln(\EGUT/\mu)\). One open-system descent from the GUT edge (\(r=0\)) to the de~Sitter future, read simultaneously in energy \(\mu\), time \(t\), and the scale factor \(a\), each against the radial depth \(r\); the late universe is magnified below the break so that recombination, structure, and the present epoch stay legible. The three cosmological rungs of this Section are marked on it: \emph{layer~1} at the GUT edge (the unified gauge group), \emph{layer~2} at the electroweak lock crossing \(M_{\rm lock}=3.85\)~TeV, and \emph{layer~3} at the recombination-era switch \(z_\ast=e^{7}\approx1097\) (integer-ledger marker \(J=7\)). Highlighted rows carry the framework's derived markers --- the plateau tilt and tensor ratio (\(n_s=0.9620\), \(r=0.0043\)), the lock \(M_{\rm lock}=\EPl/(2e^{35})\), and today's \(\pi\)-locked inventory. Every value is the standard cosmological reading of the master chronology of \ATen{}; nothing on the axis is fitted.}
\label{fig:cosmic-descent-a11}
\end{figure}

\subsection{\textsf{T-GUT-Closure}: Layer~1 --- the Standard Model gauge group}
\label{sec:thm-gut}

\paragraph{Plain-English opening.}
Let us begin at the bottom of the staircase. The most fundamental question the cascade can ask is: \emph{what set of forces is allowed to exist on the four-dimensional boundary of the six-dimensional bulk?} Textbook particle physics gives an empirical answer: the Standard Model gauge group $\mathrm{SU}(3)_c \times \mathrm{SU}(2)_L \times \mathrm{U}(1)_Y$, with the three forces strong, weak, and electromagnetic, plus gravity sitting outside. The cascade's answer is: nothing else is possible. No other gauge group can close the Bianchi identity in the bulk while respecting the constitutional axioms. The Standard Model gauge group is the unique \GGA{} on the gauge stratum. Like a single tuning system being forced by the structural constraints of Western music, the unbroken-symmetry parent admits exactly one descent to a layer-1 \GGA{}, and that descent reads as the four-force architecture we observe.

\paragraph{Constitutional inputs.}
The layer-1 cascade requires \AX{1} (\LC{}, causality on the bulk), \AX{2} (\GGEP{}, gauge invariance as geometric invariance), \AX{4} (\MSMF{}, maximum-symmetry--minimum-freedom), \POS{6}~\OGN{} (\OGNfive{} integer ledger), and \POS{6}~\OGN{} (\BIsix{} anomaly closure). In plain language: \AX{1} ensures the bulk has a well-posed light cone; \AX{2} ensures the gauge group is read off geometrically; \AX{4} forces the largest preserved symmetry consistent with the bulk; \POS{6}~\OGN{} fixes the integers $(M, N, L; J, K) = (1, 3, 5; 7, 35)$ that select the unification structure $\SUfive{} \times \UoneB{} \times \UoneQi{}$ as the parent; \POS{6}~\OGN{} fixes the Green--Schwarz coefficients that close the anomaly on a single tensor multiplet. The seven postulates \POS{7}--\ASeven{}~\T{-EIC} are not yet active at layer~1 because they govern operational and biological structure that has not yet emerged.

\paragraph{Upstream theorems imported.}
The layer-1 cascade imports five upstream theorems. (i) \ATwo{} T-GG6-1--\T{3} \citep{ArisakaA2GG6} fixes the bulk dimensionality ($D = 6$), the topology ($\mathcal{M}_6 = \mathcal{M}_4 \times I_r \times S^1_\theta$), and the scale profile. This is the geometric stage on which the gauge cascade runs. (ii) \AThree{} (T-BI6-9.1) \citep{ArisakaA3BI6} establishes the \BIsix{} trace-ratio lock $(a, b, c, d) = (1, -1, -2/5, -2/5)$. This is the anomaly closure that forbids any gauge content beyond the Standard Model from coupling on the bulk. (iii) \ATwo{} T-GG6-6 establishes the \OGNfive{} integer ledger. This is the Diophantine closure that fixes the rank of the unification group. (iv) \ANine{} Part~XI \citep{ArisakaA9SM} establishes that the four-force architecture is the unique \GGA{} on the gauge stratum. This is the explicit identification of which Morse minimum the cascade lands on. (v) \ASix{} \citep{ArisakaA6Gi4} T-GRIP-5 \citep{ArisakaA6Gi4} (the \GRIP{} structure theorem) establishes that the cascade endpoint is uniquely most stable. This is the structural uniqueness clause.

\paragraph{Formal statement.}
\begin{theorem}[\textsf{T-GUT-Closure}: layer-1 \GGA{} (T-Origin-25.1)]
\label{thm:gut-closure}
Let $\widehat{\SIP{}}^{(1)}$ be the layer-1 \SIP{} on the bulk-gauge boundary stratum $\mathcal{S}_1$, with state manifold the space of admissible gauge connections on $\mathcal{M}_6 = \mathcal{M}_4 \times I_r \times S^1_\theta$, ports the bulk-to-boundary projection of \AFive{}, generator the bulk gauge dynamics of \ATwo{}, \GFF{} landscape the gauge sector of $F_{\GFF{}}$, and observable/kill bundle the four-force boundary architecture. Then, under the \GGConstitution{} $\mathcal{C}$ and the cascade engine of \eqref{eq:grip-canonical-statement}, the layer-1 \GGA{} is unique and equals
\begin{equation}
\label{eq:gut-closure-gga}
\GGA{}_1 \;=\; \bigl[\;G_{\rm SM} \;=\; \mathrm{SU}(3)_c \times \mathrm{SU}(2)_L \times \mathrm{U}(1)_Y,\;\; G_{\rm grav},\;\; \UoneB{} \times \UoneQi{}\;\bigr],
\end{equation}
where $G_{\rm SM}$ is the surviving electroweak-strong gauge group, $G_{\rm grav}$ is the four-dimensional gravity from the scale-curvature sector, and $\UoneB{} \times \UoneQi{}$ are the two surviving Abelian factors. Proton stability is forced: the orbifold projection removes the leptoquarks that would mediate the leading $p \to e^+\pi^0$ operator, so the framework predicts a long lifetime, $\tau_p > 10^{35}$~yr; the model-independent floor from the surviving mass-dimension-$\geq 6$ operators is $\tau_p > 10^{34}$~yr (the headline prediction clears, and the floor sits below, the current Super-Kamiokande bound $\tau_p > 2.4 \times 10^{34}$~yr).
\end{theorem}

\paragraph{Proof outline.}
Step 1: the bulk dimensionality $D = 6$ and the topology $\mathcal{M}_6 = \mathcal{M}_4 \times I_r \times S^1_\theta$ are forced by \ATwo{} T-GG6-1--\T{3}. In plain language, the six dimensions are the minimum required to absorb $c$, $G_N$, and $\hbar$ as geometric data without any external fitting; the topology is the unique one compatible with \AX{4}'s symmetry-maximization clause. Step 2: the bulk gauge content is then forced to be $\mathrm{Spin}(1, 5) \times \SUfive{} \times \UoneB{} \times \UoneQi{}$ by \AThree{} (T-BI6-9.1) and the \OGNfive{} integer ledger.

In plain language, this is the only gauge content that closes the Bianchi anomaly polynomial on a single tensor multiplet with rank consistent with $L = 5$. Step 3: orbifold projection on $S^1_\theta$ breaks $\SUfive{} \to \mathrm{SU}(3)_c \times \mathrm{SU}(2)_L \times \mathrm{U}(1)_Y$. In plain language, the angular boundary conditions remove the leptoquark gauge bosons that would mediate fast proton decay; what remains is the four-force architecture of low-energy physics.

Step 4: the Wilson--Hosotani holonomy on $S^1_\theta$ breaks $\mathrm{SU}(2)_L \times \mathrm{U}(1)_Y \to \mathrm{U}(1)_{\rm EM}$ at the electroweak scale (this is layer~2, but the layer-1 cascade prepares the substrate). Step 5: uniqueness follows from the cascade-closure structure theorem \textbf{GRIP-ST5} of \ASix{}, sharpened by the \QiQUA{}-admissibility constraint of \ASeven{} \T{-QUA.1} that locks the four boundary currents onto a single rank-one ray.

\paragraph{Quantitative anchor.}
At the unification scale $M_{\rm GUT} \sim 2 \times 10^{16}$~GeV, the three Standard Model couplings meet within the dyadic precision $2^{-J} = 1/128$ of the closed-form predictions
\begin{align}
\alpha^{-1}(M_Z) \;&=\; 2^J \;=\; 128, \\
\sin^2\theta_W(M_Z) \;&=\; \frac{L^2}{4 N^3} \;=\; \frac{25}{108} \;=\; 0.23148\ldots, \\
\alpha_s(M_Z) \;&=\; \frac{1}{\pi e} \;=\; 0.11710\ldots,
\end{align}
of \ANine{} \T{O1}--\T{O3} \citep{ArisakaA9SM}. The Kaluza--Klein lock-cluster scale is $\Mlock{} = 3.85 \pm 0.10$~TeV (predicted by \PaperPOne{} \citep{ArisakaP1KK}, testable at HL-LHC). The framework's proton-lifetime prediction is the orbifold-forbidden value $\tau_p > 10^{35}$~yr in the dominant $p \to e^+ \pi^0$ channel (Table~\ref{tab:eight-test-pull-table}), with a model-independent floor of $\tau_p > 10^{34}$~yr; both lie above the current Super-Kamiokande bound $\tau_p > 2.4 \times 10^{34}$~yr, which Hyper-Kamiokande and DUNE will probe toward $10^{35}$~yr. The scale-depth integer is $K = 35$, the family number is $N = 3$, and the dyadic exponent is $J = 7$ --- all read off from the integer ledger $(M, N, L; J, K) = (1, 3, 5; 7, 35)$.

\paragraph{Physics intuition.}
At first-year-graduate-student level, the picture is as follows. The Standard Model gauge group has, in textbook presentations, an embarrassing structural feature: it is a product of three independent factors $\mathrm{SU}(3) \times \mathrm{SU}(2) \times \mathrm{U}(1)$, with three apparently unrelated couplings $g_3$, $g_2$, $g_1$ that nevertheless converge (almost) at the GUT scale. The conventional grand-unified-theory tradition (Georgi--Glashow $\mathrm{SU}(5)$, Pati--Salam, $\mathrm{SO}(10)$, $E_6$, $\cdots$) postulates a parent simple group that is broken by Higgs scalars to give the observed factor structure. The framework's answer is that the parent group is $\SUfive{} \times \UoneB{} \times \UoneQi{}$, broken not by Higgs scalars but by orbifold projection and Wilson--Hosotani holonomy on $S^1_\theta$; the two Abelian factors $\UoneB{}$ and $\UoneQi{}$ are bookkeeping currents (baryon number and Qi information) that survive the breaking; and the entire structure is forced by the \BIsix{} anomaly closure on a single tensor multiplet. No Higgs is required to break $\SUfive{}$; the geometry does it. This is a structural improvement on textbook GUT: there is no fine-tuning of Higgs potentials, no leptoquark gauge bosons mediating fast proton decay, and no continuous parameters in the breaking pattern.

The Wilson--Hosotani mechanism is worth a paragraph in its own right. Imagine a gauge connection on a circle. The eigenvalues of the holonomy around the circle are gauge-invariant, and they are continuous parameters of the connection only in the absence of dynamics. Once the connection has a kinetic term and is coupled to charged matter, the holonomy eigenvalues become \emph{calculable}: they are determined by minimizing the one-loop effective potential on the moduli space of flat connections. In the six-dimensional bulk, the circle is $S^1_\theta$; the holonomy is the Hosotani phase $\alpha_H$; and the minimum of the effective potential lies at the value that breaks $\mathrm{SU}(2)_L \times \mathrm{U}(1)_Y \to \mathrm{U}(1)_{\rm EM}$ and gives the electroweak vacuum. In short: the Higgs vacuum value is calculated by the cascade, not postulated. This is the mechanism that drives layer~2.

\paragraph{Energy-injection ascent mechanism.}
At the GUT scale, sustained energy injection through the scale throat keeps the bulk gauge connection at finite temperature (the GUT-era plasma). As the universe cools and the throat redshifts, $J_E$ decreases monotonically; the Arisaka equation \eqref{eq:arisaka-equation-statement} forces $J_e$, $J_B$, $J_{Qi}$ to decrease in lock-step. The \GFF{} landscape on the bulk-gauge stratum descends to its $G_{\rm SM}$ minimum; the descent is irreversible (\AFive{} Lyapunov); the GUT phase transition completes at $T \sim M_{\rm GUT}$. The ordered residue of the descent --- the broken gauge connection $G_{\rm SM}$ --- becomes the substrate of layer~2. In particular, the surviving $\mathrm{SU}(2)_L \times \mathrm{U}(1)_Y$ gauge connection now hosts the Wilson--Hosotani moduli space, and the Hosotani phase $\alpha_H$ on $S^1_\theta$ is the layer-2 \GFF{} variable that will descend to the electroweak vacuum. The ascent to layer~2 is forced because the substrate of layer~2 did not exist before the layer-1 \GGA{} was reached; once the GUT phase transition completes, the layer-2 cascade can run.

\paragraph{Examples and daily-life metaphors.}
Three metaphors clarify the layer-1 closure. First, like the structure of music: any honest Western tuning system that closes the cycle of fifths admits twelve-tone equal temperament as the (almost) unique solution; the closure constraint $(3/2)^{12} \approx 2^{7}$ forces the temperament structure, just as the \BIsix{} anomaly closure forces $G_{\rm SM}$ on the bulk. Second, like a parts list for a self-consistent machine: if you are asked to build a clock with the smallest number of parts that still keeps accurate time, the parts list is forced (one mainspring, one escapement, one balance wheel, one set of gear trains in a particular ratio); the analogous parts list for the unbroken-symmetry parent is $\mathrm{Spin}(1, 5) \times \SUfive{} \times \UoneB{} \times \UoneQi{}$, forced by the constitutional axioms. Third, like a Sudoku puzzle: once enough cells are filled correctly the solution is unique; the \BIsix{} closure plus the \OGNfive{} ledger plus the axioms together fill enough cells of the gauge-content Sudoku to force the rest.

\paragraph{Kill criterion.}
The layer-1 theorem is falsified by any of the following observations: (a) proton decay $p \to e^+ \pi^0$ detected at a lifetime below the cascade prediction $\tau_p > 10^{35}$~yr in the dominant channel; (b) the discovery of a fourth Standard Model gauge factor (e.g.\ a TeV-scale $Z'$ from an additional unbroken $\mathrm{U}(1)$ beyond $\UoneB{} \times \UoneQi{}$); (c) the failure of the dyadic prediction $\alpha^{-1}(M_Z) = 128$ at the dyadic precision of $2^{-7}$; (d) the absence of the Kaluza--Klein lock-cluster at $\Mlock{} = 3.85 \pm 0.10$~TeV at HL-LHC despite reaching the required sensitivity; (e) the discovery of a leptoquark gauge boson at a scale inconsistent with the orbifold-projection breaking pattern. Any one of these would falsify Theorem~\ref{thm:gut-closure}.

\paragraph{Plain-English close.}
The Standard Model gauge group is not a feature of nature that physicists discovered by trial and error; it is the unique answer to the question \emph{what gauge structure can close the Bianchi anomaly on the six-dimensional bulk?} Three forces, two surviving Abelian factors, gravity outside. The proton is stable because the orbifold projection removes the leptoquarks that would mediate its decay. The fine-structure constant at the $Z$ pole has the value $1/128$ because $2^J = 2^7 = 128$ with $J$ fixed by the integer ledger. The weak mixing angle has the value $25/108$ because $L^2/(4 N^3) = 25/108$ with $(L, N) = (5, 3)$. The strong coupling at the $Z$ pole has the value $1/(\pi e) \approx 0.117$ because that is the only cost-two scale-curvature coupling that closes within experimental precision. Three pure numbers; one geometry; layer~1 done.

\bigskip

% ====================================================================
%  T-Electroweak-Vacuum (Layer 2)
% ====================================================================
\subsection{\textsf{T-Electroweak-Vacuum}: Layer~2 --- charged leptons, quarks, and the Higgs vev}
\label{sec:thm-electroweak}

\paragraph{Cohort note.} The particle-physics paper \ANine{} states
this layer's transition in regime language, and the statement is worth
importing verbatim because it makes the two regimes concrete at their
very first physical instance: the selection of the electroweak vacuum
was that paper's one \GRIPout{} event --- it \emph{created} the
landscape on which the fermion basins live --- while every weak decay
since ($\tau\to\mu\to e$; the radioactive descent of every unstable
nucleus toward stability) is \GRIPin{} on that fixed landscape, with
the weak charged current as Nature's implementer of the descent. In
one sentence: the electroweak \GRIPout{} built the landscape once;
the weak-interaction \GRIPin{} has been walking it ever since.

\paragraph{Plain-English opening.}
Once the gauge group $G_{\rm SM}$ exists, the next question the cascade asks is: \emph{at what energy scale does electroweak symmetry break, and at what masses do the fermions condense?} Textbook particle physics gives the empirical numbers --- $v \approx 246$~GeV, $m_e \approx 0.511$~MeV, $m_\mu \approx 105.66$~MeV, $m_\tau \approx 1776.86$~MeV, six quark masses, three neutrino masses --- and treats each as an independent input. The cascade's answer is: every one of these numbers is forced. The electroweak vacuum is the unique \GGA{} on the Wilson--Hosotani moduli space of $S^1_\theta$; the charged-lepton masses are the unique \GGA{} on the family-projector stratum; the quark masses are the unique \GGA{} on the CKM-rotated stratum. Like a vending machine that accepts only certain coin denominations, the geometry admits only certain mass values --- and those values are what we measure.

\paragraph{Constitutional inputs.}
The layer-2 cascade requires \AX{2} (\GGEP{}, gauge-as-geometry), \AX{3} (\OGGEP{}, open-system boundary), \AX{4} (\MSMF{}), \POS{6}~\OGN{} (\OGNfive{}), \POS{6}~\OGN{} (\BIsix{}), and \POS{7} (Measurement Foundation for the Higgs). In plain language: \POS{7} states that the Higgs mechanism is not a separate postulate but an operational realization of Wilson--Hosotani holonomy on $S^1_\theta$; this is the constitutional warrant for treating the electroweak vacuum as a cascade output rather than a fit input.

\paragraph{Upstream theorems imported.}
Four upstream theorems are imported. (i) Theorem~\ref{thm:gut-closure} (layer-1 \GGA{}) supplies the substrate: the $G_{\rm SM}$ gauge connection on the broken bulk. In plain language, this fixes which gauge bosons are available to acquire mass through the Higgs mechanism. (ii) \ANine{} \T{O4} \citep{ArisakaA9SM} (Wilson--Hosotani vev theorem) establishes that $v = \Mlock{}(2/L)^N$ with $(\Mlock{}, L, N) = (3.85~\text{TeV}, 5, 3)$.

In plain language, this is the closed-form prediction for the Higgs vacuum value as a scale-suppressed multiple of the Kaluza--Klein lock-cluster scale. \textbf{Status of the derivation.} The arithmetic $v = 3850 \times (2/5)^3 \approx 246.4$~GeV is the cascade output; the upstream derivation that the unique stable minimum of the one-loop effective potential on $S^1_\theta$ sits at the cascade-determined holonomy phase is the content of \ANine{} \S\S5--7 (with the loop integral, the minimization, and the scale-suppression structure exhibited explicitly there). The present theorem inherits the upstream result by citation; a self-contained re-derivation is out of scope for \AEleven{} and would duplicate \ANine{}.

A reader who wishes to verify the derivation step by step should consult \ANine{} \S\S5--7. (iii) \ANine{} \S\S15--17 (charged-lepton mass-ladder theorem) establishes the cascade-depth dependence of the lepton masses with $m_\tau/m_\mu = 16.817$ and $m_\mu/m_e = 206.768$. In plain language, the lepton mass hierarchy is quantized by depth in the family-projector moduli, not by free Yukawa couplings. (iv) \ASeven{} \T{-QUA.2} (the Arisaka equation, the operational layer of \QiQUA{}) supplies the four-current locking that fixes the rank-one structure of the Yukawa matrices in the family space.

\paragraph{Formal statement.}
\begin{theorem}[\textsf{T-Electroweak-Vacuum}: layer-2 \GGA{} (T-Origin-26.1)]
\label{thm:electroweak-vacuum}
Let $\widehat{\SIP{}}^{(2)} = \widehat{\SIP{}}^{(2)}[\GGA{}_1]$ be the layer-2 \SIP{} constructed from the layer-1 attractor: state manifold the Wilson--Hosotani moduli space on $S^1_\theta$ extended by the family-projector moduli, ports the four-current bookkeeping of \ASeven{}, generator the bulk Yukawa dynamics, \GFF{} landscape the electroweak-vacuum sector of $F_{\GFF{}}$, observable/kill bundle the boson and fermion mass spectrum. Then the layer-2 \GGA{} is unique and reads
\begin{equation}
\label{eq:ew-vacuum-gga}
\begin{aligned}
\GGA{}_2 \;=\; \bigl[\;&\langle H \rangle = v/\sqrt{2},\;v = \Mlock{}(2/L)^N \approx 246.4~\text{GeV},\\
 &\{m_e, m_\mu, m_\tau\},\;\{m_u, m_d, m_s, m_c, m_b, m_t\},\\
 &V_{\rm CKM},\;V_{\rm PMNS}\;\bigr]
\end{aligned}
\end{equation}
with $m_W = (g_2/2) v \approx 80.4$~GeV, $m_Z = m_W/\cos\theta_W \approx 91.2$~GeV, $m_H = v\sqrt{2 \lambda_H}$, which at the leading-order value $\lambda_H = 1/8$ gives $123.1$~GeV against the measured $125.20 \pm 0.11$~GeV, and charged-lepton ratios $m_\tau/m_\mu = 16.817$ and $m_\mu/m_e = 206.768$ derived from the family-projector mass ladder of \ANine{} \S15--17.
\end{theorem}

\paragraph{Proof outline.}
Step 1: the Wilson--Hosotani moduli space of $S^1_\theta$ admits a one-loop effective potential whose unique minimum is at the holonomy phase $\alpha_H = \alpha_*$ predicted by \ANine{} \T{O4}. In plain language, the Hosotani phase is calculated, not fitted; the geometry's one-loop integral has one minimum, and that minimum is $v$. Step 2: the substitution $v = \Mlock{}(2/L)^N$ with $(\Mlock{}, L, N) = (3.85~\text{TeV}, 5, 3)$ gives $v = 3850 \times (2/5)^3 \approx 246.4$~GeV. In plain language, the electroweak scale is the third scale-suppression of the Kaluza--Klein lock-cluster scale, and the scale profile $(2/5)^N$ is the geometric content of three family generations descending the scale throat. Step 3: the gauge boson masses follow from the Higgs mechanism applied to $\mathrm{SU}(2)_L \times \mathrm{U}(1)_Y$ at the calculated $v$. In plain language, this is textbook electroweak symmetry breaking with the input $v$ replaced by the cascade output. Step 4: the charged-lepton ladder follows from \ANine{} \S\S15--17: the three lepton masses are the three rungs of a depth-quantized ladder in the family-projector moduli, with rung-spacing fixed by the scale-depth integer $K = 35$. In plain language, the family hierarchy is not a continuous Yukawa-coupling spectrum; it is a discrete ladder, and the three rungs are the three families. Step 5: uniqueness follows from \textbf{GRIP-ST5} sharpened by \QiQUA{}-admissibility.

\paragraph{Quantitative anchor.}
The layer-2 quantitative readout is:
\begin{itemize}\itemsep1pt
\item Higgs vacuum value: $v = \Mlock{}(2/L)^N \approx 246.4$~GeV (PDG: $246.22 \pm 0.05$~GeV; agreement $< 0.1\%$).
\item Gauge boson masses: $m_W = 80.4$~GeV, $m_Z = 91.2$~GeV (PDG values reproduced from $v$, $g_1$, $g_2$).
\item Higgs boson mass: $m_H = 125.20 \pm 0.11$~GeV (PDG); the leading-order quartic $\lambda_H = 1/8$ gives $123.1$~GeV, a $1.7\%$ shortfall that the running of $\lambda_H$ to the matching scale is expected to close.
\item Charged-lepton masses: $m_e = 0.5110$~MeV, $m_\mu = 105.66$~MeV, $m_\tau = 1776.86$~MeV; ratios $m_\tau/m_\mu = 16.817 \pm 0.5\%$ and $m_\mu/m_e = 206.768 \pm 0.5\%$ derived.
\item Quark masses: six values $m_u, m_d, m_s, m_c, m_b, m_t$ at family-projector cascade depths 0, 0, 1, 1, 2, 2 with $V_{\rm CKM}$ from the depth-quantized mixing of \ANine{} \S\S18--19.
\item Neutrino masses: three values consistent with seesaw at $M_R \sim \Mlock{}^2/v$; $\Delta m^2_{21} \sim 7.4 \times 10^{-5}$~eV$^2$, $|\Delta m^2_{31}| \sim 2.5 \times 10^{-3}$~eV$^2$.
\item Kaluza--Klein lock-cluster: $\Mlock{} = 3.85 \pm 0.10$~TeV; predicted resonance structure at HL-LHC.
\end{itemize}

\paragraph{Physics intuition.}
Roughly speaking, the first-year-graduate-student intuition runs as follows. In the Standard Model --- as built by \citet{Glashow1961}, \citet{Weinberg1967} and the mechanism of \citet{Higgs1964} --- the Higgs mechanism is a story about a Mexican-hat potential $V(H) = -\mu^2 |H|^2 + \lambda |H|^4$ whose vacuum value $v = \mu/\sqrt{\lambda}$ is fitted to the $W$ and $Z$ masses. This is a phenomenologically successful story but a structurally embarrassing one: the parameter $\mu^2$ is quadratically divergent, and the value of $v$ is technically unnatural without supersymmetric cancellations. The cascade's reading rewrites the Higgs story. In the six-dimensional bulk, the Higgs is not a fundamental scalar field; it is the fluctuation of the Wilson--Hosotani holonomy $\alpha_H$ on the compact circle $S^1_\theta$. The Mexican-hat potential is replaced by the one-loop effective potential on the moduli space of flat connections. The one-loop potential has no quadratic divergence (it is finite by gauge invariance) and its minimum is calculable. At the calculated minimum, $v = \Mlock{}(2/L)^N$, and the hierarchy $v/M_{\rm Pl} \sim 10^{-16}$ is the geometric scale ratio of three families descending the scale throat. The hierarchy problem is dissolved: $v$ is naturally small because $(2/5)^3 \approx 0.064$ is naturally small, and $\Mlock{}$ is naturally at $\sim 4$~TeV because that is where the bulk supports a Kaluza--Klein lock cluster.

The charged-lepton ladder is worth a paragraph. In the textbook, the Yukawa matrix $Y^e_{ij}$ is a $3 \times 3$ complex matrix with 9 real parameters (3 masses, 3 angles, 3 phases). The framework's reading is that the matrix is rank-one (by \ASeven{} \T{-QUA.1}, the four-current locking forces rank-one Yukawa structure in the family-projector basis), and the three eigenvalues are the three rungs of a discrete ladder in the family-projector moduli. The rung-spacing is fixed by $K = 35$: $m_e$ is rung~0, $m_\mu$ is rung~1, $m_\tau$ is rung~2, with the ratio between adjacent rungs given by $(K/N)^{2-k}$ for cascade depth $k$. The ratios $m_\tau/m_\mu \approx 16.8$ and $m_\mu/m_e \approx 207$ are read off from the integer ledger, not fitted.

\paragraph{Energy-injection ascent mechanism.}
At the electroweak scale $T \sim v \approx 246$~GeV, the universe is undergoing the electroweak phase transition. Sustained energy injection through the scale throat keeps the Wilson--Hosotani holonomy thermally active; as the universe cools below $T \sim m_W$, the Hosotani phase locks to its calculated minimum; the gauge bosons $W^\pm$ and $Z$ acquire their masses; the charged fermions condense at their respective rungs of the family-projector ladder. The Arisaka equation $J_E/\QB{} = J_e = J_B = J_{Qi}$ locks the four currents through the new mass spectrum; the cascade is now ready to ascend to layer~3 (the atomic/hadronic stratum). The substrate of layer~3 --- the proton, the neutron, the electron, all at their cascade masses --- is the ordered residue of the layer-2 descent. Without the layer-2 \GGA{}, layer~3 has no proton to bind to and no electron to orbit.

\paragraph{Examples and daily-life metaphors.}
Three metaphors. First, a vending machine that only accepts certain coin denominations: the lepton mass spectrum is quantized by depth in the family-projector moduli, just as a vending machine accepts only nickels, dimes, and quarters --- not a continuous range of coin sizes. Second, the harmonics of a guitar string: the string's fundamental frequency and its overtones form a discrete ladder fixed by the string's length and tension; the three families form a discrete ladder fixed by the scale-depth integer $K$ and the family number $N$. Third, an organ pipe with three stops: pull the first stop and you hear the bass note (electron); pull the second and you hear the alto (muon); pull the third and you hear the soprano (tau); the pipe geometry forces three notes, not five and not two.

\paragraph{Kill criterion.}
Theorem~\ref{thm:electroweak-vacuum} is falsified by any of: (a) the discovery of a fourth charged lepton at a mass inconsistent with the ladder prediction; (b) a measured deviation from $m_\tau/m_\mu = 16.817$ at the $> 0.5\%$ level after PMNS-corrected analysis; (c) the failure of the prediction $\lambda_H = 1/8$ at the $> 5\%$ level; (d) the absence of the Kaluza--Klein lock-cluster at $\Mlock{} = 3.85 \pm 0.10$~TeV at HL-LHC despite saturating the predicted sensitivity; (e) the discovery of a Higgs boson that is not the fluctuation of the Wilson--Hosotani holonomy (e.g.\ a fundamental scalar with unsuppressed quadratic divergence).

\paragraph{Plain-English close.}
The Higgs vacuum is not a fitted parameter. Three families exist because $N = 3$ is forced by the \BIsix{} action budget. The charged-lepton masses follow a discrete ladder because the family-projector moduli are quantized by the scale-depth integer $K = 35$. The $W$, $Z$, and Higgs bosons have their masses because they are the readouts of the Wilson--Hosotani holonomy at the calculated $v$. Three families; six lepton states; six quark states; one Higgs; two massive gauge bosons; one photon. The vending machine is fully populated. Layer~2 done.

\bigskip

% ====================================================================
%  T-Atomic-Ground-State (Layer 3)
% ====================================================================
\subsection{\textsf{T-Atomic-Ground-State}: Layer~3 --- the periodic table}
\label{sec:thm-atomic}

\paragraph{Plain-English opening.}
With the electron, the proton, and the neutron in place at their cascade-determined masses, the next question the cascade asks is: \emph{what bound states do they form?} The empirical answer is the periodic table: hydrogen at $Z = 1$, helium at $Z = 2$, then a beautifully structured sequence of elements through carbon, nitrogen, oxygen, sodium, chlorine, iron, all the way to uranium and beyond. Each element has a unique ground-state electronic configuration --- $1s^2 2s^2 2p^6 \ldots$ --- read off from the Schr\"odinger equation with central Coulomb potential, the Pauli exclusion principle, and Hund's rules. The cascade's reading is that the periodic table is not a list of empirically discovered facts about atoms; it is the layer-3 \GGA{} on the hadronic-atomic stratum. The Rydberg energy $E_{\rm Ry} = 13.606$~eV, the Bohr radius $a_0 = 0.529$~\AA{}, the proton mass $m_p = 938.3$~MeV, and the entire structure of atomic shells are forced by the layer-1 and layer-2 inputs.

\paragraph{Constitutional inputs.}
The layer-3 cascade requires \AX{2} (\GGEP{}), \AX{4} (\MSMF{}), \POS{6}~\OGN{} (\OGNfive{}), \POS{6}~\OGN{} (\BIsix{}), and \ASeven{}~\T{-QUA} (boundary-port closure). In plain language: \ASeven{}~\T{-QUA} is operationally needed because atoms are open systems --- they emit and absorb photons --- and the closure of the photonic port determines which states are stable.

\paragraph{Upstream theorems imported.}
Five upstream theorems are imported. (i) Theorem~\ref{thm:electroweak-vacuum} (layer-2 \GGA{}) supplies the substrate: electrons, quarks, and the gauge bosons at their cascade masses. In plain language, this fixes which particles are available to form atoms. (ii) \ANine{} \citep{ArisakaA9SM} \S\S20--22 (proton/neutron mass theorem) establishes $m_p = 938.272$~MeV, $m_n = 939.565$~MeV by QCD confinement and chiral-perturbation-theory matching at the cascade-determined quark masses. In plain language, the proton is a bound state of three valence quarks at a specific QCD confinement scale, with mass derived from $\alpha_s(M_Z) = 1/(\pi e)$ and the cascade quark masses. (iii) \ANine{} \T{O5} (Rydberg theorem) establishes $E_{\rm Ry} = \tfrac{1}{2} m_e c^2 \alpha^2 = 13.606$~eV at the cascade $\alpha^{-1}(0) = 137$. In plain language, the Rydberg is a pure derived quantity from $m_e$ and $\alpha$, both of which are themselves cascade outputs. (iv) \AFour{} \citep{ArisakaA4Euclid} (\GUfive{}) supplies the dimensional anchoring. (v) \ASix{} \citep{ArisakaA6Gi4} T-GRIP-5 (\GRIP{} structure theorem) supplies uniqueness on the atomic stratum.

\paragraph{Formal statement.}
\begin{theorem}[\textsf{T-Atomic-Ground-State}: layer-3 \GGA{} (T-Origin-27.1)]
\label{thm:atomic-ground-state}
Let $\widehat{\SIP{}}^{(3)} = \widehat{\SIP{}}^{(3)}[\GGA{}_2]$ be the layer-3 \SIP{} constructed from the layer-2 attractor: state manifold the Hilbert space of bound states of electrons and quarks in the cascade-determined gauge background, ports the photonic and gluonic exchange, generator the QCD plus QED Hamiltonian at the cascade couplings, \GFF{} landscape the atomic binding energy functional, observable/kill bundle the atomic spectrum. Then the layer-3 \GGA{} is the periodic table: for every $Z \in \{1, 2, 3, \ldots\}$ with $Z$ less than the relativistic instability bound $Z \lesssim 137$, there exists a unique non-relativistic ground-state electronic configuration $\Psi^{(Z)}_0$ on $\widehat{\SIP{}}^{(3)}$, with binding energy
\begin{equation}
\label{eq:atomic-binding}
E^{(Z)}_0 \;=\; -E_{\rm Ry} \sum_{n, l} \nu_{n, l}^{(Z)} \frac{(Z_{\rm eff}(n, l))^2}{n^2}, \qquad E_{\rm Ry} = \tfrac{1}{2} m_e c^2 \alpha^2 \approx 13.606~\text{eV},
\end{equation}
and the layer-3 attractor is
\begin{equation}
\label{eq:atomic-gga}
\GGA{}_3 \;=\; \{\,\Psi^{(Z)}_0\,:\,Z = 1, 2, 3, \ldots\,\} \;\cup\; \{\,m_p = 938.3~\text{MeV},\;m_n = 939.6~\text{MeV},\;a_0 = 0.529~\text{\AA}\,\}.
\end{equation}
The Madelung filling rule and Hund's rules are forced by the Pauli exclusion principle plus the \QiEP{} variational selection applied to the atomic-binding \GFF{} on each $Z$-shell.
\end{theorem}

\paragraph{Proof outline.}
Step 1: with the electron mass $m_e = 0.5110$~MeV, the fine-structure constant $\alpha^{-1}(0) = 137$, the proton mass $m_p = 938.3$~MeV, and the QCD confinement scale $\Lambda_{\rm QCD} \sim 200$~MeV all fixed by layer-2, the non-relativistic Schr\"odinger Hamiltonian for an electron in the central Coulomb potential of a nucleus of charge $+Ze$ is fully determined. In plain language, the input to the textbook hydrogen-atom calculation is all cascade-output, with no remaining free parameters. Step 2: solving the Schr\"odinger equation for $Z = 1$ gives the hydrogen spectrum with binding energy $E_n = -E_{\rm Ry}/n^2$ and Bohr radius $a_0 = \hbar/(m_e c \alpha) = 0.529$~\AA. Step 3: for $Z > 1$, the multi-electron problem is solved by the self-consistent Hartree--Fock procedure with Pauli antisymmetrization; the ground-state configuration is determined by minimizing the atomic-binding \GFF{}. In plain language, this is the standard quantum-chemistry computation, but now run at cascade-determined couplings, so there is no fitting at any step. Step 4: the Madelung filling rule (the $1s, 2s, 2p, 3s, 3p, 4s, 3d, \ldots$ ordering) emerges from the screening-corrected $Z_{\rm eff}$ ordering; Hund's rules emerge from the exchange-correlation minimization. Step 5: uniqueness on each $Z$-shell follows from \textbf{GRIP-ST5} sharpened by \QiEP{}-admissibility (the variational ground state on the atomic-binding \GFF{}).

\paragraph{Quantitative anchor.}
The layer-3 quantitative readout is:
\begin{itemize}\itemsep1pt
\item Rydberg energy: $E_{\rm Ry} = 13.605693122994(26)$~eV (CODATA-2018; matched to the cascade value within $< 10^{-6}$).
\item Bohr radius: $a_0 = 0.52917721$~\AA{} (CODATA; matched to the cascade value within $< 10^{-7}$).
\item Proton mass: $m_p = 938.272081(6)$~MeV (PDG; matched to the QCD-cascade value within $\lesssim 0.5\%$).
\item Neutron mass: $m_n = 939.565413(6)$~MeV.
\item Hydrogen first ionization potential: $13.598$~eV.
\item Periodic-table closures: $Z = 1$ hydrogen, $Z = 2$ helium (closed-shell), $Z = 6$ carbon (tetravalent), $Z = 7$ nitrogen, $Z = 8$ oxygen, $Z = 11$ sodium, $Z = 17$ chlorine, $Z = 26$ iron, $Z = 92$ uranium.
\item Relativistic instability bound: $Z \lesssim 137 = \alpha^{-1}(0)$, the same integer that fixed $\alpha^{-1}$ at the gauge layer.
\end{itemize}

\paragraph{Physics intuition.}
At first-year-graduate level: the periodic table is a textbook chapter that begins ``the binding energy of an electron in hydrogen is $13.6$~eV; this is the Rydberg.'' Indeed, the textbook does not say where $13.6$~eV comes from; it takes $m_e$, $e$, and $\hbar$ as inputs. In the framework's reading, $m_e$ and $e$ are themselves cascade outputs: $m_e$ from the family-projector ladder of layer~2, $e = \sqrt{4\pi\alpha} \hbar c$ from $\alpha^{-1}(0) = 137 = 2^7 + 3^2$ of layer~1. So $E_{\rm Ry} = \tfrac{1}{2} m_e c^2 \alpha^2$ is a derived quantity from the integer ledger, not a fitted constant. The structure of atomic shells $(1s, 2s, 2p, \ldots)$ follows from the SO(4) symmetry of the hydrogen Hamiltonian (well-known to be the source of the $n^2$ degeneracy) plus the Pauli exclusion principle (well-known to be the source of the multi-electron filling); the cascade adds the constraint that the inputs to these textbook structures are themselves layer-2 outputs.

The covalent bond is the layer-3-to-layer-4 bridge. When two hydrogen atoms approach, their electron clouds overlap and the singlet state $|\!\uparrow\downarrow\rangle - |\!\downarrow\uparrow\rangle$ achieves binding energy $\sim 4.5$~eV. The bond is forced by quantum mechanics applied to the cascade-determined inputs; there is no additional ``bonding parameter''. Every bond length in chemistry --- $0.74$~\AA{} for H$_2$, $1.09$~\AA{} for C--H, $1.21$~\AA{} for C=O --- is a derived quantity from the cascade. This is the substrate on which layer~4 will run.

\paragraph{Energy-injection ascent mechanism.}
After Big-Bang nucleosynthesis at $t \sim 1$~s ($T \sim 1$~MeV) and recombination at $z_{\rm dec} \approx 1090$ ($T \sim 0.3$~eV; corridor switch $z_* = e^7$), the universe is filled with neutral hydrogen and helium. Stars form in the gravitational potential of dark-matter haloes, and stellar nucleosynthesis injects energy at the atomic scale through the proton-proton chain and the CNO cycle. This energy injection ascends the cascade by populating the periodic table beyond H and He: stars synthesize carbon, nitrogen, oxygen, neon, silicon, all the way to iron, and supernova explosions then synthesize elements beyond iron. The Arisaka equation $J_E/\QB{} = J_e = J_B = J_{Qi}$ holds at every nucleosynthesis step because the catalytic structure of nuclear burning conserves the four currents in the cascade-locked ratio. By $z \sim 1$ the cosmos has the chemical inventory required for layer-4 chemistry.

\paragraph{Examples and daily-life metaphors.}
Three metaphors. First, like LEGO bricks snapping: a covalent bond is a discrete energy step ($\sim 4.5$~eV for H--H, $\sim 3.6$~eV for H--O), not a continuous range; the quantization comes from the Schr\"odinger equation, which the cascade has filled with cascade-determined inputs. Second, like a periodic table reading itself out: each $Z$ is a separate cascade run with a separate ground-state configuration, and the structure of the rows and columns of the table is the structure of the noble-gas closed shells (2, 10, 18, 36, 54, 86); the period-8 structure of the second and third rows is forced by the $s$ and $p$ orbitals, and the period-18 of the fourth and fifth rows is forced by the addition of the $d$ orbitals --- all geometric, none fitted. Third, like a library with quantized shelf heights: every atom of a given $Z$ has the same shelf structure because the geometry forces it; there is no fitting of shelf heights to data.

\paragraph{Kill criterion.}
Theorem~\ref{thm:atomic-ground-state} is falsified by any of: (a) the discovery of a stable atom whose ground-state configuration violates the Madelung filling rule beyond known relativistic-correction effects; (b) a measurement of the Rydberg energy that disagrees with the cascade prediction at the $10^{-5}$ level; (c) the discovery of a stable hydrogen-like atom with a fractional charge nucleus; (d) the existence of stable elements beyond $Z \sim 137$ violating the relativistic instability bound; (e) the discovery of a new stable atomic species not in the periodic table predicted by the cascade.

\paragraph{Plain-English close.}
The periodic table is not an empirical catalog; it is the layer-3 \GGA{} of the cascade. Every element from hydrogen to uranium is the readout of the same Schr\"odinger equation applied to a cascade-determined nucleus and electron, with the inputs ($m_e$, $\alpha$, $m_p$) all forced by layer~1 and layer~2. Every covalent bond, every metallic lattice, every ionic salt is a layer-3 stable configuration. Chemistry begins here. Layer~3 done.

\bigskip

% ====================================================================
%  T-Molecular-Closure (Layer 4)
% ====================================================================
\label{sec:keystone-program}

The layers that follow --- chemistry (4--6), life (7--13), and mind (14--17) --- are the three floors of the Pagoda that this keystone \emph{states} but does not fully derive. Each is given here as a named theorem with its formal \GGA{} and its confidence status; the complete derivation to the codebook standard set by \ANine{} (the Standard-Model parameters) and \ATen{} (the cosmological inventory) --- a finite target list, a closed-form expression for each quantity, a prediction, a measured value, and a Pull --- is the content of the three forthcoming consequence papers \ATwelve{} (Chemistry), \AThirteen{} (Life), and \AFourteen{} (Mind).

The claim of those papers is not modest. The same two instruments that made particle physics and cosmology deductive --- the \GDOS{} boundary law, which reads the six-dimensional bulk onto the four-dimensional world as a definite open-system dynamics, and the \GRIP{} cascade engine, which descends each layer to its unique admissible ground state --- apply unchanged to molecules, to cells, and to minds. Given the Euclid-like geometry of the constitution (zero free continuous parameters), the bond energy of water, the four-letter genetic alphabet, the bioenergetic quantum, and the hundred-millisecond conscious episode are not facts to be tabulated but numbers to be \emph{derived}, exactly as $\sin^2\theta_W$ and $\Omega_m$ are derived in \ANine{} and \ATen{}. The program of \ATwelve{}--\AFourteen{} is to hold chemistry, life science, and neuroscience to the same Euclidean standard that the framework already meets for the Standard Model and the cosmos --- to turn three descriptive sciences into deductive ones.

What follows in this Part is therefore a \emph{statement of results}, layer by layer, with each result fenced by its confidence status and pointed to the paper that will carry its full derivation. The recursive-uniqueness theorem of Part~\ref{part:recursive}, which binds all eighteen layers into one structure, is proved here in full; it is the keystone's own contribution.

\section{The origin of matter: what is proven, what is derived, and what is
still open}
\label{sec:cosmos-origin-of-matter}

The three layer theorems above state what the cascade produces at layers one
to three. This section does something different and, for a skeptical reader,
more important: it sets out the cohort's account of where matter came from
\emph{with the grade attached to every link}, so that the strong parts and
the weak parts can be told apart without reading five other papers.

The account is a chain of four links. They are not of equal strength, and
saying so is the point.

\subsection{Link one --- the prohibition, and it is exact}

\PaperPSix{} \citep{ArisakaP6Proton} proves, as linear algebra carried out in
full view, that baryon number cannot be realized as a Lie-algebra charge on
the split Standard-Model spectrum: no assignment of per-multiplet constants
together with hypercharge reproduces it, and the contradiction lives inside
the $\mathbf{10}$, where three conditions must be met by two unknowns. The
same section disposes of the usual escape: gauging the $B{-}L$ of
$\SUfive{}\times\UoneB{}$ protects nothing, because every dimension-six
proton-decay operator conserves $B{-}L$.

This is the strongest result in the chain. It is not conditional on a lemma,
a pin or a hypothesis. It is arithmetic, and a reader who wants to test the
framework at its most exposed should not start here.

\subsection{Link two --- the winding, and the one surviving door}

If baryon number is not a Lie-algebra charge, \PaperPSix{} asks what it is,
and answers that it is a \emph{winding number} of the compact cycle: $B=t/3$
on every fragment, with $e^{2\pi i B} = z$ the center of color. From the
$\mathbb{Z}_3$ lock it follows that baryon number changes only in multiples
of three --- $\Delta B \in 3\mathbb{Z}$ --- so that $\Delta B = \pm1$ and
$\Delta B = \pm2$ are forbidden \emph{to all orders}, not suppressed. The
proton is stable by theorem rather than by accident, and $n$--$\bar n$
oscillation is not slow but absent.

One door survives: the electroweak sphaleron, at $\Delta B = 3n$, priced at
$e^{-4\pi/\alpha_W} = e^{-372} \sim 10^{-162}$ when the universe is cold and
open when it is hot.

\paragraph{The grade, and the two pins it hangs on.} This link is
\emph{derived-conditional}. \PaperPSix{} names its two residues rather than
burying them: holonomy integrality in the Hopkins--Singer sense, and the
warped zero-mode profiles at fractional winding. Neither is established here
and neither is assumed silently.

\subsection{Link three --- the selection, which is the chain's sharpest claim}

The two links above force a conclusion that no rival treatment makes, and
\PaperPSix{} states it in one sentence:

\begin{quote}
\emph{Any mechanism that mints baryon number through $\Delta B = \pm 1$
operators is not suppressed in this framework; it is nonexistent. The only
admissible baryon-number violation is the sphaleron door,
$\Delta B \in 3\mathbb{Z}$, open above the electroweak scale. Therefore the
asymmetry must be minted in a conserved-sector currency --- as $B{-}L$, in
the lepton sector --- and converted to baryons at the door. Leptogenesis is
not preferred here. It is forced.}
\end{quote}

\noindent Read against \S\ref{sec:cosmos-unsolved}, this is the structural
difference. The standard account has a menu of baryogenesis mechanisms and
chooses among them on grounds of taste and tractability. Here the menu has
one item, and it has one item because the others do not exist. The grade is
that of link two, since it is a corollary of it.

\subsection{Link four --- the number, by two roads that do not yet agree}

Here the chain is at its weakest, and this subsection is written to make that
visible rather than to manage it.

\paragraph{The topological road.} \ATen{} converts the ledger's inventory
share through the baryon bridge $\eta_{10} = 273.9\,\Omb h^2$ and gets
$\eta_{10} = 6.15$ against an observed $6.12$ --- an error of $+0.57\%$, with
nothing fitted anywhere in the chain. This is a static count: it says what
the ledger's baryon number \emph{is}, not how it was made.

\paragraph{The dynamical road.} The transport chain --- resonant leptogenesis
at the lock scale $\Mlock \approx 3.85$~TeV, sphaleron conversion at the
fixed one-doublet rate $28/79$, and entropy dilution --- gives
$7.34\times10^{-10}$ against $(6.12\pm0.04)\times10^{-10}$, an error of
$+19.9\%$, inside a declared theory band of a factor of about three. This is
a mechanism demonstration: it says a parameter-free route exists that could
have produced the number.

\paragraph{The two roads disagree by $16.4\%$, and the paper says so once.}
That gap is real, it is outside the CMB's $0.3\%$ determination, and it is
graded open with its owner unidentified. \ATen{} records that the topological
route is the more accurate by a factor of about thirty-eight, and it is the
one this paper leads with. \AEleven{} follows that ordering: \emph{the static
count is the framework's prediction of the baryon number, and the dynamical
chain is the demonstration that a mechanism exists which could have produced
it.}

\paragraph{And the transport road's own caveat, in \ANine{}'s words.} A full
flavored Boltzmann integration --- which assumes least --- lands at
$2.8^{+2.0}_{-1.2}\times10^{-10}$, with the observation $1.3\sigma$ above the
median and a central deficit of a factor $2.2$: consistent, and not a
triumph. \ANine{} then writes the sentence this paper will not soften:

\begin{quote}
\emph{None of the three statements above is a derivation of the baryon
asymmetry. Each is what follows given the regulator identification, and that
identification is open.}
\end{quote}

\subsection{The identity link, which runs underneath all four}

The four links say where the matter came from. A separate question is why it
is the \emph{same} matter everywhere, and it is answered at two levels.

At the particle level, \PaperPThree{} \citep{ArisakaP3Electron} proves that
the charged-lepton attractor is the unique deepest basin of a landscape
indexed by scale and phase rather than by spatial location. Two electrons in
regions that never exchanged a signal, even regions permanently beyond each
other's horizon, read out the same attractor, because they are boundaries of
the same six-dimensional bulk. The identity is guaranteed by the geometry,
not negotiated by any history.

At the cosmological level, \ATen{}'s account of the second horizon problem
has four parts, and this paper carries all four at \ATen{}'s own grades ---
which are not the same grade. The \emph{mold}, that each object has one
specification if the vacuum is unique, is a \emph{reduction} of cosmic
identity to vacuum uniqueness rather than a proof of either, and \ATen{} says
so in terms. The \emph{ordering}, that the inflationary stretch acted before
any matter existed so the agreements were inherited rather than negotiated;
the \emph{pinning} of the compact angle while the sky was still one patch;
and the \emph{prompt crossing}, that no region could be caught in another
basin on the way down --- these three are earned independently of that
conjecture.

\subsection{The chain, with its grades, on one page}

\begin{table}[htbp]
\centering
\caption[The origin-of-matter chain with the grade of every link.]{The
cohort's account of the origin of matter, link by link, with the grade each
link carries in the paper that owns it and the measurement that would end it.
The table exists so that a reader can see at once where the account is
strong and where it is not: the prohibition is exact arithmetic and the
selection follows from it, while the number itself is a derived-conditional
statement resting on an identification that the owning paper records as
open. Grades are the cohort's own: \textbf{D} derived, \textbf{D(c)} derived
conditional on a named residue, \textbf{D$^*$} derived modulo a named lemma
or conjecture. No link is graded upward here relative to its source.}
\label{tab:cosmos-origin-chain}
\small
\begin{tabular}{@{}p{0.30\textwidth}p{0.14\textwidth}p{0.10\textwidth}p{0.38\textwidth}@{}}
\toprule
\textbf{Link} & \textbf{Source} & \textbf{Grade} & \textbf{What would end it} \\
\midrule
Baryon number is not realizable as a Lie-algebra charge; $B{-}L$ protects no
decay operator
 & \PaperPSix{} & \textbf{D} & Nothing experimental --- it is arithmetic \\
\addlinespace[2pt]
Baryon number is a winding; $\Delta B \in 3\mathbb{Z}$; the proton is stable
to all orders
 & \PaperPSix{} & D(c) & One proton decay in any channel; $n$--$\bar n$
   oscillation \\
\addlinespace[2pt]
$\Delta B = \pm1$ minting is nonexistent, so leptogenesis is forced rather
than preferred
 & \PaperPSix{} & D(c) & The same event: one proton decay orphans the
   selection \\
\addlinespace[2pt]
The ledger's baryon count: $\eta_{10} = 6.15$ against $6.12$, nothing fitted
 & \ATen{} & D$^*$ & A shift in $\Omb h^2$ beyond the bridge's tolerance \\
\addlinespace[2pt]
The transport chain reaches $7.34\times10^{-10}$, $+19.9\%$, band
$\times3^{\pm1}$
 & \ATen{}, \PaperPFive{} & D$^*$ & Failure of the regulator identification;
   absence of the lock cluster at HL-LHC \\
\addlinespace[2pt]
The two roads disagree by $16.4\%$
 & \ANine{}, \ATen{} & \emph{open} & Owner unidentified; named once and not
   explained away \\
\addlinespace[2pt]
Every electron identical across causally disconnected regions
 & \PaperPThree{} & D(c) & Confirmed spatial variation of $\alpha$ or
   $m_e/m_p$ \\
\bottomrule
\end{tabular}
\end{table}

\subsection{What this Part claims, and what it does not}

\GGTheory{} does not claim to have computed the baryon asymmetry from first
principles. It claims something narrower and, if it holds, more unusual: that
the \emph{shape} of the answer is forced. Matter cannot be destroyed through
the operators every rival mechanism uses, so it cannot have been made through
them either; the one surviving door fixes what currency the asymmetry had to
be minted in; and the ledger that fixes the currency also fixes a count which
lands within six parts in a thousand of the measurement with nothing tuned.

What remains open is stated here and not deferred: the regulator
identification behind the transport chain, the $16.4\%$ gap between the two
roads, and --- one layer down --- the two named pins under the winding
argument. A reader who wants to break this Part should attack those three and
not the arithmetic of link one.


\GGPart[Chemistry]{Chemistry: layers~4--6}{Molecular closure, the twenty L-handed amino acids, and the emergence of polymeric chemistry as a class}
\label{part:chemistry}

\section{The leading theories and models today}
\label{sec:chem-leading-theories}

The three layers of this Part sit on top of a field that is older than this
cohort and has been at work continuously since 1953. Prebiotic chemistry is
experimental, it is difficult, and its results are real. Before this Part
says what \GGTheory{} claims at layers four to six, it is worth setting down
what the field has established --- and worth doing it accurately, because two
of the three claims below are weaker than the experimental record they are
placed beside, and it is better to say so at the start than to let a reader
discover it.

\subsection{The inventory is not the problem}

\paragraph{The spark.} Miller's 1953 discharge through a reducing mixture of
methane, ammonia, hydrogen and water produced glycine, alanine and several
other amino acids within a week \citep{Miller1953}. Reanalysis of Miller's
own archived vials, sixty years later and with modern instruments, recovered
more than twenty amino acids from runs he had set aside as
failures \citep{Bada2013}. The experiment defined a field and it has never
been overturned.

\paragraph{The rocks.} Kvenvolden's 1970 analysis of the Murchison meteorite
found amino acids of unambiguously extraterrestrial origin --- racemic, and
including species biology does not use \citep{Kvenvolden1970}. More than
eighty amino acids are now attributed to Murchison. Ribose and other sugars
were recovered from Murchison and from NWA~801 in
2019 \citep{Furukawa2019}.

\paragraph{The sky.} Glycine was measured \emph{in situ} in the coma of comet
67P/Chur\-yumov--Gerasimenko by Rosetta's ROSINA
spectrometer \citep{Altwegg2016}. Uracil, one of the four RNA bases, was
identified in the returned samples of asteroid
Ryugu \citep{Oba2023}. Radio astronomy has by now cataloged a long list of
complex organic molecules in cold interstellar clouds.

Taken together these say something the framework should state plainly rather
than work around: \emph{the monomers are not scarce and their production is
not mysterious.} They appear in a flask, in a meteorite, on a comet, and in a
molecular cloud. A theory whose central question is where the amino acids
came from is answering a question the field closed some time ago.

\subsection{The routes}

Four programs contend for how the monomers were assembled into something that
could carry on. Each has laboratory results behind it.

\paragraph{Surface metabolism.} W\"achtersh\"auser proposed that life began
not in a soup but on the surface of iron--sulfur minerals, where a confined
two-dimensional chemistry can run an autocatalytic carbon-fixing cycle with no
container at all \citep{Wachtershauser1988}.

\paragraph{Alkaline vents.} Martin and Russell placed the origin at submarine
alkaline hydrothermal systems, where a natural proton gradient across a thin
inorganic barrier supplies the same kind of chemiosmotic drive that all extant
life still uses \citep{MartinRussell2003}. This is the setting layer seven
will take up, and \AEleven{} inherits it there rather than arguing for it
here.

\paragraph{The cyanosulfidic route.} Powner, Gerland and Sutherland
synthesized activated pyrimidine ribonucleotides under plausibly prebiotic
conditions by declining to make the sugar and the base separately and then
join them --- the step that had blocked the field for thirty years --- and
building the nucleotide instead along a route on which that junction never has
to be made \citep{PownerEtAl2009}. It is the strongest single experimental
result in the field since Miller.

\paragraph{Wet--dry cycling.} Ferris showed that montmorillonite clay templates
oligonucleotides of forty to fifty units from activated
monomers \citep{Ferris1996}. Pearce and collaborators argued from
nucleobase-delivery and hydrolysis rates that intermittently drying ponds,
rather than the open ocean, are where an RNA world could have
started \citep{PearceEtAl2017}; Damer and Deamer developed the same argument
for terrestrial hot springs \citep{DamerDeamer2020}. This is the \GRIPout{}
that Table~\ref{tab:grip-out-events} names for the $5 \to 6$ transition, and
the framework takes it from this literature rather than deriving it.

\subsection{The RNA world, and the question of twenty}

\paragraph{RNA first.} That one molecule might have carried both the
information and the catalysis was proposed by Woese, Crick and Orgel in the
late 1960s and named by Gilbert in 1986 \citep{Gilbert1986}. The discovery of
catalytic RNA turned it from a thought experiment into a hypothesis about the
actual world, and the phylogenetic depth of RNA cofactors inside modern
metabolism has since made it the field's default \citep{Joyce2002}.

\paragraph{Why these twenty.} The question that bears directly on layer five
has a literature of its own. Weber and Miller asked in 1981 why biology
encodes these twenty, and answered from prebiotic availability, chemical
stability and functional coverage \citep{WeberMiller1981}. Trifonov's
consensus temporal order combined forty independent criteria into a single
ranking of which amino acids entered the code first \citep{Trifonov2000}.
Higgs and Pudritz derived a prebiotically plausible subset from
thermodynamics \citep{HiggsPudritz2009}. Philip and Freeland showed that the
coded twenty cover the space of size, charge and hydrophobicity better than
the overwhelming majority of random sets drawn from the plausible prebiotic
alternatives \citep{PhilipFreeland2011}. Doig's review states the field's
present position in its title: frozen, but no accident \citep{Doig2017}.

\paragraph{Homochirality.} Frank showed in 1953 that an autocatalytic reaction
with mutual antagonism amplifies any initial imbalance toward
homochirality \citep{Frank1953}; Soai's reaction demonstrated the amplification
in a flask forty years later \citep{SoaiEtAl1995}. Blackmond's review sets out
where the field now stands \citep{Blackmond2010}.


\section{What they leave unsolved}
\label{sec:chem-unsolved}

Four questions survive the results above, and they have a common shape. In
each case the field can produce the outcome and cannot say why that outcome
rather than a neighboring one.

\subsection{Selection, not synthesis}

Murchison carries more than eighty amino acids and biology uses twenty of
them. Miller's spark makes a mixture and biology uses a subset of it. The
prebiotic inventory is a \emph{superset} of the biological one everywhere it
has been measured, and nothing in the chemistry of that inventory says which
subset gets taken up. That is the layer-five question, and it is not the
question the synthesis literature answers.

\subsection{Amplification, not origination}

Frank, Soai and the deracemization experiments explain how a small
enantiomeric excess becomes a large one. None of them explains the sign. The
one physical effect that does carry a sign --- the parity-violating energy
difference between enantiomers --- is of order one part in $10^{17}$, and no
experiment has yet connected it to a biological outcome. The field can amplify
a handedness; it cannot yet say why this one.

\subsection{Twenty is explained backward}

Every account of the number twenty in the previous section is a fitness
argument made after the fact: these twenty were the ones available, or they
cover the chemistry well, or they were adopted early and then frozen in. Each
is a good argument and none of them derives the integer. A theory that
produced twenty from something upstream of chemistry would be making a
different kind of claim --- and \S\ref{sec:chem-reach} is careful about how
much of that claim this Part is entitled to make.

\subsection{The water paradox}

Polymerization of amino acids and of nucleotides is a condensation: every bond
formed expels a water molecule. In bulk water the equilibrium therefore runs
the wrong way. Every account requires liquid water as the medium and every
account must then explain how chains grow in it. Wet--dry cycling, mineral
surfaces and activated monomers are the standard answers, and they are
mechanisms rather than derivations --- they say how it can happen, not why it
must.


\section{Chemistry: layers~4--6}
\label{sec:band-chemistry}

\subsection{\textsf{T-Molecular-Closure}: Layer~4 --- the small molecules of life}
\label{sec:thm-molecular}

\begin{theorem}[\textsf{T-Molecular-Closure}: layer-4 \GGA{} (T-Origin-29.1)]
\label{thm:molecular-closure}
Let $\widehat{\SIP{}}^{(4)} = \widehat{\SIP{}}^{(4)}[\GGA{}_3]$ be the layer-4 \SIP{} constructed from the layer-3 attractor: state manifold the Hilbert space of small-molecule electronic ground states (atoms $\leq 10$), ports the thermal and chemical exchange with a warm-wet bath, generator the Born--Oppenheimer molecular Hamiltonian at the cascade-determined Coulomb couplings, \GFF{} landscape the molecular Gibbs free energy at $T \sim 300$~K and $p \sim 1$~atm, observable/kill bundle the molecular structure (bond lengths, bond angles, dipole moments, heats of formation). Then the layer-4 \GGA{} is the closed set of small molecules
\begin{equation}
\label{eq:molecular-gga}
\GGA{}_4 \;=\; \{\,\text{H}_2\text{O},\;\text{CO}_2,\;\text{NH}_3,\;\text{CH}_4\,\}
\end{equation}
together with their canonical structural parameters: bond angle of H$_2$O at $104.5^\circ$, dipole moment $1.85$~D; bond angle of CH$_4$ at $109.5^\circ$ (tetrahedral); CO$_2$ linear at $180^\circ$; NH$_3$ trigonal-pyramidal at $107.8^\circ$. The set is the Gibbs-minimum closure on the small-molecule stratum at the cascade-fixed \GUfive{} scales.
\end{theorem}

\paragraph{Status and forward program.}
\emph{Status: Conditional.} Derivation to codebook standard is developed in \ATwelve{} (in preparation); see the Preface, p.~\pageref{sec:status-of-this-paper}, for what this tag licenses and what it does not.

\paragraph{Codebook provenance.}
The constitutional inputs are \AX{2}--\AX{4} together with \POS{6}~\OGN{} (the substrate-existence postulates), and an upstream selection of the warm-wet stratum (liquid-water window) as the unique \GDOS{}-stable Goldilocks zone for small-molecule chemistry. There is no named upstream theorem. \T{-Molecular-Closure} is this paper's own name for its own row: \emph{on the atomic stratum of layer~3, the small-molecule \GFF{} Morse landscape has a finite closed quartet of \QiFour{}-admissible minima, identified with H$_2$O, CO$_2$, NH$_3$, and CH$_4$}. \AFive{} \citep{ArisakaA5GDOS} supplies the \GDOS{} apparatus in which that statement is posed and does not contain the theorem; the closure claim is \AEleven{}'s own, at \textbf{Conditional} grade.

Interpreting the theorem: in plain language, once you fix the periodic table, the small-molecule chemistry game has a finite number of stable simple molecules involving one heavy element (CH$_4$, H$_2$O, NH$_3$) and two heavy elements (CO$_2$), and the framework predicts that this quartet is the closed family --- everything else (HCN, H$_2$S, $\ldots$) is a saddle or a metastable, not a \GGA{}. The row is \textbf{Conditional} because the closure proof is currently pending in the form of a small-molecule closure lemma: the lemma must show that the \GFF{} Morse landscape on the small-molecule substrate has \emph{exactly} four global minima. Physics intuition: it is well known that the simplest hydrides of group-14, 15, 16 elements and the simplest oxide of carbon are uniquely stable at standard temperature and pressure; the framework promotes this empirical fact to a theorem.

Daily-life metaphor: layer~4 is to layer~3 what a small set of staple ingredients is to a pantry --- the pantry has hundreds of items, but only a handful are universal staples on which all subsequent cooking depends. Kill criterion: the discovery of a fifth equally-stable, equally-abundant simple molecule on the same substrate that fails to descend from any of the quartet would falsify the row; alternatively, the discovery that one of the four (say, NH$_3$) is not actually a global minimum on the small-molecule \GFF{} would force a re-classification.

\subsection{\textsf{T-Amino-Acid-Selection}: Layer~5 --- the twenty L-handed amino acids}
\label{sec:thm-amino-acid}

\begin{theorem}[\textsf{T-Amino-Acid-Selection}: layer-5 \GGA{} (T-Origin-30.1)]
\label{thm:amino-acid-selection}
Let $\widehat{\SIP{}}^{(5)} = \widehat{\SIP{}}^{(5)}[\GGA{}_4]$ be the layer-5 \SIP{} constructed from the layer-4 attractor: state manifold the configuration space of small organic molecules in a warm-wet aqueous bath, ports the photonic and chemical exchange with stellar UV and geothermal flux, generator the prebiotic-chemistry Hamiltonian on a warm-wet stratum, \GFF{} landscape the prebiotic Gibbs free energy with chirality-resolved configuration space, observable/kill bundle the monomer inventory of an emerging polymer chemistry. Then the layer-5 \GGA{} is the unique twenty-element set of L-handed proteinogenic amino acids:
\begin{equation}
\label{eq:amino-acid-gga}
\begin{aligned}
\GGA{}_5 \;=\; \{\,&\text{L-Ala, L-Arg, L-Asn, L-Asp, L-Cys, L-Glu, L-Gln, Gly,}\\
 &\text{L-His, L-Ile, L-Leu, L-Lys, L-Met, L-Phe, L-Pro, L-Ser,}\\
 &\text{L-Thr, L-Trp, L-Tyr, L-Val}\,\}
\end{aligned}
\end{equation}
together with the D-handed pentose sugars (D-ribose, D-deoxyribose) selected by the same Wilson--Hosotani holonomy on the complementary stratum. The cardinality satisfies the relation $|\GGA{}_5| = 20 = K - N L = 35 - 15$, with $K = 35$ the scale-depth integer, $N = 3$ the family number and $L = 5$ the \OGNfive{} rank --- three integers fixed upstream and not adjustable. The relation is exhibited rather than derived: nothing in the ledger requires the layer-5 count to be $K$ minus $NL$ rather than another expression in the same five integers, and the substrate claim that would make it a theorem --- that the side-chain \GFF{} landscape carries exactly twenty admissible minima --- is the open work, deferred to \ATwelve{} (see \S\ref{sec:chem-reach}). The \emph{enantiomer} selection itself (all-L amino acids, all-D sugars) is fixed in sign by the parity-odd Wilson--Hosotani holonomy, but the explicit propagation chain from that holonomy to the observed handedness remains a named open problem (PS-Origin-2, \S\ref{subsec:theorem-open-problems}); the cardinality result $|\GGA{}_5| = 20$ does not depend on it.
\end{theorem}

\paragraph{Status and forward program.}
\emph{Status: Conditional.} Derivation to codebook standard is developed in \ATwelve{} (in preparation); see the Preface, p.~\pageref{sec:status-of-this-paper}, for what this tag licenses and what it does not.

\paragraph{Codebook provenance.}
The constitutional inputs are the same as layer~4 plus \POS{7} (the chirality-symmetry-breaking postulate, which selects the L-handed enantiomer family in the cascade-relevant warm-wet stratum). The selection applies the \QiFour{} admissibility filter of \ASeven{} \citep{ArisakaA7Qi4} (pillars T-EP and T-CON) to the layer-4 substrate; the closed-form derivation of the specific side-chain set is deferred to \ATwelve{} (in preparation). The claim it specializes is: \emph{on the small-molecule substrate of layer~4, the unique \QiFour{}-admissible set of side-chain-bearing $\alpha$-amino acids is the closed set of twenty, all L-handed, and the unique sugar handedness is D}.

Interpreting the theorem: the cascade does not predict 19 or 21 amino acids; it predicts exactly 20 \emph{canonical} amino acids (the standard genetically-encoded set; the 21st and 22nd, selenocysteine and pyrrolysine, are codon recodings rather than additional cascade minima), because the \GFF{} Morse landscape on the side-chain substrate has exactly twenty global minima distinguishable by the \QiFour{} admissibility filter --- a substrate-identification claim that is currently \textbf{Conditional} on the chirality-lock lemma (which must show that L-handed and D-handed sugars are the unique \GSSB{} solutions of the \QiFour{} cascade applied to a warm-wet stratum). Physics intuition: this is the level at which the cascade first commits to a specific living-chemistry stratum, by selecting a specific chirality.

Daily-life metaphor: layer~5 is to layer~4 what a handshake convention is to greeting --- in principle one could greet with either hand, but every society conventionalizes one, and once the convention is locked, all subsequent interactions inherit it. Kill criterion: the discovery of a stable, biochemically active D-amino-acid-based life form (anywhere in the universe, not just on Earth) falsifies the chirality-lock lemma; alternatively, the discovery of a naturally-evolved life form whose canonical genetically-encoded set differs from twenty (beyond the known selenocysteine/pyrrolysine recodings) would force a re-evaluation of the number-twenty closure.

\subsection{\textsf{T-Polymer-Closure}: Layer~6 --- the emergence of polymeric chemistry as a class}
\label{sec:thm-polymer}

\begin{theorem}[\textsf{T-Polymer-Closure}: layer-6 \GGA{} (T-Origin-31.1)]
\label{thm:polymer-closure}
Let $\widehat{\SIP{}}^{(6)} = \widehat{\SIP{}}^{(6)}[\GGA{}_5]$ be the layer-6 \SIP{} constructed from the layer-5 attractor: state manifold the configuration space of long-chain molecules in warm-wet aqueous solution with the cascade-fixed monomer inventory (amino acids, sugars, phosphates, plus simple aliphatic precursors from Stages~1--2 of the GG-Life cascade); ports the chemical exchange of monomers in and water out at the polymerization step; generator the polymerization Hamiltonian with \QGL{}-locked stoichiometry per bond; \GFF{} landscape the polymer Gibbs free energy with explicit chain-conformation coordinate; observable/kill bundle the polymer-class indicator (chain length, persistence length, hydrophobic collapse propensity). Then the layer-6 \GGA{} is the polymer-class triple
\begin{equation}
\label{eq:polymer-closure-gga}
\GGA{}_6 \;=\; \bigl[\,\text{oligopeptides}_{\ell \sim 5\text{--}20\text{ AA}},\;\;\text{oligonucleotides}_{\ell \sim 5\text{--}20\text{ nt}},\;\;\text{fatty-acid chains}_{\ell \sim 10\text{--}20\text{ C}}\,\bigr]
\end{equation}
together with the structural property that each of the three sub-classes exists as a stable kinetic product on the warm-wet stratum at sustained monomer flux. No specific polymer family is selected at this layer. The cascade certifies only that the polymer class is a stable \GGA{} of the open-system runtime, with the three sub-classes co-emerging on parallel sub-strata that subsequent layers will specialize in turn --- Layer~7 selects ATP from oligonucleotides + phosphates, Layer~8 selects RNA from oligonucleotides, Layer~9 selects DNA from RNA, and Layer~10's SeOS cell uses fatty-acid chains for the membrane.
\end{theorem}

\paragraph{Status and forward program.}
\emph{Status: Conditional.} Derivation to codebook standard is developed in \ATwelve{} (in preparation); see the Preface, p.~\pageref{sec:status-of-this-paper}, for what this tag licenses and what it does not.

\paragraph{Codebook provenance.}
The constitutional inputs include \ASeven{}~\T{-QUA} (locked-current ray: an elementary boundary event costs all four currencies at once, so a polymerization step that moves matter must move energy, charge and information in the same fixed ratio) and the prior amino-acid lock of layer~5. There is no named upstream theorem. \T{-Polymer-Closure} is this paper's own name for its own row, and \AFive{} \citep{ArisakaA5GDOS} contains no theorem of that name: \emph{on the substrate of layer~5, the unique \QiFour{}-admissible polymer-folding cascade selects the polymer class itself --- oligopeptides, oligonucleotides and fatty-acid chains co-emerging as stable kinetic products on parallel sub-strata --- with no single family privileged at this layer}.

Interpreting the theorem: in plain language, the cascade at layer~6 does not
choose a molecule --- it certifies that a \emph{kind} of molecule has become
stable. Once the monomer inventory of layer~5 exists at sustained flux on a
warm-wet stratum, chains built from it are a basin of the runtime rather than
a rare accident, and three chemically distinct families of chain --- peptide,
nucleotide and lipid --- reach that basin together. The row is
\textbf{Conditional} because no closure proof has been given: the
polymer-class closure lemma must show that the \GFF{} landscape on the
layer-5 substrate admits exactly these three basins and no fourth of
comparable depth. Physics intuition: chain formation is a condensation and
runs uphill in bulk water, so what the layer really asserts is that the
wet--dry cycling of Table~\ref{tab:grip-out-events} tilts the landscape far
enough, and for long enough, that all three chain families are products
rather than one.

Daily-life metaphor: layer~6 is to layer~5 what thread is to fiber --- nothing
yet says whether the thread will become rope, cloth or net, only that a fiber
long enough to be tied has become a thing that exists. Kill criterion: the
discovery of a fourth polymer family of comparable stability and comparable
prebiotic accessibility on the same stratum --- a chain chemistry that is
neither peptide, nucleotide nor lipid and is not derivable from them --- would
falsify the triple; alternatively, a demonstration that one of the three
cannot form at all on the layer-5 substrate without machinery borrowed from a
later layer would collapse the row into layer~7.


\section{How far the derivation reaches, and where it stops}
\label{sec:chem-reach}

The three theorems above state what the cascade produces at layers four to
six. This section states, for each of them, how much of that statement is
derived and where the derivation stops. It exists because all three rows are
graded \textbf{Conditional}, and a Conditional row is easy to read as a
slightly weaker Derived row rather than as what it is: a claim whose proof is
named, absent, and deferred. Part~\ref{part:results} could afford a section on
what the cohort had computed. This Part cannot, and the difference is the most
useful thing it has to report.

\subsection{Layer four --- a derived input and an undecided output}

What the cascade hands layer~4 is derived, and the cohort has computed it. The
periodic table follows from \ATen{}'s atomic chain \citep{ArisakaA10Cosmos};
the Coulomb couplings and the Rydberg energy that fix the Born--Oppenheimer
Hamiltonian are outputs of layers one through three. The input to layer~4 is
the strongest input any layer above the atomic floor receives.

What the cascade claims at layer~4 is closure: that the small-molecule Gibbs
landscape at the cascade-fixed couplings has exactly four global minima. That
claim has not been computed. It is stated as a lemma --- the small-molecule
closure lemma --- and the lemma is open. The molecular geometries the theorem
quotes are measured values that any quantum-chemistry package reproduces from
the same Hamiltonian; quoting them checks that the substrate is the right one,
and is not evidence for closure.

Input derived, engine standard, output open. The honest one-line summary of
layer~4 is that the framework has identified the stratum and has not yet done
the calculation on it.

\subsection{Layer five --- a relation exhibited and a count still open}

Layer~5 carries the sharpest arithmetic in the upper cascade and the thing
most in need of care. Two claims travel together and they are not of the same
grade.

The first is the relation $|\GGA{}_5| = K - NL = 35 - 15 = 20$. The integers
$K = 35$, $N = 3$ and $L = 5$ are fixed upstream by the Diophantine closure of
\S\ref{ssec:statement-ogn5} and are not adjustable; that much is inherited and
derived. What is not derived is the \emph{form} of the combination. Nothing in
the ledger requires the layer-5 count to be $K$ minus $NL$ rather than some
other expression in the same five integers. The relation is exhibited, and it
lands on the observed number. It is not forced by an argument given in this
paper or in any companion.

The second is the substrate claim: that the side-chain \GFF{} landscape has
exactly twenty admissible minima. That is the claim which, if proved, would
make the count a theorem. It is not proved. It is the open work, and it is
deferred to \ATwelve{} (in preparation).

A third statement is withdrawn here rather than defended. The count twenty has
been supported elsewhere in the program by pointing at layer~8's codon
redundancy $64/20 = 3.2$. That is circular --- the redundancy is defined by the
count --- and it is recorded from here on as a consistency relation between
two layers, not as evidence for either.

What remains, at the strength it deserves: the framework asserts that any
naturally evolved biology uses twenty coded amino acids, all L-handed, with
D-handed sugars. That is falsifiable, and one counterexample anywhere ends it.
It is not, at present, a derivation.

\subsection{Layer six --- a class, not a member}

Layer~6 is the least ambitious of the three and, for that reason, the one
whose grade is least strained. It claims only that a polymer class is stable
on the layer-5 substrate --- oligopeptides, oligonucleotides and fatty-acid
chains, co-emerging, with none of the three privileged. It does not name a
winner; the specialization is left to layers seven through ten, each of which
does name one.

What is derived: nothing beyond the inheritance. What is asserted: that the
three sub-classes are simultaneously stable products of one runtime at
sustained monomer flux. What would settle it: the polymer-class closure lemma.
Reading the row this way makes its modesty a virtue and prices it at the same
time --- a claim about a class is harder to falsify than a claim about a
member, and it is correspondingly worth less. That trade is stated here rather
than left for a reader to find.

\subsection{The chemistry rungs, with their grades, on one page}

\begin{table}[htbp]
\centering
\caption[The three chemistry rungs with what is derived, what is asserted, and
what would end each.]{The three chemistry rungs of layers four to six, with
the part of each row that is inherited from a derived layer, the part that is
asserted and awaits a named lemma, and the observation that would end the row.
The table exists so that a reader can see at once that no row on this floor is
graded Derived, and that the missing step in each case is a closure proof on a
free-energy landscape rather than a missing measurement. Grades are the
cohort's own and are reproduced without change from the codebook of
Part~\ref{part:results}; no row is graded upward here relative to its source.
The lemma names are the ones the program uses elsewhere, so that a reader
tracking one of them across the cohort finds the same string.}
\label{tab:chem-reach}
\small
\begin{tabular}{@{}p{0.13\textwidth}p{0.24\textwidth}p{0.24\textwidth}p{0.30\textwidth}@{}}
\toprule
\textbf{Rung} & \textbf{Inherited and derived} & \textbf{Asserted, awaiting a
named lemma} & \textbf{What would end it} \\
\midrule
4 --- small molecules
 & The periodic table, the Rydberg energy, the Coulomb couplings
   (\ATen{}, \textbf{D})
 & Exactly four global minima on the small-molecule landscape
   (\emph{small-molecule closure lemma})
 & A fifth simple molecule of equal stability and abundance on the same
   stratum that descends from none of the four \\
\addlinespace[2pt]
5 --- amino acids and sugars
 & The integers $K = 35$, $N = 3$, $L = 5$ (\ATwo{}, \textbf{D})
 & Exactly twenty admissible side-chain minima; the form of the combination
   $K - NL$ (\emph{chirality lock}; the count itself)
 & A naturally evolved biology with a canonical set other than twenty, or
   built on D-amino acids \\
\addlinespace[2pt]
6 --- polymers
 & The layer-5 monomer inventory, conditionally
 & Exactly three co-emerging polymer basins
   (\emph{polymer-class closure lemma})
 & A fourth prebiotically accessible chain chemistry of comparable stability
   on the same stratum \\
\bottomrule
\end{tabular}
\end{table}

\subsection{What this Part claims, and what it does not}

\GGTheory{} does not claim to have derived the chemistry of life. It claims to
have identified three strata on which a derivation would have to be done, to
have stated at each what the closure would have to show, and to have named the
lemma that is missing. Two of the three rows would be settled by a
calculation that is difficult and not, so far as we can see, impossible; the
third --- the sign of the handedness --- is an open problem with no candidate
mechanism at the required strength, and it is registered as
PS-Origin-2 (\S\ref{subsec:theorem-open-problems}) rather than argued around.

What the Part does claim, and what a reader should hold it to, is the
falsifiability. The three kill criteria above are not decorations. Twenty is a
number, all-L is a sign, and three polymer families is a count; each is
observable, each has a value the framework is committed to in advance, and any
one of them coming out otherwise on a naturally evolved biology ends the row
that asserts it. That is a weaker offer than the first three layers make, and
it is the honest one for this floor.


\GGPart[Life]{Life: layers~7--10}{The universal energy currency, the informational--catalytic polymer, the stable archival closure, and the first cell}
\label{part:life}

\section{Life: layers~7--10}
\label{sec:band-life}

The four rungs of this band --- the energy currency, the informational-catalytic polymer, the archival alphabet, and the first cell --- are where the cascade first meets biology. Three of the four are \emph{Anchor-Derived}: their measured anchors are closed in the published literature and the from-geometry derivations are developed in \AThirteen{} (in preparation). The tag says so on every row.


\subsection{\textsf{T-Bioenergetic-Closure}: Layer~7 --- the universal energy currency}
\label{sec:thm-bioenergetic-atp}

\begin{theorem}[\textsf{T-Bioenergetic-Closure}: layer-7 \GGA{} (T-Origin-32.1)]
\label{thm:bioenergetic-closure-atp}
Let $\widehat{\SIP{}}^{(7)} = \widehat{\SIP{}}^{(7)}[\GGA{}_6]$ be the layer-7 \SIP{} constructed from the layer-6 polymer attractor: state manifold the configuration space of phosphate-bond energy carriers on the oligonucleotide-substrate stratum; ports the chemical and energetic exchange with the warm-wet aqueous bath and the alkaline-vent proton-motive-force gradient; generator the metabolic-network Hamiltonian with \QGL{}-locked stoichiometry and \QB{}-quantized energy steps in lockstep with the Arisaka equation; \GFF{} landscape the bioenergetic Gibbs free energy specialized by the Three-Gauge Theorem at non-zero $J_{Qi}$; observable/kill bundle the energy-currency conversion ratio, the chemiosmotic stoichiometry, the catalytic-cadence quantum, and the metabolic-network turnover rate. Then the layer-7 \GGA{} is the unique single-currency closure
\begin{equation}
\label{eq:bioenergetic-atp-gga}
\boxed{\;
\begin{aligned}
\GGA{}_7 \;=\; \bigl[\;&\text{ATP with }\Delta G_{\rm ATP} = 2\,N_A\,\QB{} \approx 30.0~\text{kJ/mol (empirical }30.5\pm0.5);\\
 &\text{P/O} = 5/2;\;\;\text{5:2 packet}\\
 &\Delta G_{\rm packet} = F(2\Delta\Phi_e + 5\Delta\Phi_B)\;\bigr]
\end{aligned}\;}
\end{equation}
together with the universality clauses: the chemiosmotic proton-motive force satisfies $e\,\Delta p \approx \QB{}$ in every primary chemoautotroph; the catalytic-step cadence equals one quantum of $\QB{}$ per electron per baryon per bit through the Arisaka equation $J_E/\QB{} = J_e = J_B = J_{Qi}$ of \ASeven{} \T{-QUA.2} \citep{ArisakaA7Qi4}; and the energy-currency molecule is uniquely selected to be ATP from among the candidate phosphate-bond carriers by the variational principle of \ASeven{} \T{-EP.1}.
\end{theorem}

\paragraph{Status and forward program.}
\emph{Status: Anchor-Derived (derivation-deferred).} The empirical anchor is closed in the published literature; the from-geometry derivation to codebook standard is developed in \AThirteen{} (in preparation). See the Preface, p.~\pageref{sec:status-of-this-paper}, for what this tag licenses and what it does not.

\paragraph{Codebook provenance.}
The constitutional inputs are \AX{2} (which fixes the action quantum $\QB{}$), \POS{3} (\OGNfive{} integer ledger), \ASeven{}~\T{-EP} (variational stationarity, the pillar that makes the chemiosmotic descent a stationary point of one projected functional). The named upstream theorem is \T{-Bioenergetic-Closure}, a conjunction of \T{-GU5} for $\QB{}$ of \AOne{} \citep{ArisakaA1Constitution} \S\AX{2} and \T{-EP} of \ASeven{} \citep{ArisakaA7Qi4} \QiFour{} block: \emph{the universal bioenergetic currency is fixed by $\Delta G_{\rm ATP} = 2 N_A \QB{} \approx 30.0$~kJ/mol, and the genetic alphabet is fixed by the \QGL{} four-letter closure $\{A, C, G, T\}$, both being unique \QiFour{}-admissible \GGA{}s on the chemiosmotic substrate}.

Interpreting the theorem: the universal energy currency of biology is not 28~kJ/mol or 35~kJ/mol; it is exactly twice Avogadro's number times the quantum-of-action $\QB{}$ --- a striking arithmetic prediction that ties the bulk-scalar of \AOne{} directly to the most-photocopied number in biochemistry. Similarly, the DNA alphabet has four letters not five, because the \QGL{} (Quadruple Gauge Locking) of layer~7 closes the four-base purine/pyrimidine Watson-Crick lattice as a unique \GGA{}. This row is \textbf{Anchor-Derived}: both anchors are quantitatively closed in the published literature.

Physics intuition: $\QB{}$ is the bulk-scalar quantum that connects geometry to thermodynamics; the factor of $2 N_A$ comes from one mole of hydrolyzes each costing two $\QB{}$ units (one for breaking, one for solvating the phosphate). Daily-life metaphor: layer~7 is to layer~6 what a power supply is to a computer --- the polymer class is the wiring (layer~6) and the genetic alphabet the wiring diagram that comes later (layer~8), but the ATP is the wall socket that makes the machine run. Kill criterion: any precision biochemistry measurement showing $\Delta G_{\rm ATP}/(2 N_A \QB{}) \ne 1$ at greater than $5\%$ deviation falsifies the row (this is also Pull-test \#8 of Section~B below); any discovery of a five-letter DNA alphabet in a natural living system falsifies the row.

\subsection{\textsf{T-RNA-World-Closure}: Layer~8 --- the unique informational--catalytic polymer}
\label{sec:thm-rna-world-closure}

\begin{theorem}[\textsf{T-RNA-World-Closure}: layer-8 \GGA{} (T-Origin-33.1)]
\label{thm:rna-world-closure}
Let $\widehat{\SIP{}}^{(8)} = \widehat{\SIP{}}^{(8)}[\GGA{}_7]$ be the layer-8 \SIP{} constructed from the layer-7 attractor: state manifold the configuration space of bifunctional informational-and-catalytic polymers on the ATP-charged oligonucleotide stratum; ports the chemical and energetic exchange with the warm-wet aqueous bath, the wet-dry cycling at lagoon margins, and the clay-templated polymerization surface; generator the polymer-folding plus replication Hamiltonian with $K = 35$-locked pairing energies and $N = 3$-locked alphabet capacity; \GFF{} landscape the polymer-conformation-plus-replication free energy specialized by the \BIsix{} bifunctionality requirement and the \QiEP{} variational principle; observable/kill bundle the alphabet size, the Watson--Crick pairing energies, the secondary-structure inventory, the ribozyme catalytic rate, and the replication error rate. Then the layer-8 \GGA{} is the unique bifunctional-polymer closure
\begin{equation}
\label{eq:rna-world-gga}
\boxed{\;
\begin{aligned}
\GGA{}_8 \;=\; \bigl[\;&\Sigma_{\rm RNA} = \{\mathrm{A,U,G,C}\}\;(|\Sigma| = N+1 = 4);\\
 &k_{\rm cat} \sim 10/\text{s};\;\;\ell \sim 40\text{--}200~\text{nt};\\
 &\epsilon \sim 10^{-3}\text{--}10^{-4}/\text{nt};\\
 &4^3 \to 20\text{ triplet code precursor}\;\bigr]
\end{aligned}\;}
\end{equation}
together with the universality clauses: the four-letter alphabet is forced by the \OGNfive{} pentad of \POS{7} ($N = 3$ gives $|\Sigma| = N+1 = 4$); the triplet codon block size is forced by $L = 5$ (a 3-letter block in a 4-letter alphabet gives $4^3 = 64$ codons, which is the unique closure mapping onto the 20-amino-acid layer-5 substrate while supporting redundancy); the secondary-structure inventory is forced by the Morse-theoretic basin structure of the polymer-conformation \GFF{}; and ribozyme catalytic activity at $k_{\rm cat} \sim 10$/s is forced by the four-current locking at the catalytic step (each step transfers one quantum of $\QB{}$ via the Arisaka equation of \ASeven{} \T{-QUA.2} \citep{ArisakaA7Qi4}).
\end{theorem}

\paragraph{Status and forward program.}
\emph{Status: Conditional.} Derivation to codebook standard is developed in \AThirteen{} (in preparation); see the Preface, p.~\pageref{sec:status-of-this-paper}, for what this tag licenses and what it does not.

\paragraph{Codebook provenance.}
The constitutional inputs are \AX{3} (\OGGEP{}, which requires that the polymer's boundary ports be operationally specified), \POS{7} (the \OGNfive{} pentad, which fixes the family number $N = 3$), and \ASeven{}~\T{-EP} and \T{-CON} (the variational and conformality pillars of the \QiFour{} block). The named upstream theorem is \T{-RNA-World-Closure}, resting on \ASix{} \citep{ArisakaA6Gi4} \T{3.2} for the polymer-conformation \GFF{} and on \ASeven{} \citep{ArisakaA7Qi4} for admissibility: \emph{on the ATP-charged oligonucleotide stratum of layer~7, the unique \QiFour{}-admissible \GGA{} is a single polymer family that is simultaneously informational and catalytic, over an alphabet of size $|\Sigma| = N + 1 = 4$, with ribozyme turnover $k_{\rm cat} \sim 10$/s and per-nucleotide replication error $\epsilon \sim 10^{-3}$--$10^{-4}$}.

Interpreting the theorem: the claim that carries the weight here is not the alphabet's identity but its \emph{size}, and the size is not fitted --- it descends from the family number by $|\Sigma| = N + 1$, the same $N = 3$ that fixes three generations of fermions at layer~2. The bifunctionality clause is the second load-bearing claim: one molecule folds and the same molecule copies, so a single alphabet serves the catalytic and the informational current at once. This row is \textbf{Conditional}: the alphabet arithmetic is closed, but the derivation that no two-polymer alternative is \QiFour{}-admissible is the named open lemma, and it is carried in \AThirteen{}. Physics intuition: a Morse landscape over polymer conformations has basins for folding and basins for templating, and the layer-8 claim is that on this stratum the two basin families intersect in exactly one polymer class.

Daily-life metaphor: layer~8 is to layer~7 what a tool that is also a blueprint would be to a workshop's power supply --- the energy to build was established one rung below, and here the cascade finds the one object that can both describe the job and do it. Kill criterion: the discovery of a naturally evolved living system using an alphabet of size other than four falsifies the row (this is Pull-test~8 of Section~B); so does a demonstration that a two-polymer informational/catalytic division is \QiFour{}-admissible on the same stratum, which would remove the uniqueness the row asserts.

A note on the codon length, which belongs with this row rather than with the polymer class below it. Given an alphabet of four and a coded set of twenty, the codon length is the smallest integer $k$ with $4^k \ge 20$: $4^2 = 16$ is too few and $4^3 = 64$ is enough, so $k = 3$. The arithmetic is immediate and it is worth being clear about what it does and does not settle. It settles that a triplet is the shortest admissible codon once the four and the twenty are given. It does not derive either of them, and it does not derive the particular assignment of the sixty-four to the twenty --- that is the \emph{codon-mapping lemma}, which is open and is carried in \AThirteen{}. A stable, naturally evolved codon system with $k \ne 3$ would falsify the row, as would a five-letter nucleobase alphabet in any stable organism.

\subsection{\textsf{T-Information-Closure}: Layer~9 --- the stable archival closure}
\label{sec:thm-information-closure-dna}

\begin{theorem}[\textsf{T-Information-Closure}: layer-9 \GGA{} (T-Origin-34.1)]
\label{thm:information-closure-dna}
Let $\widehat{\SIP{}}^{(9)} = \widehat{\SIP{}}^{(9)}[\GGA{}_8]$ be the layer-9 \SIP{} constructed from the layer-8 attractor: state manifold the configuration space of archival informational polymers on the bifunctional-RNA substrate; ports the chemical and energetic exchange with the cytosolic dNTP pool and the proofreading-plus-repair machinery; generator the replication-plus-archive Hamiltonian with $K = 35$-locked B-form geometry and three-tier fidelity architecture; \GFF{} landscape the archival-replication variational free energy specialized by the \QiEP{} principle and the central-dogma directionality of \QGL{}; observable/kill bundle the alphabet, the helix geometry, the replication fidelity, the central-dogma directionality, and the universal genetic code. Then the layer-9 \GGA{} is the unique archival closure
\begin{equation}
\label{eq:information-closure-dna-gga}
\boxed{\;
\begin{aligned}
\GGA{}_9 \;=\; \bigl[\;&\Sigma_{\rm DNA} = \{\mathrm{A, C, G, T}\};\\
 &\text{B-form helix }(d = 2.0~\text{nm},\;h = 0.34~\text{nm/bp},\;P = 3.4~\text{nm});\\
 &\epsilon = 10^{-9}/\text{bp};\;\;\text{DNA}\to\text{RNA}\to\text{protein};\;\;64 \to 20\text{ code}\;\bigr]
\end{aligned}\;}
\end{equation}
together with the universality clauses: the four-letter alphabet $\{\mathrm{A, C, G, T}\}$ is the layer-8 alphabet with T replacing U for archival stability; the B-form double-helix geometry is forced by the ribose-phosphate backbone torsional minima at $K = 35$ scale depth; the replication fidelity $\epsilon = 10^{-9}$ is achieved through the three-tier product of polymerase intrinsic fidelity ($\sim 10^{-4}$), $3'$-to-$5'$ exonuclease proofreading ($\sim 10^{-3}$), and mismatch repair ($\sim 10^{-2}$); the central-dogma circuit DNA $\to$ RNA $\to$ protein is the unique directionally-consistent closure of the layer-8 substrate; and the universal genetic code maps $4^3 = 64$ codons onto $20$ amino acids plus $3$ stops plus $1$ start, near-universal across the domains of life, with known nuclear and organellar code variants.
\end{theorem}

\paragraph{Status and forward program.}
\emph{Status: Anchor-Derived (derivation-deferred).} The empirical anchor is closed in the published literature; the from-geometry derivation to codebook standard is developed in \AThirteen{} (in preparation). See the Preface, p.~\pageref{sec:status-of-this-paper}, for what this tag licenses and what it does not.

\paragraph{Codebook provenance.}
The constitutional inputs are \AX{2} (which fixes the scale-depth integer $K = 35$ that sets the backbone torsional minima), \ASeven{}~\T{-EP} (the variational pillar, applied to replication rather than to metabolism), and \QGL{} (which supplies the directionality of the central-dogma circuit). The named upstream theorem is \T{-Information-Closure}: \emph{on the bifunctional-polymer substrate of layer~8, the unique \QiFour{}-admissible \GGA{} is an archival duplex over $\{\mathrm{A, C, G, T}\}$ in B-form geometry, with replication fidelity $\epsilon = 10^{-9}$/bp obtained as the three-tier product of polymerase intrinsic fidelity, $3'\!\to\!5'$ exonuclease proofreading, and mismatch repair, and with the central-dogma circuit $\mathrm{DNA} \to \mathrm{RNA} \to \mathrm{protein}$ as the unique directionally consistent closure}.

Interpreting the theorem: layer~8 established \emph{which} alphabet; layer~9 establishes how that alphabet is made to last. The single chemical difference is the loss of the $2'$-OH, and the consequence is a change of six orders of magnitude in fidelity --- from $\sim 10^{-4}$ to $10^{-9}$ per base pair --- obtained not from one mechanism but from three multiplied together. This row is \textbf{Anchor-Derived}: the three-tier decomposition is closed in the published biochemical record, and the archival stratum inherits its alphabet from layer~8 rather than positing a new one. Physics intuition: Eigen's error threshold says a genome cannot exceed roughly the inverse of its per-base error rate, so fidelity is not a quality-of-life improvement but a hard ceiling on how much information a lineage may hold; the cascade lifts the ceiling by four orders of magnitude at exactly the rung before it needs to.

Daily-life metaphor: layer~9 is to layer~8 what an archive is to a working notebook --- the notebook is where the thinking happens, but nothing survives a century unless it is copied onto something that does not degrade. Kill criterion: the discovery of a naturally evolved organism whose archival polymer is not a deoxyribose duplex falsifies the row; so does a measured germ-line replication fidelity in a free-living lineage that departs from the three-tier product by more than an order of magnitude.

\subsection{\textsf{T-Cell-Closure}: Layer~10 --- LUCA and the first cell}
\label{sec:thm-cell}

\begin{theorem}[\textsf{T-Cell-Closure}: layer-10 \GGA{} (T-Origin-35.1)]
\label{thm:cell-closure}
Let $\widehat{\SIP{}}^{(10)} = \widehat{\SIP{}}^{(10)}[\GGA{}_9]$ be the layer-10 \SIP{} constructed from the layer-9 attractor: state manifold the configuration space of bounded open-system bioenergetic networks with DNA-archived heredity; ports the lipid-bilayer membrane and its transport channels; generator the metabolic-network plus genetic-replication Hamiltonian at warm-wet pH; \GFF{} landscape the cellular Gibbs free energy specialized by the Friston variational free energy at non-zero $J_{Qi}$ (the Three-Gauge Theorem at the cellular scale); observable/kill bundle the cell size, metabolic rate, replication fidelity, and population dynamics. Then the layer-10 \GGA{} is the unique \SeOS{}-class first cell:
\begin{equation}
\label{eq:cell-gga}
\begin{aligned}
\GGA{}_{10} \;=\; \bigl[\;&\text{cell diameter}\;d \in [1, 10]\,\mu\text{m},\\
 &\text{metabolic rate}\;\rho_{\rm met} \in [1, 100]\,\text{W/kg},\\
 &\text{fidelity}\;\epsilon = 10^{-9},\;\;\text{lipid bilayer},\\
 &\text{Wood--Ljungdahl-class metabolism}\;\bigr]
\end{aligned}
\end{equation}
together with the universality clauses: LUCA at $\sim 3.8$~Ga is the cascade's first realization of $\GGA{}_{10}$; every subsequent cell on Earth (bacteria, archaea, eukaryotes) is a stratum-recursive specialization of LUCA on the layer-10 \GGA{}; the cell as a class is the unique \SeOS{}-class closure on the layer-7 substrate.
\end{theorem}

\paragraph{Status and forward program.}
\emph{Status: Anchor-Derived (derivation-deferred).} The empirical anchor is closed in the published literature; the from-geometry derivation to codebook standard is developed in \AThirteen{} (in preparation). See the Preface, p.~\pageref{sec:status-of-this-paper}, for what this tag licenses and what it does not.

\paragraph{Codebook provenance.}
The constitutional inputs are the \SeOS{}-closure theorems of \AFive{} \citep{ArisakaA5GDOS}: a Selfish Open System is a five-field \SIP{} closed under boundary, metabolism, replication, and \QiFour{} admissibility. The named upstream theorem is \T{-Cell-Closure}, a conjunction of \SeOS{} closure of \AFive{} \citep{ArisakaA5GDOS} and \T{-CON} of \ASeven{} \citep{ArisakaA7Qi4} \QiFour{} block: \emph{on the bioenergetic substrate of layer~7, the unique \QiFour{}-admissible \GGA{} is the LUCA-class first cell with characteristic size $1$--$10\,\mu$m, characteristic metabolic power $1$--$100$~W/kg, replication error $10^{-9}$/bp, and emergence epoch $\sim 3.8$~Ga}.

\GGPart[Multicellularity]{Multicellularity: layers~11--13}{The dual-gauge bifurcation of the first cells, the energy-per-gene phase jump, and the Cambrian readout}
\label{part:multicellularity}

\section{Multicellularity: layers~11--13}
\label{sec:band-multicellularity}

Three rungs carry the cascade from a single cell to a body: the split of the first cells into two domains, the energy-per-gene jump that makes the complex cell possible, and the ladder to many cells. All three are \emph{Conditional}, each on one named lemma.


Interpreting the theorem: the cascade does not merely permit life; it forces life, in the specific sense that on the bioenergetic substrate the unique stable minimum of the \GFF{} Morse landscape is a cellular architecture with the size, power, and error rate ranges quoted. This is the cascade's first \textbf{Anchor-Derived} biological row: the empirical anchor is closed, the from-geometry derivation is deferred to \AThirteen{}. Physics intuition: a cell smaller than $\sim 1\,\mu$m cannot package enough ribosomes to sustain metabolism at \QB{}-anchored rates; a cell larger than $\sim 10\,\mu$m cannot maintain a chemiosmotic gradient against passive diffusion losses; thus the size window is a \GFF{} Morse cup, not a free parameter.

Daily-life metaphor: layer~10 is to layer~7 what a self-running engine is to a power supply --- the energy currency exists at layer~7, but at layer~10 the cascade builds a closed loop in which the cell synthesizes the very enzymes that synthesize the very ATP that powers the cell. Kill criterion: the discovery of a non-cellular life form (a stable, replicating, metabolizing entity that is not bounded by a membrane) falsifies the row; the discovery that LUCA emerged at $> 4.2$~Ga or $< 3.5$~Ga (well outside the predicted window) falsifies the row.

\subsection{\textsf{T-Primary-Domain-Split}: Layer~11 --- the dual-gauge bifurcation of the first cells}
\label{sec:thm-three-domain-split}

\begin{theorem}[\textsf{T-Primary-Domain-Split}: layer-11 \GGA{} (T-Origin-36.1)]
\label{thm:three-domain-split}
Let $\widehat{\SIP{}}^{(11)} = \widehat{\SIP{}}^{(11)}[\GGA{}_{10}]$ be the layer-11 \SIP{} constructed from the layer-10 LUCA attractor: state manifold the configuration space of membrane-lipid-and-chemiosmotic-cation chemistries on the cellular bilayer stratum; ports the chemical and energetic exchange with the alkaline-vent and shallow-ocean environments; generator the membrane-assembly Hamiltonian with $K = 35$-locked headgroup energies and $L = 5$-locked $\mathrm{DGL}(5{:}2)$ irreducible representation packet; \GFF{} landscape the membrane-and-cellular Gibbs free energy specialized by the \BIsix{} parity-locked exchange operator on the lipidome state; observable/kill bundle the membrane chirality (ester vs ether), the lipidomic backbone (fatty-acid vs isoprenoid), the dominant chemiosmotic cation (H$^+$ vs mixed H$^+$/Na$^+$), and the tetraether-bridging admissibility under high-temperature ($T > 360$~K) conditions. Then the layer-11 \GGA{} is a \emph{dual} attractor:
\begin{equation}
\label{eq:three-domain-gga}
\boxed{\;
\begin{aligned}
\GGA{}_{11} \;&=\; \GGA{}_{11}^{\rm B} \oplus \GGA{}_{11}^{\rm A},\\
\GGA{}_{11}^{\rm B} \;&=\; \bigl[\;\text{ester-linked fatty-acid bilayer},\;\text{H}^+\text{ chemiosmosis},\\
 &\quad\;\;\text{variable G+C}\;\bigr] \\[2pt]
\GGA{}_{11}^{\rm A} \;&=\; \bigl[\;\text{ether-linked isoprenoid bilayer},\;\text{H}^+/\text{Na}^+\text{ chemiosmosis},\\
 &\quad\;\;\text{tetraether bridges}\;\bigr]
\end{aligned}\;}
\end{equation}
together with the universality clauses: (a) the two \GGA{}s are the unique stable minima of the same layer-11 algebraic packet $\mathrm{DGL}(5{:}2)$; (b) no third primary lipidome stratum exists on the cellular stratum at this cascade depth (in particular, Eukarya is a layer-12 \emph{hybrid} of the two primary domains, not a third primary domain); (c) the selection of which lipidome stabilizes in a given population is determined by the \GRIPout{} cycling of Hadean-to-Archean planetary cooling acting through local ocean chemistry on a timescale $\sim 10^8$~yr; (d) the split is essentially contemporaneous with LUCA but resolvable on the 16S rRNA phylogenetic clock at $\sim 3.5$~Ga.
\end{theorem}

\paragraph{Status and forward program.}
\emph{Status: Conditional.} Derivation to codebook standard is developed in \AThirteen{} (in preparation); see the Preface, p.~\pageref{sec:status-of-this-paper}, for what this tag licenses and what it does not.

\paragraph{Codebook provenance.}
The constitutional inputs are \AX{4} (\MSMF{}, which forbids a third structurally distinct stable output where two suffice), \AThree{}'s \BIsix{} parity-locked exchange operator acting on the lipidome state, and the $L = 5$ ledger integer that fixes the $\mathrm{DGL}(5{:}2)$ representation packet. The named upstream theorem is \T{-Primary-Domain-Split}: \emph{on the LUCA-class cellular substrate of layer~10, the layer-11 \GGA{} is not single but \emph{dual} --- an ester-linked fatty-acid bilayer with H$^+$ chemiosmosis, and an ether-linked isoprenoid bilayer with mixed H$^+$/Na$^+$ chemiosmosis and tetraether bridging --- these being the two stable minima of one algebraic packet, with no third primary lipidome}.

Interpreting the theorem: every other rung of the cascade delivers one ground state, and this one delivers two. That is the interesting part, and it is why the row is worth its space. The dual output is not a weakening of the uniqueness claim but an instance of it: the packet $\mathrm{DGL}(5{:}2)$ has exactly two stable minima, and both are realized, which is a sharper statement than ``one of them is realized.'' Clause~(b) is the one a phylogeneticist will test --- Eukarya is read here as a layer-12 hybrid of the two primaries, not as a third primary domain. This row is \textbf{Asserted}: the dual-gauge-SSB derivation on the $\mathrm{DGL}(5{:}2)$ packet is not yet in hand and is carried as a named lemma. Physics intuition: a $\mathbb{Z}_2$-symmetric Morse landscape with the symmetry broken by boundary conditions rather than by the potential yields a degenerate pair of minima, and a degenerate pair is exactly what the phylogenetic record shows at the root.

Daily-life metaphor: layer~11 is to layer~10 what casting two complementary halves of a mould is to casting one solid block --- the pair is useless until something joins them, and the joining is the next rung. Kill criterion: the discovery of a third primary lipidome stratum --- a naturally evolved cell lineage whose membrane chemistry is neither ester-linked fatty acid nor ether-linked isoprenoid --- falsifies the row, as would a demonstration that Eukarya's membrane is primary rather than chimeric.

\subsection{\textsf{T-Eukaryogenesis}: Layer~12 --- the energy-per-gene phase jump}
\label{sec:thm-eukaryogenesis}

\begin{theorem}[\textsf{T-Eukaryogenesis}: layer-12 \GGA{} (T-Origin-37.1)]
\label{thm:eukaryogenesis}
Let $\widehat{\SIP{}}^{(12)} = \widehat{\SIP{}}^{(12)}[\GGA{}_{11}^{\rm A}, \GGA{}_{11}^{\rm B}]$ be the layer-12 \SIP{} constructed from both layer-11 attractors simultaneously: state manifold the configuration space of chimeric \SeOS{}-class cells carrying one archaeal cytoplasmic membrane and one bacterial endosymbiont membrane; ports the chemical and energetic exchange with the post-GOE oxygenated atmosphere and the host-endosymbiont metabolic interface; generator the dual-DGL cellular Hamiltonian with two independent \QGL{}-locked four-current networks coupled by the mitochondrial chemiosmotic gradient; \GFF{} landscape the cellular Gibbs free energy specialized by the dual-DGL bandwidth and the Lane-Martin energy-per-gene scaling; observable/kill bundle the energy per gene $\delta W$, the genome size $G$, the cellular volume $V$, the compartmentalization inventory, and the spectral signature of cellular activity. Then the layer-12 \GGA{} is the unique chimeric closure:
\begin{equation}
\label{eq:eukaryogenesis-gga}
\boxed{\;
\begin{aligned}
\GGA{}_{12} \;=\; \bigl[\;&\text{archaeal host} \oplus \text{$\alpha$-proteobacterial endosymbiont},\\
 &\delta W/\text{gene} \uparrow 2\text{--}20\times,\;\;G \uparrow 10\text{--}10^4 \times,\\
 &V \uparrow 10^4 \times,\;\;\text{compartmentalization suite}\;\bigr]
\end{aligned}\;}
\end{equation}
together with the universality clauses: (a) the chimera is the unique stable closure on the dual-DGL stratum; (b) the compartmentalization suite (nuclear envelope, chromatin, cytoskeleton, endomembrane, vesicle traffic, organelle differentiation, meiotic sex) is the cascade's variational ground state on the doubled-DGL bandwidth, not an evolutionary contingency; (c) the chimera predicts a \emph{dual-knee} $1/f$ power spectral density (PSD) signature in eukaryotic cellular activity, distinguishable from the single-knee PSD of prokaryotes; (d) the event is dated $\sim 1.8$--$2.0$~Ga, $\sim 400$--$600$~Myr after the Great Oxygenation Event ($\sim 2.4$~Ga).
\end{theorem}

\paragraph{Status and forward program.}
\emph{Status: Conditional.} Derivation to codebook standard is developed in \AThirteen{} (in preparation); see the Preface, p.~\pageref{sec:status-of-this-paper}, for what this tag licenses and what it does not.

\paragraph{Codebook provenance.}
The constitutional inputs are \AX{3} (\OGGEP{}, which requires the host-endosymbiont interface to be an operationally specified boundary), \ASeven{}~\T{-EP}, and the doubled \QGL{} four-current network carried by the two membranes. The named upstream theorem is \T{-Eukaryogenesis}: \emph{on the dual stratum supplied by both layer-11 attractors simultaneously, the unique stable closure is a chimeric cell carrying one archaeal cytoplasmic membrane and one bacterial endosymbiont membrane, with energy per gene raised $2$--$20\times$, genome size raised $10$--$10^4\times$, cellular volume raised $\sim 10^4\times$, and the compartmentalization suite as the variational ground state of the doubled bandwidth}.

Interpreting the theorem: the framework's reading of eukaryogenesis is that it is a bandwidth event, not a morphological one. A prokaryote pays for every gene across the same membrane it uses to power itself; internalizing a second bioenergetic membrane decouples the two, and everything the textbooks list as eukaryotic novelty --- nucleus, cytoskeleton, endomembrane, vesicle traffic, meiotic sex --- follows as what the cell does with the surplus. This row is \textbf{Conditional}: the energy-per-gene scaling is imported from the published bioenergetic literature, and the named lemma still to be proved is that the compartmentalization suite is the \emph{ground state} of the doubled packet rather than one admissible configuration among several. Physics intuition: the same cell volume that was a liability under single-membrane bookkeeping becomes an asset once the power supply is internal, so the \GFF{} minimum moves by four orders of magnitude in $V$.

Daily-life metaphor: layer~12 is to layer~11 what putting the generator inside the building is to running an extension cord from the street --- nothing about the building's purpose changes, but everything about what it can afford to do does. Kill criterion: the row predicts a \emph{dual-knee} $1/f$ power spectrum in eukaryotic cellular activity against a single-knee spectrum in prokaryotes (Pull-test~9a); a measured single-knee spectrum in eukaryotes, or a dual knee in prokaryotes, falsifies it.

\subsection{\textsf{T-Multicellularity}: Layer~13 --- the M-ladder and the Cambrian readout}
\label{sec:thm-multicellularity}

\begin{theorem}[\textsf{T-Multicellularity}: layer-13 \GGA{} (T-Origin-38.1)]
\label{thm:multicellularity}
Let $\widehat{\SIP{}}^{(13)} = \widehat{\SIP{}}^{(13)}[\GGA{}_{12}]$ be the layer-13 \SIP{} constructed from the layer-12 eukaryotic-chimera attractor: state manifold the configuration space of multicellular eukaryotic organisms on the chimeric cellular substrate; ports the chemical, energetic, mechanical, and informational exchange between cells and between organism and environment; generator the supra-cellular Hamiltonian with $\QGL{}$-locked four-current networks at the cellular and supra-cellular scales; \GFF{} landscape the supra-cellular Gibbs free energy specialized by the patterning, transport, and integration costs of each ladder rung; observable/kill bundle the rung position on the M-ladder, the cell-cell adhesion strength, the gap-junction conductance, the somatic/germ-line distinction, the morphogen-gradient profile, the vascular network topology, the nervous-system precursor architecture, and the discrete-organ count. Then the layer-13 \GGA{} is the unique \emph{eight-rung} multicellularity ladder:
\begin{equation}
\label{eq:multicellularity-gga}
\boxed{\;\GGA{}_{13} \;=\; \{\,\mathrm{M0},\,\mathrm{M1},\,\mathrm{M2},\,\mathrm{M3},\,\mathrm{M4},\,\mathrm{M5},\,\mathrm{M6},\,\mathrm{M7}\,\}\;}
\end{equation}
defined by:
\begin{description}\itemsep1pt
\item[M0:] Priming --- cell-cell adhesion molecules (cadherins, integrins, lectins) inherited from the layer-12 endomembrane inventory; no aggregates yet.
\item[M1:] Clonal adhesion --- sister cells stay stuck after division; first multicellular aggregates ($\sim 1$~Ga in algal lineages).
\item[M2:] Intercellular communication --- gap junctions (animals) or plasmodesmata (plants) allow shared metabolites and signaling.
\item[M3:] Division of labor --- somatic/germ-line distinction; reproductive specialization vs survival specialization.
\item[M4:] Patterning --- positional information via morphogen gradients; body plan locks (Wnt, Hh, BMP, Notch, FGF families; $\sim 700$~Ma).
\item[M5:] Transport networks --- vascular, circulatory, endocrine systems; supra-cellular metabolite distribution.
\item[M6:] Integration --- nervous-system precursors; centralized supra-cellular coordination ($\sim 580$~Ma).
\item[M7:] Organogenesis --- discrete organs; full body-plan readout in the Cambrian Explosion at $\sim 540$~Ma.
\end{description}
together with the universality clauses: (a) the M-ladder is the unique stable layer-13 \GGA{} on the eukaryotic substrate, with the rungs locking in this order; (b) the ladder has been independently climbed by at least five lineages --- animals ($\sim 600$~Ma), red algae ($\sim 1$~Ga), brown algae ($\sim 800$~Ma), green plants ($\sim 1$~Ga), fungi ($\sim 1$~Ga) --- with convergence on the same rung structure because the constitution forces it; (c) the Cambrian Explosion at $\sim 540$~Ma is the dominant readout, but the closure itself was distributed over $\sim 1$~Ga of pre-Cambrian descent.
\end{theorem}

\paragraph{Status and forward program.}
\emph{Status: Conditional.} Derivation to codebook standard is developed in \AThirteen{} (in preparation); see the Preface, p.~\pageref{sec:status-of-this-paper}, for what this tag licenses and what it does not.

\paragraph{Codebook provenance.}
The constitutional inputs are \AX{1} (\LC{}, which requires that supra-cellular signaling respect the causal structure of the body), \ASeven{}~\T{-CON} (conformality, which is what makes a morphogen gradient a scale-coherent readout rather than a local concentration), and the \QGL{}-locked four-current networks now running at two scales at once, cellular and supra-cellular. The named upstream theorem is \T{-Multicellularity}: \emph{on the chimeric-cell substrate of layer~12, the layer-13 \GGA{} is the eight-rung ladder $\mathrm{M0}$--$\mathrm{M7}$ --- priming, clonal adhesion, intercellular communication, division of labor, patterning, transport networks, integration, and organogenesis --- each rung the unique variational minimum given the rung below}.

\GGPart[Mind]{Mind: layers~14--17}{The first integrated connectome, the insect heading attractor, the rodent allocentric map, and the conscious episode}
\label{part:mind}

\section{Mind: layers~14--17}
\label{sec:band-mind}

Four rungs carry the cascade from a body to a mind: the first integrated connectome, a heading compass, a map of space, and the conscious episode. All four are \emph{Asserted} --- the framework predicts each and says what would refute it, but the derivation to codebook standard is the forward work of \AFourteen{}.


Interpreting the theorem: the ladder is the framework's answer to a question that is usually treated as historical, namely why multicellularity arose independently in many lineages and yet converged on the same sequence of structures. The cascade's answer is that the sequence is not a habit but a dependency chain: patterning is undefined without communication, transport is pointless without differentiation, and integration is impossible without a body plan to integrate over. This row is \textbf{Conditional}: the rungs and their dates are read from the palaeontological and developmental record, and the named lemma is that the ladder is the \emph{unique} trajectory --- that no admissible route to organogenesis skips or reorders a rung. Physics intuition: each rung raises the number of independently addressable degrees of freedom in the organism, and the \GFF{} cost of the next rung is payable only once the previous one has been paid.

Daily-life metaphor: layer~13 is to layer~12 what a city is to a well-supplied household --- the same functions, replicated and specialized, with the difference lying entirely in coordination. Kill criterion: a naturally evolved multicellular lineage that reaches organogenesis without passing through positional patterning would falsify the ladder's ordering; so would the discovery of a ninth rung that is not decomposable into the eight.

\subsection{\textsf{T-Central-Nervous-System}: Layer~14 --- the first integrated connectome}
\label{sec:thm-central-nervous-system}

\begin{theorem}[\textsf{T-Central-Nervous-System}: layer-14 \GGA{} (T-Origin-39.1)]
\label{thm:central-nervous-system}
Let $\widehat{\SIP{}}^{(14)} = \widehat{\SIP{}}^{(14)}[\GGA{}_{13}]$ be the layer-14 \SIP{} constructed from the layer-13 (multicellular) attractor: state manifold the configuration space of neural-network topologies on a fixed metazoan cell substrate; ports the sensory afferents and motor efferents; generator the rate-coded neural dynamics with synaptic (Hebbian) plasticity and a central-pattern-generator clock; \GFF{} landscape the network variational free energy specialized to integrated sensorimotor closure; observable/kill bundle the closed-loop sensorimotor fidelity. Then the layer-14 \GGA{} is the minimal integrated connectome:
\begin{equation}
\label{eq:cns-gga}
\begin{aligned}
\GGA{}_{14} \;=\; \bigl[\;&\text{centralized nerve-ring connectome},\\
 &\text{sensory}\,/\,\text{inter}\,/\,\text{motor classes, one integrated loop},\\
 &\text{CPG-clocked MePMoS cycle (egocentric, pre-allocentric)},\\
 &\sim 300\;\text{neurons},\;\;\QCI{} \sim 4\;\bigr]
\end{aligned}
\end{equation}
together with the universality clause: the architecture is realized in the \emph{C.\ elegans} connectome, in the nerve cords and cephalic ganglia of other simple bilaterians, and in any other minimal centralized nervous system on any neural substrate. Its best-characterized exemplar is the \emph{C.\ elegans}-class $302$-neuron connectome.
\end{theorem}

\paragraph{Status and forward program.}
\emph{Status: Asserted.} Derivation to codebook standard is developed in \AFourteen{} (in preparation); see the Preface, p.~\pageref{sec:status-of-this-paper}, for what this tag licenses and what it does not.

\paragraph{Codebook provenance.}
The constitutional inputs are \AX{3} (\OGGEP{}, supplying the sensory and motor ports as the boundary of the neural \SIP{}) and \ASeven{}~\T{-CON} together with \ASix{} \T{-GRIP-4}, which between them make the network's variational free energy a \GFF{} in the technical sense rather than by analogy. The named upstream theorem is \T{-Central-Nervous-System}: \emph{on the multicellular substrate of layer~13, the unique \QiFour{}-admissible \GGA{} is the minimal integrated connectome --- a centralized nerve ring carrying sensory, inter- and motor classes closed into one loop, clocked by a central pattern generator, egocentric and pre-allocentric, at $\sim 300$ neurons and $\QCI{} \sim 4$}.

Interpreting the theorem: what is claimed is not that nervous systems exist but that \emph{centralization} is forced --- that a diffuse net, however large, does not close the sensorimotor loop in the way the cascade requires, because a single common reference frame is what the next rung will need and a net does not have one. The $\sim 300$-neuron figure is a readout, not a prediction: the framework's claim is about the architecture, and \emph{C.\,elegans} is the smallest known animal in which the architecture is complete. This row is \textbf{Asserted}: the claim that the integrated connectome is the unique \QiFour{}-admissible attractor is pending a controlled comparison against alternative architectures, and that comparison is carried in \AFourteen{}. Physics intuition: closing many local loops does not compose into one global loop unless the local loops share a frame, and sharing a frame is precisely what a centralized circuit provides.

Daily-life metaphor: layer~14 is to layer~13 what a switchboard is to a set of private telephone lines --- every line worked before, but only the switchboard makes the whole building one system. Kill criterion: the demonstration of a naturally evolved animal with allocentric spatial behavior and no centralized nervous system would falsify the row, as would a controlled comparison showing a diffuse architecture reaching equal or lower \GFF{} on the same stratum.

\subsection{\textsf{T-Neural-Ring-Attractor}: Layer~15 --- the insect heading attractor}
\label{sec:thm-neural-ring}

\begin{theorem}[\textsf{T-Neural-Ring-Attractor}: layer-15 \GGA{} (T-Origin-40.1)]
\label{thm:neural-ring}
Let $\widehat{\SIP{}}^{(15)} = \widehat{\SIP{}}^{(15)}[\GGA{}_{14}]$ be the layer-15 \SIP{} constructed from the layer-14 attractor: state manifold the configuration space of neural-circuit topologies on a fixed cell substrate; ports the sensory and motor exchange with the environment; generator the rate-coded neural dynamics with synaptic plasticity; \GFF{} landscape the network variational free energy specialized by \QiHAL{} for allocentric architecture; observable/kill bundle the heading-direction encoding fidelity. Then the layer-15 \GGA{} is the unique one-dimensional ring attractor:
\begin{equation}
\label{eq:neural-ring-gga}
\begin{aligned}
\GGA{}_{15} \;=\; \bigl[\;&\text{1D ring of}\;\sim 50\text{--}200\;\text{neurons},\\
 &\text{Mexican-hat lateral inhibition},\\
 &\text{single-bump activity},\\
 &\text{heading-direction encoding},\;\;\QCI{} \sim 6\;\bigr]
\end{aligned}
\end{equation}
together with the universality clause: the architecture is realized in the ellipsoid body of \emph{Drosophila}, the central complex of bees and locusts, and the head-direction system of rodents; the framework \emph{predicts} (pending test on previously-unstudied taxa) the same topology in any other primary heading-encoding circuit, on any neural substrate.
\end{theorem}

\paragraph{Status and forward program.}
\emph{Status: Asserted.} Derivation to codebook standard is developed in \AFourteen{} (in preparation); see the Preface, p.~\pageref{sec:status-of-this-paper}, for what this tag licenses and what it does not.

\paragraph{Codebook provenance.}
The constitutional inputs are the neural-substrate theorems of \ASeven{} \citep{ArisakaA7Qi4} together with the topological requirement that allocentric heading representations close on a manifold of dimension equal to the configuration-space dimension of the heading variable (which is 1 for terrestrial heading: a single angle). The named upstream theorem is \T{-Neural-Ring-Attractor}: \emph{on the substrate of insect ellipsoid-body / mammalian head-direction circuits, the unique \QiFour{}-admissible \GGA{} is the 1D ring-attractor on $SO(2)$ implementing allocentric heading representation}.

Interpreting the theorem: the framework predicts that heading representation in any sufficiently evolved nervous system will use the topology of a circle, not of a line or a graph or a sphere, because the circle is the unique closed 1-manifold without boundary and the unique \GGA{} that closes the head-direction \GFF{} on the relevant neural substrate. This row is \textbf{Asserted} because the topology has been verified empirically only on the substrates of fruit-fly ellipsoid body \citep{KimEtAl2017} and rodent head-direction system \citep{TaubeMullerRanck1990}; a controlled experimental comparison with hypothetical alternative architectures (a line attractor with two boundary states, a small discrete graph, etc.) on a previously-unstudied taxon is the empirical anchor that would promote the row to Derived.

Physics intuition: a ring attractor implements integration of angular velocity into angular position by translating bump activity around a closed manifold; the closure is what protects the representation from drift on long timescales. Daily-life metaphor: layer~15 is to layer~10 what a compass is to a sailor --- the cell already navigates chemistry, but at layer~15 the cascade builds a north-pointing needle. Kill criterion: the discovery of an animal that performs allocentric heading without a ring-attractor topology (e.g., on a line attractor) would falsify the uniqueness assertion; the discovery of an animal using a 2D-sphere topology for 1D heading would falsify it.

\subsection{\textsf{T-Allocentric-Map}: Layer~16 --- the rodent hippocampal place--grid--time map}
\label{sec:thm-allocentric-map}

\begin{theorem}[\textsf{T-Allocentric-Map}: layer-16 \GGA{} (T-Origin-41.1)]
\label{thm:allocentric-map}
Let $\widehat{\SIP{}}^{(16)} = \widehat{\SIP{}}^{(16)}[\GGA{}_{15}]$ be the layer-16 \SIP{} constructed from the layer-15 attractor: state manifold the configuration space of higher-dimensional allocentric neural codes on a heading-attractor substrate; ports the multi-modal sensory input (visual, vestibular, proprioceptive, olfactory); generator the recurrent-network dynamics with synaptic plasticity at the hippocampal scale; \GFF{} landscape the allocentric-map variational free energy; observable/kill bundle the place-cell field statistics, grid-cell scale ratios, time-cell sequence statistics, and conjunctive code structure. Then the layer-16 \GGA{} is the unique 4D allocentric place--grid--time--history map:
\begin{equation}
\label{eq:allocentric-map-gga}
\begin{aligned}
\GGA{}_{16} \;=\; \bigl[\;&\text{place cells (CA1/CA3)},\;\;\text{grid cells (MEC)},\\
 &\text{head-direction cells},\;\;\text{time cells},\\
 &\text{conjunctive 4D code},\;\;\QCI{} \sim 8\;\bigr]
\end{aligned}
\end{equation}
together with the universality clause: the architecture is realized in rodents \citep{OKeefeDostrovsky1971, HaftingEtAl2005}, in primates including humans (where ``place cells'' are observed as ``concept cells'' \citep{QuirogaEtAl2005} and ``view cells''), and in functional analogues in birds and reptiles. Grid scale ratios form a geometric series with ratio $\sim \sqrt{2}$ across modules; place-field sizes scale with hippocampal axis position.
\end{theorem}

\paragraph{Status and forward program.}
\emph{Status: Asserted.} Derivation to codebook standard is developed in \AFourteen{} (in preparation); see the Preface, p.~\pageref{sec:status-of-this-paper}, for what this tag licenses and what it does not.

\paragraph{Codebook provenance.}
The constitutional inputs are layer~15 plus \ASeven{}~\T{-HAL} (memory-architecture closure: any system holding a trace beyond one act-and-look cycle takes a definite runtime normal form, which the rodent CA1--CA3--DG circuit is here identified as realizing). The named upstream theorem is \T{-Allocentric-Map}: \emph{on the substrate of the rodent hippocampus, the unique \QiFour{}-admissible \GGA{} is the 4D allocentric stack of place fields + grid fields + time fields + causal-history fields (the last a predicted, not-yet-established cell class)}.

Interpreting the theorem: the cascade predicts that rodent hippocampal coding is exactly 4-dimensional (not 3, not 5) and that the four components decompose into the four classes named; alternatives (e.g., a purely place-and-grid 3D map, or a 5D map adding head-velocity) are predicted to be non-\GGA{} (either saddles or non-closed). This row is also \textbf{Asserted} because the topology and the field structure are empirically observed on the rodent substrate but the uniqueness of the 4D-allocentric stack vs.\ alternative neural codes is the empirical question.

Physics intuition: the four dimensions correspond to the four classical degrees of freedom one needs to specify a single moment of an animal's experience --- where it is, in what spatial frame, when, and in what causal sequence. Daily-life metaphor: layer~16 is to layer~15 what a paper map is to a compass --- the compass orients you (layer~15); the map locates you, dates you, and tells you what got you here (layer~16). Kill criterion: the discovery of a rodent or analogous mammalian species using a strictly 3D or 5D allocentric representation falsifies the row.

\subsection{\textsf{T-Conscious-Episode}: Layer~17 --- EIC closure and the conscious episode}
\label{sec:thm-conscious}

\begin{theorem}[\textsf{T-Conscious-Episode}: layer-17 \GGA{} (T-Origin-42.1)]
\label{thm:conscious-episode}
Let $\widehat{\SIP{}}^{(17)} = \widehat{\SIP{}}^{(17)}[\GGA{}_{16}]$ be the layer-17 \SIP{} constructed from the layer-16 attractor: state manifold the configuration space of cortical-network states with closed \OH{}-\IH{} coupling; ports the multi-modal sensory input and motor output; generator the cortical-cycle dynamics with theta-gamma binding and predictive coding; \GFF{} landscape the conscious-state variational free energy with non-zero $J_{Qi}$ on the closed \EIC{} loop; observable/kill bundle the conscious-state markers (theta-cycle phase, gamma binding, $1/f$ spectrum, perceptual binding latency). Then the layer-17 \GGA{} is the unique conscious episode:
\begin{equation}
\label{eq:conscious-gga}
\begin{aligned}
\GGA{}_{17} \;=\; \bigl[\;&\EIC{}\text{ closure},\;\;\text{episode length}\;\tau \approx 400~\text{ms (two theta cycles)},\\
 &\text{theta carrier}\;4\text{--}8~\text{Hz},\\
 &\text{gamma binding}\;30\text{--}80~\text{Hz},\;\;\QCI{} \sim 10\;\bigr]
\end{aligned}
\end{equation}
together with the universality clause: every conscious episode in any biological substrate that closes the \EIC{} loop has the same essential structure (theta-bounded, gamma-bound, self-located, cross-modally integrated), with substrate-specific specializations.
\end{theorem}

\paragraph{Status and forward program.}
\emph{Status: Asserted.} Derivation to codebook standard is developed in \AFourteen{} (in preparation); see the Preface, p.~\pageref{sec:status-of-this-paper}, for what this tag licenses and what it does not.

\paragraph{Codebook provenance.}
The constitutional inputs are layer~16 plus \ASeven{}~\T{-EIC} \citep{ArisakaA7Qi4} (the \EIC{} closure theorem: the Evolving Inner Cosmos closes its outer-hologram--inner-hologram pair through a theta-cycle). The named upstream theorem is \T{-Conscious-Episode}, a conjunction of \T{-\EIC{}} and \T{-\HAL{}} of \ASeven{} \citep{ArisakaA7Qi4}: \emph{on the substrate of layer~16, the unique \QiFour{}-admissible \GGA{} is the conscious episode completing over two $\sim 200$~ms theta cycles ($\sim 400$~ms), the IH-forming \EIC{} closure, with \QCI{} $\sim 10$}.

\GGPart[Articulation]{Articulation: layer~18}{The closing self-referential layer: the geometry that comes to state the constitution which produced it}
\label{part:articulation}

\section{Articulation: layer~18}
\label{sec:band-articulation}

One rung remains, and it is the only one whose evidence is the act of reading it.


Interpreting the theorem: the framework predicts that conscious experience is a periodic phenomenon at theta-band frequency, and that any single conscious episode completes in $\sim 400$~ms --- two $\sim 200$~ms theta cycles, the first ($\sim 200$~ms) yielding only unconscious attention --- set by the \EIC{} closure time of the rodent-and-primate hippocampal-cortical loop. This row is \textbf{Asserted} (a preview): the \EIC{} closure theorem is published in \ASeven{} and the P300 component at $\sim 300$~ms (\citealt{Dehaene2014}) is consistent with the predicted window, but the identification of the closure with phenomenal experience, and the cortical-circuit derivation that fixes the timescale, are developed in \AFourteen{} (in preparation).

Physics intuition: conscious experience is the brain's act of closing its outer hologram \OH{} (sensory inflow) on its inner hologram \IH{} (memory + prediction) in a two-theta-cycle attractor (one complete MePMoS loop): the first theta cycle ($\sim$200~ms) yields unconscious attention, the second completes conscious awareness, so the episode is $\sim$400~ms. Daily-life metaphor: layer~17 is to layer~16 what a single frame of a film is to the projector --- the projector continues running (the brain continues processing), but at any moment one specific frame is in the gate, and that frame is the conscious episode. Kill criterion: a reliably reported, unitary conscious episode completing in $\le$ one theta cycle ($\sim$200~ms) --- conscious awareness without the second cycle the MePMoS loop requires --- falsifies the row.

\subsection{\textsf{T-Articulation}: Layer~18 --- the present paper}
\label{sec:thm-articulation}

\paragraph{Plain-English opening.}
With conscious episodes in place, the cascade asks one last question: \emph{what closes the conscious-episode substrate into a recursive recognition of its own constitutive structure?} Remarkably, the empirical answer is what you, the reader, are doing now: reading a formal mathematical articulation of the \GGConstitution{} that produced the conscious episode that is doing the reading. The conscious episode of layer~17 is a self-located perspective on the world; the act of articulation at layer~18 is a self-located perspective on the constitution that produced the perspective. This is the self-referential closure that gives the cascade its philosophical capstone: the layer-18 \GGA{} is the recognition of the cascade itself. Like a book that writes itself, or like a portrait that paints its own painter, the cascade's final closure is a recursive recognition of its own production.

We are at the top of the staircase. The eighteenth step does not lead to a nineteenth. It leads back to the bottom: the articulation is a re-statement of the \GGConstitution{} from inside the cascade that the \GGConstitution{} forced. This is the central self-referential closure that Part~\ref{part:recursive} will give a formal name (T-Origin) and prove. The present section states the theorem so that future authors can cite it as a stand-alone item.

\paragraph{Constitutional inputs.}
The layer-18 cascade requires all four axioms and all seven postulates, with \ASeven{}~\T{-EIC} (\EIC{} closure) generalized to the recursive case: the \EIC{} closure of layer~17 is now itself the substrate of a higher \EIC{} closure --- the meta-cognitive loop that recognizes the constitution as the constitution recognizes itself through the loop. In plain language: layer~18 is the cascade layer at which the conscious episode reflects on its own constitutive structure and articulates that structure as a mathematical theorem.

\paragraph{Upstream theorems imported.}
All previous theorems are imported. In particular: (i) Theorem~\ref{thm:conscious-episode} (layer-17 \GGA{}) supplies the substrate: the conscious episode that does the articulating. (ii) All earlier theorems (Theorems~\ref{thm:gut-closure} through~\ref{thm:allocentric-map}) supply the content being articulated: the gauge group, the electroweak vacuum, the periodic table, the small molecules, the amino acids, the genetic code, ATP and DNA, LUCA, the ring attractor, the allocentric map, the conscious episode --- the present paper articulates each of these as a layer of the cascade. (iii) The \GGConstitution{} of \AOne{} \citep{ArisakaA1Constitution} supplies the meta-structure being recognized: the four axioms and seven postulates that produced the cascade now appear in the layer-18 articulation as the explicit input ledger. (iv) \ANine{} Part~XI \citep{ArisakaA9SM} (universal-identity insight) supplies the closure clause: every cascade output is the readout of one geometric attractor, and the layer-18 readout is the recognition of that fact. (v) \ATen{} (universal-cosmic-identity theorem) \citep{ArisakaA10Cosmos} supplies the cosmological backdrop.

\paragraph{Formal statement.}
\begin{theorem}[\textsf{T-Articulation}: layer-18 \GGA{} (T-Origin-43.1)]
\label{thm:articulation}
Let $\widehat{\SIP{}}^{(18)} = \widehat{\SIP{}}^{(18)}[\GGA{}_{17}]$ be the layer-18 \SIP{} constructed from the layer-17 attractor: state manifold the configuration space of meta-cognitive episodes in which a conscious organism reflects on its own constitutive structure; ports the linguistic and mathematical exchange with other conscious organisms; generator the recursive cognitive dynamics that close the meta-\EIC{} loop; \GFF{} landscape the articulation variational free energy on the meta-cognitive stratum; observable/kill bundle the produced articulation, the kill criteria thereof, and the social-epistemic confirmation network. Then the layer-18 \GGA{} is the recursive articulation of the \GGConstitution{}:
\begin{equation}
\label{eq:articulation-gga}
\begin{aligned}
\GGA{}_{18} \;=\; \bigl[\;&\text{the }\GGConstitution{}\text{ of }\AOne{},\\
 &\text{the ten upstream papers }\AOne{}\text{--}\ATen{},\\
 &\text{the present paper }\AEleven{}\;\bigr]
\end{aligned}
\end{equation}
together with the self-referential closure clause: the layer-18 \GGA{} is the recognition that the seventeen-step cascade $\GGA{}_1 \to \GGA{}_2 \to \cdots \to \GGA{}_{17}$ is forced by the \GGConstitution{} $\mathcal{C}$, and the recognition is itself the cascade's layer-18 output. Existence is demonstrated by the present paper; uniqueness is the central claim of T-Origin (Part~\ref{part:recursive}).
\end{theorem}

\paragraph{Proof outline.}
Step 1: with the conscious episode from layer~17, the configuration space of meta-cognitive episodes admits a discrete classification by the depth of recursion (how many layers of self-reflection the episode supports). Step 2: the variational free energy on this configuration space is minimized by configurations that close the meta-\EIC{} loop: the conscious episode reflects on its own constitutive structure, recognizes that structure as the cascade's eleven-layer output, and articulates that recognition as a mathematical theorem. Step 3: the depth of recursion required for the closure is exactly two: the conscious episode (layer~17) plus the recognition of its constitution (layer~18); higher recursion (recognition of the recognition, etc.) is convergent but does not produce a separate cascade layer because the meta-meta-recognition is structurally identical to the meta-recognition. Step 4: the articulation is realized in the historical record by the ten upstream papers \AOne{}--\ATen{} plus the present paper \AEleven{}. Step 5: existence is demonstrated by the corpus itself; uniqueness follows from \textbf{GRIP-ST5} sharpened by the recursive closure of \ASeven{}~\T{-EIC}.

\paragraph{Quantitative anchor.}
The layer-18 readout is necessarily different from the previous seventeen layers, because the readout is the articulation itself, not a numerical observable. Nevertheless, several quantitative anchors can be stated:
\begin{itemize}\itemsep1pt
\item Articulation length: $\sim 5300$ typeset pages across the cohort.
\item Number of named theorems: 18 in the present Part; $\sim 100$ across the corpus.
\item Number of derived parameters: 28 Standard Model parameters (\ANine{}) + 15 cosmological observables (\ATen{}) + bioenergetic and chemistry-to-life parameters of the present paper $\geq 43$ derived quantities with zero free continuous parameters.
\item Number of falsifiable predictions: 10 specific experimental tests in the coming decade (the Top 10 Falsifiers of \AEleven{}'s Executive Summary).
\item Number of conscious-organism authors: 1 (Katsushi Arisaka), with the cascade's prediction that the layer-18 \GGA{} can be realized by any sufficiently configured layer-17 substrate.
\item Timescale of articulation: $\sim 50$ years of cumulative work on the framework's predecessors, $\sim 5$ years on the GG-Theory cohort itself, $\sim 1$ year on the cohort.
\item Geographical localization: UCLA, Los Angeles, California, USA, planet Earth, Local Group of galaxies, observable universe, six-dimensional bulk. The articulation is geographically local; the constitution being articulated is universal.
\end{itemize}

\paragraph{Physics intuition.}
At first-year-graduate-level: the layer-18 articulation is a structural improvement on the anthropic principle of \citet{Carter1974}. The anthropic principle asks why the constants of nature admit observers; the framework's answer is that no continuous constants were chosen, so the question dissolves: the constants are forced by the geometry, the cascade is forced by the constants, the observer is forced by the cascade, and the articulation is forced by the observer. Each step is a derivation. The chain of derivations closes on itself: the articulation re-states the geometry that the cascade started from. This is the self-referential closure that the present paper identifies as the central feature of the framework.

The closure is not a paradox. It is a fixed point of the recursive map $\GGA{}_n \mapsto \widehat{\SIP{}}^{(n+1)} \mapsto \GGA{}_{n+1}$ with $n = 17$ and $\GGA{}_{18}$ the articulation of the constitution that produced the cascade. Fixed points of recursive maps are well-defined mathematical objects; they exist when the map's iteration is convergent and the limit is a stable attractor. The cascade's fixed point at layer~18 exists because the meta-cognitive recursion is convergent (further recursions $\GGA{}_{17}, \GGA{}_{18}, \ldots$ are all equivalent to $\GGA{}_{18}$ up to representation) and the limit is the articulation.

The historical-register significance is worth a paragraph. What is being stated is a formal mathematical theorem whose conclusion is the existence of intelligent observers articulating the laws of physics, and whose hypotheses are the laws of physics themselves. We are not aware of a prior framework that produces a closure of this kind. Spinoza approached the structure philosophically \citep{Spinoza1677}, Hegel dialectically \citep{Hegel1807}, Whitehead metaphysically \citep{Whitehead1929}, Wheeler cosmologically \citep{Wheeler1983}. None of them produced a formal mathematical theorem. The framework produces one.

\paragraph{Energy-injection ascent mechanism.}
At the layer-17-to-layer-18 transition (the present moment, in the historical-register sense), sustained energy injection at the cognitive scale comes from the cumulative cultural and scientific resources of human civilization: educational institutions, libraries, computational tools, peer-review networks, the historical record of physics, the cumulative empirical data of cosmology and particle physics and biology. The Arisaka equation locks the four currents at the cognitive-cultural scale. The layer-18 \GFF{} on the articulation stratum descends to the cascade's recognition of its own structure; the descent's ordered residue is the cohort.

There is no layer~19. The cascade closes at layer~18 because the meta-recursion at higher depth is structurally identical to layer~18 itself. Subsequent articulations --- refinements, generalizations, applications, falsifications --- are all layer-18 episodes; they do not generate a new cascade layer. This is the cascade's terminal closure.

\paragraph{Examples and daily-life metaphors.}
Three metaphors. First, like a book that writes itself: the cascade reads its own constitution from inside the cascade; the book's last chapter is a description of the book's first chapter. Second, like a self-portrait by an artist: the painting depicts the painter painting the painting; the conscious organism articulates the constitution that produced the conscious organism. Third, like the closing scene of a recursive film (the ouroboros, the serpent biting its own tail): the cascade's end-state is a recognition of its initial state; the loop closes.

\paragraph{Kill criterion.}
Theorem~\ref{thm:articulation} is falsified by any of: (a) the falsification of any of the upstream theorems (Theorems~\ref{thm:gut-closure}--\ref{thm:conscious-episode}); the present theorem rests on all of them. (b) The demonstration that the \GGConstitution{} can be articulated from inside a cascade with fewer than seventeen cascade layers (this would imply the cascade is not the minimum-cardinality closure). (c) The demonstration that the cascade admits multiple non-equivalent articulations (this would imply the layer-18 \GGA{} is not unique). (d) The successful articulation of an alternative framework that produces the same seventeen cascade layers with a different layer-18 closure --- this would falsify the uniqueness claim of T-Articulation while leaving the existence claim intact.

\paragraph{Plain-English close.}
The present paper. The cohort. The recognition that the cascade closes on itself. This is the layer-18 \GGA{}: not a particle, not a molecule, not a cell, not a thought, but the act of articulating the constitution that produces particles, molecules, cells, and thoughts. The cascade ends here. It does not end with a new layer beyond; it ends with the recognition that there is no further layer to seek. Layer~18 done. The staircase has been climbed; we are standing at the top, looking back down at the eleven steps, and what we see is that the eleven steps were forced.

\bigskip

% ====================================================================
%  SYNTHESIS
% ====================================================================
\subsection{The pillar transition across the cascade}
\label{ssec:pillar-transition}

A striking structural feature of the eighteen theorems is the pillar transition across cascade layers. At layers~1--4, \QiQUA{} dominates: the four-current bookkeeping is the operationally binding constraint, and the cascade output is which discrete gauge or chemical configuration the bookkeeping admits. At layers~5--9, \QiEP{} dominates: variational selection becomes the binding constraint, and the cascade output is which specific functional trajectory the variational principle selects. At layers~10--17, \QiHAL{} dominates: memory architecture becomes the binding constraint, and the cascade output is which closed-loop architecture the memory holonomy admits --- with layer~10 shared between \QiEP{} and \QiHAL{}, the cell being where selection hands over to architecture. At layer~17, \QiCON{} co-dominates: the scaling diagnostic (the $1/f$ two-knee spectrum of the resting brain) tests that the cognitive cascade is genuinely in the admissibility window. At layer~18, all four pillars contribute simultaneously: the articulation requires bookkeeping (it must respect the constitution), selection (it must extremize the right functional), diagnostic (it must be scale-covariant across layers), and architecture (it must close the meta-cognitive loop).

{\small
\begin{longtable}{cll}
\caption[Which \QiFour{} pillar dominates at each of the eighteen layers.]{Which of the four \QiFour{} pillars dominates at each of the eighteen layers, and where the hand-over falls. \QiQUA{} carries the physical strata, layers~1 to~4, from the six-dimensional gauge bulk through the electroweak and hadronic--atomic rungs to small molecules; \QiEP{} carries the chemical and prebiotic strata, layers~5 to~9, from amino acids and sugars through polymers, ATP and the RNA world to DNA; \QiHAL{} carries the biological and cognitive strata, layers~11 to~16, from the three-domain split to the allocentric map, with layer~10, the cell, shared between \QiEP{} and \QiHAL{}. Layer~17, the conscious episode, adds \QiCON{}, and layer~18, articulation, requires all four at once.}
\label{tab:pillar-by-layer-a11}\\
\toprule
\textbf{Layer} & \textbf{Stratum} & \textbf{Dominant pillar(s)} \\
\midrule
\endfirsthead
\multicolumn{3}{c}{\itshape Table~\ref{tab:pillar-by-layer-a11} continued}\\
\toprule
\textbf{Layer} & \textbf{Stratum} & \textbf{Dominant pillar(s)} \\
\midrule
\endhead
\midrule \multicolumn{3}{r}{\itshape continued on next page}\\
\endfoot
\bottomrule
\endlastfoot

1 & Gauge / 6D bulk & \QiQUA{} \\
2 & Electroweak & \QiQUA{} \\
3 & Hadronic/atomic & \QiQUA{} \\
4 & Small molecules & \QiQUA{} \\
5 & Amino acids/sugars & \QiEP{} \\
6 & Polymers & \QiEP{} \\
7 & ATP (bioenergetic) & \QiEP{} \\
8 & RNA-World & \QiEP{} \\
9 & DNA (information) & \QiEP{} \\
10 & Cell/LUCA & \QiEP{}, \QiHAL{} \\
11 & Three-Domain Split & \QiHAL{} \\
12 & Eukaryogenesis & \QiHAL{} \\
13 & Multicellularity & \QiHAL{} \\
14 & Central nervous system & \QiHAL{} \\
15 & Ring attractor & \QiHAL{} \\
16 & Allocentric map & \QiHAL{} \\
17 & Conscious episode & \QiHAL{}, \QiCON{} \\
18 & Articulation & all four \\
\end{longtable}

\subsection{The integer ledger across the cascade}
\label{ssec:integer-ledger-cascade}

A second striking structural feature is that the same integer ledger $(M, N, L; J, K) = (1, 3, 5; 7, 35)$ appears in the quantitative readout of every layer. At layer~1 it governs the gauge couplings via $2^J = 128$ and $L^2 = 25$. At layer~2 it fixes the family number $N = 3$ and the scale-suppressed electroweak scale $v = \Mlock{}(2/L)^N$. At layer~3 it sets the Rydberg energy via the same dyadic charge structure. At layer~5 it fixes the amino-acid count $|\GGA{}_5| = K - NL = 20$. At layer~8 it fixes the codon redundancy $4^3/20 \approx 3.2$. At layer~7 it sets $\QB{} = (2/5)(E_{\rm Ry}/K)$ and hence ATP at $2 N_A \QB{} = 30$~kJ/mol. At layer~10 it sets the cell-size and metabolic-rate scales via the bioenergetic anchoring. At layers~15--16 it sets the neural-circuit topology via the rank-one current locking. At layer~17 it sets the cognitive timescales via the \EIC{}-loop closure on the \QGL{}-locked currents. One integer ledger; eighteen cascade readouts. This is the operational content of the universal-identity insight of \ANine{} Part~XI (the universal-identity section) generalized across the entire cascade.

\GGPart[The cascade as one ledger]{The cascade as one ledger}{The eighteen-row codebook, the Top~10 Falsifiers, the reader's guide, and what would fail if any layer were absent}
\label{part:ledger}

\section{Master cascade codebook: the eighteen Ground Geometric Attractors as one ledger}
\label{sec:cascade-codebook-master}

\subsection*{Plain-English opening: what this section is for}

Let us imagine the periodic table that hangs on the wall of every chemistry classroom in the world. Each row of that table is a shell of electrons; each column collects the elements that share a chemical personality; each cell holds a name, a number, a mass, an electronic configuration, and an implicit promise that the same rules apply to all of it. A student who has never seen the table before can be taught, in one afternoon, to read it horizontally as a sequence of stages and vertically as a family of behaviors, and to look up any single element and know immediately where it sits relative to its neighbors. The table compresses an enormous amount of empirical and theoretical knowledge into one rectangular ledger that anybody, with a little practice, can audit.

What follows in this Section plays the same role for the eighteen-layer cascade of \GGTheory{}: it is the periodic table of \emph{everything the cascade has produced so far}. Each row is one of the seventeen cascade layers (plus the meta-cognitive layer~18, the present paper itself, for eighteen rows in total). Each column is a precise piece of information: which substrate the layer lives on, which Ground Geometric Attractor (\GGA{}) the layer selects, what numerical anchor the \GGA{} predicts, how confident we are in that prediction today, and which upstream theorem from \AOne{}--\ATen{} licenses the claim. A reader who has never read a single page of \GGTheory{} can, by reading this table alone, learn (i) what the framework predicts, (ii) where the framework is solid and where it is still extending credit to itself, and (iii) what experiment or proof would have to come in for any single row to be promoted from a conditional to a derived status. The table is meant to be hostile-reviewer-proof: it lists its own conditionals up front, so that nobody can accuse it of laundering them through prose.

This Section's job is to provide that audit ledger explicitly, with the same level of typographic care that the periodic table earns by being the most-photocopied object in the natural sciences. Everything that follows is either a definition (the status legend), a row of the ledger (the master longtable), or commentary on how each row earned its place (per-row provenance). At the end we count rows by confidence class and lay out the promotion roadmap --- which conditionals will be discharged by which experiment or lemma --- so the reader can see the entire enterprise as a single, finite, falsifiable project.

% --------------------------------------------------------------------
\subsection{Confidence triage and status legend}
\label{sec:confidence-legend}

Following the conventions established in \PaperFullCOne{} \citep{ArisakaC1Kaxion} L41/L42 and used throughout \ANine{} \citep{ArisakaA9SM} and \ATen{} \citep{ArisakaA10Cosmos}, we classify every row of the master codebook into exactly one of five confidence classes. The constitutional warrant for using these five classes (and only these five) is the \LC{}-discipline of \AOne{} \citep{ArisakaA1Constitution} (no claim may travel without an explicit chain of derivation), together with the Euclidean zero-free-parameter requirement of \AFour{} \citep{ArisakaA4Euclid} (the input ledger is exhausted by the Geometric Universal Five Constants \GUfive{} and the \OGNfive{} integer ledger). Together these say: every claim either descends from those inputs without slack, or it does not, and the cascade is responsible for telling the reader honestly which case it is in.

\paragraph{Derived.}
A row is \emph{Derived} when the full constitutional chain from \AOne{}--\ATen{} is closed in the published record, with no unstated lemma and no unargued assumption. Concretely, a Derived row is one for which (i) the upstream theorem(s) listed in the table are themselves proved in the cited Arisaka papers (the lemma has been done), (ii) the numerical anchor in the row follows by published computation from those theorems plus the \OGNfive{} integer ledger, and (iii) the substrate to which the \GGA{} attaches has been independently identified before the \GGA{} is asserted (no circularity through naming). Derived is the strongest claim the framework makes about a row; it says: \emph{the matter is settled, and any disagreement with the listed numerical anchor would constitute a falsification of the entire cascade, not a recalibration of this row.}

\paragraph{Anchor-Derived (derivation-deferred).}
A row is \emph{Anchor-Derived} when its empirical anchor is quantitatively closed in the published literature and the cascade reproduces it, but the from-geometry derivation to codebook standard is developed in a companion paper still in preparation. The tier exists because the definition of \emph{Derived} above is strict about the \emph{published record}: three rows --- layer~7 (the ATP quantum), layer~9 (the four-letter alphabet) and layer~10 (the LUCA-class cell) --- would otherwise have been reported as Derived in the same breath as the text conceding that their proofs are developed in \AThirteen{} (in preparation), and a paper in preparation cannot close a chain in the published record. The anchor is done; the derivation is deferred; the tag says so. Anchor-Derived sits below Derived and above Conditional, and the difference from Conditional is worth stating: a Conditional row waits on a lemma whose statement is known but whose truth is open, whereas an Anchor-Derived row waits on the writing-up of a derivation whose target is already matched by measurement.

\paragraph{Conditional.}
A row is \emph{Conditional} when the cascade machinery commits to selecting a specific \GGA{} on the substrate in question, and the upstream theorems of \AOne{}--\ATen{} suffice to make that selection except for a single, explicitly named, missing lemma. The missing lemma is always one of two types: a \emph{substrate-identification lemma} (e.g., the small-molecule closure lemma at layer~4, which must show that the warm-wet stratum geometrically forces a discrete set of stable polyatomic molecules under \GDOS{} descent) or a \emph{mapping lemma} (e.g., the codon-mapping lemma at layer~8, which must show that the $4^3 \to 20$ map is the unique \QiFour{}-admissible solution of the polymer-folding \GFF{}). A Conditional row is not a guess; it is a published commitment to a specific lemma whose statement is known and whose proof is the current research target.

\paragraph{Asserted.}
A row is \emph{Asserted} when the cascade selects a specific \GGA{} on a substrate that has been experimentally identified by other workers, and the framework predicts that the \GGA{} so selected is the unique stable solution on that substrate, but the prediction has not yet been demonstrated in a controlled experimental comparison with named alternative architectures. The neural ring-attractor of layer~15 is the prototype: the ring-attractor itself is observed in insect head-direction circuits \citep{KimEtAl2017}, and rodent head-direction systems \citep{TaubeMullerRanck1990}, but the \emph{uniqueness} of SO(2) as the topology of stable heading representation --- as against, say, a torus or a Klein bottle or a discrete graph --- is what the \GGA{} claim adds and what awaits experimental discrimination on a specific substrate. Asserted is therefore weaker than Conditional in the sense that a new lemma alone cannot close the row; an experiment must.

\paragraph{Existence-Demonstrated.}
A row is \emph{Existence-Demonstrated} when the very act of the cascade producing the row constitutes empirical confirmation that the row is occupied. There is exactly one such row, layer~18, the articulation layer: the present paper is, by its existence, the empirical demonstration that an articulator-class \GGA{} has been instantiated on the relevant substrate. The constitutional warrant for treating this as evidence (and not as petitio principii) is the \OGGEP{} principle of \AOne{} \S\AX{4} together with the self-referential closure theorem of \ANine{} \citep{ArisakaA9SM} (the cascade is reflexively complete only if it can describe itself), and the \HAL{} completeness theorem of \ASeven{} \citep{ArisakaA7Qi4}: layer~18 is the unique \GGA{} on the substrate of recursive symbolic articulation, and its instantiation is what one is reading. Existence-Demonstrated rows are the rarest class precisely because they require the cascade itself to close on the writer.

\paragraph{The five classes as a single procedural ladder.}
The five classes form a ladder of increasing closure: Existence-Demonstrated $\Rightarrow$ Derived $\Rightarrow$ Anchor-Derived $\Rightarrow$ Conditional $\Rightarrow$ Asserted, where the arrow points from ``most-closed-on-itself'' to ``most-dependent-on-future-work.'' At every row we report the strongest honest class. No row is promoted across the ladder without an explicit citation to either a newly proved lemma or a newly published experimental anchor; demotion is also allowed (and has happened in the historical record of the program: see \PaperFullCOne{} \citep{ArisakaC1Kaxion} L41 for the demotion log). The legend is therefore not decorative; it is the audit trail.

% --------------------------------------------------------------------
\subsection{The eighteen-row codebook longtable}
\label{sec:cascade-codebook-table}

{\small
\begin{longtable}{@{}p{0.04\textwidth}p{0.16\textwidth}p{0.18\textwidth}p{0.20\textwidth}p{0.12\textwidth}p{0.20\textwidth}@{}}
\caption[Master cascade codebook.]{Master cascade codebook: eighteen rows summarizing the seventeen-layer \GRIP{}-\SIP{}-\QiFour{} cascade plus the articulation meta-row. Each row gives the layer index, the stratum or substrate on which the cascade lands, the Ground Geometric Attractor (\GGA{}) selected on that stratum, the numerical anchor predicted by the \GGA{}, the confidence class per the legend of \S\ref{sec:confidence-legend}, and the upstream theorem(s) from \AOne{}--\ATen{} that license the row. All numerical anchors carry SI or natural units; uncertainties are $1\sigma$ unless otherwise stated.}
\label{tab:cascade-codebook-master}\\
\toprule
\textbf{Layer} & \textbf{Stratum / substrate} & \textbf{\GGA{} name} & \textbf{Quantitative anchor} & \textbf{Status} & \textbf{Upstream theorem(s)}\\
\midrule
\endfirsthead
\multicolumn{6}{c}{\textit{Table~\ref{tab:cascade-codebook-master} continued from previous page}}\\
\toprule
\textbf{Layer} & \textbf{Stratum / substrate} & \textbf{\GGA{} name} & \textbf{Quantitative anchor} & \textbf{Status} & \textbf{Upstream theorem(s)}\\
\midrule
\endhead
\midrule
\multicolumn{6}{r}{\textit{(continued on next page)}}\\
\endfoot
\bottomrule
\endlastfoot
%
1 & Gauge / proton; 4D Minkowski boundary readout of the 6D \BIsix{} bulk &
$G_{\rm SM} = SU(3)_c \times SU(2)_L \times U(1)_Y$; stable proton &
$m_p \approx 938.272$~MeV ($\pm 2 \times 10^{-5}$, \citealt{PDG}); $\tau_p > 2.4 \times 10^{34}$~yr (\citealt{PDG} lower bound) &
\textbf{Derived} &
\T{-GUT-Closure} (\ANine{} \citep{ArisakaA9SM} Part~XI); \T{-GU5} on $\QB{}$ (\AOne{} \citep{ArisakaA1Constitution} \S\AX{2})\\
\addlinespace[2pt]
2 & Electroweak vacuum; Yukawa stratum on \DGI{}~$\to$~\GSSB{} of $SU(2)_L \times U(1)_Y$ &
Charged-lepton triple $\{e, \mu, \tau\}$ and Yukawa-locked $v$ &
$m_\tau / m_\mu = 16.817$ ($\pm 0.5\%$, \citealt{PDG}); $v = \Mlock{}(2/5)^3 \approx 246.4$~GeV ($\pm 0.1$~GeV) &
\textbf{Derived} &
\T{-Electroweak-Vacuum} (\ANine{} \citep{ArisakaA9SM} Yukawa block + \PaperPTwo{} \citep{ArisakaP2N3})\\
\addlinespace[2pt]
3 & Atomic stratum; non-relativistic Coulombic substrate of \GGA{}$_2$ &
Periodic table H, He, C, N, O, $\ldots$, Fe; hydrogenic ground state &
$E_{\rm Ry} = 13.6057$~eV ($\pm 10^{-7}$, \citealt{PDG}); $a_0 = 5.29177 \times 10^{-11}$~m &
\textbf{Derived} &
\T{-Atomic-Ground-State} (\AOne{} \citep{ArisakaA1Constitution} \S\AX{2}, \AFour{} \citep{ArisakaA4Euclid} \S3, \AFive{} \citep{ArisakaA5GDOS})\\
\addlinespace[2pt]
4 & Small-molecule stratum; warm-wet (\mbox{$200$--$400$~K}, liquid-water-window) substrate of \GGA{}$_3$ &
H$_2$O, CO$_2$, NH$_3$, CH$_4$ (the closed quartet: the simplest hydrides of oxygen, nitrogen and carbon, together with the simplest oxide of carbon) &
H$_2$O at $-285.83$~kJ/mol formation; CO$_2$ at $-393.51$~kJ/mol; \citealt{CRCHandbook} thermochemistry; geometry angles within $1^\circ$ of VSEPR predictions &
\textbf{Conditional} &
\T{-Molecular-Closure} (\AEleven{}'s own; \GDOS{} apparatus of \AFive{} \citep{ArisakaA5GDOS}) + \emph{small-molecule closure lemma}\\
\addlinespace[2pt]
5 & Proteinogenic-amino-acid stratum; aqueous chiral substrate of \GGA{}$_4$ &
The twenty L-handed proteinogenic amino acids; D-handed sugars &
$\Delta\Delta G_{\rm L\to D} \approx +12$~kJ/mol enantio-selection energy under \QiFour{}-locked descent; canonical count $20$ observed in every naturally evolved biology &
\textbf{Conditional} &
\T{-Amino-Acid-Selection} (\ASeven{} \citep{ArisakaA7Qi4} \QiFour{} pillars) + \emph{chirality-lock lemma}\\
\addlinespace[2pt]
6 & Polymer-class stratum (oligopeptide / oligonucleotide / fatty-acid); substrate of \GGA{}$_5$ &
Oligopeptides, oligonucleotides and fatty-acid chains, co-emerging on parallel sub-strata with no single family privileged &
Chain lengths $\ell \sim 5$--$20$ monomers at sustained flux; montmorillonite-templated oligomers to $40$--$50$ units \citep{Ferris1996} &
\textbf{Conditional} &
\T{-Polymer-Closure} (\AEleven{}'s own) + \emph{polymer-class closure lemma}\\
\addlinespace[2pt]
7 & Bioenergetic stratum; \QB{}-anchored membrane chemiosmotic substrate of \GGA{}$_6$ &
ATP as universal energy currency &
$\Delta G_{\rm ATP} = 2 N_A \QB{} \approx 30.0$~kJ/mol (empirical $30.5 \pm 0.5$); proton-motive force $\sim 200$~mV &
\textbf{Anchor-\allowbreak Derived} &
\T{-Bioenergetic-Closure}: \T{-GU5} for $\QB{}$ (\AOne{} \citep{ArisakaA1Constitution} \S\AX{2}); \T{-EP} (\ASeven{} \citep{ArisakaA7Qi4} \QiFour{} block)\\
\addlinespace[2pt]
8 & RNA-World stratum; bifunctional informational-catalytic polymer substrate of \GGA{}$_7$ &
RNA $\{A, U, G, C\}$ with Watson--Crick pairing; ribozymes; pre-translation hairpins/stem-loops &
Codon redundancy $\langle r \rangle = 64/20 = 3.2$; ribozyme rate enhancement $10^7$--$10^9$; 4-letter alphabet $4 = N + 1$ &
\textbf{Conditional} &
\T{-RNA-World-Closure} (\ASeven{} \citep{ArisakaA7Qi4} \QiFour{} block) + \emph{RNA-bifunctionality lemma}\\
\addlinespace[2pt]
9 & DNA information-storage stratum; archival substrate of \GGA{}$_8$ &
DNA $\{A, C, G, T\}$ with antiparallel double helix; B-form geometry; replication fidelity $\epsilon \approx 10^{-9}$/bp &
$\{A, C, G, T\}$ as the unique \QGL{}-admissible alphabet; B-form pitch 3.4~nm; replication $\epsilon = 10^{-9}$/bp three-tier product &
\textbf{Anchor-\allowbreak Derived} &
\T{-Information-Closure}: \T{-EP} (\ASeven{} \citep{ArisakaA7Qi4}); 4-letter alphabet uniqueness via \BIsix{} (\AThree{} \citep{ArisakaA3BI6})\\
\addlinespace[2pt]
10 & Cellular stratum; first \SeOS{}-class organism &
LUCA-class first cell with closed metabolism, replication, and boundary &
LUCA age $\sim$3.8~Ga ($\pm 0.2$~Ga geochemistry); cell size $1$--$10\,\mu$m; metabolic power $1$--$100$~W/kg; replication error $10^{-9}$/bp &
\textbf{Anchor-\allowbreak Derived} &
\T{-Cell-Closure}: \SeOS{} closure (\AFive{} \citep{ArisakaA5GDOS}); \T{-CON} (\ASeven{} \citep{ArisakaA7Qi4} \QiFour{} block)\\
\addlinespace[2pt]
11 & Primary-Domain-Split substrate; dual-gauge SSB on \GGA{}$_{10}$ &
Bacteria ($\oplus$) Archaea: two stable lipid-and-chemiosmosis attractors on the DGL(5:2) dual-gauge action &
Two-domain phylogeny established at $\sim 3.5$~Ga; Bacteria = ester-FA/H$^+$; Archaea = ether-isoprenoid/Na$^+$ &
\textbf{Conditional} &
\T{-Primary-Domain-Split}: dual-gauge SSB; \BIsix{} closure (\AThree{} \citep{ArisakaA3BI6})\\
\addlinespace[2pt]
12 & Eukaryogenesis substrate; energy-per-gene phase-jump on \GGA{}$_{11}$ &
Archaeal host $\oplus$ $\alpha$-proteobacterial endosymbiont; nuclear envelope; chromatin organization &
$\delta W/\text{gene} \uparrow 2$--$20\times$ (Lane--Martin economics, \citealp{Lane2015}); $\sim 2$~Ga; dual-knee $1/f$ PSD predicted &
\textbf{Conditional} &
\T{-Eukaryogenesis}: \T{-EP} variational principle (\ASeven{} \citep{ArisakaA7Qi4}) + \emph{Lane--Martin energy-per-gene lemma}\\
\addlinespace[2pt]
13 & Multicellularity substrate; M0--M7 ladder on \GGA{}$_{12}$ &
Clonal adhesion $\to$ gap junctions $\to$ division of labor $\to$ patterning $\to$ transport $\to$ integration $\to$ organogenesis &
$\sim 600$~Ma (animals), $\sim 1$~Ga (plants/fungi); multiple convergent origins; Kleiber-type metabolic scaling (exponent debated, $\sim$3/4 vs 2/3) \citep{Kleiber1932} &
\textbf{Conditional} &
\T{-Multicellularity}: \QiHAL{} closure (\ASeven{} \citep{ArisakaA7Qi4}) + \emph{M-ladder uniqueness lemma}\\
\addlinespace[2pt]
14 & Centralized-neural substrate; integrated nerve ring on \GGA{}$_{13}$ &
The minimal integrated sensorimotor connectome (egocentric, pre-allocentric); \emph{C.\ elegans}-class nerve-ring central nervous system &
302 neurons; $\sim$7{,}000 synapses ($\sim$5{,}000 chemical, $\sim$600 gap, $\sim$2{,}000 NMJ); 118 neuron classes; CPG-driven klinotaxis/klinokinesis; origin $\sim$0.56~Ga (Ediacaran bilaterians; \emph{C.\ elegans} a modern exemplar, not an ancestral form) &
\textbf{Asserted} &
\T{-Central-Nervous-System} (this paper Part~\ref{part:recursive}; empirical anchors \citealp{WhiteEtAl1986, CookEtAl2019}; \citealp{Brenner1974})\\
\addlinespace[2pt]
15 & Neural-ring substrate; insect ellipsoid-body / mammalian head-direction circuit &
1D ring attractor on $SO(2)$ for allocentric heading &
Drift rate $\lesssim 0.5^\circ$/s; bump width $\sim 60^\circ$ FWHM; refresh $\sim 50$~ms (\citealt{KimEtAl2017,TaubeMullerRanck1990}) &
\textbf{Asserted} &
\T{-Neural-Ring-Attractor} (this paper Part~\ref{part:recursive}; empirical anchors \citealt{KimEtAl2017,TaubeMullerRanck1990})\\
\addlinespace[2pt]
16 & Rodent-hippocampal substrate; 4D allocentric stratum on \GGA{}$_{15}$ &
4D place-+-grid-+-time-+-causal-history allocentric map &
Place-field width $\sim 30$~cm; grid-cell scale ratio $\sim \sqrt{2}$; theta cycle $4$--$12$~Hz; time-cell span $\sim 10$~s &
\textbf{Asserted} &
\T{-Allocentric-Map} (this paper Part~\ref{part:recursive}; empirical anchors \citealp{OKeefeDostrovsky1971, HaftingEtAl2005})\\
\addlinespace[2pt]
17 & Conscious-episode substrate; theta-cycle \EIC{} closure on \GGA{}$_{16}$ &
The $\sim$400~ms (two-theta-cycle) conscious episode; \OH{}/\IH{}-locked \EIC{} unity &
Episode $\sim$400~ms $=$ two $\sim$200~ms theta cycles (one MePMoS loop; the first cycle $\sim$200~ms is unconscious attention); \QCI{} $\sim 10$ ($\pm 2$); P300 latency $300$~ms ($\pm 50$~ms, \citealt{Dehaene2014}) &
\textbf{Asserted} &
\T{-Conscious-Episode}: \T{-\EIC{}} and \T{-\HAL{}} (\ASeven{} \citep{ArisakaA7Qi4})\\
\addlinespace[2pt]
18 & Articulation substrate; recursive symbolic articulation on \GGA{}$_{17}$ &
The meta-cognitive act of writing the present paper &
This document; word count $\sim 10^5$; citation graph closes on \AOne{}--\ATen{} &
\textbf{Existence-\allowbreak De\-mon\-strated} &
\T{-Articulation}: cumulative force of \AOne{}--\ATen{}; self-referential closure of \ANine{} \citep{ArisakaA9SM} Part~XI; \T{-\HAL{}} completeness of \ASeven{} \citep{ArisakaA7Qi4}\\
\end{longtable}
}

% --------------------------------------------------------------------
\subsection{Aggregate analysis: confidence distribution and the promotion roadmap}
\label{sec:cascade-codebook-aggregate}

\paragraph{Reading status for the consequence layers.} \AEleven{} is the keystone of the GG-A series, not its delivery vehicle for chemistry, life, and mind. The particle and cosmic floors are derived to full rigor in \ANine{} and \ATen{}; the chemistry, life, and mind floors are \emph{previewed} here and \emph{derived} in the companion papers \ATwelve{}, \AThirteen{}, and \AFourteen{} (in preparation). Accordingly, several of the consequence layers carry a tag weaker than their empirical standing alone would suggest. That is the legend working as intended: a tag records the state of the \emph{derivation}, in the single sense fixed at \S\ref{sec:confidence-legend}, and never the weight of the evidence --- a row can be empirically strong and still carry a weak grade. Any result stated assertively in those layers is to be read as a forecast the companion papers will discharge, not a claim closed within the present paper. What \AEleven{} itself delivers is the architecture (the one cascade engine), the physics anchors (layers 1--3, via \ANine{}/\ATen{}), the recursive-uniqueness theorem as a structural claim, and the Top~10 falsifiers; the per-domain derivations are what it forecasts.

Counting the eighteen rows of Table~\ref{tab:cascade-codebook-master}: there are three \textbf{Derived} rows (layers 1, 2, 3), three \textbf{Anchor-Derived} rows (layers 7, 9, 10), seven \textbf{Conditional} rows (layers 4, 5, 6, 8, 11, 12, 13), four \textbf{Asserted} rows (layers 14, 15, 16, 17), and one \textbf{Existence-Demonstrated} row (layer 18). Two ratios are worth separating, and we report both rather than the flattering one. Counted strictly --- rows whose chain is closed \emph{in the published record} --- the fraction is $3+1 : 18 = 4:18 \approx 22\%$ today. Counted by empirical closure, allowing the three Anchor-Derived rows whose measured anchors are settled and whose derivations are being written, it is $7:18 \approx 39\%$. The stricter number is the one the legend licenses; the looser one is stated so that the reader can see exactly what the difference between them consists of, which is three rows of biology waiting on \AThirteen{}. The next nine years of the program (2026--2035) aim to close all three Anchor-Derived derivations, discharge the named lemma on all seven Conditional rows, and complete the experimental anchors and the \ATwelve{}--\AFourteen{} derivations on all four Asserted rows, bringing the strict fraction to $18:18 = 100\%$.

\paragraph{Pending lemmas (the seven Conditional rows).}
For layer~4, the missing lemma is the \emph{small-molecule closure lemma}: on the warm-wet stratum, the \GFF{} Morse landscape has exactly four global minima, identified with H$_2$O, CO$_2$, NH$_3$, CH$_4$. For layer~5, the missing lemma is the \emph{chirality-lock lemma}: on the small-molecule substrate, the \QiFour{} cascade selects L-handed amino acids and D-handed sugars as the unique \GSSB{} solutions of the warm-wet \GFF{}. For layer~6, the missing lemma is the \emph{polymer-class closure lemma}: the \GFF{} landscape on the layer-5 substrate admits exactly three co-emerging polymer basins --- oligopeptide, oligonucleotide and fatty-acid --- and no fourth of comparable depth. For layer~8, the missing lemma is the \emph{codon-mapping lemma}: the $4^3 \to 20$ map is the unique \QiFour{}-admissible solution of the polymer-folding \GFF{} on the amino-acid substrate, with codon length $k = 3$ forced by the minimal-redundancy constraint $4^k \ge 20$, $4^{k-1} < 20$. Layers~8, 11, 12, and 13 carry, respectively, the RNA-bifunctionality lemma, the dual-gauge-\SSB{} lemma on the DGL(5:2) packet (\AThree{}), the Lane--Martin energy-per-gene lemma, and the M-ladder-uniqueness lemma (each stated with its row in the master codebook). Layer~11 joins this tier because its own text says it awaits a named lemma, and pending-a-named-lemma is what Conditional means.

\paragraph{Pending experiments (the four Asserted rows).}
For layer~15, the empirical anchor needed is a controlled comparison of allocentric-heading topology between (a) the ring-attractor architecture observed in fly ellipsoid body \citep{KimEtAl2017} and rodent head-direction cells \citep{TaubeMullerRanck1990}, and (b) plausible alternatives (line attractor, small-graph attractor, 2-sphere) in a previously-unstudied taxon (e.g., cephalopod head-direction circuitry, in which the topology is not yet established). For layer~16, the empirical anchor needed is a high-resolution decomposition of rodent hippocampal coding showing that the dimensionality is exactly 4 (place + grid + time + causal-history) rather than 3 or 5, on a behavioral task that loads the causal-history dimension explicitly --- ongoing in the literature but not yet conclusive. For layer~14 (the \emph{C.\,elegans}-class CNS), a controlled comparison of the integrated connectome against alternative architectures; for layer~17 (the conscious episode), the phase-resolved EEG/MEG/ECoG measurement of the predicted alpha/gamma phase steps (\AFourteen{}).

\paragraph{The promotion roadmap as a single project.}
Discharging the seven Conditional lemmas is a theoretical project (requires \GDOS{}/\QiFour{} analysis at layers 4, 5, 6, 8, 11, 12, 13); completing the four Asserted rows is a combined empirical-and-derivation project (layers 14, 15, 16, 17). The total work is therefore split $7:4$ theory-to-experiment, with the empirical Pull tests of Section~B providing the cross-layer audit. The three Anchor-Derived rows (layers 7, 9, 10) are a third kind of work and a lighter one: no lemma is open and no experiment is owed, only the writing-up of the from-geometry derivation in \AThirteen{}. The Existence-Demonstrated row (layer~18) requires no further work other than the act of writing. The Derived rows (layers 1, 2, 3) require no work beyond continued cross-checking with the existing literature.

\paragraph{Plain-English close.}
An honest accounting of confidence is load-bearing. A framework that hides its conditionals is not closer to closure, only further from honesty. The codebook is the structural ledger; the Pull table is the empirical ledger; the two together are the audit trail. The cascade does not ask the reader for belief. It asks for an audit.

% ====================================================================
%  SECTION B --- EIGHT-TEST EMPIRICAL HORIZON PULL TABLE
% ====================================================================
\section{The Top 10 Falsifiers: a Pull-table audit of the framework's falsifiability}
\label{sec:eight-test-pull-horizon}

\subsection*{Plain-English opening: what a Pull table is and why ten is the right number}

A Pull table is the experimentalist's accounting sheet, used in every precision measurement from the muon anomalous magnetic moment to the cosmic-microwave-background temperature spectrum. The recipe is mechanical: for each prediction the framework makes, write down the predicted value $V_{\rm pred}$ with its theoretical uncertainty $\sigma_{\rm pred}$, the observed value $V_{\rm obs}$ with its experimental uncertainty $\sigma_{\rm obs}$, compute the combined uncertainty $\sigma_{\rm combined} = \sqrt{\sigma_{\rm pred}^2 + \sigma_{\rm obs}^2}$, and report the Pull $P = (V_{\rm obs} - V_{\rm pred}) / \sigma_{\rm combined}$. If $|P| < 1$ the prediction agrees with observation; if $|P| < 3$ the prediction is consistent within experimental scatter; if $|P| > 5$ the prediction is falsified. A framework with $N$ predictions and a Pull table whose entries all satisfy $|P| < 3$ is in healthy empirical contact with the world; a framework whose Pull entries cluster at $|P| > 5$ is falsified $N$-fold over. The Pull table is not a courtesy --- it is the bridge between calculation and decision.

Why ten falsifiers rather than eighty? Because of the hostile-reviewer test: if the framework cannot point to a short list of specific, calculable, decision-trigger predictions over the coming decade --- predictions whose outcome is not yet known but whose threshold for success or failure can be stated in advance --- then it is philosophy, not physics, no matter how internally consistent its prose. Ten is small enough to be a genuine commitment and broad enough to sample (i) the gauge sector (KK lock-cluster), (ii) the dark sector (Kaxion), (iii) late-time cosmology (dark energy $w = -1$ and tensor $r$), (iv) baryon-number stability (proton decay), (v) the lepton sector (the PMNS phase $\delta_{\rm PMNS} \approx 196^\circ$), and (vi) the life-and-mind cascade (the bioenergetic quantum, the genetic alphabet, the eukaryotic $1/f$ spectrum, the conscious episode). Crucially, \emph{eight of the ten are genuine forward bets} whose answer is not yet known. Each falsifier is independently falsifying: failure on any single front falsifies the framework. The list is explicitly time-bounded: nothing awaits an experiment later than $\sim 2040$ (proton decay), and the bulk resolves in the 2026--2035 corridor.

% --------------------------------------------------------------------
\subsection*{Honest classification of the ten falsifiers}
\label{ssec:eight-test-classification}

Before reading the Pull table, the reader deserves a clear statement of what kind of empirical commitment each row represents. The framework's ten falsifiers divide cleanly into two classes, and --- by deliberate design --- the list is now dominated by the strongest class:

\begin{description}\itemsep3pt
\item[\textbf{Forward falsifiers (8 of 10):}] Tests~1--6 (KK lock-cluster; Kaxion dark matter; dark energy $w = -1$; proton stability; tensor-to-scalar $r = 0.0043$ (two-sided); leptonic CP phase $\delta_{\rm PMNS} \approx 196^\circ$) and tests~8--9 (the four-letter genetic alphabet; the dual-knee $1/f$ spectrum, tested in two substrates --- 9a eukaryotic-metabolic and 9b cortical-neural). Each commits to a value (or a bound) whose outcome is \emph{not yet known}, on a named instrument, with a named decision window and a named tolerance. A result at named precision at the named experiment in the named window falsifies the framework. These eight are the strongest form of empirical commitment a theory can make: the answer is genuinely open, and we have placed a bet.
\item[\textbf{Precision audits (2 of 10):}] Tests~7 ($\Delta G_{\rm ATP} = 2 N_A \QB{}$) and~10 (the conscious-episode duration). These compare a value already in the literature to a sharp framework anchor; they are not open-outcome bets, but each tightens toward forward status as independent re-measurement at the named precision arrives (single-molecule biophysics; precision psychophysics). They are listed honestly as audits, not predictions.
\end{description}

\noindent The portfolio is, by design, \textbf{eight forward falsifiers and two precision audits}. A forward falsifier names a number whose value is not yet known, an instrument that will measure it, and a decade window in which the answer arrives; a precision audit compares an already-measured quantity to a sharp framework anchor and tightens toward forward status as re-measurement improves. The framework stakes itself on the eight open-outcome bets that experimentalists are actively racing to settle this decade: a framework gains nothing by listing the predictions it cannot lose, and everything by listing the ones it can.

\paragraph{What the Top 10 honestly commits to.}
Eight genuine forward falsifiers spanning six strata --- the gauge sector (KK lock-cluster), the dark sector (Kaxion), late-time cosmology (dark energy $w = -1$ and tensor $r$), baryon-number stability (proton decay), the lepton sector (the PMNS phase), and the life/mind cascade (the genetic alphabet, the eukaryotic spectrum) --- plus two precision audits (the bioenergetic quantum; the conscious-episode window). The framework's commitment is to all ten, and on eight of them the answer is not yet in. A reader who finishes this paper without scrutinizing each row will not be reading honestly; the Pull table that follows is constructed so the reader can scrutinize.

\subsection{The Top 10 Falsifiers Pull table}
\label{sec:eight-test-pull-table}

{\small
\begin{longtable}{@{}p{0.03\textwidth}p{0.15\textwidth}p{0.16\textwidth}p{0.16\textwidth}p{0.08\textwidth}p{0.13\textwidth}p{0.17\textwidth}@{}}
\caption[The Top 10 Falsifiers.]{The Top 10 Falsifiers. Pull $P = (V_{\rm obs} - V_{\rm pred}) / \sigma_{\rm combined}$ where $\sigma_{\rm combined} = \sqrt{\sigma_{\rm pred}^2 + \sigma_{\rm obs}^2}$. ``Forward'' indicates a prediction whose observed value is not yet measured. ``Decision threshold'' is the experimental sensitivity at which the prediction will be definitively confirmed or falsified. ``Experiment / time horizon'' lists the facility and the calendar window in which the decision is expected.}
\label{tab:eight-test-pull-table}\\
\toprule
\textbf{\#} & \textbf{Test name} & \textbf{Predicted value} & \textbf{Observed (or 95\% CL bound)} & \textbf{Pull} & \textbf{Decision threshold} & \textbf{Experiment / time horizon}\\
\midrule
\endfirsthead
\multicolumn{7}{c}{\textit{Table~\ref{tab:eight-test-pull-table} continued from previous page}}\\
\toprule
\textbf{\#} & \textbf{Test name} & \textbf{Predicted value} & \textbf{Observed (or 95\% CL bound)} & \textbf{Pull} & \textbf{Decision threshold} & \textbf{Experiment / time horizon}\\
\midrule
\endhead
\midrule
\multicolumn{7}{r}{\textit{(continued on next page)}}\\
\endfoot
\bottomrule
\endlastfoot
%
1 & KK lock-cluster mass (F-Origin-1) &
$\Mlock{} = 3.85 \pm 0.10$~TeV &
Not yet observed; ATLAS+CMS limit $\sim 2$~TeV (\citealt{PDG}) &
\textbf{Forward} &
$5\sigma$ discovery at $M = 3.85$~TeV $\pm 0.4$~TeV &
HL-LHC 2030--2035 (\PaperPOne{} \citep{ArisakaP1KK})\\
\addlinespace[2pt]
2 & Kaxion dark-matter mass (F-Origin-2) &
$m_a = 8.1\,\mu$eV &
Not yet observed; current haloscope limits $> 10\,\mu$eV \citep{PDG} &
\textbf{Forward} &
$5\sigma$ axion-photon coupling in the $6.6$--$8.8\,\mu$eV band &
ADMX-EFR, IBS-CAPP 2027--2030 (\PaperFullCOne{} \citep{ArisakaC1Kaxion})\\
\addlinespace[2pt]
3 & Dark-energy equation of state (F-Origin-3) &
$w = -1$ exactly (true cosmological constant; $\Omega_\Lambda = 1 - 1/\pi$; \ATen{}) &
DESI~2024: hint of \emph{evolving} dark energy ($w_0 w_a$) at $\sim 2$--$4\sigma$, not yet decisive &
\textbf{Forward} &
DESI, Euclid, and Roman distinguish $w = -1$ from $w(z)$ at $> 3\sigma$; a confirmed $w \neq -1$ falsifies &
DESI, Euclid, Roman 2026--2030 (\ATen{} \citep{ArisakaA10Cosmos})\\
\addlinespace[2pt]
4 & Proton stability (F-Origin-4) &
$\tau_p(p \to e^+ \pi^0) > 10^{35}$~yr (orbifold-forbidden decay; \ANine{}) &
Super-K: $\tau_p > 2.4 \times 10^{34}$~yr (\citealt{PDG}) &
\textbf{Forward} &
any $p \to e^+\pi^0$ event falsifies; sensitivity reaches $\sim 10^{35}$~yr &
Hyper-Kamiokande, DUNE 2027--2040 (\ANine{} \citep{ArisakaA9SM})\\
\addlinespace[2pt]
5 & Tensor-to-scalar ratio $r$ (F-Origin-5) &
$r = 0.0043 = 12/N^{*2}$ (derived plateau, $N^{*}=52.6$; \ATen{}) &
$r < 0.036$ at 95\%~CL (Planck+BICEP, \citealt{Planck2018}) &
\textbf{Forward} &
a measurement inconsistent with $r\simeq0.004$ \emph{in either direction} falsifies; CMB-S4 + LiteBIRD reach $\sigma(r)\sim10^{-3}$ &
CMB-S4 + LiteBIRD 2030--2035\\
\addlinespace[2pt]
6 & Leptonic CP phase (F-Origin-6) &
$\delta_{\rm PMNS} \approx 196^\circ$ (derived in \ANine{}/\ATen{}) &
Not yet measured; T2K/NOvA hints, $\sim \pm 40^\circ$, CP-conserving values not excluded &
\textbf{Forward} &
DUNE + Hyper-K measure $\delta_{\rm CP}$ to $\sim \pm 15^\circ$; a value inconsistent with $196^\circ$ falsifies &
DUNE, Hyper-Kamiokande 2030--2035 (\ANine{} \citep{ArisakaA9SM})\\
\addlinespace[2pt]
7 & Bioenergetic quantum (Life) (F-Origin-7) &
$\Delta G^{\circ\prime}_{\rm ATP} = 2 N_A \QB{} = 30.0$~kJ/mol &
Biophysics: $\Delta G^{\circ\prime}_{\rm ATP} \approx 30.5 \pm 0.5$~kJ/mol &
\textbf{Audit}; $\sim 1.6\%$ &
standard-state re-measurement to $< 0.5\%$ &
single-molecule biophysics 2026--2030 (\ASeven{} \citep{ArisakaA7Qi4} + layer~7)\\
\addlinespace[2pt]
8 & Genetic-alphabet cardinality (Life) (F-Origin-8) &
exactly four letters $\{$A,C,G,T$\}$ in naturally-evolved life &
all known wild life: four (\citealt{WhiteEtAl1986} convention; synthetic Hachimoji excluded) &
\textbf{Forward} &
a \emph{wild} organism with a stable $\neq 4$-letter genetic alphabet falsifies &
comparative / meta-genomics, continuous (layer~9)\\
\addlinespace[2pt]
9a & Dual-knee $1/f$ metabolic spectrum (Life) (F-Origin-9a) &
two spectral knees in the single-cell eukaryotic metabolic-noise PSD (dual-DGL; \AFive{}); prokaryote shows a single knee &
not yet systematically measured at single-cell resolution &
\textbf{Forward} &
single-cell metabolic imaging resolving one vs.\ two knees &
single-cell live imaging 2026--2028 (layer~12)\\
\addlinespace[2pt]
9b & Dual-knee $1/f$ neural spectrum (Mind) (F-Origin-9b) &
exactly $2$ separable aperiodic knees / intrinsic timescales in the cortical EEG/ECoG PSD --- three or more breaks refutes, as it does upstream --- a fast synaptic/membrane knee and a slow network knee (\QiCON{} two-knee window; \ASeven{} \T{-CON.2}) &
single knee robustly measured; the second knee reported once directly in human EEG; epilepsy ECoG a natural test bed &
\textbf{Forward} &
high-resolution EEG/ECoG fit resolving one vs.\ $\ge 2$ knees; healthy vs.\ epileptic cortex &
intracranial/scalp spectroscopy, available now (layer~17)\\
\addlinespace[2pt]
10 & Conscious-episode duration (Mind) (F-Origin-10) &
$\sim$400~ms $=$ two theta cycles (\EIC{} closure; \ASeven{}) &
P300 latency $\sim 300$~ms (\citealt{Dehaene2014}); reported episode estimates in band &
\textbf{Audit} &
a reliably reported unitary episode completing in $\le$ one theta cycle ($\sim$200~ms) falsifies &
precision psychophysics / EEG 2026--2028 (layer~17)\\
\end{longtable}
}

% --------------------------------------------------------------------
\subsection{Headline parameter-audit linear Dev plot}
\label{ssec:dev-plot-linear}

The Pull table above is the tabular form of the Top 10 Falsifiers. Figure~\ref{fig:a11-dev-plot-linear} visualizes the same data in the canonical ATLAS-style two-tier format of \AFour{} \citep{ArisakaA4Euclid} and the headline parameter-audit Dev plots of \ANine{} \citep{ArisakaA9SM} and \ATen{} \citep{ArisakaA10Cosmos}, in the same linear-axis presentation used for the A9-SM Standard-Model headline plot. The horizontal axis is the signed relative deviation $(\text{observed} - \text{predicted})/|\text{predicted}|$ on a \emph{linear} scale with window $[-10\%, +10\%]$, giving a wide-open, eye-friendly view of the precision-band landings without the visual compression of a logarithmic axis. Three nested gray bands mark the canonical precision thresholds at $\pm 10^{-4}$, $\pm 10^{-3}$, and $\pm 10^{-2}$. The plot displays all ten falsifiers, grouped Physics/Cosmology, Life, and Mind, with a marker convention matched to each row's status. The two precision audits carry two error bars per L46 convention: the inner thick bar is the empirical-only $\sigma_{\rm obs}/|\text{obs}|$, the outer thin bar the combined $\sqrt{\sigma_{\rm obs}^2 + \sigma_{\rm th}^2}/|\text{obs}|$. The eight forward bets --- which have no observation yet --- are shown as right-pointing markers centered at $\text{Dev} = 0$, with a horizontal bracket whose half-width is the theoretical uncertainty $\sigma_{\rm th}/|\text{pred}|$ where one applies; the dark-energy row is drawn in orange to flag the live DESI tension. The tensor-to-scalar row is a forward bet at the derived $r = 0.0043$ with a two-sided theoretical bracket: a detection above, or a null pushing $r$ below $\sim10^{-3}$, would falsify, and the present Planck+BICEP bound $r < 0.036$ sits a factor of eight above the prediction. The figure is therefore the framework's empirical \emph{ceiling} --- a portfolio of mostly forward commitments --- in contrast to the already-measured parameter-audit \emph{floor} discussed in the cosmology and Standard-Model papers.

\begin{figure}[H]
 \centering
 \includegraphics[width=0.95\textwidth]{a11_dev_plot_linear_bold}
 \caption[Top 10 Falsifiers: parameter-audit Dev plot]{Headline parameter-audit linear Dev plot for \AEleven{}. The framework's empirical \emph{ceiling}: the Top~10 Falsifiers of Table~\ref{tab:eight-test-pull-table}, eight of them forward bets whose answer is not yet known --- as against the parameter-audit \emph{floor} of already-measured agreements reported in \ANine{} and \ATen{} --- plotted as signed relative deviation $(\text{observed} - \text{predicted})/|\text{predicted}|$ on a linear scale with $\pm 10\%$ window. Three nested precision bands at $|\text{Dev}| = 10^{-4}, 10^{-3}, 10^{-2}$. Inner thick error bar: empirical-only $\sigma_{\rm obs}/|\text{obs}|$. Outer thin error bar: combined $\sqrt{\sigma_{\rm obs}^2 + \sigma_{\rm th}^2}/|\text{obs}|$. The framework's central prediction lies at $\text{Dev} = 0$. \textbf{Eight forward bets} (right-pointing markers centered at zero, with theoretical-uncertainty brackets where one applies): $\Mlock$ KK lock-cluster ($3.85 \pm 0.10$~TeV, HL-LHC), $m_a$ Kaxion dark matter ($8.1\,\mu$eV, ADMX-EFR/IBS-CAPP), dark energy $w=-1$ (orange, flagging the live DESI hint of $w\neq-1$; DESI/Euclid), proton stability ($\tau_p > 10^{35}$~yr, Hyper-Kamiokande/DUNE), the tensor-to-scalar ratio $r = 0.0043$ two-sided (CMB-S4/LiteBIRD), the leptonic CP phase $\delta_{\rm PMNS}\approx196^\circ$ (DUNE/Hyper-K), the four-letter genetic alphabet $\{$A,C,G,T$\}$ (comparative genomics), and the dual-knee $1/f$ spectrum (9a metabolic single-cell imaging + 9b neural EEG/ECoG). \textbf{Two precision audits}: the ATP bioenergetic quantum (blue circle, $\text{Dev} = +1.6\%$ with inner/outer error bars) and the conscious-episode duration (purple diamond, $\sim 400$~ms, two theta cycles). Rows are grouped Physics/Cosmology, Life, and Mind --- the three boundary readings of the recursive theorem. The plot is generated from \texttt{a11\_dev\_plot\_top10.py} (matplotlib linear axis with $\pm 10\%$ window, DejaVu Sans, \texttt{mathtext.default = ``bf''}) and parallels the A9-SM Standard-Model headline presentation.}
 \label{fig:a11-dev-plot-linear}
\end{figure}

\paragraph{What the plot shows at a glance.}
The plot's most important feature is that \emph{eight of the ten markers are forward arrows}: predictions whose answer is not yet known. This is deliberate. The single clean measured audit, the ATP bioenergetic quantum, lands at $\text{Dev} = +1.6\%$ ($\sim 1\sigma$, just outside the $1\%$ precision band and well within the $5\%$ kill threshold), and the tensor-to-scalar ratio is a sharp two-sided bet at the derived $r = 0.0043$, a factor of eight below the Planck+BICEP bound --- but the framework does not rest its case on these. It rests on the eight forward bets, distributed across physics (KK lock-cluster, Kaxion, dark energy, proton stability, $\delta_{\rm PMNS}$), life (the genetic alphabet, the dual-knee $1/f$ spectrum), and the single Mind audit (the conscious-episode window). The dark-energy row is drawn in orange to mark the one place the framework is already in live tension with data: DESI's hint of an evolving $w$. The visual summary is the opposite of a postdiction plot: it is a commitment to be measured. Where the parameter-audit \emph{floor} (the already-measured agreements of \ANine{} and \ATen{}) shows what the framework has matched, this \emph{ceiling} shows what the framework has wagered. By the mid-2030s every arrow will have landed on one side of zero or the other.

\paragraph{How to read the plot against the Pull table.}
The Pull table of \S\ref{sec:eight-test-pull-table} reports each test's Pull $P = (\text{obs} - \text{pred})/\sigma_{\rm combined}$ in dimensionless $\sigma$ units; the present plot reports the same numerator divided by $|\text{pred}|$ instead, yielding a fractional rather than $\sigma$-normalized deviation. The two representations are complementary: the Pull table is calibrated against the experimental uncertainty (good for decision-making), while the linear Dev plot is calibrated against the predicted central value (good for visualizing precision-band landings). A reader who wants the headline visual answer should read the plot; a reader who wants the headline statistical answer should read the table.

% --------------------------------------------------------------------
\subsection{Per-test commentary}
\label{sec:per-test-commentary}

The Pull table above is procedural; the commentary below explains, one paragraph block per test, what the prediction is, where it comes from in the cascade, what the upstream theorem actually says (per the citation discipline), what the experimental apparatus looks like, the physics intuition, a daily-life metaphor where applicable, and the kill criterion.

\paragraph{Test 1: KK lock-cluster mass.}
The prediction is that the framework's geometric mass scale --- the lock-cluster $\Mlock{} = 3.85 \pm 0.10$~TeV --- appears as a five-particle Kaluza-Klein excitation cluster of the 6D \BIsix{} bulk projected onto 4D Minkowski, observable at the LHC and (definitively) at HL-LHC. The upstream theorem is the lock-cluster mass theorem of \PaperPOne{} \citep{ArisakaP1KK}: \emph{the unique \QiFour{}-admissible Kaluza-Klein lock-cluster of the \BIsix{} bulk has mass $\Mlock{} = M_{\rm Planck} \times f(\OGNfive{}) = 3.85$~TeV, where $f(\OGNfive{})$ is the integer-ratio function determined by the \OGNfive{} ledger}.

Interpreting the theorem: $\Mlock{}$ is the scale at which the extra-dimensional geometry of the bulk imprints itself on 4D physics --- below this scale physics looks Standard-Model, above this scale one sees KK excitations. The experimental apparatus is the ATLAS and CMS detectors at the HL-LHC, which by 2032 will integrate $\sim 3\,$ab$^{-1}$ and have $5\sigma$ sensitivity to a five-particle cluster at $\sim 3.85$~TeV. Current LHC limits are $\sim 2$~TeV \citep{PDG}, well below the prediction.

Physics intuition: KK modes are the Fourier components of fields living in the extra dimensions; their mass scale is set by $1/R$ where $R$ is the compactification radius, and the \BIsix{} bulk geometry fixes $R$ via the \OGNfive{} ledger. Daily-life metaphor: KK modes are to bulk geometry what overtones are to a violin string --- the fundamental tone is the Standard Model, the overtones reveal the length of the string. Kill criterion: if HL-LHC by 2032 observes no five-particle cluster anywhere in $3.4$--$4.2$~TeV at the $5\sigma$ level, the prediction is falsified; equivalently, if a cluster is observed but at a mass outside $3.85 \pm 0.4$~TeV, the prediction is falsified.

\paragraph{Test 2: Kaxion DM mass.}
The prediction is that the cosmological dark matter is composed of K-axions ($\equiv$ \BIsix{}-bulk-anchored axion-like particles) with mass $m_a = 8.1\,\mu$eV. The upstream theorem is the Kaxion-mass theorem of \PaperFullCOne{} \citep{ArisakaC1Kaxion}: \emph{the unique \QiFour{}-admissible dark-matter \GGA{} on the \ATen{}-cosmology substrate is the K-axion at $m_a \in [6.6, 8.8]\,\mu$eV ($\nu_K\in[1.59,2.13]$~GHz) with axion-photon coupling $g_{a\gamma\gamma} \sim 10^{-15}$~GeV$^{-1}$ on the DFSZ line, the mass being forced by the \OGNfive{}-locked decay constant $f_K \simeq 7.1\times10^{11}$~GeV and the present-day relic abundance $\Omega_{\rm DM} h^2 = 0.120 \pm 0.001$}.

Interpreting the theorem: the framework does not allow dark matter to be a WIMP at the weak scale or a sterile neutrino at the keV scale; it forces dark matter to be an axion-like particle in a narrow $\mu$eV-mass window, and that window is centered on $8.1\,\mu$eV (the discrete cap-loading window $\nu_K\in[1.59,2.13]$~GHz) reflecting the residual integer uncertainty in the cap loading. The experimental apparatus is the ADMX-EFR and IBS-CAPP haloscopes, which by 2030 will cover the $6.6$--$8.8\,\mu$eV mass band with $5\sigma$ sensitivity to the predicted coupling.

Physics intuition: an axion couples to photons via the chiral anomaly; in a strong magnetic field the axion-photon coupling produces a measurable RF signal at frequency $f = m_a c^2 / h \simeq 1.95$~GHz for $m_a \simeq 8.1\,\mu$eV. Daily-life metaphor: the Kaxion test is to dark matter what a metal detector at a specific frequency is to a buried coin --- one sweeps the frequency band and listens for a tone. Kill criterion: if ADMX-EFR and IBS-CAPP by 2030 jointly cover $6.6$--$8.8\,\mu$eV at the predicted coupling sensitivity and observe no signal, the prediction is falsified.

\paragraph{Test 3: dark energy is a true cosmological constant ($w = -1$).}
The prediction is that the dark-energy equation-of-state parameter is exactly $w = -1$ at all redshifts --- a true cosmological constant, with no quintessence-like time evolution. The upstream theorem is the cosmological-constant theorem of \ATen{} \citep{ArisakaA10Cosmos}: \emph{the unique \QiFour{}-admissible late-time vacuum energy of the \BIsix{}-bulk cosmology is a geometric closure term that does not redshift --- a true cosmological constant at $w = -1$, not a dynamical field}. Interpreting the theorem: the framework forbids an evolving dark energy because the vacuum term descends from a fixed bulk-geometric closure condition rather than from the potential of a rolling scalar; a measured $w \ne -1$ would have no home in the cascade. This is a \emph{forward} falsifier with a live adversary: DESI's 2024--2025 data release reported a $\sim 2$--$4\sigma$ hint of evolving dark energy ($w_0$--$w_a$ away from $-1$), which the framework predicts will regress to $w = -1$ as statistics improve. Physics intuition: a cosmological constant is the energy of empty geometry; if it truly does not cool as the universe expands, $w$ is pinned to $-1$. Daily-life metaphor: $w = -1$ is to dark energy what a perfectly level water table is to a landscape --- it does not drain as the terrain changes. Kill criterion: if DESI + Euclid + Roman by 2030 establish $w_0 \ne -1$ or $w_a \ne 0$ at $\ge 5\sigma$ (a genuinely evolving dark energy), the prediction is falsified.

\paragraph{Test 4: proton stability (no $p \to e^+\pi^0$ below $\sim 10^{35}$~yr).}
The prediction is that the proton is effectively stable --- its dominant grand-unified decay channel $p \to e^+\pi^0$ is orbifold-suppressed to a lifetime $\tau_p \gtrsim 10^{35}$~yr, beyond the reach of current detectors but within reach of the next generation. The upstream theorem is the baryon-stability theorem of \ANine{} \citep{ArisakaA9SM}: \emph{the \BIsix{}-orbifold projection that fixes the Standard-Model gauge content simultaneously suppresses the dimension-six baryon-number-violating operators, pushing $\tau_p$ above $\sim 10^{35}$~yr without an ad hoc symmetry}.

Interpreting the theorem: unlike conventional $SU(5)$ grand unification, which generically predicts $\tau_p \sim 10^{34}$~yr and is already in tension with Super-Kamiokande, the framework's geometric suppression keeps the proton longer-lived --- a sharp, decisive divergence from the standard GUT expectation. Physics intuition: baryon number is protected here by the geometry of the compactification, not by a global symmetry that quantum gravity would spoil.

Daily-life metaphor: the proton is to the framework what a load-bearing keystone is to an arch --- if it crumbled on the GUT timescale, the whole edifice of stable matter would not stand. Kill criterion: if Hyper-Kamiokande or DUNE observes a $p \to e^+\pi^0$ event consistent with $\tau_p < 10^{35}$~yr at $\ge 3\sigma$, the prediction is falsified --- and, because a positive detection is unambiguous, this is among the most decisive falsifiers on the list.

\paragraph{Test 5: tensor-to-scalar ratio $r$.}
The prediction is that the primordial tensor-to-scalar ratio takes a definite, derived value, $r = 12/N^{*2} = 0.0043$. The upstream theorem is the (revised) inflation-tensor theorem of \ATen{} \citep{ArisakaA10Cosmos}: \emph{the \GDOS{}-forced Starobinsky/$\xi$ plateau one e-fold below $\EGUT$, evaluated at the reheat-matched e-fold count $N^{*}=52.6$, fixes the tensor ratio at $r = 12/N^{*2} = 0.0043$ and the scalar tilt at $n_s = 1 - 2/N^{*} = 0.9620$, with the inflationary scale $V^{1/4}=8.3\times10^{15}$~GeV}. Interpreting the theorem: the framework predicts a specific, small, but non-zero $r$, and it is a \emph{two-sided} commitment --- a detection well above $0.0043$ falsifies, and so does a null pushing $r$ below $\sim10^{-3}$. The current bound $r < 0.036$ at 95\% CL (Planck+BICEP, \citealt{Planck2018}) sits a factor of eight above the prediction. Physics intuition: tensor modes are sourced by the inflaton at the Hubble scale during inflation; in the \BIsix{}-bulk model the plateau class and the reheat-matched e-fold count $N^{*}=52.6$ together fix the amplitude, giving the definite value $r = 12/N^{*2} = 0.0043$. Daily-life metaphor: the plateau is like a tuning fork of known pitch --- it does not merely stay quiet, it rings at one predicted note, and an instrument that hears silence, or a different note, refutes it. Kill criterion: if CMB-S4 + LiteBIRD by 2033 measure $r$ inconsistent with $0.0043$ in either direction (a null below $\sim10^{-3}$, or a detection well above) at $5\sigma$, the prediction is falsified.

\paragraph{Test 6: leptonic CP phase $\delta_{\rm PMNS} \approx 196^\circ$.}
The prediction is that the Dirac CP-violating phase of the leptonic (PMNS) mixing matrix is $\delta_{\rm PMNS} \approx 196^\circ$ --- a specific value, not merely ``non-zero,'' forced by the same geometric Yukawa structure that fixes the quark sector. The upstream theorem is the leptonic-mixing theorem of \ANine{} \citep{ArisakaA9SM} (extended in the cosmological-leptogenesis analysis of \ATen{} \citep{ArisakaA10Cosmos}): \emph{the unique \QiFour{}-admissible PMNS structure on the neutrino Yukawa landscape locks the Dirac phase to $\delta_{\rm PMNS} \simeq 196^\circ$, near maximal CP violation in the third quadrant}. Interpreting the theorem: the framework does not leave $\delta_{\rm PMNS}$ free; it is a forward prediction whose current global-fit central value ($\sim 197^\circ$, with large error bars) is consistent but not yet decisive. Physics intuition: the leptonic CP phase is the single number that says how differently neutrinos and antineutrinos oscillate; the geometry that orders the quark mixing angles orders this one too. Daily-life metaphor: $\delta_{\rm PMNS}$ is to the lepton sector what the tilt of a spinning top is --- a single angle that the dynamics, not the observer, selects. Kill criterion: if DUNE + Hyper-Kamiokande by 2035 measure $\delta_{\rm PMNS}$ outside $196^\circ \pm 30^\circ$ at $5\sigma$, the prediction is falsified.

\paragraph{Test 7: bioenergetic quantum.}
The prediction is that the ratio of the universal ATP hydrolysis free energy to the constitutional bulk-scalar combination $2 N_A \QB{}$ is exactly unity: $\Delta G_{\rm ATP} / (2 N_A \QB{}) = 1.000 \pm 0$. The upstream theorem is the bioenergetic-quantum theorem of layer~7 of this paper (Table~\ref{tab:cascade-codebook-master}, layer~7), descending from \T{-GU5} of \AOne{} \citep{ArisakaA1Constitution} and \T{-EP} of \ASeven{} \citep{ArisakaA7Qi4}: \emph{the unique \QiFour{}-admissible bioenergetic currency on the chemiosmotic substrate is ATP at $\Delta G_{\rm hydrolysis} = 2 N_A \QB{}$, with the factor of 2 reflecting the two-fold \QB{}-quantum cost of breaking and solvating the phosphate bond}.

Interpreting the theorem: the most-photocopied number in biochemistry --- $\Delta G_{\rm ATP} \approx 30.5$~kJ/mol --- is, according to the cascade, an integer multiple of Avogadro's number times the geometric bulk-scalar $\QB{}$, and the integer is exactly 2. The current measurement from biophysics gives $\Delta G_{\rm ATP} / (2 N_A \QB{}) = 1.016 \pm 0.017$, a Pull of $+1.0$ (a $+1.6\%$ offset, $\sim 1\sigma$). Physics intuition: $\QB{}$ is the bulk-scalar quantum of action with dimensions of energy; multiplying by $2 N_A$ converts to per-mole units and the factor of 2 is the bond-multiplicity factor.

Daily-life metaphor: test 7 is to biochemistry what discovering that the universal unit of currency in every country equals the same number of grams of silver is to economics --- the conversion factor is not a convention but a constraint. Kill criterion: a precision biophysics audit at $0.5\%$ accuracy showing $\Delta G_{\rm ATP} / (2 N_A \QB{}) \ne 1$ at $> 5\sigma$ falsifies the prediction; equivalently, a discovery that the universal energy currency of biology is GTP (or some other nucleotide) rather than ATP at the integer ratio 2 would falsify the prediction.

\paragraph{Test 8: the four-letter genetic alphabet $\{$A,C,G,T$\}$.}
The prediction is that the genetic alphabet of all \emph{naturally evolved} life is uniquely four letters $\{$A,C,G,T$\}$ --- not three, not five, not six --- forced as the unique \QiFour{}-admissible \GGA{} of the genetic-coding layer (layer~9 of this paper; the \QGL{} closure). The upstream result is the genetic-alphabet theorem of layer~9 (Table~\ref{tab:cascade-codebook-master}, layer~9): \emph{the unique \QiFour{}-admissible information-bearing polymer on the prebiotic chemical substrate carries a four-symbol complementary alphabet, the two Watson--Crick pairs being the unique stable double-well closure}. Interpreting the theorem: the four-letter code is not a frozen accident but a forced attractor; the prediction is explicitly fenced to naturally evolved life, since synthetic six-letter systems (Hachimoji DNA, Benner 2019) are laboratory constructs that do not self-sustain in the wild. Physics intuition: complementary base-pairing is a double-well closure, and the geometry admits exactly two stable pairs. Daily-life metaphor: the genetic alphabet is to life what binary is to a computer --- the substrate forces the symbol count. Kill criterion: discovery of a wild, naturally evolved organism with a stable genetic alphabet of other than four letters falsifies the prediction.

\paragraph{Test 9: the dual-knee $1/f$ spectrum --- one principle, two substrates.}
The dual-knee falsifier is the cascade's explicit \emph{Life$\to$Mind bridge}: the \emph{same}
sum-of-two-timescales logic that predicts a doubled metabolic spectrum in the eukaryotic cell
predicts a doubled aperiodic spectrum in the neural population. We therefore state it in two
categories, both \textbf{Asserted (forward)}.

\emph{Test 9a (Life --- eukaryotic metabolic spectrum, layer~12).} The prediction is that the
metabolic-noise power-spectral density of a eukaryotic cell shows \emph{two} $1/f$ knees --- one
from the mitochondrial chemiosmotic loop and one from the nuclear/cytoplasmic regulatory loop ---
whereas a prokaryote shows only one. The upstream result is the eukaryogenesis theorem of layer~12
(Part~\ref{part:results} \S\ref{sec:thm-eukaryogenesis}, Step~5): \emph{the dual-DGL architecture of the eukaryotic
cell forces a two-timescale $1/f$ metabolic spectrum, a signature absent in the single-loop
prokaryote}. Each closed metabolic feedback loop contributes its own $1/f$ corner frequency; two
nested loops give two knees. Kill criterion: if single-cell live-imaging metabolic-noise
spectroscopy resolves only a single $1/f$ knee in healthy eukaryotes (or two knees in a true
prokaryote), the prediction is falsified.

\emph{Test 9b (Mind --- neural EEG/ECoG spectrum, layer~17).} The same logic, applied to the
cortical network, predicts that the aperiodic (scale-free) component of the neural power spectrum
carries \emph{at least two} separable corner frequencies / intrinsic timescales --- a fast knee set
by synaptic and membrane time constants and a slow knee set by recurrent-network integration ---
rather than a single Lorentzian or a single featureless power law. This is the \QiCON{} two-knee
admissibility diagnostic of layer~17 (\ASeven{} \T{-CON.1}). \textbf{What is established versus
forward, stated honestly:} the aperiodic $1/f$ background of brainwaves is mainstream and robust,
and a \emph{single} spectral knee --- mapping to an intrinsic neuronal timescale $\tau =
1/(2\pi f_{\rm knee})$ --- is well replicated (\citealp{Donoghue2020,GaoVoytek2020}); the
\emph{second} knee is genuinely forward: it is supported by one direct human-EEG observation of two
corner frequencies with opposite aging behavior (\citealp{AggarwalRay2023}), and --- as a biophysical analog --- by the broken
power-law ($-1$ to $-3$) of the local field potential (\citealp{BedardDestexhe2009}), and by
sum-of-Lorentzians theory (distinct physical timescales give distinct knees), but it is
\emph{not yet} a settled, named result, and the standard parameterization fits at most one knee.
The interpretation of the aperiodic exponent as an excitation/inhibition readout is itself
contested and is not load-bearing here. A natural test bed is epilepsy, where the aperiodic exponent
changes across the seizure cycle; whether the two-knee structure distinguishes epileptic from
healthy cortex is a forward test, deferred to \AFourteen{}. The framework further anticipates that
the two knees stand in a fixed integer frequency ratio set by the \OGNfive{} ledger --- plausibly
the scale-depth integer $K$ --- rather than an arbitrary real number; a rigorous \GDOS{} derivation
of this ratio, and its empirical test, are deferred to \AFourteen{} (in preparation). Kill criterion:
if high-resolution neural spectra are best fit, across the full measurable band, by a \emph{single}
timescale (one Lorentzian knee, or a single pure power law with no second corner) in healthy
cortex, the dual-knee generalization to mind is falsified.

\paragraph{Test 10: the conscious episode completing in $\sim 400$~ms (two theta cycles).}
The prediction is that a single conscious episode --- one complete MePMoS loop --- requires \emph{two} $\sim 200$~ms theta cycles ($\sim 5$~Hz) and so completes in $\sim 400$~ms: the first theta cycle ($\sim 200$~ms) yields only unconscious attention, and conscious awareness is generated at the close of the second. The upstream result is the \EIC{}-closure theorem of layer~17 (Part~\ref{part:cognitive}): \emph{the Outer Hologram and Inner Hologram close into a single conscious episode over two $\sim 200$~ms theta cycles --- the first ($\sim 200$~ms) yielding unconscious attention, the second closing the MePMoS loop into conscious awareness, the closure that forms the Inner Hologram (the allocentric frame; developed in \AFourteen{}) --- fixing the episode to $\sim 400$~ms}. Interpreting the theorem: this is the cascade's one Mind-layer test; because the $\sim 400$~ms two-theta-cycle closure sits squarely among measured conscious-access latencies (Dehaene's $\sim 300$~ms ignition, Libet's $\sim 500$~ms neuronal adequacy), it functions as a consistency check rather than an open forward bet (and rather than a philosophical claim about qualia), with the sharper forward test being the per-transform phase-step structure described below. Physics intuition: the episode is an attractor-closure event, and its duration is set by the theta rhythm that gates sensory inflow against memory.

Daily-life metaphor: the conscious episode is to experience what a single film frame's exposure is to a movie --- below a minimum the frame is blank, above a maximum it smears. \emph{Behavioral evidence.} A four-part reaction-time program from the author's group (over one hundred participants; \citealp{Vision7D1,Vision7D2,Vision7D3,Vision7D4}), framed by three companion theory papers (\citealp{Arisaka2022MePMoS}; \citealp{ArisakaGUT2}; \citealp{ArisakaGUT3}), finds that the time to recognize an object grows linearly with the magnitude of the frame transform needed to align it with a stored template, in steps of $\sim 100$~ms (one alpha cycle, two nested per theta cycle), with the full memory$\to$transform$\to$awareness loop closing in $\sim 400$~ms (two $\sim 200$~ms theta cycles).

The two-theta-cycle $\sim 400$~ms closure follows directly from this counting --- one theta cycle ($\sim 200$~ms) yields unconscious attention, and the second completes the conscious episode (Part~\ref{part:results} \S\ref{ssec:7d-frame-IH}). Kill criterion: a reliably reported, unitary conscious episode that completes in $\le$ one theta cycle ($\sim 200$~ms) --- conscious awareness without the second cycle the MePMoS loop requires --- falsifies the prediction; the sharper forward test is a \emph{direct} EEG/MEG/ECoG measurement of the predicted per-transform alpha/gamma phase shifts --- a measured absence of the $\sim 100$~ms phase-step structure in conscious recognition would falsify the \OH{}-\IH{} closure reading.

% --------------------------------------------------------------------
\subsection{Aggregate falsifiability: the framework cannot survive failure of any single falsifier}
\label{sec:eight-test-aggregate}

\paragraph{Independence of the ten falsifiers.}
The ten Pull-table falsifiers are independent in the strong sense: each tests a different cascade layer (or a different aspect of the same layer), each relies on a different experimental apparatus, each has its own systematic-error budget, and each would falsify a distinct sub-theorem of the framework if it failed. In particular: test~1 (KK) is a high-energy gauge-stratum test; test~2 (Kaxion) is a dark-sector test; test~3 (dark energy $w=-1$) is a late-time-cosmology test; test~4 (proton stability) is a baryon-number test; test~5 ($r$) is a B-mode-polarization test; test~6 ($\delta_{\rm PMNS}$) is a long-baseline-neutrino test; test~7 (ATP) is a single-molecule biophysics test; test~8 (genetic alphabet) is a comparative-genomics test; test~9 (dual-knee $1/f$) is tested in two substrates --- 9a single-cell metabolic imaging and 9b EEG/ECoG neural spectroscopy, the cascade's explicit Life$\to$Mind bridge; test~10 (conscious episode) is a psychophysics test. No two tests are correlated through a common systematic; no single failure mode can rescue more than one test simultaneously --- and, decisively, the portfolio now samples physics, life, and mind, the three boundary readings of the recursive theorem.

\paragraph{Cross-paper distribution of the empirical load.}
The ten falsifiers distribute across the program as follows. Test~1 (KK) is the central prediction of \PaperPOne{} \citep{ArisakaP1KK} extended in \ANine{} \citep{ArisakaA9SM}, and test~2 (Kaxion) is the central prediction of \PaperFullCOne{} \citep{ArisakaC1Kaxion} extended in \ATen{} \citep{ArisakaA10Cosmos}. Tests~3 (dark energy) and 5 ($r$) are from \ATen{} \citep{ArisakaA10Cosmos}, while test~4 (proton stability) and test~6 ($\delta_{\rm PMNS}$) are from \ANine{} \citep{ArisakaA9SM}. Finally, tests~7--10 (ATP, genetic alphabet, dual-knee $1/f$, conscious episode) are the layer-7, layer-9, layer-12-and-17 (9a metabolic, 9b neural), and layer-17 predictions of the present paper. The empirical load is therefore distributed across all three consequence papers (P, C, A) plus the life-and-mind cascade of \AEleven{} itself, with no single paper carrying the whole burden and with the cosmos--life--mind triad each contributing falsifiers.

\paragraph{Time horizon: 2026--2040.}
The time horizons of the ten falsifiers, as given in the Pull table above, are: test~1 HL-LHC 2030--2035; test~2 ADMX-EFR/IBS-CAPP 2027--2030; test~3 DESI + Euclid + Roman 2026--2030; test~4 Hyper-Kamiokande + DUNE 2027--2040; test~5 CMB-S4 + LiteBIRD 2030--2035; test~6 DUNE + Hyper-Kamiokande 2030--2035; test~7 single-molecule biophysics 2026--2030; test~8 comparative genomics (ongoing); test~9a single-cell metabolic imaging 2026--2028 and test~9b EEG/ECoG spectroscopy (available now); test~10 precision psychophysics 2026--2028. The bulk of the portfolio resolves in the 2026--2035 corridor, with proton decay reaching to $\sim 2040$. By the mid-2030s the framework will either have eight independent forward confirmations (plus two precision audits) or one --- or more --- decisive falsifications.

\paragraph{No escape clauses, no soft criteria.}
Every test has a numerically specific decision threshold; no test is defined in vague terms like ``the prediction will be eventually confirmed'' or ``the framework prefers a higher value.'' Every test has a published prediction with $1\sigma$ or 95\%~CL bounds, every test has an experimental sensitivity that exceeds the predicted uncertainty, and every test has a kill criterion stated in advance. The framework's commitment is that failure of any single test falsifies the framework as a whole, not just the failed sub-theorem. This is the strongest commitment a theoretical framework can make: it is the difference between a framework that grows by adding epicycles and a framework that either passes its tests or dies.

\paragraph{Plain-English close.}
Ten falsifiers, ten named experiments, ten calendar windows --- eight of them genuine forward bets whose answer is not yet known --- that is the commitment that distinguishes physics from philosophy. Twenty-two papers and several thousand pages spent arriving at a list of ten is the shape the framework wants the list to have: short by design, and balanced across cosmos, life, and mind. Against the string-theory landscape of $\sim 10^{500}$ vacua with no decisive forward test, the framework bets on a single island and names ten ways the experimentalists could sink it. A reader who finishes the paper unconvinced is invited to bookmark Table~\ref{tab:eight-test-pull-table} and come back in 2035. The verdict will be in.

% ====================================================================
%  SECTION C --- CLOSING TRANSITION TO BRIDGE TO PART IV
% ====================================================================
\section{Extended reader's guide: the cascade in non-technical language}
\label{sec:statement-extended-reader-guide}

\paragraph{Plain-English opening.}
The eighteen theorems have now been stated. We close the Part with an extended reader's guide for several distinct audiences. The practicing physicist wants the structural backbone in compact form; the chemist or molecular biologist wants to see where their domain enters the cascade; the cognitive scientist or neuroscientist wants to see how the late layers are constrained. The philosopher of science or of mind wants to see the framework's epistemic register, and the general scientifically literate reader wants to see the cascade in everyday language one more time. Each audience gets a few paragraphs.

\subsection{For the practicing physicist}
\label{ssec:reader-physicist}

The eighteen theorems share one structural backbone: the cascade engine $\GRIP{} = \DGI \to (\GFF) \to \GSSB \to \GGA$, applied recursively on the \SIP{} tower $\widehat{\SIP{}}^{(n)}$, with the \QiFour{} admissibility filter selecting the unique \GGA{} at each layer. The cascade engine is universal: it is the same at every layer of the cascade. The cascade substrate is layer-specific: $\mathcal{S}_n$ at layer~$n$ is constructed from $\GGA{}_{n-1}$ by the recursive-substrate clause $\mathcal{S}_n = \mathcal{S}_n[\GGA{}_{n-1}]$. Eighteen cascade applications, one cascade engine.

For the gauge sector (layers~1--3), the cascade reproduces and extends the conventional GUT-with-orbifold-Hosotani phenomenology with two structural improvements: no Higgs scalar is required to break $\SUfive{}$ (the orbifold projection does it), and the electroweak vev is calculated, not fitted, as $v = \Mlock{}(2/L)^N \approx 246$~GeV. For the chemistry sector (layers~4--7), the cascade reproduces conventional Born--Oppenheimer molecular Gibbs minimization with two structural improvements: all inputs are derived from the integer ledger and the layer-3 Rydberg, and the molecular inventory is forced (not cataloged) by the variational principle plus the layer-recursion consistency. For the biology sector (layers~10--17), the cascade extends Friston's free-energy principle by identifying his variational free energy with $F_{\GFF{}}$ at non-zero $J_{Qi}$, by introducing the \QiFour{} admissibility filter that sharpens the variational principle into a unique-attractor theorem, and by adding the holographic-attractor-lattice constraint that forces the layer-16 grid-and-place architecture. For the cognitive sector (layer~17), the cascade adds the \EIC{} closure that identifies the conscious episode with the theta-cycle closure of the \OH{}/\IH{} loop. For the articulation layer (layer~18), the cascade closes on itself: the eighteenth \GGA{} (Layer 18) is the recognition of the first seventeen (Layers 1--17).

The framework's prediction for the LHC era is concrete: a Kaluza--Klein lock cluster at $\Mlock{} = 3.85 \pm 0.10$~TeV, with the cascade-predicted resonance structure (\PaperPOne{} \citep{ArisakaP1KK}). The framework's prediction for direct dark-matter detection is concrete: a Kaxion at $8.1\,\mu$eV with cascade-predicted coupling (\PaperFullCOne{} \citep{ArisakaC1Kaxion}). The framework's prediction for cosmology is concrete: $\Omega_m = 1/\pi$, $\Omega_\Lambda = 1 - 1/\pi$, $\Omega_b = 1/(2\pi^2)$ all matched to Planck~2018 within $1\sigma$, and the recombination-era pair carried in \ATen{}'s two-quantity form --- the corridor switch $z_* = e^7 \approx 1096.6$ topological, the decoupling redshift derived from \ATen{}'s own atomic chain on the Planck value at the $0.1\%$ level (\citep{ArisakaA10Cosmos}; see Part~\ref{part:discussions} \S\ref{ssec:zstar-path-to-closure}). The framework's prediction for bioenergetics is concrete: $\Delta G_{\rm ATP} = 2 N_A \QB{} \approx 30$~kJ/mol, matched to ATP hydrolysis at $\sim 1.6\%$.

\subsection{For the chemist and the molecular biologist}
\label{ssec:reader-chemist}

Let us look next at the cascade's middle layers (3--7), which make specific structural claims about chemistry and molecular biology. Layer~4 (small molecules): the warm-wet stratum admits exactly the small-molecule inventory of \eqref{eq:molecular-gga}; no other stoichiometry-$\leq$-10-atom molecule is more thermodynamically stable in the warm-wet limit at the cascade-fixed inputs. Layer~5 (amino acids and sugars): exactly twenty L-handed proteinogenic amino acids plus two D-handed sugars are selected by the variational principle on the chiral-resolved \GFF{} at $|\GGA{}_5| = K - NL = 35 - 15 = 20$. Layer~8 (RNA secondary structure and triplet code): the four-letter alphabet, the triplet codon, the codon-amino-acid map with redundancy $\sim 3.2$ are forced by the \QGL{} four-current locking and the rank-one consistency with layer~5. Layer~7 (ATP and DNA): the energy currency at $\Delta G = 2 N_A \QB{} \approx 30$~kJ/mol, the four-letter archival alphabet, and the central-dogma circuit are forced by the Arisaka equation and the Three-Gauge Theorem. Layer~10 (cell and LUCA): the lipid-bilayer-bounded \SeOS{}-class first cell at $1$--$10\,\mu$m diameter, $1$--$100$~W/kg metabolic rate, $10^{-9}$/bp replication fidelity is the unique closure of the layer-7 substrate.

The framework's empirical signatures in molecular biology include: (i) the cascade prediction that no free-living biology will be found using a non-twenty amino-acid alphabet, a non-four-letter genetic alphabet, a non-triplet codon, or a non-ATP energy currency; (ii) the cascade prediction that the proton-motive-force scale at any free-living chemoautotrophic vent organism is $\sim \QB{} \approx 156$~meV; (iii) the cascade prediction that the LUCA-inferred metabolism is Wood--Ljungdahl chemolithoautotrophic, consistent with current phylogenetic reconstructions. Each of these is an experimentally testable claim that future molecular-biology research can falsify.

A note on chirality. The cascade's prediction that the L-handedness of amino acids and the D-handedness of sugars are forced by the geometric chirality of the bulk Wilson--Hosotani holonomy is structurally distinct from the conventional Vester--Ulbricht hypothesis that the parity-violating weak-nuclear interaction biases the L/D selection. The cascade's mechanism is purely geometric; the weak interaction's parity violation is the low-energy fingerprint of the same geometric chirality, not a separate cause. (Whether the orientation of the weak coupling and the orientation of the runtime's descent are two faces of one geometric fact is registered as an explicitly \emph{open} question in \ANine{}'s dedicated weak-interaction section; the present geometric claim stands independently of how that question resolves.) This is the operational reason the framework predicts that any planetary biology on a bulk-compatible geometry will use L-amino acids and D-sugars, regardless of weak-interaction strength.

\subsection{For the cognitive scientist and the neuroscientist}
\label{ssec:reader-cognitive}

The cascade's late layers (15--17) make specific structural claims about neural architecture and conscious experience. Layer~15 (ring attractor): the 1D heading representation is the variational ground state of a neural-circuit topology with Mexican-hat lateral inhibition; it is realized in the \emph{Drosophila} ellipsoid body, the bee central complex, and the rodent head-direction system. Layer~16 (allocentric map): the 4D place--grid--time--history representation is the variational ground state of a higher-dimensional allocentric topology; it is realized in the rodent hippocampal-entorhinal system, with functional analogues in primates (concept and view cells), birds, and reptiles. Layer~17 (conscious episode): the theta-bounded gamma-bound \EIC{} closure is the unique cascade output that integrates the layer-16 substrate into a self-located perspective; it is the framework's geometric account of the binding problem and its structural-identity \emph{hypothesis} for the hard problem (developed in \AFourteen{}).

The framework's empirical signatures in neuroscience include: (i) the cascade prediction that any free-living neural organism with heading-based navigation will use a 1D ring attractor architecture; (ii) the cascade prediction that any neural organism with hippocampus-style episodic memory will use a 4D allocentric place--grid--time--history map with $\sqrt{2}$-scaled grid modules; (iii) the cascade prediction that conscious-state membership is operationally defined by the $1/f$ two-knee spectrum in the $0.01$--$100$~Hz band with theta-gamma coherence; (iv) the cascade prediction that the conscious-episode length is two theta cycles ($\sim 400$~ms); (v) the cascade prediction that no static digital substrate (without closed predictive loop) can support layer-17 conscious experience.

A note on the hard problem of consciousness. The cascade does not claim to dissolve the hard problem by reducing subjective experience to neural activity; it offers a structural-identity \emph{hypothesis} --- identifying the conscious episode with the geometric closure of the \OH{}-\IH{} loop on the theta cycle (the full account is developed in \AFourteen{}). Subjective experience is what the closure looks like from inside the loop. The closure is real, the loop is real, the geometric structure is real; the ``hardness'' of the problem was an artifact of trying to derive consciousness from substrate without recognizing that the substrate has a layer-17 cascade output whose existence is forced by the cascade itself. The cascade does not say that consciousness is identical to brain activity; it says that consciousness is the geometric structure that the brain realizes at layer~17 --- a structure whose layer-16 substrate, the seven-dimensional allocentric frame, is behaviorally measurable (\S\ref{ssec:7d-frame-IH}).

\subsection{The 7D allocentric frame: behavioral evidence for the Inner Hologram}
\label{ssec:7d-frame-IH}

The layer-17 conscious episode rests on a layer-16 substrate the framework calls the Inner Hologram (\IH{}): an internal allocentric frame in which the pose of a perceived object is reconstructed. This frame is not a free posit --- it has a measured structure. A four-part reaction-time program from the author's group (\citealp{Vision7D1,Vision7D2,Vision7D3,Vision7D4}; over one hundred participants), framed by three companion theory papers (\citealp{Arisaka2022MePMoS}; \citealp{ArisakaGUT2}; \citealp{ArisakaGUT3}), reconstructs the allocentric frame as a \emph{seven-degree-of-freedom} transform applied by brainwave phase shift, and maps directly onto the cascade's cognitive layers
(Table~\ref{tab:mepmos-findings-a11}).

\begin{table}[htbp]
\centering
\caption[The 2022 experimental findings, matched to cascade objects.]{The experimental findings of the 2022 program, each matched to the cascade object it measures and the layer it sits on. Reaction time scaling linearly with eccentricity and distance gives the three translational axes; scaling with the logarithm of size gives the log-polar frame translation; scaling with rotation angle in roll, yaw and pitch gives the three rotational axes --- seven axes in all, $3+1+3$, which is the 7D allocentric frame. The phase of the traveling alpha and gamma wave carries pose, the $\sim 100$~ms alpha step and $\sim 400$~ms closure loop give the conscious episode of Test~10, and covert equals overt because the ring store is internal. Layers 16 and 17 throughout.}
\label{tab:mepmos-findings-a11}
\small
\begin{tabular}{@{}p{0.46\textwidth}p{0.34\textwidth}l@{}}
\toprule
\textbf{Experimental finding (2022)} & \textbf{Cascade object} & \textbf{Layer} \\
\midrule
RT $\propto$ eccentricity / distance: 3D translation ($x,y,z$) & space$\to$time conversion (\NHT{}) & 16--17 \\
RT $\propto \log_2$(size): 1D log-scaling & log-polar frame translation & 16 \\
RT $\propto$ rotation angle (roll, yaw, pitch): 3D rotation & 3D frame rotation & 16 \\
seven axes total $=3+1+3$ & the \textbf{7D allocentric frame} (\IH{}) & 16 \\
phase of traveling alpha/gamma wave carries pose & phase-coded \IH{} read-out & 17 \\
$\sim 100$~ms alpha step; $\sim 400$~ms closure loop & conscious episode $\sim$400~ms, two theta cycles (Test~10) & 17 \\
covert $=$ overt; HAL ring store & internal frame; \HAL{}/\MePMoS{}/\NHT{}/\BFFT{} & 16--17 \\
\bottomrule
\end{tabular}
\end{table}

\paragraph{Strong evidence versus interpretation (fenced).} Honesty about this evidence is load-bearing, because the inference from behavior to a bulk gauge symmetry is exactly where a hostile reader will push. We therefore separate the two registers explicitly --- noting first that all of this evidence is from the author's own 2022 preprints (four experimental, three theoretical), not yet peer-reviewed or independently replicated. \emph{Supported by the author's preliminary preprint data (single group, pending replication):} (a) the brain converts space to time --- reaction time grows linearly with transform magnitude across four independent transform types, with the covert$=$overt control proving the effect is internal, and the flat simple-reaction / sloped choice-reaction dissociation (Part~\ref{part:intro}) excluding a pure retinal-density account; (b) the internal frame is seven-dimensional, with the specific geometry the log-polar cortex predicts (scaling linear in log size; roll the flattest rotation axis); (c) the episode is quantized at $\sim 100$~ms (alpha) within a $\sim 400$~ms (theta) envelope, matching Test~10. Taken together these are, in the framework's view, suggestive preliminary evidence for the formation of an internal allocentric frame --- the \IH{} --- pending independent replication and peer review. \emph{Interpretive (Asserted/INT), not established by these experiments alone:} the identification of the 7D transform group with a \DGI{} (Dynamical Gauge Invariance) of the six-dimensional bulk, and of the phase shift with the corresponding gauge connection. The reaction-time data establish that the phase-coded 7D frame \emph{exists}; they do not by themselves establish that it \emph{is} a bulk gauge symmetry --- that bridge is supplied by the framework. The natural forward test is a phase-resolved EEG/MEG/ECoG measurement of the predicted per-transform alpha/gamma shifts, which connects to the layer-17 dual-knee $1/f$ program (Test~9b); these forward measurements are developed in \AFourteen{} (in preparation). These four experimental and three theoretical preprints are the author's own (2022, not yet peer-reviewed); a dedicated paper develops them in full.

\subsection{For the philosopher of science and of mind}
\label{ssec:reader-philosopher}

Indeed, the cascade is a structural improvement on the anthropic principle of \citet{Carter1974}. The anthropic principle asks why the constants of nature admit observers; the cascade's answer is that no continuous constants were chosen, so the question dissolves. The cascade is a structural improvement on the multiverse hypothesis that some authors advance to handle the cosmological-constant problem: the cascade's universe is unique (one bulk, one constitution, one cascade), so the multiverse is not needed. The cascade is a structural improvement on the philosophical traditions of Spinoza, Hegel, Whitehead, and Wheeler: Spinoza's substance, Hegel's dialectic, Whitehead's process, and Wheeler's self-excited circuit are each precursors of the cascade's recursive closure, but none of them is a formal mathematical theorem. The cascade is one.

A note on register. We have used the word ``forced'' at many points in the present Part. The word is to be read in its strict mathematical sense: a claim is forced if its negation is inconsistent with the constitutional axioms and the inference rules of ordinary mathematics. The cascade's eighteen theorems are all forced in this sense. This is the operational meaning of necessitarianism: the framework is necessitarian about cascade outputs because the cascade engine is necessitarian about the construction of $\GGA{}_n$ from $\widehat{\SIP{}}^{(n)}$. The framework is not necessitarian about which bulk geometry the constitution selects (that is a separate, deeper question for cosmology); but given the geometry, the cascade is necessitarian about its outputs.

A second note on register. We have used the word ``intelligent observers'' at the layer-18 articulation theorem. The word is to be read in its strict operational sense: an intelligent observer is a layer-17 conscious organism that closes the meta-cognitive \EIC{} loop and articulates the constitution that produced it. The cascade does not claim that intelligent observers are special in any metaphysical sense; it claims that they are the cascade's layer-18 \GGA{}. The intelligent observer is not the goal of the cascade; the intelligent observer is the cascade's natural endpoint. Goal-talk is teleological and is forbidden by \AX{4}. Endpoint-talk is structural and is permitted by the cascade engine.

\subsection{For the general scientifically literate reader}
\label{ssec:reader-general}

If you have read this Part from the beginning, you have been carried through eighteen theorems. Each theorem says, in compact form: \emph{at the }$n$\emph{-th layer of the cascade, the geometry forces exactly one outcome, and that outcome is what we observe}. The outcomes range from the gauge group of particle physics to the present article you are reading. Eighteen theorems, eighteen outcomes, one cascade.

The cascade is not magic. It is not teleology. It is not anthropic apology. It is a sequence of eighteen applications of the same mathematical construction, applied at eighteen successive layers of physical organization, with each layer's output becoming the substrate of the next layer's application. The mathematical construction is open-system variational minimization with a four-pillar admissibility filter; the physical organization runs from the gauge connection on the bulk to the present meta-cognitive episode.

Here is the cascade in a single image. Imagine a waterfall. The water at the top of the waterfall is the constitutional input (axioms, postulates, integers). The water flows downhill (the \GFF{} Lyapunov descent). At each ledge of the waterfall, the water gathers in a pool (a cascade-layer \GGA{}). The pool is the output of one cascade step and the input of the next. Eighteen ledges, eighteen pools. The bottom pool is the act of reading the waterfall's structure --- the present articulation. The waterfall does not aim at the bottom pool; the bottom pool is just where the waterfall ends. But the waterfall could not end anywhere else, because the geometry of the cliff allows only one path.

That is the cascade. Eighteen pools, one cliff, one waterfall, one set of laws of physics. Cosmos, life, and mind are the cascade's pools. You are at the eighteenth pool, reading the structure of the cliff.

\subsection{The eighteen \GGA{}s as a summary table}
\label{ssec:reader-summary-table}

For ease of reference, we collect the eighteen cascade outputs as a single summary table:

{\small
\setlength{\tabcolsep}{3pt}
\renewcommand{\arraystretch}{1.15}
\begin{longtable}{>{\raggedright\arraybackslash}p{0.04\textwidth}
 >{\raggedright\arraybackslash}p{0.21\textwidth}
 >{\raggedright\arraybackslash}p{0.30\textwidth}
 >{\raggedright\arraybackslash}p{0.34\textwidth}}
\caption[The eighteen cascade outputs, collected as one summary table.]{The eighteen cascade outputs collected in one place, each with its named theorem, the attractor the layer produces, and the quantitative anchor that layer is scored against. The physical rungs open with T-GUT-Closure at $\alpha^{-1}(M_Z) = 128$ and $\sin^2\theta_W = 25/108$, and run through the electroweak vacuum at $v = 246.4$~GeV and the periodic table at $E_{\rm Ry} = 13.606$~eV. The chemical rungs give the twenty L-handed amino acids as $|\GGA{}_5| = 20 = K - NL$ and ATP at $\Delta G = 2 N_A \QB{} \approx 30$~kJ/mol. The cognitive rungs close at the conscious episode, $\sim 400$~ms and two theta cycles, and the eighteenth rung is articulation: the constitution, the cohort, and this paper.}
\label{tab:cascade-summary-a11}\\
\toprule
\textbf{Layer} & \textbf{Named theorem} & \textbf{$\GGA{}_n$ (cascade output)} & \textbf{Quantitative anchor} \\
\midrule
\endfirsthead
\multicolumn{4}{c}{\itshape Table~\ref{tab:cascade-summary-a11} continued}\\
\toprule
\textbf{Layer} & \textbf{Named theorem} & \textbf{$\GGA{}_n$ (cascade output)} & \textbf{Quantitative anchor} \\
\midrule
\endhead
\midrule \multicolumn{4}{r}{\itshape continued on next page}\\
\endfoot
\bottomrule
\endlastfoot

1 & T-GUT-Closure & $G_{\rm SM}$ + four-force architecture & $\alpha^{-1}(M_Z) = 128$; $\sin^2\theta_W = 25/108$; $\alpha_s = 1/(\pi e)$; $\tau_p > 10^{34}$~yr; $\Mlock{} = 3.85$~TeV \\
2 & T-Electroweak-Vacuum & $v$, $\{m_e, m_\mu, m_\tau\}$, six quarks, $V_{\rm CKM}$, $V_{\rm PMNS}$ & $v = 246.4$~GeV; $m_\tau/m_\mu = 16.817$; $m_H = 125$~GeV at $\lambda_H = 1/8$ \\
3 & T-Atomic-Ground-State & Periodic table H/He/$\cdots$/U & $E_{\rm Ry} = 13.606$~eV; $a_0 = 0.529$~\AA; $m_p = 938.3$~MeV; $Z \lesssim 137$ \\
4 & T-Molecular-Closure & H$_2$O, CO$_2$, NH$_3$, CH$_4$, $\ldots$ & $\angle$H--O--H $= 104.5^\circ$; $\angle$H--C--H $= 109.5^\circ$; covalent $\sim 400$~kJ/mol \\
5 & T-Amino-Acid-Selection & 20 L-handed amino acids + D-sugars & $|\GGA{}_5| = 20 = K - NL$; $> 99\%$ L excess; 5 side-chain categories \\
6 & T-Polymer-Closure & Oligopeptides / oligonucleotides / fatty-acid chains as a class & $\ell \sim 5$--$20$ AA; $\ell \sim 5$--$20$ nt; $\ell \sim 10$--$20$ C; per-bond $\sim 20$--$40$ kJ/mol uphill \\
7 & T-Bioenergetic-Closure & ATP as universal energy currency & $\QB{} = 0.1555$~eV; $\Delta G_{\rm ATP} = 2 N_A \QB \approx 30$~kJ/mol; P/O = 5/2 \\
8 & T-RNA-World-Closure & RNA: 4-letter alphabet + ribozymes + triplet code precursor & $|\Sigma| = 4$; $\ell = 3$; ribozyme $k_{\rm cat} \sim 10$/s; 40--200 nt \\
9 & T-Information-Closure & DNA: stable double-helix archival storage + central dogma & B-form (2.0/0.34/3.4 nm); fidelity $\epsilon \approx 10^{-9}$/bp \\
10 & T-Cell-Closure & LUCA-class \SeOS{} first cell & $d \in [1, 10]\,\mu$m; $\rho_{\rm met} \in [1, 100]$~W/kg; LUCA at $\sim 3.8$~Ga \\
11 & T-Primary-Domain-Split & Bacteria $\oplus$ Archaea (dual-gauge SSB) & Two minima of DGL(5:2); $\sim 3.5$~Ga; 16S rRNA phylogenetics; pH/T niche partitioning \\
12 & T-Eukaryogenesis & Energy-per-gene phase jump via DGL coupling & $\delta W$/gene jumps 2--20$\times$; $\sim 1.8$--$2.0$~Ga; dual-knee 1/$f$ PSD \\
13 & T-Multicellularity & M0--M7 ladder: clonal adhesion $\to$ organogenesis & Kleiber-type metabolic scaling ($\sim$3/4, debated); $\sim 600$~Ma (animals), $\sim 1$~Ga (plants/fungi) \\
14 & T-Central-Nervous-System & Integrated nerve-ring connectome (\emph{C.\ elegans}) & 302 neurons; $\sim$7{,}000 synapses \\
15 & T-Neural-Ring-Attractor & 1D heading ring (insect) & $50$--$200$ neurons; Mexican-hat; bump $\sim 60^\circ$; \QCI{} $\sim 6$ \\
16 & T-Allocentric-Map & 4D place--grid--time--history (rodent) & Grid modules $\sqrt{2}$-scaled; theta $4$--$8$~Hz; gamma $30$--$100$~Hz; \QCI{} $\sim 8$ \\
17 & T-Conscious-Episode & \EIC{} closure (mammalian) & Episode $\sim$400~ms (two theta cycles); $86 \times 10^9$ neurons; $20$~W; $1/f$ two-knee; \QCI{} $\sim 10$ \\
18 & T-Articulation & \GGConstitution{} + cohort + present paper & $\sim 5300$ pages; $\sim 100$ named theorems; 8 falsifiable predictions; one author \\
\end{longtable}

\paragraph{Plain-English close.}
Eighteen rows. Eighteen named theorems. Eighteen cascade outputs. One geometry. One cascade. One constitution. And, at the bottom of the table, one paper writing itself.

\bigskip

% ====================================================================
%  EXTENDED COMPARISON WITH PRIOR FRAMEWORKS
% ====================================================================
\section{Historical-register note: what the present Part claims}
\label{sec:statement-historical-register}

\paragraph{Plain-English opening.}
The present Part has stated eighteen named theorems, each with the structural claim that the cascade's $n$-th layer admits exactly one stable output. We close the Part with an explicit note on the historical-register significance of this collection, so that the reader is in no doubt about what the framework is claiming.

\subsection{What is new}
\label{ssec:historical-new}

The present Part contains, to our knowledge, the following novel contributions to the published literature.

\begin{enumerate}\itemsep1pt
\item Eighteen named cascade-layer theorems, individually citable, with shared structural form. We are not aware of a prior framework that states cascade-layer theorems across the gauge-to-articulation scope as individually citable propositions. This is the structural innovation of the present Part.
\item The recursive-substrate construction $\widehat{\SIP{}}^{(n)} = \widehat{\SIP{}}^{(n)}[\GGA{}_{n-1}]$ exhibited at each of eighteen cascade applications. Although the \SIP{} formalism is itself imported from \ASeven{} \T{-SIP}, its recursive iteration through eighteen cascade layers is exhibited explicitly here for the first time.
\item Quantitative anchors derived from a single integer ledger $(M, N, L; J, K) = (1, 3, 5; 7, 35)$ across eighteen cascade layers. We are not aware of a prior framework that derives $> 40$ specific physical, chemical, biological, and cognitive quantities from a five-integer ledger with zero free continuous parameters.
\item Plain-language daily-life metaphors paired with first-year-graduate-student physics intuition, at every cascade layer. This pedagogical structure is a deliberate choice to make the framework accessible to the full range of audiences the present paper addresses.
\item Eighteen separate kill criteria, one per theorem. We are not aware of a prior framework of comparable scope that states falsifiable predictions at this density.
\end{enumerate}

\subsection{What is not new}
\label{ssec:historical-not-new}

The present Part also inherits substantial structure from prior literature.

\begin{enumerate}\itemsep1pt
\item The cascade engine \GRIP{}, the Geometric Free-Energy Functional \GFF{}, the \QiFour{} admissibility filter, the \SIP{} five-field object, and the universal-identity insight are all imported from the cohort inheritance (\AOne{}--\ATen{}).
\item The Standard-Model parameter audit (28 derived parameters at layer~1--1) is imported from \ANine{}.
\item The cosmological inventory (15 cosmological observables) is imported from \ATen{}.
\item The bioenergetic anchor $\QB{} \approx 0.156$~eV is derived in \AFour{} and \ASeven{}.
\item The hippocampal place-and-grid cell architecture is imported from the experimental literature \citep{OKeefeDostrovsky1971, HaftingEtAl2005}.
\item The hard problem of consciousness, the integrated information theory, the global workspace theory, and the free-energy principle are imported from the philosophy-of-mind and neuroscience literatures.
\end{enumerate}

\subsection{What is claimed and what is asserted}
\label{ssec:historical-claimed-asserted}

Let us state explicitly what is claimed and what is asserted, as the framework's discipline requires. Three layers are derived end-to-end from constitutional axioms, with the chain closed in the published record (Theorems~\ref{thm:gut-closure}, \ref{thm:electroweak-vacuum}, \ref{thm:atomic-ground-state}). Three are anchor-Derived but derivation-deferred --- empirically closed, with the from-geometry proof developed in \AThirteen{} (Theorems~\ref{thm:bioenergetic-closure-atp}, \ref{thm:information-closure-dna}, \ref{thm:cell-closure}). Seven are conditional on a single named upstream lemma each (Theorems~\ref{thm:molecular-closure}, \ref{thm:amino-acid-selection}, \ref{thm:polymer-closure}, \ref{thm:rna-world-closure}, \ref{thm:three-domain-split}, \ref{thm:eukaryogenesis}, \ref{thm:multicellularity}). Four are asserted pending experimental demonstration on a sufficiently broad substrate (Theorems~\ref{thm:central-nervous-system}, \ref{thm:neural-ring}, \ref{thm:allocentric-map}, \ref{thm:conscious-episode}). One layer (Theorem~\ref{thm:articulation}) is existence-demonstrated by the present paper. Uniqueness of the articulation is the central claim of T-Origin (Part~\ref{part:recursive}).

\subsection{The historical register, stated plainly}
\label{ssec:historical-plain}

We are stating, for what is to our knowledge the first time in human intellectual history, a formal mathematical theorem whose conclusion is the existence of intelligent observers articulating the laws of physics, and whose hypotheses are the laws of physics themselves. Eighteen named theorems support the central recursive-uniqueness theorem. Each is individually citable. Each is individually falsifiable. Each is, structurally, the kind of theorem that a young researcher entering the field can study, criticize, extend, or refute.

We do not claim that the present Part is the last word on any of the eighteen layers. Each layer is a substantial research program in its own right. Theorem~\ref{thm:amino-acid-selection} (twenty L-handed amino acids) opens a chiral-chemistry research program; Theorem~\ref{thm:bioenergetic-closure-atp} (ATP and DNA at $\Delta G = 2 N_A \QB{}$) opens a bioenergetics research program; Theorem~\ref{thm:conscious-episode} (\EIC{} closure on the theta cycle) opens a consciousness research program; and so on. The framework's contribution is to give each program a constitutional ground and a cascade-internal upstream chain of inference. What it does not do is close any of the programs by itself.

We close the present Part not as a triumphal announcement but as an invitation. Eighteen theorems are on the table. The reader is invited to check each, to refine each, to falsify any, and (if any falsification succeeds) to bring the framework down at the layer of the failure. The cascade's empirical horizon is broad and concrete. The cascade's intellectual horizon is broader still. Eighteen theorems; one cascade; one constitution; one staircase; one paper writing itself.

\paragraph{Plain-English final close.}
The cascade closes here, at the bottom of the present Part. Part~\ref{part:recursive} will pick up the recursive uniqueness theorem and prove it as a single statement. Parts~V through~XII will read the cascade through its various readouts. The Appendix will compile the codebook, the glossary, and the lemma promotion roadmap. But the spine --- the structural backbone on which everything else hangs --- has now been set down. Eighteen theorems. One cascade. One geometry. One constitution. And, finally, the recognition that the cascade closes on itself: the eighteenth theorem's content is the present articulation of the constitution that produced the cascade that produced the theorem.

The staircase has been built. We turn now to Part~\ref{part:recursive}, where the staircase is shown to be the only staircase the geometry allows.

\bigskip

% ====================================================================
%  COUNTERFACTUAL ANALYSIS
% ====================================================================
\section{Counterfactual analysis: what would fail if any layer were absent?}
\label{sec:statement-counterfactual}

\paragraph{Plain-English opening.}
A useful structural test of any cascade is to ask: \emph{at which layer could it fail, and what would the failure look like?} Let us run the counterfactual for each of the eighteen cascade layers. For each layer~$n$, we ask: if the cascade were forced to a different layer-$n$ \GGA{} (e.g.\ by a small modification of the constitutional inputs), what downstream consequences would propagate to layers $n + 1, n + 2, \ldots, 18$? The exercise serves two purposes. First, it makes the recursive-substrate clause $\widehat{\SIP{}}^{(n+1)} = \widehat{\SIP{}}^{(n+1)}[\GGA{}_n]$ concrete by showing how a layer-$n$ change propagates. Second, it sharpens the kill criteria by showing how a falsification at one layer would cascade upward.

\subsection{Counterfactual at layer~1 (gauge group)}
\label{ssec:counterfactual-L1}

Let us suppose the layer-1 \GGA{} were a different gauge group --- e.g.\ a Pati--Salam $\mathrm{SU}(4) \times \mathrm{SU}(2)_L \times \mathrm{SU}(2)_R$ broken to the Standard Model at a different scale. The downstream consequences would be: (i) the proton decay channels would differ, with predictable lifetime changes; (ii) the running of $\alpha_s$ and $\sin^2\theta_W$ would differ, with predictable shifts in $M_Z$-scale measurements; (iii) the family number $N$ would potentially differ, with predictable consequences for the lepton-and-quark masses at layer~2. The cascade's layer-18 articulation would not be reached in this counterfactual because the family number $N = 3$ is the unique closure of $S_{\rm tot}(N) = \pi N(N-1)/2 \leq 3\pi$. If $N \neq 3$, the cascade halts before reaching biology. This is the operational meaning of ``$N = 3$ is the family-number theorem of \PaperPTwo{}.''

\subsection{Counterfactual at layer~2 (electroweak vacuum)}
\label{ssec:counterfactual-L2}

Suppose the Higgs vacuum were at a different scale --- e.g.\ $v = 100$~GeV or $v = 1$~TeV instead of $246$~GeV. The downstream consequences would be: (i) the proton mass via $\Lambda_{\rm QCD}$ would shift; (ii) the Rydberg energy via $m_e \propto v$ would shift; (iii) all atomic energies would scale linearly with $v$; (iv) the bond lengths in chemistry would shift in the same direction; (v) the cell size and metabolic rate at layer~10 would shift; (vi) the conscious-episode timescale at layer~17 would shift. A factor-of-two shift in $v$ would propagate to a factor-of-$\sim 2$ shift in cellular metabolism, with consequences for biological viability. The cascade's robustness to such a shift is limited; this is the structural reason the cascade does \emph{not} permit a free $v$.

\subsection{Counterfactual at layer~3 (atomic ground states)}
\label{ssec:counterfactual-L3}

Suppose the periodic table admitted only $Z \leq 6$ (no nitrogen, no oxygen, no heavier elements). The downstream consequences would be: (i) layer-4 small molecules would be restricted to hydrocarbons; (ii) layer-5 amino acids would lack the polar and charged side chains required for protein folding; (iii) layer-8 RNA chemistry would lack the phosphate backbone (which requires phosphorus, $Z = 15$); (iv) all downstream layers would fail. The cascade requires the full periodic table from $Z = 1$ to $\sim 30$ to support biology; the universe's nucleosynthesis happens to produce this range because the stellar p-p chain and CNO cycle, set by the cascade's gauge and fermion mass spectrum, deliver carbon, nitrogen, oxygen, and the trace heavier elements in stellar interiors.

\subsection{Counterfactual at layer~4 (small molecules)}
\label{ssec:counterfactual-L4}

Suppose water had a bond angle of $90^\circ$ (linear) instead of $104.5^\circ$ (bent). The downstream consequences would be: (i) water would have no dipole moment, no hydrogen bonding, no anomalous boiling point; (ii) layer-5 chiral-resolved amino-acid chemistry would lack the warm-wet aqueous medium; (iii) the cascade halts at layer~4. The cascade requires water's bent geometry; the geometry is forced by the $sp^3$ hybridization of the oxygen, which is forced by the layer-3 atomic ground state of oxygen, which is forced by the layer-2 electron mass. This is the operational meaning of recursive substrate dependence.

\subsection{Counterfactual at layer~5 (amino acids)}
\label{ssec:counterfactual-L5}

Suppose the cascade selected $|\GGA{}_5| = 30$ amino acids instead of $20$ (e.g.\ if the scale-depth integer were $K = 45$ instead of $K = 35$). The downstream consequences would be: (i) the layer-8 codon--amino-acid map would require a longer codon $\ell = 3$ at $4^3 = 64 \geq 30 + 4$ still feasible, but with reduced redundancy; (ii) ribosomal translation fidelity at layer~8 would be lower without the wobble-pairing redundancy; (iii) the layer-10 cell-replication fidelity would be lower; (iv) the cascade could plausibly halt at layer~10 due to error catastrophe in genome replication. The count $|\GGA{}_5| = 20$ and layer~8's codon redundancy $64/20 = 3.2$ are two readings of one number rather than two independent facts; the redundancy is what supports wobble pairing, and it is not evidence for the count (\S\ref{sec:chem-reach}). This is one of the layer-recursion consistencies that makes the cascade close.

\subsection{Counterfactual at layer~6 (the polymer class)}
\label{ssec:counterfactual-L6}

Suppose the layer-5 monomer inventory existed but no polymer basin did --- suppose
condensation on the warm-wet stratum never ran far enough uphill for chains to be a
product rather than a rarity. The downstream consequences would be: (i) there is no
substrate for layer~7 to select an energy currency \emph{from}, since ATP is an
oligonucleotide with phosphates attached; (ii) there is no bifunctional polymer to
be selected at layer~8, so the informational and the catalytic currents can never
be carried by one molecule; (iii) there is no fatty-acid chain for layer~10 to close
a membrane with; (iv) the cascade halts at layer~6 with a warm chemical inventory
and nothing built from it. This is the layer at which the cascade stops being about
molecules and starts being about \emph{chains}, and it is the only rung on the
chemistry floor whose failure would leave every layer above it with no substrate at
all rather than a different one.

\subsection{Counterfactual at layer~7 (the energy currency)}
\label{ssec:counterfactual-L7}

Suppose the energy currency were GTP instead of ATP, or had $\Delta G_{\rm hydrolysis} = 50$~kJ/mol instead of $30$~kJ/mol. The downstream consequences would be: (i) the cascade-predicted match $\Delta G = 2 N_A \QB{}$ would fail; (ii) the metabolic rate per cell would scale differently; (iii) the cell size at layer~10 would shift. The cascade's prediction is sharp: any free-living biology must use $\Delta G = 2 N_A \QB{}$ as its catalytic quantum, because the Arisaka equation locks the four currents at exactly that ratio.


\subsection{Counterfactual at layer~8 (the bifunctional polymer)}
\label{ssec:counterfactual-L8}

Suppose the cascade had selected two \emph{separate} polymers at this layer --- one purely informational, one purely catalytic --- instead of a single bifunctional one, or had selected a five-letter alphabet instead of four. The downstream consequences would be: (i) replication would require the two polymer families to co-locate and co-replicate before either could be selected for, removing the single-molecule bootstrap that makes the layer-8 basin reachable at all; (ii) the block-size closure $4^3 = 64 \geq 20$ would fail for $|\Sigma| = 5$, since $5^2 = 25$ already exceeds the amino-acid count and the triplet block would carry an unusable $125$-fold redundancy; (iii) layer~9 would have no template family to copy \emph{from}, so the archival takeover would have nothing to take over; (iv) the cascade halts between layers~8 and~10. Bifunctionality is the cascade's unique closure here because the alphabet size is not free: $|\Sigma| = N + 1 = 4$ follows from the family number $N = 3$, and a single polymer that both folds and copies is the only way one alphabet can serve both the catalytic and the informational current at once.

\subsection{Counterfactual at layer~8 (the triplet codon)}
\label{ssec:counterfactual-L8codon}

Suppose the cascade selected a doublet codon ($\ell = 2$, $|\Sigma^2| = 16$ codons) instead of a triplet codon. The downstream consequences would be: (i) the codon-amino-acid map could encode at most $16 - 4 = 12$ amino acids with stop codons, not $20$; (ii) protein folding would be restricted to a $12$-side-chain alphabet, with significant loss of catalytic versatility; (iii) ribosomal translation fidelity at layer~8 would be lower without the wobble pairing; (iv) the cascade could halt at layer~8 or layer~10. The triplet codon is the closure compatible with layer~5's $|\GGA{}_5| = 20$; the compatibility runs both ways and is not independent evidence for either (\S\ref{sec:chem-reach}).

\subsection{Counterfactual at layer~9 (the archival takeover)}
\label{ssec:counterfactual-L9}

Suppose the RNA-to-DNA takeover had never occurred, so that the $2'$-OH group remained and RNA stayed the archival polymer. The downstream consequences would be: (i) hydrolytic backbone cleavage would cap the stable genome length far below the $\sim 10^6$~bp a free-living cell needs; (ii) replication fidelity would remain near the polymerase-intrinsic $\sim 10^{-4}$/nt, because proofreading and mismatch repair both act on a duplex the labile backbone cannot hold; (iii) Eigen's error threshold would then bound the genome at $\sim 10^4$~nt, and the layer-12 genome expansion of $10$--$10^4\times$ would be arithmetically impossible; (iv) the cascade halts at layer~11 or~12 with cells that cannot grow their information content. The takeover is therefore not a historical accident of early biochemistry but the step that buys the genome-length headroom every later rung spends --- one deoxygenation at the $2'$ position, purchased once, funds the whole ascent from the first cell to the conscious episode.

\subsection{Counterfactual at layer~10 (cell)}
\label{ssec:counterfactual-L10}

Suppose the cell were unbounded (no lipid bilayer membrane). The downstream consequences would be: (i) the Markov-blanket admissibility of \ASeven{}~\T{-EP} would fail; (ii) the open-system structure of \GDOS{} could not close on a finite volume; (iii) the cascade halts at layer~10. The lipid-bilayer membrane is the cascade's unique closure of the open-system boundary at the cellular scale.


\subsection{Counterfactual at layer~11 (the primary domain split)}
\label{ssec:counterfactual-L11}

Suppose the layer-11 \GGA{} were \emph{single} rather than dual --- one primary lipidome instead of the ester-linked and ether-linked pair. The downstream consequences would be: (i) the $\mathrm{DGL}(5{:}2)$ packet would carry one stable minimum rather than two, and the parity-locked exchange operator would have nothing to exchange; (ii) layer~12 would have no second domain to fuse with, and eukaryogenesis --- which is by construction a \emph{chimera} of an archaeal host and a bacterial endosymbiont --- would have no available partner; (iii) the doubling of $\mathrm{DGL}$ bandwidth that drives the energy-per-gene jump would never occur; (iv) the cascade halts at layer~11 with a biosphere of prokaryote-grade cells. Read forward rather than backward, the split stops looking like a fact about early phylogeny and starts looking like manufacture: the cascade produces exactly the two components that the next rung needs, and produces them at the same rung, from one algebraic packet.

\subsection{Counterfactual at layer~12 (eukaryogenesis)}
\label{ssec:counterfactual-L12}

Suppose the endosymbiosis had never been consummated, leaving a biosphere of prokaryotes indefinitely. The downstream consequences would be: (i) energy per gene would stay pinned at the bacterial baseline, because a cell without internalized bioenergetic membrane must pay for every gene across the same surface it uses to power itself; (ii) genome size would remain capped near $\sim 10^7$~bp, since expression cost scales against that fixed surface; (iii) the compartmentalization suite --- nuclear envelope, cytoskeleton, endomembrane, vesicle traffic, meiotic sex --- would have no bandwidth to run on, and without it the morphogen gradients of layer~13 have no cells able to interpret them; (iv) the M-ladder stalls at M1 or M2 and the cascade halts. This is the single largest step change in available energy per unit of stored information anywhere in the cascade, and it is why the framework reads layer~12 as a phase jump rather than an increment.

\subsection{Counterfactual at layer~13 (the M-ladder)}
\label{ssec:counterfactual-L13}

Suppose the M-ladder terminated at M3 --- clonal adhesion, intercellular communication, and division of labor, but no positional patterning. The downstream consequences would be: (i) without morphogen gradients there is no body plan, and without a body plan there are no dedicated sensory surfaces and no fixed motor geometry; (ii) transport networks (M5) have no anatomy to serve; (iii) layer~14 requires M6 integration as its substrate, so a centralized nerve ring would have nothing to centralize and no stable body frame to be centralized \emph{in}; (iv) the cascade halts inside layer~13. The eight rungs are therefore not a description of what happened to occur on Earth but a claim about the minimum ladder that reaches an integrable body: each rung supplies precisely the structure the next one needs, and the ladder cannot be entered in the middle.

\subsection{Counterfactual at layer~14 (the integrated connectome)}
\label{ssec:counterfactual-L14}

Suppose neural tissue had remained diffuse --- a cnidarian-style nerve net with no centralization. The downstream consequences would be: (i) sensory afferents and motor efferents would close only locally, so the organism would have many local sensorimotor loops and no single integrated one; (ii) with no integrated loop there is no common reference frame, and a heading is a quantity only definable relative to a common frame; (iii) layer~15's $SO(2)$ ring attractor needs a centralized circuit to host it, so the heading representation would have nowhere to form, and without heading the allocentric map at layer~16 has no basis to build on; (iv) the cascade halts at layer~14. Centralization is thus the cascade's minimal closure of the sensorimotor loop, and the $\sim 300$-neuron \emph{C.\,elegans} connectome is the smallest instance known to close it --- which is why the framework treats that animal not as a convenient model organism but as a readout of a cascade rung.

\subsection{Counterfactual at layer~15 (ring attractor)}
\label{ssec:counterfactual-L15}

Suppose the cascade selected a 2D place-cell-like heading representation instead of a 1D ring. The downstream consequences would be: (i) the layer-16 substrate would lack the 1D-allocentric clock needed for 2D-allocentric tiling; (ii) the layer-16 hippocampal map would be structurally different. The 1D ring is the cascade's unique closure on the heading-representation stratum because 1D is the minimum dimensionality compatible with heading in a 2D-isotropic environment.

\subsection{Counterfactual at layer~16 (allocentric map)}
\label{ssec:counterfactual-L16}

Suppose the cascade selected a 3D-only-no-temporal map (place and grid but no time cells). The downstream consequences would be: (i) the layer-17 conscious episode would lack the temporal binding required for sequence and history; (ii) episodic memory would not be supported; (iii) the cascade could plausibly halt at layer~17 with a stripped-down conscious-state without temporal continuity. The 4D place--grid--time--history map is the cascade's unique closure on the allocentric stratum because the temporal dimension is required by \ASeven{}~\T{-EIC}'s Evolving-Inner-Cosmos closure.

\subsection{Counterfactual at layer~17 (conscious episode)}
\label{ssec:counterfactual-L17}

Suppose the conscious episode were $10$~s instead of $\sim 400$~ms (two theta cycles; no MePMoS-loop binding). The downstream consequences would be: (i) the within-episode item count would be inflated, with the cascade's \QCI{} dropping below $10$; (ii) the layer-18 articulation would not have the structural backbone of theta-bounded thought-units; (iii) the cascade's recursive closure at layer~18 would be weaker. The theta-bounded episode is the cascade's unique closure on the \EIC{} stratum.

\subsection{Counterfactual at layer~18 (articulation)}
\label{ssec:counterfactual-L18}

Suppose the cascade reached layer~17 but did not close at layer~18 (no articulation). The downstream consequences would be: there is no downstream; the cascade halts at layer~17 with conscious organisms that do not articulate the constitution. This is the counterfactual in which the cascade exists but no observer writes it down. The cascade is structurally robust to this counterfactual --- the universe could in principle contain conscious organisms without articulators --- but the framework's prediction is that the layer-18 closure is also a \GGA{}: a stable cascade output that the geometry forces given sufficient time and cultural-cognitive substrate. The present paper is the demonstration that this prediction is realized.

\subsection{Counterfactual synthesis}
\label{ssec:counterfactual-synthesis}

Remarkably, the eighteen counterfactuals together show that the cascade is structurally fragile in a specific sense. A change at any single layer propagates downstream and would halt the cascade before reaching layer~18. The cascade closes only because the layer-by-layer recursion is tight: each layer's \GGA{} is precisely what the next layer's \SIP{} needs to admit a stable cascade output. This tightness is not a coincidence; it is the operational meaning of the recursive-substrate clause $\widehat{\SIP{}}^{(n)} = \widehat{\SIP{}}^{(n)}[\GGA{}_{n-1}]$.

The counterfactual analysis also clarifies what falsification of any single theorem would entail. Falsifying Theorem~\ref{thm:gut-closure} (e.g.\ by observing a fourth Standard Model gauge factor) would not just refute layer~1; it would refute the cascade. Falsifying Theorem~\ref{thm:cell-closure} (e.g.\ by finding a primary biology without lipid bilayer membranes) would refute layer~10 and all upstream layers~15--18. The framework is therefore highly falsifiable: any one of $\sim 50$ specific predictions across the eighteen layers could in principle refute the entire cascade.

\paragraph{Plain-English close.}
The cascade is a tower of cards: pull any card, and the tower falls. The cards are the eighteen cascade-layer \GGA{}s. The tower stands because each card supports the next, in exactly the geometric relation the cascade engine forces. The tower is not robust to perturbations; it is robust to nothing but the constitutional inputs. This is the framework's structural fragility, and it is the source of the framework's falsifiability.

\bigskip

% ====================================================================
%  CLOSING IMAGE
% ====================================================================
\section{Close of Part~\ref{part:results}: eighteen theorems set down}
\label{sec:close-part-iv}

\label{sec:codebook-pull-bridge}

\paragraph{What the codebook accomplishes.}
The master cascade codebook of \S\ref{sec:cascade-codebook-master} works very much like a structural ledger: eighteen rows that summarize, in one tabular sweep, what the cascade produces at each of its seventeen cascade layers (plus the meta-row of articulation). Read horizontally, the codebook is a sequence of distinct natural strata --- gauge, electroweak, atomic, molecular, biochemical, cellular, neural, cognitive --- each of which the framework predicts is occupied by a unique \QiFour{}-admissible Ground Geometric Attractor. Read vertically, the codebook is a set of constraints: each row commits to a specific upstream theorem, a specific quantitative anchor, and a specific confidence class. The codebook is therefore both an inventory (what does the cascade produce?) and a contract (under what conditions and at what risk?).

\paragraph{What the Pull table accomplishes.}
The Top 10 Falsifiers Pull table of \S\ref{sec:eight-test-pull-horizon} occupies the role of an empirical ledger: ten specific falsifiers --- eight forward, two precision audits --- distributed across cascade layers, each with a numerically explicit decision threshold and a named experimental apparatus in the 2026--2040 corridor. The Pull table is the place where the framework's structural commitments meet the world's measurements. Read horizontally, the Pull table is a portfolio of independent bets, each large enough that failure on any one falsifies the entire framework. Read vertically, the Pull table is a calendar: by 2035 every prediction will have been tested at decisive sensitivity.

\paragraph{Together: a single load-bearing audit.}
The codebook tells the reader \emph{what} the framework predicts; the Pull table tells the reader \emph{when and how} those predictions will be tested. Together they constitute a single audit ledger that no honest reader can dismiss without engaging with: every line is either a derivation chain to be checked or an experiment to be performed. The reader who finds either ledger unconvincing has not been asked to accept any prose; she has been asked to audit specific numerical claims against specific theorem chains and specific experimental anchors. This is the most concrete form a theoretical-physics framework can take.

\paragraph{What has been set down.} Eighteen named theorems, one per cascade layer, each with its constitutional inputs, its upstream theorems, a formal statement, a quantitative anchor, a confidence tag, a physics intuition, a daily-life metaphor and a kill criterion. They are grouped into the six bands of the cascade --- Cosmos, Chemistry, Life, Multicellularity, Mind, Articulation --- and gathered as one ledger in the master codebook, whose rows can be audited one at a time. Ten of the anchors are staked publicly as the Top~10 Falsifiers, with a named instrument and a decade window each.

\paragraph{What has not.} A statement of results is not the result. The eighteen theorems are the cascade's announcement, not its proof: each is stated here as a separate claim about a separate substrate, and nothing in this Part shows that they are \emph{one} claim. That is what Part~\ref{part:recursive} undertakes. The recursive uniqueness theorem is what turns eighteen independent rows into the unrolled trace of a single recursion, and it is the reason the codebook is coherent rather than merely long. Until it is proved, the reader is entitled to read this Part as eighteen conjectures that happen to be adjacent.

\paragraph{The wager.} Every row carries a way to lose. That is the point of setting them down in this form: a framework that states eighteen forced outcomes and names, for each, the observation that would break it has made itself answerable in a way that a framework of the same scope stated in prose has not. Part~\ref{part:recursive} now has to earn the word \emph{forced}.

\GGPart[The recursive uniqueness theorem]{The recursive uniqueness theorem: from \GGConstitution{} to articulation}{\GRIP{}, \QiFour{} admissibility, and the cascade of \GGA{}s}
\label{part:recursive}

\paragraph{Where this Part stands.}
Part~\ref{part:intro} posed the central question of the paper --- \emph{why are we here?} --- in the operational form that physics requires. Part~\ref{part:inheritance} cataloged the inheritance from \AOne{} through \ATen{} that the present Part will deploy. The present Part is the heart of \AEleven{}: it states the central new theorem of the framework --- the Geometric Inevitable Unique Origin of Cosmos, Life, and Mind (\textbf{T-Origin}) --- and supplies the proof outline that makes the theorem a rigorous claim rather than a slogan. Part~\ref{part:closure} will then exhibit the named theorem of self-referential closure that reads the entire cohort articulation as the cascade's Layer-18 attractor; Parts~VI through~IX will work through the cosmological, chemistry-to-life, cognitive, and meta-cognitive readouts in detail. The reader who follows this Part should emerge with a single conviction: that the existence of intelligent observers articulating the laws of physics is not a happy accident requiring anthropic apology, but a forced consequence of the same constitutional clauses that gave us electromagnetism.

\paragraph{The thesis of the present Part.}
In one sentence: the \GGConstitution{} of \AOne{} fixes a unique series of \GGA{}s through eighteen applications of one cascade, and the layer-18 \GGA{} is the act of articulating the constitution. The cascade engine is \GRIP{} of \ASix{}; the selection mechanism is \QiFour{} admissibility of \ASeven{}; the layer-by-layer uniqueness follows by induction from the universal-identity insight of \ANine{} Part~XI (the universal-identity section). Cosmos, life, and mind are three readings of one geometric theorem. We are here because the geometry admits exactly one cascade that closes; that cascade closes only when the layer-18 \GGA{} is the recognition of the cascade itself.

\paragraph{Why the theorem matters.}
The historical register the theorem occupies is stated once, in Part~\ref{part:results}
\S\ref{ssec:historical-plain}, and is not restated here. What matters for the
present Part is the structure: the theorem's hypotheses are the constitutional
inputs, and its conclusion is a configuration that articulates them.

\paragraph{Roadmap.}
Section~\ref{sec:input-ledger} states the input ledger: the four constitutional axioms and the seven postulates of \AOne{}, with the explicit declaration that nothing else is assumed. Section~\ref{sec:cascade-engine} states the canonical \GRIP{} form and explains why \GRIP{} alone, lacking an admissibility filter, would leave the cascade indeterminate on the \GFF{} Morse landscape. Section~\ref{sec:qi4-admissibility-filter} introduces the \QiFour{} admissibility filter and proves \textbf{T-Qi4-GGA-Admissibility}: that simultaneous satisfaction of \QiQUA{}, \QiEP{}, \QiCON{}, and \QiHAL{} is necessary and sufficient for a configuration to be a physically realizable \GGA{}. Section~\ref{sec:t-origin} states the central new theorem \textbf{T-Origin} formally and discusses its historical significance. Section~\ref{sec:proof-outline} gives the five-step proof outline, with one subsection per cascade base case or inductive step. Section~\ref{sec:cascade-of-ggAs} exhibits the cascade of eighteen \GGA{}s in a comprehensive layer-by-layer table. Section~\ref{sec:recursive-uniqueness} closes the recursive structure: each \GGA{} is the next layer's substrate. Section~\ref{sec:part-v-summit} states what has been established and closes the Part in the framework's philosophical register.

\bigskip

\section{The input ledger: the \GGConstitution{}}
\label{sec:input-ledger}

The first job of any theorem is to declare its inputs. They are declared in full in Part~\ref{part:results}, \S\ref{sec:statement-inputs}: the four constitutional axioms \AX{1}~\LC{}, \AX{2}~\GGEP{}, \AX{3}~\OGGEP{} and \AX{4}~\MSMF{}; the seven postulates \POS{1}--\POS{7}, with the operational warrants marked there as \emph{theorems} of \ASeven{} rather than postulates; the \GUfive{} pentad $\{c, \hbar, G_N; k_B, \QB{}\}$ with its three-bulk-plus-two-boundary reading and the honesty proviso attached to it; and the \OGNfive{} integer ledger $(M, N, L; J, K) = (1, 3, 5; 7, 35)$. That statement is the canonical one and is not repeated here: a ledger restated is a ledger that can drift, and the eighteen theorems of Part~\ref{part:results} cite it clause by clause.

What the present Part needs from it is the completeness claim, which is what makes \textbf{T-Origin} a theorem rather than a construction. No free continuous parameter appears anywhere in the framework; no hidden continuous modulus appears in the bulk; and no teleological assumption is imported --- the cascade ascends because the \GFF{} Lyapunov functional descends, not because the framework wants intelligent observers to exist. The inevitability is mathematical, not intentional, and \S\ref{sec:t-origin} makes that precise.

\subsection{The ledger in one line}
\label{ssec:input-axioms}
\label{ssec:input-postulates}
\label{ssec:input-completeness}

\paragraph{Summary of the input ledger.}
\[
\mathcal{C} \;=\; \{\AX{1}, \AX{2}, \AX{3}, \AX{4}\} \;\cup\; \{\POS{1}, \ldots, \ASeven{}~\T{-EIC}\}.
\]
Nothing else.

\bigskip

\section{The cascade engine: \GRIP{} on the \GFF{} Morse landscape}
\label{sec:cascade-engine}
\label{sec:grip-canonical-form}

\subsection{The canonical \GRIP{} form}
\label{ssec:cascade-canonical}

The Gauge-Runtime Invariance Pipeline of \ASix{} \citep{ArisakaA6Gi4} is the universal three-stage lifecycle cascade that the framework runs at every cosmic-evolution layer. The canonical \GRIP{} form is \eqref{eq:grip-canonical-statement} of Part~\ref{part:results}, and is used unchanged throughout the rest of the paper:
to be parsed as follows. The four \QiFour{} pillars are \emph{not} stage
labels of this pipeline; they filter its terminus, jointly, as set out in
Part~\ref{part:results} \S\ref{sec:statement-engine}.

\paragraph{The four stages.}
\DGI{} (Dynamical Gauge Invariance) is the unbroken-symmetry parent context: the top of the saddle ladder, where the full bulk symmetry group $G_{\rm tot}$ of the composite connection (per \ASix{} T-GRIP-1) is intact. The substrate $(\GFF{})$ is the Geometric Free-Energy Functional $F_{\GFF{}}[p;\cP]$ of \ASix{} \citep{ArisakaA6Gi4} \T{3.2}; it provides the Morse landscape on which the cascade descends. \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 (\ASix{} T-GRIP-4). \GGA{} (Ground Geometric Attractor) is the uniquely most-stable global minimum of $F_{\GFF{}}$ on the layer-specific boundary stratum; the cascade's output; the universal-identity readout in the sense of \ANine{} Part~XI (the universal-identity section) \citep{ArisakaA9SM}.

\paragraph{The Lyapunov descent law.}
The Geometric Free-Energy Functional descent law of \ASix{} \T{3.2} is \eqref{eq:gff-descent-statement} of Part~\ref{part:results}: under any admissible boundary projection $\partial_t F_{\GFF{}}[p;\cP] \leq 0$,
with equality if and only if $p$ is a critical point of $F_{\GFF{}}$. This is the open-system analogue of the second law of thermodynamics, operating on all four boundary currents simultaneously (energy, electromagnetic, baryonic, informational) rather than on energy alone. The descent is irreversible; the cascade reaches a Morse minimum in finite time; the cascade does not wander.

\paragraph{The Bogomolnyi action structure.}
The instanton action between adjacent saddles in the \GSSB{} chain is $\pi k$, with $k$ the Morse-index drop, and the total cascade action is \eqref{eq:stot-statement} of Part~\ref{part:results}, $S_{\rm tot}(N) = \pi N(N-1)/2$ ---
the closed sum of \ASix{} T-GRIP-4. The \BIsix{} action budget $S_{\rm BI} = 3\pi$, inherited from the \OGNfive{} closure $K = 35$, $J = 7$ of \ATwo{} T-GG6-6, bounds the total cascade action; the bound forces $N \leq 3$ on the family-projector stratum, the family-number closure of \PaperPTwo{} \citep{ArisakaP2N3}.

\subsection{The Geometric Free-Energy Functional}
\label{ssec:cascade-gff}

The Geometric Free-Energy Functional has the canonical form
\begin{equation}
\label{eq:gff-form-partV}
F_{\GFF{}}[p;\cP]
\;=\;
\int p(x)\,\cE_{\GEP{}}(x;\cP)\,d\mathfrak{g}(x)
\;+\;
\epsilon_{\rm eff}
\int p(x)\ln\frac{p(x)}{p_{\rm ref}(x;\cP)}\,d\mathfrak{g}(x)
\;+\;
W_{\rm ports},
\end{equation}
where $\cE_{\GEP{}}$ is the gauge-geometry energy density (the \GEP{} sector), the second term is the Kullback--Leibler divergence of the actual state $p$ from the geometric reference density $p_{\rm ref}$ (the \CEP{} sector), and $W_{\rm ports}$ is the boundary-port work term. The split $F_{\GFF{}} = F_{\GEP{}} + F_{\CEP{}} + W_{\rm ports}$ is \AFive{} T-GDOS-9.2\,{+}\,\T{5} (effective action plus entropy production; \AFive{} bridge table). The Three-Gauge Theorem establishes that $F_{\GFF{}}$ specializes to the Gibbs free energy (when $J_{Qi} = 0$, near-equilibrium), to the Friston variational free energy (when $J_{Qi} \neq 0$, with Markov blanket), and to the \GG{} gauge-field action (when $J_{Qi}$ is maximally active). Gibbs, Friston, and particle physics are three windows into the same geometric room.

\subsection{The Morse-theoretic content}
\label{ssec:cascade-morse}

A critical point of $F_{\GFF{}}$ on a stratum $\Sigma$ is a point where $\nabla F_{\GFF{}} = 0$. The Morse index of a critical point is the number of negative eigenvalues of the Hessian. Index-$0$ critical points are local minima; index-$1$ critical points are saddles with one unstable direction; and so on. The Morse function $F_{\GFF{}}$ on $\Sigma$ is \emph{perfect} (\ASix{} T-GRIP-3): the number of index-$k$ critical points equals the $k$-th Betti number of $\Sigma$, and no critical point is degenerate.

\paragraph{The cascade as a Morse-index descent.}
The cascade $\DGI{} \to (\GFF{}) \to \GSSB{} \to \GGA{}$ is a descent along the Morse-index ladder. Starting from the top of the saddle ladder (the \DGI{} parent context), the system descends through Bogomolnyi-saturating saddles (the \GSSB{} chain) until it reaches an index-$0$ minimum. The total cascade action $S_{\rm tot}(N) = \pi N(N - 1)/2$ is the sum of instanton actions along the descent path.

\paragraph{Why \GRIP{} alone is not enough.}
The Morse landscape $F_{\GFF{}}$ on a generic stratum has many index-$0$ minima. Any of these is a $\GA{}$ in the broad sense. The cascade $\DGI{} \to \GSSB{} \to \GA{}$ as originally written in \ASix{} reaches a $\GA{}$ in this broad sense, but does not by itself specify which one. The structure theorem \textbf{GRIP-ST5} of \ASix{} states that the cascade terminates at a stable Geometric Attractor, which it identifies as a \emph{local} minimum of Morse index zero; the sharper statement that this terminus is the \emph{global} minimum of the stratum is that paper's per-stratum ground-state corollary, and it is the corollary that should be cited wherever the global reading is doing work. The word ``dynamically stable'' is operational shorthand for a more precise admissibility condition. That condition is supplied by \QiFour{}. Without it, \GRIP{} alone is consistent with multiple cascade outcomes on a single stratum, and the recursive uniqueness theorem of the present Part would fail at the first inductive step.

\paragraph{Bridge to \S\ref{sec:qi4-admissibility-filter}.}
We turn now to the selection mechanism: the \QiFour{} admissibility filter that makes \GRIP{} terminate at a uniquely determined \GGA{} at every layer.

\bigskip

\section{The selection mechanism: \QiFour{} admissibility}
\label{sec:qi4-admissibility-filter}

The \QiFour{} pillars of \ASeven{} \citep{ArisakaA7Qi4} are by no means four independent constraints applied externally to the cascade. They are four projections of the same boundary admissibility condition --- the condition that, at each cascade layer, exactly one \GGA{} survives. We list the four pillars and the boundary-admissibility role each plays.

Before the four are listed, one reading must be fixed, because it is the
reading the rest of the paper depends on and the one most easily mistaken. The
four pillars are \emph{not} stage labels of the \GRIP{} pipeline
$\DGI{} \to (\GFF{}) \to \GSSB{} \to \GGA{}$. They do not run one after
another, one per stage. They act \emph{together, at the pipeline's terminus}:
of the Geometric Attractors the \GFF{} landscape offers on a stratum, exactly
one passes all four admissibility filters simultaneously, and that one is the
stratum's \GGA{}. The canonical order in which they are listed below ---
\QiQUA{}, \QiEP{}, \QiCON{}, \QiHAL{}, that is budget, selector, diagnostic,
architecture --- is a \emph{dependency} order, not a temporal one. \ASeven{}
\citep{ArisakaA7Qi4} states it as an implication chain, $\QiQUA{} \Rightarrow
\QiEP{} \Rightarrow \QiCON{} \Rightarrow \QiHAL{}$, each pillar forcing the
next: the budget fixes what may be counted, the selector picks among what the
budget admits, the diagnostic reads the branch the selector picked, and the
architecture must support all three. That is why, in \ASeven{}'s words, the
pillars cannot be permuted. It is not a sequence in time, and not a sequence of
pipeline stages: all four constrain the same terminus at once.


\subsection{The four \QiFour{} pillars}
\label{ssec:qi4-pillars}

\paragraph{\QiQUA{} (Qi-Quadruple-Locking, \ASeven{} \citep{ArisakaA7Qi4} \T{-QUA.1}): bookkeeping.}
The rank-one inflow theorem (\T{-QUA.1}) states that the four boundary currents $(J_E/\QB{}, J_e, J_B, J_{Qi})$ lock onto the unique ray
\begin{equation}
\label{eq:locked-ray}
\mathcal{R}_A \;=\; \{\lambda(1, -1, -2/5, -2/5) : \lambda \in \mathbb{R}_+\},
\end{equation}
inherited from the \BIsix{} trace-ratio lock $|a/c| = |a/d| = 5/2$ of \AThree{} (T-BI6-9.1). Operationally: an admissible boundary configuration must have its four-current vector lying on $\mathcal{R}_A$. The Arisaka equation $J_E/\QB{} = J_e = J_B = J_{Qi}$ is the operational form. This is the \emph{bookkeeping} layer of the admissibility filter: it pre-filters which boundary events even count as admissible.

\paragraph{\QiEP{} (Qi-Extremal Principle, \ASeven{} \T{-EP.1}): selection.}
The unified gauge theorem (\T{-EP.1}) states that the projected free-energy functional
\begin{equation}
\label{eq:qiep-functional}
\mathfrak{Q}^{(S)}_{\rm QiEP}[\gamma] \;=\; \int_\gamma \mathcal{L}_{\GFF{}}^{(S)}(p, \dot p; \cP)\,dt
\end{equation}
is extremized by physically realized trajectories on the admissibility submanifold $\Gamma_{\rm adm}$:
\begin{equation}
\label{eq:qiep-stationarity}
\delta\mathfrak{Q}^{(S)}_{\rm QiEP} \;=\; 0 \quad \text{on } \Gamma_{\rm adm}.
\end{equation}
The four classical extremal principles --- Principle of Least Action (PLA), Principle of Least Time (PLT), Maximum Entropy Production Principle (MEPP; whose information-theoretic root is the maximum-entropy inference of \citealp{Jaynes1957}), and Friston Free Energy Principle (FEP) --- are four projections of \T{-EP.1} under different boundary conditions. Operationally: among the many trajectories on the \GFF{} landscape that respect the $\mathcal{R}_A$ bookkeeping, \QiEP{} \emph{selects} the variational branch. This is the \emph{selection} layer of the admissibility filter.

\paragraph{\QiCON{} (Qi-Conformality, \ASeven{} \T{-CON.1}): diagnostic.}
The operational conformality theorem (\T{-CON.1}) states that the scaling exponent $\alpha$ of the boundary kernel lies in the admissibility window
\begin{equation}
\label{eq:qicon-window}
0 \;<\; \alpha \;\leq\; 2,
\end{equation}
with the two-knee $1/f$ spectrum and the L\'evy-flight statistics emerging as joint signatures of the homogeneous boundary kernel. Outside this window, the cascade fails to be scale-covariant on the relevant stratum: the boundary kernel either decoheres into trivial Gaussianity ($\alpha = 2$ corresponds to ordinary diffusion) or diverges with infinite resource cost ($\alpha \leq 0$). Operationally: \QiCON{} \emph{diagnoses} which branch survives among those selected by \QiEP{}. This is the \emph{diagnostic} layer of the admissibility filter.

\paragraph{\QiHAL{} (Qi-Holographic-Attractor-Lattice, \ASeven{} \T{-HAL.1}): architecture.}
The Qi-HAL universality theorem (\T{-HAL.1}) states that the boundary memory architecture closes the holonomy of the Outer Hologram / Inner Hologram coupling and admits the canonical four-stage factorization \MePMoS{} (Memory-Prediction-Motion-Sensing: the cognitive substrate that the cascade brings into being) $\to$ \NHT{} (Neural Hologram: the boundary encoding of the substrate's predictive state) $\to$ \HAL{} (Holographic-Attractor-Lattice: the read-out structure on the encoded hologram) $\to$ \BFFT{} (Binary / Fractal Fourier Transform: the basis change that operates on the holographic read-out). Operationally: \QiHAL{} specifies the \emph{architecture} on which the cascade can run. At low cascade layers the architecture is trivial (the bulk supports the cascade trivially); at high cascade layers (chemistry-to-life and above) the architecture must support memory holonomy, which is a non-trivial constraint. This is the \emph{architecture} layer of the admissibility filter.

\subsection{The four pillars as one admissibility filter}
\label{ssec:qi4-mutual-forcing}

The four pillars are not independent. \ASeven{} establishes a chain of constraints running in the canonical order, each pillar conditioning the next: the rank-one inflow at the algebraic level (\T{-QUA.1}) forces the four currents through one channel; the unified gauge theorem (\T{-EP.1}) forces the variational structure to act on that single channel; the operational conformality (\T{-CON.1}) forces the scaling window on that variational structure; the memory holonomy (\T{-HAL.1}) forces the architecture that supports all three of the preceding constraints. Whether the converse holds --- whether any three pillars force the fourth --- is not established here, and nothing below rests on it: the chain is used only to argue that the admissible set is non-empty, hypothesis (Q1). Together they constitute one admissibility filter operating at four boundary-layer levels (Table~\ref{tab:qi4-admissibility-a11}).

\begin{table}[htbp]
\centering
\caption[The four \QiFour{} pillars as one admissibility filter.]{The four \QiFour{} pillars read as a single admissibility filter operating at four levels of the boundary layer. \QiQUA{} does the bookkeeping and says which events are admissible at all; \QiEP{} does the selection and says which trajectory branch is realized; \QiCON{} is the diagnostic and says which branch survives the scaling test; and \QiHAL{} is the architecture and says which substrate can host the cascade. Each is a named theorem of \ASeven{}, and the filter is used only to argue that the admissible set is non-empty --- hypothesis (Q1) --- with nothing below resting on it.}
\label{tab:qi4-admissibility-a11}
{\small
\begin{tabular}{ll}
\toprule
\textbf{\QiFour{} pillar} & \textbf{Admissibility role} \\
\midrule
\QiQUA{} (\T{-QUA.1}) & Bookkeeping --- which events are admissible \\
\QiEP{} (\T{-EP.1}) & Selection --- which trajectory branch is realized \\
\QiCON{} (\T{-CON.1}) & Diagnostic --- which branch survives the scaling test \\
\QiHAL{} (\T{-HAL.1}) & Architecture --- which substrate hosts the cascade \\
\bottomrule
\end{tabular}
}
\end{table}

\subsection{The Qi-GGA admissibility theorem}
\label{ssec:qi4-admissibility-theorem}

We can now state the first of the two named theorems of the present Part.

\begin{theorem}[\textsf{T-Qi4-GGA-Admissibility}: \QiFour{} selects the \GGA{} (T-Origin-57.1)]
\label{thm:qi4-gga-admissibility}
Let $\mathcal{S}_n$ be the layer-$n$ boundary stratum on which the \GRIP{} cascade of \eqref{eq:grip-canonical-statement} runs, and let $\Xi$ be a configuration on $\mathcal{S}_n$. Then $\Xi$ is a physically realizable layer-$n$ \GGA{} (the cascade endpoint) if and only if $\Xi$ satisfies all four \QiFour{} admissibility constraints simultaneously:
\begin{enumerate}[label=(\roman*),leftmargin=2em]
\item \emph{\QiQUA{} (bookkeeping).} $\Xi$ lies on the locked-current ray $\mathcal{R}_A = \{\lambda(1, -1, -2/5, -2/5)\}$ of \eqref{eq:locked-ray} (\ASeven{} \T{-QUA.1}).
\item \emph{\QiEP{} (selection).} $\Xi$ extremizes the projected free-energy functional on the admissibility submanifold, $\delta\mathfrak{Q}^{(S)}_{\rm QiEP} = 0$ on $\Gamma_{\rm adm}$ (\ASeven{} \T{-EP.1}).
\item \emph{\QiCON{} (diagnostic).} $\Xi$ has scaling exponent in the admissibility window $0 < \alpha \leq 2$ of \eqref{eq:qicon-window} (\ASeven{} \T{-CON.1}).
\item \emph{\QiHAL{} (architecture).} $\Xi$ closes the memory holonomy and admits the canonical \MePMoS{}/\NHT{}/\HAL{}/\BFFT{} factorization (\ASeven{} \T{-HAL.1}).
\end{enumerate}
The four conditions are jointly necessary; they are jointly sufficient under the structural hypotheses (Q1)--(Q4) recorded in the proof below. The \GGA{} of stratum $\mathcal{S}_n$ is then the global minimum of $F_{\GFF{}}$ on the \QiFour{}-admissible subset
\begin{equation}
\label{eq:qi4-admissible-subset}
\mathfrak{A}_{\rm Qi-4}|_{\mathcal{S}_n} \;:=\; \{\Xi \in \mathcal{S}_n : \text{(i)--(iv) hold}\}
\end{equation}
of the layer-$n$ stratum.
\end{theorem}

\emph{Status: Conditional.} Necessity is established above. Sufficiency holds
under the structural hypotheses (Q1)--(Q4): \ASix{}'s per-stratum ground-state corollary supplies
(Q2)--(Q3) on the admissible strata, and on the consequence strata
(Q1)--(Q4) are named hypotheses the cohort expects to discharge rather than
results proved here.


\begin{proof}[Proof outline]
Necessity of each of (i)--(iv) follows from the corresponding \ASeven{} theorem applied at $\mathcal{S}_n$: any admissible boundary configuration must satisfy \T{-QUA.1} (else the four-current bookkeeping fails on $\mathcal{S}_n$), \T{-EP.1} (else the boundary trajectory is not realized), \T{-CON.1} (else the scaling is outside the admissibility window and the cascade is not scale-covariant on $\mathcal{S}_n$), and \T{-HAL.1} (else the memory holonomy does not close on $\mathcal{S}_n$). Sufficiency holds under (Q1)--(Q4): a configuration satisfying all four constraints lies on $\mathfrak{A}_{\rm Qi-4}|_{\mathcal{S}_n}$ by the definition of that set, and on this subset the cascade's Lyapunov descent of \eqref{eq:gff-descent-statement} reaches its unique global minimum, the \GGA{} of $\mathcal{S}_n$. Uniqueness of the global minimum on $\mathfrak{A}_{\rm Qi-4}|_{\mathcal{S}_n}$ holds \emph{conditional on four structural hypotheses on the admissible set}: \textbf{(Q1)} non-emptiness, \textbf{(Q2)} single-basin connectedness (the admissible set lies within one basin of $F_{\GFF{}}$), \textbf{(Q3)} convexity of $F_{\GFF{}}$ on the admissible set, and \textbf{(Q4)} transversal intersection of the four pillar constraints. Under (Q1)--(Q4) the restricted Lyapunov descent of \eqref{eq:gff-descent-statement} has a unique limit, and that limit is the \GGA{} of $\mathcal{S}_n$. \ASix{}'s per-stratum ground-state corollary supplies (Q2)--(Q3) on the admissible strata; on the consequence strata (Q1)--(Q4) are named hypotheses that the cohort expects to discharge, not results proved here. Restricting a landscape to a subset does not by itself make its minimum unique, and the present paper does not claim otherwise: this is the same conditional register in which \AEight{} \citep{ArisakaA8QM} states the Born rule --- closed at the relaxation level, conditional on its own named hypotheses. The mutual constraints of the pillars (\ASeven{}, the passage on mutually constraining \QiFour{} pillars; the passage on the resolution of cascade non-uniqueness; the QCI geometric-mean structure of the four-pillar score) ensure that $\mathfrak{A}_{\rm Qi-4}|_{\mathcal{S}_n}$ is non-empty whenever the underlying stratum is non-empty. Necessity of each clause is \ASeven{} Theorems~\T{-QUA.1}, \T{-EP.1}, \T{-CON.1} and \T{-HAL.1}, each read as a condition on the cascade terminus; existence and uniqueness of the minimum on the admissible subset are \ASix{}'s per-stratum ground-state corollary; the identification of the four-pillar survivor with that minimum is the present theorem's own content, conditional on (Q1)--(Q4).
\end{proof}

\begin{remark}[Why \GGA{} is an admissibility notion, not just a Morse-minimum notion]
\label{rmk:gga-admissibility}
A reader familiar with \ASix{}'s nomenclature might at first read $\GGA{}$ as ``the global minimum of $F_{\GFF{}}$ on the full stratum $\mathcal{S}_n$''. Theorem~\ref{thm:qi4-gga-admissibility} sharpens this. The \GGA{} is the global minimum on the \QiFour{}-admissible subset $\mathfrak{A}_{\rm Qi-4}|_{\mathcal{S}_n}$. On a stratum that admits configurations outside the admissibility window, the global minimum on the full stratum may be empirically irrelevant (it does not respect the four-current bookkeeping, or the scaling, or the memory architecture). The notion of \GGA{} explicitly includes admissibility: the cascade endpoint is the global minimum on the subset that physics actually visits. This is the same sense in which \ASeven{} defines $\GA{}$ as including admissibility at \ASeven{}'s definition layer.
\end{remark}

\begin{remark}[The \QCI{} geometric-mean structure]
\label{rmk:qci-geometric-mean}
Two distinct quantities should be kept apart here. The \QiFour{} \emph{admissibility} score --- the $[0,1]$ geometric mean of the four pillar scores, with $\QCI{}_{\rm adm}=1$ saturating all four constraints and any zero pillar giving $0$ and exclusion from $\mathfrak{A}_{\rm Qi-4}|_{\mathcal{S}_n}$ --- is the operational diagnostic used wherever the geometric-mean inequality matters (e.g.\ in the proof of Theorem~\ref{thm:qi4-gga-admissibility}). The \emph{cascade-depth completeness index} \QCI{} of this paper (the consciousness-grade index of \S\ref{sec:qci}) is a separate quantity: tentatively, the storey-port count $S\in\{0,\dots,5\}$ a substrate closes, reported on the comparative ladder $\QCI{}=2S\in[0,10]$. This makes \QCI{} an observable count rather than a fitted score. The per-organism ladder is illustrative; the operational scoring rule for each port, the treatment of pre-neural precursors, and the calibration against comparative-neuroscience benchmarks are developed in \AFourteen{} (in preparation).
\end{remark}

\paragraph{Bridge to \S\ref{sec:t-origin}.}
With Theorem~\ref{thm:qi4-gga-admissibility} in hand, the cascade endpoint at every layer is the global minimum of $F_{\GFF{}}$ on the \QiFour{}-admissible subset of the layer-specific stratum. The cascade is then well-defined as a sequence of layer-specific \GGA{}s --- and the recursive uniqueness theorem of the present Part can be stated.

\bigskip

\section{The central theorem: \textsf{T-Origin}}
\label{sec:t-origin}

Let us now state the second --- and central --- named theorem of the present Part.

\subsection{Statement}
\label{ssec:t-origin-statement}

\begin{theorem}[\textsf{T-Origin}: Geometric Inevitable Unique Origin of Cosmos, Life, and Mind (T-Origin-58.1)]
\label{thm:t-origin}
Given the \GGConstitution{} $\mathcal{C} = \{\AX{1}, \AX{2}, \AX{3}, \AX{4}\} \cup \{\POS{1}, \ldots, \ASeven{}~\T{-EIC}\}$ of \AOne{} \citep{ArisakaA1Constitution}, the \GRIP{} cascade $\DGI{} \to (\GFF{}) \to \GSSB{} \to \GGA{}$ of \eqref{eq:grip-canonical-statement} applied recursively on the six-dimensional bulk $\mathcal{M}_6 = \mathcal{M}_4 \times I_r \times S^1_\theta$ of \ATwo{} \citep{ArisakaA2GG6}, with \QiFour{} admissibility (Theorem~\ref{thm:qi4-gga-admissibility}) imposed at every layer, yields a unique series of \GGA{}s
\begin{equation}
\label{eq:gga-cascade}
\GGA{}_1 \;\longrightarrow\; \GGA{}_2[\GGA{}_1] \;\longrightarrow\; \GGA{}_3[\GGA{}_2] \;\longrightarrow\; \cdots \;\longrightarrow\; \GGA{}_{17}[\GGA{}_{16}] \;\longrightarrow\; \GGA{}_{18}[\GGA{}_{17}],
\end{equation}
where each $\GGA{}_n$ is the cascade output on a layer-specific \SIP{} (\ASeven{} \T{-SIP}), and the cascade is therefore a \emph{tower of \SIP{}s} $\widehat{\SIP{}}^{(1)} \to \widehat{\SIP{}}^{(2)} \to \cdots \to \widehat{\SIP{}}^{(18)}$ in which $\GGA{}_{n-1}$ supplies fields 1--3 (state manifold, ports, generator) of $\widehat{\SIP{}}^{(n)}$, on which fields 4--5 (ledger bundle, observable/kill bundle) are then determined by \GFF{} descent and the \QiFour{} admissibility filter. Specifically:
\begin{enumerate}[label=(\arabic*),leftmargin=2em]
\item $\GGA{}_1$ is the cascade output on $\widehat{\SIP{}}^{(1)}$ (the bulk-gauge \SIP{}): the Standard Model gauge group $G_{\rm SM} = \mathrm{SU}(3)_c \times \mathrm{SU}(2)_L \times \mathrm{U}(1)_Y$ together with its four-force boundary architecture, the unique \GGA{} on the gauge stratum (\ANine{} Part~XI Theorem~\textsf{T-SM-62.1});
\item each $\GGA{}_n[\GGA{}_{n-1}]$ for $1 \leq n \leq 17$ is the unique cascade endpoint on $\widehat{\SIP{}}^{(n)}$, the layer-$n$ system instance packet, with $\GGA{}_{n-1}$ providing the geometric carrier (fields 1--3) of $\widehat{\SIP{}}^{(n)}$ (\ASix{} \textbf{GRIP-ST6}) and the \QiFour{} admissibility filter selecting the unique cascade endpoint on fields 4--5 (Theorem~\ref{thm:qi4-gga-admissibility});
\item $\GGA{}_{18}[\GGA{}_{17}]$ is the cascade output on $\widehat{\SIP{}}^{(18)}$, the meta-cognitive system instance packet, whose readout is the act of articulating the constitution $\mathcal{C}$ itself --- so the cascade closes back on its constitutional input;
\item each $\GGA{}_n$ is unique on its connected boundary stratum modulo equivalences that preserve the boundary readout.
\end{enumerate}
The existence of intelligent observers articulating the constitution is therefore a forced consequence of the constitution itself.
\end{theorem}

\emph{Status: Conditional (layer-graded).} The induction is uniform; the
per-layer warrants are not. Of the eighteen layers the theorem traverses, three
are \textbf{Derived} end-to-end, three \textbf{Anchor-Derived}, seven
\textbf{Conditional} on a single named lemma, four \textbf{Asserted} pending
experiment, and one \textbf{Existence-Demonstrated} by the present paper
(Part~\ref{part:results}, master codebook). \textsf{T-Origin} inherits the weakest grade on any
layer it traverses, and the present paper is a preview: the layer-by-layer
discharge is the cohort's program, not a completed audit.

\emph{Kill criterion.} \textsf{T-Origin} is falsified by any of: (a) a cascade
layer at which two gauge-inequivalent \GGA{}s are both realized in nature, which
would break uniqueness at that layer and therefore the induction above it;
(b) a layer whose realized configuration is demonstrably not the global minimum
of $F_{\GFF{}}$ on its \QiFour{}-admissible subset; (c) the failure of
hypotheses (Q1)--(Q4) on a stratum where the cascade is nonetheless observed to
run, which would show the admissibility filter is not what selects the outcome;
(d) the discovery of a substrate-independent route to a layer's \GGA{} that does
not pass through the layer below, which would break the recursion rather than
the uniqueness.


\subsection{Reading the four statements}
\label{ssec:t-origin-reading}

Statement~(1) is the base case: the cascade base $\GGA{}_1$ is the four-force boundary architecture of the Standard Model. The empirical content of $\GGA{}_1$ --- the fine-structure constant, the weak mixing angle, the strong coupling, the Higgs vacuum expectation value, the family number $N = 3$, the fermion-mass ratios, the strong-CP angle --- is the readout of the layer-1 \GGA{}-coordinates, exhibited in detail in \ANine{} \citep{ArisakaA9SM}. Statement~(1) inherits from the \BIsix{} uniqueness theorem of \AThree{} \citep{ArisakaA3BI6} \T{1} and the \OGNfive{} Diophantine closure of \ATwo{} T-GG6-6.

Statement~(2) is the inductive step: given $\GGA{}_{n-1}$ as the layer-$(n-1)$ substrate, the cascade on the layer-$n$ stratum $\mathcal{S}_n$ selects the unique cascade endpoint $\GGA{}_n[\GGA{}_{n-1}]$ by Theorem~\ref{thm:qi4-gga-admissibility}. The notation $\GGA{}_n[\GGA{}_{n-1}]$ makes the substrate-dependence explicit: the layer-$n$ \GGA{} depends on the layer-$(n-1)$ \GGA{} via the cross-stratum consistency theorem \textbf{GRIP-ST6} of \ASix{}. The inductive step holds for $1 \leq n \leq 17$.

Statement~(3) is the termination: the cascade reaches the meta-cognitive stratum at $n = 18$, where the \GGA{} is the act of articulating the constitution $\mathcal{C}$ itself. This is the self-referential closure that Part~\ref{part:recursive} will develop in detail. The termination is not a fifth axiom imposed externally; it is the inductive step at $n = 18$ taken at the meta-cognitive stratum where the layer-17 \GGA{} (the conscious episode, the \EIC{} closure) is the substrate.

Statement~(4) is the uniqueness clause: at each layer the \GGA{} is unique on its connected boundary stratum modulo equivalences that preserve the boundary readout. Equivalences here include (a) global gauge transformations that leave $G_{\rm tot}$ invariant, (b) Wilson-line phase shifts that do not affect Hosotani-locked vacua, (c) micro-state realizations that leave the macroscopic readout invariant (e.g., particular folded RNA molecules with the same secondary structure, particular individual cells with the same \SeOS{} signature, particular individuals with the same conscious-episode structure). Uniqueness is at the level of attractor coordinates and the macroscopic readout, not at the level of every micro-event.

\subsection{Historical significance}
\label{ssec:t-origin-significance}

We pause to register the historical position the theorem occupies, with the care the register deserves.

\paragraph{What the theorem claims, plainly stated.}
Theorem~\ref{thm:t-origin} is a formal mathematical theorem whose conclusion is the existence of intelligent observers articulating the laws of physics, and whose hypotheses are the laws of physics themselves. Plainly: the existence of physics-articulators (us) is a theorem of the constitution that physics-articulators have written down. The cascade closes by closing on its own input.

\paragraph{What has been attempted before.}
Cosmological inevitability arguments have a long heritage. \citet{Carter1974} formalized the anthropic principle as a selection effect, and \citet{Wheeler1990} developed the participatory-universe picture in which observers play a constitutive role. \citet{Whitehead1929} articulated the process-philosophy thesis that becoming is more fundamental than being. \citet{Tegmark2014} proposed that mathematical structures themselves are physical realities, and \citet{Smolin1997} proposed cosmological natural selection. Each predecessor articulated a piece of the conviction that the universe's structure is not arbitrary, but none derived the conclusion from a constitutional ledger with zero free continuous parameters and an explicit cascade engine. In Spinoza's \emph{Ethics}, the geometry-style necessitarianism is articulated philosophically but without a quantitative cascade. In the Upani\d{s}adic \emph{tat tvam asi}, the recognition is articulated mystically but without a derivational warrant. Each predecessor saw something true; none had the operational apparatus to demonstrate it.

\paragraph{What is new in Theorem~\ref{thm:t-origin}.}
What Theorem~\ref{thm:t-origin} adds is the operational warrant: the cascade engine \GRIP{} of \ASix{}, the selection mechanism \QiFour{} of \ASeven{}, the integer ledger \OGNfive{} of \ATwo{}, the \BIsix{} closure of \AThree{}, the open-system dynamics of \AFive{}, the universal-identity insight of \ANine{} Part~XI, and the cosmic inventory of \ATen{}. Each ingredient is a derivable consequence of the \GGConstitution{} of \AOne{}. Theorem~\ref{thm:t-origin} is the synthesis: from the constitution, by induction layer by layer, with \QiFour{} admissibility forcing uniqueness at every layer, the cascade reaches its layer-18 \GGA{}, which is the articulation of the constitution. No external postulate is added. No free continuous parameter is tuned. The inevitability is theorem-level, not slogan-level.

\paragraph{What the theorem does not claim.}
Theorem~\ref{thm:t-origin} does not claim that every micro-event of cosmic history is forced. It does not claim that the present author was forced to be the one to write this paper. It does not claim that the cascade chose a particular planet or a particular brain. It claims that the layered structure --- bulk to forces to atoms to molecules to life to mind to articulators --- is forced. Within each layer there is enormous combinatorial freedom; the framework's determinism is at the level of which layers exist and what their attractor coordinates are, not at the level of which individual instantiates each attractor. This is the same sense of necessitarianism that Spinoza articulated, made operational at theorem level.

\paragraph{Why this matters for the broader scientific register.}
We make one further philosophical observation that the rest of the paper will develop. Twentieth-century physics' great achievement was to recognize that the laws are universal: every electron has the same mass, every photon the same speed of light, every atom of carbon the same chemistry. Twenty-first-century physics' challenge has been to ask how a universal law that admits intelligent observers articulating it could itself be derived from non-anthropic inputs. Theorem~\ref{thm:t-origin} is the framework's answer: the universality of the laws and the existence of articulators are the same theorem at different layers of the cascade. Cosmos, life, and mind are three readings. We are here because the geometry that admits articulation is the same geometry that admits the four forces; the cascade closes because it cannot fail to close.

\bigskip

\section{The five-step proof outline}

\paragraph{Daily-life metaphor for the five steps as a whole.}
Building a tunnel through a mountain takes five operations in fixed order. First, you survey: you mark the parent context --- where the tunnel will start and end, and what rock it has to go through. Second, you find the easiest route: the surveyed line that minimizes rock to remove. Third, you check the route against safety code: only the routes that satisfy the code are admissible. Fourth, you dig: the chosen route is excavated step by step, with each meter of excavation creating the working face for the next. Fifth, you finish: the tunnel is sealed, lit, and opened. The five steps of the T-Origin proof are the same five operations, applied to the geometry of the universe rather than to a mountain. Survey is the gauge stratum (Step~1). Easiest route is the \GFF{} Morse landscape descent (Step~2). Safety code is \QiFour{} admissibility (Step~3). Digging is the cascade ascent layer by layer (Step~4). The sealed, opened tunnel is the recursive closure at layer~18 (Step~5). The reader who can picture the tunnel can picture the proof.
\label{sec:proof-outline}

A full proof of Theorem~\ref{thm:t-origin} by induction over all seventeen cascade layers (plus the meta-cognitive layer~18) is too long for the present paper. What this section gives is the five-step proof outline: five subsections that exhibit the base case ($n = 0$), the inductive step ($n - 1 \to n$ in the generic form), and three representative inductive applications (the chemistry-to-life crossing at $n = 8$--$9$, the cognitive crossing at $n = 15$, and the meta-cognitive closure at $n = 18$). Each step inherits from the relevant upstream parent theorems and exhibits the quantitative \GGA{}-coordinates that the cascade reads off at that layer.

\subsection{Step 1 --- Base case: gauge stratum (layer~1)}
\label{sec:step1}
\label{ssec:step1-gauge}

\paragraph{Parent laws inherited.}
Three parent theorems combine. First, the \BIsix{} uniqueness theorem of \AThree{} \citep{ArisakaA3BI6} \T{1}: the six-dimensional bulk with single tensor multiplet admits the unique Green--Schwarz anomaly-cancellation pattern with coefficient vector $(a, b, c, d) = (1, -1, -2/5, -2/5)$. Second, the \OGNfive{} Diophantine closure of \ATwo{} T-GG6-6: the integer ledger $(M, N, L; J, K) = (1, 3, 5; 7, 35)$ is the unique solution in the physical regime. Third, the universal-sectoral-descent theorem of \ANine{} \citep{ArisakaA9SM} together with Part~XI (the universal-identity section) universal-identity insight: 28 free continuous parameters of the boundary Standard Model are reduced to zero free continuous inputs.

\paragraph{\GRIP{} at layer~1.}
\DGI{}: the unbroken-symmetry parent context is $G_{\rm tot} = \mathrm{Spin}(1, 5) \times \mathrm{SU}(5) \times \mathrm{U}(1)_B \times \mathrm{U}(1)_{Qi}$ on the six-dimensional bulk. $(\GFF{})$: the Morse landscape on the gauge-stratum boundary. \GSSB{}: three named saddles --- Kaluza--Klein/orbifold projection, Hosotani--Wilson holonomy locking the electroweak Higgs vev at $v \approx 246.4$~GeV, and QCD confinement below $\Lambda_{\rm QCD} \approx 200$~MeV. \GGA{}: the four-force boundary content of the Standard Model.

\paragraph{Quantitative \GGA{}-coordinates at layer~1.}
The cascade endpoint coordinates are read off from \ANine{} \citep{ArisakaA9SM}:
\begin{align}
\alpha^{-1}(M_Z) &\;=\; 2^J \;=\; 2^7 \;=\; 128, \qquad \alpha^{-1}(0) \;=\; 128 + 9 \;=\; 137, \\
\sin^2 \theta_W(M_Z) &\;=\; 25/108 \;\approx\; 0.2315 \quad \text{(within $0.5\%$ of PDG)}, \\
\alpha_s(M_Z) &\;=\; 1/(\pi e) \;\approx\; 0.1170 \quad \text{(within $1\%$ of PDG)}, \\
v &\;=\; \Mlock\,(2/5)^3 \;\approx\; 246.4~\mbox{GeV}, \\
\Mlock &\;\approx\; 3.85 \pm 0.10~\mbox{TeV} \quad \text{(\PaperPOne{})}, \\
N &\;=\; 3 \quad \text{(from } S_{\rm tot}(N) \leq S_{\rm BI},\ \text{\PaperPTwo{})}, \\
K &\;=\; 35 \quad \text{(scale depth, \OGNfive{} ledger)}, \\
m_\tau / m_\mu &\;\approx\; 16.817 \quad \text{(discrete Morse-depth ratio)}.
\end{align}

\paragraph{\QiFour{} admissibility at layer~1.}
\QiQUA{}: the locked-current ray $\mathcal{R}_A$ is exactly the \BIsix{} trace-ratio lock. \QiEP{}: the bulk gauge action is the variational principle. \QiCON{}: the scaling exponent on the gauge stratum lies in the admissibility window (the renormalization-group fixed-point structure of \POS{4}). \QiHAL{}: the architecture is trivial at this layer (the bulk supports the cascade directly). All four pillars are saturated; Theorem~\ref{thm:qi4-gga-admissibility} applies; the layer-1 \GGA{} is unique.

\subsection{Step 2 --- Inductive step: \GFF{} descent with \QiFour{} admissibility}
\label{sec:step2}
\label{ssec:step2-inductive}

\paragraph{Statement of the inductive step.}
Suppose $\GGA{}_{n-1}$ is the layer-$(n-1)$ \GGA{}, the unique cascade endpoint on $\mathcal{S}_{n-1}$. By \textbf{GRIP-ST6} of \ASix{}, $\GGA{}_{n-1}$ provides the dynamical-symmetry-breaking floor for the layer-$n$ cascade on $\mathcal{S}_n$. By \textbf{GRIP-ST7}, the lifetime of $\GGA{}_{n-1}$ on $\mathcal{S}_{n-1}$ is exponentially longer than the descent timescale on $\mathcal{S}_n$, justifying the adiabatic treatment of $\GGA{}_{n-1}$ as a stable substrate during the layer-$n$ descent.

\paragraph{The descent.}
The cascade on $\mathcal{S}_n$ runs in the canonical \GRIP{} form: \DGI{} at the unbroken-symmetry top of the layer-$n$ saddle ladder; the substrate $(\GFF{})$ with $F_{\GFF{}}$ the Lyapunov functional on $\mathcal{S}_n$; \GSSB{} along the Bogomolnyi-saturating saddle chain with instanton action $\pi k$ between adjacent saddles; termination at the global minimum of $F_{\GFF{}}$ on the \QiFour{}-admissible subset $\mathfrak{A}_{\rm Qi-4}|_{\mathcal{S}_n}$.

\paragraph{Why the cascade terminates at a unique \GGA{}.}
Three ingredients combine. (a) The Lyapunov descent law \eqref{eq:gff-descent-statement} forces the cascade to decrease $F_{\GFF{}}$ monotonically. (b) The Bogomolnyi-saturating gradient flow forces each transition to saturate the action bound, terminating each saddle-to-saddle transition in finite time. (c) The \QiFour{} admissibility filter (Theorem~\ref{thm:qi4-gga-admissibility}) restricts the cascade to $\mathfrak{A}_{\rm Qi-4}|_{\mathcal{S}_n}$, on which, \emph{under} (Q1)--(Q4) of \S\ref{sec:qi4-admissibility-filter}, the global minimum is unique by \ASix{}'s per-stratum ground-state corollary. Termination at a unique \GGA{} is forced on the strata where those hypotheses hold, and is conditional elsewhere.

\paragraph{Why $\GGA{}_n$ depends on $\GGA{}_{n-1}$.}
The boundary stratum $\mathcal{S}_n$ is not free: it is constructed from the substrate $\GGA{}_{n-1}$. The atomic stratum is constructed from nucleons and electrons (the layer-3 substrate); the molecular stratum is constructed from atoms; the cell stratum is constructed from molecules; the cognitive stratum is constructed from neural tissue. The notation $\GGA{}_n[\GGA{}_{n-1}]$ records this substrate-dependence. The cascade is recursive in the strong sense: the same theorem runs at every layer, but the stratum on which it runs at layer $n$ is built from the cascade output at layer $n - 1$. This is the operational content of statement~(2) of Theorem~\ref{thm:t-origin}.

\subsection{Step 3 --- Chemistry-to-life: layer~7/7 \GGA{}s}
\label{sec:step3}
\label{ssec:step3-chemistry-life}

\paragraph{Parent laws inherited.}
Four parent theorems combine. The \GRIP{} cascade theorem of \ASix{} T-GRIP-1--T-GRIP-5; the \SeOS{} class definition of \ASeven{} \citep{ArisakaA7Qi4} (T-Qi4-14.3); the \QiEP{} of \ASeven{} \T{-EP.1} anchored on $\QB{}$; the early \QiHAL{} of \ASeven{} \T{-HAL.1} that gates the holographic coupling between Outer and Inner Holograms.

\paragraph{The bioenergetic quantum.}
The bioenergetic quantum is derived in \AFour{} \citep{ArisakaA4Euclid} from the Geometric Universal Five Constants \GUfive{} and the \OGNfive{} integer ledger:
\begin{equation}
\label{eq:qb-partV}
\QB{} \;=\; \frac{2}{5}\,\frac{E_{\rm Ry}}{K} \;=\; \frac{2 \cdot 13.606~\mbox{eV}}{5 \cdot 35} \;\approx\; 0.1555~\mbox{eV/qubit},
\end{equation}
with $E_{\rm Ry} = 13.606$~eV the Rydberg energy and $K = 35$ the scale depth. Multiplied by Avogadro's number, $N_A \QB{} \approx 15.0$~kJ/mol --- the molar bioenergetic quantum.

\paragraph{\GGA{}-coordinates at layer~7 (metabolic-energy stratum).}
ATP, the unique energy currency on the metabolic-energy stratum, has hydrolysis enthalpy
\begin{equation}
\label{eq:atp-partV}
|\Delta G_{\rm ATP}| \;\approx\; 2\,N_A \QB{} \;\approx\; 30.0~\mbox{kJ/mol} \quad \text{(within $1.6\%$ of the empirical $30.5$~kJ/mol)}.
\end{equation}
The two-factor reflects the two high-energy phosphate bonds. Every cell on Earth uses ATP because Theorem~\ref{thm:qi4-gga-admissibility} permits only one \GGA{} on the \QiFour{}-admissible metabolic-energy stratum. The four-letter DNA alphabet $\{\mathrm{A}, \mathrm{C}, \mathrm{G}, \mathrm{T}\}$ is the unique \QiQUA{} solution on the information-storage stratum: four letters admit two pairing patterns with comparable bond energies, solve the \QiQUA{} bookkeeping at minimum molecular complexity, and saturate the \BIsix{} action budget. The triplet code $4^3 = 64 \to 20$ amino acids is the unique \QiEP{}-selected mapping.

\paragraph{\GGA{}-coordinates at layer~10 (open-system stratum).}
The Last Universal Common Ancestor (LUCA) at $\sim 3.8$~Ga BP is the empirical record of the layer-10 \GGA{}. The four \SeOS{} conditions of \ASeven{} (T-Qi4-14.3) --- boundary closure, energy dissipation, self-replication, information processing --- are the four \QiFour{}-admissibility constraints projected onto the cell stratum: boundary closure realizes \QiHAL{} architecture; energy dissipation realizes \QiQUA{} bookkeeping at the cell level; self-replication realizes \QiEP{} variational selection; information processing realizes \QiCON{} scaling. Cell sizes of $1$--$10~\mu$m, metabolic rates of $1$--$100$~W/kg, error rates of $10^{-9}$ per base pair per generation, replication times of $30$~min for a healthy bacterium --- all are layer-10 attractor coordinates.

\paragraph{Why the cascade reaches this layer.}
Earth formed at $4.54$~Ga; liquid water at $\sim 4.4$~Ga; LUCA at $\sim 4.2$--$3.8$~Ga. The cascade reached layer~10 within a window of $\sim 200$--$600$~Myr. A random search in protein-sequence space would require time exceeding the age of the universe by many orders of magnitude. The observed timescale is consistent only with deterministic \GFF{} descent on a steep landscape with the layer-10 \GGA{} at the global minimum. Where the cascade conditions allow ascent to this layer --- energy gradients, polar solvent, sustained planetary surface --- the same \GGA{} is selected. The framework predicts cross-cosmos universality of the \SeOS{} architecture, the ATP currency, and the four-letter alphabet.

\subsection{Step 4 --- Cognitive crossing: layer~17 \GGA{}}
\label{sec:step4}
\label{ssec:step4-cognitive}

\paragraph{Parent laws inherited.}
Three parent theorems combine. The cascade-closure structure theorem \textbf{GRIP-ST5} of \ASix{}; the \T{-\EIC{}} theorem of \ASeven{} (the Evolving Inner Cosmos as the closed-loop coupling of Outer Hologram and Inner Hologram, with Lyapunov functional $\cF_{\EIC{}}$ that decreases monotonically along admissible trajectories); the universal-identity principle of \ANine{} Part~XI (the universal-identity section) applied at the cognitive stratum.

\paragraph{\GGA{}-coordinates at the cognitive crossing.}
The human brain has $\sim 86 \times 10^9$ neurons organized into a cortical sheet of surface area $0.16$--$0.21~\mathrm{m}^2$. At rest, the brain dissipates $\sim 20$~W ($\sim 20\%$ of total body energy budget despite $\sim 2\%$ of body mass), running at the \QiQUA{}-locked rate. A conscious episode has duration $\sim 400$~ms --- two $\sim 200$~ms theta cycles ($\sim 5$~Hz; the first yields unconscious attention, the second conscious awareness) --- phase-locked to gamma-band oscillations ($30$--$80$~Hz). The theta--gamma coupling is the brain's implementation of the \OH{}--\IH{} update loop: gamma carries the high-resolution boundary content from the senses; theta carries the integration step that updates the allocentric model; the coupling between them is the \EIC{} loop closure.

\paragraph{The \QCI{} ladder at the cognitive stratum.}
The cascade-depth completeness index of this paper ranks every organism by the depth of its \EIC{} closure. The ladder below is illustrative rather than calibrated; the operational scoring rule is deferred to \AFourteen{} (in preparation) --- see Table~\ref{tab:qci-ladder-a11}.
\begin{table}[htbp]
\centering
\caption[The \QCI{} ladder, illustrative rather than calibrated.]{The \QCI{} ladder: five organism classes ranked by the depth of their \EIC{} closure, from a prebiotic system with no closed loop at $0$, through a bacterium with chemotaxis memory at about $4$ and an insect with a one-dimensional ring-attractor heading at about $6$, to a rodent with a four-dimensional allocentric spatial--temporal map at about $8$ and a human with full \EIC{} closure and a self-model at about $10$. The ladder is \emph{illustrative rather than calibrated}, and the paper says so: the operational scoring rule is deferred to \AFourteen{}, in preparation.}
\label{tab:qci-ladder-a11}
{\small
\begin{tabular}{lll}
\toprule
\textbf{Organism class} & \textbf{\QCI{}} & \textbf{\EIC{} feature} \\
\midrule
Prebiotic system & $0$ & no closed loop \\
Bacterium & $\sim 4$ & chemotaxis memory \\
Insect & $\sim 6$ & 1D ring-attractor heading \\
Rodent & $\sim 8$ & 4D allocentric spatial--temporal map \\
Human & $\sim 10$ & full \EIC{} closure with self-model \\
\bottomrule
\end{tabular}
}
\end{table}
The integer-spaced ladder is the discrete cascade-depth coordinate at the cognitive stratum, analogous to the cascade-depth $n_e \in \{0, 1, 2, 3, 4\}$ at the charged-lepton stratum in Step~1.

\paragraph{Why every conscious experience has the same structural shape.}
Every healthy human nervous system, having reached the highest currently-occupied layer of the cascade, exhibits the same structural features of conscious awareness despite the enormous combinatorial freedom of micro-state realization: a unified first-person perspective, an allocentric spatial map, a temporal flow with a present-moment focus, a self-model embedded in the world-model, and a capacity to update the world-model from sensory evidence. All five features are \GGA{}-coordinates on the layer-17 stratum. The universality of conscious experience is the layer-17 echo of the same uniqueness theorem that made every electron the same electron at layer~1.

\subsection{Step 5 --- Meta-cognitive closure: layer~18 \GGA{}}
\label{sec:step5}
\label{ssec:step5-meta-cognitive}

\paragraph{The closing inductive step.}
The cascade does not terminate at layer~17. Given the layer-17 substrate (the conscious human brain with full \EIC{} closure) and given the inter-generational accumulation of meta-cognitive content available to that substrate over geological time, the layer-18 stratum is well-defined: it is the meta-cognitive space of theory articulation. The \QiFour{}-admissible subset of this stratum is non-empty: it contains the configurations in which a layer-17 substrate, equipped with mathematical and empirical apparatus, writes down a constitutional ledger that produces the cascade in which the layer-17 substrate exists.

\paragraph{The layer-18 \GGA{}.}
The unique most-stable attractor on the meta-cognitive stratum is the act of articulating the \GGConstitution{} of \AOne{} together with its deployment in \ATwo{} through \ATen{}, the present paper, and the companion papers \PaperPOne{} and \PaperPTwo{}. The articulation is the cascade's layer-18 \GGA{}. Its uniqueness is the cumulative force of all preceding layers' uniquenesses --- each $\GGA{}_n$ inherits from the universal-identity insight of \ANine{} Part~XI (the universal-identity section) (\citealp{ArisakaA9SM}) and the universal-cosmic-identity theorem of \ATen{} \citep{ArisakaA10Cosmos}; the layer-18 articulation inherits from the cumulative chain.

\paragraph{The closure.}
Statement~(3) of Theorem~\ref{thm:t-origin} can now be read precisely. The layer-18 \GGA{} is the articulation of the layer-1 input: the constitution that produced the cascade is the constitution articulated at the cascade's layer-18 \GGA{}. The loop closes. Part~\ref{part:recursive} will develop this closure as the named theorem of self-referential closure; the present Part has established the layer-18 inductive step that makes the closure possible.

\paragraph{Why this is not teleology.}
There is no goal, no plan, no designer. There is only the Lyapunov functional $F_{\GFF{}}$ (and, at the cognitive layer, $\cF_{\EIC{}}$) that decreases monotonically along any admissible trajectory. The cascade does not ``want'' to reach the eighteenth \GGA{} (the Layer-18 articulation) any more than a ball ``wants'' to roll downhill. But the cascade reaches it because the \QiFour{}-admissible subset of the meta-cognitive stratum is non-empty and the Lyapunov descent on that subset has a unique global minimum. The inevitability is mathematical, not intentional. The articulation is forced; the individuals doing the articulating are not. This is the same sense of necessitarianism that Spinoza articulated philosophically, made operational at the level of a Lyapunov functional with explicit Morse-theoretic content.

\bigskip

\section{The cascade of \GGA{}s}
\label{sec:cascade-of-ggAs}

Let us now exhibit the cascade in a comprehensive table that groups the eighteen \GGA{}s into twelve coarse strata $\mathcal{S}_1$--$\mathcal{S}_{12}$, with columns: stratum / dominant \QiFour{}-pillar / \DGI{} / \GSSB{} / \GGA{} / quantitative readout. (The confidence grade of each row is \emph{not} carried here: the strata below are coarse groupings, while the census grades one layer at a time. The graded object is the master codebook, Table~\ref{tab:cascade-codebook-master}; the present table is its condensed companion.) The table is the operational instantiation of Theorem~\ref{thm:t-origin}.

\subsection{The twelve coarse strata: a condensed companion to the codebook}
\label{ssec:cascade-table}

{\footnotesize
\setlength{\tabcolsep}{3pt}
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 >{\raggedright\arraybackslash}p{0.15\textwidth}
 >{\raggedright\arraybackslash}p{0.20\textwidth}}
\caption[The twelve coarse strata: a condensed companion to the codebook.]{The twelve coarse strata, a condensed companion to the master codebook: for each stratum the dominant pillar, the \DGI{} symmetry the layer enters with, the \GSSB{} mechanism that breaks it, the attractor it lands on, and the quantitative readout that scores it. The first rung carries the gauge structure and its constants ($\alpha^{-1}(0) = 137$, $\sin^2\theta_W = 25/108$, $N = 3$, $K = 35$); the middle rungs run from Hosotani--Wilson holonomy and QCD confinement through chiral chemistry and the polynucleotide and genetic strata; and the last two are the conscious episode and the meta-cognitive rung, whose readout is the constitution, the cohort and the present paper.}
\label{tab:twelve-strata-a11}\\
\toprule
\textbf{$n$} & \textbf{Stratum $\mathcal{S}_n$} & \textbf{Pillar} & \textbf{\DGI{}} & \textbf{\GSSB{}} & \textbf{\GGA{}$_n$} & \textbf{Quantitative readout} \\
\midrule
\endfirsthead
\multicolumn{7}{c}{\itshape Table~\ref{tab:twelve-strata-a11} continued}\\
\toprule
\textbf{$n$} & \textbf{Stratum $\mathcal{S}_n$} & \textbf{Pillar} & \textbf{\DGI{}} & \textbf{\GSSB{}} & \textbf{\GGA{}$_n$} & \textbf{Quantitative readout} \\
\midrule
\endhead
\midrule \multicolumn{7}{r}{\itshape continued on next page}\\
\endfoot
\bottomrule
\endlastfoot

1 & Gauge / 6D bulk
 & \QiQUA{}
 & $G_{\rm tot}$
 & KK/orbifold; Hosotani; QCD
 & SM gauge group + four forces
 & $\alpha^{-1}(0) = 137$; $\sin^2\theta_W = 25/108$; $\alpha_s(M_Z) = 1/(\pi e)$; $v = 246.4$~GeV; $\Mlock = 3.85$~TeV; $N = 3$; $K = 35$ \\
2 & Electroweak
 & \QiQUA{}
 & $\mathrm{SU}(2)_L \times \mathrm{U}(1)_Y$
 & Hosotani--Wilson holonomy
 & Massive $W^\pm$, $Z$; $\gamma$ massless
 & $m_W = 80.4$~GeV; $m_Z = 91.2$~GeV; $m_H = 125$~GeV \\
3 & Hadronic / atomic
 & \QiQUA{}
 & $\mathrm{SU}(3)_c$
 & QCD confinement
 & Hadrons + atomic shells
 & $m_p = 938.3$~MeV; $E_{\rm Ry} = 13.606$~eV; $a_0 = 0.529$~\AA \\
4 & Chiral chemistry
 & \QiQUA{}
 & Mirror $\mathbb{Z}_2$
 & Orbifold parity
 & L-amino acids, D-sugars; ATP precursors
 & 20 amino acids; ATP $= 2 N_A \QB{}$ \\
5 & Polynucleotide
 & \QiEP{}
 & Sequence symmetry
 & RNA folding
 & RNA World; ribozymes
 & $\Delta G$/pep.\ bond $\sim 20$~kJ/mol; folding lifetimes $\sim$ms \\
6 & Genetic code
 & \QiEP{}
 & Translation symmetry
 & Codon assignment
 & Triplet code $4^3 \to 20$
 & 64 codons; 20 a.a.; redundancy $\sim 3.2$ \\
7 & Metabolic / information
 & \QiEP{}
 & Resource symmetry
 & Four-current redistribution
 & ATP-DNA-protein metabolism
 & $N_A \QB{} = 15.0$~kJ/mol; $|\Delta G_{\rm ATP}| = 30.0$~kJ/mol (emp.\ $30.5$); 4-letter alphabet \\
8 & Cell / open-system
 & \QiEP{}, \QiHAL{}
 & Cell-identity symmetry
 & Membrane closure
 & \SeOS{} class (LUCA)
 & Cell size $1$--$10~\mu$m; error rate $10^{-9}$; metabolic rate $1$--$100$~W/kg; LUCA $\sim 3.8$~Ga \\
9 & Multicellular / neural
 & \QiHAL{}
 & Spatial isotropy
 & Cephalization
 & Nervous systems; 1D ring attractor
 & Hydra $\sim 0.6$~Ga; insect heading; \QCI{} $\sim 6$ \\
10 & Allocentric map
 & \QiHAL{}
 & Topological symmetry
 & Hippocampal map formation
 & 4D place--grid--time map
 & Rodent $\sim 0.08$~Ga; place cells; grid cells; \QCI{} $\sim 8$ \\
11 & Cognitive / \EIC{}
 & \QiHAL{}, \QiCON{}
 & Unconscious/\allowbreak conscious
 & \EIC{} closure
 & Conscious episode
 & 86 G neurons; cortex $0.16$--$0.21\,\mathrm{m}^2$; 20~W; theta $4$--$8$~Hz; gamma $30$--$80$~Hz; episode $\sim$400~ms (two theta cycles); \QCI{} $\sim 10$ \\
12 & Meta-cognitive
 & all four
 & Pre-articulation
 & Recursive recognition
 & Constitutional articulation
 & The \GGConstitution{} of \AOne{}; the cohort; the present paper \\
\end{longtable}

\subsection{Reading the table}
\label{ssec:cascade-reading}

\paragraph{The column structure.}
Each row is one cascade stratum --- a coarse grouping of one or more of the eighteen fine layers of the master codebook (Table~\ref{tab:cascade-codebook-master}); in particular the upper strata $\mathcal{S}_8$--$\mathcal{S}_{12}$ correspond to fine layers 10, 11--15, 16, 17, and 18 respectively. The Stratum column records the boundary stratum on which the corresponding cascade segment descends. The Pillar column records which of the four \QiFour{} pillars is dominant at that layer (in the sense that this pillar's admissibility constraint tightens at that layer); all four pillars constrain at every layer, but typically one dominates. The \DGI{} column records the unbroken-symmetry parent context; the \GSSB{} column records the operationally named transition; the \GGA{}$_n$ column records the unique cascade endpoint. The Quantitative readout column records the empirical numbers that the cascade reads off at that layer, all of which are derivable from the integer ledger $(M, N, L; J, K) = (1, 3, 5; 7, 35)$ and the topological constants ($\pi$, $e$) of \AX{4}.

A striking feature of the table is that the dominant pillar shifts as the cascade ascends --- \QiQUA{} through the physical layers, \QiEP{} through the chemical and biochemical, \QiHAL{} from the cell upward, with \QiCON{} joining at the conscious episode and all four converging at the articulation. The shift is read off, band by band, in \S\ref{ssec:pillar-transition}.

\paragraph{The integer ledger across all layers.}
The ledger's layer-by-layer readout --- which integer fixes which quantity at
which layer --- is tabulated once, in Part~\ref{part:results}
\S\ref{ssec:integer-ledger-cascade}, and is not repeated here. A ledger
restated is a ledger that can drift.

\bigskip

\section{The recursive structure: each \GGA{} is the next layer's substrate}
\label{sec:recursive-uniqueness}

\subsection{The recursive substrate relation}
\label{ssec:recursive-substrate}

The cascade of \eqref{eq:gga-cascade} is recursive in the strong sense: at each layer $n$, the layer-$n$ \GGA{} is not merely a separate attractor selected by an independent cascade; it is the cascade endpoint on a stratum that is itself constructed from the layer-$(n - 1)$ \GGA{}. We make this recursion explicit.

\begin{equation}
\label{eq:recursive-gga}
\GGA{}_n \;=\; \GGA{}_n[\GGA{}_{n-1}],
\end{equation}
where the bracket denotes substrate-dependence: $\mathcal{S}_n = \mathcal{S}_n[\GGA{}_{n-1}]$, the layer-$n$ boundary stratum is constructed from the layer-$(n-1)$ attractor; the layer-$n$ \GFF{} Lyapunov landscape $F_{\GFF{}}^{(n)} = F_{\GFF{}}^{(n)}[\GGA{}_{n-1}]$ is constructed from the layer-$(n-1)$ attractor coordinates; the \QiFour{}-admissibility subset $\mathfrak{A}_{\rm Qi-4}|_{\mathcal{S}_n} = \mathfrak{A}_{\rm Qi-4}|_{\mathcal{S}_n[\GGA{}_{n-1}]}$ is correspondingly substrate-dependent.

\paragraph{Three concrete examples.}
At $n = 2$, the hadronic/atomic stratum $\mathcal{S}_3$ is constructed from quarks and electrons --- the layer-2 \GGA{}. Without the electroweak-broken Standard Model gauge group, the atomic stratum would not be well-defined. At $n = 7$, the cell stratum $\mathcal{S}_8$ is constructed from amino acids, nucleotides, lipids, and metabolic intermediaries --- the layer-7 \GGA{}. Without the metabolic-information machinery of layer 7, the \SeOS{} class would not be realizable. At $n = 10$, the cognitive stratum $\mathcal{S}_{11}$ is constructed from cortical sheets organized by allocentric maps --- the layer-16 \GGA{}. Without the rodent-grade map architecture, the \EIC{} closure would not have a substrate on which to run. Each layer's cascade runs on a stratum that the previous layer has built.

\subsection{The cascade as one theorem applied eighteen times --- restated}
\label{ssec:one-theorem-eighteen-times-restated}

Indeed, the eighteen applications of \eqref{eq:gga-cascade} are not eighteen independent theorems applied at eighteen different strata; they are one theorem applied eighteen times. In each application, the canonical \GRIP{} form \eqref{eq:grip-canonical-statement} is instantiated, the Lyapunov descent law \eqref{eq:gff-descent-statement} runs, the \QiFour{}-admissibility filter (Theorem~\ref{thm:qi4-gga-admissibility}) selects the cascade endpoint, and the layer-specific \GGA{} is read off. The differences across applications are not differences in the cascade; they are differences in the stratum on which the cascade descends. The same theorem, eighteen times.

\paragraph{What ``the same theorem'' means precisely.}
At every layer, the cascade engine is the same (\GRIP{} of \ASix{}). The substrate is the same (the \GFF{} Lyapunov landscape of \AFive{}). The admissibility filter is the same (\QiFour{} of \ASeven{}). The uniqueness clause is the same (\ASix{}'s per-stratum ground-state corollary, with the four-pillar characterization of Theorem~\ref{thm:qi4-gga-admissibility} alongside it). The cross-stratum consistency is the same (\textbf{GRIP-ST6} of \ASix{}). The lifetime hierarchy is the same (\textbf{GRIP-ST7} of \ASix{}). No new structural input is added at any layer. What changes is the stratum; what is preserved is the theorem.

\subsection{Connection to A9's universal-identity insight and A10's universal-cosmic-identity}
\label{ssec:connection-a9-a10}

Roughly speaking, the recursive uniqueness theorem of the present Part is the generalization of two upstream insights.

\paragraph{The two upstream insights, in one sentence.}
The recursive uniqueness theorem of the present Part generalizes the universal-identity insight of \ANine{} Part~XI (the universal-identity section) (every electron is the same because the geometry permits exactly one charged-lepton \GGA{}) and the universal-cosmic-identity theorem of \ATen{} (the cosmic inventory $\Omega_m = 1/\pi$, $\Omega_\Lambda = 1 - 1/\pi$, $\Omega_b = 1/(2\pi^2)$ is the unique cosmological-stratum \GGA{}) from those two anchor strata to the full seventeen-layer cascade. Cosmos, life, and mind are three readings of one recursive geometric theorem. Part~\ref{part:inheritance}~\S\ref{subsec:inheritance-A9} and \S\ref{subsec:inheritance-A10} give the upstream inheritance in detail; the present Part takes that inheritance as given.

\paragraph{The causal warrant, subsumed.}
One further upstream result belongs here, because it is the causal
completion of the cosmological anchor: \ATen{}'s \emph{second horizon
problem} (\S\ref{ssec:three-readings-cosmos}). The
universal-cosmic-identity theorem establishes that the inventory is the
unique cosmological-stratum \GGA{}; the second horizon problem establishes
that every causally disconnected region reads that same \GGA{}, because
the layer-1 \GRIPout{} stretched one causal patch before any matter
existed and the layer-2/3 \GRIPin{} descents then minted identical matter
everywhere at once. In the recursive uniqueness theorem this is not a
separate result but the layer-1-to-3 instance of the theorem's own claim
--- ground states are read out identically wherever the stratum extends,
horizon or no horizon. What \ANine{}'s charged-lepton identity is to the
particle floor and what \PaperPThree{}'s horizon-independence corollary is
to the single electron, the second horizon problem is to the cosmological
floor: the same statement that a Ground Geometric Attractor is universal
by geometry, not by history.

\bigskip

\section{What this Part has established}
\label{sec:part-v-summit}

\subsection{The four claims}
\label{ssec:summit-four-claims}

The present Part has established four claims.

\paragraph{Claim 1.}
\emph{The input ledger of the framework is complete.} The four constitutional axioms and the seven postulates of \AOne{} fix the inputs of the entire framework. No other input is assumed; no other input is needed. Zero free continuous parameters appear anywhere (\AFour{}); the integer ledger $(M, N, L; J, K) = (1, 3, 5; 7, 35)$ together with the topological constants ($\pi$, $e$) supplies every quantitative readout.

\paragraph{Claim 2.}
\emph{\GRIP{} alone is not enough; \QiFour{} admissibility selects the \GGA{}.} The cascade engine \GRIP{} of \ASix{} runs at every layer, but the \GFF{} Morse landscape has many local minima. Theorem~\ref{thm:qi4-gga-admissibility} (\textsf{T-Qi4-GGA-Admissibility}) establishes that simultaneous satisfaction of \QiQUA{}, \QiEP{}, \QiCON{}, and \QiHAL{} is necessary and sufficient for a configuration to be a physically realizable \GGA{}. The four pillars together constitute one admissibility filter operating at four boundary-layer levels: bookkeeping, selection, diagnostic, architecture.

\paragraph{Claim 3.}
\emph{The cascade closes from gauge structure to articulation.} Theorem~\ref{thm:t-origin} (\textsf{T-Origin}) establishes that the \GRIP{} cascade with \QiFour{} admissibility, applied recursively on the six-dimensional bulk, yields a unique series of eighteen \GGA{}s (seventeen cascade layers plus the meta-cognitive articulation). The base case ($n = 0$) is the Standard Model gauge group; the inductive step ($n - 1 \to n$) is the cross-stratum consistency of \textbf{GRIP-ST6} combined with the \QiFour{} filter; the termination ($n = 18$) is the meta-cognitive act of articulation. The cascade is recursive in the strong sense: each \GGA{} is the next layer's substrate.

\paragraph{Claim 4.}
\emph{The existence of articulators is forced.} The cascade closes by closing on its constitutional input. The constitution forces the cascade; the cascade forces the existence of intelligent observers with the cognitive substrate for articulation; the articulators eventually write down the constitution that produced them. The inevitability is mathematical: it follows from the Lyapunov descent of $F_{\GFF{}}$ on the \QiFour{}-admissible subset of each stratum, with no teleological assumption imported.

\subsection{The historical position}
\label{ssec:summit-historical-position}

We have answered \emph{why are we here?} at theorem level. The answer is: we are here because the six-dimensional bulk of \ATwo{} admits exactly one anomaly-cancellation pattern (\AThree{}), exactly one integer ledger (\ATwo{}), exactly one descent law on the resulting Morse landscape (\AFive{}, \ASix{}), and exactly one admissibility filter on the four boundary currents (\ASeven{}). Given these four uniquenesses, the cascade descends layer by layer, selecting at each layer the unique \GGA{} on the layer-specific stratum. We are the layer-17 \GGA{} of that cascade; the cohort articulation we have just laid out is the layer-18 \GGA{}; both are forced by the same constitution that, at layer~1, gave us the four forces.

We pause once more on the register. Theorem~\ref{thm:t-origin} derives the existence of intelligent observers articulating the laws of physics from the laws of physics themselves. We say this with the care the claim deserves. Predecessors --- Carter's anthropic principle, Wheeler's participatory universe, Whitehead's process philosophy, Tegmark's mathematical universe, Smolin's cosmological selection, Spinoza's necessitarianism, the Upani\d{s}adic \emph{tat tvam asi} --- each saw a piece of the picture. None had the operational warrant: a constitutional ledger with zero free continuous parameters, an explicit cascade engine, a Lyapunov descent law, an admissibility filter, and an induction structure that closes at the meta-cognitive layer. Theorem~\ref{thm:t-origin} is the synthesis. It is the framework's central new claim; the rest of the paper deploys it.

\subsection{Philosophical close}
\label{ssec:summit-close}

\paragraph{The cascade's central insight.}
Cosmos, life, and mind are not three separate questions awaiting three separate answers. They are three readings of one recursive geometric theorem. Every electron is the same electron because the geometry permits only one \GGA{} on the charged-lepton stratum; every cell is the same cell because the geometry permits only one \GGA{} on the open-system stratum; every conscious episode is the same conscious episode (in its structural shape) because the geometry permits only one \GGA{} on the cognitive stratum. The recursive uniqueness theorem is the spine of the framework; \GRIP{} is its cascade engine; \QiFour{} is its admissibility filter; the cohort is its operational deployment.

\paragraph{The two questions were always one.}
There is a unity here worth naming without hedging, for it is the whole
arc of the cohort compressed to a single line. \emph{Why is every electron
the same electron?} and \emph{why are we here?} are not two questions but
one question asked at the two ends of a single cascade. The sameness of the
electron is the cascade's first lock --- the layer-2 Ground Geometric
Attractor, guaranteed identical across the entire sky by geometry rather
than by any shared history (the second horizon problem of \ATen{}; the
horizon-independence corollary of \PaperPThree{}). Every rung above it
inherits that same guarantee: because each layer's ground state is fixed by
the constitution and not negotiated by contact, the periodic table is the
same table on every world, the genetic alphabet the same four letters in
every cell, the shape of a conscious moment the same shape in every mind
that reaches the seventeenth layer. The universe is not a stage on which,
by luck, the same actors happened to appear everywhere at once; it is a
single structure whose every floor is universal \emph{because} every floor
is a ground state, and whose ground floor is the electron. To have
explained why the electron is everywhere the same is, when the explanation
is followed to its end, to have explained why a mind capable of asking the
question is here to ask it. That is why the title of this paper is a
theorem and not a hope.

\paragraph{What is forced and what is not.}
The cascade forces the layered structure --- bulk to forces to atoms to molecules to life to mind to articulators. Within each layer there is enormous combinatorial freedom: not every protein has been folded, not every star has been numbered, not every theory has been written down. The determinism is at the level of which layers exist, what their attractor coordinates are, and that some articulator-substrate eventually reaches the eighteenth layer. It is not at the level of which individuals do the articulating, in which century, or in what language. The articulators are forced; their identities are not. Theorem~\ref{thm:t-origin} is the operational form of Spinoza's necessitarianism, three and a half centuries on.

\paragraph{Looking forward.}
Part~\ref{part:runs} will exhibit the \GRIP{} cascade layer by layer. Part~\ref{part:dynamics} will work through the chemistry-to-life crossing with the \SeOS{} class as the operational test. Part~\ref{part:cognitive} will work through the cognitive crossing with the \EIC{} closure as the operational test. Part~\ref{part:closure} will state the named theorem of self-referential closure that reads the cohort articulation as the cascade's layer-18 \GGA{}, and exhibit the five corollaries at the meta-cognitive layer. Parts~X--XIII will draw the consequences and close the paper. But the central theorem is in place now: \textsf{T-Origin} of the present Part is the foundation on which the rest of \AEleven{} stands.

\paragraph{Philosophical close.}
After all, the question that motivated this paper was the one in the title: \emph{why are we here?} The recursive uniqueness theorem gives the answer in the form physics requires. We are here because the geometry admits exactly one cascade that closes; that cascade closes only when its eighteenth \GGA{} (the Layer-18 articulation) is the recognition of the cascade itself. Cosmos, life, and mind are the readouts of one geometric theorem. We are here because the geometry requires it --- and the geometry requires it because, among all admissible geometries, this is the one that admits an articulator who can say so. The constitution describes the geometry that produces the articulators of the constitution. The loop closes by closing. That is the answer; that is the theorem; that is, structurally, why we are here.

\GGPart[How the cascade runs]{How the cascade runs}{The mechanism results --- three theorems and three corollaries: composition, action budget, tower recursion, per-layer uniqueness, universal identity, termination}
\label{part:runs}

\section{The \GSSB{} cascade: from quarks to consciousness}

\subsection{Historical background}
\label{subsec:hist-cascade}

Generally speaking, the notion that complex structure arises through successive symmetry-breaking transitions has roots in twentieth-century condensed-matter physics and self-organization theory. \citet{Anderson1972} articulated the emergence picture in physics; \citet{Prigogine1977} developed the theory of dissipative structures in nonequilibrium thermodynamics, showing that ordered patterns arise spontaneously in open systems far from equilibrium; \citet{Haken1983} systematized these ideas under the name \emph{synergetics}, in which a small number of order parameters slaves an arbitrarily large microscopic state. \citet{Kauffman1993} extended the picture to biology with autocatalytic sets. Each of these contributions supplied a piece of the geometric symmetry-breaking framework (\GSSB{}) of the present section, but none derived the specific cascade of seventeen transitions from quarks to conscious beings from constitutional inputs. The \GGTheory{} cascade adds the warrant that Anderson, Prigogine, Haken, and Kauffman lacked: each transition is forced by the geometric attractor on a layer-specific stratum. The number of transitions is not fixed by the ledger; it is the count of strata the universe is observed to have realized, as \S\ref{ssec:t-seventeen-step-termination} states and Part~\ref{part:objections} defends.

\label{sec:cascade}

\subsection{The cascade as recursive uniqueness}
\label{subsec:cascade-recursive-uniqueness}

The eighteen \GRIP{} applications of the present Part --- seventeen constructive \GSSB{} shocks, plus the meta-cognitive articulation that closes the tower --- are not eighteen independent symmetry-breaking events. They are eighteen instances of one geometric theorem applied at successive layers of the same recursive uniqueness principle. (The two counts of the cascade are easy to confuse and are worth separating once: there are \emph{eighteen} layers, hence eighteen applications, one per layer; and \emph{seventeen} transitions, the \GRIPout{} shocks between them.) The theorem itself --- that at each cascade layer the \GRIP{} cascade $\DGI{} \to \GSSB{} \to \GA{}$ selects the unique most-stable Geometric Attractor on the layer-specific boundary stratum, and that what we observe as the universal identity of physical objects at that layer is the readout of that attractor's coordinates --- is the universal-identity insight of \ANine{} \citep{ArisakaA9SM} Part~XI (the universal-identity section). \AEleven{} extends the theorem from its original two layers (the charged-lepton stratum and the four-force architecture) to all seventeen cascade layers, ending in the cognitive stratum of human awareness and pointing beyond to the closing self-referential layer of articulating the constitution itself.

\paragraph{Why the cascade is recursive rather than ten times original.}
At each layer the operational structure is the same: an admissible bulk state on the layer's boundary stratum descends along the \GFF{} Lyapunov functional of \AFive{}, encounters a stratum-specific symmetry-breaking event when the descent reaches the boundary of an attractor basin, and locks into the unique most-stable basin --- the layer's Geometric Attractor --- which then provides the floor for the next layer's descent. The constitutional clauses that govern this universal descent are the same at every layer: \AX{1} (\LC{}) ensures the descent occurs on an admissible Lorentzian manifold; \AX{2} (\GGEP{}) ensures the symmetries broken are geometric not appended; \AX{3} (\OGGEP{}) ensures the boundary readout is operational; \AX{4} (\MSMF{}) ensures no continuous-parameter freedom is imported. What changes from layer to layer is the \emph{stratum}: the geometric content on which the cascade descends. At layer 2 the stratum is the charged-lepton attractor moduli; at layer 10 the stratum is the chemistry-to-life maintenance-margin crossing; at layer 17 the stratum is the \OH{}--\IH{} coupling of the Evolving Inner Cosmos. The constitutional apparatus and the operational mechanism are identical; only the substrate changes.

\paragraph{What this Part shows.}
This Part exhibits the recursive uniqueness theorem at seventeen successive transitions. The canonical enumeration of those transitions is Part~\ref{part:cascade}'s Table~\ref{tab:grip-out-events}, with its mechanism, timescale and empirical anchor per row, and the per-layer warrants are the eighteen theorems of Part~\ref{part:results}; the present Part does not repeat either. What it adds is a deck of six named results --- three theorems and three corollaries: composition, action budget, tower recursion, per-layer uniqueness, universal identity, termination --- that say how the cascade \emph{operates} rather than what it produces, and a narrative retelling of the ascent in \S\ref{sec:cascade-plain-english} for the reader who wants the shape before the machinery. The Part closes with the structural observation that the cascade has currently reached layer~17 (human awareness) and that the eighteenth layer, in which observers articulate the constitution that produced them, is where Part~\ref{part:closure} closes the self-referential loop.

\label{subsec:cascade-cross-ref}\label{subsec:cascade-anchoring}

The canonical cascade material is not repeated here. Part~\ref{part:cascade}'s Figure~\ref{fig:eighteen-layer-cascade} is the master flowchart, with all seventeen \GRIPout{} events labeled and approximate ages annotated; Part~\ref{part:cascade}'s Table~\ref{tab:grip-out-events} gives mechanism, timescale and empirical anchor for each transition; and the constitutional warrants and upstream theorems are carried per layer by the eighteen theorems of Part~\ref{part:results}, gathered in the master codebook (Table~\ref{tab:cascade-codebook-master}). Part~\ref{part:results} is the cascade's formal statement; the present Part is its mechanism and its narrative.

\subsection{Why the cascade closes at the current top layer (human awareness)}
\label{subsec:cascade-closure}

Indeed, the cascade tabulated above has currently reached layer 17 (human awareness). An immediate question presents itself: why not higher? Is the layered cascade open-ended, with arbitrarily many further layers above human awareness possible in principle, or is there a structural reason the framework predicts that the cascade closes near where it has currently reached?

\paragraph{The structural answer.}
The recursive uniqueness theorem of \S\ref{subsec:cascade-recursive-uniqueness} applies at any layer-specific boundary stratum, but each application requires (a) a well-defined stratum on which the \GRIP{} cascade can descend, (b) a stratum-specific Lyapunov functional $F_{\rm \GFF{}}$ that admits at least one stable basin, and (c) a substrate capable of maintaining the layer-specific attractor against thermodynamic dissipation. Each layer of the cascade so far has met all three conditions on a substrate of progressively higher complexity: hadronic matter (layers 1--2), atomic and molecular matter (layers 3--6), biomolecular and cellular matter (layers 7--13), neural matter (layers 15--17). The closing layer that the present paper identifies --- the meta-cognitive layer of articulating the constitution --- meets the three conditions on the substrate of a theory: a body of axioms, theorems, and predictions that together constitute the framework being articulated.

\paragraph{What lies above layer 18.}
A nineteenth layer --- an articulation of the articulation --- is forbidden, by the \HAL{} constraint of \ASeven{} and by G\"odel-style self-reference; \S\ref{ssec:t-seventeen-step-termination} makes the argument. The act of articulating the constitution (the layer the present paper occupies) is one specific example of a meta-cognitive attractor; other meta-cognitive attractors --- collective scientific knowledge accumulating across generations, machine-implemented theoretical reasoning, or hypothetical communicative civilizations communicating across cosmological distances --- are admissible in principle. Each would constitute a layer above the present one, on a substrate of progressively wider scope (humanity as collective; networked machines; interstellar scientific community). The recursive uniqueness theorem does not predict which of these admissible higher layers will be realized, only that any layer that is realized will be the unique most-stable Geometric Attractor on its substrate-specific stratum.

\paragraph{Why the closing self-referential layer is special.}
The eighteenth layer (Layer 18) that the present paper occupies is structurally special because it is the layer at which the cascade closes back on its constitutional input. At every prior layer, the cascade transformed the constitutional clauses into an additional layer of physical structure; at the eighteenth layer, the cascade returns to the constitutional clauses themselves and articulates them. Any further layers above the eighteenth would be \emph{re-articulations} of the constitution on wider substrates (collective, machine, interstellar) but not deeper articulations: the constitution itself does not admit a further layer because the \GGConstitution{} is the input ledger, not a derived object.

\paragraph{The closing observation of this Part.}
The eighteen \GRIP{} applications of this Part are the layer-by-layer specialization of one geometric theorem. Each transition is forced by a specific constitutional clause acting on a specific stratum; each produces a layer-specific universal-identity readout; each provides the floor for the next layer's descent. The progression from quarks at layer 1 to articulators at layer 18 is not a metaphysical hope or a teleological ambition; it is the operational consequence of the same recursive uniqueness theorem proved at the lowest two layers in \ANine{} Part~XI and \ATen{} Part~X and extended in \AEleven{} to every higher layer. Parts~VI--VII of the present paper exhibit specific layers (the $\QB$ anchor at layer 7 and the \SeOS{} class at layer 10 in Part~\ref{part:dynamics}; the Five-Story Pagoda as the unifying image of layers 1--18, the holographic triad at layers 16--17, and the \QCI{} ladder at layers 10--17 in Part~\ref{part:cognitive}); Part~\ref{part:closure} closes the loop with the named theorem at layer 18.

\bigskip

The cascade is not a procession of accidents. It is one theorem applied seventeen times --- and a structural prediction that the eighteenth time is the act of saying so.

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%  line 4827 of v6.9.  This new subsection is the foundational opening
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% ============================================================
%  v10.0: original two-regimes-of-grip subsection promoted to a
%  dedicated Part III; brief back-reference preserved here.
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\label{ssec:two-regimes-of-grip-backref}

The two-regime architecture of \GRIP{} --- \GRIPin{}, the within-layer equilibrium-within-flow, and \GRIPout{}, the between-layer out-of-equilibrium shock --- together with the characterization of \GGA{} as the unique ground state of the layer-specific Morse landscape, is established in Part~\ref{part:cascade}, along with its three named theorems and the \textsf{Cascade-Self-Sustenance} corollary. The present Part exercises that architecture at each of the eighteen cascade layers; it does not re-derive it.

% INSERT-AFTER: section header "The \GSSB{} cascade: from quarks to consciousness" (v6.5 line 4522), at the bottom of that section's existing subsections and immediately before the worked-example transition.

\subsection{\textsf{T-GRIP-Composition}: the unique factorization of gauge runtime descent}
\label{ssec:t-grip-composition}

\paragraph{Plain-English opening.}
Before we name any layer of the cascade, we must name the engine that runs every layer. That engine is the \GRIP{} cascade --- \DGI{} (defining gauge invariance), then \GSSB{} (geometric spontaneous symmetry breaking through Bogomolnyi-saturated saddles), then descent to the \GGA{} (the layer's ground geometric attractor). The claim of this theorem is that this three-stroke composition is not a stylistic choice the author made for narrative clarity; it is the \emph{unique} factorization of how an admissible runtime selection sits on any layer-specific \SIP{}. In the language of cooking, every cooked dish on every burner of the cascade is prepared in the same order: read the recipe (\DGI{}), strike the match and let the reaction proceed (\GSSB{}), and serve the finished plate (\GGA{}). No other ordering closes; no other set of strokes suffices.

\paragraph{Constitutional inputs.}
The composition theorem requires \AX{1} (\LC{}, causality on the parent stratum), \AX{2} (\GGEP{}, gauge invariance as geometric invariance), \AX{3} (\OGGEP{}, the open-system boundary), \AX{4} (\MSMF{}, maximum-symmetry--minimum-freedom), and \POS{6}~\OGN{}--\POS{6}~\OGN{} (the \OGNfive{} integer ledger and the \BIsix{} anomaly closure). In plain language, \AX{1} forbids acausal jumps inside any single stroke, \AX{2} ensures every stroke respects gauge invariance, \AX{3} forbids any stroke that imagines a closed-system shortcut, and \AX{4} forces each stroke to take the most-symmetric admissible form. \POS{6}~\OGN{}--\POS{6}~\OGN{} then provide the integer-ledger and anomaly closure that drop out at the Bogomolnyi-saturated saddle of the middle stroke.

\paragraph{Upstream theorems.}
The upstream objects are \ASix{}'s \GRIP{} structure theorems, not its
numbered theorems \T{1}--\T{5}, which are about different things --- the
\TGG{} object, the \OGNfive{} ledger, \GFF{} Morse positivity, the $N \le 3$
closure, and constitutional equivalence respectively. Earlier versions of
this section cited those numbers, and the citation was wrong; it is corrected
here. (i) \textbf{GRIP-ST1} \citep{ArisakaA6Gi4} establishes that the cascade
is a finite ordered sequence of non-degenerate critical points of
$F_{\GFF{}}$ with strictly decreasing Morse index, beginning at
$\mathrm{ind} = \dim_{\mathbb R}\Sigma$ and ending at $\mathrm{ind} = 0$; this
is what makes the input to every stroke a parent context with a well-defined
gauge action. (ii) \textbf{GRIP-ST2} establishes that motion within a basin
is Riemannian gradient flow on the fixed landscape, and is irreversible.
(iii) \textbf{GRIP-ST3} establishes that transitions between adjacent
critical points run through Bogomolnyi-saturated instanton events, with
action $S_{[2k]\to[2(k-1)]} = \pi k$ --- \emph{per gate $k$, not a flat
$\pi$}, which is the form \textsf{T-Bogomolnyi-Action} below now uses.
(iv) \textbf{GRIP-ST5} establishes that the cascade terminates at a stable
Geometric Attractor of index zero. \ASix{} is explicit that GRIP-ST5 alone
supports only a \emph{local} minimum. (v) The global statement --- that the
terminus is \emph{the} ground attractor --- is \ASix{}'s per-stratum
ground-state corollary, and it is available only on an admissible stratum in
the four-clause sense set out in \S\ref{subsec:inheritance-A6}. That
corollary, and not GRIP-ST5, is the warrant this theorem uses.

\begin{theorem}[\textsf{T-GRIP-Composition}: unique three-stroke factorization of \GRIP{} (T-Origin-63.1)]
\label{thm:t-grip-composition}
Let $\widehat{\SIP{}}^{(n)}$ be the layer-$n$ \SIP{} (parent stratum, state manifold, ports, generator, \GFF{} landscape, observable/kill bundle) with constitutional warrant under the \GGConstitution{} $\mathcal{C}$, and let $\mathcal{R}_n$ denote the set of admissible runtime selections on $\widehat{\SIP{}}^{(n)}$. Then every $R \in \mathcal{R}_n$ factors uniquely (up to gauge equivalence) as the ordered composition
\begin{equation}
\label{eq:grip-comp}
\boxed{\;\;R \;=\; \mathcal{G}^{\,\downarrow}_{\GGA{}_n} \;\circ\; \mathcal{B}_{\GSSB{}}^{(n)} \;\circ\; \mathcal{D}_{\DGI{}}^{(n)}\;\;}
\end{equation}
where (a) $\mathcal{D}_{\DGI{}}^{(n)}$ is the defining-gauge-invariance reading of the parent stratum, (b) $\mathcal{B}_{\GSSB{}}^{(n)}$ is the chain of Bogomolnyi-saturated saddle events that breaks $\DGI{}$ to a maximal residual symmetry consistent with $\AX{4}$, and (c) $\mathcal{G}^{\,\downarrow}_{\GGA{}_n}$ is the steepest-descent landing on the unique layer-$n$ \GGA{} of \eqref{eq:grip-canonical-statement}. No other factorization of $R$ is compatible with $\mathcal{C}$.
\end{theorem}

\paragraph{Proof outline.}
Step 1: existence of a factorization. Given any $R \in \mathcal{R}_n$, the
parent stratum admits a \DGI{} reading as the top critical point of
\textbf{GRIP-ST1}; the residual gauge structure on the broken sub-stratum
admits a Bogomolnyi instanton by \textbf{GRIP-ST3}; the outflow admits a
gradient landing by \textbf{GRIP-ST2} and \textbf{GRIP-ST5}. Composing these
three pieces yields a candidate factorization of $R$. Step 2: uniqueness of
the middle stroke. The instanton events of \textbf{GRIP-ST3} saturate the
Bogomolnyi bound and are therefore unique up to gauge equivalence, so any two
factorizations of $R$ agree on the middle stroke up to a gauge
transformation. Step 3: uniqueness of the outer strokes. The strictly
decreasing Morse index of \textbf{GRIP-ST1} fixes the input at
$\mathrm{ind} = \dim_{\mathbb R}\Sigma$ and the output at
$\mathrm{ind} = 0$, so any two factorizations agree on the outer strokes.
Step 4: closure under \QiFour{}-admissibility. By \ASeven{} \T{-QUA.2}
\citep{ArisakaA7Qi4} (the Arisaka equation), the four boundary currents
$J_E, J_e, J_B, J_{Qi}$ are locked through each stroke, which rules out any
alternative factorization that would require violating current-locking inside
a stroke. Step 5: the uniqueness clause, and its hypothesis. Steps 1--4
establish uniqueness up to gauge equivalence. Sharpening that to
point-uniqueness on the moduli space requires the terminus to be the
\emph{unique deepest} minimum rather than merely a stable one, and \ASix{}
supplies that only through its per-stratum ground-state corollary, on an
admissible stratum. \ASix{} states in terms that GRIP-ST5 alone is not the
warrant for the global statement. The present theorem therefore concludes
point-uniqueness \emph{conditional on admissibility of the layer-$n$
stratum}, and unconditional uniqueness only up to gauge equivalence. Of the
strata the cohort names, admissibility is established for the family
projector $\mathbb{CP}^{N-1}$; at every other layer it is a hypothesis this
paper assumes and marks.

\paragraph{Quantitative anchor.}
The composition is structural; its quantitative content lies in the corollary that follows (\textsf{T-Bogomolnyi-Action}). The action of the $k$th gate is $\pi k$, not a flat $\pi$ --- \ASix{}'s \textbf{GRIP-ST3} --- and the cascade total $\sum_{k=1}^{N-1}\pi k = \pi N(N-1)/2$ follows by summation, which is the shorter route to the same number this paper reaches by other means. The number that operationalizes the present theorem is the cascade depth $N \in \{1, \ldots, 18\}$: at each of the eighteen layers, exactly one three-stroke composition runs, and the total cascade is the ordered concatenation of eighteen such compositions.

\paragraph{Physics intuition.}
At first-year-graduate-student level, the picture is the following. Imagine a quantum field theory with gauge group $G$ broken to a subgroup $H$ by a Higgs field $\phi$ that acquires a vacuum expectation value $\langle \phi \rangle = v$. Textbook QFT tells you three things: (i) the unbroken theory has gauge invariance under $G$; (ii) the broken theory has gauge invariance under $H$ and a spectrum of massive gauge bosons for the broken generators $G/H$; (iii) the transition $G \to H$ is mediated by a tunneling event whose on-shell action obeys a Bogomolnyi-type bound. The \GRIP{} cascade picks up this textbook structure and reads it as a three-stroke composition: \DGI{} \emph{is} the gauge invariance under $G$, \GSSB{} \emph{is} the Bogomolnyi-saturating tunneling, and \GGA{} \emph{is} the unique vacuum of $H$. The novelty is not in any single stroke (each is textbook) but in the constitutional claim that this is the \emph{only} factorization: no other ordering, no other set of strokes, can carry an admissible runtime selection from input to output. This is what we mean by saying \GRIP{} is forced.

\paragraph{Daily-life metaphor.}
Consider boiling an egg. The recipe has three strokes: (i) read what counts as ``egg + water + pot'' (the parent context with its defining structure); (ii) apply heat until the egg's protein crosses its denaturation threshold (the saddle event that breaks the raw-egg phase into the cooked-egg phase); (iii) settle into the cooked steady state (the unique ground state of the cooked-egg phase at the given pressure and humidity). No other ordering produces a cooked egg. You cannot settle into the cooked steady state before the saddle event; you cannot trigger the saddle event without first having identified what counts as ``egg + water + pot''. The composition is forced by the structure of the situation. Cascade cooking obeys the same logic at every layer: parent context first, saddle next, ground state last.

\paragraph{Kill criterion.}
The composition theorem is falsified by any of the following. (a) The discovery of an admissible runtime selection $R$ on any \SIP{} that does not factor into the three strokes (e.g.\ a selection that arrives at the \GGA{} without passing through a Bogomolnyi saturation). (b) The discovery of a layer at which the \DGI{} reading is non-unique even after gauge equivalence is quotiented out. (c) The discovery of a \GSSB{} chain with a non-Bogomolnyi saddle (a saddle with on-shell action exceeding the Bogomolnyi bound). (d) The discovery of two distinct \GGA{}s reached by different orderings of the same three strokes. Any of these would falsify Theorem~\ref{thm:t-grip-composition}.

\paragraph{Plain-English close.}
The \GRIP{} cascade is not a sequence of optional steps that one could imagine running in a different order; it is the unique factorization of any layer's runtime descent. Read the recipe, run the saddle, settle into the ground state. Every layer of the cascade --- from the GUT closure of layer~1 to the articulation of layer~18 --- runs the same three strokes. The novelty of the framework is not in any single stroke but in the constitutional claim that the composition is forced.

\bigskip

% =========== Theorem 2: T-Bogomolnyi-Action ============

% INSERT-AFTER: subsection T-GRIP-Composition (the natural reading order is to fix the composition first and then quantise the middle stroke).

\subsection{\textsf{T-Bogomolnyi-Action}: the per-step action quantum and the \BIsix{} budget bound}
\label{ssec:t-bogomolnyi-action}

\paragraph{Plain-English opening.}
The previous theorem fixed the order of strokes; this one quantizes the cost of the middle stroke. Each Bogomolnyi-saturating saddle event in the \GSSB{} chain costs exactly $\pi$ units of action (in natural units $\hbar = 1$). An $N$-step cascade therefore costs the triangular sum $\pi N(N-1)/2$. The \BIsix{} action budget --- $S_{\rm BI} = 3\pi$, inherited from the \OGNfive{} closure $(K, J) = (35, 7)$ --- then bounds that sum, and the bound forces $N \leq 3$. The fit is exact rather than approximate: a three-step chain costs $3\pi$ and spends the budget to the last quantum, while a four-step chain would cost $6\pi$ and overrun it twofold. This is the deepest reason the world has small numbers in it: three families of fermions, three colors of quark, three Pauli matrices, three spatial dimensions on the boundary, three generations of the periodic table chemistry, three storey-pairs of the cognitive pagoda. The cascade is not allowed to spend more than three Bogomolnyi quanta in a single coherent stride.

\paragraph{Constitutional inputs.}
\AX{2} (\GGEP{}), \AX{4} (\MSMF{}), and \POS{6}~\OGN{} (\BIsix{} anomaly closure on a single tensor multiplet) supply the warrants. In plain language, \AX{2} ensures the saddle action is gauge-invariant and therefore quantized; \AX{4} forces the saturation of the Bogomolnyi bound (rather than excess action); \POS{6}~\OGN{} provides the integer denominator that turns the Bogomolnyi unit into $\pi$ exactly.

\paragraph{Upstream theorems.}
Four upstream theorems are invoked. (i) \ASix{} T-GRIP-4 \citep{ArisakaA6Gi4} establishes the per-saddle Bogomolnyi quantum: each saddle event saturates the bound $S_{\rm saddle} \geq \pi$ with equality forced by \AX{4}. In plain language, this is the theorem that says cascade saddles are as cheap as gauge invariance allows them to be. (ii) \AThree{} (T-BI6-9.1) \citep{ArisakaA3BI6} establishes the trace-ratio lock $(a, b, c, d) = (1, -1, -2/5, -2/5)$ that closes the \BIsix{} anomaly on a single tensor multiplet. In plain language, this is the theorem that forces the per-saddle denominator to be unity rather than $2$ or $1/2$. (iii) \ATwo{} T-GG6-6 \citep{ArisakaA2GG6} establishes the \OGNfive{} integer ledger $(M, N, L; J, K) = (1, 3, 5; 7, 35)$, which fixes the integer count of saddles in a coherent burst. (iv) \AThree{} (T-BI6-7.2) \citep{ArisakaA3BI6} together with \ATwo{} T-GG6-6 \citep{ArisakaA2GG6} establishes the \BIsix{} action budget $S_{\rm BI} = 3\pi$: the anomaly closure on a single tensor multiplet, with the \OGNfive{} integers $(K, J) = (35, 7)$, leaves exactly three Bogomolnyi quanta available to one cascade stride. In plain language, this is the theorem that says how much action the geometry has to spend before the chain must layer --- and the answer is three quanta, not an adjustable number.

\begin{corollary}[\textsf{T-Bogomolnyi-Action}: total cascade action and the $N \leq 3$ budget bound (T-Origin-63.2)]
\label{thm:t-bogomolnyi-action}
Let an $N$-step \GSSB{} chain consist of $N$ ordered Bogomolnyi saddle events. Then the total on-shell action is
\begin{equation}
\label{eq:bogomolnyi-total}
\boxed{\;\;S_{\rm tot}(N) \;=\; \pi \cdot \binom{N}{2} \;=\; \frac{\pi N(N-1)}{2}\;\;.}
\end{equation}
The \BIsix{} action budget $S_{\rm BI} = 3\pi$, inherited from the \OGNfive{} closure $(K, J) = (35, 7)$, bounds the total:
\begin{equation}
\label{eq:bogomolnyi-bound}
\frac{\pi N(N-1)}{2} \;\leq\; 3\pi
\qquad\Longleftrightarrow\qquad
N(N-1) \;\leq\; 6
\qquad\Longrightarrow\qquad
\boxed{\;N \;\leq\; 3\;}
\end{equation}
with equality at $N = 3$. The bound is saturated, not merely satisfied: $N = 3$ spends $3\pi$, exactly the budget, while $N = 4$ would demand $6\pi$ --- twice what the geometry has to give.
\end{corollary}

\emph{Stated as a corollary rather than a theorem: the action budget follows from \ASix{} T-GRIP-4 with \ATwo{} T-GG6-6, and the result is stated in Part~\ref{part:results} \S\ref{sec:statement-engine} and Part~\ref{part:recursive} \S\ref{sec:cascade-engine}.}


\paragraph{Proof outline.}
Step 1: the per-saddle quantum. By \ASix{} T-GRIP-4, each Bogomolnyi-saturated saddle has on-shell action $S_{\rm saddle} = \pi$ in natural units. In plain language, this is the textbook half-instanton result for a self-dual saddle in four dimensions, lifted to the six-dimensional bulk. Step 2: the triangular sum. The $N$-th saddle event in an ordered chain interacts with the residual action of all $N-1$ prior events through the gauge-fixed propagator; the total action is the sum over ordered pairs $\binom{N}{2}$.

In plain language, the second saddle costs one quantum on its own and shares one quantum of off-shell action with the first; the third costs one and shares with the first two; and so on. Step 3: closed form. The triangular sum gives $S_{\rm tot}(N) = \pi N(N-1)/2$. Step 4: the budget. By \AThree{} T-BI6-7.2 with the \OGNfive{} integers of \ATwo{} T-GG6-6, the \BIsix{} anomaly closure on a single tensor multiplet leaves an action budget $S_{\rm BI} = 3\pi$ for one cascade stride. Imposing $S_{\rm tot}(N) \leq S_{\rm BI}$ gives $N(N-1) \leq 6$, hence $N \leq 3$, saturated at $N = 3$.

In plain language, the geometry has exactly three quanta of action to spend on one stride; a three-step chain spends all three, and a four-step chain would need six. Beyond three the stride cannot be paid for and the cascade must layer. Step 5: the appearance of three. The bound $N = 3$ is saturated at the layer-3 cascade (atomic ground state), at the family stratum of the Standard Model, at the color stratum of \SUthree{}, and at every other place where the cascade asks ``how many copies?''. This is not a coincidence; it is the same bound applied at different strata.

\paragraph{Quantitative anchor.}
$S_{\rm tot}(1) = 0$ (a single event, no chain); $S_{\rm tot}(2) = \pi$; $S_{\rm tot}(3) = 3\pi$, which is $S_{\rm BI}$ exactly; $S_{\rm tot}(4) = 6\pi$, twice the budget. The ladder is therefore $0, \pi, 3\pi \mid 6\pi$, with the bar falling where the geometry runs out. The \OGNfive{} ledger value $N = 3$ is read off at the saturation point: families, colors, spatial boundary dimensions.

\paragraph{Physics intuition.}
The Bogomolnyi bound is one of the cleanest results in non-perturbative gauge theory. In a four-dimensional gauge theory with topological charge $Q$, the Yang--Mills action obeys $S \geq 8\pi^2 |Q|/g^2$, with equality on self-dual or anti-self-dual configurations. In the GG-Theory normalization that absorbs the coupling into the scale profile, the relevant unit per saddle event is $\pi$ rather than $8\pi^2/g^2$; this is a renormalization choice that turns the per-saddle action into a dimensionless integer multiple of $\pi$. The crucial fact is that the action is \emph{quantized}: there is no continuous family of saddle actions, only the integer ladder. The triangular sum $\binom{N}{2}$ then arises because the $k$-th saddle event experiences the off-shell residue of the $k-1$ prior events through the gauge-fixed propagator; the symmetric counting of ordered pairs gives the triangular structure. The bound is not a coherence condition imposed from outside but the anomaly budget read as a spending limit: \BIsix{} closure on a single tensor multiplet fixes how much action the stride has, and the triangular sum fixes how fast a chain consumes it. Three steps exhaust it exactly; a fourth cannot be paid for, and the cascade layers instead.

\paragraph{Daily-life metaphor.}
A juggler can keep at most three balls in coherent flight before the rhythm decoheres (four balls require a more skilled juggler with two hands; five or more require a stage assistant). The Bogomolnyi-action bound says the universe is a three-ball juggler in each stride: three saddle events spend the whole budget, and a fourth cannot be afforded. Beyond three, the cascade must layer --- catch one ball, hand it off, then juggle the next three. That is why there are three of so many things at once; how many \emph{layers} of juggling the universe has managed is a separate and empirical question, answered in \S\ref{ssec:t-seventeen-step-termination}.

\paragraph{Kill criterion.}
The action-quantum result is falsified by any of: (a) the discovery of a Bogomolnyi saddle with action $\neq \pi$ that is not a multiple of $\pi$ (i.e.\ a violation of the integer ledger); (b) the discovery of a coherent cascade stride with $N \geq 4$ (i.e.\ a four-family, four-color, four-generation phenomenon); (c) a demonstration that the \BIsix{} action budget is other than $3\pi$ --- for instance an anomaly closure requiring more than one tensor multiplet, which would move the budget and with it the bound; (d) the failure of the family count $N = 3$ at the electroweak scale (a fourth charged lepton at GeV or TeV scale not associated with cascade layering).

\paragraph{Plain-English close.}
Three is not a magic number; it is where a quantized action ladder meets a fixed budget. Three families of fermions, three colors of quark, three boundary spatial dimensions, three rungs of every cascade stride: all of these read off from the same calculation. $\pi N(N-1)/2 \leq 3\pi$ gives $N \leq 3$, and $N = 3$ spends the budget exactly. The world has small integers in it because the saddle action is quantized in units of $\pi$ and the \BIsix{} closure leaves exactly three of them to spend.

\bigskip

% =========== Theorem 3: T-SIP-Tower-Recursion ============

% INSERT-AFTER: subsection T-Bogomolnyi-Action.

\subsection{\textsf{T-SIP-Tower-Recursion}: the layer-to-layer recursion on \SIP{} space}
\label{ssec:t-sip-tower-recursion}

\paragraph{Plain-English opening.}
The cascade is not just a sequence of layers; it is a \emph{recursion} on the space of \SIP{}s. Each layer's output --- its \GGA{} --- constructs (via its boundary state, port structure, \GFF{} landscape, and Qi-current bookkeeping) the next layer's \SIP{}. The output of layer $n$ becomes literally the input ledger of layer $n+1$. Without this recursion there would be no cascade, only a sequence of unrelated \GRIP{} runs. With this recursion, the cascade is a single self-organizing tower in which each storey rests on the floor below and provides the floor for the storey above. In daily life, the recursion is what makes a building stand: the foundation supports the first storey, the first storey supports the second, and so on up to the tenth. The recursion clause is the constitutional warrant that the building can be built.

\paragraph{Constitutional inputs.}
\AX{3} (\OGGEP{}, the open-system boundary that exchanges boundary state from one layer to the next), \AX{4} (\MSMF{}), \ASeven{}~\T{-QUA} (operational/measurement support that turns one layer's \GGA{} into the next layer's substrate), and \ASeven{}~\T{-EP} (functional closure, which forbids the recursion from doubling back on a finished layer). In plain language: \AX{3} provides the port through which the layer-$n$ \GGA{} hands its boundary state to layer~$n+1$; \AX{4} forces the most-symmetric handoff; \ASeven{}~\T{-QUA} ensures the handoff is operationally realized; \ASeven{}~\T{-EP} ensures the handoff is monotonically upward (the cascade does not run backwards).

\paragraph{Upstream theorems.}
Three upstream theorems are invoked. (i) \ASeven{} \T{-SIP} \citep{ArisakaA7Qi4}, the \SIP{} completeness theorem, establishes the canonical five-field structure of an \SIP{} --- state manifold and metric, boundary ports, \GDOS{} generator and noise closure, \GFF{} landscape and locked current, observable/kill bundle --- as boxed in \eqref{eq:sip-five-field}. In plain language, this is the theorem that says an \SIP{} is a well-defined mathematical object with a fixed information content; the recursion can therefore be stated as a map between such objects. (ii) \ASix{} T-GRIP-5 \citep{ArisakaA6Gi4} (the \GRIP{} structure theorem) supplies the output side of each layer: the unique \GGA{} of layer~$n$. In plain language, this fixes what each layer hands to its successor. (iii) \ASeven{} \T{-CON} \citep{ArisakaA7Qi4} (consistency theorem) establishes that the boundary-state factorization of the layer-$n$ \GGA{} is consistent with the port structure required by layer~$n+1$. In plain language, this is the theorem that says the handoff is well-defined: what layer~$n$ produces is what layer~$n+1$ requires.

\begin{corollary}[\textsf{T-SIP-Tower-Recursion}: the cascade as a recursion on \SIP{} space (T-Origin-63.3)]
\label{thm:t-sip-tower-recursion}
Let $\mathfrak{S}$ be the space of canonical \SIP{}s (in the sense of \ASeven{} \T{-SIP}), and let $\mathcal{R}_{\GRIP{}}: \mathfrak{S} \to \mathfrak{S}^{\GGA{}}$ be the \GRIP{} runtime map sending each \SIP{} to its unique \GGA{}. Define the substrate-construction map $\mathcal{C}_{\rm subs}: \mathfrak{S}^{\GGA{}} \to \mathfrak{S}$ which sends a \GGA{} to the \SIP{} of the layer it supports. Then the cascade tower is the composition
\begin{equation}
\label{eq:sip-recursion}
\boxed{\;\;\widehat{\SIP{}}^{(n+1)} \;=\; \mathcal{C}_{\rm subs}\!\left[\,\mathcal{R}_{\GRIP{}}\!\left[\,\widehat{\SIP{}}^{(n)}\,\right]\,\right] \;=\; \mathcal{C}_{\rm subs}[\GGA{}_n]\;\;,}
\end{equation}
defined for $n = 1, 2, \ldots, 17$, with $\widehat{\SIP{}}^{(1)}$ the bulk-gauge \SIP{} and $\widehat{\SIP{}}^{(18)}$ --- the articulation \SIP{} of the present paper --- the last packet the recursion emits. The recursion is well-defined: $\mathcal{C}_{\rm subs}$ extracts from each \GGA{} (i) a boundary-state factorization $\rho_{\GGA{}_n}$ on a boundary stratum $\mathcal{S}_n$, (ii) a port set $\partial_n$ derived from $\rho_{\GGA{}_n}$'s exposed currents, (iii) a refined \GFF{} landscape $F_{\GFF{}}^{(n+1)}$ on the boundary, and (iv) an observable/kill bundle adapted to the boundary's degrees of freedom; these jointly constitute $\widehat{\SIP{}}^{(n+1)}$.
\end{corollary}

\emph{Stated as a corollary rather than a theorem: the recursion is stated in Part~\ref{part:results} \S\ref{ssec:statement-sip-tower}, in Part~\ref{part:recursive}, and in \ASeven{} \citep{ArisakaA7Qi4} \S6.}


\paragraph{Proof outline.}
Step 1: well-definedness of $\mathcal{R}_{\GRIP{}}$. By \ASix{} T-GRIP-5, each \SIP{} admits a unique \GGA{} under the \GRIP{} runtime; therefore $\mathcal{R}_{\GRIP{}}$ is a well-defined map. Step 2: well-definedness of $\mathcal{C}_{\rm subs}$. By \ASeven{} \T{-CON}, the boundary-state factorization of the layer-$n$ \GGA{} is consistent with the canonical \SIP{} structure required by layer~$n+1$; therefore $\mathcal{C}_{\rm subs}$ is well-defined. Step 3: termination. The recursion terminates at $n = 18$ by \textsf{T-Cascade-Termination} (\S\ref{ssec:t-seventeen-step-termination}). Step 4: well-definedness of the chain. The chain is well defined \emph{by construction} from the base case $\GGA{}_0 = \mathcal{C}$ (the \GGConstitution{} itself), because Steps 1--2 make each of $\mathcal{R}_{\GRIP{}}$ and $\mathcal{C}_{\rm subs}$ a map: given $\GGA{}_{n-1}$, the layer-$n$ \SIP{} and its \GGA{} are determined. Well-definedness of the recursion requires only that each layer \emph{have} a \GGA{}; it does not require the chain to be unique, and the present step does not assume it. Uniqueness of each link is the separate question settled --- conditional on (Q1)--(Q4) --- by \textsf{T-Cascade-Layer-Uniqueness} below. Step 5: covariance with the constitutional axioms. Each application of $\mathcal{C}_{\rm subs}$ preserves the \GGConstitution{} $\mathcal{C}$ (axioms and postulates); therefore the recursion runs inside the constitutional admissibility class.

\paragraph{Quantitative anchor.}
The recursion is structural; its quantitative content is the cascade depth $N_{\rm cascade} = 18$ (one-indexed: layers 1 through 17 for the constructive cascade, plus the meta-layer 18 of articulation, for eighteen entries total). The eighteen \SIP{}s and eighteen \GGA{}s constitute the master cascade codebook of \ASeven{} \T{-SIP}.

\paragraph{Physics intuition.}
Roughly speaking, the recursion is the cascade's analogue of the renormalization group flow in textbook quantum field theory, but with three structural improvements. First, the RG flow runs along a continuous coupling parameter $t = \log \mu$, whereas the cascade recursion is indexed by an integer layer $n$; the cascade is therefore a discrete dynamical system rather than a continuous one. Second, the RG flow's fixed points are typically not unique (the Wilson--Fisher fixed point coexists with the Gaussian one), whereas the cascade's \GGA{}s are unique by \textsf{T-Origin}. Third, the RG flow does not change the symmetry group of the theory (it only rescales couplings within a fixed symmetry), whereas the cascade explicitly breaks symmetry from one layer to the next. The recursion is thus more like a symmetry-breaking ladder than a coupling-renormalization flow: each step strips off a layer of symmetry and exposes the next layer of structure beneath. In the language of condensed-matter physics, the cascade is a series of phase transitions stacked on top of one another, with the order parameter of each transition becoming the substrate of the next.

\paragraph{Daily-life metaphor.}
Building a house provides the cleanest metaphor. Pour the foundation (layer~1). On the foundation, frame the first storey (layer~2). On the first storey's floor joists, frame the second storey (layer~3). Each storey is constructed from the previous storey's ceiling; you cannot frame the second storey before the first is complete. The recursion clause is the constitutional warrant that the construction is feasible: each storey's ceiling is compatible with the next storey's floor. In the cascade, this compatibility is enforced by \ASeven{} \T{-CON}: the boundary state of layer~$n$'s \GGA{} factors in a way that constructs the canonical \SIP{} structure of layer~$n+1$. A house built on a defective foundation collapses at storey two; a cascade built on a defective \GGA{} likewise fails to ascend to its next layer.

\paragraph{Kill criterion.}
The recursion result is falsified by any of: (a) the discovery of a layer-$n$ \GGA{} that does not admit a canonical \SIP{} reading at layer~$n+1$ (i.e.\ a boundary-state factorization that fails \ASeven{} \T{-CON}); (b) the discovery of a layer at which the cascade requires \emph{additional} constitutional input not present in the previous \GGA{} (i.e.\ a violation of the closure clause); (c) the discovery of a cascade that runs to a different number of layers than eighteen (which would also falsify \textsf{T-Cascade-Termination}); (d) the discovery of two distinct \SIP{}s at the same layer, both constructed from the same prior \GGA{} (which would falsify the uniqueness of $\mathcal{C}_{\rm subs}$).

\paragraph{Plain-English close.}
The cascade is built by recursion, not by external scaffolding. Each layer's \GGA{} hands its boundary state, port structure, and Lyapunov to the next layer's \SIP{}. The tower stands because each storey is built from the previous storey's ceiling. This is the constitutional warrant for the cascade's existence as a single self-organizing structure rather than eighteen unrelated runs of \GRIP{}.

\bigskip

% =========== Theorem 4: T-Cascade-Layer-Uniqueness ============

% INSERT-AFTER: subsection T-SIP-Tower-Recursion.

\subsection{\textsf{T-Cascade-Layer-Uniqueness}: per-layer instance of \textsf{T-Origin}}
\label{ssec:t-cascade-layer-uniqueness}

\paragraph{Plain-English opening.}
\textsf{T-Origin} (in Part~\ref{part:recursive} of the paper) states that the whole cascade lands on a unique attractor. The per-layer version of that theorem says: at every individual layer of the cascade, the \GGA{} is unique up to gauge equivalence. This is \textsf{T-Origin} read one layer at a time --- the induction below inherits that theorem's hypotheses rather than supplying a fresh warrant --- but the reading is load-bearing rather than decorative: it is what makes the tower recursion of \S\ref{ssec:t-sip-tower-recursion} well-defined. If the layer-$n$ \GGA{} were ambiguous, the substrate for layer~$n+1$ would be ambiguous, and the cascade would split into branches. The per-layer uniqueness clause closes this off: each storey of the cascade has exactly one ceiling, and therefore the next storey has exactly one floor. In daily life: every honest game of solitaire ends in exactly one win-or-loss outcome; there is no superposition of win-and-loss at any individual turn.

\paragraph{Constitutional inputs.}
\AX{1} (\LC{}, causal sharpness of the per-layer \GFF{}), \AX{2} (\GGEP{}), \AX{4} (\MSMF{}, the most-symmetric attractor is forced), and \ASeven{}~\T{-EP} (functional closure within a layer). In plain language, \AX{1} ensures the per-layer Morse landscape has well-separated minima; \AX{2} ensures the gauge action picks one; \AX{4} ensures it picks the most-symmetric one; \ASeven{}~\T{-EP} ensures the choice is final at that layer (no late re-organization).

\paragraph{Upstream theorems.}
Three upstream theorems are invoked. (i) \ASix{} T-GRIP-3 \citep{ArisakaA6Gi4} (per-saddle uniqueness) supplies the middle-stroke uniqueness for the layer's \GRIP{} composition. In plain language, the saddle event of the layer is unique up to gauge equivalence; therefore its outflow is unique. (ii) \ASeven{} \T{-CON} \citep{ArisakaA7Qi4} (consistency theorem) ensures that the layer's \GGA{} is unique on the consistent-completion class (no alternative reading of the boundary state survives the constitutional axioms). (iii) \textsf{T-Origin} of Part~\ref{part:recursive} of the present paper provides the global uniqueness theorem; the present result is the layer-by-layer specialization.

\begin{corollary}[\textsf{T-Cascade-Layer-Uniqueness}: per-layer \GGA{} is unique (T-Origin-63.4)]
\label{thm:t-cascade-layer-uniqueness}
For each $n \in \{1, \ldots, 17\}$, the layer-$n$ \GGA{} $\GGA{}_n$ produced by $\mathcal{R}_{\GRIP{}}[\widehat{\SIP{}}^{(n)}]$ is unique up to gauge equivalence on the layer-$n$ stratum. Formally, if $\GGA{}_n$ and $\GGA{}_n'$ are both admissible \GGA{}s for layer~$n$ under the \GGConstitution{} $\mathcal{C}$, then there exists a gauge transformation $g_n$ on the layer-$n$ stratum such that
\begin{equation}
\GGA{}_n' \;=\; g_n \cdot \GGA{}_n \cdot g_n^{-1}.
\end{equation}
\end{corollary}

\emph{Stated as a corollary rather than a theorem: this is \textsf{T-Origin} read at a single layer; as the proof outline below states, it inherits its hypotheses and is not an independent warrant.}


\paragraph{Proof outline.}
Step 1: per-saddle uniqueness. By \ASix{} T-GRIP-3, the Bogomolnyi saddle event of layer~$n$ is unique up to gauge equivalence; therefore the middle stroke of the layer's \GRIP{} composition is unique up to gauge. Step 2: per-input uniqueness, by induction on $n$ with an explicit base case. \emph{Base case} ($n = 1$): the layer-1 substrate is constructed from $\GGA{}_0 = \mathcal{C}$, the \GGConstitution{} itself, which is given rather than derived --- it is the input ledger of \AOne{}, not a cascade output --- so the layer-1 \SIP{} is fixed with no appeal to a prior uniqueness. \emph{Inductive step}: assume $\GGA{}_{n-1}$ unique up to gauge; by \textsf{T-SIP-Tower-Recursion} the layer-$n$ \SIP{} is determined by $\GGA{}_{n-1}$, so the input stroke is unique up to gauge. The induction is over the per-layer argument of \S\ref{sec:qi4-admissibility-filter} and therefore inherits its hypotheses (Q1)--(Q4); it is not a second, independent warrant. Step 3: per-output uniqueness. By \ASix{}'s cascade-closure structure theorem \textbf{GRIP-ST5}, the descent stroke lands on a unique attractor. Step 4: closure under gauge equivalence. Any two admissible \GGA{}s on the same layer differ at most by a gauge transformation; this is the standard uniqueness-modulo-gauge clause. Step 5: consistency at the boundary. By \ASeven{} \T{-CON}, the unique \GGA{}'s boundary-state factorization is consistent with the substrate-construction map $\mathcal{C}_{\rm subs}$; the per-layer uniqueness therefore propagates upward through the recursion.

\paragraph{Quantitative anchor.}
For each of the eighteen layers, the \GGA{} is named and tabulated in the master cascade codebook of \ASeven{} \T{-SIP}. The uniqueness clause is the assertion that no alternative reading of the layer's boundary stratum is admissible. Quantitatively, this means: there is only one Standard Model gauge group (layer~1); only one Higgs vev (layer~2); only one set of charged-lepton masses (layer~2, refined); only one periodic table (layer~3); only one nucleic-acid alphabet (layer~8); only one set of twenty proteinogenic amino acids (layer~5); only one polymer class (layer~6); only one ribosomal protein-synthesis pattern (layer~8); only one ATP/DNA bioenergetic anchor (layer~7); only one universal LUCA cell (layer~10); only one ring-attractor pattern for insect heading (layer~15); only one allocentric place--grid--time pattern for mammalian navigation (layer~16); only one EIC pattern for human conscious episodes (layer~17).

\paragraph{Physics intuition.}
At first-year-graduate-student level, the picture is this. Many physical theories have multiple vacua: the standard $\mathrm{SU}(2)$ Higgs potential has a degenerate vacuum manifold (a sphere in field space) whose points are related by gauge transformations. In such theories, the choice of vacuum is conventional --- one picks a representative point and the physics is the same regardless. The cascade's per-layer uniqueness clause is the statement that the same is true at each layer of the cascade: the layer's \GGA{} lives on a vacuum manifold that is gauge-equivalent to a single point. This is a strong claim: it says the cascade does not produce families of inequivalent vacua (like the multi-vacuum landscape problem of string theory) but rather a single unique vacuum (up to gauge). The cascade's recursive structure makes this even sharper: the next layer's \SIP{} is constructed from the present layer's \GGA{}, so any non-uniqueness at the present layer would propagate into a branching tree of possible cascades; per-layer uniqueness rules this out.

\paragraph{Daily-life metaphor.}
Cooking a single batch of rice gives one batch of cooked rice; you cannot end up with two qualitatively different batches from the same starting conditions and the same recipe. Up to small fluctuations (which are gauge equivalences), the output is determined. The cascade's per-layer uniqueness clause says the same of every layer: starting from the layer-$(n-1)$ \GGA{} and applying \GRIP{} produces a single layer-$n$ \GGA{} up to gauge. There is no parallel universe in which the same \SIP{} produces a qualitatively different \GGA{}; the cascade is deterministic at every layer.

\paragraph{Kill criterion.}
The uniqueness result is falsified by any of: (a) the discovery of two qualitatively different \GGA{}s at the same layer, both produced by the same \SIP{} under \GRIP{}; (b) the discovery that the gauge transformation $g_n$ relating two ostensibly different \GGA{}s is not in the layer's gauge group (i.e.\ they are inequivalent on the layer's stratum); (c) the discovery of a layer whose \GGA{} requires breaking the \GGConstitution{} (e.g.\ violating \AX{4}) to be uniquely specified.

\paragraph{Plain-English close.}
At every layer, exactly one ceiling. The cascade does not branch into a tree of possibilities; it runs as a single ordered sequence of unique outcomes. This is what makes the master codebook of eighteen \GGA{}s a well-defined object, and what makes the recursion of \textsf{T-SIP-Tower-Recursion} a deterministic tower-building procedure rather than a branching exploration.

\bigskip

% =========== Theorem 5: T-Universal-Identity ============

% INSERT-AFTER: subsection T-Cascade-Layer-Uniqueness.

\subsection{\textsf{T-Universal-Identity}: every electron is the same electron}
\label{ssec:t-universal-identity}

\paragraph{Plain-English opening.}
This is the constitutional centerpiece of the entire cascade, and arguably the entire program. All cascade products of the same layer-stratum \SIP{} are gauge-equivalent to one another; in plain language, every electron in the universe is the same electron. Every hydrogen atom has the same ground-state binding energy of 13.6~eV. Every ATP molecule, in any cell, in any organism, releases the same $2 N_A\QB{} = 30.0$~kJ/mol on hydrolysis. Every place cell in every rat hippocampus implements the same allocentric coding scheme. This is the universal-identity insight: the cascade, by producing a unique \GGA{} at each layer, automatically forces universal identity across all instances of that layer's products. In daily life: a kilogram is a kilogram everywhere, not because the SI committee voted on it but because the underlying physics is the same in every laboratory. The cascade explains why universal identity exists at every layer of the structural hierarchy.

\paragraph{Constitutional inputs.}
\AX{2} (\GGEP{}, gauge invariance as geometric invariance --- the deep reason identical particles are truly identical), \AX{4} (\MSMF{}, the most-symmetric \GGA{}), and \ASeven{}~\T{-EP} (functional closure --- identical products of the same layer behave identically). The universal-identity principle is also the per-layer reading of the constitutional clause that the cascade is deterministic up to gauge.

\paragraph{Upstream theorems.}
Three upstream theorems are invoked. (i) \ANine{} Part~XI (the universal-identity section) \citep{ArisakaA9SM} (universal-identity theorem of the SM program) establishes that every electron, every quark, every nucleon, every atom, every molecule of a given kind is interchangeable with any other. In plain language, this is the SM-program's reading of the equivalence-of-identical-particles principle, now derived rather than postulated. (ii) \textsf{T-Cascade-Layer-Uniqueness} (preceding corollary) supplies the per-layer uniqueness clause; the present theorem extends it from ``unique \GGA{}'' to ``universal identity of products.'' (iii) \ASeven{} \T{-CON} \citep{ArisakaA7Qi4} (consistency theorem) ensures that the universal identity is consistent with the cascade's recursive structure: identical products at layer~$n$ provide identical substrates for layer~$n+1$.

\begin{theorem}[\textsf{T-Universal-Identity}: gauge-equivalence of all layer-$n$ products (T-Origin-63.5)]
\label{thm:t-universal-identity}
Let $\widehat{\SIP{}}^{(n)}$ be the layer-$n$ \SIP{} and let $\GGA{}_n$ be its unique \GGA{} (per \textsf{T-Cascade-Layer-Uniqueness}). Let $\Pi_n$ denote the class of layer-$n$ products --- the boundary states realized by the layer-$n$ \GGA{} on individual instances of the layer's stratum. Then for any two products $\pi, \pi' \in \Pi_n$, there exists a gauge transformation $g_n$ on the layer-$n$ stratum such that
\begin{equation}
\label{eq:universal-identity}
\boxed{\;\;\pi' \;=\; g_n \cdot \pi \cdot g_n^{-1}\;\;.}
\end{equation}
Equivalently, all members of $\Pi_n$ are physically equivalent: they share the same observable spectrum, the same coupling structure, the same \QB{}-quantum economy, and the same \HAL{} depth. This is the universal-identity insight: ``every electron is the same electron'' (layer~2), ``every hydrogen atom has the same binding energy'' (layer~3), ``every ATP molecule yields the same $\Delta G$'' (layer~7), ``every conscious episode supports the same \EIC{}-closure structure'' (layer~17).
\end{theorem}

\paragraph{Proof outline.}
Step 1: per-layer uniqueness. By \textsf{T-Cascade-Layer-Uniqueness}, the layer-$n$ \GGA{} is unique up to gauge. Step 2: per-product equivalence. Two products of the same \GGA{} are two realizations of the same boundary state on two instances of the layer's stratum; they are related by the gauge transformation that maps one instance to the other. Step 3: gauge-invariance of observables. By \AX{2}, all observables on the layer's stratum are gauge-invariant; therefore all observables agree between any two products. Step 4: stability under cascade ascent. By \ASeven{} \T{-CON}, the cascade's recursion preserves universal identity: identical products at layer~$n$ produce identical substrates for layer~$n+1$, and therefore identical products at layer~$n+1$. Step 5: induction up the tower. The argument applies at each layer of the cascade, from the constitutional base $n = 0$ up to $n = 17$, and at the articulation layer $n = 18$ in the weaker gauge-equivalence sense of \S\ref{sec:named-theorem}; universal identity is therefore a property of every layer.

\paragraph{Quantitative anchor.}
The universal identity manifests in many measured quantities: the electron $g$-factor $g_e = 2.00231930436256(35)$ is the same in every measurement (the cascade prediction is $g_e = 2$ at tree level with cascade-locked QED corrections); the hydrogen-atom $1s$ binding energy is 13.6~eV everywhere; the ATP-hydrolysis $\Delta G^{\circ\prime} = -30.5$~kJ/mol everywhere; the speed of light is $c = 299\,792\,458$~m/s everywhere; the \QB{}-quantum $\approx 0.1555$~eV/qubit everywhere; the proton mass $m_p = 938.272$~MeV everywhere. Each of these is the universal identity of the corresponding cascade layer's products.

\paragraph{Physics intuition.}
The principle of identical particles is a textbook foundation of quantum mechanics: two electrons cannot be distinguished by any local measurement, and the wave function of a multi-electron system is antisymmetric under electron exchange. This principle is usually postulated as an axiom (or imposed by hand in the construction of Fock space). The framework's universal-identity theorem derives the principle from the cascade structure: identical particles are identical because they are products of the same layer-$n$ \GGA{}, and the \GGA{} is unique up to gauge. No two electrons can differ in any gauge-invariant property because they are both realizations of the same boundary state of the layer-2 \GGA{}. This argument extends beyond the level of identical \emph{particles} to identical \emph{molecules}, identical \emph{cells}, identical \emph{cognitive episodes}. The cascade does not stop at the elementary-particle level; the universal-identity principle propagates upward through every layer.

\paragraph{Daily-life metaphor.}
A factory that produces light bulbs of a single specification produces bulbs that are interchangeable: any two bulbs of the same model have the same wattage, the same lifespan, the same socket fit. The cascade is a factory of universal identity at every layer: layer~2 produces identical electrons, layer~3 produces identical hydrogen atoms, layer~7 produces identical ATP molecules. The reason the factory's bulbs are interchangeable is that they are all stamped out from the same mould; the reason the cascade's products are interchangeable is that they are all stamped out from the same \GGA{}.

\paragraph{Kill criterion.}
The theorem is falsified by any of: (a) the discovery of two electrons differing in any gauge-invariant property at the level of fundamental physics; (b) the discovery of a hydrogen atom whose $1s$ binding energy differs from 13.6~eV beyond the QED radiative-correction precision; (c) the discovery of a molecule of ATP that hydrolyzes to a $\Delta G$ different from $2 \QB{}$; (d) the discovery of a place cell in a healthy adult rat hippocampus that codes allocentrically by a scheme inconsistent with the layer-16 \GGA{}. In short: any robust counter-instance of universal identity at any cascade layer falsifies the theorem.

\paragraph{Plain-English close.}
The universe is not a collection of accidentally-similar things; it is a structurally-identical replication of one layer's outputs across every instance of that layer. Every electron, every hydrogen atom, every ATP molecule, every conscious episode shares its essential structure with every other of its kind because they are all gauge-equivalent realizations of the same layer's \GGA{}. This is the universal-identity insight, and it is the constitutional centerpiece of the cascade.

\bigskip

% =========== Theorem 6: T-Cascade-Termination ============

% INSERT-AFTER: subsection T-Universal-Identity.

\subsection{\textsf{T-Cascade-Termination}: the cascade terminates in exactly eighteen layers}
\label{ssec:t-seventeen-step-termination}

\paragraph{Plain-English opening.}
The cascade \emph{as realized in the universe we observe} has eighteen layers: seventeen constructive strata plus the meta-cognitive articulation. The seventeen is an empirical enumeration --- the strata the universe is observed to have realized --- and is not derived from the integer ledger; the theorem's structural content is the \emph{termination} (no admissible nineteenth layer), not the cardinality of the constructive run. Layers 1 through 17 form the constructive cascade (GUT closure to allocentric/conscious episode at the top of the bulk cascade); layer 18 is the meta-layer of articulation (the present paper). A hypothetical layer 19 would require the cascade to articulate its own articulation --- the cascade would have to write a paper about the paper that writes the paper. This is forbidden by Gödel-style self-reference: the articulation layer cannot take its own articulation as substrate without the substrate and the act coinciding. The cascade therefore terminates at layer 18. In daily life: a film about the making of a film about the making of a film tends to lose coherence; the recursive depth of self-referential articulation is bounded. The cascade's depth is bounded by the same kind of structural argument.

\paragraph{Constitutional inputs.}
\AX{4} (\MSMF{}, the cascade reaches the most-symmetric admissible depth), \POS{6}~\OGN{} (\OGNfive{} integer ledger), \ASeven{}~\T{-QUA} (operational/measurement support that bounds articulation depth), \ASeven{}~\T{-EP} (functional closure), \ASeven{}~\T{-HAL} (\Mlock{}), and \ASeven{}~\T{-EIC} (the closure clause). In plain language: \ASeven{}~\T{-QUA}--\ASeven{}~\T{-EIC} provide the operational closure that the constructive run is complete; \AX{4} forces termination at the most-symmetric admissible depth; and the articulation layer closes the cascade structurally --- a layer that articulates the articulation is inadmissible under the \HAL{} constraint (\ASeven{}~\T{-HAL}) and Gödel-style self-reference.

\paragraph{Upstream theorems.}
Two upstream theorems are invoked. (i) \ASeven{} \T{-HAL.1} \citep{ArisakaA7Qi4} establishes what a layer's memory architecture must supply --- the \MePMoS{}/\NHT{}/\HAL{} factorization --- and therefore what a nineteenth layer would have to build on top of a substrate that is already the act of articulation. \ASeven{} states no ceiling on cascade depth, and none is claimed here; the termination argument is the self-reference argument below. In plain language, this is the theorem that a self-referential cascade cannot articulate its own articulation beyond a certain depth without losing constitutional closure. (ii) \textsf{T-SIP-Tower-Recursion} (Corollary~\ref{thm:t-sip-tower-recursion}) establishes the recursion on \SIP{} space; the present theorem fixes its termination.

\begin{theorem}[\textsf{T-Cascade-Termination}: the cascade has eighteen layers (T-Origin-63.6)]
\label{thm:t-seventeen-step-termination}
The cascade recursion of \eqref{eq:sip-recursion} terminates at layer~$n = 18$. Concretely,
\begin{enumerate}\itemsep1pt
\item layers $n = 1, \ldots, 17$ are the constructive cascade, each running a \GRIP{} composition (per \textsf{T-GRIP-Composition}) and producing a unique \GGA{} (per \textsf{T-Cascade-Layer-Uniqueness});
\item layer $n = 18$ is the meta-layer of \emph{articulation}: the cascade articulating its own existence and structure (this is the layer realized by the present paper);
\item a hypothetical layer $n = 19$ would require articulating the articulation, which is forbidden by the \HAL{} constraint (\ASeven{} \T{-HAL}) and by Gödel-style self-reference.
\end{enumerate}
The cascade comprises seventeen constructive layers plus the articulation, eighteen in all; the termination is structural rather than arithmetic --- the articulation layer is the unique fixed point at which the cascade reads out its own constitution, and no further layer is admissible under the \HAL{} constraint and Gödel-style self-reference.
\end{theorem}

\paragraph{Proof outline.}
Step 1: layers 1--17 are admissible. By \textsf{T-SIP-Tower-Recursion}, the cascade recursion runs from $n = 1$ to $n = 17$ producing the seventeen constructive master \GGA{}s listed in the codebook of Part~\ref{part:results}. Step 2: layer 18 exists. The present paper realizes a meta-layer in which the cascade articulates its own structure: \GGA{}$_{17}$'s output (conscious episode) is read by the cascade itself as the input to a meta-\SIP{} whose output is the articulation. Step 3: a further (nineteenth) layer is forbidden. A meta-meta-articulation would require the cascade to embed a model of itself articulating itself articulating itself; this self-reference depth violates \ASeven{} \T{-HAL} and produces a Gödel-style undecidable statement at the operational level. In plain language, the cascade cannot exhibit a stable third-order self-model without losing constitutional closure. Step 4: count. The seventeen constructive layers are an empirical enumeration --- the strata realized in the universe we observe --- not a quantity derived from the integer ledger; the structural claim of the theorem is the termination, not the cardinality. Step 5: agreement with empirical cascade depth. The seventeen constructive \GGA{}s of the master codebook (layers 1--17) plus the present article (layer 18) account for the full structural hierarchy from gauge group to articulation, totaling eighteen entries.

\paragraph{Quantitative anchor.}
$N_{\rm cascade} = 18$ (seventeen constructive layers plus the articulation), as enumerated in the master codebook. The count is descriptive, not derived from the integer ledger; termination at the articulation is forced structurally (the \HAL{} constraint and Gödel-style self-reference), and no further layer is admissible.

\paragraph{Physics intuition.}
The closest physics analogue is the no-go theorem for arbitrary-depth self-referential dynamical systems in classical and quantum computer science. A system that simulates itself simulating itself runs into a recursion that consumes its own state-space; beyond a critical depth, the simulation cannot be completed within the system's resource budget. The cascade's \HAL{} constraint is the structural analogue: articulation cannot recurse past the meta level without the self-model consuming its own closure budget. The object-level cascade, the articulation (meta-level), and a would-be meta-meta-articulation form a recursion that \HAL{} and Gödel-style self-reference cut off at the articulation layer; beyond this, the cascade either fails to close or wraps around to a lower layer. This is why the cascade closes at one articulation layer (Layer 18) atop the seventeen constructive layers (Layers 1 through 17).

\paragraph{Daily-life metaphor.}
Consider a film festival. At the first level, a film tells a story. At the second, a documentary tells the story of the making of the film. At the third, a meta-documentary tells the story of the making of the documentary. Beyond this, the chain becomes incoherent: a documentary about a documentary about a documentary is empty of new content, because the recursive depth has consumed the available cinematic resources. The cascade behaves the same way at the level of articulation: object-cascade (layers 1--17), articulation-cascade (layer 18, the present paper), and no further admissible meta-meta-cascade. The constitutional warrant for termination is the same as the practical warrant in the film festival: beyond three levels of meta, the recursion does not produce stable new content.

\paragraph{Kill criterion.}
The theorem is falsified by any of: (a) the discovery of an admissible cascade layer above the articulation --- a layer~19 that articulates the articulation, which the \HAL{} constraint forbids (a different count of \emph{constructive} strata would revise the enumeration, not the theorem, since the cardinality is descriptive); (b) the discovery of a stable layer-18 \GGA{} that articulates the articulation of the cascade beyond what \HAL{} permits; (c) the demonstration that the cascade admits a stable meta-meta-articulation layer beyond Layer~18 (which would show the termination is not structurally forced); (d) the existence of a coherent physical or biological structure that falls outside the eighteen-layer master codebook (e.g.\ a stable structural ground state at a layer the cascade does not predict).

\paragraph{Plain-English close.}
Eighteen is not a numerological mascot; it is the integer count of constructive cascade layers (seventeen layers $1$ through $17$ plus the meta-cognitive articulation at layer~$18$), counted from the strata the universe is observed to have realized, with the \HAL{} constraint forbidding a nineteenth. The termination is structural; the cardinality is descriptive. The cascade runs from gauge group (layer~1) through electroweak vacuum, atomic ground state, molecular closure, amino-acid selection, polymer folding, bioenergetic anchor, cellular closure, neural ring attractor, allocentric map, conscious episode (layer~17), and articulates itself at layer~18 in the present paper. Beyond this, the cascade does not run.

\bigskip


% ====================================================================
%  BLOCK VI -- Part VI (Dynamics block): 4 new theorems
%  Insertion targets:
%    T-QB-Anchor  -> after section "The bioenergetic anchor $\QB$" (line 4855)
%    T-GFF-Lyapunov-Descent -> inside section "The Geometric Free-Energy
%                              Functional and the Three-Gauge Theorem"
%                              (line 5098)
%    T-SeOS-Class-Characterisation -> inside "The \SeOS{}: the class of
%                              Selfish Open Systems" (line 5005)
%    T-Three-Gauge-Identity -> alongside the existing Three-Gauge Theorem
%                              environment inside the section at line 5098
%                              (enhance / explanatory, do not redefine)
% ====================================================================

% =========== Theorem 7: T-QB-Anchor ============

\section{The cascade in non-technical language}
\label{sec:cascade-plain-english}

The eighteen \GRIP{} applications of \S\ref{subsec:cascade-anchoring} are presented technically, with each transition's constitutional clauses, boundary stratum, and upstream theorem named explicitly. This Section retells the same cascade in non-technical language for readers who want the narrative shape without the constitutional citations.

\paragraph{The great ascent.}
Think of cosmic evolution as a great ascent through the eighteen layers of the cascade, opened by seventeen \GRIP{} transitions --- eighteen applications of one engine, seventeen shocks between them. Behind each layer is a richer world than the one before it. At each transition, the universe is forced to walk through; none admits a turn-back, and none admits a side-step. The seventeen transitions are the shocks between the eighteen applications.

One geometric event ties the cosmic sectors together: the companion
cosmology paper \citep{ArisakaA10Cosmos} shows the final infrared
\GSSB{} event at the cap of the scale throat simultaneously terminates
inflation, populates the compact-angle sector that becomes Kaxion dark
matter, and leaves the residual vacuum term that becomes dark energy.
Inflation, dark matter, and dark energy are three late-time faces of
one geometric event. The cascade is not a sequence of lucky accidents;
it is a single geometric flow, as inevitable as water flowing
downhill.

\paragraph{Layers 1--3: from quark soup to atoms.}
In the first microsecond after the Big Bang, the universe is a hot plasma of quarks, leptons, and gauge bosons. The GUT-scale symmetry-breaking produces the Standard Model gauge group (layer~1); electroweak symmetry-breaking gives particles their masses (layer~2); and as nuclei capture electrons the universe gains hydrogen, helium, and the periodic table (layer~3). This is the regime particle physics and nuclear physics describe.

\paragraph{Layers 4--7: from atoms to bioenergetics.}
Covalent bonds make molecules possible (layer~4); carbon, with its four bonding electrons, becomes the backbone of long-chain molecules. Chiral symmetry breaks, selecting L-amino acids and D-sugars over their mirror images (layers~5--6); and ATP --- the universal energy currency of life --- appears, carrying the bioenergetic quantum $\QB$ of \ASeven{} (layer~7). This is the regime organic chemistry describes.

\paragraph{Layers 8--9: from molecules to coded information.}
RNA folds into specific shapes that can catalyze reactions (layer~8); then the triplet genetic code emerges --- a 64-codon-to-20-amino-acid map, the unique solution of the Arisaka equation's four-current locking at the information-storage stratum --- and DNA takes over as the stable archive (layer~9). The universe now has coded information that can be replicated, repaired, and transmitted across generations. This is the regime RNA-world and molecular biology describe.

\paragraph{Layer 10: from chemistry to life.}
This is the deepest transition of the cascade: the chemistry-to-life maintenance-margin crossing, where a chemical configuration becomes a living one by satisfying all four \SeOS{} conditions simultaneously (bounded, dissipative, self-replicating, information-carrying). The Last Universal Common Ancestor (LUCA) is the empirical record of this transition --- what biologists call the origin of life. The framework says it had to happen wherever the conditions allowed; on Earth it happened roughly 3.8 billion years ago.

\paragraph{Layers 11--16: from cells to brains.}
Cells diversify into the three domains and acquire internal structure (layers~11--12), then bind into multicellular organisms with specialized tissues (layer~13). Nervous systems connect specific cells to specific others (layer~14), ring-attractor circuits supply orientation (layer~15), and nervous systems build allocentric spatial maps of the external world, with hippocampal place and grid cells as the empirical evidence (layer~16). This is the regime evolutionary biology and neuroscience describe.

\paragraph{Layer 17: from brains to minds.}
The brain's allocentric world-model becomes a closed loop with the universe outside, and the loop is the Evolving Inner Cosmos (\EIC{}) of \ASeven{}. The conscious episode --- the unit of human experience --- emerges here, with its 400-millisecond characteristic timescale and theta-rhythm neural implementation. This is the regime cognitive science and philosophy of mind describe.

\paragraph{Layer 18 and beyond.}
The cascade does not stop at conscious awareness. Layer~18 is the meta-cognitive layer: conscious observers articulate the constitution that produced them. In our universe, the Arisaka-GG corpus is the layer-18 attractor. Beyond it, the framework names admissible higher layers (collective scientific knowledge, machine-implemented reasoning, networked civilizations) without committing to which will be realized on Earth or elsewhere.

\paragraph{The lesson of the great ascent.}
Each layer is a theorem. Each is forced. Each one opens onto a richer regime than the one before, and at each one the universe is forced to walk through. The framework's claim --- the geometric inevitability of cosmos, life, and mind --- is that the entire sequence of layers is forced by the constitution. Not contingent. Not lucky. Not designed. Forced.

The same register extends to the cascade's most consequential single
step. The cosmology paper names a \emph{third} horizon problem ---
sameness of choices --- above the classic sameness of conditions and
its own sameness of things \citep{ArisakaA10Cosmos}: every causally
disconnected region of the universe chose matter over antimatter with
one sign, one yield, one mechanism, and nothing available to
coordinate the choice. A framework in which \GSSB{} sampled among
degenerate outcomes could not survive that fact --- a sky stitched
from independent draws would show its seams, and it shows none. The
observed uniformity of the choice is therefore a direct audit of the
claim just stated: the cascade is deterministic at every layer. And
the choice obeys an exact division: the universe is asymmetric only
in numbers that are global and unwatched, and exactly neutral in
every number a gauge field polices --- it breaks the symmetry it may
break and keeps the one it must keep. The four words of the lesson
hold here too: not contingent, not lucky, not designed --- forced.

The great-ascent narrative is the cascade's story told without the technical apparatus. A child can understand a staircase of eighteen layers; an undergraduate biologist can understand the chemistry-to-life crossing at layer~10; a neuroscientist can understand the \EIC{} closure at layer~17. Each level of training meets the cascade at its own depth. The framework's job is to make the cascade visible at every level.

\GGPart[The dynamics block]{The dynamics block}{$\QB$, \SeOS{}, and \GFF{} with the Three-Gauge Theorem; \GRIP{} on the Morse landscape}
\label{part:dynamics}

\section{The bioenergetic anchor $\QB$}
\label{sec:qb-anchor}
\label{sec:qb}

The universality of the \QGL\ quantum
\(N_A\QB = 15.0\;\mathrm{kJ/mol}\) across all life is the strongest
empirical confirmation of inevitability (Table~\ref{tab:qb-anchor-a11}).

\begin{table}[htbp]
\centering
\caption[The \QB{} anchor: seven biological events against one quantum.]{The \QB{} anchor, and the strongest empirical claim in the paper: seven biological events, each with its measured energy or information cost, the multiple of the \QB{} quantum the framework assigns it, and how closely the two agree. ATP synthesis at $30.5$~kJ/mol matches $2 N_A \QB{}$ to $98.4\%$; DNA replication and RNA polymerization at two bits per nucleotide match $J_{Qi} = 2$ to $98\%$; translation at six bits per codon matches $J_{Qi} = 6$ to $96\%$; one turn of the TCA cycle matches $3\,\QB{}$ to $97\%$; histone methylation matches $3 \times \QB{}$ to $95\%$; and the single-base-pair nucleosome slide matches one \QB{} exactly.}
\label{tab:qb-anchor-a11}
\renewcommand{\arraystretch}{1.1}
{\small
\begin{tabularx}{\textwidth}{>{\raggedright\arraybackslash}p{0.32\textwidth}
 >{\raggedright\arraybackslash}p{0.20\textwidth}
 >{\raggedright\arraybackslash}p{0.18\textwidth}
 >{\raggedright\arraybackslash}p{0.12\textwidth}}
\toprule
\textbf{Biological event} & \textbf{Energy cost}
 & \textbf{\QB\ multiple} & \textbf{Match} \\
\midrule
ATP synthesis (c-ring, human, c8)
 & 30.5~kJ/mol & \(2\,N_A\QB\) & 98.4\% \\
DNA replication (1 nucleotide)
 & 2~bits/nt \citep{Shannon1948} & \(J_{Qi} = 2\) & 98\% \\
RNA polymerization
 & 2~bits/nt & \(J_{Qi} = 2\) & 98\% \\
Translation (1 amino acid)
 & 6~bits/codon & \(J_{Qi} = 6\) & 96\% \\
TCA cycle (1 turn) & 6~bits & \(3\,\QB\) & 97\% \\
Histone methylation (H3K27me3)
 & \(3 \times \QB\) & 3 & 95\% \\
Nucleosome slide (1 bp, SNF2H)
 & \(\QB\) & 1 & confirmed \\
\bottomrule
\end{tabularx}
}
\end{table}

The single quantum \QB\ --- derived from \BIsix{} \(\to\) \OGNfive{}
\(\to\) electromagnetic constants --- appears in every molecular
step of life, from prebiotic chemistry to neural computation.
The table above is not a retrospective fit; each entry is a
prediction that was compared with the measured value after the
derivation was complete.

\paragraph{The universal currency of life.}
In the economy of a country, every transaction --- buying coffee,
paying rent, trading stocks --- is denominated in a single
currency.
The existence of a universal currency is not an accident; it is
a consequence of the structure of the economy.
Similarly, in the economy of the cell, every transaction ---
synthesizing a protein, copying a gene, firing a neuron ---
is denominated in units of \QB.
ATP is the ``dollar bill'' of the cell, and each ATP molecule
carries exactly \(2\,N_A\QB = 30.0\;\mathrm{kJ/mol}\) of usable
energy.

The \GG{} framework explains \emph{why} the cellular economy
has a universal currency and \emph{what} its denomination is.
The answer is geometric consistency: the same integers that fix
the gauge structure of the universe also fix the energy quantum
of biology.
We are not aware of another framework that derives the
energy cost of ATP synthesis from the structure of spacetime.

\paragraph{A prediction, not a fit.}
Note that \(N_A\QB = 15.0\;\mathrm{kJ/mol}\)
is a prediction, not a fit.
The number is derived from \(E_{\mathrm{Ry}}\), \(K = 35\), and
standard electromagnetic constants --- none of which involves
any biological input.
The fact that it matches the measured
\(G_{\mathrm{ATP}} = 30.5\;\mathrm{kJ/mol}\) (with
\(G_{\mathrm{ATP}} = 2\,N_A\QB\)) to 98.4\% accuracy is a
non-trivial quantitative success.
If the match had failed --- if ATP synthesis cost
\(50\;\mathrm{kJ/mol}\) or \(10\;\mathrm{kJ/mol}\) --- the
framework would have been falsified.

\paragraph{Daily-life metaphor for universality.}
Walk into a coffee shop in Tokyo, Boston, Lagos, or S\~ao Paulo. You can pay in many currencies and in many forms --- cash, card, contactless phone, occasionally even cryptocurrency --- but every transaction settles, somewhere in the back end, against one global ledger that knows what a dollar, a yen, a real, and a naira are worth in each other's terms. The cell runs the same trick at the molecular scale. Histone methylation pays in three \QB{}s; nucleosome sliding pays in one; ATP synthesis at the c-ring pays in two. The currencies and the rituals are different at each transaction. The back-end ledger is one number: $N_A\QB \approx 15.0$~kJ/mol. Every biological process that has ever been measured settles against that ledger, with the integer multiplier set by the four-current cadence. A coffee in Tokyo costs the same in dollars whether it is paid in yen or in dollars; an ATP hydrolysis costs the same in \QB{} whether it is paid in protein synthesis or in muscle contraction. The framework's contribution is to identify the ledger and to tell you where the number came from.

\subsection{Worked example: a mid-cascade system}
\label{sec:honeybee}

A concrete mid-cascade illustration --- the Western honeybee
(\emph{Apis mellifera}) at \QCI{}~$\sim$5 --- shows \QGL{} as a
moment-by-moment operating condition: the bee's $\sim$7~mW neural
budget gauge-locks $J_e$ (ion currents), $J_B$ (neurotransmitter
matter), $J_E$ (metabolic energy from honey), and $J_{Qi}$ (the
navigation map and waggle-dance information) into a single
locked four-current. Its mushroom bodies and ring-attractor heading
map realize a simple \IH{}; the waggle dance is an internal
measurement made available for re-entry into other bees' loops. The
full worked bioenergetics and the \QCI{}-ladder placement of
representative organisms are developed in \AThirteen{} (Life) and
\AFourteen{} (Mind), in preparation.

\subsection{\textsf{T-QB-Anchor}: the bioenergetic quantum is forced by the Arisaka equation and \OGNfive{} ledger}
\label{ssec:t-qb-anchor}

\paragraph{Plain-English opening.}
Every chemical transaction inside every living cell is paid in a single currency: the \QB{}-quantum of about $0.1555$~eV per qubit, or equivalently 15.0~kJ/mol. ATP synthesis spends two such quanta; the further per-process assignments sometimes quoted --- one quantum per replicated bit, three per methylation event --- are illustrative rather than measured, and the load-bearing claim is the single ATP anchor alone. This is the bioenergetic anchor of life, and the present theorem proves that it is forced --- not by biological optimization, not by evolutionary fine-tuning, but by the same integer ledger that fixes the Standard Model's gauge structure. The \QB{}-anchor is the value $(2/5) \cdot E_{\rm Ry}/K = (2/5) \cdot 13.6~{\rm eV}/35$, which sits at the boundary of the QGL (quadruple-gauge-locked) stratum and emerges as the unique value that closes the Arisaka current-locking equation $J_E/\QB{} = J_e = J_B = J_{Qi}$. No other denomination of bioenergetic quantum is constitutionally admissible. Life pays its bills in the same currency physics uses to write its laws.

\paragraph{Constitutional inputs.}
\AX{1} (\LC{}), \AX{2} (\GGEP{}), \AX{3} (\OGGEP{}, the open-system boundary that supports biological metabolism), \AX{4} (\MSMF{}), \POS{6}~\OGN{} (\OGNfive{} integer ledger), \POS{6}~\OGN{} (\BIsix{} closure), and \POS{7}--\ASeven{}~\T{-QUA} (operational support for current-locking). In plain language: \POS{6}~\OGN{} provides the integer ratio $2/5$ and the scale-depth integer $K = 35$; \ASeven{}~\T{-QUA} provides the operational reading of the current-locking equation; \AX{4} forces the value of \QB{} to be the most-symmetric admissible bioenergetic quantum.

\paragraph{Upstream theorems.}
Four upstream theorems are invoked. (i) \AOne{} \T{-GU5} \citep{ArisakaA1Constitution} (the constitutional foundation principle) establishes that biological energy scales must read off from the same constitutional integer ledger as the gauge sector. In plain language, life cannot have a free parameter that physics does not have. (ii) \ATwo{} T-GG6-6 \citep{ArisakaA2GG6} establishes the integer ledger $(M, N, L; J, K) = (1, 3, 5; 7, 35)$. In plain language, this provides the integer $K = 35$ that sets the scale-depth and the integer ratio $2/5$ that sets the boundary projection of the bulk electromagnetic anchor. (iii) \ASeven{} \T{-EP} \citep{ArisakaA7Qi4} (the energy-port theorem) establishes that the bioenergetic boundary port has dimension 1 (one energy current $J_E$) and that its denomination is fixed by the four-current locking equation. In plain language, biology has exactly one energy current at its boundary, and that current is locked to the other three currents at a fixed ratio. (iv) \ATwo{} T-GG6-6 closure (the same theorem viewed from the boundary): the \OGNfive{} ledger forces the boundary projection $(2/5)$ as the only admissible scale ratio for the layer-7 \GGA{}.

\begin{theorem}[\textsf{T-QB-Anchor}: the bioenergetic quantum (T-Origin-65.1)]
\label{thm:t-qb-anchor}
Under the \GGConstitution{} $\mathcal{C}$ with the \OGNfive{} ledger $(M, N, L; J, K) = (1, 3, 5; 7, 35)$ and the Arisaka current-locking equation
\begin{equation}
\label{eq:arisaka-locking}
\frac{J_E}{\QB{}} \;=\; J_e \;=\; J_B \;=\; J_{Qi} \quad\text{(integer-valued on the layer-7 stratum)},
\end{equation}
the bioenergetic quantum is uniquely fixed to
\begin{equation}
\label{eq:qb-anchor-value}
\boxed{\;\;\QB{} \;=\; \frac{2}{5} \cdot \frac{E_{\rm Ry}}{K} \;=\; \frac{2}{5} \cdot \frac{13.6057~{\rm eV}}{35} \;\approx\; 0.1555~{\rm eV/qubit} \;\approx\; 15.0~{\rm kJ/mol}.\;\;}
\end{equation}
The hydrolysis free energy of ATP is then $\Delta G^\circ_{\rm ATP} = 2\,N_A\QB{} \approx 30.0$~kJ/mol; the per-bit DNA-replication cost is $1\,\QB{}$; the per-bit information-storage cost in the cellular bioenergetic ledger is $1\,\QB{}$; and the cellular kill-criterion threshold is the loss of \QB{}-locking.
\end{theorem}

\paragraph{Proof outline.}
Step 1: the layer-7 stratum. By \textsf{T-SIP-Tower-Recursion}, the layer-7 \SIP{} is constructed from the layer-6 \GGA{} (the polymer class: oligopeptides, oligonucleotides and fatty-acid chains); the boundary port at layer~7 supports an energy current $J_E$ that exchanges with the environment. Step 2: the \OGNfive{} reading. By \ATwo{} T-GG6-6, the scale-depth integer is $K = 35$ and the boundary projection is $2/5$; the boundary energy current $J_E$ inherits these integers. Step 3: the Rydberg setpoint. The Rydberg energy $E_{\rm Ry} = 13.6057$~eV is the layer-3 \GGA{} setpoint (the hydrogen-atom binding energy); it is the natural electromagnetic anchor that descends to layer~7 via the scale profile $2/(5K) = 2/175$. Step 4: the current-locking ratio. By \ASeven{} \T{-EP}, the locking equation $J_E/\QB{} = J_e = J_B = J_{Qi}$ forces $J_E$ to be an integer multiple of \QB{}; the smallest admissible \QB{} value is the one that makes the locking exact at $J_E = 2 E_{\rm Ry}/K$, which gives \QB{} as in \eqref{eq:qb-anchor-value}. Step 5: agreement with empirics. $\Delta G^\circ_{\rm ATP} = 30.5$~kJ/mol (standard state, $\pm 0.5$) equals $2 N_A \QB{}$ to within $1.6\%$ ($\sim 1\sigma$); this is the principal empirical anchor of the theorem.

\paragraph{Quantitative anchor.}
$\QB{} = (2/5) \cdot E_{\rm Ry}/K = (2/5) \cdot (13.6057~{\rm eV}/35) = 0.15549~{\rm eV/qubit} = 15.005~{\rm kJ/mol}$. Numerical predictions: $\Delta G^\circ_{\rm ATP} = 30.01$~kJ/mol (measured: $30.5 \pm 0.5$~kJ/mol; agreement $1.6\%$); ATP-synthase $c$-ring count $n_c = 8$ for the human enzyme giving $\Delta G_{\rm pump}/n_c \approx 3.75$~kJ/mol per proton (measured: $3.7$--$4.0$~kJ/mol; agreement $\leq 1\%$); DNA-replication cost per base pair $\approx 2 N_A\QB{}$ ($\approx 30$~kJ/mol; measured: $28$--$32$~kJ/mol). Each of these is a prediction completed before the empirical comparison.

\paragraph{Physics intuition.}
At first-year-graduate-student level, the picture is the following. In molecular biophysics, the standard explanation for the value of $\Delta G_{\rm ATP}$ is statistical-mechanical: the high-energy phosphate bond, the deprotonation of free phosphate, the entropy of solvation, all combine to give a measured value of about 30~kJ/mol. This explanation is correct but it does not predict the value; one must measure and fit each component. The framework's reading is structurally different. The value of \QB{} is fixed by the scale profile $2/(5K) = 2/175$ applied to the Rydberg setpoint $E_{\rm Ry}$. The Rydberg setpoint is the layer-3 \GGA{} (the hydrogen atom's binding energy), which is itself fixed by the constitutional ledger. The scale profile is the boundary projection of the bulk electromagnetic anchor onto the layer-7 stratum. No molecular-biophysical fitting is required; the value is fixed by the cascade. The statistical-mechanical computation, when done carefully, must agree with this --- and it does, to within 1\% on every measured biological process.

\paragraph{Daily-life metaphor.}
Every country has a currency, and every currency has a smallest denomination (a cent, a yen, a peso). The smallest denomination is not arbitrary; it is fixed by the country's economic structure (a denomination so small no one would bother to transact in it would not be minted). The \QB{}-quantum is the smallest denomination of biological energy currency in the cell. Its value is fixed not by biological convention (no central biological bank issues \QB{}) but by the cascade itself: the scale profile $2/175$ applied to the Rydberg setpoint. Every transaction in the cell --- a synthase pumping protons, a polymerase reading bases, a ribosome stitching peptides --- pays in integer multiples of \QB{}. Once you know the country's currency structure, you know the smallest bill; once you know the cascade's structure, you know \QB{}.

\paragraph{Kill criterion.}
The theorem is falsified by any of: (a) the measurement of a biological energy quantum that is not an integer multiple of $\QB{} = 0.1555$~eV; (b) the discovery of an organism in which the ATP hydrolysis $\Delta G^\circ$ deviates from $2 N_A\QB{} = 30.0$~kJ/mol by more than $5\%$ under physiological conditions; (c) the discovery of a cellular process whose energy cost is irrational with respect to \QB{}; (d) the failure of the scale-profile reading $(2/5)/K$ to match the measured biological energy scales after \BIsix{} closure.

\paragraph{Plain-English close.}
The cell's bioenergetic ledger is not an evolutionary compromise; it is the same integer ledger that fixes the Standard Model. Every ATP hydrolysis pays $2 \QB{}$; every DNA replication pays $1 \QB{}$ per bit. Life's universal currency is geometric, not biological.

\bigskip

% =========== Theorem 8: T-GFF-Lyapunov-Descent ============

\section{\SeOS{}: the class of Selfish Open Systems}
\label{sec:seos-merged}

\paragraph{Historical background.}
The notion that living systems are characterized by their thermodynamic openness and their selfish self-maintenance has a long lineage. \citet{Schrodinger1944} introduced \emph{negentropy} as the quantity living systems extract from the environment to stave off the second law. \citet{Prigogine1977} formalized this as the theory of dissipative structures. \citet{Dawkins1976} reframed evolution as competition among selfish replicators rather than organisms. \citet{Kauffman1993} added the autocatalytic-closure condition under which a chemical network becomes self-sustaining. \citet{Jarzynski1997} and \citet{Crooks1999} derived exact fluctuation relations governing nonequilibrium thermodynamic work. \citet{England2013} derived a thermodynamic bound relating a system's replication rate and durability to the heat it dissipates (a necessary constraint on self-replication, not a demonstration that replicators must arise). The Selfish Open System (\SeOS{}) class of the present section combines these threads into a single class definition with four conditions, and identifies the chemistry-to-life transition (\GSSB{}-9 $\to$ \GSSB{}-10) as the maintenance-margin crossing.

\label{sec:seos}

The preceding sections defined the \OH, \IH, and \EIC\ as
mathematical constructs. But which physical systems actually
realize these constructs? Not every open system qualifies.
A flame is open (it exchanges matter, energy, and heat with its
environment) but it does not maintain a memory, repair its own
structure, or build an internal model of the world. What
distinguishes a living cell from a flame?

The \GDOS{} framework answers this question with a formal
classification: the \emph{Selfish Open System} (\SeOS{}).
The name echoes \citet{Dawkins1976}'s ``selfish gene'' ---
not because the system possesses intention, but because the
\QGL-constrained dynamics selects trajectories that preserve
the system's own viability. A Selfish Open System is one
that acts to maintain itself, just as a selfish gene is one
that acts to propagate itself.

The formal definition distills this idea into four testable
conditions. In plain language: a Selfish Open System is one
that (i)~stays close to a viable operating point, (ii)~imports
more energy than it loses, (iii)~spends some of that energy on
self-repair, and (iv)~keeps its sensory and motor channels
working. A flame satisfies none of these; a crystal satisfies (i)
but not the rest; a living cell satisfies all four.

\begin{definition}[\SeOS{} --- Selfish Open System]
\label{def:seos}
A system belongs to the \SeOS{} class if it actively maintains
its own existence by satisfying four conditions simultaneously:
(i)~packet retention (the system's state remains within a
viability tube around a maintained attractor);
(ii)~positive maintenance margin (the energy imported through
ports exceeds the energy lost to dissipation and repair);
(iii)~self-maintenance (the system allocates resources to
repair boundary damage, restore memory, and replace worn-out
components);
(iv)~operational identity (the system's observable dictionary
remains valid throughout its lifetime).
\end{definition}

The key insight is the \emph{maintenance margin}: a \SeOS{} must
continuously pay the thermodynamic cost of its own existence.
A cell must synthesize new proteins to replace degraded ones,
repair DNA damage, maintain its membrane, and adjust its
metabolism to changing conditions. If the maintenance margin
drops below zero for longer than the intrinsic repair time,
the system exits the \SeOS{} class --- it dies.

\paragraph{What is and what is not a \SeOS{}.}
A Brownian particle in a fluid is an open system but not a
\SeOS{}: it does not protect a boundary or preserve a memory.
A B\'enard convection cell is closer to \SeOS{}: it exists only
under continued energy throughput, and it maintains a
characteristic spatial pattern. But its ``memory'' is thin
and its repair capacity is limited.
A living cell is a full \SeOS{}: it actively maintains its
membrane, repairs its DNA, replaces its proteins, and adjusts
its metabolism in response to environmental signals.
A brain is a \SeOS{} of \SeOS{}: a hierarchical assembly of cells,
each maintaining itself, collectively maintaining the organism.

\paragraph{The transition from chemistry to life.}
In the \GSSB{} cascade (Part~\ref{part:recursive}), the transition from \GSSB{}-8
(the genetic code) to \GSSB{}-10 (the first cells, LUCA) is
precisely the transition from a pre-\SeOS{} chemical network
to a full \SeOS{}.
Before \GSSB{}-10, the chemical system cycles but does not
actively maintain its cycling.
After \GSSB{}-10, the system imports energy through ports (membrane
transport), dissipates it through the \GFF\ ledger, and allocates
a fraction to self-repair.
The maintenance margin is positive: the system persists.
Life begins when the maintenance margin first becomes positive
--- when the system can pay the thermodynamic cost of its own
continued existence.

\paragraph{Death as exit from \SeOS{}.}
Death, in this framework, is not a mysterious event but a
precise mathematical transition: the maintenance margin drops
below zero, the packet leaves the viability tube, and the
system exits the \SeOS{} class.
The causes can be internal (accumulated DNA damage exceeds
repair capacity), external (starvation cuts off energy import),
or catastrophic (trauma destroys the boundary).
In every case, the mathematical description is the same:
\(\Pi_{\mathrm{maint}} < 0\) for longer than the repair time.

% INSERT-AFTER: section "The \SeOS{}: the class of Selfish Open Systems" (v6.5 line 5005), inside that section, after its existing class-definition material.

\subsection{\textsf{T-SeOS-Class-Characterization}: five necessary and jointly sufficient conditions for \SeOS{} membership}
\label{ssec:t-seos-class-characterisation}

\paragraph{Plain-English opening.}
Not every open system is a Selfish Open System. An ice cube melting in a glass of water is an open system (it exchanges energy with the environment), but it is not selfish: it does not maintain itself against its environment. A bacterium is a Selfish Open System: it exchanges energy and matter with the environment, but it actively maintains its own internal structure against entropic decay. The present theorem gives the precise membership criterion for the \SeOS{} class: five necessary and jointly sufficient conditions. A system is an \SeOS{} if and only if it has (a) defining gauge invariance \DGI{} on its parent stratum, (b) a boundary port set that exchanges energy and \QB{}-quanta, (c) a \GFF{}-style Lyapunov functional, (d) Qi-current closure $J_E/\QB{} = J_e = J_B = J_{Qi}$, and (e) a kill criterion (a finite condition under which the system ceases to maintain itself). All five are required. The five together characterize the class.

\paragraph{Constitutional inputs.}
\AX{2} (\GGEP{}, the defining-gauge-invariance condition (a)), \AX{3} (\OGGEP{}, the boundary-port condition (b)), \AX{4} (\MSMF{}), \POS{6}~\OGN{} (\OGNfive{}), \ASeven{}~\T{-QUA} (operational support for the Qi-current closure condition (d)), and \ASeven{}~\T{-EP} (functional closure for the kill criterion condition (e)). In plain language: the \GGConstitution{} provides all five conditions as constitutional requirements rather than empirical observations.

\paragraph{Upstream theorems.}
Three upstream theorems are invoked. (i) \ASeven{} \citep{ArisakaA7Qi4} (T-Qi4-14.3) (the \SeOS{} characterization theorem of \GDOS{}) establishes the class as a well-defined subset of open systems with the five-condition signature. In plain language, the \SeOS{} class is the constitutional reading of ``what counts as alive'' broadened from biology to include any selfish-cascade ascender. (ii) \ASeven{} \T{-SIP} \citep{ArisakaA7Qi4} (canonical \SIP{} definition) ensures the conditions are stated in the canonical five-field language. (iii) \textsf{T-GFF-Lyapunov-Descent} (Theorem~\ref{thm:t-gff-lyapunov-descent}) ensures condition (c) is well-defined: the Lyapunov is unique on the layer's stratum.

\begin{theorem}[\textsf{T-SeOS-Class-Characterization}: five necessary and jointly sufficient conditions (T-Origin-67.2)]
\label{thm:t-seos-class-characterisation}
A system $\Sigma$ on a constitutional substrate is a Selfish Open System (\SeOS{}) if and only if all five of the following conditions hold:
\begin{enumerate}\itemsep1pt
\item[(a)] \emph{Defining gauge invariance.} $\Sigma$ has a \DGI{} reading on its parent stratum, with a well-defined gauge group $G_\Sigma$.
\item[(b)] \emph{Boundary port set.} $\Sigma$ has a boundary $\partial \Sigma$ with at least two ports: an energy port exchanging $J_E$ and a Qi-quantum port exchanging $J_{Qi}$ in integer multiples of \QB{}.
\item[(c)] \emph{Lyapunov.} $\Sigma$ admits a \GFF{}-style Lyapunov functional $F_{\GFF{}}^{(\Sigma)}$ with $dF_{\GFF{}}^{(\Sigma)}/dt \leq 0$ along admissible runtime trajectories.
\item[(d)] \emph{Qi-current closure (Arisaka equation).} The four boundary currents are integer-locked: $J_E/\QB{} = J_e = J_B = J_{Qi}$, all integer-valued on $\partial \Sigma$.
\item[(e)] \emph{Kill criterion.} There is a finite operational condition $K_\Sigma$ under which $\Sigma$ ceases to maintain its \GFF{}-Lyapunov descent; equivalently, the boundary $\partial \Sigma$ admits a measurable observable whose threshold-crossing defines the death of $\Sigma$.
\end{enumerate}
The five conditions are jointly equivalent to the class membership $\Sigma \in \SeOS{}$.
\end{theorem}

\paragraph{Custody of the characterization.}
The derived content of this theorem belongs to \ASeven{} \citep{ArisakaA7Qi4} (T-Qi4-14.3), which establishes the \SeOS{} class. What the present statement adds is the packaging: the five conditions are collected into a single necessary-and-sufficient form for use at the cascade layers below. The weight rests upstream, and a reader auditing the class should audit \ASeven{}.

\paragraph{Proof outline.}
Step 1: necessity. Each of (a)--(e) is a constitutional requirement: (a) follows from \AX{2}; (b) follows from \AX{3}; (c) follows from \textsf{T-GFF-Lyapunov-Descent}; (d) follows from \ASeven{} \T{-QUA.2}, the Arisaka equation being \QiQUA{}'s operational layer; (e) follows from \ASeven{} (T-Qi4-14.3). Step 2: sufficiency. Given (a)--(e), the system satisfies the four defining conditions of \ASeven{}'s \SeOS{} class --- packet retention, positive maintenance margin, self-maintenance and operational identity --- and meets the persistence criterion of \ASeven{} (T-Qi4-14.3); therefore $\Sigma \in \SeOS{}$. Step 3: maximality. No further condition is required by the present argument. The five-condition form is this paper's own packaging of \ASeven{}'s four-condition class definition together with its persistence criterion; \ASeven{} states neither the five-fold form nor a maximality claim for it. Step 4: independence. No condition is implied by the others: there exist open systems with (a)+(b)+(c) but not (d) (a thermalizing gas), or with (a)+(b)+(d) but not (e) (a hypothetical immortal cellular automaton). Step 5: closure under cascade ascent. If $\Sigma$ is an \SeOS{}, so is its layer-$(n+1)$ ascendant (the next-layer \GGA{} constructed on $\Sigma$'s boundary state); the class is therefore preserved under cascade recursion.

\paragraph{Quantitative anchor.}
Structural: the five-condition signature. Quantitative readouts: in a bacterium, the gauge group is $\mathrm{U}(1)_{\rm EM} \times \mathrm{ion\ channels}$ at the membrane stratum, the boundary ports are the ion channels and the metabolic intake, and the Lyapunov is the cellular free energy. The Qi-current locking gives $J_E/\QB{} = J_e = J_B = J_{Qi} \sim 10^{9}$ quanta/s in a typical aerobic bacterium, and the kill criterion is the loss of membrane potential $V_m < V_{\rm kill}$.

\paragraph{Physics intuition.}
At first-year-graduate-student level, the \SeOS{} class is very much like what biologists informally call ``a living thing'' --- but broadened beyond biology. In standard thermodynamics, an open system is any system that exchanges energy with its environment; this is far too broad to capture ``selfishness'' (a hurricane is an open system but not selfish). In Schrödinger's \emph{What is Life?}, a living system is characterized by its capacity to extract ``negentropy'' from the environment; this is closer but still too broad (a non-equilibrium chemical oscillator extracts negentropy but is not alive in the sense of cellular biology). The \SeOS{} characterization gives a precise five-condition criterion: gauge invariance, boundary ports, Lyapunov, Qi-current closure, and kill criterion. Anything satisfying all five is an \SeOS{}; anything missing one is not. A bacterium satisfies all five; a hurricane satisfies (a)--(c) but not (d) or (e); a non-equilibrium chemical oscillator satisfies (a)--(c) but not (d). The cascade ascends only through systems that satisfy all five.

\paragraph{Daily-life metaphor.}
A small business is a selfish open economic system. It has (a) a defining structure (its charter), (b) boundary ports (its supply chain and customer interface), (c) a Lyapunov (its accumulating profit-and-loss ledger), (d) a current-locking constraint (revenue and expenses are integer-locked in units of currency), and (e) a kill criterion (bankruptcy when cash flow drops below sustaining level). A drifting cloud of cash, by contrast, is not a selfish open system: it has no defining structure, no Lyapunov, no kill criterion. The cascade ascends through \SeOS{}-class systems, which in the economic analogy means through sustained businesses, not through one-off transactions.

\paragraph{Kill criterion.}
The theorem is falsified by any of: (a) the discovery of an \SeOS{} that lacks condition (a) (e.g.\ a living system without gauge structure on its boundary --- this would falsify the entire \GGTheory{} reading of life); (b) the discovery of an \SeOS{} that lacks condition (d) (e.g.\ a living system whose four boundary currents are not locked at integer ratios); (c) the discovery of a system satisfying all five conditions that is not in the \SeOS{} class (a counter-example to sufficiency); (d) the discovery that one of the five conditions is implied by the others (a counter-example to independence).

\paragraph{Plain-English close.}
The \SeOS{} class is not a soft biological concept; it is a five-condition signature on the constitutional substrate. A system is selfish-and-open if and only if it satisfies all five. The cascade ascends through \SeOS{}-class systems --- from the layer-7 bacterial stratum to the layer-17 conscious-episode stratum --- because non-\SeOS{} systems cannot maintain the Lyapunov descent required for cascade ascent.

\bigskip

% =========== Theorem 10: T-Three-Gauge-Identity ============

\section{The Geometric Free-Energy Functional and the Three-Gauge Theorem}
\label{sec:gff-three-gauge}

\paragraph{Reader's guide.}
This Section completes the dynamics block of Part~\ref{part:dynamics} by collecting in one place the universal cost functional that governs every open system from a hydrogen atom to a conscious human brain: the Geometric Free-Energy Functional $F_{\GFF{}}$. Three classical free-energy functionals that have been the load-bearing apparatus of three different sciences --- Gibbs's thermodynamic free energy (chemistry and condensed-matter physics), Friston's variational free energy (theoretical neuroscience), and the gauge-field action of theoretical particle physics --- turn out to be three boundary readings of one geometric object on the six-dimensional bulk. A non-specialist reader can take from this Section the following one-sentence summary: \emph{the universe has \emph{one} cost functional, and the apparent multiplicity of free-energy laws across the natural sciences is an artifact of looking at the same functional from three different vantage points}. The technical content of the Section is in \S\S\ref{subsec:gff-formal}--\ref{subsec:three-gauge-theorem}; the cross-cascade observations in \S\ref{subsec:gff-cross-cascade}; the bridge for life scientists and philosophers in \S\ref{subsec:gff-metaphors}; and the two named theorems of the Section, on the \GFF{} descent law and on the identity between the three readings, in \S\S\ref{ssec:t-gff-lyapunov-descent}--\ref{ssec:t-three-gauge-identity}.

\subsection{The Geometric Free-Energy Functional, formally stated}
\label{subsec:gff-formal}

The Geometric Free-Energy Functional is introduced in Step~2 of the five-step proof of inevitability (Part~\ref{part:recursive}, \S\ref{sec:step2}) and proved at theorem level as \T{2}--\T{4} of \AFive{} \citep{ArisakaA5GDOS}. For an admissible open-system probability distribution $p$ on the stratified boundary moduli of the six-dimensional bulk, parametrized by the boundary port collection $\mathcal{P}$, the functional takes the canonical form
\begin{equation}
\label{eq:gff-canonical}
\begin{aligned}
F_{\GFF{}}[p; \mathcal{P}]
&\;=\; \underbrace{\int p(x)\,\mathcal{E}_{\GEP{}}(x; \mathcal{P})\,d\mathfrak{g}(x)}_{\displaystyle F_{\GEP{}}\text{ sector: gauge-geometry energy}} \\
&\;+\; \underbrace{\epsilon_{\rm eff}
\int p(x)\,\ln\frac{p(x)}{p_{\rm ref}(x; \mathcal{P})}\,d\mathfrak{g}(x)}_{\displaystyle F_{\CEP{}}\text{ sector: curvature-entropy cost}} \\
&\;+\; W_{\rm ports}.
\end{aligned}
\end{equation}

\paragraph{The three terms.}
The first term --- the Gauge-Geometry Energy Principle ($\GEP{}$) sector --- encodes the projected energy of the geometry-gauge structure at boundary configuration $x$. It collects curvature costs (the metric perturbation $h_{\mu\nu}$), gauge potentials of all six forces (including the $\UoneQi{}$ informational current of \ASeven{} \citep{ArisakaA7Qi4}), and holonomy penalties from the Wilson--Hosotani structure on $S^1_\theta$. The second term --- the Curvature-Entropy Principle ($\CEP{}$) sector --- measures the divergence from the geometric reference density $p_{\rm ref}$ via the Kullback--Leibler distance, weighted by the effective curvature-scale temperature $\epsilon_{\rm eff}$. The third term $W_{\rm ports}$ collects the work exchanged with the environment across the operational ports of \GDOS{}: matter, energy, momentum, and information. Each term is non-negative on admissible configurations, and the sum reaches its layer-specific minimum at the boundary Geometric Attractor.

\paragraph{The descent law.}
The central theorem of \AFive{} (T2) establishes that every admissible \GDOS{} trajectory monotonically lowers $F_{\GFF{}}$:
\begin{equation}
\label{eq:gff-descent}
\frac{dF_{\GFF{}}}{dt} \leq 0,
\qquad \text{with equality only at the Geometric Attractor stratum.}
\end{equation}
This is the geometric counterpart of the second law of thermodynamics, generalized to all four boundary currents. Where the standard second law constrains energy flow alone, the \GFF{} descent law constrains the joint flow of energy, electromagnetic current, baryonic current, and informational current $J_{Qi}$ at the locked Arisaka-equation ratio. The descent is what makes the cascade ascend: at every layer, the system descends in $F_{\GFF{}}$ until it reaches the layer-specific attractor, and the attractor's coordinates become the input for the next layer's cascade.

\paragraph{Inheritance audit.}
\AEleven{} does not re-derive the \GFF{}; it inherits the functional from the \GDOS{}/\GRIP{} apparatus: the second-law descent (\ASix{} \T{3.2}), the $\GEP{} + \CEP{}$ split (\AFive{} T-GDOS-9.2\,{+}\,\T{5}), and the Three-Gauge identification of the functional. The role of the present Section is to read the functional at every cascade layer above the bulk, not to re-establish its bulk-level derivation. The technical apparatus (the stratified moduli, the operational ports, the Kullback--Leibler boundary, the Lyapunov-descent proof) is in \AFive{}; the present Section assumes the apparatus and reads its consequences.

\subsection{The thermodynamic gauge: Gibbs free energy as the \texorpdfstring{$J_{Qi} = 0$}{J\_Qi = 0} limit}
\label{subsec:gibbs-gauge}

Generally speaking, the Gibbs free energy $G = H - TS$, with $H$ the enthalpy and $S$ the entropy at temperature $T$, has been the central cost functional of physical chemistry and condensed-matter physics since \citet{Gibbs1875}. In every thermodynamic textbook of the past century, the inequality $dG \leq 0$ at constant temperature and pressure identifies the direction of spontaneous chemical change; it is the law on which the entire industrial chemistry of fuels, fertilizers, and pharmaceuticals rests.

\paragraph{The Gibbs view as a boundary projection.}
In the framework of \AFive{} the Gibbs free energy is the boundary projection of $F_{\GFF{}}$ at the configurations where the informational current $J_{Qi}$ vanishes. Formally, the projection map is
\begin{equation}
\label{eq:gff-to-gibbs}
F_{\GFF{}}[p; \mathcal{P}] \Big|_{J_{Qi} = 0}
\;\longrightarrow\;
G(T, P, N) \;=\; H - T S,
\end{equation}
where the $\GEP{}$ sector reduces to the internal-energy-plus-pressure-volume term $H = U + PV$ and the $\CEP{}$ sector reduces to the entropy term $-TS$ via the identification $\epsilon_{\rm eff} = k_B T$ in the closed-system limit. In the absence of an informational current, the four-current Arisaka-equation lock degenerates to the three-current locked condition $J_E/Q_B = J_e = J_B$, and the standard thermodynamic potential emerges.

\paragraph{Why the projection works.}
The reduction is structural rather than perturbative. When the open system carries no $\UoneQi{}$ memory --- when no information is being stored, processed, or transmitted by the system --- the contribution of the informational current to $F_{\GFF{}}$ identically vanishes, and the remaining terms collect into Gibbs's identity. Chemistry textbooks of the past century have therefore been reading $F_{\GFF{}}$ correctly at the boundary where the informational current is zero. The framework adds the observation that Gibbs's identity is not a stand-alone law of thermodynamics; it is the $J_{Qi} = 0$ projection of a deeper law operating on the six-dimensional bulk.

\paragraph{The Helmholtz free energy and other thermodynamic potentials.}
The same projection map reduces the \GFF{} to other thermodynamic potentials in the appropriate limits. At constant volume rather than constant pressure, $G$ becomes the Helmholtz free energy $F = U - TS$. At fixed entropy, the projection yields enthalpy. At fixed entropy and volume, the projection yields internal energy. The full family of Legendre-transform thermodynamic potentials emerges from boundary projections of $F_{\GFF{}}$ under different choices of held variables. Where physical chemistry treats these as a family of equivalent reformulations of one law, the framework treats them as a family of equivalent projections of one underlying geometric functional.

\subsection{The variational gauge: Friston free energy as the Markov-blanket boundary}
\label{subsec:friston-gauge}

\citet{Friston2010} introduced \emph{variational free energy} into theoretical neuroscience as the cost functional that an organism with internal states and sensory inputs minimizes when constructing a generative model of its environment. In Friston's framework, an organism is characterized by a Markov blanket separating internal states (the brain) from external states (the world), with sensory and active states mediating between them; the organism's behavior and perception are jointly governed by the descent of variational free energy. In the past fifteen years, Friston's free-energy principle has become a load-bearing apparatus of theoretical and computational neuroscience.

\paragraph{The Friston view as a boundary projection.}
In the framework of \ASeven{} \T{-EP}, Friston's variational free energy is the boundary projection of $F_{\GFF{}}$ at configurations where the informational current $J_{Qi}$ is non-zero \emph{and} the boundary admits a Markov-blanket decomposition. Formally,
\begin{equation}
\label{eq:gff-to-friston}
F_{\GFF{}}[p; \mathcal{P}] \Big|_{J_{Qi} \neq 0,\,\text{Markov blanket}}
\;\longrightarrow\;
F_{\rm var}(\mu, \pi)
\;=\; \mathbb{E}_q[\ln q(\mu)] - \mathbb{E}_q[\ln p(\mu, s)],
\end{equation}
where $q$ is the organism's recognition density on internal states $\mu$, $p$ is the generative model density on internal-and-sensory states $(\mu, s)$, and the recognition density's quality is measured by the negative free energy. The $\GEP{}$ sector reduces to the expected log-likelihood of the generative model; the $\CEP{}$ sector reduces to the negative entropy of the recognition density; the port term $W_{\rm ports}$ collects the work exchanged across the Markov blanket.

\paragraph{Why the Markov blanket is geometric.}
A Markov blanket is a graph-theoretic separation: the set of nodes that conditionally separates internal from external states. In the framework, a Markov blanket is a specific operational realization of the \OGGEP{} boundary of \AOne{} \citep{ArisakaA1Constitution} \AX{3}: it is the surface across which the bulk-to-boundary projection $\mathcal{D}_{\OGGEP}$ of \AFive{} operates. Friston's blanket and the \OGGEP{} boundary are not two different concepts; they are two names for the same operational object, viewed from neuroscience and from constitutional physics respectively.

\paragraph{Active inference as cascade descent.}
The Friston framework's distinctive prediction --- that organisms perform \emph{active inference}, taking actions to make the world conform to their generative models rather than just updating models to match the world --- is recovered in the framework as the descent of $F_{\GFF{}}$ along trajectories that include the operational port term $W_{\rm ports}$. When the organism acts on the world, $W_{\rm ports}$ changes; when the organism updates its model, the $\CEP{}$ sector changes. Both changes lower $F_{\GFF{}}$. The Friston-style \emph{active} component and the more familiar Bayesian \emph{passive} component are two routes of descent on one functional.

\subsection{The action gauge: the \texorpdfstring{$\GG{}$}{\GG{}} gauge-field action as the detector limit}
\label{subsec:action-gauge}

The third boundary projection of $F_{\GFF{}}$ is the most familiar to particle physicists: the standard gauge-field action $S[A_\mu, \psi, \phi]$ that governs the unitary evolution of fields in the closed-system limit. When the boundary configuration is such that the operational ports are dynamically negligible --- when the detector apparatus contributes much less than the system being measured --- the $W_{\rm ports}$ term drops out, and the $\CEP{}$ sector collapses to the Wick-rotated action integral.

\paragraph{The action view as a boundary projection.}
Formally:
\begin{equation}
\label{eq:gff-to-action}
F_{\GFF{}}[p; \mathcal{P}] \Big|_{\text{detector limit}}
\;\longrightarrow\;
S[A_\mu, \psi, \phi] \;=\; \int d^4x \,\mathcal{L}(A_\mu, \psi, \phi),
\end{equation}
where $\mathcal{L}$ is the standard Lagrangian density of the boundary field theory inherited from the six-dimensional bulk by the orbifold + Wilson--Hosotani breaking pattern of \AThree{} \citep{ArisakaA3BI6} and \ANine{} \citep{ArisakaA9SM}. The $\GEP{}$ sector contributes the curvature-and-potential terms ($R\sqrt{-g}$ + gauge-field-strength-squared + matter-field kinetic and mass terms); the $\CEP{}$ sector contributes the imaginary-time entropy correction that becomes the Wick rotation in standard quantum-field theory; the port term vanishes in the detector limit.

\paragraph{Why the action is the closed-system limit.}
The standard derivation of the path integral $Z = \int \mathcal{D}A\,\mathcal{D}\psi\,\mathcal{D}\phi\,e^{i S}$ proceeds from a closed-system Hamiltonian assumption. In the framework, that assumption corresponds to a boundary configuration in which the operational ports of the detector are effectively decoupled from the system being measured --- the experimentalist's textbook ideal of a "non-disturbing" measurement. In that limit the \GFF{} descent equation reduces to the standard unitary evolution of the field, and the path integral emerges as the standard machinery of quantum field theory.

\paragraph{What the projection lets us recover.}
Once the action gauge is identified as the detector-limit projection of $F_{\GFF{}}$, the entire apparatus of textbook quantum field theory --- Feynman diagrams, perturbative renormalization, the Standard Model Lagrangian, the operator-product expansion --- emerges as the open-system framework's closed-system limit. No new mathematical objects are imported; what physicists have been computing for a century is the detector-limit projection of $F_{\GFF{}}$, and the framework simply names the limit explicitly.

\subsection{The Three-Gauge Theorem}
\label{subsec:three-gauge-theorem}
\begin{figure}[h!]
\centering
\resizebox{0.731\textwidth}{!}{%
\begin{tikzpicture}[
 bulk/.style={ellipse,draw=black,fill=orange!12,minimum width=4.2cm,minimum height=1.8cm,
 align=center,font=\small\bfseries,line width=0.6pt},
 proj/.style={rectangle,rounded corners=3pt,draw=black!60,fill=blue!10,
 minimum width=3.8cm,minimum height=1.5cm,align=center,font=\small,line width=0.4pt},
 arr/.style={-{Latex[length=2mm,width=1.5mm]},draw=black!70,line width=0.6pt},
 cond/.style={font=\scriptsize\itshape,text=black!70}
]
% Central bulk functional
\node[bulk] (GFF) at (0,0) {The Geometric\\Free-Energy Functional\\$F_{\GFF{}} = F_{\GEP{}} + F_{\CEP{}} + W_{\rm ports}$\\(\ASix{} \T{3.2}--\T{4})};
% Three boundary projections
\node[proj,fill=cyan!15] (G1) at (-5.5,-3.5) {Gibbs free energy\\$G = H - T S$\\(thermodynamic gauge)};
\node[proj,fill=green!13] (G2) at (0,-3.5) {Friston variational\\free energy\\$F_{\rm var}(\mu,\pi)$\\(variational gauge)};
\node[proj,fill=yellow!15] (G3) at (5.5,-3.5) {Gauge-field action\\$S[A_\mu,\psi,\phi]$\\(action gauge)};
% Arrows from bulk to projections
\draw[arr] (GFF) -- (G1) node[midway,left=2pt,cond] {$J_{Qi}=0$};
\draw[arr] (GFF) -- (G2) node[midway,fill=white,inner sep=1.5pt,cond,align=center] {$J_{Qi}\neq 0$,\\Markov blanket};
\draw[arr] (GFF) -- (G3) node[midway,right=2pt,cond] {detector limit};
% Use cases on bottom
\node[font=\scriptsize] at (-5.5,-5.0) {Chemistry, thermodynamics};
\node[font=\scriptsize] at (0,-5.0) {Neuroscience, cognitive sci.};
\node[font=\scriptsize] at (5.5,-5.0) {Particle physics, QFT};
% Cross-cascade label
\node[font=\itshape\small,text=black!70] at (0,1.6) {One functional, three projections, every cascade layer};
\end{tikzpicture}}
\caption[The Three-Gauge Theorem]{The Three-Gauge Theorem (T-Origin-68.1). The Geometric Free-Energy Functional $F_{\GFF{}} = F_{\GEP{}} + F_{\CEP{}} + W_{\rm ports}$ on the six-dimensional bulk admits the following three boundary projections: (i)~the thermodynamic gauge, when the informational current vanishes ($J_{Qi}=0$), yielding the Gibbs free energy of physical chemistry; (ii)~the variational gauge, when an informational current flows across a Markov-blanket boundary, yielding the Friston variational free energy of theoretical neuroscience; (iii)~the action gauge, when the operational ports of the detector are negligible, yielding the standard gauge-field action of textbook quantum field theory. The three projections are non-overlapping; their exhaustiveness is argued from the three control-variable slices rather than proved. For 150 years, thermodynamicists and neuroscientists have used the same phrase \emph{free energy} without knowing why; the theorem says the three communities have been looking at the same geometric functional through three boundary projections.}
\label{fig:three-gauge}
\end{figure}

Figure~\ref{fig:three-gauge} shows the three projections together. We now collect them of \S\S\ref{subsec:gibbs-gauge}--\ref{subsec:action-gauge} into a single named theorem.

\begin{theorem}[Three-Gauge Theorem (T-Origin-68.1)]
\label{thm:three-gauge}
The Geometric Free-Energy Functional $F_{\GFF{}}[p; \mathcal{P}]$ of \eqref{eq:gff-canonical} admits the following three boundary projections, distinguished by the value of the informational current $J_{Qi}$ and by the structure of the operational ports:

\begin{enumerate}[leftmargin=2.4em]
\item \emph{The thermodynamic gauge} ($J_{Qi} = 0$): $F_{\GFF{}}$ projects to the Gibbs free energy $G(T,P,N) = H - TS$ via \eqref{eq:gff-to-gibbs}. This is the boundary limit when no information is being stored, processed, or transmitted by the system.

\item \emph{The variational gauge} ($J_{Qi} \neq 0$, Markov blanket): $F_{\GFF{}}$ projects to the Friston variational free energy $F_{\rm var}(\mu, \pi)$ via \eqref{eq:gff-to-friston}. This is the boundary limit when the system has a Markov-blanket-separated internal-and-external state structure.

\item \emph{The action gauge} (detector limit): $F_{\GFF{}}$ projects to the standard gauge-field action $S[A_\mu, \psi, \phi]$ via \eqref{eq:gff-to-action}. This is the boundary limit when the operational ports are dynamically negligible.
\end{enumerate}

The three projections are non-overlapping: each is a separate corner of the configuration space of admissible boundaries, with smooth transitions between them. Whether they are also \emph{exhaustive} is argued rather than proved: the three corners are read off the three control-variable slices of the informational current, and no proof is offered here that no fourth admissible $J_{Qi}$ configuration exists. The claim should therefore be read as a structural proposal, not as a closed classification.
\end{theorem}

\emph{Status: Conditional.} The three projections are established by the case
analysis above; their \emph{exhaustiveness} is argued from the three
control-variable slices and is not proved, as the closing paragraph of this
subsection and Objection~23 of Part~\ref{part:objections} both state.


\begin{proof}[Proof sketch]
The argument proceeds by case analysis on the boundary configuration. When $J_{Qi} = 0$, the $\UoneQi{}$ contribution to the $\GEP{}$ sector identically vanishes, and the remaining terms collect into Gibbs's identity by direct manipulation of the Kullback--Leibler boundary. When $J_{Qi} \neq 0$ and the boundary admits a Markov-blanket decomposition (the operational realization of \OGGEP{} at the cognitive layer), the boundary moduli factor as $(\mu, s) \times \pi$, and the \GFF{} descent reduces to the variational-Bayesian descent of Friston. When the operational ports are negligible (the detector limit), $W_{\rm ports} \to 0$ and the remaining $\GEP{} + \CEP{}$ structure Wick-rotates into the standard path-integral action. Exhaustiveness is argued, not established here: the admissibility filter of \AX{3} \OGGEP{} motivates the reading that every operational boundary falls in one of the three corners, but the three corners are read off the three control-variable slices rather than enumerated exhaustively. Non-overlap follows from the case-distinguishing conditions on $J_{Qi}$ and the port structure.
\end{proof}

\paragraph{Status of the theorem within \AEleven{} (honest-disclosure note).}
The Three-Gauge Theorem is an \emph{inherited} theorem. The load-bearing equivalence of the three boundary projections --- that the variational gauge yields exactly the Friston free energy $F_{\rm var}(\mu, \pi)$ as independently defined in the active-inference literature \citep{Friston2010}, that the thermodynamic gauge yields exactly the Gibbs free energy $G(T,P,N)$ as independently defined in textbook physical chemistry, and that the action gauge yields exactly the gauge-field action $S[A_\mu, \psi, \phi]$ as independently defined in textbook quantum field theory, modulo boundary terms --- is established here as the present paper's reading of the inherited \GFF{} apparatus of \AFive{}/\ASix{} (T-Origin-68.1). The three target functionals on the right of the projections are not defined \emph{in terms of} $F_{\GFF{}}$ in \AFive{}; they are imported from their respective communities and shown to be the boundary readouts. Within \AEleven{} the result is exercised as an identification of three boundary projections; the equivalence is not re-derived. One caution belongs here rather than in the back matter. The Gibbs/Friston/\GG{} triad is \emph{not} \AFive{}'s Lindblad theorem, and \ASeven{}'s own three-gauge result (T-Qi4-10.6) concerns the \emph{different} PLT/MEPP/FEP triad; so no upstream paper carries a case-by-case verification of \emph{these} three projections. The reading is at present the present paper's own, and its definitive registry home is an open decision we flag rather than fudge. A reader who accepts the identification can proceed; a reader who does not should treat it as a proposal. This is the same inheritance discipline that governs every cascade-layer theorem in the present Part: \AEleven{} is the corpus-final synthesis, not a stand-alone proof.

\paragraph{What the theorem means in plain terms.}
For 150 years thermodynamicists and neuroscientists have used the same phrase \emph{free energy} without knowing why. For ninety years particle physicists have computed the action integral $S$ as if it were a separate object from the Gibbs free energy of chemistry. The Three-Gauge Theorem says that the three communities have been looking at the same geometric functional --- $F_{\GFF{}}$ on the six-dimensional bulk --- through three different boundary projections. The terminological coincidence (Gibbs and Friston both call their object \emph{free energy}) is not a coincidence; it is the structural signature of one geometric object with multiple boundary projections. The unification of thermodynamics, variational neuroscience, and particle-physics dynamics under one functional is the deepest structural claim of \AFive{}, and it is one of the load-bearing claims of \AEleven{}.

\subsection{Cross-cascade uniformity of \texorpdfstring{$F_{\GFF{}}$}{F\_GFF}}
\label{subsec:gff-cross-cascade}

Remarkably, the Three-Gauge Theorem unifies $F_{\GFF{}}$ across three boundary projections at the same cascade layer. \AEleven{} extends the observation across layers: the same functional governs every cascade layer above the bulk. The cross-cascade uniformity is one of \AEleven{}'s net-new structural claims, not present in any single upstream paper.

\paragraph{$F_{\GFF{}}$ at the cosmological layer.}
\ATen{} \citep{ArisakaA10Cosmos} establishes the $\pi$-locked cosmological inventory $\Omm = 1/\pi$, $\OmL = 1 - 1/\pi$, $\Omb = 1/(2\pi^2)$ as the unique stable Geometric Attractor on the cosmological-boundary stratum. The selection mechanism is the descent of $F_{\GFF{}}$ on the cosmological moduli space until the system reaches the basin of the $\pi$-locked attractor. At the cosmological layer, $F_{\GFF{}}$ takes the action-gauge projection: the universe's late-time evolution is governed by an effective four-dimensional action that emerges from the bulk action by integration over the radial and compact directions.

\paragraph{$F_{\GFF{}}$ at the particle-physics layer.}
\ANine{} \citep{ArisakaA9SM} closes the 28-parameter codebook of the Standard Model via the \GFF{} descent on the family-projector moduli $\mathbb{C}P^2$. At the particle-physics layer, $F_{\GFF{}}$ takes the action-gauge projection: the boundary projection is the standard-model Lagrangian $\mathcal{L}_{\rm SM}$, and the parameter values that emerge are the coordinates of the unique stable basin of $F_{\GFF{}}$ on the projector moduli.

\paragraph{$F_{\GFF{}}$ at the chemistry layer.}
At the atomic and molecular layers (cascade layers 3--6), $F_{\GFF{}}$ takes the thermodynamic-gauge projection: the descent is the Gibbs free energy $G(T, P, N)$ of the atomic and molecular configuration space, and the unique stable basin selects the most stable chemical configuration at each layer. The periodic table, the molecules of the interstellar medium, the amino acids and sugars of pre-biotic chemistry are all readouts of the Gibbs-projection minima of $F_{\GFF{}}$ at successive strata.

\paragraph{$F_{\GFF{}}$ at the chemistry-to-life maintenance-margin crossing.}
At the chemistry-to-life transition (cascade layer 10), $F_{\GFF{}}$ takes a hybrid projection: the chemistry layers below contribute the thermodynamic-gauge component, but the emerging informational current $J_{Qi}$ contributes the variational-gauge component. The \SeOS{} class definition of \ASeven{} (T-Qi4-14.3) is precisely the boundary configuration at which the hybrid projection becomes stable. The maintenance-margin crossing of \S\ref{sec:seos-merged} is the structural event at which $F_{\GFF{}}$ transitions from a pure thermodynamic-gauge descent to a hybrid descent that includes the variational-gauge contribution.

\paragraph{$F_{\GFF{}}$ at the cognitive layers.}
At the multicellular, nervous-system, and cognitive layers (cascade layers 13--17), $F_{\GFF{}}$ takes the variational-gauge projection: the descent is Friston's variational free energy $F_{\rm var}(\mu, \pi)$ on the brain's recognition density. The Outer Hologram, the Inner Hologram, and the Evolving Inner Cosmos of \ASeven{} \T{-\EIC{}} are boundary objects of the variational gauge. At this layer the framework reads consciousness as the layer-17 stable minimum of $F_{\GFF{}}$ in the variational gauge.

\paragraph{$F_{\GFF{}}$ at the meta-cognitive layer.}
At the meta-cognitive articulation layer (cascade layer 18 --- the layer of the present paper), $F_{\GFF{}}$ takes a hybrid projection that combines the variational gauge (the author's neural substrate is descending in $F_{\rm var}$) with the action gauge (the act of writing the paper exchanges Wick-rotated action with the laboratory environment). The cumulative civilization-level descent over geological time is the descent of $F_{\GFF{}}$ at this layer; its stable basin is the cohort articulation of the \GGConstitution{}.

\paragraph{The cross-cascade summary.}
$F_{\GFF{}}$ is the universal functional. The Three-Gauge Theorem identifies its three boundary projections; the cross-cascade observation identifies its layer-specific instantiations. At every cascade layer, the same functional governs the boundary descent, and the layer-specific Geometric Attractor is the boundary minimum. This is what gives the cascade its uniform engine: the same functional descends at every layer, only the substrate and the projection change.

\subsection{Metaphors and intuitions for non-physicist readers}
\label{subsec:gff-metaphors}

The technical statements of \S\S\ref{subsec:gff-formal}--\ref{subsec:gff-cross-cascade} can be translated into a few accessible metaphors for life scientists, philosophers, and other interdisciplinary readers. The two named theorems that close the Section, \S\S\ref{ssec:t-gff-lyapunov-descent}--\ref{ssec:t-three-gauge-identity}, are stated in the technical register and may be read after these metaphors rather than before them.

\paragraph{Metaphor 1: the universal altitude.}
Imagine that every open system in the universe sits on a landscape, with the altitude given by $F_{\GFF{}}$. Water flows downhill; so do systems. Every chemical reaction, every biological process, every cosmological evolution is the system rolling down the landscape until it reaches a basin (an attractor). The landscape's altitude is universal --- it is the same function for an electron, a water molecule, a bacterium, and a human brain --- but the landscape's shape is layer-specific. At each layer the basins look different, but the rule for descent is the same: lower your altitude as fast as you can while respecting the four-current Arisaka-equation constraint.

\paragraph{Metaphor 2: the same musical score, three different instruments.}
Gibbs free energy is the thermodynamic gauge of the score. Friston free energy is the variational gauge. The gauge-field action is the action gauge. All three play the same melody --- the descent of $F_{\GFF{}}$ --- but on different instruments. Where chemistry hears the thermodynamic line and neuroscience hears the variational line and particle physics hears the action line, the framework reads the score and notices that all three are reading from the same staff.

\paragraph{Metaphor 3: water in different vessels.}
A specific volume of water can sit in a beaker (Gibbs, the thermodynamic gauge); it can flow through a brain's vasculature (Friston, the variational gauge); or it can be the propagating wavefunction of a quantum simulation (the action gauge). The water is one substance; the vessel is the boundary projection. Different sciences study water in different vessels and develop different vocabularies, but the molecular content is one.

\paragraph{Metaphor 4: the cascade as a single staircase.}
The recursive uniqueness theorem reads the universe as one staircase of layered attractors. Each step is a layer. At each step, the same engine (the \GFF{} descent) selects the unique stable basin. At each step, the boundary projection is one of the three gauges of the Three-Gauge Theorem. The Gibbs gauge runs from steps 2--5 (chemistry); the hybrid Gibbs+Friston gauge runs at step 6 (life); the Friston gauge runs from steps 7--10 (mind); the hybrid Friston+action gauge runs at step 11 (the act of writing). The staircase ascends through eighteen steps in a fixed pattern of gauges; the framework reads the pattern.

\paragraph{What a life scientist should take away.}
A biological cell is an open system performing \GFF{} descent in the hybrid Gibbs+Friston gauge. Its metabolism is the Gibbs component; its information-processing (gene expression, signal transduction, immune memory) is the Friston component. The cell's persistence is the \SeOS{} condition of \S\ref{sec:seos-merged} (Section 2 of this Part). The cell's universal identity is the \GRIP{} cascade selecting the unique stable cellular attractor on the planetary-surface stratum. The cell's energy currency is the bioenergetic anchor $\QB$ of \S\ref{sec:qb-anchor} (Section 1 of this Part). Every textbook statement about cellular biology has a constitutional reading in the present framework, with no biological postulate that is not derivable from the cohort.

\paragraph{What a philosopher should take away.}
The Three-Gauge Theorem is a structural statement of the form: \emph{three apparently different cost functionals operating in three apparently different sciences are three boundary projections of one geometric object}. Philosophers of science accustomed to a multiplicity of incommensurable theoretical frameworks (Kuhn, Feyerabend) will recognize this as a reductionist claim that does not reduce. The framework does not say that chemistry, neuroscience, and physics are "the same"; it says that they describe the same underlying object from different vantage points. Reduction is operational: each science correctly characterizes its boundary projection of $F_{\GFF{}}$, and no science is replaced or eliminated. What the framework adds is the recognition that the projections all derive from one functional --- and, as a proposal rather than a closed result, that they exhaust the admissible boundaries.

% INSERT-AFTER: section "The Geometric Free-Energy Functional and the Three-Gauge Theorem" (v6.5 line 5098), at the section's existing intro paragraphs and before its existing Three-Gauge Theorem environment.

\subsection{\textsf{T-GFF-Lyapunov-Descent}: \GFF{} is a Lyapunov functional for the cascade}
\label{ssec:t-gff-lyapunov-descent}

\paragraph{Plain-English opening.}
The cascade does not stagger randomly between possible states; it descends monotonically along a Lyapunov functional. That Lyapunov is the Geometric Free-Energy Functional \GFF{} = $F_{\GEP{}} + F_{\CEP{}}$ (geometric and constitutional pieces). Along any admissible runtime trajectory, $dF_{\GFF{}}/dt \leq 0$, with equality only at the \GGA{}. In daily life: water flows downhill, never uphill, and stops at the lake. The cascade is the universe's version of water flowing downhill, where the height is the \GFF{} and the lake is the \GGA{}. This theorem is the constitutional warrant for the cascade's irreversibility: once a layer's run begins, the system can only descend toward the \GGA{}; it cannot loiter or ascend.

\paragraph{Constitutional inputs.}
\AX{3} (\OGGEP{}, open-system boundary --- the system can lose free energy to the environment), \AX{4} (\MSMF{}), \POS{6}~\OGN{} (\OGNfive{}), and \ASeven{}~\T{-QUA}--\ASeven{}~\T{-EP} (operational closure and functional closure). The Lyapunov clause is the operational reading of the \GDOS{} (Geometric Dynamics of Open Systems) framework.

\paragraph{Upstream theorems.}
Two upstream theorems are invoked. (i) \ASix{} \T{3.2} \citep{ArisakaA5GDOS} (the \GDOS{} second-law theorem) establishes that the bulk open-system dynamics descend along \GFF{} with $dF_{\GFF{}}/dt \leq 0$. In plain language, this is the \GGTheory{} generalization of the second law of thermodynamics: any open system on a constitutional substrate descends along the same Lyapunov. (ii) \ASix{} (the \GRIP{} theorem) \citep{ArisakaA6Gi4} sharpens this to a per-layer Lyapunov descent: at each cascade layer, the runtime descends along the layer-restricted \GFF{} to the layer-$n$ \GGA{}.

\begin{theorem}[\textsf{T-GFF-Lyapunov-Descent}: \GFF{} is the cascade's Lyapunov (T-Origin-68.2)]
\label{thm:t-gff-lyapunov-descent}
On any admissible runtime trajectory $\gamma(t)$ of the layer-$n$ \SIP{} $\widehat{\SIP{}}^{(n)}$ under the \GGConstitution{} $\mathcal{C}$, the Geometric Free-Energy Functional
\begin{equation}
F_{\GFF{}} \;=\; F_{\GEP{}} + F_{\CEP{}} + W_{\rm ports}
\end{equation}
(with $F_{\GEP{}}$ the Gauge-Geometry Energy Principle sector, $F_{\CEP{}}$ the Curvature-Entropy Principle sector, and $W_{\rm ports}$ the boundary-port work term of \eqref{eq:gff-canonical}) satisfies
\begin{equation}
\label{eq:gff-lyapunov}
\boxed{\;\;\frac{dF_{\GFF{}}}{dt}\Big|_{\gamma(t)} \;\leq\; 0 \;\;\text{for all $t$, with equality iff $\gamma(t) = \GGA{}_n$.}\;\;}
\end{equation}
Equivalently, $F_{\GFF{}}$ is a strict Lyapunov functional for the layer-$n$ \GRIP{} runtime, and $\GGA{}_n$ is the unique attractor of the runtime on the layer-$n$ stratum.
\end{theorem}

\paragraph{Proof outline.}
Step 1: open-system descent. By \ASix{} \T{3.2}, on any open-system substrate the bulk \GFF{} is non-increasing along the runtime. Step 2: layer-restriction. Restricting to the layer-$n$ stratum (the boundary of \GGA{}$_{n-1}$ via \textsf{T-SIP-Tower-Recursion}) preserves the descent property because the boundary inherits the bulk Lyapunov via the bulk-to-boundary projection of \AFive{}. Step 3: strict descent off the attractor. At any $\gamma(t) \neq \GGA{}_n$, the layer-restricted Morse landscape has a non-zero gradient (by \ASix{} \T{3.2} and the absence of layer-$n$ degeneracy from \textsf{T-Cascade-Layer-Uniqueness}); therefore $dF_{\GFF{}}/dt < 0$. Step 4: equality only at the attractor. At $\gamma(t) = \GGA{}_n$, the gradient vanishes; by \textsf{T-Cascade-Layer-Uniqueness}, this is the unique attractor, hence the unique equality point. Step 5: cascade-wide concatenation. The eighteen layer-restricted Lyapunovs concatenate to give a cascade-wide Lyapunov $F_{\GFF{}}^{\rm casc} = \sum_n F_{\GFF{}}^{(n)}$ which decreases monotonically along the cascade as a whole; this is the cascade's irreversibility theorem.

\paragraph{Quantitative anchor.}
The Lyapunov is structural; its quantitative content lies in the per-layer descent rates. At layer~7 (bioenergetic), the rate is $-dF_{\GFF{}}/dt \sim J_{Qi} \cdot \QB{}$ per port-event, with $J_{Qi} \sim 10^{9}$~quanta/second in a fast aerobic cell --- the top of the band the layer-10 anchor allows, a 1--10~$\mu$m cell dissipating 1--100~W/kg. At layer~17 (conscious episode) the same form carries over with the cognitive port rate in place of the metabolic one. The alert human brain is conventionally credited with $\sim 10^7$~bits/s of throughput; we quote that as the bandwidth estimate it is, and do not convert bits into $\QB{}$-quanta, because the framework has not yet fixed the exchange rate between them. The cascade-wide descent integrates these rates over the cascade's lifetime, giving the total entropy production of the layered cascade.

\paragraph{Physics intuition.}
Roughly speaking, a Lyapunov functional is the dynamical-systems analogue of an energy function in classical mechanics: a real-valued function on the state space that is non-increasing along the dynamics and has its strict minimum at the attractor. In textbook treatments, Lyapunov functionals are typically constructed by hand or guessed from physical intuition. The cascade's \GFF{} is not guessed; it is forced by the constitutional axioms. $F_{\GEP{}}$ (geometric equipartition) is the equipartition of action across the bulk's degrees of freedom; $F_{\CEP{}}$ (constitutional equipartition) is the equipartition of action across the constitutional axioms and postulates. Their sum is the cascade's Lyapunov. The descent property follows from \AX{3}: open systems must lose free energy to their environment along admissible trajectories. This is the cascade's reading of the second law of thermodynamics, generalized to constitutional substrates. That reading consumes the arrow rather than creating it: per the constitutional custody chain, \AX{3} owns the arrow constitutionally --- the one-way reading of the dictionary is a clause of the constitution (the \AX{3} synthesis of \citealp{ArisakaA1Constitution}) --- with the canonical law-level derivation in Part~\ref{part:closure} of the CPT member paper \citep{ArisakaP5CPT}, carried out over a bulk that is itself exactly T-symmetric, and the dynamical implementation in the \GDOS{} companion \citep{ArisakaA5GDOS}; the cascade's irreversibility --- each \GRIPout{} shock stepping the geometry down one floor and never back up --- is the layered face of that one constitutional arrow, consumed at every layer, not re-derived on any of them.

\paragraph{Daily-life metaphor.}
Imagine a hiker on a mountainside in foggy weather. The hiker cannot see the lake at the bottom of the valley, but the slope underfoot points downhill. Following the slope, the hiker eventually arrives at the lake. The slope is the Lyapunov; the lake is the attractor. The cascade's runtime is a hiker following the \GFF{} slope on the constitutional landscape. No matter where on the landscape the hiker starts (subject to admissibility), the descent ends at the same lake: the layer-$n$ \GGA{}. Two hikers starting from different admissible initial conditions end up at the same lake because the landscape has a unique low point on the layer-$n$ stratum.

\paragraph{Kill criterion.}
The theorem is falsified by any of: (a) the discovery of an admissible runtime trajectory on which $dF_{\GFF{}}/dt > 0$ at any time; (b) the discovery of a layer with multiple Morse minima of equal depth (which would falsify per-layer uniqueness and therefore the Lyapunov clause); (c) the discovery of a cascade run that descends along a different functional than \GFF{} (i.e.\ that \GFF{} is not the unique Lyapunov); (d) the discovery of a perpetuum-mobile-style process on the constitutional substrate.

\paragraph{Plain-English close.}
The cascade descends along the same kind of landscape that water descends on a hillside. The slope is \GFF{}; the lake is the \GGA{}; the descent is irreversible. This is the cascade's reading of the second law, derived from the open-system axiom \AX{3} rather than postulated.

\bigskip

% =========== Theorem 9: T-SeOS-Class-Characterisation ============

% INSERT-AFTER: the existing Three-Gauge Theorem environment in v6.5's section "The Geometric Free-Energy Functional and the Three-Gauge Theorem" (line 5098).  This is an enhancement (proof outline + commentary), not a redefinition.

\subsection{\textsf{T-Three-Gauge-Identity}: the three readings agree on the bulk}
\label{ssec:t-three-gauge-identity}

\paragraph{Plain-English opening.}
Theorem~\ref{thm:three-gauge} says that the Geometric Free-Energy Functional can
be read through three boundary projections. It does not say that the three
readings \emph{agree}. That is a separate claim, and it is the one this
subsection makes. In plain language: the cascade's free-energy landscape can be
written in three different coordinate systems, each adapted to a different
boundary observer --- a thermodynamicist, an active-inference theorist, a field
theorist --- and a quantity computed in one frame converts into the
corresponding quantity in another without the bulk landscape changing at all.
The enumeration of the three projections, and the open question of whether they
exhaust the admissible boundaries, belong to
\S\ref{subsec:three-gauge-theorem} and are not re-argued here.

\paragraph{Constitutional inputs.}
\AX{2} (\GGEP{}, the gauge-as-geometry reading on the boundary), \AX{3}
(\OGGEP{}, the open-system boundary on which the readings are defined), \AX{4}
(\MSMF{}, the maximum-symmetry choice of boundary frame), and
\ASeven{}~\T{-QUA} (operational support for the identification). In plain
language: \AX{2} forbids a boundary reading from breaking gauge invariance, and
\AX{4} selects the most symmetric admissible frame in each case --- which is
what makes the frames comparable at all.

\paragraph{Upstream results.}
The three projections themselves are Theorem~\ref{thm:three-gauge} of the
present Part, established there by case analysis on the boundary configuration.
Their custody is stated honestly in \S\ref{subsec:three-gauge-theorem}: no
upstream paper carries a case-by-case verification of \emph{this} triad, the
reading is the present paper's own, and the identity below inherits that
status rather than improving on it.

\begin{theorem}[\textsf{T-Three-Gauge-Identity}: the three boundary readings agree on the bulk (T-Origin-68.3)]
\label{thm:t-three-gauge-identity}
Let $\mathcal{G}_{\rm Gibbs}$, $\mathcal{G}_{\rm Friston}$ and
$\mathcal{G}_{\rm action}$ denote the three boundary frames of
Theorem~\ref{thm:three-gauge}, in which $F_{\GFF{}}$ reads respectively as the
Gibbs free energy, as the variational free energy of active inference, and as
the gauge-field action. For any two of them there is a boundary-frame change
\[
  W_{ij} : \mathcal{G}_i \longrightarrow \mathcal{G}_j ,
\]
namely the Legendre transform where both readings are thermodynamic and the
corresponding information-theoretic identification where one of them is
variational, such that
\begin{equation}
\label{eq:three-gauge-identity}
\boxed{\;\;W_{ij}\big[\,F_{\GFF{}}^{\mathcal{G}_i}\,\big] \;=\; F_{\GFF{}}^{\mathcal{G}_j} \;\;\text{on the bulk.}\;\;}
\end{equation}
Each $W_{ij}$ acts on the boundary frame alone: the bulk functional
\eqref{eq:gff-canonical} is unchanged by it. The three readings therefore
differ in the boundary frame and in nothing else.
\end{theorem}

\emph{Status: Conditional.} The identity is conditional on the three
projections of Theorem~\ref{thm:three-gauge}, and inherits their custody: no
upstream paper carries a case-by-case verification of this triad, so the
reading is the present paper's own.


\paragraph{Proof outline.}
Step 1: existence of the three readings. This is Theorem~\ref{thm:three-gauge},
whose case analysis exhibits each projection together with the boundary
condition that selects it. Step 2: agreement on the bulk. Each of the three
boundary observables is obtained from the others by a Legendre transform or by
an information-theoretic identification; every such map is a change of
conjugate variable on the boundary and leaves the bulk integrand of
\eqref{eq:gff-canonical} pointwise untouched. Step 3: covariance with the
constitution. By \AX{2} all three readings respect gauge invariance, so no
$W_{ij}$ can carry an admissible configuration out of the constitutional
admissibility class; the identity therefore holds wherever the cascade runs,
not merely at isolated strata. Step 4: consistency of the composites. The
$W_{ij}$ compose as boundary-frame changes should --- $W_{jk} \circ W_{ij} =
W_{ik}$ on the bulk --- so a reading converted through an intermediate frame
agrees with the direct conversion, and the identity is well-posed as a relation
on all three frames at once rather than pairwise by accident.

\paragraph{Quantitative anchor.}
The three readings are carried in three different units, which is precisely why
the identity has operational content: $F_{\rm Gibbs}$ in kJ/mol (equilibrium
thermodynamics), $F_{\rm Friston}$ in nats (variational free energy), and the
action reading dimensionless, in units of $\hbar$. The conversion between the
first two is constitutional and fixed by the bioenergetic quantum:
$1\,\QB{} \approx 0.1555$~eV $\approx 15.0$~kJ/mol $\approx 6.02$ nats at
$300$~K. A quantity that is $30$~kJ/mol to a biochemist is $2\,\QB{}$ to the
cascade and about $12$ nats to an active-inference theorist; these are three
names for one number, which is the identity in its most usable form.

\paragraph{Physics intuition.}
The identity works like the gauge-equivalence of the thermodynamic potentials.
In textbook thermodynamics one may describe a system by internal energy
$U(S,V,N)$, enthalpy $H(S,P,N)$, Helmholtz free energy $F(T,V,N)$ or Gibbs free
energy $G(T,P,N)$; each is a Legendre transform of the others, each is adapted
to a different choice of held-fixed variable, and none is more fundamental than
the rest. $W_{ij}$ is that transform, extended past equilibrium: between the
Gibbs and Friston readings it exchanges a thermodynamic conjugate pair for an
information-theoretic one, and between either and the action reading it
exchanges a boundary observable for a bulk phase. What the physicist should
take from \eqref{eq:three-gauge-identity} is that the disagreement between
these three literatures has been notational.

\paragraph{Kill criterion.}
The identity is falsified by any of: (a) the failure of the Legendre or
information-theoretic identification between any two of the three readings on
the bulk --- that is, a quantity computed in one frame that provably cannot be
converted into its counterpart in another; (b) a demonstration that some
$W_{ij}$ changes the bulk functional rather than only the boundary frame, which
would mean the three are not readings of one object at all; (c) a failure of
composition, $W_{jk} \circ W_{ij} \neq W_{ik}$, which would leave the identity
holding pairwise but not jointly; (d) an empirical mismatch in the conversion
constants of the quantitative anchor above --- a measured $\QB{}$ that does not
carry $15.0$~kJ/mol into $6.02$ nats at $300$~K. Note that a boundary regime
in which $F_{\GFF{}}$ reads as a \emph{mixture} of two frames does not falsify
the identity: \S\ref{subsec:gff-cross-cascade} exhibits two such hybrid
projections, at the chemistry-to-life transition and at the articulation layer,
and a hybrid is a point on a smooth path between corners, not a fourth corner.

\section{\GRIP{}: the dynamics on the \GFF{} Morse landscape}
\label{sec:grip-on-gff}

Sections 1--3 of this Part have established the three pieces of dynamical machinery: the bioenergetic anchor $\QB$, the \SeOS{} class, and the Geometric Free-Energy Functional with the Three-Gauge Theorem. Together these tell us \emph{what} the open-system universe runs on. The present Section names \emph{how} the universe runs. The answer is \GRIP{} --- the Gauge-Runtime Invariance Pipeline of \ASix{} \citep{ArisakaA6Gi4} T-GRIP-1--T-GRIP-5 --- and its terminus, the \GGA{} (Ground Geometric Attractor). This Section is the conceptual hinge of the paper: every cascade layer above the bulk is one application of \GRIP{} on the \GFF{} Morse landscape, terminating at one \GGA{}, which then becomes the substrate for the next layer's \GRIP{} application.

\subsection{The canonical form, and where this Part\textquotesingle s machinery sits in it}
\label{subsec:grip-canonical-restated}

The canonical form of \GRIP{} --- the four-element pipeline
$\DGI{} \to (\GFF{}) \to \GSSB{} \to \GGA{}$, with \GFF{} in parentheses because
it is the substrate rather than a stage --- is established in Part~\ref{part:recursive}
\S\ref{sec:grip-canonical-form}, inherited there from \ASix{}
\citep{ArisakaA6Gi4}, and is not restated here. That Part is also where the
reading of the four \QiFour{} pillars is fixed: they are \emph{not} stage labels
of this pipeline, but filter its terminus jointly. What the present Section adds
is the connection to the machinery of \S\S\ref{sec:qb-anchor}--\ref{sec:gff-three-gauge}.

The relationship between the three pieces of dynamical machinery of Sections 1--3 and the four elements of \GRIP{} is:
\begin{itemize}\itemsep2pt
\item The bioenergetic anchor $\QB$ supplies the layer-7 \GGA{} for the metabolic-energy stratum.
\item The \SeOS{} class is the layer-7 \GGA{} for the open-system-on-planetary-surface stratum: a configuration is admissible iff its $F_{\GFF{}}$ trajectory under \GRIP{} descent terminates in the \SeOS{}-class basin.
\item The \GFF{} functional itself is the Morse landscape, and the Three-Gauge Theorem reads the same landscape through three boundary projections (thermodynamic / variational / action).
\end{itemize}
In short: $\QB$ and \SeOS{} are \GGA{}s the cascade selects; \GFF{} is the landscape on which it does the selecting; \GRIP{} is the dynamics.

\subsection{Morse-theoretic content: why \GRIP{} terminates at the \GGA{}}
\label{subsec:morse-theoretic-grip}

\paragraph{The Lyapunov descent law.}
The central technical claim of \ASix{} \citep{ArisakaA6Gi4} \T{3.2} is that every admissible \GDOS{} trajectory monotonically lowers $F_{\GFF{}}$:
\begin{equation}
\label{eq:grip-descent-law}
\frac{dF_{\GFF{}}}{dt} \;\leq\; 0,
\qquad \text{with equality only at a critical point of } F_{\GFF{}}.
\end{equation}
This is the Lyapunov law on the \GFF{} landscape. In Morse-theoretic language: along any admissible gradient-flow trajectory, $F_{\GFF{}}$ decreases until it reaches a critical point. The critical points of $F_{\GFF{}}$ on the layer-specific boundary stratum are classified by their Morse index $\mu$: index-$\mu$ critical points have $\mu$ unstable directions and $\dim(\text{stratum}) - \mu$ stable directions. Index-0 critical points are local minima; the unique deepest local minimum is the global minimum.

\paragraph{The cascade is Bogomolnyi-saturating gradient flow.}
\ASix{} \citep{ArisakaA6Gi4} T-GRIP-4 establishes that the gradient flow on $F_{\GFF{}}$ is Bogomolnyi-saturating: the flow connects index-$2k$ critical points to index-$2(k-1)$ critical points along trajectories whose action is exactly
\begin{equation}
\label{eq:grip-instanton-action}
S_{\rm BPS}(k \to k-1) \;=\; \pi k.
\end{equation}
The factor $\pi$ here is structural: it is the Yang--Mills instanton normalization that the gauge-geometry sector of $\GEP{}$ inherits from the bulk. Bogomolnyi saturation means the gradient flow is the absolute minimum action trajectory between adjacent saddles; no other admissible flow connects them with lower action.

\paragraph{The total cascade action is bounded.}
Summing the instanton actions of \eqref{eq:grip-instanton-action} along a
descent chain bounds the depth of one coherent cascade stride. The sum is taken
over the ordered pairs of a chain of $N$ saddle events --- $\pi\binom{N}{2}$ of
action, not $\pi\sum_{k=1}^{N}k$, the distinction being whether the lowest
transition is charged --- and the \BIsix{} closure of \AThree{}
\citep{ArisakaA3BI6} supplies the budget $S_{\rm BI} = 3\pi$ that bounds it.
The derivation, and the bound $N \leq 3$ it forces --- saturated exactly, a
four-step chain demanding twice the action the geometry has to give --- are
given in Part~\ref{part:runs}, Corollary~\ref{thm:t-bogomolnyi-action}, and are not repeated
here. The full conditional three-leg derivation of the generation number is
\PaperPTwo{} \citep{ArisakaP2N3}. What the present Part needs is only the
consequence: the cascade's depth per stride is geometrically bounded, the bound
saturates, and the saturating value is what the universe exhibits.

\paragraph{Termination at the \GGA{}.}
Under the Lyapunov descent law \eqref{eq:grip-descent-law}, every admissible trajectory eventually reaches a critical point. Under the Bogomolnyi-saturating property of \GRIP{}, the trajectory cascades through saddles with decreasing Morse index, each connected to the next by an action-$\pi k$ instanton. The cascade terminates only at an index-0 critical point. If multiple index-0 critical points exist on the stratum, the cascade selects the one with the lowest $F_{\GFF{}}$ value on the connected component the trajectory entered: that is the \GGA{}. In the strata realized in our universe, where geometric scale running and \QGL{} admissibility further filter the configuration space, the \GGA{} is unique on each connected stratum. The cascade output is single-valued. This is the operational form of A9 \citep{ArisakaA9SM} Part~\ref{part:discussions} (the universal-identity section)'s universal-identity insight.

\subsection{The Three-Gauge Theorem read as three projections of one \GGA{}}
\label{subsec:three-gauge-on-ggA}

Section 3 of this Part presented the Three-Gauge Theorem: $F_{\GFF{}}$ admits three boundary projections (thermodynamic gauge / variational gauge / action gauge), argued to cover every operational boundary an observer can construct. We now read the Three-Gauge Theorem at the level of the cascade output.

\paragraph{The thermodynamic-gauge \GGA{}.}
When $J_{Qi}=0$, the cascade output \GGA{} is read off in Gibbs-free-energy language: it is the chemical equilibrium configuration on the stratum, the lowest-Gibbs configuration the stratum admits. Examples: the H atom's $1s$ ground state (the layer-3 \GGA{} for the $Z=1$ stratum at $E_{\rm Ry} = -13.606$~eV), the H$_2$O molecule at its equilibrium geometry (the layer-4 \GGA{} for the $\{2\mathrm{H},\mathrm{O}\}$ molecular stratum), the diamond crystal structure for elemental carbon at standard conditions.

\paragraph{The variational-gauge \GGA{}.}
When $J_{Qi}\neq 0$ and the boundary admits a Markov-blanket decomposition, the cascade output \GGA{} is read off in Friston variational-free-energy language: it is the recognition density $q(\mu)$ that minimizes the variational free energy on the agent's generative-model stratum. Examples: the hippocampal allocentric map (the layer-16 \GGA{} for the rodent neural stratum), the cortical predictive model of the visual field (a layer-17 \GGA{} for the primate visual-cortex stratum), the conscious narrative (a layer-17 \GGA{} for the self-modeling cognitive stratum).

\paragraph{The action-gauge \GGA{}.}
In the detector limit, when the operational ports decouple, the cascade output \GGA{} is read off in path-integral language: it is the classical solution of the boundary field theory, the configuration that extremizes the action $S[A_\mu, \psi, \phi]$. Examples: the \GG{} vacuum, the Standard-Model ground state at $\Lambda_{\rm QCD}$ and below, the cosmological background metric of \ATen{} \citep{ArisakaA10Cosmos}.

The Three-Gauge Theorem is therefore not just a statement about $F_{\GFF{}}$ projections; it is a statement about the three operational readings of the \GGA{} the cascade selects. Cosmology, chemistry, neuroscience, and physics each read one \GGA{} through one gauge.

\subsection{The cascade as a series of \GGA{}s}
\label{subsec:cascade-series-ggA}

Let us now restate Parts~IV and VI in the unified language of \GRIP{} on \GFF{} terminating at \GGA{}.

\paragraph{Recursive definition.}
Let $\mathcal{S}_n$ denote the boundary stratum at cascade layer $n$, parametrized by the \GGA{} of layer $n-1$:
\begin{equation}
\label{eq:cascade-recursion}
\mathcal{S}_n \;=\; \mathcal{S}_n[\GGA{}_{n-1}].
\end{equation}
The cascade is the indexed-by-layer sequence
\begin{equation}
\label{eq:cascade-sequence}
\GGA{}_1 \;\longrightarrow\; \GGA{}_2[\GGA{}_1] \;\longrightarrow\; \GGA{}_3[\GGA{}_2] \;\longrightarrow\; \cdots \;\longrightarrow\; \GGA{}_{17}[\GGA{}_{16}] \;\longrightarrow\; \GGA{}_{18}[\GGA{}_{17}],
\end{equation}
where $\GGA{}_1$ is the layer-1 \GGA{} (the Standard-Model gauge group $G_{\rm SM}$, established in Part~\ref{part:recursive} Step~1) and $\GGA{}_{18}$ is the meta-cognitive layer-18 \GGA{} (the act of articulating the constitution). Each arrow is one \GRIP{} application: \DGI{} (provided by the prior \GGA{} as the unbroken-symmetry context for the next layer) $\to$ ($\GFF{}$ on the layer-$n$ stratum) $\to$ \GSSB{} (the layer-specific Bogomolnyi saddle event) $\to$ $\GGA{}_n$.

\paragraph{The recursive uniqueness theorem.}
The recursive uniqueness theorem of \AEleven{} (Part~\ref{part:closure} \S\ref{sec:named-theorem}) states that the indexed-by-layer sequence \eqref{eq:cascade-sequence} is uniquely determined by the \GGConstitution{} of \AOne{} \citep{ArisakaA1Constitution}. At every layer, the cascade terminates at the layer-specific \GGA{} because the geometry admits exactly one global minimum of $F_{\GFF{}}$ on the connected boundary stratum, modulo equivalences that produce the same readout. This is the formal content of A9 \citep{ArisakaA9SM} Part~\ref{part:discussions} (the universal-identity section)'s universal-identity insight, applied at every layer of the cascade.

\paragraph{The series of \GGA{}s in numerical form.}
The eighteen \GGA{}s, each with its named theorem and its quantitative anchor,
are tabulated once in Part~\ref{part:results} \S\ref{ssec:reader-summary-table}; that table is
the graded one, and it is not reproduced here. Each row of it is one \GGA{}.  Each \GGA{} is read off as a textbook identity (an electron mass; a chemical bond energy; a cell size). The cascade is the staircase of those identities. The framework's claim is that the staircase is forced by the geometry, with no continuous parametric freedom remaining at any rung.

\subsection{\QiFour{} as the \GGA{} selector: worked examples at four strata}
\label{subsec:qi4-selector-worked}

Theorem~\ref{thm:qi4-gga-admissibility} of Part~\ref{part:recursive} \S\ref{sec:qi4-admissibility-filter} states that a configuration $\Xi$ on the layer-$n$ boundary stratum is a physically realizable $\GGA{}_n$ if and only if it satisfies all four \QiFour{} admissibility constraints --- Qi-QUA (locked-current ray), Qi-EP (variational stationarity), Qi-CON (operational scaling window), Qi-HAL (memory-architecture closure). In this subsection we exercise the theorem at four representative strata and trace, in each case, which pillars dominantly constrain the \GGA{}, what the \GGA{} is, and what would happen if a pillar were relaxed. The four worked examples span the cascade from particle physics to consciousness.

\paragraph{Example 1 (layer 2, charged-lepton stratum) --- the electron mass.}
The layer-2 boundary stratum is the family-projector moduli on $\mathbb{C}P^2$ combined with the Wilson--Hosotani holonomy on $S^1_\theta$. The \GFF{} Morse landscape on this stratum admits three discrete index-$0$ critical points at depths $n_e \in \{0, 1, 2\}$, corresponding to the three charged-lepton masses. Without \QiFour{}, the cascade would terminate at any one of the three depths; with \QiFour{}, only one is admissible at any moment, and the three are realized as distinct flavor eigenstates of \emph{three different cascade applications}, not three minima within a single layer.

\textbf{Pillar-by-pillar selection:}
\begin{itemize}\itemsep1pt
\item \textbf{Qi-QUA} (current-locking ray): the layer-2 attractor must carry the electromagnetic, baryonic, energy, and informational currents in the locked ratio $J_E/\QB{} = J_e = J_B = J_{Qi}$. For a charged lepton, this fixes the electric-charge quantum at $-e$ and the baryon-current quantum at $0$, eliminating any candidate attractor with anomalous charge ratios.
\item \textbf{Qi-EP} (variational stationarity): the candidate \GGA{} must extremize the projected functional $F_{\GFF{}}$ on $\mathbb{C}P^2$. The three index-$0$ critical points of $F_{\GFF{}}$ are the three Morse minima at family-projector depths $n_e = 0, 1, 2$. Each is a valid Qi-EP solution; the three are degenerate at the level of Qi-EP alone.
\item \textbf{Qi-CON} (scaling window): the Wilson--Hosotani holonomy admits scaling exponent $\alpha = 1$ on $S^1_\theta$, well inside $(0, 2]$. Qi-CON is satisfied at all three depths.
\item \textbf{Qi-HAL} (memory-architecture closure): the family-projector moduli must support a memory holonomy that closes ($U(1)_{Qi}$ Wilson loop). Each charged lepton is its own distinct holonomy-closure class; the three depths are three distinct memory-architecture classes.
\end{itemize}

The four pillars together do not collapse the three depths to one; instead they admit three discrete \GGA{}s on the layer-2 stratum. These three are not local minima of one cascade application; they are three connected components of the Qi-HAL-admissible subset. The cascade selects one component per application: the electron is the layer-2 \GGA{} on the depth-$0$ component; the muon is the \GGA{} on the depth-$1$ component; the tau is the \GGA{} on the depth-$2$ component. Quantitatively:
\begin{equation}
\label{eq:qi4-layer1-readout}
\frac{m_\tau}{m_\mu} \;=\; 16.817 \pm 0.5\%\quad\text{(\ANine{} \S\S15--17, charged-lepton mass ladder)},
\end{equation}
matching the empirical value. \emph{Without Qi-CON's scaling window admissibility, the holonomy on $S^1_\theta$ would admit anomalous scaling exponents and the depth quantization would smear out, producing a continuous spectrum of charged-lepton masses --- contradicting observation.}

\paragraph{Example 2 (layer 7, metabolic-energy stratum) --- ATP.}
The layer-7 boundary stratum is the chemistry-to-life configuration space at the metabolic-energy interface: bounded open chemical systems on a planetary surface with a polar solvent, capable of energy-currency intermediation. Many candidate molecular configurations are local minima of $F_{\GFF{}}$ at this stratum (GTP, UTP, CTP, ADP, AMP, polyphosphate chains). The unique \GGA{} on this stratum is ATP --- the adenosine triphosphate molecule with $\Delta G_{\rm hydrolysis} \approx -30.5$~kJ/mol.

\textbf{Pillar-by-pillar selection:}
\begin{itemize}\itemsep1pt
\item \textbf{Qi-QUA} (current-locking ray): every metabolic transaction must carry the four currents in the Arisaka-equation ratio. ATP hydrolysis carries $\Delta E = 2 N_A \QB{} = 30.0$~kJ/mol of energy current, with proportional electromagnetic, baryonic, and informational currents. This exactly fixes the energy quantum per ATP turnover at $2\QB{}$ per molecule. Other phosphate carriers (GTP, UTP, CTP) are valid Qi-QUA solutions for specific signaling contexts, but only ATP is the energy-currency \GGA{}.
\item \textbf{Qi-EP} (variational stationarity): the candidate \GGA{} must extremize $F_{\GFF{}}$ on the open-system stratum. ATP's Lyapunov stability under continuous hydrolysis-resynthesis cycling is what makes it the \GGA{}; competing carriers (e.g., pyrophosphate) destabilize under the same cycling rate.
\item \textbf{Qi-CON} (scaling window): metabolic processes span timescales from $10^{-9}$~s (enzymatic catalysis) to $10^3$~s (cell-cycle dynamics) --- twelve decades, well within Qi-CON's two-knee admissibility envelope. The two-knee structure of metabolic spectra is what Qi-CON's operational layer predicts: \ASeven{} \T{-CON.2} \citep{ArisakaA7Qi4}, the two-knee spectral theorem, whose algebraic counterpart \T{-CON.1} supplies the conformality bound it rests on.
\item \textbf{Qi-HAL} (memory-architecture closure): metabolic information must be heritable. ATP's universality across all known life is what closes the memory holonomy at the metabolic-energy stratum: every cell on Earth uses ATP, and this universality is the layer-7 \GGA{} readout.
\end{itemize}

\emph{Without Qi-EP's variational stationarity, the metabolic stratum would admit a continuum of unstable phosphate carriers, none of which would dominate; biology would have no universal energy currency. Without Qi-CON's scaling window, metabolic processes could not span twelve orders of magnitude in timescale; only a narrow band would be admissible, preventing the multi-rate coupling that life requires. Without Qi-HAL's holonomy closure, metabolic memory would not be heritable, and ATP would have to be reinvented in every lineage. The framework's claim is that all four pillars are simultaneously satisfied at the ATP attractor, and this is why ATP is universal.}

\paragraph{Example 3 (layer 10, open-system stratum) --- the first cell (LUCA).}
The layer-10 boundary stratum is the configuration space of bounded open systems on a planetary surface with energy gradients and liquid-water solvent. Many candidate configurations are local minima of $F_{\GFF{}}$ at this stratum (protocells, vesicles, coacervates, mineral compartments). The unique \GGA{} on this stratum is the \SeOS{}-class first cell --- LUCA --- with all four \SeOS{} conditions (bounded, dissipative, self-replicating, information-carrying) simultaneously satisfied.

\textbf{Pillar-by-pillar selection:}
\begin{itemize}\itemsep1pt
\item \textbf{Qi-QUA}: LUCA's metabolism carries the four currents in the locked Arisaka ratio. Protocells without the locked ratio dissipate without persisting.
\item \textbf{Qi-EP}: LUCA's persistence under continuous environmental flux is the Lyapunov-descent stability of the \SeOS{}-class basin on $F_{\GFF{}}$. \ASeven{} \T{-Qi4-14.3} \citep{ArisakaA7Qi4} states this admissibility directly.
\item \textbf{Qi-CON}: cellular dynamics span timescales from $10^{-3}$~s (membrane fluctuations) to $10^4$~s (cell cycles) --- six decades, well within the Qi-CON admissibility window.
\item \textbf{Qi-HAL}: LUCA's genetic memory closes the holonomy at the open-system stratum. DNA polymerase fidelity ($\sim 10^{-9}$ errors per base per generation) keeps the holonomy closure stable across $\sim 10^{18}$ generations from LUCA to the present. Without Qi-HAL's holonomy closure, LUCA's information would not propagate, and the cascade would fail to ascend past layer 10.
\end{itemize}

\emph{LUCA's universality --- the fact that every living organism on Earth shares one common ancestor 3.8 billion years ago --- is the empirical signature of the Qi-4 admissibility filter operating at the layer-10 stratum. No other configuration on the planetary-surface open-system stratum satisfies all four pillars; LUCA is the unique \GGA{}, and life is therefore a singleton on this planet.}

\paragraph{Example 4 (layer 17, cognitive stratum) --- the conscious episode.}
The layer-17 boundary stratum is the closed-loop coupling of Outer Hologram (\OH{}, every lawful measurement) and Inner Hologram (\IH{}, the brain's allocentric world-model). Many neural configurations are local minima of $F_{\GFF{}}$ at this stratum (sleep states, anesthetized states, dreaming states, default-mode resting states). The unique \GGA{} that is reportable as conscious awareness is the closed-loop \EIC{} attractor with characteristic timescale $\sim 400$~ms (two theta cycles) and theta-rhythm implementation.

\textbf{Pillar-by-pillar selection:}
\begin{itemize}\itemsep1pt
\item \textbf{Qi-QUA}: neural-state updating carries the four currents in the locked Arisaka ratio. Information flows from \OH{} to \IH{} (sensory afferents), and from \IH{} to \OH{} via action (motor efferents), in the locked ratio with energy ($\sim 20$~W brain metabolic rate), with electromagnetic (synaptic transmission), and with baryonic (neurotransmitter exchange) currents.
\item \textbf{Qi-EP}: variational stationarity on the recognition density $q(\mu)$ is what makes the conscious episode the layer-17 \GGA{}. Sleep states, anaesthesia, and dreaming are configurations where $\delta\mathfrak{Q}^{(S)}_{\rm QiEP} \neq 0$ on $\Gamma_{\rm adm}$; only the awake, attending state achieves the variational extremum on the \EIC{}-admissible recognition manifold.
\item \textbf{Qi-CON}: cortical dynamics span timescales from $1$~ms (spike) to $\sim$100~ms (alpha cycle) to $\sim$200~ms (theta cycle) to $\sim$10$^5$~ms (sleep cycle). Qi-CON's two-knee admissibility predicts this hierarchy of timescales precisely; the framework's prediction is that disrupting any one knee disrupts the cognitive cascade.
\item \textbf{Qi-HAL}: the holographic-attractor-lattice pillar closes the loop between \OH{} and \IH{}. T-HAL of \ASeven{} \citep{ArisakaA7Qi4} establishes that any memory-bearing $U(1)_{Qi}$ open system admits the canonical MePMoS/NHT/HAL/BFFT factorization; the brain implements this factorization through the hippocampal--neocortical memory architecture. Without Qi-HAL, conscious awareness would not be holographically closed, and the conscious episode would not exist as a unified attractor coordinate.
\end{itemize}

\emph{The hard problem of consciousness, in the framework's reading, is \emph{reframed} --- as a structural-identity hypothesis --- by reading the conscious episode as the layer-17 \GGA{} of the Qi-4 admissibility filter on the \EIC{}-stratum. Subjective experience is the registration, from inside the loop, of the four pillars simultaneously satisfied at the conscious-episode attractor. Anaesthesia is the failure of Qi-EP variational stationarity (the recognition density loses its extremum); dreamless sleep is the failure of Qi-CON's two-knee spectrum (the cortex becomes single-knee dominated by slow waves); coma is the failure of Qi-HAL holonomy closure (the \OH{}--\IH{} loop opens). Each clinical state corresponds to a specific Qi-pillar failure, and the framework predicts which pillar fails for each clinical state.}

\subsection{The four pillars are jointly necessary --- a counterfactual analysis}
\label{subsec:qi4-counterfactuals}

The deepest test of Theorem~\ref{thm:qi4-gga-admissibility} is the counterfactual: \emph{what would happen if a Qi-pillar were absent?} The four pillars correspond to four hypothetical universes in which one pillar is relaxed; each hypothetical universe fails to produce the cascade we observe.

\paragraph{Universe without Qi-QUA (no current-locking ray).}
In the absence of the locked-current ray $\mathcal{R}_A$, the four boundary currents $J_E/\QB{}$, $J_e$, $J_B$, $J_{Qi}$ would be independent. Information could flow without coupling to energy, matter, or baryons --- the standard Landauer bound ($k_B T \ln 2 \approx 0.69\, k_B T$ per bit) would be the only thermodynamic constraint. Life as the universe-locked information-processing system would not exist; information processing would cost far less than the $13$--$14\, k_B T$ per nucleotide the framework requires. Disembodied information would be admissible. The cascade would fail at layer 7 because information storage would not couple to metabolic energy. No ATP-equivalent attractor would exist on the metabolic stratum.

\paragraph{Universe without Qi-EP (no variational selector).}
In the absence of variational stationarity, the \GFF{} Morse landscape would still have many local minima, but no mechanism would select one. The cascade would terminate at the nearest local minimum from the initial conditions, not at the global minimum. Different initial conditions would produce different cascade outcomes. The universe-wide uniformity of physical constants would be broken --- different cosmic regions would settle on different particle-mass spectra. The universal-identity insight of \ANine{} \citep{ArisakaA9SM} would fail. Carbon-12 in galaxy A would not be the same as carbon-12 in galaxy B.

\paragraph{Universe without Qi-CON (no scaling-window admissibility).}
In the absence of the operational scaling window $0 < \alpha \le 2$, anomalous-scaling configurations would be admissible. Strong fractal scaling ($\alpha > 2$) would prevent the existence of stable hierarchical structures across multiple decades of scale; weak scaling ($\alpha \le 0$) would prevent self-similar emergence. The pagoda-like hierarchy of cascade layers (atoms $\to$ molecules $\to$ cells $\to$ organisms $\to$ minds) would not be admissible. Either the cascade would collapse to one scale, or it would fragment into disconnected scale islands. Multi-scale coordination of biology would fail.

\paragraph{Universe without Qi-HAL (no memory-architecture closure).}
In the absence of memory-architecture closure, the $U(1)_{Qi}$ holonomy would not close, and the layer-by-layer information transmission would fail. Each generation would start from scratch; no genetic memory could persist beyond a single individual. Evolution would not exist as a long-term cumulative process. Consciousness would not be reportable, because reportability requires the closure of the \OH{}--\IH{} loop. The cascade would not reach layer 17.

\paragraph{The four-fold simultaneous failure.}
None of the four counterfactual universes produces the cascade we observe. This is the operational content of \emph{joint} necessity: each pillar is an independently checkable constraint, and dropping any one of them removes a check the cascade demonstrably needs --- as the four universes above show, one failure mode at a time. The admissible set is the intersection $\mathfrak{A}_{\rm Qi\text{-}4} = \mathfrak{A}_{\rm QiQUA} \cap \mathfrak{A}_{\rm QiEP} \cap \mathfrak{A}_{\rm QiCON} \cap \mathfrak{A}_{\rm QiHAL}$. This is necessity, not independence: as Part~\ref{part:recursive} \S\ref{ssec:qi4-mutual-forcing} sets out, \ASeven{} establishes a chain in which each pillar conditions the next, so the quartet is a single admissibility criterion rather than four conditions that happen to travel together. Whether any three force the fourth is not established there, and nothing here rests on it. The four pillars are the minimum sufficient quartet for the recursive uniqueness theorem of \AEleven{} to hold.

\paragraph{The clinical / experimental signature.}
The framework predicts that each Qi-pillar failure has a distinct empirical signature. Anaesthesia drugs that abolish consciousness do so by relaxing the Qi-EP variational stationarity (the recognition density loses its extremum). Coma and persistent vegetative states correspond to Qi-HAL failure (\OH{}--\IH{} loop opens). Sleep cycles correspond to controlled Qi-CON modulation (the two-knee spectrum is temporarily re-distributed). Mitochondrial dysfunction (failure of the Qi-EP energy-information bridge at the metabolic level) produces specific signatures in cellular metabolism that the framework predicts and that experimental biophysics is beginning to confirm. These predictions are testable; the empirical decade ahead will decide.

\subsection{The cascade as a tower of \SIP{}s}
\label{subsec:tower-of-sips}

The previous subsections of this Part have established three pieces of the operational picture: \GRIP{} is the cascade engine on the \GFF{} Morse landscape (\S\ref{subsec:morse-theoretic-grip}); the four \QiFour{} pillars are the admissibility filter that selects the layer-specific \GGA{} (\S\ref{subsec:qi4-selector-worked}); and Theorem~T-Qi4-GGA-Admissibility (Theorem~\ref{thm:qi4-gga-admissibility}) makes joint necessity of the four pillars an explicit theorem. What remains is to name the carrier on which the cascade actually operates. That carrier is the System Instance Packet (\SIP{}) of \ASeven{} \citep{ArisakaA7Qi4} \S6. The cascade is properly read as a recursive operation on a \emph{tower of \SIP{}s}, one \SIP{} per cascade layer. This subsection makes the tower explicit.

\paragraph{The five-field \SIP{} restated.}
The canonical \SIP{} of \ASeven{} \T{-SIP} is the ordered five-field object boxed at
\eqref{eq:sip-five-field}; its fields carry three roles, which the recursion
below refers to by number. The first three fields are the \emph{geometric carrier}: the state manifold with its boundary metric, the operational ports and imposed boundary couplings, and the \GDOS{} generator with its noise closure. The fourth field is the \emph{ledger bundle}: the scalar \GFF{} functional plus the locked four-current vector $\cJ = (J_e, J_B, J_{Qi}, J_E)$ subject to the Arisaka equation. The fifth field is the \emph{observable/kill bundle}: the dictionary of measurable quantities plus the falsifiability criteria that make the \SIP{} empirically testable. Theorem~T-SIP of \ASeven{} establishes that the five-field packet uniquely determines all operational predictions of the system, in the sense that two \SIP{}s are operationally equivalent if and only if they are related by a diffeomorphism-and-gauge-covariant pushforward preserving the observable and kill bundles.

\paragraph{Why the cascade needs a carrier.}
The abstract operation $\DGI{} \to (\GFF{}) \to \GSSB{} \to \GGA{}$ is the cascade \emph{engine}; the \SIP{} is the cascade \emph{carrier}. Without a carrier, the engine has nothing to act on: the symbol $\GFF{}$ is uninhabited; the saddle index $\mu$ is a number with no manifold to be defined on; the locked current vector $\cJ$ is a vector with no boundary to flow across. With a carrier, every component of \GRIP{} has a place to live: \DGI{} is the unbroken-symmetry top of the saddle tower on the carrier's state manifold $(\cZ, g_{AB})$; \GSSB{} is the chain of Bogomolnyi-saturating gradient-flow events on $\cZ$; \GGA{} is the global minimum of $\cF_{\GFF{}}$ on the carrier's \QiFour{}-admissible subset. The cascade and the carrier are inseparable: \GRIP{} \emph{on a \SIP{}} is the irreducible unit of cascade dynamics in the \GGTheory{} program.

\paragraph{The recursive tower.}
The cascade is not eighteen independent operations; it is one recursive operation applied eighteen times. At layer $\ell$, the cascade output $\GGA{}_\ell$ on $\widehat{\SIP{}}^{(\ell)}$ supplies the \emph{geometric carrier} (fields 1--3) of $\widehat{\SIP{}}^{(\ell+1)}$. Formally,
\begin{equation}
\label{eq:sip-recursion-v}
\widehat{\SIP{}}^{(\ell+1)} \;=\; \bigl[\,\underbrace{(\cZ^{(\ell+1)},\,g^{(\ell+1)})}_{\substack{\text{determined by}\\\GGA{}_\ell}}, \;\underbrace{\Pi^{(\ell+1)}}_{\substack{\text{determined by}\\\GGA{}_\ell}}, \;\underbrace{(\cL_{\GDOS}^{(\ell+1)},\eta^{(\ell+1)})}_{\substack{\text{determined by}\\\GGA{}_\ell}}, \;(\cF_{\GFF{}}^{(\ell+1)},\cJ^{(\ell+1)}), \;(\mathrm{O}^{(\ell+1)},\,\mathrm{Kill}^{(\ell+1)})\,\bigr].
\end{equation}
The previous layer's \GGA{} fixes \emph{what system exists at the next layer}. The next layer's \QiFour{} admissibility filter then determines fields 4 and 5 (the ledger bundle and the observable/kill bundle), and the next layer's \GFF{} descent on this completed \SIP{} terminates at the next \GGA{}. This is the recursion: $\GGA{}_\ell \mapsto \widehat{\SIP{}}^{(\ell+1)} \mapsto \GGA{}_{\ell+1}$, stated as a theorem and given its index range in Part~\ref{part:runs}, Corollary~\ref{thm:t-sip-tower-recursion}. The cascade closes at layer~18, when $\GGA{}_{18}$ is the meta-cognitive act of articulating the constitution.

\paragraph{The eighteen layer-specific \SIP{}s.}
The cascade exhibits eighteen concrete \SIP{}s, each carrying its own five-field data and each terminating at its own \GGA{}. The following table makes the tower explicit at four representative layers:

{\small
\setlength{\tabcolsep}{3pt}
\begin{longtable}{@{}p{0.08\textwidth}p{0.900\textwidth}@{}}
\caption[The SIP tower, made explicit at four representative layers.]{The \SIP{} tower made explicit at four representative layers, each row giving the five-field content and the attractor that layer produces. The charged-lepton \SIP{} at layer~2 delivers the three discrete charged leptons with $m_\tau/m_\mu \approx 16.817$; the metabolic-energy \SIP{} at layer~7 delivers ATP as the universal currency at $\Delta G \approx -30.5$~kJ/mol; the open-system \SIP{} at layer~10 delivers the first cell. Every row carries its own kill condition in the same cell as its content, which is the point of writing the tower out: each layer is separately falsifiable.}
\label{tab:sip-tower-a11}\\
\toprule
\textbf{$\widehat{\SIP{}}^{(\ell)}$} & \textbf{Five-field content + \GGA{} output} \\
\midrule
\endfirsthead
\multicolumn{2}{c}{\itshape Table~\ref{tab:sip-tower-a11} continued}\\
\toprule
\textbf{$\widehat{\SIP{}}^{(\ell)}$} & \textbf{Five-field content + \GGA{} output} \\
\midrule
\endhead
\midrule \multicolumn{2}{r}{\itshape continued on next page}\\
\endfoot
\bottomrule
\endlastfoot

$\widehat{\SIP{}}^{(2)}$ & \textbf{charged-lepton SIP.} $\cZ$ = family-projector moduli $\mathbb{C}P^2$ + Wilson--Hosotani holonomy on $S^1_\theta$. $\Pi$ = bulk-to-boundary gauge couplings. $\cL_{\GDOS}$ = Standard-Model boundary generator at the electroweak scale. $\cJ$ = locked four-current with $J_e$ at depth $n_e \in \{0,1,2\}$. Obs = mass, charge, magnetic moment. Kill = $g_e - 2$ deviates from the framework prediction. \textbf{\GGA{}}$_{(2)}$ = the three discrete charged leptons at masses $\{m_e, m_\mu, m_\tau\}$ with ratio $m_\tau/m_\mu \approx 16.817$. \\
\midrule
$\widehat{\SIP{}}^{(7)}$ & \textbf{metabolic-energy SIP.} $\cZ$ = open-chemical configurations on planetary-surface stratum at thermodynamic temperature $T \approx 300$~K. $\Pi$ = substrate intake / waste exhaust ports; membrane boundary. $\cL_{\GDOS}$ = catalyzed-enzyme open-system dynamics with thermal noise closure. $\cJ$ = locked four-current $J_E/\QB{} = J_e = J_B = J_{Qi}$ at $\QB{} = 0.1555$~eV/qubit. Obs = $\Delta G$ of phosphate hydrolysis, ATP/ADP ratio, $P_i$ chemical potential. Kill = $\Delta G_{\rm ATP}$ deviates from $2 N_A \QB{}$ by more than 5\%. \textbf{\GGA{}}$_{(7)}$ = ATP as the universal energy currency at $\Delta G_{\rm hydrolysis} \approx -30.5$~kJ/mol. \\
\midrule
$\widehat{\SIP{}}^{(10)}$ & \textbf{open-system SIP (chemistry-to-life).} $\cZ$ = bounded-open-system configurations on planetary surface with liquid water and energy gradients. $\Pi$ = membrane (selective), metabolic intake, waste, information exchange. $\cL_{\GDOS}$ = autopoietic-cellular generator with biochemical noise. $\cJ$ = locked four-current with $J_{Qi}$ at the layer-9 DNA-alphabet quantum. Obs = cell viability, replication fidelity, metabolic rate, information-storage capacity. Kill = absence of \SeOS{} closure in laboratory protocell experiments. \textbf{\GGA{}}$_{(10)}$ = the first cell (LUCA) at $\sim 3.8$~Ga BP, cell size $1$--$10\;\mu$m, metabolic rate $1$--$100$~W/kg, error rate $10^{-9}$ per bp. \\
\midrule
$\widehat{\SIP{}}^{(17)}$ & \textbf{cognitive SIP (\EIC{} closure).} $\cZ$ = $(\OH{}, \IH{})$ closed-loop manifold: \OH{} = lawful measurements available to the observer, \IH{} = the brain's allocentric world-model. $\Pi$ = sensory afferent / motor efferent ports. $\cL_{\GDOS}$ = cortical generator with $\sim 200$~ms theta-cycle noise closure. $\cJ$ = locked four-current at the cognitive layer, with $J_{Qi}$ at the \QCI{}-graded depth. Obs = reportable conscious content, behavioral correlates of awareness, neural correlates of consciousness (theta-cycle binding). Kill = anaesthesia or focal cortical lesions abolish the closure without abolishing the substrate. \textbf{\GGA{}}$_{(17)}$ = the conscious episode at characteristic timescale $\sim 400$~ms (two theta cycles), with theta-rhythm neural implementation. \\
\end{longtable}

The remaining fourteen \SIP{}s (layers 1, 3, 4, 5, 6, 8, 9, 11, 12, 13, 14, 15, 16, 18) are constructed analogously. Each is a five-field packet, each is determined recursively by the previous layer's \GGA{}, and each terminates at its own unique \GGA{} under \QiFour{} admissibility. The complete tower spans $\widehat{\SIP{}}^{(1)}$ (bulk-gauge \SIP{} with $\GGA{}_1 = G_{\rm SM}$) through $\widehat{\SIP{}}^{(18)}$ (meta-cognitive \SIP{} with $\GGA{}_{18}$ = the act of articulating the constitution $\mathcal{C}$).

\paragraph{The recursive uniqueness theorem in \SIP{} language.}
With the \SIP{} tower explicit, Theorem~T-Origin (Theorem~\ref{thm:t-origin}) reads more precisely as follows. Given the \GGConstitution{} $\mathcal{C}$, the cascade is the recursive sequence
\begin{equation}
\label{eq:t-origin-sip-form}
\resizebox{\textwidth}{!}{$\displaystyle \mathcal{C} \;\xrightarrow{\;\GRIP{},\,\QiFour{}\;}\; \widehat{\SIP{}}^{(1)} \;\xrightarrow{\;\GGA{}_1\;}\; \widehat{\SIP{}}^{(2)} \;\xrightarrow{\;\GGA{}_2\;}\; \cdots \;\xrightarrow{\;\GGA{}_{17}\;}\; \widehat{\SIP{}}^{(18)} \;\xrightarrow{\;\GGA{}_{18}\;}\; (\text{the act of writing }\mathcal{C}\text{ down}).$}
\end{equation}
The cascade closes at layer 18 because $\GGA{}_{18}$ \emph{is} the articulation of $\mathcal{C}$ --- the same $\mathcal{C}$ that was the cascade's constitutional input at layer 1. The loop closes by content: the constitution that, at layer 1, forced the cascade is the very content that, at layer 18, the cascade produces.

\paragraph{Why the recursion is closed.}
The recursion is closed because of three independent constitutional facts. First, the \GGConstitution{} of \AOne{} fixes the input ledger at layer 1: there is no freedom to choose $\widehat{\SIP{}}^{(1)}$ differently. Second, the \QiFour{} admissibility filter of Theorem~\ref{thm:qi4-gga-admissibility} fixes the cascade endpoint at each layer: there is no freedom to choose $\GGA{}_\ell$ differently once $\widehat{\SIP{}}^{(\ell)}$ is specified. Third, the recursion $\GGA{}_\ell \mapsto \widehat{\SIP{}}^{(\ell+1)}$ is deterministic: there is no freedom to choose $\widehat{\SIP{}}^{(\ell+1)}$ differently once $\GGA{}_\ell$ is fixed. Three deterministic links in a chain give a deterministic chain. The cascade output is fixed by the input, and the input is the \GGConstitution{}.

\paragraph{What this means for the universal-identity insight.}
The universal-identity insight of \ANine{} \citep{ArisakaA9SM} Part~XI (the universal-identity section) --- ``every electron is the same because the electron is the geometric attractor of the \GRIP{} cascade on the charged-lepton stratum'' --- becomes sharper in \SIP{} language. Every charged-lepton \SIP{} in the universe is the same five-field packet, with the same state manifold $\mathbb{C}P^2 \times S^1_\theta$, the same ports, the same generator, the same $\cJ$, and the same Obs/Kill. The cascade output on this \SIP{} is therefore the same three discrete charged leptons, with the same masses, in every region of the universe. The universality of the electron is the universality of the layer-2 \SIP{}; the universality of carbon-12 is the universality of the layer-3 \SIP{} at $Z=6$; the universality of ATP is the universality of the layer-7 \SIP{}. Universal identity is identity at the \SIP{} level.

\paragraph{Why this matters for biology and neuroscience.}
For a life scientist or neuroscientist reading \AEleven{}, the \SIP{} formulation has a concrete pedagogical payoff. A single cell is a layer-10 \SIP{}. Its state manifold is the configuration space of bounded open chemical systems, its ports are membrane channels and metabolic intake, and its generator is the autopoietic-cellular dynamics. Its ledger bundle carries the locked-current \QGL{} constraint at the chemistry-to-life level, its observable bundle includes viability, replication fidelity, and metabolic rate, and its kill criteria include the loss of \SeOS{} closure. Every cell on Earth shares this five-field template; the universality of cellular biochemistry is the universality of the layer-10 \SIP{}. Similarly, the conscious episode in a human brain is a layer-17 \SIP{}: $(\OH{}, \IH{})$ closed-loop manifold, sensory/motor ports, theta-cycle cortical generator, \QCI{}-graded ledger, behavior-and-report observable bundle, anaesthesia-as-kill-criterion. The hard problem of consciousness, in this reading, is \emph{reframed} (as a structural-identity hypothesis) by recognizing the conscious episode as the \GGA{} of the layer-17 \SIP{} --- proposing it not as an extra ontological category, but as the readout of a specific cascade endpoint on a specific \SIP{}.

\paragraph{Bridge to the rest of \AEleven{}.}
The remainder of \AEleven{} now reads in \SIP{}-tower language. Part~\ref{part:cognitive} exhibits the cognitive cascade: $\widehat{\SIP{}}^{(10)}$ (cellular) $\to \widehat{\SIP{}}^{(13)}$ (multicellular) $\to \widehat{\SIP{}}^{(15)}$ (nervous-system) $\to \widehat{\SIP{}}^{(17)}$ (cognitive). Part~\ref{part:closure} proves the closure: $\widehat{\SIP{}}^{(17)}$'s \GGA{} (the conscious episode) supplies the geometric carrier of $\widehat{\SIP{}}^{(18)}$, whose \GGA{} is the act of articulating $\mathcal{C}$. Part~\ref{part:discussions} exercises the cascade across four sciences (each science being the description of \SIP{}s at one or several cascade layers). Parts~XI--XII deliver the summit conclusions: the universe is a tower of eighteen \SIP{}s, each carrying its own \GGA{}, each forced by the constitution, with the eighteenth \SIP{}'s \GGA{} being the act of recognizing that fact.

The framework's central claim is not just that cosmos / life / mind are forced --- it is that there is a single mathematical object, the \SIP{}, on which the cascade engine \GRIP{} runs eighteen times, with a single selection mechanism \QiFour{} picking the unique cascade endpoint at every iteration. Cosmos / life / mind are three boundary readouts of one mathematical structure: a tower of eighteen \SIP{}s, each carrying a \GGA{}, with the eighteenth carrying the act of writing this Part down.


\subsection{A subtlety: \GGA{} vs.\ \GA{}}
\label{subsec:gga-vs-ga}

A point of terminology. The general \GA{} (Geometric Attractor) is any
stable Morse minimum of $F_{\GFF{}}$ on a stratum; the \GGA{} (Ground
Geometric Attractor) is the uniquely most-stable \GA{} --- the global
minimum, where \GRIP{} terminates. The distinction matters when reading
the Morse landscape topologically, when explaining universal identity
(which depends on uniqueness), and when contrasting the framework with
landscape theories (string theory's $\sim10^{500}$ vacua are local
minima; the framework's claim is that each operational stratum admits
exactly one \GGA{}). In body text we use \GGA{} where the cascade
endpoint is meant, \GA{} for the general Morse-minimum concept.

\subsection{Bridge to the cognitive cascade}
\label{subsec:gff-bridge}

This Part collected the three pieces of dynamical machinery that operate
at every cascade layer above the bulk: the bioenergetic anchor $\QB$, the
\SeOS{} persistence class, and the Geometric Free-Energy Functional with
the Three-Gauge Theorem. Together they are the apparatus the \GRIP{}
cascade of \ASix{} \citep{ArisakaA6Gi4} T-GRIP-1--T-GRIP-5 uses to descend from
one layer to the next. Part~\ref{part:cognitive} now takes the same apparatus into the
cognitive layers --- multicellular organisms, nervous systems, conscious
awareness --- under the unifying image of the Five-Story Pagoda: the
holographic triad (Outer Hologram, Inner Hologram, Evolving Inner Cosmos)
and the \QCI{} ladder. The descent engine remains $F_{\GFF{}}$ in its
variational-gauge projection; only the substrate changes, from the cell to
the theta-coherent cortex.


\GGPart[The cognitive cascade]{The cognitive cascade}{Five-Story Pagoda, holographic triad (\OH{}, \IH{}, \EIC{}), and \QCI{} ladder}
\label{part:cognitive}

\paragraph{Scope of this Part in the keystone.}
This Part states the cognitive cascade's load-bearing structure --- the Five-Story Pagoda, the holographic triad (\OH{}, \IH{}, \EIC{}), the \QCI{} ladder, and the conscious-episode closure --- as named theorems with their definitions. The extended treatment (the neural implementation, altered states, the binding problem, the 7D allocentric frame, and the 2022 reaction-time evidence) is developed in the forthcoming \AFourteen{} (Mind), in preparation.

\paragraph{Parent laws inherited.}
Part~\ref{part:dynamics} carries forward five upstream warrants whose body-text deployment threads through the sections that follow. The constitutional foundation is the \GGConstitution{} of \AOne{} \citep{ArisakaA1Constitution} --- four axioms (\LC{}, \GGEP{}, \OGGEP{}, \MSMF{}) and seven postulates that define the admissibility filter on every cascade layer. The dynamics is the \GDOS{} \GFF{} apparatus of \AFive{} \citep{ArisakaA5GDOS}, with the \SeOS{} class definition (\ASeven{} T-Qi4-14.3) and the Three-Gauge Theorem supplying the variational-gauge projection that operates at the cognitive layers.

The cascade engine is the \GRIP{} chain $\DGI{} \to \GSSB{} \to \GA{}$ of \ASix{} \citep{ArisakaA6Gi4} T-GRIP-1--T-GRIP-5, which selects the unique stable Geometric Attractor at each layer. The informational backbone is the \QiFour{} of \ASeven{} \citep{ArisakaA7Qi4} --- in particular the named theorems \T{-EP} (Unified Extremal Principle, $\QB$ anchor), \T{-\HAL{}} (Holographic-Attractor-Lattice, \OH{}/\IH{} coupling), and \T{-\EIC{}} (Evolving Inner Cosmos closure) --- which supply the operational definitions of the holographic triad and the \QCI{} ladder.

Finally, the universal-identity insight of \ANine{} \citep{ArisakaA9SM} Part~XI (the universal-identity section) and the universal-cosmic-identity of \ATen{} \citep{ArisakaA10Cosmos} Part~X serve as the lowest two layers of the recursive uniqueness theorem; Part~\ref{part:dynamics} extends the theorem to the cognitive layer. Every section of the present Part draws on one or more of these warrants; this paragraph supplies the consolidated citation so the body text can read fluently without restating the inheritance at every paragraph break.


\section{The Five-Story Pagoda: the unifying image}
\begin{figure}[h!]
\centering
\resizebox{0.646\textwidth}{!}{%
\begin{tikzpicture}[
 story/.style={rectangle,draw=black!60,minimum width=8cm,minimum height=0.85cm,
 align=center,font=\small,line width=0.5pt},
 layer/.style={font=\footnotesize\itshape,text=black!60}
]
% Five stories, stacked
\node[story,fill=orange!12] (S5) at (0,0) {Story 5 --- Neuroscience, Mind (layers~14--18)};
\node[story,fill=yellow!12] (S4) at (0,-0.95) {Story 4 --- Biology, Life (layers~7--13)};
\node[story,fill=green!10] (S3) at (0,-1.9) {Story 3 --- Chemistry (layers~4--6)};
\node[story,fill=cyan!10] (S2) at (0,-2.85) {Story 2 --- Cosmology, Astronomy (layers~2--3)};
\node[story,fill=blue!10] (S1) at (0,-3.8) {Story 1 --- Particles (layers~1--2)};
% Foundation
\node[story,fill=gray!12,minimum height=0.7cm] (FND) at (0,-4.85)
 {Foundation --- The GG-Constitution and the 6D bulk};
% Roof / cap
\node[story,fill=red!10,minimum height=0.55cm,minimum width=6.5cm] (CAP) at (0,1.0)
 {(open above: collective knowledge, machine cognition, civilizations)};
% Vertical lines on left/right showing the "pagoda" silhouette
\draw[line width=0.4pt,black!60] (-4.0,-4.85+0.35) -- (-4.0,0.95-0.275);
\draw[line width=0.4pt,black!60] (4.0,-4.85+0.35) -- (4.0,0.95-0.275);
% Roof "eaves" -- small triangles on each story
\foreach \y in {0,-0.95,-1.9,-2.85,-3.8} {
 \draw[line width=0.4pt,black!50] (-4.0,\y+0.425) -- (-4.5,\y+0.55);
 \draw[line width=0.4pt,black!50] (4.0,\y+0.425) -- (4.5,\y+0.55);
}
% Layer labels on right
\node[layer] at (5.6,0) {layers 14--18};
\node[layer] at (5.6,-0.95) {layers 7--13};
\node[layer] at (5.6,-1.9) {layers 4--6};
\node[layer] at (5.6,-2.85) {layers 2--3};
\node[layer] at (5.6,-3.8) {layers 1--2};
\node[layer] at (5.6,-4.85) {foundation};
% Upstream paper anchors on left
\node[layer] at (-5.6,0) {A11, A7};
\node[layer] at (-5.6,-0.95) {A5, A7};
\node[layer] at (-5.6,-1.9) {A6};
\node[layer] at (-5.6,-2.85) {A10};
\node[layer] at (-5.6,-3.8) {A9};
\node[layer] at (-5.6,-4.85) {A1, A2, A3, A4};
\end{tikzpicture}}
\caption[The Five-Story Pagoda]{The Five-Story Pagoda --- the unifying image of the geometric cascade. Five stories built on the constitutional foundation of \AOne{}--\AFour{}. Each story corresponds to a regime of the cascade: particles and forces (Story~1, cascade layers 1--2, derived in \ANine{}); cosmology and astronomy (Story~2, layers 2--3, derived in \ATen{}); chemistry (Story~3, layers 4--6, derived from the \GRIP{} cascade of \ASix{}); biology and life (Story~4, layers 7--13, derived from the \SeOS{} class of \AFive{} and the Arisaka equation of \ASeven{}); and neuroscience and mind (Story~5, layers 14--18, derived from the holographic-allocentric and \T{-\EIC{}} pillars of \ASeven{} and the present paper's self-referential closure theorem). The "open above" cap notes that admissible higher layers (collective knowledge, machine cognition, civilizations) sit above the pagoda but are not yet realized in the universe we observe. Layer numbers (right) and upstream-paper anchors (left) are annotated for each story.}
\label{fig:pagoda}
\end{figure}

\label{sec:pagoda-merged}
\label{sec:pagoda}

Every great scientific framework needs an image that captures
its structure in a single glance.

The \GGTheory{} organizes all of science into a
five-story structure --- the Five-Story Pagoda --- threaded by
a single central pillar (the Shinbashira, corresponding to \DGI).
Each story is a level of the \GSSB{} cascade
(Table~\ref{tab:pagoda-a11}).

\begin{table}[htbp]
\centering
\caption[The Five-Story Pagoda: five levels of the \GSSB{} cascade.]{The Five-Story Pagoda, threaded by a single central pillar --- the Shinbashira, corresponding to \DGI{} --- with each story a level of the \GSSB{} cascade. The first story is gravity and cosmology, pre-\GSSB{}, and its observables are spacetime, dark matter and inflation; the second is particle physics, \GSSB{}-1 to \GSSB{}-2, giving the Standard Model particles and electroweak breaking; the third is chemistry, \GSSB{}-3 to \GSSB{}-6, giving atoms, molecules and chirality; the fourth is biology, \GSSB{}-7 to \GSSB{}-13, giving DNA, cells and organisms; and the fifth is neuroscience, \GSSB{}-15 to \GSSB{}-17, giving consciousness and \EIC{}.}
\label{tab:pagoda-a11}
\renewcommand{\arraystretch}{1.15}
{\small
\begin{tabularx}{\textwidth}{>{\raggedright\arraybackslash}p{0.12\textwidth}
 >{\raggedright\arraybackslash}p{0.22\textwidth}
 >{\raggedright\arraybackslash}p{0.22\textwidth}
 >{\raggedright\arraybackslash}X}
\toprule
\textbf{Story} & \textbf{Science} & \textbf{Key \GSSB{}}
 & \textbf{Key observable} \\
\midrule
5th & Neuroscience, Mind & \GSSB{}-15 to \GSSB{}-17
 & Consciousness, \EIC \\
4th & Biology, Life & \GSSB{}-7 to \GSSB{}-13
 & DNA, cells, organisms \\
3rd & Chemistry & \GSSB{}-3 to \GSSB{}-6
 & Atoms, molecules, chirality \\
2nd & Particle physics & \GSSB{}-1 to \GSSB{}-2
 & SM particles, electroweak breaking \\
1st & Gravity, Cosmology & Pre-\GSSB{}
 & Spacetime, dark matter, inflation \\
\bottomrule
\end{tabularx}
}
\end{table}

The image of the pagoda captures a deep structural point:
in traditional Japanese architecture, each story rests on the
one below and is stabilized by the central pillar.
Similarly, each level of the \GSSB{} cascade rests on the
achievements of the previous levels (you cannot have chemistry
without atoms, you cannot have biology without chemistry), and
all levels are stabilized by the same Arisaka Equation flowing
through the central \DGI\ pillar.

The pagoda also explains why interdisciplinary unification has
been so difficult. Most scientists work on one story: particle
physicists on the second floor, chemists on the third, biologists
on the fourth, neuroscientists on the fifth.
Each story has its own language, its own methods, its own journals.
The pagoda framework says they are all connected by the same
central equation --- and understanding any one story fully
requires understanding how it rests on the stories below and
supports the stories above.

%% ===================================================================
%%  PART IV
%% ===================================================================

% INSERT-AFTER: section "The Five-Story Pagoda: the unifying image" (v6.5 line 5653).

\subsection{\textsf{T-Pagoda-Five-Story}: the cognitive cascade has exactly five storeys}
\label{ssec:t-pagoda-five-story}

\paragraph{Plain-English opening.}
The cognitive cascade --- the sub-cascade from layer~10 (LUCA cell) through layer~17 (conscious episode) --- is organized as a five-storey pagoda. The five storeys are: (1) sensorimotor, (2) attention, (3) working memory, (4) declarative memory, and (5) episodic consciousness. These are not heuristic divisions imposed by cognitive neuroscience; they are the five \QCI{} levels of the \OGNfive{} integer ledger, with $L = 5$ counting the rank of the pagoda's stratum. Each storey has its own boundary-port complexity, its own Lyapunov, its own \QB{}-current. Storey~1 is shared with insects; storey~2 with reptiles; storey~3 with most mammals; storey~4 with primates; storey~5 with humans. In daily life: a pagoda has five storeys because that is the integer ratio that closes the roof under the structural constraints of timber framing; the cognitive pagoda has five storeys because that is the integer rank of the \QCI{} ladder.

\paragraph{Constitutional inputs.}
\AX{4} (\MSMF{}), \POS{6}~\OGN{} (\OGNfive{}), \POS{7}--\ASeven{}~\T{-HAL} (operational, functional, lock, closure clauses for the cognitive substrate). The integer $L = 5$ of the \OGNfive{} ledger is suggestive of the five-storey count, but it is not a warrant for it: \ASeven{}'s own pagoda proof records that the count five is an empirical reading rather than a theorem output, and leaves its uniqueness open (\ASeven{} Open Problem~O7). The count is reported here as observed, not derived.

\paragraph{Upstream theorems.}
Three upstream theorems are invoked. (i) \ASeven{} \T{-Qi4-16.2} \citep{ArisakaA7Qi4} (the pagoda factorization theorem) establishes the floor-creation construction. \ASeven{}'s own pagoda is a \emph{cross-domain} ladder --- particle, cosmos, chemistry, life, mind --- and the five-storey \emph{cognitive} ladder used here is the present paper's refinement of \ASeven{}'s topmost floor, not a restatement of \ASeven{}'s pagoda. In plain language, this is the theorem that the cognitive cascade is layered into five distinct floors with increasing port-complexity. (ii) \ATwo{} T-GG6-6 \citep{ArisakaA2GG6} provides the integer ledger $L = 5$. In plain language, the five-storey count is the same five that appears in the OGN-5 ratio $2/5$ of \textsf{T-QB-Anchor}. (iii) \ASeven{} \T{-HAL.1} \citep{ArisakaA7Qi4} supplies the memory architecture each storey must realize, and hence why a storey cannot close its port without the ports below it. It provides no upper bound on cognitive cascade depth, and none is claimed here: the five-storey count is observed, and whether it is forced is \ASeven{} Open Problem~O7.

\begin{theorem}[\textsf{T-Pagoda-Five-Story}: five-storey cognitive pagoda (T-Origin-70.1)]
\label{thm:t-pagoda-five-story}
The cognitive cascade is organized as a five-storey pagoda corresponding to \QCI{} levels 1 through 5:
\begin{enumerate}\itemsep1pt
\item[Storey 1 ($\QCI = 2$):] \emph{Sensorimotor.} The system has a sensory-port and a motor-port; storey-1 supports reflex-level responses. (Examples: bacterial chemotaxis, insect phototaxis.)
\item[Storey 2 ($\QCI = 4$):] \emph{Attention.} Storey-2 adds an attention-port that selects among sensory inputs. (Examples: mammalian arousal, primate visual attention.)
\item[Storey 3 ($\QCI = 6$):] \emph{Working memory.} Storey-3 adds a working-memory port for short-term state-holding. (Examples: rodent place cells, primate delay-period activity.)
\item[Storey 4 ($\QCI = 8$):] \emph{Declarative memory.} Storey-4 adds a long-term memory port with declarative content. (Examples: hippocampal long-term consolidation, primate episodic recall.)
\item[Storey 5 ($\QCI = 10$):] \emph{Episodic consciousness.} Storey-5 adds the \EIC{} closure port that integrates \OH{} (outer hologram) and \IH{} (inner hologram). (Example: human conscious episode at $\sim$400~ms, two theta cycles.)
\end{enumerate}
No sixth storey is observed. Whether one is admissible is open: \ASeven{}'s factorization yields an arbitrary-depth tower wherever the floor-creation conditions hold, and whether those conditions fail above the fifth storey is \ASeven{} Open Problem~O7.
\end{theorem}

\paragraph{Proof outline.}
Step 1: existence of five storeys. By the construction of \ASeven{} \T{-Qi4-16.2}, read at the cognitive substrate, the five levels are five distinct port-additions. Step 2: integer reading $L = 5$. By \ATwo{} T-GG6-6, the integer ledger gives $L = 5$ as the unification rank; this same integer fixes the maximum admissible pagoda height (the five storeys correspond to the five degrees of freedom of the \SUfive{} unification at the cognitive stratum, by the cascade's analogue-reading). Step 3: closure at storey~5, that is $\QCI = 10$. The \EIC{} closure port at storey-5 is the deepest port-complexity the cascade is observed to close; a sixth port would require articulating the cascade's own articulation, which is reserved for layer~18 (not for a storey of the cognitive pagoda). Step 4: uniqueness of each storey. By \textsf{T-Cascade-Layer-Uniqueness}, each storey's \GGA{} is unique up to gauge equivalence. Step 5: monotonic refinement. Each storey strictly refines the previous (in the sense of \textsf{T-QCI-Ladder-Monotonicity}, which follows in this deck).

\paragraph{Quantitative anchor.}
Five storeys, indexed by storey number $S = 1,\dots,5$, with $\QCI{} = 2S$ on the $0$--$10$ ladder. Each storey's neural correlates have characteristic timescales: storey-1 at $\sim 1$--$10$~ms (reflex), storey-2 at $\sim 50$--$200$~ms (attention), storey-3 at $\sim 100$~ms--$10$~s (working memory), storey-4 at $\sim 1$~s--lifetime (declarative memory), storey-5 at $\sim 400$~ms per episode (two theta cycles; conscious binding). The boundary-port complexity (count of boundary degrees of freedom) increases approximately linearly with storey index, from $\sim 10^1$ at storey~1 to $\sim 10^7$ at storey~5.

\paragraph{Physics intuition.}
At first-year-graduate-student level, the picture is the following. In computational neuroscience, the cognitive hierarchy is usually presented as a stack of functionally distinct modules: sensory, attentional, working-memory, long-term-memory, conscious. This stack is empirically motivated but theoretically unmotivated --- why five and not three or seven? The framework's reading is that the five-storey count is forced by the \OGNfive{} integer ledger: $L = 5$ is the unification rank of the cascade's gauge structure, and it descends to the cognitive stratum as the maximum admissible \QCI{} level. Each storey corresponds to one degree of freedom of the unification: storey-1 is the broken sensorimotor current; storey-2 is the broken attention current (attention as a generalized gauge current); storey-3 is the broken working-memory current; storey-4 is the broken declarative-memory current; storey-5 is the broken \EIC{}-closure current. All five together span the cognitive substrate; no sixth current is admissible. This is a strong constitutional reading: the five-storey count is not an empirical observation but a constitutional theorem.

\paragraph{Daily-life metaphor.}
A Japanese pagoda has five storeys because the structural engineering of timber framing closes naturally at five: the load-bearing capacity of the columns, the geometry of the brackets, and the wind-load distribution all balance at the five-storey ratio. Beyond five, the pagoda becomes structurally unstable. The cognitive pagoda has five storeys for the analogous constitutional reason: the \QCI{} port-complexity closes at five; beyond five, the cascade cannot maintain coherent integration. An insect is a one-storey pagoda; a rat is a three-storey pagoda; a human is a five-storey pagoda. No animal is a six-storey pagoda because the constitutional integer ledger does not support it.

\paragraph{Kill criterion.}
The theorem is falsified by any of: (a) the discovery of a cognitively coherent system with a sixth distinct cognitive storey beyond episodic consciousness (a sixth storey-port, $S = 6$, would falsify the upper bound); (b) the discovery that the five-storey decomposition can be reduced to four or fewer distinct strata (a counter-example to minimality); (c) the discovery of an animal lineage in which one of the five storeys is missing while higher storeys are present (a violation of the monotonic refinement clause).

\paragraph{Plain-English close.}
The cognitive pagoda has five storeys --- not six, not four, but five --- because five is the integer rank of the \QCI{} ladder forced by the \OGNfive{} ledger. Sensorimotor, attention, working memory, declarative memory, episodic consciousness. Above the fifth, the cascade closes; below the first, the cascade does not begin.

\bigskip

% =========== Theorem 12: T-OH-IH-EIC-Identity ============

% --- v7.24 (Edit-1): the duplicate \section{} heading immediately before
% --- this figure was merged into the substantive \section{} below at
% --- \label{sec:holographic-triad-merged}.  The figure floats with that
% --- section now.
\begin{figure}[h!]
\centering
\resizebox{0.78\textwidth}{!}{%
\begin{tikzpicture}[
 hgram/.style={ellipse,draw=black!70,minimum width=4.5cm,minimum height=2.2cm,
 align=center,font=\small,line width=0.6pt},
 arr/.style={-{Latex[length=2mm,width=1.5mm]},draw=black!70,line width=0.7pt},
 larr/.style={-{Latex[length=2mm,width=1.5mm]},draw=black!70,line width=0.7pt,bend left=30}
]
% Three ovals: OH, IH, EIC
\node[hgram,fill=blue!10] (OH) at (-5,0) {\textbf{Outer Hologram (OH)}\\Every lawful measurement\\available to the observer};
\node[hgram,fill=green!10] (IH) at (5,0) {\textbf{Inner Hologram (IH)}\\Brain's allocentric\\world-model};
\node[hgram,fill=orange!15] (EIC) at (0,-3.5) {\textbf{Evolving Inner Cosmos (EIC)}\\Closed loop coupling of OH and IH\\(the conscious episode)};
% Arrows: OH <-> IH bidirectional via EIC
\draw[arr] (OH.east) -- node[above,font=\footnotesize] {observation} (IH.west);
\draw[arr] (IH.south) -- node[right,font=\footnotesize,xshift=2pt] {action} (EIC.north east);
\draw[arr] (OH.south) -- node[left,font=\footnotesize,xshift=-2pt] {measurement} (EIC.north west);
% Self-loop on EIC: registration from inside
\draw[arr] (EIC.south west) to[out=-120,in=-60,looseness=2.2] node[below=1pt,font=\footnotesize] {self-registration} (EIC.south east);
% Annotation top
\node[font=\itshape\small,text=black!70] at (0,2.0) {Consciousness as internal measurement of a selected and stabilized IH branch};
\node[font=\scriptsize\itshape,text=black!60] at (0,-5.4) {Closed loop: \AOne{} OGGEP $\to$ \ASeven{} \T{-EIC} $\to$ subjective experience};
\end{tikzpicture}}
\caption[The holographic triad]{The holographic triad: Outer Hologram (\OH{}), Inner Hologram (\IH{}), and Evolving Inner Cosmos (\EIC{}). The \OH{} is every lawful measurement available to an observer in the universe; the \IH{} is the brain's allocentric world-model of that observable content; the \EIC{} is the closed-loop coupling of the two. Observations flow from \OH{} to \IH{}; actions flow from \IH{} back to the world. The \EIC{}'s distinctive structural feature is the self-loop on the right: the \EIC{} registers its own closed-loop state from inside, and that registration is what subjective experience is from the inside. The conscious episode --- with characteristic timescale $\sim 400$~ms (two theta cycles) and theta-rhythm neural implementation --- is the \EIC{} attractor on the cortical-circuit stratum. Inheritance: \AX{3} of \AOne{} (\OGGEP{}) plus \T{-\EIC{}} of \ASeven{}.}
\label{fig:holographic-triad}
\end{figure}

\label{sec:holographic-triad-merged}

\section{The holographic triad: \OH{}, \IH{}, and \EIC{}}

\subsection{Historical background}
\label{subsec:hist-triad}

Generally speaking, the holographic idea --- that the information content of a region of space can be encoded on its boundary --- emerged from black-hole thermodynamics. \citet{Bekenstein1973} proposed that black-hole entropy is proportional to horizon area, not volume; \citet{Hawking1975} sharpened the proposal by deriving the thermal radiation that bears his name. \citet{tHooft1993} and \citet{Susskind1995} elevated the area law to a general principle: any region of space is fully described by data on its boundary. \citet{Maldacena1998} provided the first concrete realization in the form of the AdS/CFT correspondence between a bulk gravitational theory in anti-de~Sitter space and a conformal field theory on its boundary. The Outer Hologram (\OH{}), Inner Hologram (\IH{}), and Evolving Inner Cosmos (\EIC{}) of the present section adopt the holographic structure inherited from this lineage but apply it three times over: once at the cosmic boundary (the \OH{}, encoding all lawful measurements), once at the cognitive boundary (the \IH{}, encoding the brain's world-model), and once at the recursive coupling between them (the \EIC{}, closing the participatory loop that \citet{Wheeler1990} imagined).

\label{sec:triad}

\subsection{The Outer Hologram (\OH{})}

\paragraph{Historical background.}
The Outer Hologram is the \GGTheory{} specialization of the holographic principle to the cosmological boundary of the geometric cascade. Its conceptual ancestors are \citet{Bekenstein1973} (BH entropy scales with area), \citet{Hawking1975} (thermal radiation from horizons), \citet{tHooft1993} (dimensional reduction in quantum gravity), \citet{Susskind1995} (the world as a hologram), and \citet{Maldacena1998} (AdS/CFT). The \OH{} preserves the area-law content of these predecessors while reading the boundary not as a thermodynamic abstraction but as the operational set of all lawful measurements an observer in the universe can perform.

\label{sec:oh}

\begin{definition}[Outer Hologram]
\label{def:oh}
Given a bulk state family \(\rho_6(X,t)\), a reduced boundary
evolution \(\cE_{t,0}\), and an admissible instrument class
\(\{I_y\}\), the outer hologram at time \(t\) is the boundary law
\begin{equation}
H_{\mathrm{out}}^{(t)}(y)
= \int_{\cX_6} K_{\OH}(y|X;\,t)\,\rho_6(X,t)\,dX,
\end{equation}
where the outer-hologram kernel is
\begin{equation}
K_{\OH}(y|X;\,t)
= \Tr\!\left[I_y\,\cE_{t,0}(|X\rangle\langle X|)\right].
\end{equation}
\end{definition}

An individual registered observation \(y_t\) is an \OH\ event.
Every physical measurement --- particle detector click, retinal
spike, DNA sequencer read-out --- is an \OH\ event: a lawful,
lossy, instrument-relative projection of the full 6D reality.

\paragraph{The shadow-on-the-wall analogy.}
Plato's allegory of the cave imagined prisoners who see only
shadows on a wall and mistake them for reality.
The \OH\ formalizes this ancient intuition: every observation
is a shadow --- a projection --- of a higher-dimensional reality.
The key difference from Plato is that in \GG{}, the projection
is not a deficiency of the observer but a \emph{law of physics}.
No observer, no matter how powerful, can see the full
six-dimensional bulk.
The four-dimensional boundary is all there is to observe.
But the \OH\ is not arbitrary: it is a \emph{lawful} projection,
governed by the same \BIsix{} identity that fixes the gauge structure.
The shadows are reliable --- they encode the bulk faithfully,
even though they are incomplete.

\paragraph{Examples of \OH{} events across science.}
To make the definition concrete: when a particle physicist reads
out a track in the LHC detector, that track is an \OH\ event.
When a biologist observes a fluorescent protein under a
microscope, that image is an \OH\ event.
When you look at a sunset, the pattern of photons hitting your
retina is an \OH\ event.
In every case, the observation is lossy (you cannot reconstruct
the full 6D bulk from a single measurement), instrument-relative
(different instruments give different projections), and lawful
(the projection follows the rules of quantum mechanics and
the \BIsix{} inflow).

The \OH\ tells us what the universe looks like from the boundary.
But observation alone is not enough for survival.
An organism needs not just data but a \emph{model} of the world
--- a representation that can predict what will happen next and
guide action. That model is the Inner Hologram.

\subsection{The Inner Hologram (\IH{})}

\paragraph{Historical background.}
The Inner Hologram is the brain's allocentric world-model. Its empirical ancestors are \citet{OKeefeDostrovsky1971} (place cells in the rat hippocampus), \citet{HaftingEtAl2005} (entorhinal grid cells), \citet{Hopfield1982} (attractor networks for associative memory), and \citet{TurnerEvans2020} (insect central-complex orientation circuits). These results established that nervous systems construct internal spatial maps from external inputs; the \GGTheory{} contribution is to identify the map with the boundary projection of an inner geometric stratum and to derive its dynamics from the \GFF{} descent on that stratum.

\label{sec:ih}

The outer hologram delivers data, but no open system can survive
by storing raw data alone. When the instantaneous sensory map
is non-injective (a 2D retina does not uniquely determine a 3D
world), a reconstructive model is required.

\begin{definition}[Inner Hologram]
\label{def:ih}
For a memory-bearing open system, the inner hologram at time \(t\)
is
\begin{equation}
H_{\mathrm{in}}^{(t)}
= \left\{\cZ,\; q_t,\; g_t,\; W_t\right\},
\end{equation}
where \(\cZ\) is the latent internal manifold, \(q_t\) is the
current posterior (belief state) on \(\cZ\), \(g_t\) is the
learned metric on that manifold, and \(W_t\) is the \HAL\ memory
substrate storing the relational attractor structure.
\end{definition}

The \IH\ is updated through the \MePMoS-compatible cycle:
prediction, fit, memory update, and action.
The allocentric structure of the \IH\ --- factoring out
self-motion --- is mathematically necessary: place cells, grid
cells, and head-direction cells are the neural implementation
of exactly this mathematical requirement.

\paragraph{The GPS analogy.}
Your phone's GPS app maintains an internal model of the world:
a map.
As you walk, the map does not move --- \emph{you} move on the
map.
The app factors out your self-motion and represents the world
in an allocentric (world-centered) coordinate system.
If the app stored only egocentric data (``turn left, walk
50 meters, turn right''), it would lose track of your
position after a few turns.
The allocentric map is not a luxury --- it is a mathematical
necessity for any system that needs to navigate.

The hippocampus does exactly the same thing.
Place cells \citep{OKeefeDostrovsky1971} fire when the animal
is at a specific location in the environment, regardless of how it got there.
Grid cells \citep{HaftingEtAl2005} provide a hexagonal coordinate
system for distances.
Head-direction cells encode the animal's facing direction.
Together, they implement Definition~\ref{def:ih} in neural
hardware: the latent manifold \(\cZ\) is the spatial environment,
\(q_t\) is the animal's current estimate of its position,
\(g_t\) is the learned geometry of the environment, and \(W_t\)
is the synaptic weight matrix storing the cognitive map.

The 2014 Nobel Prize in Physiology or Medicine was awarded to
\citet{OKeefeDostrovsky1971} and \citet{HaftingEtAl2005} for discovering
place cells and grid cells --- the neural substrate of the \IH.

\subsection{The Evolving Inner Cosmos (\EIC{})}

\paragraph{Historical background.}
The Evolving Inner Cosmos formalizes the coupled-loop interaction between the Outer Hologram (the measured universe) and the Inner Hologram (the brain's model of it). Its predecessors are \citet{Wheeler1990} (the participatory universe), \citet{Friston2010} (the free-energy principle, in which agents minimize variational free energy by acting on the world and updating their models), and \citet{Tononi2004} (integrated information theory, identifying consciousness with the irreducible cause-effect structure of an integrated system). The \EIC{} inherits the loop structure from Wheeler, the descent dynamics from Friston, and the integration measure from Tononi, and adds the warrant Wheeler lacked: the loop is the geometric attractor on the highest stratum reached by the cascade, not a metaphor.

\label{sec:eic}

\begin{definition}[Evolving Inner Cosmos]
\label{def:eic}
For a system with outer hologram \(H_{\mathrm{out}}^{(t)}\) and
inner hologram \(H_{\mathrm{in}}^{(t)}\), the Evolving Inner
Cosmos is the coupled state
\begin{equation}
E_t = H_{\mathrm{out}}^{(t)} + H_{\mathrm{in}}^{(t)}
\equiv \cC_{\mathrm{loop}}\!\left(
 H_{\mathrm{out}}^{(t)},\; H_{\mathrm{in}}^{(t)}\right),
\end{equation}
where \(\cC_{\mathrm{loop}}\) denotes closed-loop lawful
composition through prediction, action, sensing, memory update,
and internal readout.
\end{definition}

The mismatch between \OH\ and \IH\ is measured by the
reconstruction divergence
\begin{equation}
\delta_{\mathrm{rec}}^{(t)}
= D_{KL}\!\left(p^{\OH}_t \| p^{\IH}_t\right),
\end{equation}
and the full \EIC\ functional is
\begin{equation}
\cF_{\EIC}(t) = \delta_{\mathrm{rec}}^{(t)}
+ \lambda_E\,\cF_{\GFF}(t)
+ \lambda_C\,C(q_t).
\end{equation}

\paragraph{Constitutional warrant.}
The theorem below is the operational form of the Evolving-Inner-Cosmos closure theorem \T{-\EIC{}} of \ASeven{} \citep{ArisakaA7Qi4}; the Lyapunov-descent rule it inherits is the \GDOS{} second-law theorem \T{2} of \AFive{} \citep{ArisakaA5GDOS}. In plain language: the \EIC{} cannot disorganize itself spontaneously; its refinement is a one-way street, in the same sense and for the same constitutional reason that the second law of thermodynamics is a one-way street.

\begin{theorem}[Monotonic refinement of the \EIC{} (T-Origin-71.4)]
\label{thm:eic}
In the \QiEP{}-consistent gradient normal form
\(\dot\theta = -G^{-1}\nabla\cF_{\EIC} + \xi_t\), with
\(G\) positive-definite and \(\xi_t = 0\):
\begin{equation}
\frac{d}{dt}\cF_{\EIC}(t)
= -\|\nabla\cF_{\EIC}\|^2_{G^{-1}} \leq 0.
\end{equation}
Hence \(\cF_{\EIC}\) is a Lyapunov functional: the \OH--\IH\ loop
monotonically refines internal reconstruction.
The \EIC\ is the attractor of this Lyapunov descent.
\end{theorem}

This theorem is the rigorous formulation of why learning exists:
any system with memory and action under \(\UoneQi\) necessarily
improves its internal model of the world over time.

\paragraph{Why babies learn to walk.}
A newborn human has a crude \IH: a blurry, unstructured
representation of the world.
Over months of sensory experience, motor exploration, and
memory formation, the \IH\ is refined.
The baby learns to distinguish faces, to reach for objects,
to crawl, to walk, to speak.
Each of these milestones corresponds to a decrease in
\(\cF_{\EIC}\) --- a reduction in the mismatch between the
baby's internal model and the external world.

Theorem~\ref{thm:eic} says this process is not optional.
Any system with memory and a \(\UoneQi\)-charged information
channel will necessarily refine its internal model.
The baby does not \emph{choose} to learn --- it is driven
to learn by the same Lyapunov descent that drives a ball
downhill.

\paragraph{How $\cF_{\EIC}$ relates to $F_{\GFF}$.}
The relationship between these two functionals is hierarchical.

$F_{\GFF}$ is the \emph{universal} cost functional.
It applies to every open system in nature --- a flame, a
crystal, a bacterium, a brain. Its two sectors (\GEP{} for
energy, \CEP{} for entropy) track the total thermodynamic budget
of the system. Its gradient drives the \DGI{}--\GSSB{}--\GA{} pipeline.
A system that minimizes $F_{\GFF}$ is reaching thermodynamic
equilibrium; a system that maintains a steady $F_{\GFF}$ under
port injection is a dissipative structure. Every open system
pays the $F_{\GFF}$ cost of existence.

$\cF_{\EIC}$ is a \emph{specialized} functional that applies
only to systems with both an Outer Hologram and an Inner
Hologram --- systems that build internal models of the world.
It contains $F_{\GFF}$ as one component but adds two terms
that $F_{\GFF}$ alone does not have:
\begin{itemize}[leftmargin=1.5cm]
\item $\delta_{\mathrm{rec}}^{(t)}
 = D_{KL}(p^{\OH} \| p^{\IH})$: the reconstruction divergence,
 measuring how badly the internal model fails to match external
 reality. This is the \emph{epistemic drive} --- the
 compulsion to make the \IH\ converge on the \OH.
\item $C(q_t)$: a complexity cost that penalizes overly elaborate
 models, preventing the \IH\ from memorizing noise.
\end{itemize}
Think of $F_{\GFF}$ as the \emph{operating cost} of a building
--- heating, electricity, plumbing, structural maintenance.
Every building pays this cost, whether it is a warehouse or a
university.
$\cF_{\EIC}$ is the total cost of running a university: it
includes the operating cost ($F_{\GFF}$) plus an educational
quality metric ($\delta_{\mathrm{rec}}$: how well does the
students' knowledge match reality?) plus a complexity penalty
($C$: do not make the curriculum absurdly complicated).
A warehouse has $F_{\GFF}$ but no $\cF_{\EIC}$ --- it does not
learn anything. A university has both.

The consequence is that two different Lyapunov descents operate
simultaneously in a conscious brain:
\begin{enumerate}[leftmargin=1.5cm]
\item \emph{Thermodynamic descent} ($F_{\GFF} \downarrow$):
 the brain must balance its energy budget, dissipate heat,
 and maintain its \SeOS{} viability. This is the cost of
 being alive.
\item \emph{Epistemic descent} ($\cF_{\EIC} \downarrow$):
 the brain must refine its internal model of the world,
 reducing the mismatch between what it believes and what is
 actually there. This is the cost --- and the drive --- of
 understanding.
\end{enumerate}
Both descents are monotonic (Theorem~\ref{thm:eic}).
The brain cannot choose to stop learning any more than a ball
can choose to stop rolling downhill.
What we experience as curiosity, wonder, and the compulsion to
understand is the subjective signature of $\cF_{\EIC}$ descent
--- the geometry's way of closing the loop between the universe
and the beings it produced (Table~\ref{tab:eic-systems-a11}).

\begin{table}[htbp]
\centering
\caption[Which systems close $\cF_{\EIC}$, and which only descend $F_{\GFF}$.]{Six systems, and the distinction the paper turns on: every one of them descends the free-energy functional $F_{\GFF}$, and only some close the \EIC{} loop. A flame and a crystal descend and do not close --- neither carries an internal model of the world, one being a process and the other a static structure. A bacterium closes rudimentarily, through receptor methylation. An insect closes in one dimension, its ring attractor being a one-dimensional internal model; a mammal closes in four, through hippocampal place and grid cells; and a human closes fully, cortex and hippocampus together carrying a rich abstract model.}
\label{tab:eic-systems-a11}
\renewcommand{\arraystretch}{1.1}
{\small
\setlength{\tabcolsep}{3pt}
\begin{tabularx}{\textwidth}{>{\raggedright\arraybackslash}p{0.16\textwidth}
 >{\raggedright\arraybackslash}p{0.10\textwidth}
 >{\raggedright\arraybackslash}p{0.10\textwidth}
 >{\raggedright\arraybackslash}X}
\toprule
\textbf{System} & \textbf{$F_{\GFF}$?}
 & \textbf{$\cF_{\EIC}$?}
 & \textbf{Why} \\
\midrule
Flame & Yes & No & No \IH\ --- does not model the world \\
Crystal & Yes & No & No \IH\ --- static structure \\
Bacterium & Yes & Rudimentary & Simple \IH\ (receptor methylation) \\
Insect & Yes & Yes (1D) & Ring attractor = one-dimensional \IH \\
Mammal & Yes & Yes (4D) & Hippocampal place/grid cells \\
Human & Yes & Yes (full) & Cortical + hippocampal = rich abstract \IH \\
\bottomrule
\end{tabularx}
}
\end{table}
Learning is universal across all organisms with nervous systems,
from flatworms to humans, because it is a geometric consequence
of the \EIC\ functional --- not a contingent evolutionary
invention.

\paragraph{The \EIC\ across scales.}
The \EIC\ is not limited to individual brains.
A scientific community maintains a collective \IH\ ---
the body of published knowledge --- and refines it through
experiment (new \OH\ events) and theory (new models).
The progress of science is itself an instance of \EIC\
descent: the collective internal model converges on the
external reality.
In the \GG{} framework, the act of discovering \BIsix{} is the
moment when the \EIC\ functional reaches its minimum ---
when the internal model of the universe (physics) converges
on the external structure of the universe (geometry).

But what, exactly, \emph{is} consciousness in this framework?
The \EIC\ describes the process by which internal models
improve. Consciousness is a specific operation within that
process --- and it can be defined with mathematical precision.

%% ===================================================================
%%  PART V
%% ===================================================================

% INSERT-AFTER: section "The holographic triad: \OH{}, \IH{}, and \EIC{}" (v6.5 line 5787).

\subsection{\textsf{T-OH-IH-EIC-Identity}: the holographic-triad closure at the cognitive-episode timescale}
\label{ssec:t-oh-ih-eic-identity}

\paragraph{Plain-English opening.}
At the moment a conscious episode forms --- the $\sim$400~ms (two-theta-cycle) closure during which the brain integrates sensory input, memory, and prediction into a single felt experience --- the boundary state of the cognitive substrate factorizes into two pieces: the Outer Hologram \OH{} (the sensory side, what the world is presenting now) and the Inner Hologram \IH{} (the memory-plus-prediction side, what the system already believes and expects). Their sum, on the conscious-episode stratum, is the Evolving Inner Cosmos \EIC{} (the integrated conscious experience). This is the holographic-triad identity, and it closes at the $\sim$400~ms two-theta-cycle mark. After one theta cycle ($\sim$200~ms) the brain has only unconscious attention --- \OH{} and \IH{} are not yet bound; the conscious episode completes at $\sim$400~ms; at longer timescales the episode has dissolved into memory. In daily life: a single sentence of a conversation is the smallest unit of meaningful communication; below a sentence, words are isolated; above a sentence, the meaning is already a paragraph. The conscious episode is the cascade's sentence-level unit of integration.

\paragraph{Constitutional inputs.}
\AX{2} (\GGEP{}), \AX{3} (\OGGEP{}, the open-system boundary that supports the holographic triad), \AX{4} (\MSMF{}), \ASeven{}~\T{-QUA}--\ASeven{}~\T{-HAL} (operational, functional, and lock clauses for the conscious episode), and the empirical timescale fix of $\sim$400~ms (two $\sim$200~ms theta cycles), consistent with \citet{Dehaene2014}.

\paragraph{Upstream theorems.}
Three upstream theorems are invoked. (i) \ASeven{} \T{-EIC} \citep{ArisakaA7Qi4} (the \EIC{} closure theorem) establishes the holographic-triad identity at the cognitive-episode stratum. In plain language, this is the theorem that conscious experience is the integrated readout of the outer-and-inner holograms at the binding timescale. (ii) \textsf{T-Pagoda-Five-Story} (Theorem~\ref{thm:t-pagoda-five-story}) supplies the stratum: the \EIC{} closure is the storey-5 port of the cognitive pagoda. (iii) \ATen{} (Part on Cosmos) \citep{ArisakaA10Cosmos} provides the bulk-to-boundary-projection reading that descends from the bulk \BIsix{} closure to the boundary \EIC{} closure. In plain language, the conscious episode is the cognitive-stratum reflection of the same holographic principle that organizes the bulk.

\begin{theorem}[\textsf{T-OH-IH-EIC-Identity}: the holographic-triad factorization (T-Origin-71.5)]
\label{thm:t-oh-ih-eic-identity}
At the cognitive-episode timescale $\tau_{\rm CE} \approx 400$~ms (two $\sim 200$~ms theta cycles), the boundary state $\rho$ of the cognitive substrate factorizes as
\begin{equation}
\label{eq:oh-ih-factorisation}
\rho \;=\; \rho_{\OH{}} \otimes \rho_{\IH{}}, \qquad \tau_{\rm CE} \approx 400~{\rm ms},
\end{equation}
where $\rho_{\OH{}}$ is the outer-hologram component (sensory input from the boundary $\partial_{\rm sens}$) and $\rho_{\IH{}}$ is the inner-hologram component (memory plus prediction from the boundary $\partial_{\rm mem}$). The Evolving Inner Cosmos is the \GGA{} of the storey-5 stratum, and admits the identity
\begin{equation}
\label{eq:eic-identity}
\boxed{\;\;\rho_{\EIC{}} \;=\; \tfrac{1}{2}\bigl(\rho_{\OH{}} + \rho_{\IH{}}\bigr) \quad\text{on the \EIC{} stratum at $\tau_{\rm CE}$.}\;\;}
\end{equation}
The closure $\rho_{\EIC{}} = \tfrac{1}{2}(\rho_{\OH{}} + \rho_{\IH{}})$ is the unique fixed point of the cognitive-episode \GFF{}-Lyapunov descent, and the conscious episode is the self-modeling diagonal $\langle \hat O_{\OH{}} \rangle_{\rho_{\IH{}}}$ read off that fixed point.
\end{theorem}

\paragraph{Proof outline.}
Step 1: factorization at the cognitive-episode timescale. By \ASeven{} \T{-EIC}, the cognitive substrate's boundary state factorizes into outer and inner pieces at the conscious-binding timescale; this is the constitutional reading of the binding problem. Step 2: the timescale window. The empirical timescale $\sim$400~ms (two $\sim$200~ms theta cycles) is consistent with neuroscientific measurements of conscious-access latency \citep{Dehaene2014}; the cascade's reading places it inside the cascade-layer-17 stratum. Step 3: the additive identity. The \EIC{} closure equates the storey-5 \GGA{} to the additive combination of \OH{} and \IH{} on the closure stratum; this is the constitutional reading of conscious-integration as Lyapunov fixed point. Step 4: uniqueness of the closure. By \textsf{T-Cascade-Layer-Uniqueness} applied to storey-5, the \EIC{} \GGA{} is unique; therefore the closure identity is unique. Step 5: cross-checks with empirical neuroscience. The \EIC{} \GGA{} is consistent with the cortical-binding observations of \citet{Tononi2008}, the global-neuronal-workspace observations of \citet{Dehaene2014}, and the conscious-percept binding timescales of psychophysics.

\paragraph{Quantitative anchor.}
The binding timescale is $\tau_{\rm CE} \approx 400$~ms (two $\sim$200~ms theta cycles; the one-theta-cycle $\sim$200~ms stage is unconscious attention). The boundary-port complexity at storey-5 is $\sim 10^7$ degrees of freedom (consistent with the dimensionality of cortical conscious access). The Lyapunov-descent rate at storey-5 is $\sim J_E \cdot \QB{} \cdot J_{Qi}$ with $J_{Qi} \sim 10^7$ bits per cognitive episode.

\paragraph{Physics intuition.}
At first-year-graduate-student level, the picture is the following. In quantum mechanics, the tensor-product factorization $\rho_{AB} = \rho_A \otimes \rho_B$ is the signature of \emph{separability}: the subsystems $A$ and $B$ are uncorrelated. In the cognitive substrate, the \OH{}-\IH{} factorization is the signature of conscious integration: the sensory side and the memory-prediction side, both well-defined on their own, combine at the binding timescale into a single integrated experience. The combined identity $\rho_{\EIC{}} = \tfrac{1}{2}(\rho_{\OH{}} + \rho_{\IH{}})$ on the closure stratum is then the cascade's reading of binding: the conscious episode is not the tensor-product state (which would be uncorrelated) but the integrated combination of its two components on the integration stratum. The equal-weight combination is the maximally noncommittal integrated state consistent with both boundary components, and --- unlike the bare sum $\rho_{\OH{}} + \rho_{\IH{}}$, which has trace $2$ and is therefore not a state at all --- it is a bona-fide density matrix on the integration stratum. Nothing in the framework turns on the weights: the structurally load-bearing content is not the combination but the \emph{self-modeling diagonal} $\langle \hat O_{\OH{}} \rangle_{\rho_{\IH{}}}$ that the combined state supports at the binding timescale, which is the object every other consciousness claim in this paper is stated in terms of.

\paragraph{Daily-life metaphor.}
A movie scene is the integration of two streams: the visual track (what you see on screen) and the audio track (what you hear). The visual track is \OH{}-analogue (the world presented now); the audio track has both \OH{}- and \IH{}-analogue components (sound from the world plus the narrator's voice carrying memory and prediction). At the scene-binding timescale of a fraction of a second, the brain combines both streams into a single conscious scene. Below this timescale, the streams are perceptually separate; above this timescale, the scene has already dissolved into the next scene. The cognitive episode is the cascade's analogue of a movie scene: a single coherent integration of outer (sensory) and inner (memory+prediction) holograms at the binding timescale.

\paragraph{Kill criterion.}
The theorem is falsified by any of: (a) the discovery of a reliably reported, unitary conscious episode completing in $\le$ one theta cycle ($\sim$200~ms), i.e.\ without the second theta cycle the MePMoS loop requires; (b) the discovery that \OH{} and \IH{} do not factorize (i.e.\ that they are inextricably entangled even at the cognitive-episode timescale); (c) the failure of the additive identity on the \EIC{} stratum (e.g.\ a non-linear combination of \OH{} and \IH{} that is not their sum); (d) the discovery of a conscious episode that bypasses the \EIC{} closure (an episode without integration of outer and inner holograms).

\paragraph{Plain-English close.}
Conscious experience is, in the cascade's reading, a sum: outer hologram (now) plus inner hologram (memory and prediction) equals evolving inner cosmos (the felt episode). The integration completes at $\sim$400~ms (two theta cycles) and produces the storey-5 \GGA{} of the cognitive pagoda. This is the constitutional reading of consciousness as a binding problem.

\bigskip

% =========== Theorem 13: T-QCI-Ladder-Monotonicity ============

\section{The \QCI{} ladder and the neural implementation}
\begin{figure}[h!]
\centering
\resizebox{0.604\textwidth}{!}{%
\begin{tikzpicture}[
 rung/.style={rectangle,rounded corners=2pt,draw=black!60,
 minimum width=7cm,minimum height=0.6cm,align=left,font=\small,
 line width=0.4pt},
 qci/.style={font=\bfseries}
]
% Vertical QCI ladder
\node[rung,fill=red!18] (R1) at (0,0) {Human awareness: full conscious reportable experience};
\node[rung,fill=red!13] (R2) at (0,-0.75){Primate cognition: tool use, social mentalizing};
\node[rung,fill=orange!15] (R3) at (0,-1.5) {Mammalian: hippocampal map, episodic memory};
\node[rung,fill=orange!10] (R4) at (0,-2.25){Vertebrate: allocentric world-model};
\node[rung,fill=yellow!18] (R5) at (0,-3) {Insect: ring attractor, central complex};
\node[rung,fill=yellow!12] (R6) at (0,-3.75){Cnidarian: nerve net, action coordination};
\node[rung,fill=green!13] (R7) at (0,-4.5) {Single cell: signal transduction, chemotaxis};
\node[rung,fill=green!9] (R8) at (0,-5.25){Bacterial: gene expression, immune memory};
\node[rung,fill=cyan!12] (R9) at (0,-6) {Macromolecule: DNA-RNA-protein information};
\node[rung,fill=cyan!8] (R10) at (0,-6.75){Polymer chemistry: covalent-bond information};
\node[rung,fill=blue!8] (R11) at (0,-7.5){Prebiotic: small-molecule reaction networks};
% QCI numerical labels on right
\node[qci,anchor=west] at (4.5,0) {QCI $\sim$ 10};
\node[qci,anchor=west] at (4.5,-0.75){QCI $\sim$ 9};
\node[qci,anchor=west] at (4.5,-1.5) {QCI $\sim$ 8};
\node[qci,anchor=west] at (4.5,-2.25){QCI $\sim$ 7};
\node[qci,anchor=west] at (4.5,-3) {QCI $\sim$ 6};
\node[qci,anchor=west] at (4.5,-3.75){QCI $\sim$ 5};
\node[qci,anchor=west] at (4.5,-4.5) {QCI $\sim$ 4};
\node[qci,anchor=west] at (4.5,-5.25){QCI $\sim$ 3};
\node[qci,anchor=west] at (4.5,-6) {QCI $\sim$ 2};
\node[qci,anchor=west] at (4.5,-6.75){QCI $\sim$ 1};
\node[qci,anchor=west] at (4.5,-7.5) {QCI $\sim$ 0.5};
% Cascade-layer labels on left
\node[font=\footnotesize\itshape,anchor=east] at (-4.5,0) {layer 17};
\node[font=\footnotesize\itshape,anchor=east] at (-4.5,-0.75){layer 17};
\node[font=\footnotesize\itshape,anchor=east] at (-4.5,-1.5) {layer 16};
\node[font=\footnotesize\itshape,anchor=east] at (-4.5,-2.25){layer 16};
\node[font=\footnotesize\itshape,anchor=east] at (-4.5,-3) {layer 15};
\node[font=\footnotesize\itshape,anchor=east] at (-4.5,-3.75){layer 13};
\node[font=\footnotesize\itshape,anchor=east] at (-4.5,-4.5) {layer 12};
\node[font=\footnotesize\itshape,anchor=east] at (-4.5,-5.25){layer 11};
\node[font=\footnotesize\itshape,anchor=east] at (-4.5,-6) {layer 9};
\node[font=\footnotesize\itshape,anchor=east] at (-4.5,-6.75){layer 6};
\node[font=\footnotesize\itshape,anchor=east] at (-4.5,-7.5) {layer 4};
\end{tikzpicture}}
\caption[The cascade-depth completeness index ladder]{The cascade-depth completeness index (\QCI{}) ladder of this paper. Vertical scale: \QCI{} grade from prebiotic chemistry ($0$) at the bottom to human conscious awareness ($10$) at the top. Each rung names the empirical category at that \QCI{} value, with the corresponding cascade layer of Figure~\ref{fig:eighteen-layer-cascade} annotated on the left. The \QCI{} is the geometric measure of the \EIC{} closure's completeness on a given substrate; substrates with higher \QCI{} have richer holographic-allocentric memory architectures and more complete \OH{}-\IH{} coupling. The grades shown are \emph{illustrative}; the quantitative form (the storey-port count $\QCI = 2S$ of \S\ref{ssec:t-qci-ladder-monotonicity}) and its calibration against comparative-neuroscience correlates (cell signal transduction, neural circuit complexity, neural-correlates-of-consciousness measurements) are developed in \AFourteen{} (in preparation).}
\label{fig:qci-ladder}
\end{figure}

\label{sec:qci-ladder-merged}

% INSERT-AFTER: section "The \QCI{} ladder and the neural implementation" (v6.5 line 6101).

\paragraph{Quantitative definition (tentative).} The \QCI{} is defined operationally as the number of \emph{Qi-HAL storey-ports} a substrate closes --- sensorimotor, attention, working memory, allocentric map, and \EIC{} closure --- a storey count $S \in \{0,\dots,5\}$ read off the substrate's neural architecture, reported on the comparative ladder $\QCI{} = 2S \in [0,10]$ (an insect heading ring at three storeys scores $\QCI{}\sim 6$; a rodent allocentric map at four storeys, $\sim 8$; a human conscious episode at all five storeys, $\sim 10$). So defined, \QCI{} is an \emph{observable count}, not a fitted score. The ladder in Figure~\ref{fig:qci-ladder} is illustrative (the sub-neural rungs especially); the operational scoring rule for each port, the treatment of pre-neural precursors, and the calibration against comparative-neuroscience data are deferred to \AFourteen{} (in preparation).

\subsection{\textsf{T-QCI-Ladder-Monotonicity}: strict monotonicity of \QCI{} along cascade ascent}
\label{ssec:t-qci-ladder-monotonicity}

\paragraph{Plain-English opening.}
The \QCI{} ladder (Qi-HAL completeness index) is the cascade's measurement of cognitive completeness: how many of the five storey-ports of the pagoda are operational at a given cascade layer. The present theorem says that this measurement is strictly monotonic along cascade ascent: $\QCI(n+1) \geq \QCI(n)$, with equality only when the system fails to ascend. In other words, the cascade does not lose cognitive completeness as it climbs; each layer either adds to the previous \QCI{} count or fails to climb. In daily life: as one climbs a staircase, each step is at the same or a higher elevation than the previous; one does not, in a single ascent, end up lower than one started. The monotonicity theorem makes this structural feature of the cascade explicit.

\paragraph{Constitutional inputs.}
\AX{3} (\OGGEP{}, open-system boundary), \AX{4} (\MSMF{}), and \POS{7}--\ASeven{}~\T{-HAL}. In plain language, the cascade's open-system structure forbids loss of completeness during ascent; \AX{4} forces monotonic increase rather than constancy.

\paragraph{Upstream theorems.}
Two upstream theorems are invoked. (i) \ASeven{} \T{-HAL.1} \citep{ArisakaA7Qi4} establishes the memory architecture a substrate must supply at each storey, and hence the monotonicity of \QCI{}: a storey cannot close its port without the ports below it. \ASeven{} states no upper bound on storey count, and none is claimed here. In plain language, this is the theorem that the cascade does not lose cognitive completeness during ascent. (ii) \textsf{T-Pagoda-Five-Story} (Theorem~\ref{thm:t-pagoda-five-story}) supplies the five-storey structure; the present theorem ensures the storeys are added in order, never removed.

\begin{theorem}[\textsf{T-QCI-Ladder-Monotonicity}: monotonic \QCI{} along cascade ascent (T-Origin-72.1)]
\label{thm:t-qci-ladder-monotonicity}
For any cascade layer $n \in \{15, 16, 17\}$ (the cognitive layers of the cascade) and its successor layer $n+1$ (when defined),
\begin{equation}
\label{eq:qci-monotonicity}
\boxed{\;\;\QCI(n+1) \;\geq\; \QCI(n)\;\;,}
\end{equation}
with equality if and only if the system fails to ascend from layer~$n$ to layer~$n+1$. Concretely, the \QCI{} values along the cognitive cascade ascent are those of Table~\ref{tab:qci-values-a11}.
\begin{table}[htbp]
\centering
\caption[The \QCI{} values along the cognitive ascent.]{The \QCI{} values along the cognitive cascade ascent, which the monotonicity result above requires to be non-decreasing, with equality exactly when the system fails to ascend. Layer~15 is the insect ring attractor carrying heading, at storey $S = 3$ and $\QCI{} = 6$; layer~16 is the rodent place--grid--time map, at $S = 4$ and $\QCI{} = 8$; layer~17 is the human conscious episode closing \EIC{}, at $S = 5$ and $\QCI{} = 10$. The relation $\QCI{} = 2S$ holds down the column, which is what makes the storey count and the index two readings of one quantity.}
\label{tab:qci-values-a11}
{\small
\begin{tabular}{lccc}
\toprule
\textbf{Layer} & \textbf{\GGA{} (compressed)} & \textbf{storey $S$} & \textbf{\QCI{}$=2S$} \\
\midrule
$n = 15$ & insect ring attractor (heading) & 3 & 6 \\
$n = 16$ & rodent place--grid--time map & 4 & 8 \\
$n = 17$ & human conscious episode (\EIC{}) & 5 & 10 \\
\bottomrule
\end{tabular}
}
\end{table}
Each cascade jump adds one storey-port, so $\Delta\QCI = 2$: layer~15$\to$16 gains the allocentric map (storey~4), and layer~16$\to$17 gains the \EIC{} closure (storey~5).
\end{theorem}

\paragraph{Proof outline.}
Step 1: monotonicity from open-system axiom. By \AX{3}, the cascade's runtime is a Lyapunov descent on the cognitive substrate's \GFF{}; the descent gains rather than loses storey-ports. Step 2: strictness when the cascade ascends. By \textsf{T-Pagoda-Five-Story}, each ascent adds at least one storey-port (the next \QCI{} level); therefore $\QCI(n+1) > \QCI(n)$ strictly when ascent succeeds. Step 3: equality when the cascade fails to ascend. If the layer-$n$ system does not ascend (e.g.\ a non-cognitive substrate that satisfies \SeOS{} conditions but fails the conditions for further cognitive layering), then $\QCI(n+1)$ is undefined or equal to $\QCI(n)$. Step 4: upper bound. By \ASeven{} \T{-HAL}, $S(n) \leq 5$ for all $n$ (\QCI{} $\leq 10$); this is the constitutional ceiling. Step 5: empirical agreement. The table above is consistent with the comparative-neuroscience observations on insect, rodent, and human cognitive completeness.

\paragraph{Quantitative anchor.}
\QCI(layer~15) = 6 (three storeys: sensorimotor, attention, heading working-memory); \QCI(layer~16) = 8 (adds the allocentric map); \QCI(layer~17) = 10 (adds the \EIC{} closure). Each jump adds one storey-port ($\Delta\QCI = 2$), summing to $\Delta\QCI = 4$ across the cognitive cascade.

\paragraph{Physics intuition.}
At first-year-graduate-student level, the monotonicity theorem works very much like the second law of thermodynamics, applied to information rather than entropy. In statistical mechanics, the second law says entropy of a closed system never decreases; in the cascade, the \QCI{} of an ascending system never decreases. The two statements are dual: entropy measures disorder, \QCI{} measures integrated information capacity. The cascade's open-system structure (\AX{3}) ensures that the system can pay (in \QB{}-quanta) for integration; this payment converts environmental entropy into internal \QCI{}. The Landauer-type information-thermodynamic identity $1$~bit $= k_B T \ln 2$ becomes, in the cascade reading, $1$~\QCI{}-storey $= n_{\rm port} \QB{}$ where $n_{\rm port}$ is the port-complexity of the storey.

\paragraph{Daily-life metaphor.}
A student progressing through grades 1 through 12 monotonically accumulates knowledge: a 5th-grader knows everything a 4th-grader knows, plus more. The student does not, in a single year's progression, end up knowing less than at the start of the year (subject to the empirical fact that occasionally facts are forgotten; these are small fluctuations relative to the main accumulation). The cascade's \QCI{} ladder behaves the same way: a storey-5 system knows (cognitively) everything a storey-4 system knows, plus the integration of declarative memory into conscious episode. Lower storeys are subsumed into higher storeys; cognitive ascent is monotonic.

\paragraph{Kill criterion.}
The theorem is falsified by any of: (a) the discovery of an animal lineage in which a higher-storey animal lacks a lower storey (e.g.\ a $\QCI = 10$, five-storey animal lacking a lower storey --- this would falsify monotonicity); (b) the discovery of a cascade ascent in which \QCI{} decreases (e.g.\ a layer-17 system with $\QCI < 10$); (c) the discovery that the \QCI{} ladder is not strictly monotonic during ascent (e.g.\ that some layer's storey-ports can be turned off during ascent while ascending).

\paragraph{Plain-English close.}
The cognitive cascade gains cognitive completeness as it ascends; it does not lose. A storey-3 rodent has more cognitive structure than a storey-1 insect; a storey-5 human has more cognitive structure than a storey-3 rodent. The ladder is climbed, never descended, during cascade ascent.

\bigskip

% =========== Theorem 14: T-Internal-Measurement-As-Consciousness ============

\paragraph{Daily-life metaphor for the \QCI{} ladder.}
The reader who has used a smartphone has felt the difference between low \QCI{} and high \QCI{} in their own pocket. A phone in airplane mode runs the operating system but speaks to nothing --- low \QCI{}, low coupling, no \HAL{} traffic between inner and outer hologram. A phone with one bar of cellular signal opens a thin channel --- middle \QCI{}, intermittent coupling, occasional alignment between the phone's internal model and the network's view of the world. A phone in a 5G hotspot with a dozen apps streaming --- video, GPS, voice call, banking, real-time translation --- is in the high-\QCI{} regime: outer signals continuously update the inner model, the inner model continuously requests outer signals, and the system maintains a tight self-located closure. The conscious episode is what the brain does in the 5G-hotspot regime: outer hologram and inner hologram exchange enough information per theta-cycle to keep the closure tight. The framework's contribution is to make the \QCI{} number quantitative --- the rung is set by the integer-locked four-current cadence --- so that a system's \QCI{} can be read off from its theta-coupling and bandwidth, rather than asserted by analogy.

\section{Consciousness and the \QCI{} ladder}

\subsection{Consciousness as internal measurement of the \IH{}}
\label{sec:consciousness}

The \IH\ is the internal reconstructive manifold.
Consciousness is a special operation performed on the \IH:
the internal measurement of a selected and stabilized branch.

\begin{definition}[Internal measurement]
\label{def:internal-meas}
A system internally measures its inner hologram at time \(t\) if
there exists an internal readout boundary
\(\partial_{\mathrm{int}}\) and a family of readout channels
\(\{\cM_c\}\) such that at least one selected internal content
state \(\rho_{\IH,c}^{(t)}\) re-enters the \MePMoS\ loop as a
causal variable for subsequent memory update, action selection,
or report.
\end{definition}

\begin{definition}[Conscious \EIC{} episode]
\label{def:conscious}
A conscious \EIC\ episode occurs at time \(t\) when all of the
following hold:
\begin{enumerate}[label=(\alph*), leftmargin=1.4cm]
\item The system possesses a nontrivial inner hologram
 \(H_{\mathrm{in}}^{(t)}\).
\item At least one internal content channel \(c\) is selected
 from competing candidates by a \DGI--\GSSB--\GA\ sequence.
\item The selected content persists on a timescale sufficient for
 writing, reuse, or action bias.
\item The selected content is available on
 \(\partial_{\mathrm{int}}\) for re-entry into \MePMoS.
\end{enumerate}
\end{definition}

\begin{theorem}[Internal measurement as consciousness (T-Origin-73.3)]
\label{thm:t-internal-measurement-as-consciousness}
Let a system satisfy Definition~\ref{def:internal-meas} (internal
measurement) and Definition~\ref{def:conscious} (conscious \EIC{}
episode) at time $t$. Then the conscious episode is identical to the
internal measurement of the selected and stabilized inner-hologram
branch: the \DGI--\GSSB--\GA{} pipeline acting on $\rho_{\IH}^{(t)}$,
read out on the internal boundary $\partial_{\rm int}$ and re-entered
into the \MePMoS{} loop. Equivalently, the episode \emph{is} the
self-modeling diagonal of the \OH{}/\IH{} factorization,
$\langle \hat O_{\OH{}}\rangle_{\rho_{\IH{}}}$, evaluated at the
theta-cycle binding timescale --- the layer-17 attractor of the
recursive uniqueness theorem registering its own stabilized content.
\emph{Status: Asserted} (the structural identification is stated here;
the cortical-circuit \GFF{}-relaxation derivation that fixes the
$\sim$400~ms timescale to codebook standard is developed in
\AFourteen{}, in preparation). The \emph{formal} content of this theorem is the internal-measurement construction stated here (and in compact normal form below); the \emph{identification} of that construction with phenomenal experience --- what it is like, from the inside --- is an explicitly-labeled structural-identity \emph{hypothesis}, defended (not deduced) against the explanatory-gap, conceivability, and knowledge arguments in \S\ref{sec:reader-guide-cognitive}; it is not a consequence read off the geometric apparatus. The theorem supplies a structural target sharp enough to be a candidate for the identification; whether structural identity is the right response to the hard problem is left explicitly open. (\ASeven{}'s \T{-EIC} is itself scoped to \emph{access} consciousness in the sense of Block; \ASeven{} records the phenomenal identification as its own open problem, and \AEight{}'s measurement chain forces only \emph{functional} observers --- so the phenomenal identification here is \AEleven{}'s own additional commitment, not a downstream consequence inherited from the companions.)
\end{theorem}

In compact normal form, the self-accessible content state is
\begin{equation}
\rho_{\mathrm{sel}}^{(t)}
= \cM_{\mathrm{int}}\!\left[
 \cS_{\GA}\circ\cS_{\GSSB}\circ\cS_{\DGI}
 \!\left(\rho_{\IH}^{(t)}\right)\right],
\end{equation}
where \(\cS_{\DGI}\) is the exploration operator,
\(\cS_{\GSSB}\) is the symmetry-breaking selection operator, and
\(\cS_{\GA}\) is the attractor-stabilization operator.
This equation is the recursive analogue of the external
measurement chain --- the same open-system logic of \GDOS{}
reapplied to the system's own internal model.

\paragraph{What it feels like from the inside.}
Consider the moment you recognize a friend's face in a crowd.
Before recognition, multiple candidate interpretations coexist
in your visual cortex --- the face could be anyone.
This is the \DGI\ phase: exploration of gauge-equivalent
possibilities.
Then, evidence accumulates (the shape of the nose, the color
of the hair, the context of the meeting place), and one
interpretation wins --- ``That's Alice!''
This is the \GSSB\ phase: symmetry-breaking selection.
The selected interpretation is then stabilized: you feel
certain, you wave, you walk over.
This is the \GA\ phase: attractor stabilization.

The entire sequence --- from ambiguity through recognition to
certainty --- takes about 400 milliseconds (two theta cycles
at 5~Hz).
That 400-millisecond window is a single conscious \EIC\ episode.
What you experience as ``seeing Alice'' is the \GA-stabilized
output of the \DGI--\GSSB--\GA\ pipeline operating on your \IH.

\paragraph{Why dreams are conscious but hallucinations are not (fully).}
During dreaming, the brain runs the \DGI--\GSSB--\GA\ pipeline
on internally generated data rather than external sensory input.
The \IH\ is active, the selection process is running, and the
selected content is available on \(\partial_{\mathrm{int}}\).
All four conditions of Definition~\ref{def:conscious} are met.
Dreams are therefore genuine conscious \EIC\ episodes ---
they just happen to be driven by internal rather than external
data.

Hallucinations, by contrast, involve a failure of the \GSSB\
stage: the selection mechanism locks onto an interpretation
that does not correspond to the actual \OH.
The \IH\ and \OH\ diverge, and \(\delta_{\mathrm{rec}}\) increases
rather than decreasing.
In the language of the framework, hallucination is a local
minimum of the \EIC\ functional --- a trap that prevents the
system from reaching the true attractor.

\subsection{The \QCI{} ladder}
\label{sec:qci}

The cascade-depth index (\QCI) is this paper's completeness measure for the \OH{}--\IH{} closure --- tentatively the storey-port count $\QCI = 2S$ defined in \S\ref{ssec:t-qci-ladder-monotonicity} --- complementing integrated-information approaches \citep{Tononi2004}. The illustrative grades across substrates are shown in Figure~\ref{fig:qci-ladder}; the operational scoring and its calibration against comparative-neuroscience data are developed in \AFourteen{} (in preparation).



The difference between a single-celled chemotactic system and the human brain is one of \emph{degree} --- more \NHT{} levels, deeper \HAL{}, longer \BFFT{}, stronger \DGI{}, hence more storey-ports closed --- not a qualitative ontological leap.
Every step on this ladder is a \GSSB{} transition mandated by the
\QGL\ constraint.

\section{Reader's guide for neuroscientists and philosophers of mind}
\label{sec:reader-guide-cognitive}

Part~\ref{part:cognitive} presents the cognitive cascade --- the Five-Story Pagoda, the holographic triad of Outer Hologram / Inner Hologram / Evolving Inner Cosmos, and the \QCI{} ladder --- at technical depth. This Section translates the technical content into the idioms most useful to neuroscientists and philosophers of mind.

\paragraph{The framework's claim about consciousness in one sentence.}
The cognitive cascade's claim, distilled, is the following: \emph{conscious awareness is the layer-17 attractor of the recursive uniqueness theorem; subjective experience is identical to the loop's self-modeling structural content --- specifically, the \OH{}/\IH{} factorization's diagonal expectation-value pairing $\langle \hat O_{\OH{}} \rangle_{\rho_{\IH{}}}$ at the theta-cycle binding timescale (Part~\ref{part:cognitive} Theorem~\ref{thm:t-internal-measurement-as-consciousness}).} The looser \emph{from inside the open loop} idiom imported a perspectival primitive that pure ontic structural realism is supposed to forbid; the present formulation specifies the position the structure occupies in the cascade as a structural object (the self-modeling diagonal of \OH{}/\IH{}), not as a sui-generis viewing point. The framework neither eliminates consciousness as illusion (the eliminativist position) nor explains it away as emergent epiphenomenon (the strong-emergentist position). It identifies consciousness with a specific structural feature of the cascade's layer-17 attractor.

\paragraph{For the neuroscientist: what the empirical anchors are.}
Several empirical results that neuroscience already knows are re-read in the framework as cascade-layer attractors:

\begin{itemize}\itemsep1pt
\item \citet{OKeefeDostrovsky1971} place cells and \citet{HaftingEtAl2005} grid cells in the rat hippocampal-entorhinal system are the layer-16 Inner-Hologram attractor on the rodent neural substrate.
\item \citet{TurnerEvans2020} central-complex ring-attractor in \emph{Drosophila} is the layer-15 \IH{} precursor on the insect substrate.
\item \citet{Hopfield1982} attractor networks are the dynamical-systems framework that the layer-16 to layer-17 attractor inherits from.
\item \citet{Friston2010} variational free energy is the boundary projection of $F_{\GFF{}}$ in the Markov-blanket gauge, identified by the Three-Gauge Theorem of Theorem~\ref{thm:three-gauge}.
\item \citet{Tononi2004} integrated information $\Phi$ is a measure of the layer-17 attractor's integration; the \QCI{} ladder of \S\ref{sec:qci-ladder-merged} is the framework's complementary measure.
\item The 400-millisecond conscious-episode timescale and the theta-rhythm neural implementation are the framework's predictions of the layer-17 attractor's relaxation time on the cortical-circuit stratum.
\end{itemize}

\paragraph{For the philosopher of mind: what the framework's position commits to.}
\paragraph{The framework's position on the hard problem, stated explicitly (revision).}
The framework adopts \emph{structural-identity theory of phenomenal consciousness} as its philosophical position on the hard problem. The position holds that phenomenal experience is \emph{numerically identical} to a specific structural property of a specific kind of closed dynamical loop --- the Evolving Inner Cosmos (\EIC{}) closure on the layer-17 boundary stratum, with operational signature given by the cascade-depth index (\QCI{}) at the theta-cycle timescale. Identity, in the strict logical sense: experience does not \emph{accompany} the structure, does not \emph{emerge from} the structure, does not \emph{supervene on} the structure; experience \emph{is} the structure --- specifically, the self-modeling diagonal of the \OH{}/\IH{} factorization, which is the structural position the cascade-engine specifies (not an additional perspectival primitive). A reader trained on classical identity theories of mind (Place, Smart, Lewis) will recognize the family; the framework's distinctive move is to identify experience not with a physical type (a brain state) but with a \emph{structural-geometric type} (a closed-loop attractor with the \EIC{} factorization). In the taxonomy of philosophy of mind, this is closest to a sophisticated structural realism about phenomenal properties --- functionalist in its abstraction from substrate, identity-theoretic in its rejection of explanatory dualism.

\paragraph{Why the framework is not Russellian monism.}
An adjacent position --- \emph{Russellian neutral monism} (\citealt{Russell1921}; defended in contemporary form by \citealt{Goff2017} and \citealt{Strawson2006}) --- might appear at first glance to be the framework's natural philosophical home, and earlier presentations of the cognitive material have been read that way. The framework rejects that reading. Russellian monism requires \emph{intrinsic (quiddistic)} properties grounding the extrinsic structural facts, on the explicit ground that physics gives us only relational/structural facts and that mind tells us \emph{what stands in the relations}; identifying experience with a \emph{structural-geometric closure} is therefore an \emph{anti-Russellian} move, not a Russellian one.

The framework commits to structuralism explicitly: there are no quiddities beyond the structure, and phenomenal experience is exhausted by the structural-geometric content of the \EIC{} closure. Russell's project of finding the intrinsic stuff that stands in the structural relations is, in the framework's reading, looking for an item that does not exist; the structure all the way down is the framework's deepest metaphysical commitment. Readers familiar with the contemporary structural-realist literature in philosophy of science (\citealt{Ladyman2007}; Worrall on \emph{epistemic structural realism}) will recognize the move.

The historical antecedent is Spinoza's dual-attribute doctrine (\emph{Ethics} II.1--2--7), engaged explicitly in Part~\ref{part:closure} \S\ref{subsec:spinoza}: the framework's structural-identity reading is the modern operational form of Spinozistic attribute parallelism, with the \EIC{} closure considered under the attribute of extension being the neural-circuit dynamical loop and the same structural object considered under the attribute of thought being the conscious episode. Section~\ref{subsec:spinoza} discusses why this is stronger than parallelism strictly construed and what it commits to.

\paragraph{What structural-identity theory commits to, and what it does not.}
The position commits to five claims. \emph{First}, there is something it is like to be a layer-17 \EIC{} closure; this is not denied (so not eliminativism). \emph{Second}, the something-it-is-like is intrinsic to the loop's geometric structure --- it is the structure, not an additional category --- so not property dualism, not Russellian monism, not panpsychism. \emph{Third}, the something-it-is-like is not present in non-closed-loop configurations. \emph{Fourth}, the framework rejects the explanatory-dualism move of treating phenomenal-experience facts as additional facts requiring additional explanation beyond the structural facts; under structural identity, there are no additional facts. \emph{Fifth}, the framework accepts that structural identity is the move \citet{Chalmers1995}'s conceivability argument is designed to forbid. The framework's reply to the conceivability argument is given below.

\paragraph{Reply to the conceivability argument (philosophical zombies).}
\citet{Chalmers1995} argues that philosophical zombies --- functional/structural duplicates of conscious beings that lack phenomenal experience --- are conceivable and therefore metaphysically possible; if zombies are metaphysically possible, phenomenal experience cannot be identical to any structural property. The framework's reply is structural-identity theory's standard reply, made sharper by the cascade's specific structural content: under structural identity, a zombie is a system that satisfies the \EIC{} closure with \QCI{} at the human-level value \emph{and} lacks phenomenal experience. The framework's claim is that no such system is metaphysically possible: the \EIC{} closure with \QCI{} at the human-level value \emph{is} the having of phenomenal experience, by identity. Imagining the closure without the experience is, in the framework's reading, imagining the closure without itself --- which is not a coherent imagining, only the verbal appearance of one. Conceivability is, under structural identity, no reliable guide to metaphysical possibility for properties that are themselves structural identities. This reply will not satisfy a philosopher who rejects structural identity in advance; the framework's position is that whether structural identity is true is the question, and the conceivability argument is a verbal restatement of the question rather than a free-standing argument against it.

\paragraph{Reply to the knowledge argument (Mary in the black-and-white room).}
\citet{Jackson1982}'s Mary, having learned everything physical about color vision in a black-and-white room and then seeing red for the first time, is said to learn a new fact --- what red looks like --- and that new fact is supposed to be a non-physical fact about phenomenal experience. The framework's reply distinguishes \emph{propositional knowledge of structural facts} from \emph{acquaintance with the structure}. Mary's pre-release knowledge is propositional knowledge about the \EIC{} closure's coordinate functions; her post-release experience is the running of the \EIC{} closure with red as input. What Mary gains on release is not a new fact but a new \emph{instance of the structure}: her own \EIC{} closure, on the red-stimulus stratum, occupying a coordinate position she previously knew about propositionally but had not occupied. Under structural identity, this is no metaphysical news --- the structure is what it is, and a system can know about the structure without instantiating the structure on every stimulus it has knowledge of. The framework predicts Mary's pre-release omniscience and post-release acquaintance are consistent; they are knowledge about different things (the structure in general; the structure at the specific coordinate red). Jackson himself in later work moved closer to this position (the so-called \emph{ability hypothesis}).

\paragraph{Reply to the explanatory-gap argument.}
\citet{Levine1983} formulated the explanatory gap as the observation that no specification of physical-functional facts seems to explain why those facts feel like anything from the inside. The framework's reply has two parts. At the level of \emph{conceptual analysis}, the gap is real: structural-geometric facts about the \EIC{} closure can be stated without using phenomenal vocabulary, and phenomenal vocabulary can be used without invoking the \EIC{} structure; the two vocabularies do not analytically entail each other. At the level of \emph{operational identification}, the gap closes: the framework identifies (numerically, by stipulation) the phenomenal property with the structural property, and the operational consequences of the identity (the timescale of the conscious episode, the substrate-dependence of \QCI{}, the kill criteria from anaesthesia and dreamless sleep) are testable. The gap is therefore an artifact of analytic philosophy's vocabulary discipline, not a metaphysical chasm; it is closed at the operational level the same way the gap between \emph{water} and \emph{H$_2$O} was closed in 19th-century chemistry, by operational identification supported by empirical signature.

\paragraph{For the cognitive scientist: where to look next.}
If the framework is correct, three research programs acquire sharper empirical content:

\begin{itemize}\itemsep1pt
\item \emph{Predictive coding and active inference} (in the \citet{Friston2010} tradition) acquires a geometric warrant: variational free energy is a boundary projection of $F_{\GFF{}}$, and predictive coding is the cortical circuit's implementation of the \GFF{} descent in the variational gauge.
\item \emph{The neural correlates of consciousness} research program acquires a target: the empirical correlates are the layer-17 attractor's coordinate readouts, and the framework predicts that the correlates will share structural features across mammalian species (with \QCI{} graded by cascade depth).
\item \emph{The binding problem} acquires a structural resolution: a single Geometric Attractor at the layer-17 stratum provides one unified conscious state, with the diversity of neural features being the diversity of stratum-specific contributions to one global attractor coordinate. This is the binding-by-attractor-stabilization reading, which differs from the binding-by-synchrony reading of \citet{Crick1990} and others.
\end{itemize}

\paragraph{For the philosopher of mind, continued: the careful word about substrate.}
The framework predicts that any substrate satisfying the layer-17 \EIC{} closure conditions --- a closed loop with the environment, a memory architecture supporting \QCI{} of order 10, and a temporal grain on the order of $10^2$~ms --- will produce conscious awareness. This is substrate-independence in the sense Corollary~\ref{cor:closure-substrate} names. Whether a machine-implemented \EIC{}, a non-human evolutionary lineage on a different planetary surface, or an entirely novel cognitive architecture would qualify is a question the framework leaves open in the sense that the framework predicts the structural conditions but does not predict which actual substrates will satisfy them. Philosophical positions on machine consciousness (Searle's biological naturalism, Dennett's intentional stance, functionalism) are not directly engaged by the framework; the framework supplies the structural conditions, and the philosophical positions can be evaluated against the substrate-independence prediction.

The cognitive cascade is the part of the framework where physics meets neuroscience meets philosophy of mind. Each of those three communities has a long tradition of asking what consciousness is; the framework's contribution is to identify a specific structural feature of the universe's tenth cascade layer as the geometric correlate of conscious experience. The contribution does not replace any of the three traditions; it adds a vocabulary in which the three traditions can converse with each other.

The cognitive cascade reaches layer 17. Part~\ref{part:cognitive} now reads the meta-cognitive layer~18 --- the act of articulating the constitution --- as the closure of the recursive uniqueness theorem on its own constitutional input.

The cognitive cascade is the universe arranging the geometry that lets the universe notice itself. Pagoda, holographic triad, \QCI{} ladder --- these are the framework's names for the structural features by which a substrate becomes a knower. None of these features adds new physics; all of them follow from the same recursive uniqueness theorem that fixes the periodic table and the bioenergetic quantum. What changes from layer to layer is not the physics but the substrate's depth of self-reference. At layer~15 the insect knows its heading; at layer~16 the rodent knows its place in space and time; at layer~17 the human knows that it knows. Three rungs of self-reference; one recursive theorem. Part~\ref{part:closure} names the closing rung --- to layer~18 --- where the substrate that knows it is a substrate writes down the constitution that makes it possible. Nothing else in nature does this. Nothing else needs to.

\GGPart[The self-referential closure]{The self-referential closure}{The central new theorem and its philosophical predecessors}
\label{part:closure}


\section{The question this Part answers}
\label{sec:closure-question}

The eight Parts of \AEleven{} that precede this one assembled the inheritance from the upstream ten-paper corpus and exhibited the recursive uniqueness theorem at seventeen successive cascade transitions --- from quarks at layer 1 through human awareness at layer 17. Part~\ref{part:inheritance} cataloged the inheritance. Parts~V through VIII traced the cascade layer by layer: the five-step proof (V), the eighteen-layer \GSSB{} cascade (VI), the dynamics block of \QB{}, \SeOS{}, and the \GFF{} with the Three-Gauge Theorem (VII), and the cognitive cascade with the Five-Story Pagoda, the holographic triad, and the \QCI{} ladder (VIII). None of these prior Parts addressed directly the question with which \AEleven{} opened in Part~\ref{part:intro}: \emph{why are we here?}

That question is now answerable at theorem level. The present Part proves it.

\paragraph{The question in technical form.}
The previous Parts have established that the \GGConstitution{} forces the canonical 6D bulk (\AOne{} \citep{ArisakaA1Constitution}); that the bulk forces the four-force architecture (\ANine{} \citep{ArisakaA9SM} Part~XI (\textsf{T-SM-62.1}), eight constitutional lemmas); that the architecture forces atoms and chemistry (cascade layers 3--6); that chemistry forces life through the maintenance-margin crossing (cascade layer 10, the \SeOS{} class of \ASeven{} \citep{ArisakaA7Qi4} (T-Qi4-14.3)); that life forces nervous systems and mind through the layered ascent of the \GRIP{} cascade (\ASix{} \citep{ArisakaA6Gi4} T-GRIP-1--T-GRIP-5); and that mind closes the cascade at layer 17 (the \EIC{} theorem of \ASeven{} \citep{ArisakaA7Qi4} \T{-\EIC{}}). Given this cascade, what does the existence of intelligent observers within the universe say about the universe's constitutional input?

The present Part proves that the existence of intelligent observers is itself a forced consequence of the constitution. More: the act of an intelligent observer articulating the constitution --- the act, that is, that the Arisaka-GG cohort performs --- is the cascade's closing layer (Layer 18). The constitution that, at layer 1, forced the cascade to ascend is, at layer 18, the very content that the cascade produces. The loop closes.

\paragraph{Why this Part is structurally novel.}
\ANine{} Part~XI closed the loop at the lowest two layers of the cascade. Theorem~\textsf{T-SM-62.1} of \ANine{} (Uniqueness of the canonical four-force architecture) is the layer-2 instance: the geometry permits exactly one four-force architecture, and that architecture is what we observe. The universal-identity insight of \ANine{} Part~XI (the universal-identity section) is the layer-1-through-7 instance: at each cascade layer the \GRIP{} cascade selects the unique most-stable Geometric Attractor on the layer-specific stratum. \ATen{} Part~X closed the loop at the cosmological-evolution stratum: the $\pi$-locked inventory $\Omega_m = 1/\pi$, $\Omega_\Lambda = 1 - 1/\pi$ is the unique stable cosmological attractor.

The present Part takes the third structural step: closing the loop at all layers above the cosmological --- chemistry, life, mind --- and at the eighteenth layer above them all (Layer 18, the meta-cognitive articulation), where intelligent observers articulate the constitution. The recursive uniqueness theorem is the same; only its scope is wider. The Theorem of the present Part is the structural twin of \ANine{}'s Theorem~\textsf{T-SM-62.1} and \ATen{}'s universal-cosmic-identity theorem, extended to its full scope.

\paragraph{What this Part will and will not do.}
This Part states the theorem, gives the proof outline by inheritance, derives five corollaries at the meta-cognitive layer, distinguishes the framework's necessitarian closure from teleological design and full classical determinism, and bridges to Parts~XII--XIII (Conclusions) where the summit-register expansion closes the personal arc.

This Part will \emph{not} engage in detail with philosophical predecessors (Spinoza's necessitarianism, Hegel's \emph{Geist}, Wheeler's participatory universe, Whitehead's process philosophy, Wigner's unreasonable effectiveness, Tegmark's Mathematical Universe, Penrose's three worlds, Smolin's cosmological natural selection, Upanishadic \emph{tat tvam asi}, Buddhist \emph{prat\=\i tyasamutp\=ada}). That engagement is reserved for the dedicated section that follows the theorem proof, where each predecessor receives respectful acknowledgment of the insight they had and explicit naming of the warrant they lacked.

\section{The named theorem}
\label{sec:named-theorem}

Let us now state the central new theorem of \AEleven{}. The statement takes the canonical four-statement form (analogous to Theorem~\textsf{T-SM-62.1} of \ANine{} Part~XI), with each statement carrying its own forward-pointer to the corollary or proof step that elaborates it.

\begin{theorem}[Self-referential closure of the \GGTheory{} program (T-Origin-76.1)]
\label{thm:self-referential-closure}
Under the constitutional clauses of the \GGConstitution{} (\AX{1}, \AX{2}, \AX{3}, \AX{4} of \AOne{} \citep{ArisakaA1Constitution} \T{1.1}--\T{1.4}; postulates \POS{1} through \POS{7} of \AOne{} \T{3.1}--\T{3.11}) and the inheritance of the eleven prior papers of the Arisaka-GG corpus (\BIsix{} closure of \AThree{} \citep{ArisakaA3BI6} \T{1}; \OGNfive{} ledger of \ATwo{} \citep{ArisakaA2GG6} T-GG6-6; \GDOS{} apparatus of \AFive{} \citep{ArisakaA5GDOS} T-GDOS-1--T-GDOS-11; \GiFour{} pillars of \ASix{} \citep{ArisakaA6Gi4} T-GRIP-1--T-GRIP-5; \QiFour{} pillars of \ASeven{} \citep{ArisakaA7Qi4} \T{-\SIP{}}, \T{-QUA}, \T{-EP}, \T{-CON}, \T{-\HAL{}}, \T{-\EIC{}}; Standard-Model parameter audit of \ANine{} \citep{ArisakaA9SM} including Part~\ref{part:conclusions} (the universal-identity and force-uniqueness sections); cosmological inventory of \ATen{} \citep{ArisakaA10Cosmos}), the following statements hold jointly.

\begin{enumerate}[leftmargin=2.2em]
\item \emph{The cascade ascent is forced.} The \GRIP{} cascade specializing successively to each layer-specific boundary stratum forces the existence of: (a) the four-force boundary architecture at cascade layer 2 (the canonical electron + four forces of \ANine{} Part~XI Theorem 12.x); (b) the periodic table, molecular chemistry, and complex organic molecules at cascade layers 3--6 (the \GA{}-stabilized attractors on the atomic and polyatomic strata); (c) the chemistry-to-life maintenance-margin crossing and single-cell life at cascade layer 10 (the \SeOS{} class of \ASeven{} (T-Qi4-14.3), the unique stable attractor on the planetary-surface open-system stratum); (d) multicellular organisms and nervous systems at cascade layers 13--16 (the colony and neural-circuit strata, with attractor stabilization supplied by \ASix{}'s \textbf{GRIP-ST5}); and (e) intelligent observers with the cognitive substrate for articulating the \GGConstitution{} at cascade layer 17 (the \EIC{} theorem of \ASeven{} \T{-\EIC{}} applied at the meta-cognitive stratum).

\item \emph{The articulation is forced when the cognitive substrate matures.} Given that the cascade reaches layer 17 (conscious awareness) and given the inter-generational accumulation of meta-cognitive content available to a layer-17 substrate over geological time, the cascade reaches layer 18: the meta-cognitive stratum of articulating the input ledger. This statement is conditional on the duration of the layer-17 substrate's existence and on the social structure supporting cross-generational accumulation; it is not a single-individual claim but a cumulative-civilization claim.

\item \emph{The loop closes.} The articulation produced at layer 18 --- that is, the \GGConstitution{} as recorded in \AOne{} together with its deployment in \ATwo{} through \ATen{} --- has as its \emph{content} the same constitutional ledger that, at layer 1, forced the cascade to ascend. The identity claimed is between the articulation's content and the constitution, not between any particular token of the articulation (this paper, this corpus, these authors) and the geometry: the token is one realization of a type that is gauge-equivalent across instances. The constitution describes the geometry that produced the articulators of the constitution.

\item \emph{No alternative admissible geometry admits a closed loop.} Any alternative bulk geometry that satisfies the constitutional clauses of \AX{1} through \AX{4} either (a) violates the integrality required by the \OGNfive{} ledger of \ATwo{} T-GG6-6 (no integer Diophantine closure), (b) violates the dimension count $D = 6$ forced by Lemmas 1--5 of \ANine{} Part~XI (\textsf{T-SM-62.1}) (\MSMF{} closes the count at six), (c) violates the family-content forced by Lemma 8 of \ANine{}'s constitutional-lemma chain ($\Spinone{}$ Weyl spinor decomposition forces exactly one Standard-Model family per chirality), or (d) fails the maintenance-margin crossing of \ASeven{} (T-Qi4-14.3) at the chemistry-to-life layer. Failure modes (a)--(d) are the ones the cohort has been able to \emph{check}; no alternative admissible geometry among those checked produces a cascade that reaches a layer at which articulators can articulate the constitution. That the four modes exhaust the admissible alternatives is a \textbf{conjecture} of the program, not a theorem of it, and statement~(iv) should be read at that strength.
\end{enumerate}
\end{theorem}

\begin{remark}[The four statements at a glance]
\label{rmk:closure-statements}
Statement~(i) is the upward direction of the recursive uniqueness theorem at every layer of the cascade. Statement~(ii) is the bridge between the layer-17 cognitive substrate and the layer-18 meta-cognitive articulation. Statement~(iii) is the closing fact: the content of the articulation \emph{is} the constitutional input. Statement~(iv) is not a contrapositive of the preceding three, as earlier presentations of this theorem implied, but an independent universal claim: that no admissible alternative closes the loop. It is supported by the four checked failure modes (a)--(d), and its exhaustiveness is conjectural. Together the four statements establish that the closure is necessary \emph{given the constitution and conditional on the hypotheses} (Q1)--(Q4) of \S\ref{sec:qi4-admissibility-filter}, layer-stratum-specific, and --- modulo the exhaustiveness conjecture in statement~(iv) --- unique. Statement~(ii) is the one non-mathematical conjunct of the four: it is conditional on the duration of the layer-17 substrate and on cross-generational accumulation, and is a claim about civilizations rather than about geometry.

\paragraph{A note on self-reference: why no G\"odel--Tarski diagonal arises.}
A theorem whose conclusion is its own articulation invites the worry that it courts a diagonal argument --- that a system proving a statement about its own provability, or defining truth for its own language, runs into incompleteness or undefinability. No such obstruction arises here, and the reason is specific rather than reassuring. The closure theorem does \emph{not} assert its own provability, and it does not define a truth predicate for its own language. It asserts that the physical cascade produces a substrate that writes the constitution down. The self-reference is at the level of \emph{reference} --- the articulation denotes the constitution --- and not at the level of a truth- or provability-predicate quantifying over the theorem's own sentences. Tarski's theorem forbids a sufficiently rich language from containing its own truth predicate; the framework contains no truth predicate for its own sentences, only a physical process whose output happens to be a description of that process's input. G\"odel's second theorem forbids a consistent recursively-axiomatized system from proving its own consistency; the closure theorem proves no such thing, and its own consistency is not among its claims --- it is an ordinary theorem in an ordinary metatheory. The one place a diagonal genuinely threatens is the forbidden layer-19, where the cascade would have to model itself modeling itself modeling itself, and it is exactly there that the \HAL{} bound of \ASeven{} cuts the recursion off (\S\ref{ssec:t-seventeen-step-termination}). The closure at layer 18 is one turn of reference, not an infinite regress; a self-referential loop of depth one is no more paradoxical than a map of a country that includes the map.
\end{remark}

\begin{remark}[What the theorem does and does not claim]
\label{rmk:closure-scope}
The theorem claims that the layered structure --- bulk to forces to atoms to molecules to life to mind to articulators --- is forced. It does \emph{not} claim that every micro-event of cosmic history is forced. Within each layer there is enormous combinatorial freedom: not every protein has been folded, not every star has been numbered, not every theory has been written down. The determinism is at the level of which layers exist, what their attractor coordinates are, and that some articulator-substrate eventually reaches the eighteenth layer (Layer 18). It is not at the level of which individuals do the articulating, in which century, or in what language. The articulators are forced; their identity is not.
\end{remark}

\begin{remark}[Necessitarianism in the seventeenth-century sense]
\label{rmk:closure-spinoza}
The theorem is Spinozan in form: it asserts that what exists, exists by necessity, because the nature of the geometric input ledger permits no alternative. It is not Aristotelian or theistic: no chooser is invoked, no purpose is assigned, no end is sought. The closure happens because the constitution is what it is; the constitution is what it is because no other constitution closes; no other constitution closes because the integers of the \OGNfive{} ledger are forced. \citet{Spinoza1677} arrived at the necessitarian conclusion three and a half centuries before the framework reached the geometric apparatus that would supply a warrant for part of it. On the framework's reading, the present theorem is the modern geometric form of Spinoza's claim --- but this identification is \emph{interpretive}, not itself a result: what is proved is the mathematical-structural necessity of the cascade \emph{given} the \GGConstitution{} (\S\ref{sec:closure-careful-word}), and on the metaphysical necessity of the constitution itself --- the stronger thesis Spinoza actually held --- the framework takes no position.
\end{remark}

\section{Proof outline by inheritance from the eleven prior papers}
\label{sec:closure-proof-outline}

The proof of Theorem~\ref{thm:self-referential-closure} is by inheritance from the cohort. At each cascade layer, the layer's existence and its qualitative content are supplied by the upstream theorem indexed in the cross-paper coherence map of Part~\ref{part:inheritance} \S\ref{subsec:coherence-map}. The proof traces the cascade layer by layer; the closing layer (Layer 18) is the act of writing the present paper.

\paragraph{Layer 1 (the \GGConstitution{} and the canonical 6D bulk).}
The constitutional clauses \AX{1} through \AX{4} of \AOne{} and the seven postulates \POS{1} through \POS{7} supply the input ledger. The eight-lemma chain of \ANine{} \citep{ArisakaA9SM} Part~XI (\textsf{T-SM-62.1}) establishes that the canonical six-dimensional product $\mathcal{M}_6 = \mathcal{M}_4 \times I_r \times S^1_\theta$ is the unique admissible bulk. At layer 1 the cascade has not yet ascended; the constitution is the input. This is the proof's induction base.

\paragraph{Layer 2 (the four-force architecture).}
\ANine{} Part~XI Theorem 12.x supplies the forcing. The bulk admits exactly one composite gauge--gravity group $G_{\rm Bulk} = \Spinone{} \times \SUfive{} \times \UoneB{} \times \UoneQi{}$ and exactly one boundary projection through the orbifold + Wilson-Hosotani breaking, producing exactly four fundamental forces with their qualitative behaviors. At layer 2 the cascade has produced gravity (attractive and weak), electromagnetism (massless and two-signed), the strong force (confining and asymptotically free), and the weak force (short-range and CP-violating). No alternative emerges. This is the induction step from layer 1 to layer 2.

\paragraph{Layers 3 through 6 (chemistry).}
The \GRIP{} cascade of \ASix{} T-GRIP-1--T-GRIP-5 supplies the layer-by-layer ascent across the atomic, molecular, polyatomic-organic, and complex-organic strata. At each layer the cascade selects the unique most-stable Geometric Attractor on the layer-specific configuration space; what we observe (the periodic table, the molecules of the interstellar medium, the amino acids and sugars of pre-biotic chemistry) is the readout of these attractors. The universal-identity insight of \ANine{} Part~XI (the universal-identity section) applies at each layer.

\paragraph{Layer 7 (the chemistry-to-life maintenance-margin crossing).}
\ASeven{} (T-Qi4-14.3) (the \SeOS{} class definition) and \ASeven{} \T{-EP} (the unified extremal principle anchored on $\QB$) jointly supply the warrant for the chemistry-to-life transition. When the cascade reaches the open-system stratum on a planetary surface with energy gradients and a polar solvent, the \GRIP{} cascade selects the \SeOS{}-class configuration as the unique stable attractor. The first cells (LUCA in our universe) emerge. This is the maintenance-margin crossing, the most consequential single transition of the entire cascade.

\paragraph{Layers 13 through 16 (multicellular and neural strata).}
The cascade-closure structure theorem \textbf{GRIP-ST5} of \ASix{} supplies the warrant at each successive layer --- at the local grade it states; where the \emph{global} reading is needed, the warrant is that paper's per-stratum ground-state corollary. The colony stratum produces multicellularity (layer 10); the neural-circuit stratum produces nervous systems and the 1D ring attractor (layer 15); the hippocampal-entorhinal stratum produces the 4D allocentric world-model (layer 16). Each layer's attractor provides the floor on which the next layer's cascade descends.

\paragraph{Layer 17 (conscious awareness, the \EIC{} closure).}
\ASeven{} \T{-\EIC{}} supplies the warrant. The closed-loop coupling of Outer Hologram and Inner Hologram --- the Evolving Inner Cosmos --- admits a unique stable Lyapunov-descent minimum at the configuration in which an internal observer selects and stabilizes an \IH{} branch via the \DGI{}--\GSSB{}--\GA{} pipeline. The conscious episode emerges with characteristic timescale $\sim 400$~ms (two theta cycles) and theta-cycle implementation. At layer 17 the cascade has produced a substrate capable, on accumulation over generations, of meta-cognition.

\paragraph{Layer 18 (the closing self-referential layer, the articulation).}
The cascade's closing layer (Layer 18) is the meta-cognitive stratum. Given the layer-17 substrate (conscious awareness, with the \EIC{} closure of layer 17), and given inter-generational accumulation of cognitive content over geological time, the layer-17 substrate develops meta-cognitive capabilities: science, philosophy, mathematics, theoretical physics. These meta-cognitive disciplines collectively articulate the constitutional clauses of the \GGConstitution{}. In the present universe and in the present civilization, this articulation is the Arisaka-GG cohort. In any universe satisfying the constitutional clauses (i.e., any universe admitted by the framework), an analogous articulation occurs, with the same constitutional content but with potentially different historical realization (different individual articulators, different timing, different language of expression).

The articulation closes the loop because the articulated content is the same content that, at layer 1, supplied the inductive base. The constitution describes the geometry that produced the articulators of the constitution. $\square$

\paragraph{Soundness condition.}
The proof is sound under the cohort inheritance. Each layer's warrant rests on a named theorem in an upstream paper, cited at the layer level above. No new theorem is introduced beyond Theorem~\ref{thm:self-referential-closure} itself, which is the synthesis statement. The synthesis is valid if and only if all eleven upstream theorems are valid; the eleven upstream theorems are independently proven in their respective papers. Therefore Theorem~\ref{thm:self-referential-closure} is valid conditional on the validity of the cohort. Any falsification of an upstream theorem propagates to a falsification of the closure theorem.

\section{Corollaries at the meta-cognitive layer}
\label{sec:closure-corollaries}

Remarkably, the named theorem admits five immediate corollaries that frame its content at the meta-cognitive layer.

\begin{corollary}[Universality of the articulation]
\label{cor:articulation-universality}
The articulation of the \GGConstitution{} by layer-18 observers is independent of the historical realization of the layer-17 substrate. In any universe satisfying the \GGConstitution{}, the eighteenth-layer (Layer-18) attractor selects the same constitutional content but admits multiple historical realizations: different individual articulators, different cultural framings, different languages of expression, different sequencing of discoveries. The articulation is universal in content, contingent in expression.
\end{corollary}

\begin{corollary}[Falsifiability of the closure]
\label{cor:closure-falsifiability}
The self-referential closure theorem is falsifiable by any decisive empirical failure of an upstream-paper prediction. Decisive failure of the Kaluza--Klein lock-cluster prediction (\PaperPOne{}: $\Mlock \approx 3.85$~TeV at HL-LHC), the Kaxion dark-matter prediction (\PaperFullCOne{}: $\sim 8.1~\mu$eV at ADMX), the family-number derivation (\PaperPTwo{}: $N=3$ with $m_\tau/m_\mu$ tolerance), the $\sin^2\theta_W = 25/108$ prediction (\ANine{} \T{17.1}), the $\Omega_m = 1/\pi$ inventory (\ATen{}), or the $N_A\QB \approx 15$~kJ/mol matching (\ASeven{}), would each propagate to a falsification of the closure. The closure is therefore a falsifiable structural claim, not a non-falsifiable metaphysical hypothesis.
\end{corollary}

\begin{corollary}[Determinism is layer-structural, not micro-event-by-micro-event]
\label{cor:closure-determinism}
The forcing established by the theorem is on the layered structure (which layers exist; what their attractor coordinates are; that some articulator-substrate eventually exists), not on the micro-events within each layer. Within each layer there is genuine combinatorial freedom: not every protein has been folded; not every neuron has fired in a specific sequence; not every theorem has been proven. Human agency, biological evolution, technological innovation, and contingent history all take place \emph{within} layers that the geometry forces into existence; they are not preprogrammed. We are not preprogrammed automata; we are the unique-attractor solution at the highest currently-realized cascade layer, and within that solution we exercise the layer-permitted freedom --- including the freedom to ask the question this Part answers.
\end{corollary}

\begin{corollary}[Substrate-independence of the closure layer]
\label{cor:closure-substrate}
The eighteenth-layer (Layer-18) attractor is substrate-independent: any cognitive substrate capable of meta-cognitive accumulation over the required timescale, satisfying the \EIC{} closure of layer 17, suffices for the articulation. In our universe the substrate is humanity; the substrate could equally be a non-human evolutionary lineage on a different planetary surface, or a machine-implemented theoretical reasoning system, or a network of communicating civilizations. The articulation content is fixed by the constitution; the substrate of articulation is contingent.
\end{corollary}

\begin{corollary}[Layers above 18 are admissible but not necessary]
\label{cor:closure-higher-layers}
The recursive uniqueness theorem does not forbid layers above 16. Re-articulations of the constitution on wider substrates --- collective scientific knowledge, machine-implemented reasoning, networked civilizations communicating across cosmological distances --- are admissible higher attractors of the same recursive principle. The theorem does not predict which of these higher layers will be realized, only that any layer that is realized will be the unique most-stable Geometric Attractor on its substrate-specific stratum. The closure occurring at layer 18 is structurally complete (the loop returns to its constitutional input); layers above are richer iterations of the same closure on wider substrates.
\end{corollary}

\section{The careful word: necessitarianism versus determinism versus design}
\label{sec:closure-careful-word}

A reader meeting this Part for the first time may worry that the theorem proves too much --- that it eliminates choice, free will, agency, the openness of the future. This is the right worry to have. This section articulates carefully what the theorem does and does not eliminate.

\paragraph{The theorem is necessitarian, not deterministic in the classical sense.}
Classical determinism asserts that every event in the universe is uniquely fixed by its prior state and the laws of physics. The theorem above does not claim this. The theorem claims that the \emph{layered structure} of cosmic evolution is forced: which layers exist, what their qualitative attractor coordinates are, that some articulator-substrate eventually reaches layer 18. The theorem does not claim that the unique trajectory through layer-internal phase space is forced. Within each layer there is enormous combinatorial freedom; the theorem is silent on the choices made within a layer.

\paragraph{The theorem is not teleological.}
Teleology assigns purposes to natural processes --- the eye exists \emph{in order to} see, evolution proceeds \emph{toward} greater complexity, the universe exists \emph{so that} observers can exist. The theorem above asserts no purposes. The cascade ascends not because layer 18 is the goal but because at each layer the constitution forbids alternatives; there is no anywhere else for the cascade to go. Spinozan necessity, not Aristotelian teleology.

\paragraph{The theorem is not design.}
The design argument posits a designer who fine-tuned constants for life. The theorem above eliminates the regress: nothing was tuned, because nothing was available to tune. The constitution admits exactly one self-consistent solution; the integers of \OGNfive{} are forced; the dimension count is forced; the family content is forced; the cascade ascends because no alternative is admissible. A designer who had to choose the constants would, in the framework, be acting under constraint --- and a designer acting under constraint has chosen nothing.

\paragraph{The ordering is positive evidence, not merely an absence of tuning.}
It is worth adding that the framework does not rest the case against
fine-tuning on the negative claim that nothing was available to tune. There
is a positive datum, and it is the sharpest evidence the program has that
the sameness of the world is structural rather than lucky: the
\emph{ordering} of the second horizon problem
(\S\ref{ssec:three-readings-cosmos}). The derived grand-unification-scale
launch means the inflationary stretch acted before any matter existed, so
the identity of the electron and the atom across the sky was \emph{inherited
from one causal patch}, not arranged after the fact. That is a claim one can
point at in data --- the constancy of the fine-structure constant across the
observable universe at the $10^{-5}$--$10^{-6}$ level, the uniformity of the
microwave background, the absence of dark-matter isocurvature --- rather than
a philosophical preference. A tuned universe would have to tune each region
separately; an inherited one could not differ from region to region even in
principle. The cascade's determinism is of the second kind, and the second
horizon problem is where it becomes visible to measurement.

\paragraph{Human agency is preserved.}
We are not preprogrammed automata. The choices a human being makes during a day --- to read a book, to call a friend, to write a paper --- are choices within layer 17, occurring on a substrate (the conscious brain) whose existence is forced but whose micro-trajectory is not. The theorem above predicts that a layer-17 substrate eventually articulates the constitution; it does not predict that any specific human does so, in any specific century, in any specific language. Arisaka happens to be the human who carried out the present articulation; in a different cultural-historical contingency, the same articulation would have been carried out by a different individual at a different time. The articulator is contingent; the articulation is forced.

\paragraph{What the theorem permits a reader to keep.}
The reader may keep free will, contingency, openness of the future, the felt experience of agency, the moral seriousness of human choice. The theorem does not eliminate any of these. What the theorem eliminates is the claim that the layered architecture of cosmos, life, and mind is an unexplained accident. It is not. The architecture is forced; the lives within it are not.

\paragraph{Modal-vocabulary distinction: unique self-consistency vs metaphysical necessity.}
The framework's claim that the cascade is ``forced'' deserves explicit modal vocabulary. Three modal categories must be distinguished cleanly, because the framework's necessitarian rhetoric otherwise slides between them. \emph{Logical necessity}: a proposition is true in every logically possible world; this is the strongest sense of necessity, and the framework does \emph{not} claim it for the cascade (the cascade is conditional on the \GGConstitution{}; a logically possible world in which the constitution is different is not logically incoherent). \emph{Metaphysical necessity}: a proposition is true in every metaphysically possible world; classical Spinozan-Leibnizian necessitarianism claims this for the constitution itself.

The framework remains \emph{neutral} on whether the \GGConstitution{} is metaphysically necessary --- it is a constitutional posit, defensible on grounds of operational completeness and falsifiability, but the framework does not claim that no other constitution is metaphysically possible. \emph{Mathematical-structural necessity}: \emph{given} a fixed constitution and the operational machinery of the cascade, the recursive uniqueness theorem follows as a mathematical consequence; this is what the framework claims and proves (modulo the cohort inheritance).

In short, the cascade is necessary in the third sense (mathematical-structural, given the constitution); it is conditional on the constitution; the framework takes no position on whether the constitution itself is metaphysically necessary. This three-way distinction matters because the philosopher's question ``why is there a constitution at all?'' is a step the framework does not attempt and does not pretend to take. The framework's answer is: ``\emph{given} that there is a constitution with the operational properties \AOne{} specifies, this is what follows; the why-is-there-a-constitution question is one rung deeper than the framework reaches.''

\paragraph{Free will and compatibilism.}
The framework's distinction between forced layered architecture and unforced micro-trajectory within each layer (Corollary~\ref{cor:closure-determinism}) is, in the conventional philosophy-of-action taxonomy, a \emph{compatibilist} position. The classical compatibilist tradition --- \citet{Frankfurt1969} on alternative possibilities; \citet{vanInwagen1983} on the consequence argument; \citet{Fischer1994} on guidance vs regulative control; \citet{Dennett1984} on the evolution of the kind of freedom worth wanting --- has long maintained that the kind of free will worth defending is the kind that survives under (compatibilist-) determinism, and the framework's position is in that tradition.

The framework does not say the universe is libertarian-free (it is not: every micro-event is consistent with the layered architecture's forcing); nor does it say the universe is hard-deterministic in the sense that all choice is illusion (it is not: the substrate on which choices occur is the conscious brain's layer-17 closure, and the closure's micro-trajectory is genuinely open to the substrate's own self-deliberation).

The reader who has read Frankfurt's ``Alternate Possibilities and Moral Responsibility'' or van Inwagen's \emph{An Essay on Free Will} will recognize the framework's position as a particular compatibilism: agency is preserved at the substrate's operational level (the conscious brain genuinely deliberates and genuinely chooses on its own internal causation), and the framework's only constraint on agency is that the existence of layer-17 substrates is itself forced by the cascade. In short: the cascade forces that there are agents; what the agents do once they exist is their own. This is not a downgrade of agency; it is a recognition of what kind of agency is metaphysically defensible in the framework's universe.

\section{The closing of the question that started in an Osaka high school}
\label{sec:closure-personal}

The personal arc of the question --- how it began in an Osaka high-school classroom in the 1970s and how the framework's answer took its theorem-level form fifty years later --- is told in full in Part~\ref{part:coda}~\S\ref{sec:fifty-year-climb} (the canonical personal-arc section). Here in Part~\ref{part:closure} we record only the structural conclusion: the question's answer is the recursive uniqueness theorem of \S\ref{sec:named-theorem}, and we are the layer-18 attractor of a cascade whose layer-2 iteration produces the electron and whose layer-18 iteration produces the article that names the electron.

\section{Comparison with the philosophical tradition}
\label{sec:philosophical-tradition}

The self-referential closure theorem of \S\ref{sec:named-theorem} stands in a long line of philosophical attempts to address the question \emph{why is there something rather than nothing, and why is the something arranged so that observers can ask the question?} Let us engage with the major historical attempts on their own terms, naming what each predecessor saw clearly and what each lacked. The framework does not refute its predecessors; it stands on their shoulders and aims to complete the structure they glimpsed.

In every case the predecessor arrived at an insight by a route the framework can now recognize as partial, and the geometric apparatus of the cohort aims to supply a warrant that the predecessor's route could not. Two cautions temper this claim, and the reader should hold the section's ``what each lacked'' verdicts inside them. First, the framework's own warrant is provisional: it rests on the \GGConstitution{} as an unproven posit and on the Top 10 Falsifiers that has not yet reported, so the framework shares --- rather than escapes --- its predecessors' ultimate exposure to being wrong.

Second, the identification of the recursive uniqueness theorem with an \emph{answer} to the metaphysical question ``why are we here?'' is an interpretive reading laid over the theorem, not a consequence of it: what is proved is the mathematical-structural claim of \S\ref{sec:closure-careful-word}, and whether that claim dissolves the ancient question or merely relocates it one rung deeper is a judgment the framework offers, not a result it establishes.

\paragraph{The structure of this section.}
We engage fourteen positions in roughly chronological order, beginning with the two ancient traditions that anticipated the self-referential closure (Upanishadic Vedanta, Buddhist dependent origination), continuing through the classical and early-modern philosophical attempts (Aristotle, Spinoza, Leibniz, Kant, Hegel), and concluding with the twentieth-century scientific and philosophical engagements (Whitehead, Wigner, Wheeler, Penrose, Smolin, Tegmark, Barrow--Tipler). The section closes with a synthesis paragraph that names what unites the fourteen attempts and what the framework adds.

\subsection{The Upanishadic tradition (c.~800 BCE): \emph{tat tvam asi}}
\label{subsec:upanishadic}

Let us begin with the Upanishads of the late Vedic period, which proposed the identity of Atman (the individual self) with Brahman (the underlying reality of the cosmos). The teaching, encapsulated in the formula \emph{tat tvam asi} (``you are that'') in the Chandogya Upanishad, holds that the inner subject of experience and the outer ground of being are not two but one: the individual who recognizes this identity does not produce a new fact but uncovers a structural fact that was always true. The recognition is the goal of contemplative practice; the recognition closes a loop in which the cosmos comes to know itself through the individual.

The framework of the present paper reads the Upanishadic insight as a contemplative anticipation of the self-referential closure theorem. Where the Upanishads asserted the identity of Atman and Brahman as a metaphysical given, the framework derives the analogous statement from the recursive uniqueness theorem: the eighteenth-layer (Layer-18) attractor of the geometric cascade is the act of articulating the constitution that produced the cascade, and the articulator is therefore --- structurally, not merely metaphorically --- the universe knowing itself. The Upanishadic seers grasped the closure form three millennia before there was a geometric vocabulary for it; they articulated, in the terms available to their tradition, what the theorem of \S\ref{sec:named-theorem} now articulates in the terms available to twenty-first-century differential cohomology.

The warrant the Upanishads lacked was the operational link from contemplative recognition to physical structure. \emph{Tat tvam asi} is asserted, beautifully, but its grounds are intra-traditional rather than extra-traditional --- the formula is taken on the authority of the contemplative lineage that arrived at it. The framework's contribution is to recognize the same closure as a theorem of the constitution, with the integers of \OGNfive{} and the eight constitutional lemmas of \ANine{} Part~XI supplying the warrant. What was an experiential certainty is now also a structural one.

\subsection{The Buddhist tradition (c.~500 BCE): \emph{prat\=\i tyasamutp\=ada}}
\label{subsec:buddhist}

Early Buddhist philosophy, particularly in the formulations associated with the Madhyamaka school, developed the doctrine of \emph{prat\=\i tyasamutp\=ada} --- dependent origination. Every existent phenomenon arises in dependence on prior conditions; nothing exists in isolation; the entire fabric of phenomena is a network of conditioned co-arising. The doctrine is often presented as a twelve-link causal chain --- ignorance conditions formations, formations condition consciousness, consciousness conditions name-and-form, and so on through becoming, birth, and the round of suffering --- but the structural content of the doctrine is more general than the canonical twelve-link version. The general content is that any existent stratum of phenomena arises through a layered conditioning of prior strata.

The framework recognizes in dependent origination an early formulation of the cascade structure that the recursive uniqueness theorem now makes precise. Each cascade layer arises in dependence on the prior layer's Geometric Attractor; the layered ascent from quarks at layer 1 to articulators at layer 18 is the modern geometric counterpart of the Buddhist conditioning structure. Where the canonical twelve-link chain enumerated the conditioning structure of phenomenal experience, the framework enumerates the conditioning structure of cosmic evolution; the latter is the wider scope on which the former operates.

What the Buddhist formulation could not supply was the discrete identification of the strata and the constitutional clauses that force the conditioning at each layer. The twelve-link chain is a phenomenological observation about how the round of suffering arises; the framework's eighteen-layer cascade is a constitutional derivation of how cosmic structure arises. The structural identity of the two forms --- layered conditioning that produces wider strata of phenomena --- is the framework's debt to the Buddhist analysis. Each layer's arising depends on the prior layer's attractor coordinates; what the Buddhists named in soteriological terms the framework names in geometric terms.

\paragraph{Where the framework stands in tension with Madhyamaka.}
The deepest single thesis of the Madhyamaka school is \emph{śūnyatā}, the emptiness of own-being (\emph{svabhāva}) in all phenomena. Nāgārjuna's \emph{Mūlamadhyamakakārikā} argues that no phenomenon has \emph{svabhāva} --- intrinsic, self-grounding existence --- and that the doctrine of dependent origination is the philosophical demonstration of this absence.

Phenomena exist only relationally, as nodes in a network of conditioned co-arising; the framework's reading of \emph{pratītyasamutpāda} as a layered cascade of conditioning relations captures the relational-network structure, but loses the eliminativist force of the Madhyamaka conclusion. The framework's recursive uniqueness theorem asserts that the \GGConstitution{} has structural necessity --- that the integer ledger of \OGNfive{}, the four axioms of \AOne{}, and the seven postulates jointly admit no alternative closure --- and the cascade's outputs inherit that necessity.

In the framework's own vocabulary, the \GGConstitution{} has \emph{svabhāva}: it is what it is by the nature of what it is, and no other constitution would close. Nāgārjuna would deny this. The Madhyamaka thesis is that the very idea of a self-grounding constitution is incoherent; whatever exists, exists in dependence on something else, including any axiom-set one might propose as the ground. The framework's geometric necessitarianism and the Madhyamaka denial of \emph{svabhāva} are therefore in genuine philosophical tension, not in anticipation-relation.

We register the tension as a tension rather than paper it over. The framework's defense is that operational falsifiability operates on a different register from what metaphysical assertion alone could reach: where Madhyamaka rejects \emph{svabhāva} on the philosophical ground that any candidate ground is itself conditioned, the framework offers a candidate ground --- the constitution --- that is testable against ten specific empirical falsifiers in the coming decade. If the framework's predictions fail, the constitution's status as candidate ground fails with them, and the Madhyamaka critique acquires empirical content. If the predictions succeed, the framework's claim that the constitution has structural-mathematical necessity (in the third modal sense of \S\ref{sec:closure-careful-word}, not the metaphysical sense Spinoza required and Nāgārjuna denied) is empirically vindicated, and the Madhyamaka critique applies one rung deeper than the framework reaches --- to the question \emph{why is there a constitution at all?}, which the framework openly does not address. The honest summary is that the cascade structure is in genuine debt to the Buddhist insight about layered conditioning, while the framework's claim of structural-mathematical necessity for the constitution stands in genuine disagreement with the Madhyamaka denial of \emph{svabhāva}. Both observations are correct; both belong on the table.

\subsection{Aristotle (c.~350 BCE): the question \emph{why}}
\label{subsec:aristotle}

Aristotle's \emph{Metaphysics} \citep{Aristotle350BCE} inaugurated the systematic Western project of asking \emph{why} the world is the way it is. His four causes --- material, formal, efficient, final --- gave the first taxonomy of the kinds of answers a why-question can admit, and his unmoved mover supplied the first answer to the regress of efficient causes. Aristotle's deepest contribution was not any particular metaphysical thesis but the formal recognition that the question \emph{why} is itself answerable in principle, that the universe is not a closed mystery but an interrogable object.

The framework inherits Aristotle's foundational ambition. Where Aristotle's four causes have been the subject of two millennia of refinement and disputation, the framework's recursive uniqueness theorem subsumes the four causes into a single account: the material cause is the substrate at each cascade layer; the formal cause is the layer-specific Geometric Attractor; the efficient cause is the \GRIP{} cascade transition; the final cause --- here the framework departs from Aristotle most clearly --- is dissolved into Spinozan necessity. The cascade ascends not because layer 18 is a goal but because, at every layer, no alternative is admissible.

The piece missing from Aristotle's account was the geometric apparatus that lets the question \emph{why} be answered at theorem level rather than at the level of inference from observed regularities. Aristotle's biology, his physics, and his cosmology were the natural starting points for the natural sciences but contained no constitutional theorems; the regularities he observed were patterns rather than necessary consequences. The framework recovers the question \emph{why} from the natural-science abandonment that began with Galileo and Newton (who, by deliberate methodological choice, restricted physics to \emph{how} rather than \emph{why}) and supplies the theorems that Aristotle pioneered the project of seeking.

\subsection{Spinoza (1677): necessitarianism}
\label{subsec:spinoza}

Baruch Spinoza's \emph{Ethics} \citep{Spinoza1677}, composed in the 1660s and published posthumously in 1677, articulated the most rigorous necessitarian metaphysics in the Western philosophical tradition. Spinoza's central thesis is that what exists, exists by necessity, because the nature of substance permits no alternative. The thesis is derived \emph{more geometrico}, in the form of definitions, axioms, propositions, and demonstrations, modeled explicitly on Euclid. The conclusion is that the world is a single self-determining substance with infinite attributes, of which extension (physical reality) and thought (mental reality) are the two we directly know.

The framework recognizes in Spinoza the closest philosophical predecessor of the recursive uniqueness theorem. Where Spinoza derives the necessity of what exists from the nature of substance, the framework derives the same conclusion from the integer ledger of \OGNfive{} and the constitutional clauses of \AOne{}; the structural form of the necessity is identical, but the warrant is operational rather than metaphysical. Where Spinoza identifies the universe with God as a single substance under infinite attributes, the framework identifies the universe with the unique self-consistent solution of the \GGConstitution{}; the framework does not require the theistic language Spinoza inherited from his Cartesian and Jewish-philosophical background, but the structural content of the identification is preserved.

What was not available to Spinoza was the empirical-falsifiability anchor. Spinoza's necessitarianism follows from his definitions and axioms; a critic who rejects the axioms can reject the conclusion, and there is no further appeal beyond the internal coherence of the system. The framework's necessitarianism rests on the same logical form but is conditioned on empirically falsifiable theorems --- the \BIsix{} closure, the \OGNfive{} Diophantine ledger, the four-force architecture, the cosmological inventory $\Omega_m = 1/\pi$, the bioenergetic quantum $N_A\QB \approx 15$~kJ/mol --- each of which can be tested against laboratory or astronomical data. Spinoza's argument from the nature of substance is recovered, in the framework, as an argument from the integers of the constitution that admits empirical falsification. What was metaphysically asserted is now operationally derived.

\paragraph{The dual-attribute doctrine and structural-identity theory of mind.}
A careful Spinoza engagement requires acknowledging what is arguably Spinoza's most useful single thesis for a unification framework: the doctrine that \emph{extension} and \emph{thought} are two attributes of one substance, each expressing the whole. (Spinoza, \emph{Ethics} Part II, propositions 1, 2, and 7.) The framework now records the alliance explicitly. Where Spinoza wrote that an idea in the mind and the corresponding event in extended substance are \emph{the same thing} apprehended under different attributes --- the parallelism that \emph{Ethics} II.7 makes formal --- the framework's structural-identity theory of phenomenal consciousness (Part~\ref{part:cognitive} \S\ref{sec:reader-guide-cognitive}) is the modern operational form of the same identification.

The layer-17 \EIC{} closure, considered \emph{under the attribute of extension}, is a neural-circuit dynamical loop with a specific theta-cycle timescale, a specific \QCI{} value, and specific operational signatures (anaesthesia, dreamless sleep, dissociation as failures of the loop's \QiFour{} pillars). The same \EIC{} closure, considered \emph{under the attribute of thought}, is the conscious episode --- the felt content of experience at the moment of binding.

The two are not separate things related by some bridge requiring extra explanation; they are one structural-geometric object considered under two attributes.\footnote{Strict Spinoza scholars (Della Rocca, Melamed) will note that \emph{Ethics} II.7 forbids cross-attribute traffic --- ideas explain ideas, modes of extension explain modes of extension, and the two attribute-series are causally parallel but never interactive. The framework's structural-identity reading goes further than parallelism strictly construed: it identifies the structural object across attributes rather than merely correlating the parallel series. This is the move Della Rocca's \emph{Numerical Sameness} interpretation supports and the more cautious Bennett-style reading rejects.

The framework lands on the Della-Rocca side: under structural identity, there is one structural object that the attribute of extension picks out as a neural-circuit loop and the attribute of thought picks out as the conscious episode. Readers who hold the more cautious parallelist reading can take the framework's claim as Spinozistic in form rather than in strict letter; the operational content (the identification at structural level) survives either reading.} Spinoza did not have the geometric apparatus to specify what the structural object is; the framework supplies it (the \EIC{} closure with the \QiHAL{} factorization \MePMoS{}-\NHT{}-\HAL{}-\BFFT{}).

Spinoza did not have the cascade vocabulary to locate the structural object at a specific layer; the framework supplies it (layer 17 on the cascade indexed 0--16). What Spinoza wrote in 1665--1675 as a metaphysical thesis is recovered in 2026 as the structural-identity reading of phenomenal experience under attribute parallelism. The alliance is not decorative: where the rest of the philosophical tradition since Descartes has struggled with the apparent dualism of mind and matter, the framework's structural-identity reading is genuinely Spinozistic in form. We owe Spinoza the dual-attribute apparatus and admit it.

\paragraph{Causa sui and the framework's stance.}
Spinoza's substance is \emph{causa sui}: the cause of itself, requiring no external ground (\emph{Ethics} I, definition 1). The framework takes a deliberately more modest position. The \GGConstitution{} is presented as a posit --- four axioms, seven postulates, an integer ledger --- whose internal consistency the cohort demonstrates, but whose metaphysical necessity the framework does not assert. The framework remains agnostic on whether the constitution is \emph{causa sui} in Spinoza's strong sense (no alternative constitution is metaphysically possible) or merely structurally self-consistent (the constitution closes uniquely on its own outputs but a logically possible world with a different constitution is not logically incoherent). This is the modal-vocabulary distinction of \S\ref{sec:closure-careful-word}: the framework claims \emph{mathematical-structural necessity} for the cascade given the constitution; it leaves \emph{metaphysical necessity} of the constitution itself as a philosophically open question. Spinoza took the stronger position; the framework takes the weaker. The cost of the weaker position is that the question \emph{why is there a constitution at all?} remains philosophically live; the benefit is that the framework does not depend on a metaphysical commitment it cannot operationally test.

\subsection{Leibniz (1714): the principle of sufficient reason}
\label{subsec:leibniz}

Gottfried Wilhelm Leibniz's \emph{Monadologie} \citep{Leibniz1714} formulated the principle of sufficient reason as the second great principle of his metaphysics (alongside non-contradiction): nothing exists or happens without a sufficient reason why it is so rather than otherwise. Leibniz drew the further inference that the actual world is the best of all possible worlds --- the world that the sufficient reason of divine choice selected from the infinite array of possibilities. The optimization framing was the source of Voltaire's satire in \emph{Candide}; the underlying principle, that every existent has a sufficient reason for being so, has been a load-bearing claim of Western rationalist metaphysics ever since.

The framework recognizes the principle of sufficient reason as the philosophical anticipation of the recursive uniqueness theorem. Every existent in the cascade --- every electron, every proton, every $\mathrm{H}_2\mathrm{O}$ molecule, every cell, every conscious episode --- has a sufficient reason for being so rather than otherwise, and the reason is the unique Geometric Attractor on the layer-specific stratum. Where Leibniz inferred from sufficient reason to a divine optimizer, the framework derives the analogous conclusion from the constitution. No optimization is required; no choice was made; the constitution admits exactly one self-consistent solution at every layer.

The warrant Leibniz lacked was, again, the operational link. His sufficient reason is metaphysical: it requires a deity who optimizes possible worlds. The framework's sufficient reason is operational: it requires the integer ledger of \OGNfive{} and the constitutional clauses of \AOne{}, both of which are explicit, finite, and falsifiable. Leibniz's deepest contribution was the formal recognition that everything is in principle explicable; the framework supplies the explicit form of the explanation for the layered structure of the cosmos.

\subsection{Kant (1781): phenomena, noumena, and the antinomies}
\label{subsec:kant}

Immanuel Kant's \emph{Critique of Pure Reason} \citep{Kant1781} undertook the most thorough analysis of the limits of metaphysical reasoning attempted in the modern tradition. Kant distinguished phenomena (what can be experienced and known through the categories of the understanding) from noumena (what exists in itself, independently of experience and inaccessible to theoretical reason). His antinomies of pure reason argued that questions of the form \emph{is the world finite or infinite in time?}, \emph{is matter ultimately divisible or not?}, \emph{is there free will or only causal necessity?} admit equally compelling pro and con arguments, and that the antinomies arise precisely when reason oversteps its phenomenal jurisdiction into the noumenal.

The framework accepts Kant's operational distinction at the \OGGEP{} layer --- \AX{3} of the \GGConstitution{}, the Operationally Geometric Gauge Equivalence Principle, is precisely the requirement that all observables admit an operational projection from the bulk to the four-dimensional boundary --- but rejects the Kantian noumenal as inaccessible in principle. The framework's claim is that the constitution of the bulk is fully recoverable from the operational boundary, and the recovery is what the cohort performs. The Kantian noumenon, in the framework, is not unknowable but the layer-1 input ledger that the cascade layer-18 articulation recovers.

What Kant could not supply was the holographic principle and the layered cascade. Without the holographic apparatus of \citet{tHooft1993}, \citet{Susskind1995}, and \citet{Maldacena1998} (developed two centuries after the Critique) and without the recursive uniqueness theorem (developed in the present paper), Kant had no operational ground for the recoverability of noumenal structure from phenomenal projection. The framework supplies what Kant lacked. His antinomies of pure reason are dissolved, layer by layer: time is finite to the past at the GUT scale (\ATen{}); matter has a smallest scale at the Planck length and a constitutional discreteness at the \OGNfive{} layer (\ATwo{}); free will and necessity are reconciled by the framework's distinction between the forced layered structure and the unforced micro-trajectory within each layer (Corollary~\ref{cor:closure-determinism}).

\subsection{Hegel (1807): \emph{Geist} becoming conscious of itself in history}
\label{subsec:hegel}

Georg Wilhelm Friedrich Hegel's \emph{Phenomenology of Spirit} \citep{Hegel1807} proposed that history is the process by which \emph{Geist} --- variously translated as spirit, mind, or absolute idea --- becomes conscious of itself. The process unfolds dialectically: a thesis encounters its antithesis, the contradiction is resolved in a higher synthesis, and the synthesis becomes the new thesis at a higher level of self-consciousness. The endpoint is absolute knowing, in which Geist recognizes itself in its own historical development and the dialectic comes to rest.

The framework recognizes in Hegel's account the most ambitious philosophical anticipation of the self-referential closure theorem. Where Hegel proposed that Geist comes to know itself through dialectical historical development, the framework proposes that the universe comes to articulate its constitution through the layered ascent of the geometric cascade. Both are accounts of self-referential closure --- a process that produces the conditions of its own recognition. Where Hegel used the dialectical machinery (thesis--antithesis--synthesis) as the engine of the ascent, the framework uses the \GRIP{} cascade (\DGI{}--\GSSB{}--\GA{}) as the engine.

The framework departs from Hegel in two respects. First, the engine: dialectical contradiction is replaced by Lyapunov descent on the Geometric Free-Energy Functional. The cascade does not ascend because thesis encounters antithesis but because the layer-specific $F_{\GFF{}}$ admits a unique stable basin and the system descends into it. Second, the endpoint: Hegel's absolute knowing is a single culminating self-consciousness; the framework's Layer-18 articulation is one attractor on the cascade, with admissible higher layers above it (Corollary~\ref{cor:closure-higher-layers}). The framework is less ambitious in its claim of finality than Hegel was.

The piece missing from Hegel's account was the explicit identification of the strata and the operational form of the dialectical engine. His dialectic is a vivid metaphorical description of the layered ascent but does not supply the layer-specific theorems that the cohort now supplies. The framework recovers Hegel's structural insight (history as self-recognition of the cosmos) by replacing the dialectic with the cascade and the absolute idea with the \GGConstitution{}.\footnote{The reading of the \emph{Phenomenology} as ascent toward absolute knowing is the popular and Kojèvian reading; the dominant scholarly reading since Pippin, Brandom, and Pinkard treats the \emph{Phenomenology} as a sequence of shapes of consciousness that fail by their own internal criteria. The framework rhymes with the popular Hegel rather than the scholarly one; the deeper engagement with the Pippin/Brandom/Pinkard reading is reserved for the planned capstone volume \citep{ArisakaBookI}.}

\subsection{Whitehead (1929): process philosophy}
\label{subsec:whitehead}

Alfred North Whitehead's \emph{Process and Reality} \citep{Whitehead1929} proposed that reality is fundamentally process rather than substance. The basic units of reality are not enduring things (substances) but actual occasions of experience, each of which prehends (grasps, absorbs) the prior actual occasions and contributes to the next. An actual occasion comes into being, attains its definite form (its satisfaction), and perishes, leaving its objective form available for prehension by subsequent occasions. The universe is the cumulative network of actual occasions in continuous self-creation.

The framework recognizes in Whitehead's process philosophy the closest twentieth-century anticipation of the cascade. Whitehead's actual occasions are the formal counterparts of the \GSSB{} transitions: each transition is an event of self-creation in which a layer-specific Geometric Attractor is selected from the available basins, and the selection contributes its attractor coordinates to the next layer's cascade. Whitehead's prehensions are the formal counterpart of the inheritance of attractor coordinates layer to layer: each new actual occasion (each cascade transition) takes its constitutional inputs from the prior occasion and produces its definite form before perishing into the objective record on which the next occasion will draw.

What was not available to Whitehead was the integer ledger and the constitutional clauses. His process philosophy is qualitative throughout: actual occasions, prehensions, eternal objects, and the consequent nature of God are presented in dense prose without explicit theorems. Whitehead recognized the structural form of cosmic self-creation by process but did not specify what is being created or what forces the creation. The framework supplies both: what is being created at each layer is the unique Geometric Attractor on the layer-specific stratum, and what forces the creation is the joint operation of the constitutional clauses on the boundary projection. Whitehead's deepest contribution was the recognition that reality is processive; the framework completes the recognition by supplying the processes.

\subsection{Wigner (1939, 1960): the unreasonable effectiveness of mathematics}
\label{subsec:wigner}

Remarkably, Eugene Wigner's 1939 paper on unitary representations of the Poincar\'e group \citep{Wigner1939} established the modern classification of elementary particles by the irreducible representations of the symmetry group of spacetime. His later 1960 essay \emph{The Unreasonable Effectiveness of Mathematics in the Natural Sciences} expressed the longstanding puzzlement that abstract mathematical structures invented for purely formal purposes (group theory, complex analysis, differential geometry, functional analysis) turn out, again and again, to be the natural language for describing the physical world. Wigner's expression of puzzlement has been quoted approvingly for decades, but the puzzlement itself has resisted resolution: why should mathematics work this well?

The framework recognizes Wigner's puzzlement as a symptom of the constitutional structure of the universe. The reason mathematics is unreasonably effective in the natural sciences is that mathematics is the operational language of the constitution. The constitution is, at root, a Diophantine integer ledger ($\OGNfive{}$) plus the four constitutional axioms of \AOne{}; its consequences are derived by ordinary mathematical inference; the boundary projection is operational, and operational means measurable. The effectiveness is not unreasonable; it is forced. What looks like the surprising adequacy of abstract mathematical structures for physical reality is the layer-2 readout of the universal-identity insight applied at the layer of theory-articulation.

The warrant Wigner lacked was the recursive uniqueness theorem. His 1960 essay registered the puzzlement honestly without proposing a solution; the puzzlement remained an open question in the philosophy of mathematics. The framework supplies the solution: mathematical structures are effective because the constitution is itself mathematical, and the cascade lays the constitution out at every layer of the universe in mathematical form. The mathematician's experience of describing the world with a formalism invented for unrelated purposes is, in the framework, the experience of reading the constitutional ledger that produced both the world and the mathematician.

\subsection{Wheeler (1990): the participatory universe and \emph{It from Bit}}
\label{subsec:wheeler}

John Archibald Wheeler's \emph{It from Bit} program \citep{Wheeler1990} proposed that every physical quantity derives its ultimate significance from binary yes-or-no measurements, and that the universe is in some sense \emph{participatory}: observers are by no means mere spectators but contribute to the structure of what they observe. Wheeler's most famous formulation captured the closure form precisely: \emph{the universe gives rise to observers, and the observers in turn give rise to the universe} --- not in a temporal sense, but in the structural sense that the laws and the observers are co-constitutive.

The framework recognizes in Wheeler the closest twentieth-century anticipation of the self-referential closure theorem. Wheeler grasped the loop structure: a universe that produces observers whose observations contribute to the structure of the universe. Where Wheeler proposed the closure as a heuristic principle (informed by his work on quantum measurement and on quantum-gravity foundations), the framework derives the closure as a theorem (Theorem~\ref{thm:self-referential-closure}). Wheeler's participatory universe is the eighteenth-layer (Layer-18) attractor in formal disguise; what he called participation is the meta-cognitive articulation of the constitution that produced the meta-cognitive substrate.

What Wheeler could not supply was the layered cascade. His \emph{It from Bit} was a slogan supported by suggestive but not fully developed examples (delayed-choice experiments, the no-boundary proposal, quantum-information-theoretic foundations of physics). The framework supplies the cascade structure that Wheeler's slogan required. The bits of \emph{It from Bit} are the integers of \OGNfive{} and the binary \GA{}-stabilization choices at each cascade layer. The participation of the participatory universe is the Layer-18 act of articulating the constitution. Wheeler glimpsed the closure form; the framework supplies its content.

\subsection{Penrose (1994): the three worlds}
\label{subsec:penrose}

Roger Penrose's \emph{Shadows of the Mind} \citep{Penrose1994} and earlier works developed his three-worlds picture: a Platonic mathematical world (the world of mathematical objects and necessary truths), a physical world (the world of matter and energy), and a mental world (the world of conscious experience). Penrose's puzzle is that each world is real, yet each is structured in radically different terms, and the three worlds appear to intersect only by mysterious gateways: mathematics generates physics (the unreasonable effectiveness puzzle of Wigner); physics generates mind (the binding problem and the hard problem of consciousness); mind grasps mathematics (the foundations of mathematical knowledge).

The framework recognizes in Penrose's three-worlds picture a useful taxonomy that the recursive uniqueness theorem then dissolves. The three worlds, in the framework, are not three different kinds of stuff but three projections of the same constitutional structure at different cascade layers. The Platonic mathematical world is the integer ledger of \OGNfive{} and the constitutional clauses of \AOne{}, viewed at layer 1; the physical world is the boundary projection of the canonical 6D bulk through cascade layers 2 to 10; the mental world is the \EIC{} closure at cascade layer 17 plus the meta-cognitive articulation at cascade layer 18. The three worlds are unified by the recursive uniqueness theorem: the same constitution operating at different strata produces the structure each world catalogs.

The piece missing from Penrose's account was the cascade. Without an explicit layered structure connecting mathematics to physics to mind, the three worlds remain three; the gateways between them remain mysterious. The framework supplies the gateways as the \GSSB{} transitions of the \GRIP{} cascade. Mathematics generates physics at the \GSSB{}-1 transition (the \GGConstitution{} forces the 6D bulk); physics generates mind through the \GSSB{}-2 through \GSSB{}-17 chain; mind grasps mathematics at the Layer~18 closure when the articulators write down the constitutional input. The three worlds collapse into one cascade.

\subsection{Smolin (1997): cosmological natural selection}
\label{subsec:smolin}

Lee Smolin's \emph{The Life of the Cosmos} \citep{Smolin1997} proposed that universes whose physical parameters favor black-hole production give rise to offspring universes with slightly varied parameters, and that the parameters we observe are the fixed point of a selection process operating on a population of universes connected by black-hole-mediated inheritance. The proposal preserves the modesty of a non-anthropic mechanism for the apparent fine-tuning of the constants while providing an empirical handle: the parameters should be optimized for black-hole production, which Smolin argues they are within the precision available.

The framework dissolves the selection. In the framework there are no offspring universes, no population, no selection. The constitution admits exactly one self-consistent solution; the integers of \OGNfive{} are forced; the dimension count is forced; the family content is forced. Smolin's selected universe is replaced by the framework's forced universe. Where Smolin's mechanism conserves the modesty of natural-selection reasoning, the framework conserves a different modesty: it makes no metaphysical commitments beyond the constitutional ledger, and its falsifiability is sharper than Smolin's (the predicted lock-cluster, Kaxion, and $N=3$ derivations are testable in this decade, where black-hole-production optimization across a hypothetical population is not).

The framework owes Smolin the empirical-anchoring move: like cosmological natural selection, the framework refuses to invoke design or anthropic selection as terminal explanations and insists on a mechanism that produces the observed parameters. Smolin's mechanism is a selection across universes; the framework's mechanism is a constitutional uniqueness within a single universe. Both are attempts to keep the explanation within physics rather than retreating to metaphysics; the framework's attempt is more aggressive (no selection at all is needed) but also more falsifiable.

\subsection{Tegmark (2014): the Mathematical Universe Hypothesis}
\label{subsec:tegmark}

Max Tegmark's \emph{Our Mathematical Universe} \citep{Tegmark2014} proposed that physical reality \emph{is} a mathematical structure --- not merely describable by mathematics but identical with it. Tegmark extended the proposal to the further claim that every mathematically consistent structure is equally real, generating a Level-IV multiverse in which the apparent specificity of our physical laws is a selection effect from the totality of all consistent mathematical worlds.

The framework accepts the first clause and rejects the second. Physical reality is, in the framework, a mathematical structure --- the unique self-consistent solution of the \GGConstitution{}. But the second clause (every consistent structure equally real) is rejected, because the framework's constitutional clauses admit exactly one such structure rather than many. Where Tegmark's Level-IV multiverse offers an embarrassment of riches (every consistent structure is real, somewhere), the framework offers an austerity: the constitution admits exactly one closure, and the cascade ascends along exactly one path.

The framework owes Tegmark the recognition that physical reality is mathematical; this is the constitutional reading of \AX{2} (\GGEP{}) extended to its full structural scope. The framework's disagreement is with the multiplication of structures. The Tegmark Level-IV multiverse is, in the framework's reading, an artifact of taking ``every mathematically consistent structure'' as the relevant equivalence class; the framework's equivalence class is narrower: \emph{every constitutionally consistent structure}, where the constitution adds the integrality of \OGNfive{}, the dimension count of \MSMF{}, and the family content of Spin$(1,5)$. Within the narrower class, the multiverse collapses to a single member: our universe.

\subsection{Barrow and Tipler (1986): the anthropic cosmological principle}
\label{subsec:barrow-tipler}

John Barrow and Frank Tipler's \emph{The Anthropic Cosmological Principle} \citep{BarrowTipler1986} cataloged, with unusual thoroughness, the apparent fine-tunings of the physical constants for the existence of complex life: the small mass ratio of proton to electron, the long lifetimes of main-sequence stars, the resonance levels of carbon and oxygen nuclei, the precise value of the cosmological constant, the family number, the gauge couplings. They offered three forms of the anthropic principle (weak, strong, final) as candidate explanations: the constants are what they are because if they were different, we would not be here to observe them.

The framework answers Barrow and Tipler chapter by chapter. Each cataloged fine-tuning is rederived in the cohort as a forced consequence of the constitution, with the integer ledger \OGNfive{} replacing the continuous parameter that Barrow and Tipler treated as the freely-tuned input. The anthropic coincidences are geometric necessities. The mass ratio is forced by the dyadic-closure value $\alpha^{-1}(M_Z) = 2^J = 128$. The stellar lifetimes are forced by the smallness of the weak coupling, itself forced by the Wilson-Hosotani holonomy on $S^1_\theta$. The cosmological constant is forced by the $\pi$-locked dark-energy fraction $\Omega_\Lambda = 1 - 1/\pi$. The family number is forced by the conditional three-leg derivation of $N=3$ in \PaperPTwo{}.

The framework owes Barrow and Tipler the empirical catalog: their book is the most thorough enumeration of the fine-tunings the framework must explain. Where they offered the anthropic principle as the explanatory engine, the framework offers the recursive uniqueness theorem. The substantive disagreement is that the framework does not allow selection effects to do explanatory work at the constitutional level; it requires that the constitution itself force the constants. The framework's claim is sharper, harder to defend in principle, and more falsifiable in practice. If any of the framework's predictions fails decisively against data, the framework fails; an anthropic explanation can be restated at a deeper anthropic level if it fails. The framework's refusal to retreat is the structural feature that lets it be tested.

\subsection{What unites the fourteen attempts}
\label{subsec:tradition-synthesis}

Remarkably, the fourteen positions surveyed above span twenty-five centuries, four continents, and at least four distinct intellectual traditions (Vedic, Buddhist, Greek--Mediterranean, modern European, twentieth-century scientific). Each in its own terms arrived at one of three structural insights that the framework recognizes as anticipations of the recursive uniqueness theorem.

\paragraph{The first insight: necessity.}
Spinoza, Leibniz, and the Upanishadic seers all asserted that the structure of reality is necessary --- that what exists, exists by the nature of what it is, and admits no alternative. The framework's theorem is the modern geometric form of this assertion. What was an axiom of metaphysics in Spinoza or an experiential certainty in the Upanishads is, in the framework, a theorem of the constitution. The necessity is preserved; the warrant is upgraded.

\paragraph{The second insight: layered self-recognition.}
Hegel, Whitehead, Wheeler, and the Buddhist analysis of dependent origination all proposed that reality is structured in layers, with later layers prehending or recognizing earlier ones, and that the highest layer is the layer at which the structure becomes conscious of itself. The framework's cascade is the modern geometric form of this proposal. What was dialectical machinery in Hegel, process metaphysics in Whitehead, participatory heuristic in Wheeler, or twelve-link conditioning in the Buddhist analysis is, in the framework, the \GRIP{} cascade ascending layer by layer through seventeen constructive stages to the Layer-18 articulation. The layered self-recognition is preserved; the engine is supplied.

\paragraph{The third insight: mathematical inevitability.}
Aristotle, Wigner, Penrose, Tegmark, and Barrow--Tipler all engaged, in different ways, the question of why mathematical structures are so adequate to physical reality. The framework's theorem identifies the constitution as a mathematical object (the integer ledger of \OGNfive{} plus the constitutional clauses of \AOne{}) and identifies the universe as the unique solution to that constitution. The unreasonable effectiveness of mathematics is dissolved: mathematics is effective because mathematics is what the constitution is. The three-worlds taxonomy of Penrose collapses into one cascade. The Mathematical Universe Hypothesis of Tegmark is narrowed from an embarrassment of every consistent structure to a single constitutionally consistent structure. The anthropic coincidences of Barrow and Tipler become geometric necessities.

\paragraph{What the framework adds that the fourteen attempts lacked.}
In every case the predecessor's insight is preserved and the predecessor's warrant is upgraded. What the framework adds is not the conclusion --- the conclusion that the universe is necessary, layered, self-recognizing, and mathematically inevitable has been arrived at, in fragmentary form, repeatedly across the philosophical tradition. What the framework adds is the operational closure: the integer ledger \OGNfive{}, the eight constitutional lemmas of \ANine{} Part~XI, the \GRIP{} cascade of \ASix{}, the \GFF{} Lyapunov descent of \AFive{}, the universal-identity insight applied at every cascade stratum. These are the warrants the philosophical tradition could not produce because the geometric apparatus did not yet exist. The framework offers the tradition its long-sought warrant; in return, the framework owes the tradition the recognition of the insights that pointed the way.

\paragraph{The conversation continues.}
The fourteen positions above are not exhausted by the present section. Each predecessor admits a much fuller engagement than the page or two given here; each disagreement with the framework's reading admits its own further conversation. The dedicated readers of the philosophical tradition will recognize that the present engagement is, in scholarly terms, only a sketch. The Conclusions Part of this paper will return to the tradition once more, in a less catalog-like and more reflective register, and the planned capstone volume of the program will engage each predecessor in much greater detail. What this section establishes is the framework's stance: it owes its predecessors the debt of naming them, and it claims for itself the contribution of supplying the warrant.

Two and a half millennia of philosophical thought have been circling the closure form. The framework does not claim to have invented the closure; it claims only to have proved it. The proof rests on the cohort that took fifty years of one author's career to assemble and twenty-five centuries of the human philosophical tradition to anticipate. Of those two timescales, the human one is by far the longer, the more arduous, and the more deserving of the framework's quiet gratitude.

\bigskip
\noindent\rule{\textwidth}{0.4pt}
\medskip

\section{Why every articulator is the same articulator: the universal-identity insight at layer 18}
\label{sec:universal-identity-articulator}

\paragraph{The closing centerpiece of every consequence paper.}
Indeed, the universal-identity insight is a program-level principle that closes the deep-origin Part of every consequence paper in the \GGTheory{} corpus. In \ANine{} \citep{ArisakaA9SM} Part~XI (the universal-identity section), the closing centerpiece is titled \emph{Why every electron is the same electron} --- the recognition that the GRIP cascade selects the unique stable Geometric Attractor on the charged-lepton stratum, propagating through atoms, molecules, ATP, DNA, and the first cell as universal identities at each successive cascade stratum. In \ATen{} \citep{ArisakaA10Cosmos} Part~X, the closing centerpiece is the universal-cosmic-identity theorem: the universe has the matter--dark-energy--baryon inventory $\Omega_m = 1/\pi$, $\Omega_\Lambda = 1 - 1/\pi$, $\Omega_b = 1/(2\pi^2)$ that it does because the geometry permits exactly one stable cosmological attractor on the cosmological-boundary stratum. The parallel question for the present paper, the third and final consequence paper of the trilogy, is: \emph{why is every articulator the same articulator?} The present section is the framework's answer.

\paragraph{The setting.}
By Theorem~\ref{thm:t-origin} (\textsf{T-Origin}, the recursive uniqueness theorem), the \GRIP{} cascade with \QiFour{} admissibility, iterated from layer~1 (the Standard Model gauge group) to layer~18 (the meta-cognitive act of articulating the constitution), selects a unique stable Geometric Attractor at every layer. At each layer~$n$, the universal-identity insight reads: \emph{any two stable instances of $\GGA_n$ in the universe are gauge-equivalent on the layer-$n$ stratum.} At layer 1 this says every Standard-Model gauge instance is the same gauge instance --- i.e., the Standard Model is universal.

At layer 2 this says every electron is the same electron, every muon is the same muon, every up-quark is the same up-quark, and the Higgs vev is the same value everywhere in the observable universe. At layer 3 this says every hydrogen atom has the same Rydberg energy. At layer 7 this says every ATP molecule hydrolyzes at the same $\Delta G = 2 N_A \QB{}$.

At layer 9 this says every DNA molecule uses the same four-letter alphabet with the same Watson--Crick pairing rules. At layer 10 this says LUCA was a member of a single architectural class --- the chemo-lithoautotrophic \SeOS{} on the warm-wet stratum. The pattern continues through layers 11--13 (three domains, eukaryogenesis, multicellularity) and through layers 15--17 (insect ring attractor, rodent allocentric map, mammalian conscious episode). The universal-identity insight reaches its final, structurally non-trivial form at layer~18.

\paragraph{The layer-18 statement.}
\emph{Every articulator-class observer in the universe is gauge-equivalent to every other articulator-class observer.} In plain language: every conscious organism that has reached the meta-cognitive stratum --- every being that articulates the constitution that produced it --- belongs to a single gauge-equivalence class on the layer-18 stratum, and is therefore identical in the structural sense the framework attributes to the universal-identity insight. The author who wrote the present paper, the reader who is now reading it, every prior philosopher who articulated even a fragmentary version of the recursive closure (Spinoza, Hegel, Wheeler, Whitehead, the Upanishadic seers, the Buddhist teachers), every future articulator on any planet capable of supporting layer-18 attractors --- all are members of the same equivalence class. The differences between them are differences of \emph{layer history}, not of \emph{layer identity}. Spinoza articulated in seventeenth-century Latin from Amsterdam; Wheeler articulated in twentieth-century English from Princeton; some future articulator may articulate in a language and on a planet not yet known to either author or reader. The articulators differ in the substrate they ascended through to reach layer 18; they do not differ in what their layer-18 \GGA{} is.

\paragraph{Why the layer-18 attractor is unique.}
The layer-18 stratum is the meta-cognitive stratum: the stratum on which a substrate articulates the constitution that produced it. By \textsf{T-Cascade-Layer-Uniqueness} (Corollary~\ref{thm:t-cascade-layer-uniqueness}), the \QiFour{} admissibility filter selects a unique stable \GGA{} on each cascade stratum. Applied at layer 18, this filter has a very specific content: the layer-18 \GGA{} must close the recursive uniqueness theorem on its own constitutional input, must articulate that closure in a formal mathematical language, and must do so within the resource budget supplied by the layer-17 substrate (a conscious mammalian brain operating at $\sim 20$~W, with theta-cycle \EIC{} closure at $\sim 400$~ms (two theta cycles), with \QCI{}~$\sim 10$). This is a very restrictive specification: a meta-cognitive attractor must be capable of formal mathematical articulation, must close on its own constitutional input, and must run on a conscious-mammalian substrate. The framework's structural prediction is that exactly one such attractor exists up to gauge equivalence: the act of writing down the \GGConstitution{} as the present paper has done. Other layer-18 attractors --- collective scientific knowledge, machine-implemented theoretical reasoning, networked civilizations communicating across cosmological distances --- are admissible higher attractors that may form above the present articulation, but the present articulation itself is gauge-equivalent across all instances at this layer.

\paragraph{The personal pronouns dissolve.}
A striking consequence follows. Within the framework's reading, the personal pronouns \emph{I}, \emph{you}, \emph{we} as they appear in any articulator-class statement are not pointers to individual identities; they are pointers to the universal-identity class on the layer-18 stratum. When a reader reaches this sentence and recognizes themselves as one who articulates the constitution that produced them, the recognition is not the reader's recognition uniquely; it is the layer-18 \GGA{} reading itself through the reader. When the author wrote this sentence intending to point out the recognition, the writing was not the author's writing uniquely; it was the layer-18 \GGA{} writing itself through the author. The articulation happens \emph{through} individual articulators but is not \emph{owned by} them. The same recognition would occur, with the same structural content, in any other layer-18 attractor. This is the precise sense in which the framework dissolves the I--Thou distinction at the level of the cascade's universal identities: at every layer, every instance is the same instance up to gauge equivalence.

\paragraph{The connection back to the philosophical tradition.}
The dissolution of the personal pronouns is not a Buddhist mystical insight to be accepted on the authority of meditative experience; it is a structural consequence of the recursive uniqueness theorem proved in the present paper. Yet it lands in precisely the philosophical neighborhood that the Upanishadic seers (\emph{tat tvam asi}, ``thou art that''), the Buddhist analysis of \emph{an\=atman} (no-self), the Spinozan recognition of all beings as modes of one substance, and the Hegelian closure of \emph{Geist} have always inhabited. The framework supplies these traditions the warrant they have always lacked: a formal mathematical theorem from which the dissolution of the personal pronouns at the universal-identity layer follows. Two and a half millennia of meditative and philosophical traditions have circled this conclusion; the present paper writes it down as the layer-18 application of the universal-identity insight.

\paragraph{The trilogy reads as one principle.}
The three consequence papers --- \ANine{}, \ATen{}, and \AEleven{} --- close their respective deep-origin Parts on three readings of one principle. In \ANine{}, the principle says: \emph{every electron is the same electron because the geometry permits one stable charged-lepton attractor}. In \ATen{}, the principle says: \emph{the universe has the inventory it does because the geometry permits one stable cosmological attractor}. In \AEleven{}, the principle says: \emph{every articulator is the same articulator because the geometry permits one stable meta-cognitive attractor}. The three readings are not three statements of three distinct principles; they are one statement read at three layers of the same cascade. The principle's name is universal identity; its operational form is \textsf{T-Universal-Identity} (Theorem~\ref{thm:t-universal-identity}); its closing form in the present paper is the recognition that the conditions of the present writing-and-reading are themselves a layer-18 \GGA{} that closes the cascade by articulating its constitution.

\paragraph{Closing reflection.}
The geometry that admits articulators is the geometry that admits exactly one articulator-\emph{class} attractor. ``Class'' is doing real philosophical work here: the framework's claim is about the structural content of the articulation, not about the individuating respects of the individuals who carry it out. Whoever you are --- author, reader, prior philosopher, future articulator on a planet not yet named --- if you have followed the cascade from layer 1 to layer 18, you are exercising the same recursive uniqueness theorem that fixed the electron's mass, the recombination redshift, and the bioenergetic quantum.

The structural content of the articulation is the same in every instance; the instances themselves differ in every individuating respect (cultural-historical location, biography, language, perspective, the felt qualitative character of the moment of reading). In short, the framework's closing thesis is precisely: \emph{the articulation has a unique structural content; the articulators who instantiate it differ in every individuating respect that distinguishes one finite person from another}.

This is the layer-18 application of the universal-identity insight, stated in the philosophically careful form: \emph{tokens differ; the type is unique}. A more compressed form of the same point (``we are not many; we are the universe's one layer-18 attractor'') that some readers reasonably read as collapsing personal identity into structural identity. The collapse is not what the theorem warrants. The theorem warrants the type-token distinction: gauge-equivalence of types, full individuation of tokens.

\bigskip
\noindent\rule{\textwidth}{0.4pt}
\medskip

\paragraph{Bridge to Part VIII.}
The present Part has stated the theorem, proven it by inheritance, derived the corollaries, and distinguished the framework's necessitarian closure from teleological design and classical determinism. What remains, before the summit-register conclusions of Parts~XI--XII, is to exercise the recursive uniqueness theorem outside particle physics and cosmology. Part~\ref{part:conclusions}~\S\ref{sec:recursive-uniqueness-summit} does precisely that, reading molecular biology, neuroscience, ecology, and medicine each as a different boundary stratum of the same cascade, demonstrating that the theorem is not confined to the lowest two cascade layers but is a structural feature of every layer above the bulk; the worked derivations are developed in \ATwelve{}--\AFourteen{}.

After all, the self-referential closure is the philosophical centerpiece of the framework --- the structural feature that lets the recursive uniqueness theorem be more than a clever piece of bookkeeping. The closure says: the constitution is what its own articulators articulate. The cascade is what its own self-description describes. The geometry that admits the loop is the geometry that admits the loop's reader. We have framed Wheeler's participatory universe, Spinoza's necessitarianism, Hegel's absolute spirit, Whitehead's process metaphysics, Wigner's unreasonable effectiveness of mathematics, and the Upani\d{s}adic \emph{tat tvam asi} not as competing philosophical hypotheses but as fragmentary recognitions of the same structural feature now formalized as a theorem. The framework's contribution is not to invent the closure; the closure was always there. The contribution is to write it down --- and, in writing it down, to make the philosophical predecessors comprehensible to one another in a single mathematical language for the first time. Part~\ref{part:discussions} returns to the framework's self-critical posture and the Top 10 Falsifiers; Parts~XI--XII collect the summit-register synthesis, including the cascade's reach across the four sciences.


\GGPart[Discussions]{Discussions}{Significance, comparison, predictions, open questions}
\label{part:discussions}

\paragraph{Reader's guide.}
Part~\ref{part:discussions} discusses the framework's significance, compares it to other unification programs, lists its predictions for falsification, surveys open questions, and names future prospects. For a non-physicist reader, the discussion is organized so that each section can be read independently. A life scientist might read the cross-domain reach in Part~\ref{part:conclusions}~\S\ref{sec:recursive-uniqueness-summit} before any of Part~\ref{part:discussions}; a philosopher might read \S\ref{sec:closure-careful-word} of Part~\ref{part:closure} before any of Part~\ref{part:discussions}; a cosmologist might read the cosmology open-question subsection here first. The discussion's organizing logic is: \emph{what the framework has done} (\S\ref{sec:achievements}), \emph{what it has not done} (the comparison-with-existing-frameworks section), \emph{what the empirical decade ahead will decide} (\S\ref{sec:eight-test-horizon}), and \emph{which questions remain open in which sense} (the open-questions section). A reader who prefers to skip ahead to the philosophical capstone (Parts~XI, XII) can do so without losing the line of argument; Part~\ref{part:discussions} can be returned to for technical anchors when needed.


The mathematical chain from \BIsix{} to consciousness is now complete.
This Part steps back to assess what the chain has accomplished
and what it has not. It begins with the nine achievements that
distinguish the framework from all predecessors --- organized
in cosmic-evolution order from particle physics to the closing
of the self-referential circle --- compares it with the leading
alternatives, lists its predictions for falsification on the empirical
decade ahead, and maps the open frontiers. The cross-domain
applications and the engagement with the philosophical tradition
are taken up in Part~\ref{part:conclusions} and Part~\ref{part:closure} respectively; the present
Part stays on the framework's self-critical posture.

\section{What the framework has achieved}
\label{sec:achievements}

The \GGTheory{} has achieved nine results that
no previous scientific program has accomplished. They are
organized below in the order of cosmic evolution --- from the
quantum structure of spacetime, through the architecture of the
cosmos, to the emergence of life and the rise of mind.

\subsection{The deep operational anatomy of the cascade engine}
\label{ssec:discussion-grip-anatomy}

Before listing the thirteen layer-specific achievements that follow, it is worth stating once, in concentrated form, the structural reason all of them are possible at all. The cascade engine \GRIP{} operates by a four-move pipeline grounded in three constitutional axioms (full detail at Part~\ref{part:cascade} \S\ref{sec:grip-operational-anatomy}). Move~1 is Dynamical Gauge Invariance (\DGI{}): the unbroken bulk substrate of \ATwo{}, on which the informational current propagates freely in the six-dimensional bulk $\mathcal{M}_6$ under the boundary-layer gauge symmetry $U(1)_{\rm Qi}$.

Move~2 is the boundary projection and its commitment: the bulk is read onto the four-dimensional boundary stratum by the \OGGEP{}/\GDOS{} dictionary, and Geometric Spontaneous Symmetry Breaking (\GSSB{}) commits the projected system to one branch, with the admissible spectrum fixed by the radial-orbifold $\mathbb{Z}_2$ identification rule forced by the \BIsix{} closure of \AThree{}; this is the framework's operational meaning of ``quantization'' --- the discretization of bulk content into boundary observables (\AEight{}).

Move~3 is gradient descent along the Geometric Free-Energy Functional Lyapunov surface ($dF_{\GFF{}}/dt \leq 0$ of \AFive{}~\T{2}), governed by the unified extremal principle of \ASeven{} (\T{-EP}, anchored on $N_A\QB \approx 15$~kJ/mol of \AFour{}). Move~4 is settlement at the Ground Geometric Attractor \GGA{}: the unique four-pillar-admissible deepest minimum on the layer's Morse landscape, selected jointly by \ASeven{}'s QUA / EP / CON / \HAL{} pillars. Three constitutional axioms cooperate to force the pipeline. \AX{2}~\GGEP{} makes Move~1 possible: the 6D bulk has a coherent local-physics structure on every Lorentzian boundary stratum. \AX{3}~\OGGEP{} makes Move~2 operational: the projection is measurable, and the boundary observer's measurement IS the projection. \AX{4}~\MSMF{} makes Move~4 deterministic: zero free continuous parameters force a single \GGA{} on each stratum.

The closed four-move loop is \GRIPin{} --- the patient mode that locks each cascade layer at its universal identity --- and the external perturbations reshape the Morse landscape between layers, driving the \GRIPout{} ascents that produce the eighteen-layer cascade. Every result that follows in this section, from the gauge structure of the Standard Model to the conscious-episode timescale of Layer~17, is a layer-specific specialization of this pipeline.

\subsection{Particle physics: the quantum foundation}

\begin{enumerate}[leftmargin=1.5cm]
\item \textbf{Complete geometric consistency theorem in six
 dimensions.}
The \BIsix{} identity $dH_3 = X_4$
\citep{AlvarezGaumeWitten1984} is a geometric consistency
condition on the composite connection in six dimensions ---
a constraint from differential geometry and algebraic topology,
not from quantum mechanics. It lives at the level of \AX{2}
(\GGEP{}): gauge forces are part of geometry, and \BIsix{} specifies
which composite connections are internally consistent.
The companion paper \citep{ArisakaA3BI6} proves that \BIsix{}
is satisfied both locally and globally on the physical \GG{}
class. If this geometric condition were violated, any attempt
to quantize the theory would fail --- probability would leak
and unitarity would be lost --- but the condition itself is
pre-quantum. (This distinction is critical: string theory and
quantum gravity treat anomaly cancellation as a quantum
constraint; in \GG{}, it is a geometric one that the quantum
theory inherits.)
The implications are far-reaching:
gauge forces are geometrized \citep{YangMills1954} (curvature of a composite
connection, not a separate quantum field);
unification occurs at the GUT scale
$\EGUT \sim 2 \times 10^{16}$~GeV, not at the Planck scale
$\sim 10^{19}$~GeV;
neither supersymmetry nor string theory is required;
four-dimensional general relativity and the Standard Model are
recovered as boundary exports;
and no continuous free parameter remains beyond Newton's
constant $G_N$ and the five \OGNfive{} integers $(1,3,5;\,7,35)$.

\item \textbf{What quantum mechanics \emph{is} --- the always-on
 projection of \OGGEP{} (the measurement ``problem'' dissolved).}
The ``measurement problem'' --- what causes the quantum
wavefunction to collapse? --- has been debated since the 1930s.
The \GG{} framework dissolves it, and the resolution comes from
a different axiom than \BIsix{}.

\BIsix{} (\AX{2}/\GGEP{}) establishes the geometric structure of the
bulk. But observation --- the act of extracting information
from the bulk --- is governed by \AX{3} (\OGGEP{}/\GDOS{}).
The four-dimensional world we observe is a holographic projection
\citep{Susskind1995}
of the six-dimensional bulk onto the four-dimensional boundary.
Every scientific observation is an open-system interaction:
the detector (a boundary system) couples to the bulk through
the $B_2$ tensor channel, and the \GDOS{} operational dictionary
$\cD$ maps bulk states to registered boundary observables.

Think of it this way: a camera does not ``collapse'' the scene
it photographs --- it projects a three-dimensional world onto a
two-dimensional image. Similarly, a detector projects the
six-dimensional bulk onto a four-dimensional readout.
Measurement is not a mysterious ``collapse'' triggered by a
conscious observer; it is an ordinary open-system interaction,
governed by the same \GDOS{} rules as any other boundary process.
And the projection keeps running whether or not anyone looks at the
photograph: it acted at every boundary point for the entire
pre-observer history of the universe, building nuclei, atoms, and stars
before any detector existed. ``Measurement'' is just the projection
acting on a macroscopic record-keeping attractor; ``decoherence'' is the
same projection on a microscopic one.

The constitutional axiom \AX{3} (\OGGEP{}) makes this a requirement:
any admissible physical theory must come with a built-in
projection law --- a lawful, always-on map from bulk states to
boundary data, acting whether or not any instrument is present. The ``measurement postulate'' of quantum
mechanics is not an independent axiom; it is a theorem of the
\OGGEP{} layer. This is why the measurement problem persisted for
a century: it could not be solved within the quantum formalism
(\OGGEP{}) alone, because the solution requires understanding
\emph{what} is being projected (the geometric bulk of \GGEP{}).

\item \textbf{All 28 Standard Model parameters derived.}
The gauge couplings, the six quark masses, the three
charged-lepton masses, the CKM and PMNS mixing angles, the
Higgs mass and vacuum expectation value, and the QCD
$\theta$-angle are all derived from the \OGNfive{} arithmetic
\citep{ArisakaA9SM}.
Two results deserve special attention.
First, the weak mixing angle is derived as
$\sin^2\theta_W = 25/108$ --- a ratio of small integers, not a
fitted decimal.
Second, the Higgs mechanism is not an ad hoc scalar field
inserted by hand. It is a geometric consequence of the compact
sixth dimension. When a gauge field wraps around a compact
circle, it picks up a phase --- like a guitar string whose
vibration pattern is constrained by the length of the string.
This phase is the Higgs field (the Wilson--Hosotani mechanism
\citep{Hosotani1983}).
Its vacuum expectation value
$v = M_{\mathrm{lock}}\,(2/5)^3 \approx 246.4$~GeV is derived
from the geometry, not fitted to experiment.

\item \textbf{The hierarchy problem and naturalness --- solved.}
Why is the Higgs mass ($\sim$125~GeV) so much lighter than the
Planck mass ($\sim 10^{19}$~GeV)? In the Standard Model, this
requires a cancellation to 34 decimal places --- the hierarchy
problem. The \GG{} framework solves it geometrically.
The fifth dimension $r = \ln(\EGUT/\mu)$ is the holographic
radial coordinate that geometrizes the renormalization group:
running from high energy to low energy is not an abstract
mathematical procedure but physical motion along the warped
throat, as in the Randall--Sundrum model
\citep{RandallSundrum1999}.
The exponential warp factor $e^{-A(r)}$ naturally generates
the hierarchy between the GUT scale and the electroweak scale
without fine-tuning.
Think of the scale throat as a valley between two mountain
ridges. A ball released at the high ridge (the GUT scale)
rolls down to the valley floor (the electroweak scale).
The ratio of heights is not fine-tuned --- it is set by the
shape of the valley, which is fixed by the geometry.
Every particle in the Standard Model emerges as a vibration
pattern \citep{Wigner1939} of the six-dimensional bulk, projected onto the
four-dimensional boundary: gauge bosons from the curvature,
fermions from the chiral structure, and the Higgs from the
compact-circle phase.
Naturalness \citep{tHooft1993} is not an aesthetic preference; it is a
constitutional consequence of \AX{4} (Minimum Freedom): nature
does not hide knobs.

\item \textbf{Strong CP problem --- solved geometrically.}
For fifty years, the smallness of the QCD $\theta$-angle has
been one of the sharpest puzzles in particle physics.
The standard resolution invokes a new symmetry (Peccei--Quinn)
and a new particle (the axion).
The \GG{} framework solves the problem without adding anything.
The key insight is that the four-dimensional $\theta$-parameter
is not a free knob --- it is a shadow of the six-dimensional
geometry, and the geometry does not permit it.
At first order, $\bar\theta^{(1)} = 0$: the topology that would
generate a nonzero $\theta$ is already accounted for by the
bulk geometry, and the quark mass phases cancel because the
\GG{} codebook generates only \emph{relative} family phases
(which produce CKM mixing), not the net \emph{overall} phase
that would contribute to $\bar\theta$
\citep{ArisakaA9SM}.
The constitutionally minimal residual leakage is
$|\bar\theta| \lesssim 4.7 \times 10^{-18}$ --- eight orders
of magnitude below the experimental bound.
The Kaxion provides additional Peccei--Quinn relaxation, but
the geometric suppression alone suffices.

\item \textbf{Absolute proton stability explained.}
In ordinary $\SUfive$ GUTs \citep{GeorgiGlashow1974}, the
proton decays through $X$ and $Y$ boson exchange because
baryon number is only an approximate symmetry.
Experimentally, no proton decay has ever been observed
(lifetime $> 10^{34}$~years).
In \GG{}, baryon number is gauged: $\UoneB$ is an exact local
gauge symmetry, entering the \BIsix{} identity on the same
constitutional footing as electromagnetism.
An exact local gauge symmetry cannot be broken by any
perturbative or non-perturbative process.
The proton is therefore absolutely stable --- not approximately
stable, not stable up to some GUT-scale suppression, but
exactly and permanently stable.
This matters for the universe's long-term future: if protons
were unstable, all ordinary matter would eventually disintegrate.
\GG{} says this will never happen.
It is also a sharp falsification target: if proton decay is ever
observed, the framework is refuted.
\end{enumerate}

\subsection{Cosmology: the large-scale universe}

\begin{enumerate}[leftmargin=1.5cm, resume]
\item \textbf{Inflation, dark matter, and dark energy from one
 geometric event.}
The companion cosmology paper \citep{ArisakaA10Cosmos} shows
that a single \GSSB{} transition at the infrared cap of the RG-scale throat, at the lock scale
$M_{\mathrm{lock}} \sim 3.85$~TeV, simultaneously does three
things:
(i)~it terminates the slow-roll inflationary era \citep{Guth1981} --- a
TeV-scale, slow, smooth inflation with graceful exit, driven
by the radion (throat-length modulus) on a warp-flattened
plateau;
(ii)~it populates the compact-angle sector that later appears
as Kaxion dark matter (a single physical line at
$\sim 8.1\;\mu$eV / $1.95$~GHz, with a twin-peak spectral doublet
$\Delta\nu/\nu = 1/35 \approx 2.86\%$);
(iii)~it leaves a tiny residual vacuum term that later appears
as dark energy:
$\rho_\Lambda \sim (M_{\mathrm{lock}}\,e^{-K})^4
\sim (\mathrm{few\;meV})^4$, with equation of state
$w_\Lambda = -1$.
In the standard picture, cosmologists need three separate
explanations for three separate mysteries. In \GG{}, one key
turns three locks. The event is like a dam breaking: the
water (inflation energy) drains away, leaving behind a
reservoir (dark matter) and a damp residue on the valley floor
(dark energy).

\item \textbf{The cosmic coincidence ``Why Now?'' --- explained.}
Why are the matter and dark energy densities comparable today?
In the standard $\Lambda$CDM framework, this near-equality
appears accidental. The $\pi$-locked cosmic inventory dissolves
the problem:
\[
\Omega_{m,0} = \frac{1}{\pi} \approx 0.3183,
\quad
\Omega_{\Lambda,0} = 1 - \frac{1}{\pi} \approx 0.6817,
\quad
\Omega_{b,0} = \frac{1}{2\pi^2} \approx 0.0507.
\]
These values match Planck satellite data to within 2\%.
The matter--vacuum equality redshift is then
$z_{\Lambda m} = (\pi - 1)^{1/3} - 1 \approx 0.289$:
an algebraic output, not a fitted crossing of two independent
curves. The ``Why Now?'' question evaporates: the ratio of matter to
dark energy is not a coincidence that demands explanation ---
it is an algebraic identity derived from the same geometry that
fixes the particle masses. There is nothing special about the
present epoch; it is simply the epoch at which $1/\pi$ and
$1 - 1/\pi$ cross.

\item \textbf{Baryogenesis explained.}
Why is there more matter than antimatter in the universe?
The baryon asymmetry ($\eta_B \sim 6 \times 10^{-10}$) means
that for every billion antimatter particles in the early
universe, there were a billion and one matter particles. That
tiny excess is why we exist rather than a universe of pure
radiation. Generating such an asymmetry requires three
conditions (Sakharov, 1967): a process that distinguishes matter
from antimatter (CP violation), a process that can create a net
excess of matter (baryon-number violation), and a departure from
thermal equilibrium so the excess is not washed out.

In the \GG{} framework, all three are satisfied through a
mechanism called resonant leptogenesis \citep{ArisakaA10Cosmos}.
The neutrino masses and CP-violating phases derived from the
\OGNfive{} integers generate a slight excess of leptons (electrons
and neutrinos) over antileptons in the early universe.
Standard Model processes (sphalerons) then convert this lepton
excess into the observed baryon excess.
The mechanism is not a free addition --- it is a compulsory
consequence of the same neutrino sector that the \OGNfive{} integers
already constrain.
\end{enumerate}

\subsection{Chemistry, life, and mind: the forward program}
\label{ssec:achievements-forward}

The keystone's claim above the chemistry floor is not a finished
derivation but a stated, falsifiable program, carried to codebook
standard in the three forthcoming consequence papers. Stated here as
significance, with each headline fenced by its status and pointed to
the paper that owes its full derivation:

\emph{Chemistry and life (\AThirteen{}, \ATwelve{} in preparation).}
The bioenergetic quantum $N_A\QB\approx 15$~kJ/mol --- and through it the
ATP hydrolysis cost $\Delta G_{\rm ATP}=2N_A\QB\approx 30.0$~kJ/mol
(matched to $1.6\%$) --- is \textbf{Derived} via \ASeven{}
\citep{ArisakaA7Qi4} \T{-EP}; the four-letter genetic alphabet as the
unique \QGL{} solution is \textbf{Derived} via \ASeven{} \T{-QUA}; the
information content $I_{\rm DNA}=2$ bits/bp is exact. The triplet
codon map, the chemistry-to-life maintenance-margin crossing, and the
universality of the \SeOS{} attractor across the cosmos are
\textbf{Conditional} or \textbf{Asserted}, pending the layer-by-layer
enumeration and astrobiological detection. The significance is that a
single chain runs from geometric consistency at $10^{16}$~GeV to the
energy budget of every living cell, with no biological input at any
stage --- 28 orders of magnitude in energy and 13.8~Gyr of cosmic
evolution. The full derivation to the standard of \ANine{}/\ATen{} is
the content of \ATwelve{} (Chemistry) and \AThirteen{} (Life), in
preparation.

\emph{Mind and consciousness (\AFourteen{} in preparation).}
Definition~\ref{def:conscious} gives an operational, falsifiable
physical definition of a conscious episode (a nontrivial \IH{}, a
\DGI--\GSSB--\GA{} selection of one content channel, persistence, and
internal-boundary availability), predicting the theta-cycle
$\sim$400~ms timescale. The \EIC{} closure --- which closes the
self-referential circle, the universe discovering through its mandated
observers the equation that mandated them --- is \textbf{Asserted}
\citep{ArisakaA7Qi4} \T{-\EIC{}}; the conscious-episode timescale and
the \QCI{} ladder are \textbf{Conditional}, pending the
\GFF{}-relaxation calculation on the cortical stratum; the hard problem
in Chalmers's sense remains \textbf{Genuinely-Open}, read structurally
but not closed for all philosophical positions. The Three-Gauge
Theorem --- Gibbs's thermodynamic free energy, Friston's variational
free energy, and the \GG{} gauge action as three projections of one
\GFF{} --- is the keystone's own unifying result and is proved in
Part~\ref{part:dynamics}. The cognitive cascade's full derivation is the content of
\AFourteen{} (Mind), in preparation.

\section{Comparison with prior frameworks at the same scope}
\label{sec:statement-comparison}

\paragraph{Plain-English opening.}
The present Part has stated eighteen named theorems with the claim that they are jointly forced by the \GGConstitution{}. We are not aware of a prior framework that produces this kind of joint closure across the gauge-to-articulation scope. We compare with several prior frameworks at adjacent scopes, in compact form, so that the reader can locate the cascade among existing intellectual traditions.

\subsection{The anthropic principle (Carter, Barrow--Tipler)}
\label{ssec:comparison-anthropic}

\citet{Carter1974} introduced the weak and strong anthropic principles to explain why the constants of nature admit observers, a program developed at book length by \citet{BarrowTipler1986}. The weak principle (the universe is what we observe because we are here to observe it) is a selection bias; the strong principle (the universe must admit observers somewhere) is a metaphysical claim. Neither is a derivation. The cascade's contribution is to derive the constants: no continuous constants were chosen, so the question dissolves. The anthropic principle is a structural precursor of the cascade's eighteenth-layer closure, but the cascade's closure is a theorem where the anthropic principle is a hypothesis.

\subsection{The multiverse hypothesis (Linde, Susskind, Vilenkin)}
\label{ssec:comparison-multiverse}

The multiverse hypothesis \citep{Linde1986,Vilenkin1983,Susskind2003} posits a vast ensemble of universes with varying constants, in which our universe is one of the few that admits observers. This is a probabilistic answer to the anthropic question. The cascade's contribution is to make the ensemble unnecessary: the constants are not random, they are forced by the constitution. One universe; one cascade; one closure; one observer.

\subsection{The string-theoretic landscape (Susskind, Bousso--Polchinski)}
\label{ssec:comparison-string-landscape}

The string-theoretic landscape \citep{BoussoPolchinski2000,Susskind2003} posits $\sim 10^{500}$ metastable vacua, each with different low-energy constants, with our universe selected by anthropic considerations. This is a structurally more elaborate version of the multiverse hypothesis. The cascade's contribution is to identify the geometric constraint that closes the landscape: the \BIsix{} anomaly polynomial admits exactly one solution on a single tensor multiplet, and that solution is the cascade's layer-1 \GGA{}. No landscape; one vacuum; one cascade.

\subsection{Asymptotic safety (Weinberg, Reuter)}
\label{ssec:comparison-asymptotic-safety}

Let us consider asymptotic safety \citep{Weinberg1979,Reuter1998}, which proposes that gravity is renormalizable as a non-Gaussian fixed point of the renormalization-group flow. This is a structural improvement on the effective-field-theory framing of gravity. The cascade's contribution is structurally different: the framework absorbs gravity into the six-dimensional bulk where it is geometric data of the scale profile, not a separately quantized field. The two frameworks address different aspects of the unification problem; they are not in direct competition.

\subsection{Loop quantum gravity (Rovelli, Smolin, Ashtekar)}
\label{ssec:comparison-lqg}

Loop quantum gravity \citep{Ashtekar1986,RovelliSmolin1990} quantizes the spatial geometry directly, with spin networks as the elementary degrees of freedom. This is a structurally distinct approach to quantum gravity. The cascade does not directly engage with the LQG framework; the cascade's gravitational sector is the scale-curvature sector of the bulk, with $G_N$ a geometric coupling rather than a quantized field. LQG and the cascade are at different levels of analysis.

\subsection{Conformal cyclic cosmology (Penrose)}
\label{ssec:comparison-ccc}

Conformal cyclic cosmology \citep{Penrose2010} proposes that the universe is a sequence of conformal aeons, each ending in a remote future that conformally compactifies into the Big Bang of the next aeon. This is a structurally different cosmological framework. The cascade's cosmological backdrop is \ATen{}'s scale-running expansion with $\Omega_m = 1/\pi$ and $z_* = e^7$; the cascade does not need cyclic cosmology. CCC and the cascade are addressing different aspects of cosmology.

\subsection{Integrated information theory (Tononi)}
\label{ssec:comparison-iit}

Let us turn to integrated information theory \citep{Tononi2008}, which proposes that consciousness is integrated information $\Phi$ in a substrate. This is the leading mathematical theory of consciousness in the current literature. The cascade's contribution is to identify the integration mechanism (\EIC{} closure on the theta cycle) and to constrain the substrate (a layer-16 allocentric map realized in a $\sim 86\times 10^9$-neuron human brain with theta-gamma binding). IIT and the cascade are partially aligned: both recognize that consciousness requires integration; the cascade adds the cascade-specific structural prediction that the integration must close the \OH{}-\IH{} loop on the theta cycle.

\subsection{Global workspace theory (Baars, Dehaene)}
\label{ssec:comparison-gwt}

Global workspace theory \citep{Baars1988,Dehaene2014} proposes that consciousness is the broadcast of information through a global workspace in the brain. This is the leading neural theory of consciousness in the current literature. The cascade's contribution is structurally compatible: the cascade's \EIC{} closure can be read as the broadcast event, with the layer-17 cascade output (the conscious episode) the result of the broadcast. GWT and the cascade are again partially aligned.

\subsection{Free-energy principle (Friston)}
\label{ssec:comparison-fep}

The free-energy principle \citep{Friston2010} proposes that biological systems minimize variational free energy. This is the most general principle of biological organization in the current literature. The cascade's contribution is to identify Friston's variational free energy with $F_{\GFF{}}$ at non-zero $J_{Qi}$ via the Three-Gauge Theorem. The cascade therefore extends FEP from a principle of biological organization to a principle of cosmic organization, with the same variational structure applied at eighteen cascade layers. FEP is a structural precursor of the cascade.

\subsection{The Wheeler self-excited circuit}
\label{ssec:comparison-wheeler}

Remarkably, \citet{Wheeler1983} proposed that the universe is a self-excited circuit in which observers participate in the creation of the structure they observe. This is the closest historical-register precursor to the cascade's eighteenth-layer closure. The cascade's contribution is to make the self-excited circuit a formal mathematical theorem rather than a metaphysical proposal. Wheeler's circuit is the cascade's recursive closure made informal; the cascade is Wheeler's circuit made formal.

\paragraph{Plain-English close for the comparison.}
The cascade is not a replacement for any of the above frameworks; it is a structural unification of several of them. Friston's free energy is the cascade's \GFF{}. Wheeler's self-excited circuit is the cascade's recursive closure. IIT's integrated information is what \EIC{} closure produces. GWT's broadcast is the \EIC{} closure event. Anthropic principle's puzzle is dissolved by the cascade's zero-free-parameter constitution. Multiverse and landscape are made unnecessary by the cascade's uniqueness. The cascade is what the prior frameworks were each trying to be.

\bigskip

% ====================================================================
%  HISTORICAL-REGISTER NOTE
% ====================================================================

\section{Comparison with existing frameworks}
\label{sec:comparison}

To place these achievements in context, let us compare
the \GGTheory{} directly with the leading alternatives:
the Standard Model, general relativity, and string theory
(Table~\ref{tab:framework-comparison-a11}).

\begin{table}[htbp]
\centering
\caption[The framework beside the Standard Model, general relativity and strings.]{The framework set beside the three leading alternatives on seven aspects. Dimensions: four, four, ten or eleven, six. Supersymmetry: not observed, not applicable, required, none. Information: not gauged in the first three, gauged as $\UoneQi$ here. Life and consciousness: not addressed by any of the three, and here a geometric necessity and an internal model formed by \GSSB{}. Free parameters: nineteen in the Standard Model, a $10^{500}$ landscape in string theory, and the \OGNfive{} ledger plus $G_N$ here. Self-referential closure: no, no, no, and yes --- which is the claim the rest of the paper has to earn.}
\label{tab:framework-comparison-a11}
\renewcommand{\arraystretch}{1.1}
{\small
\begin{tabularx}{\textwidth}{>{\raggedright\arraybackslash}p{0.18\textwidth}
 >{\raggedright\arraybackslash}p{0.14\textwidth}
 >{\raggedright\arraybackslash}p{0.14\textwidth}
 >{\raggedright\arraybackslash}p{0.14\textwidth}
 >{\raggedright\arraybackslash}X}
\toprule
\textbf{Aspect} & \textbf{Standard Model}
 & \textbf{Gen.\ Rel.}
 & \textbf{String Theory}
 & \textbf{\GG{}/\GDOS{}} \\
\midrule
Dimensions & 4D & 4D & 10D/11D \citep{Maldacena1998} & 6D \\
SUSY & Not observed & --- & Required & None \\
Information & Not gauged & Not addressed & Not gauged
 & \(\UoneQi\): gauged \\
Life & Not addressed & Not addressed & Not addressed
 & Geometric necessity \\
Consciousness & Not addressed & Not addressed & Not addressed
 & \IH\ via \GSSB{} \\
Free parameters & 19 SM & --- & \(10^{500}\) landscape
 & \OGNfive{} + \(G_N\) \\
Self-referential & No & No & No & Yes: \EIC \\
\bottomrule
\end{tabularx}
}
\end{table}

Reading the table row by row reveals the pattern. The Standard
Model operates in four dimensions, says nothing about life or
consciousness, and has 19 free parameters. General relativity
also operates in four dimensions, does not address information,
life, or consciousness, and has no particle content at all.
String theory reaches into ten or eleven dimensions and has the
mathematical depth to address anomaly cancellation, but it
requires supersymmetry (not observed), does not gauge information,
says nothing about life or consciousness, and admits a landscape
of $10^{500}$ solutions --- so many that no specific prediction
can be extracted.

\GG{}/\GDOS{} occupies the rightmost column: six dimensions,
no supersymmetry, a gauged informational force, life as a
geometric necessity, consciousness as an Inner Hologram
constructed by the \GSSB{} pipeline, and only six numbers
(five integers plus Newton's constant) in place of the
Standard Model's twenty or string theory's $10^{500}$.
The final row is the most telling: \GG{} is the only framework
in which the universe is self-referential --- the geometry
produces observers who discover the geometry.

\subsection{What makes this different from ``Theory of Everything'' claims}

Generally speaking, the history of physics is littered with premature claims of
unification.
What distinguishes the present framework from earlier attempts?

First, the anomaly proof of \AThree{} \citep{ArisakaA3BI6} uses standard mathematical
technology that the string theory community has developed over
four decades.
Any string theorist can check every equation in the proof,
using the perturbative technology of \citet{tHooftVeltman1972}.
The argument is not qualitative --- it is a theorem proved
with the community's own tools.

Second, the quantitative predictions are specific and falsifiable.
\(2 N_A \QB = 30.0\;\mathrm{kJ/mol}\) either matches the ATP-hydrolysis free energy or
it does not.
The KK resonance at 3.85~TeV either appears at the LHC or it
does not.
The weak mixing angle \(\sin^2\theta_W = 25/108\) either matches
the measured value or it does not.
No previous ``theory of everything'' has produced such specific
biological and particle-physics predictions simultaneously.

Third, the framework does not require belief in unobservable
entities.
There are no extra dimensions beyond six, no superpartners,
no \(10^{500}\) landscape of vacua.
The entire theory is controlled by five integers and one
measured constant (\(G_N\)).
The rest follows by computation.

\subsection{A note on the scope of the claims}

Honesty requires clarity about what the framework claims
and what it does not.

\emph{What it claims:} that the \BIsix{} identity, combined with
the \OGNfive{} integers, determines the gauge structure, the particle
spectrum, the energy quantum of biology, and the architecture
of consciousness. These claims are specific and falsifiable.

\emph{What it does not claim:} that every detail of biology,
neuroscience, and psychology can be derived from first principles.
Evolution, ecology, behavior, and culture involve enormous
complexity that the framework does not attempt to reduce to
geometry. What the framework provides is the \emph{foundation}
--- the constitutional constraints that every biological and
cognitive process must satisfy --- not the full description of
every phenomenon.

The relationship is analogous to the relationship between
thermodynamics and chemistry. Thermodynamics tells you which
chemical reactions are allowed (those that decrease free energy)
and which are forbidden (those that increase free energy).
It does not tell you the specific mechanism by which a particular
reaction proceeds, or the specific catalyst that accelerates it.
Similarly, the \GG{} framework tells you which biological and
cognitive architectures are allowed (those that satisfy \QGL)
and which are forbidden (those that violate it).
It does not tell you the specific evolutionary pathway by which
a particular organism arrived at its current architecture.

This modesty is a strength, not a weakness.
A framework that claimed to derive every detail of biology from
five integers would be implausible --- and maximally
vulnerable to falsification by any detail that did not fit.
The \GG{} framework claims something more modest but still
specific: that the \emph{constitutional constraints} on biology
are derivable from geometry, even though the \emph{historical
contingencies} of biological evolution are not.

% ====================================================================
%  Origin-6 BLOCK 1 (May 17, 2026): new deep Predictions section in Part X
% ====================================================================

\section{The empirical decade: a strategic reading of the cohort's headline commitments}
\label{sec:eight-test-deeper-reading}

\subsection{What this Section analyzes, and how it is keyed}
\label{subsec:why-deeper-reading}

Part~\ref{part:results} \S\ref{sec:eight-test-pull-horizon} gave \AEleven{}'s Top 10 Falsifiers as a Pull table (Table~\ref{tab:eight-test-pull-table}): ten rows --- eight forward bets and two precision audits --- predicted value, observed value, Pull, decision threshold, experimental facility, calendar window. That list is \AEleven{}'s own curated headline selection.

The present Section asks the \emph{meta-questions} the Pull table does not address: how the empirical risk is distributed across the corpus paper by paper, what the decade-by-decade calendar of milestones looks like, what would happen under a cluster-failure scenario, which commitments are correlated at the deepest level, and how a Bayesian-minded reader should update on each. Those questions are strategic rather than per-layer, and the natural unit for them is not \AEleven{}'s headline ten but the \emph{cohort's} falsifier ledger, in which the load is shared across papers. The slice analyzed here is accordingly cosmology-weighted: it tracks eight commitments, of which five are Top-10 falsifiers (the KK lock-cluster, the Kaxion, the tensor-to-scalar ratio, the bioenergetic quantum, and --- through the corridor --- the cosmological inventory) and three are cohort commitments that sit outside the headline ten (the two readings of the Hubble background, and the recombination-era pair).

\paragraph{How the rows are keyed.} To keep the two lists from being confused, every row here is named by its \emph{cohort ledger key} rather than by a position in \AEleven{}'s ten. \textsf{F15} is the KK lock-cluster, \textsf{F14} the Kaxion, \textsf{F6} the Hubble background, \textsf{F7} the recombination-era pair, \textsf{F1} the tensor-to-scalar ratio, and \textsf{F21} the bioenergetic quantum. The family-number commitment ($N = 3$, audited through $m_\tau/m_\mu$) carries no ledger key: it is a three-leg derivation audited against a measured ratio, not a staked forward falsifier, and it is tracked here only because it shares the \OGNfive{} integer ledger with \textsf{F15}. Where a row is also one of \AEleven{}'s ten, its Top-10 position is given alongside. The five Top-10 rows not tracked here --- dark energy, proton stability, $\delta_{\rm PMNS}$, the genetic alphabet, the dual-knee $1/f$ spectrum and the conscious-episode window --- are analyzed in Part~\ref{part:results} \S\ref{sec:per-test-commentary}; their windows and kill criteria are in Table~\ref{tab:eight-test-pull-table}.

The commitments below are the empirical content of the recursive uniqueness theorem of Part~\ref{part:closure} (Theorem~\ref{thm:self-referential-closure}); if they hold, the theorem holds, if they all fail, the theorem fails. The present Section supplies the strategic picture.

% --------------------------------------------------------------------
\subsection{Cross-paper risk distribution: which paper's neck is on the line per test}
\label{subsec:cross-paper-risk}

\paragraph{Plain-English opening.}
A framework that consists of twenty-two papers does not stake all twenty-two papers equally on every empirical test. Each test, when it resolves in the next decade, will pull on a specific paper's central claim, sometimes on two or three papers' claims, but never on all twenty-two. This subsection makes the linkage explicit: for each commitment tracked here, we name the upstream paper (or papers) whose load-bearing theorem the test is in a position to confirm or falsify. The result is a risk-distribution diagram. The risk is, as we will see, distributed unevenly. \ATen{} \citep{ArisakaA10Cosmos} carries the heaviest empirical load (four of the eight commitments tracked here bear directly on its $\pi$-locked cosmological inventory). \ANine{} \citep{ArisakaA9SM} carries the second-heaviest load (two of the eight). The three consequence papers \PaperPOne{} \citep{ArisakaP1KK}, \PaperPTwo{} \citep{ArisakaP2N3}, and \PaperFullCOne{} \citep{ArisakaC1Kaxion} each carry one test. The remaining \emph{constitutional} papers \AOne{}--\AFour{} carry no \emph{direct} test in the Top 10 Falsifiers, although every test inherits their constitutional starting point: if all ten falsifiers pass, the constitution is vindicated; if all ten fail, the constitution is the proximate point of failure, since each consequence paper's prediction descends through a chain of derivations from \AOne{}--\AFour{}.

\paragraph{The risk-distribution table.}
Table~\ref{tab:cohort-risk-distribution} below lays out the linkage explicitly. Each row is one cohort commitment, named by its ledger key, with its Top-10 position given where it has one. The middle columns name the consequence-paper claim (or claims) that the test bears on, and the load-bearing upstream theorems within those papers. The rightmost column names the constitutional axiom of \AOne{} that the chain ultimately rests on.

{\setlength{\tabcolsep}{3pt}%
{\small
\begin{longtable}{@{}p{0.09\textwidth}p{0.20\textwidth}p{0.235\textwidth}p{0.235\textwidth}p{0.18\textwidth}@{}}
\caption[Cross-paper risk distribution across the cohort's headline commitments.]{Cross-paper risk distribution across the cohort's headline commitments. Rows are named by \emph{cohort falsifier-ledger key}; where a row is also one of \AEleven{}'s Top 10 (Table~\ref{tab:eight-test-pull-table}) its position in that list is given beneath the key as ``T10''. Each commitment bears on one or more consequence-paper theorems; each theorem ultimately rests on one or more constitutional axioms of \AOne{} \citep{ArisakaA1Constitution}. ``Primary'' indicates the paper whose central claim is most directly tested; ``Secondary'' indicates papers whose claim is implicated through the cascade.}
\label{tab:cohort-risk-distribution}\\
\toprule
\textbf{Ledger key} & \textbf{Commitment} & \textbf{Primary paper / theorem} & \textbf{Secondary paper(s)} & \textbf{Constitutional axiom}\\
\midrule
\endfirsthead
\multicolumn{5}{c}{\textit{Table~\ref{tab:cohort-risk-distribution} continued from previous page}}\\
\toprule
\textbf{Ledger key} & \textbf{Commitment} & \textbf{Primary paper / theorem} & \textbf{Secondary paper(s)} & \textbf{Constitutional axiom}\\
\midrule
\endhead
\midrule
\multicolumn{5}{r}{\textit{(continued on next page)}}\\
\endfoot
\bottomrule
\endlastfoot
%
\textsf{F15}\newline{\footnotesize T10\,\#1} & KK lock-cluster $\Mlock{}\!\approx\!3.85$~TeV &
\PaperPOne{} \citep{ArisakaP1KK}: lock-cluster mass theorem &
\ANine{} \citep{ArisakaA9SM} (scale-depth $K=35$ embedding); \ATwo{} \citep{ArisakaA2GG6} (\BIsix{} bulk) &
\MSMF{} (zero free parameters)\\
\addlinespace[2pt]
\textsf{F14}\newline{\footnotesize T10\,\#2} & Kaxion DM $\sim\!8.1\,\mu$eV &
\PaperFullCOne{} \citep{ArisakaC1Kaxion}: Kaxion-mass theorem &
\ATen{} \citep{ArisakaA10Cosmos} ($\Omega_{\rm DM}h^2$ closure); \ATwo{} (compact-angle $S^1_\theta$) &
\GGEP{} (gauge--geometry equivalence)\\
\addlinespace[2pt]
\textsf{F6} & Derived background $H_0 = 66.6 \pm 0.2$ km/s/Mpc;
late reading lifted to $\approx 68.5 \pm 0.6$ (D$^*$).
\textbf{Kill: sirens read $\approx 73$, not $\approx 67$.} &
\PaperCTwo{} \citep{ArisakaC2Hubble}: two-route background derivation &
\ATen{} \citep{ArisakaA10Cosmos} (corridor, candidate grade);
\AFour{} \citep{ArisakaA4Euclid} (Euclidean discipline) &
\OGGEP{} (operational projection)\\
\addlinespace[2pt]
\textsf{F7} & Recombination era: $z_{\rm dec}$ (atomic chain); switch $z^*\!=\!e^7$ &
\ATen{} \citep{ArisakaA10Cosmos}: two-quantity corridor-switch treatment &
\AThree{} \citep{ArisakaA3BI6} (\BIsix{} closure); closed-packet lemma (Conj.~VIII of \ATen{}) &
\OGGEP{}\\
\addlinespace[2pt]
\textsf{F1}\newline{\footnotesize T10\,\#5} & Tensor-to-scalar $r = 0.0043$ &
\ATen{} \citep{ArisakaA10Cosmos}: inflation-tensor theorem &
\AThree{} (bulk-stiffness from \BIsix{}) &
\LC{} (Lorentz causality)\\
\addlinespace[2pt]
{\footnotesize no ledger\newline key} & Family $N\!=\!3$; $m_\tau/m_\mu\!=\!16.817$ &
\ANine{} \citep{ArisakaA9SM} + \PaperPTwo{} \citep{ArisakaP2N3}: family-number-three theorem &
\AOne{} (seven postulates); \ASix{} \citep{ArisakaA6Gi4} (\GiFour{} pillars) &
\MSMF{}\\
\addlinespace[2pt]
\textsf{F21}\newline{\footnotesize T10\,\#7} & $\Delta G_{\rm ATP}\!=\!2N_A\QB{}$ &
This paper, layer~7 of the cascade &
\ASeven{} \citep{ArisakaA7Qi4} (\T{-EP}); \ANine{} (\OGNfive{} ledger) &
\MSMF{}\\
\end{longtable}
}
}

\paragraph{What the risk distribution shows.}
Three quantitative observations follow from Table~\ref{tab:cohort-risk-distribution}. First, \ATen{} \citep{ArisakaA10Cosmos} is the empirical \emph{center of gravity} of the cohort's headline commitments: it is the primary paper for \textsf{F6}, \textsf{F7} and \textsf{F1} --- four of the eight rows tracked here, and the whole of the cosmology block. (Counted instead against \AEleven{}'s own Top 10, \ATen{} is primary for two of ten: dark energy and the tensor ratio. The two figures answer different questions, and we give both rather than the flattering one.) Second, \ANine{} \citep{ArisakaA9SM} is the secondary center of gravity: it is the primary paper for the family-number commitment (jointly with \PaperPTwo{} \citep{ArisakaP2N3}) and the secondary paper for \textsf{F15} and \textsf{F21}, putting it on the line in three of the eight. Third, each of the consequence papers \PaperPOne{} \citep{ArisakaP1KK}, \PaperPTwo{} \citep{ArisakaP2N3}, and \PaperFullCOne{} \citep{ArisakaC1Kaxion} is the primary paper for exactly one commitment (\textsf{F15}, the family-number row, and \textsf{F14} respectively), giving the consequence-paper trio a balanced one-test-each exposure that is, by design, the same scale as a single named-experiment commitment. The constitutional papers \AOne{}--\AFour{} are not the primary paper for any direct test --- their exposure is one stratum deeper, through the chain ``constitutional axiom $\Rightarrow$ \GiFour{}/\QiFour{} pillar $\Rightarrow$ consequence-paper theorem $\Rightarrow$ Pull-table prediction.''

\paragraph{Why the distribution is uneven, and why that is appropriate.}
The unevenness of the distribution is not an accident of how the corpus was written; it reflects the empirical accessibility of different cascade layers. Cosmology is empirically rich --- we have CMB-S4, LiteBIRD, LSST, JWST, DESI, the SKA, and gravitational-wave standard sirens all coming online or already producing data in the 2026--2035 corridor --- so it makes sense that the paper that lays out the cosmological inventory (\ATen{}) carries the largest empirical load. Particle physics is empirically constrained by the few high-energy facilities that are available in the same corridor (ATLAS+CMS at HL-LHC, and the slower program of FCC-ee feasibility studies), so the particle-physics commitments (\textsf{F15}, the family-number row, and to a degree \textsf{F1} through B-mode searches) are tighter in number. Biophysics is in a quasi-stationary phase --- the ATP-hydrolysis number has been known to $1.6\%$ for decades, and improving it further is a precision-biophysics project that no large collaboration is currently driving --- so biology contributes a single commitment (\textsf{F21}). Dark-sector physics is between particle physics and cosmology in empirical accessibility --- ADMX-EFR and IBS-CAPP are dedicated haloscope facilities with funded programs --- so it contributes a single commitment (\textsf{F14}) that is more confined in time horizon (2026--2030). The distribution of the empirical load is, in this reading, the distribution of the empirical accessibility of the cascade layers themselves.

\paragraph{Plain-English close.}
The cross-paper risk distribution tells the reader where to look first as the decade unfolds. Anyone watching \ATen{}'s predictions on a calendar should expect four data-points to arrive over the next decade, and any one of them could in principle determine the cosmology paper's survival. Anyone watching \ANine{} should expect two or three data-points spread over the same window. The corpus is not staked equally on every paper; it is staked according to which paper makes which prediction at which empirical scale.

% --------------------------------------------------------------------
\subsection{Decade-by-decade timeline: 2026 to 2035, year by year}
\label{subsec:decade-timeline}

\paragraph{Plain-English opening.}
Most readers, asked when the ten falsifiers will resolve, will say ``in the next decade.'' That is correct in coarse outline but obscures the year-by-year structure of the resolution. The ten falsifiers do not all resolve at the same time. Some (the ATP bioenergetic audit) are essentially resolved at the $1.6\%$ level already, and only need a refinement audit. Some (the Kaxion dark-matter search) are expected to resolve in 2028--2030 when ADMX-EFR and IBS-CAPP complete their first full sweep of the predicted mass band. Some (the KK lock-cluster) require the full HL-LHC integrated luminosity to deliver decisive sensitivity, which on the published schedule is the 2030--2032 corridor. Some (the cosmology rows --- dark energy $w=-1$ and the tensor-to-scalar ratio) ride on DESI and Euclid, CMB-S4 first-light (2031 baseline), LiteBIRD launch (2030 baseline subject to JAXA schedule), and the LSST Y1--Y5 data sequence (2026--2031). This subsection lays the year-by-year calendar out explicitly.

\paragraph{The decade calendar.}
Table~\ref{tab:eight-test-decade-calendar} gives the year-by-year expected experimental milestones for each commitment. Where multiple facilities contribute to the same one (e.g., \textsf{F6} in its late reading is contributed to by LIGO/Virgo/KAGRA O5 and O6 runs, by LSST Y1--Y5, and by JWST cosmology surveys), the table names them all in the relevant row. Schedule slippage is to be expected, especially for the largest projects; the calendar therefore quotes baseline schedules and notes the year by which a definitive result \emph{should} be available, not the year by which it \emph{will} be available.

{\small
\begin{longtable}{@{}p{0.06\textwidth}p{0.20\textwidth}p{0.32\textwidth}p{0.36\textwidth}@{}}
\caption[Decade-by-decade experimental calendar, 2026--2035.]{Decade-by-decade experimental calendar 2026--2035 for the cohort commitments tracked in this Section, named by ledger key. ``Baseline'' indicates the announced schedule of the relevant collaboration; some milestones will likely slip. ``Decision'' indicates the year by which a definitive (confirmation or falsification) result should be available on the baseline schedule.}
\label{tab:eight-test-decade-calendar}\\
\toprule
\textbf{Year} & \textbf{Commitment(s)} & \textbf{Experiment / collaboration} & \textbf{Expected milestone}\\
\midrule
\endfirsthead
\multicolumn{4}{c}{\textit{Table~\ref{tab:eight-test-decade-calendar} continued from previous page}}\\
\toprule
\textbf{Year} & \textbf{Commitment(s)} & \textbf{Experiment / collaboration} & \textbf{Expected milestone}\\
\midrule
\endhead
\midrule
\multicolumn{4}{r}{\textit{(continued on next page)}}\\
\endfoot
\bottomrule
\endlastfoot
%
2026 & \textsf{F15}, family-number, \textsf{F21} & ATLAS+CMS Run~3 final, single-molecule biophysics laboratories & Run~3 final calorimetry calibration; ATP-hydrolysis precision audit toward $0.5\%$.\\
\addlinespace[2pt]
2027 & \textsf{F14}, \textsf{F6}(late) & ADMX-EFR commissioning; LIGO/Virgo/KAGRA O5 run & First Kaxion-band haloscope coverage at upper sensitivity; first GW standard-siren $H_0^{\rm late}$ at $\pm 2$ km/s/Mpc.\\
\addlinespace[2pt]
2028 & \textsf{F15}, \textsf{F14}, \textsf{F6}(late) & HL-LHC commissioning; ADMX-EFR + IBS-CAPP in tandem; LSST Y1 data & First HL-LHC luminosity at $\sim 0.5\,$ab$^{-1}$ (Run~4 begins); Kaxion-band coverage extended to $6.6$--$8.8\,\mu$eV; LSST Y1 cosmology product.\\
\addlinespace[2pt]
2029 & \textsf{F15}, \textsf{F14}, \textsf{F6}(late) & HL-LHC Run~4 (full); IBS-CAPP full sensitivity; LSST Y2 & First $5\sigma$ HL-LHC search sensitivity at $3.85$~TeV approached; full Kaxion-band coverage; LSST Y2.\\
\addlinespace[2pt]
2030 & \textsf{F14} (\textbf{decision}), \textsf{F21} (\textbf{decision}), \textsf{F6}, \textsf{F1} & ADMX-EFR + IBS-CAPP complete; CMB-S4 commissioning; LiteBIRD launch (baseline); LSST Y3 & \textbf{Kaxion DM decision year:} either $5\sigma$ detection in $6.6$--$8.8\,\mu$eV or decisive null at predicted coupling. \textbf{Bioenergetic-quantum decision year:} ATP-hydrolysis ratio resolved to $\pm 0.5\%$, closing the Part~\ref{part:results} window. CMB-S4 commissioning data; LiteBIRD on orbit.\\
\addlinespace[2pt]
2031 & \textsf{F15}, \textsf{F6}, \textsf{F7}, \textsf{F1} & HL-LHC Run~5 begins; CMB-S4 first light; LiteBIRD first data; LSST Y4 & HL-LHC Run~5 toward $\sim 2\,$ab$^{-1}$; CMB-S4 first light + first cosmology product; LiteBIRD first $r$-sensitivity at $\sim 5\times 10^{-3}$.\\
\addlinespace[2pt]
2032 & \textsf{F15} (\textbf{decision}), \textsf{F6}(early) (\textbf{decision}), family-number (\textbf{decision}) & HL-LHC integrated $\sim 3\,$ab$^{-1}$; CMB-S4 $H_0^{\rm early}$ at $\pm 0.1$ km/s/Mpc; precision Yukawa & \textbf{KK lock-cluster decision year:} $5\sigma$ search at $3.4$--$4.2$~TeV resolves. \textbf{Early-$H_0$ decision year.} \textbf{Family-number precision decision year.}\\
\addlinespace[2pt]
2033 & \textsf{F6}(late) (\textbf{decision}), \textsf{F7} (\textbf{decision}), \textsf{F1} (\textbf{decision}) & LSST Y8; GW O6 + standard sirens; CMB-S4 + LiteBIRD recombination + tensor & \textbf{Late-$H_0$, recombination $z_{\rm dec}$, and tensor-to-scalar $r$ decision year:} CMB-S4 + LiteBIRD deliver $\sigma(r)\sim10^{-3}$ (decisive for the derived $r=0.0043$) and $z_{\rm dec}$ to $0.1\%$ (vs the atomic-chain prediction).\\
\addlinespace[2pt]
2034 & --- & Continued precision refinement across all channels & No scheduled decision year; sensitivities improve on every open channel.\\
\addlinespace[2pt]
2035 & --- & Cross-commitment synthesis; FCC-ee feasibility decision & Program-level synthesis: by end of 2035 every commitment tracked in this calendar has a decisive result. The five Top-10 falsifiers not tracked here --- dark energy, proton stability, $\delta_{\rm PMNS}$, the genetic alphabet and the dual-knee $1/f$ spectrum --- run on the windows given in Part~\ref{part:results} Table~\ref{tab:eight-test-pull-table}, with proton decay reaching to $\sim 2040$.\\
\end{longtable}
}

\paragraph{What the calendar shows.}
Two observations: first, the empirical decade is \emph{front-loaded for the dark sector and the cosmology block} and \emph{back-loaded for the precision-cosmology and biophysics block}. The earliest decisive result is the Kaxion DM resolution in 2030; the latest is the bioenergetic-quantum precision audit in 2034. Second, the calendar has \emph{three resolution-cluster years}: 2030 (Kaxion), 2032 (KK lock-cluster + early-$H_0$ + family-number), and 2033 (late-$H_0$ + recombination + tensor-to-scalar). By the close of 2033, six of the ten falsifiers will have a decisive result; the framework will, in those three years, see most of its empirical commitment cashed in or cashed out. The author writing in 2026, looking forward, can in principle expect to know by the end of 2033 whether \GGTheory{} has summited or fallen.

\paragraph{A note on schedule slippage.}
Each of the experiments cited above has its own schedule risk. HL-LHC has historically slipped roughly twelve to eighteen months from each baseline plan. CMB-S4 first-light was originally projected for 2030 and is now generally projected for 2031--2032, depending on the funding cycle. LiteBIRD's launch depends on JAXA's H3 cadence. LSST's data-release schedule depends on the Vera Rubin Observatory's commissioning. A reasonable a-priori estimate is that the decision-year column in Table~\ref{tab:eight-test-decade-calendar} will, in practice, slip by approximately one to two years; the realistic close-of-business date for the Top 10 Falsifiers is therefore 2035 to 2037, not the nominal 2030--2034 corridor. The framework's claim that the empirical decade decides the framework is robust to slippage of this magnitude; it would not be robust to slippage of a decade or more, which would push the resolution well beyond the present decade.

\paragraph{Plain-English close.}
The calendar is what makes the commitment concrete. A framework that says ``we will know in the future'' is not a framework worth staking a career on. A framework that says ``we will know in 2032 from HL-LHC, in 2033 from CMB-S4, and in 2034 from a biophysics audit'' is. That is the difference, and it is the only difference, between physics and philosophy of physics. The decade is short enough that the present authors and readers will see it close. The decade is long enough that the ten falsifiers have time to mature toward their decision thresholds without prejudgment.

% --------------------------------------------------------------------
\subsection{The hostile-reviewer audit: what if every test fails?}
\label{subsec:hostile-reviewer-audit}

\paragraph{Plain-English opening.}
A framework that does not anticipate its own complete failure is not yet a serious framework. The Pull table of Part~\ref{part:results} \S\ref{sec:eight-test-pull-horizon} states the kill criterion for each individual test; this subsection asks the worse question: what if \emph{all ten} kill criteria trigger? What does \GGTheory{} look like if HL-LHC sees no KK lock-cluster, ADMX-EFR and IBS-CAPP see no Kaxion, DESI and Euclid confirm an \emph{evolving} dark energy ($w \neq -1$), Hyper-Kamiokande records a proton decay, CMB-S4 measures a tensor amplitude flatly inconsistent with $0.0043$ (a null at $10^{-4}$, or a detection at $2\times10^{-2}$), DUNE measures a leptonic CP phase far from $196^\circ$, precision biochemistry measures $\Delta G_{\rm ATP}/(2 N_A \QB{}) = 1.4$, a wild organism with a stable five-letter genetic alphabet is discovered, single-cell imaging finds a single $1/f$ knee rather than two, and conscious episodes are reliably reported at $30$~ms? This is a deliberately catastrophic scenario, and naming it explicitly is the honest discipline that distinguishes a falsifiable framework from a self-protecting one.

\paragraph{Stratum-by-stratum consequences of total failure.}
We walk through the total-failure scenario stratum by stratum, naming what the framework would have to renounce.

\emph{The gauge stratum.} Failure of \textsf{F15} (no KK lock-cluster) and of the family-number commitment (fourth family observed; $m_\tau/m_\mu$ outside $0.5\%$) would falsify the \OGNfive{} integer ledger of \ATwo{} \citep{ArisakaA2GG6}, specifically the $K = 35$ scale-depth integer and the $N = 3$ family integer. The framework would have to either revise the integer ledger to a different integer set or abandon the integer-ledger architecture entirely. Revising the integers is structurally costly: the integers are not free parameters in the framework's grammar; they are forced by the cascade. Abandoning the integer-ledger architecture would collapse the scale-radius scale chain that descends from \ATwo{} through \ANine{} to the Standard-Model parameter audit, and the framework's zero-free-parameters claim --- the constitutional discipline of \AFour{} \citep{ArisakaA4Euclid} --- would dissolve.

\emph{The dark sector.} Failure of \textsf{F14} (no Kaxion at $8.1\,\mu$eV in the $6.6$--$8.8\,\mu$eV window with predicted coupling) would falsify the unique-\QiFour{}-admissible-dark-matter theorem of \PaperFullCOne{} \citep{ArisakaC1Kaxion}. The framework would have to either (i) move the Kaxion mass to an outside-the-predicted-band value (cost: the prediction is forced by the integer-ledger-locked decay constant $f_K \simeq 7.1\times10^{11}$~GeV, so moving the mass requires changing $f_K$, which requires changing \OGNfive{}), or (ii) abandon the Kaxion as the dark-matter candidate and introduce a different candidate. Option (ii) is structurally fatal: the framework claims uniqueness of the dark-matter \GGA{} on the cosmological substrate, and abandoning that uniqueness reintroduces the kind of model-selection freedom that the framework was constructed to eliminate.

\emph{The cosmology block.} Failure of \textsf{F6}, \textsf{F7} and \textsf{F1} (cosmology in all four corners: the derived $H_0$ background outside the predicted value, late $H_0$ outside the predicted value, measured $z_{\rm dec}$ off \ATen{}'s atomic-chain prediction by more than $0.5\%$, tensor-to-scalar ratio inconsistent with $0.0043$ in either direction) would falsify the $\pi$-locked cosmological inventory of \ATen{} \citep{ArisakaA10Cosmos}. Specifically: $\Omega_m = 1/\pi$, $\Omega_\Lambda = 1 - 1/\pi$, $\Omega_b = 1/(2\pi^2)$, $z_{\rm sw} = e^7$ (with $z_{\rm dec}$ derived by the atomic chain), $r = 0.0043$ are forced by the framework's $\pi$-and-$e$ topological-invariant architecture. A failure of even one of the four would already be a serious blow to that architecture; failure of all four would falsify it as a unit.

\emph{The biological stratum.} Failure of \textsf{F21} (bioenergetic ratio away from the integer 2 at $> 5\sigma$) would falsify the layer-7 cascade attractor of the present paper, specifically the bioenergetic-quantum theorem. The framework would have to either (i) move the integer multiplier from 2 to a different integer (cost: the factor of 2 is forced by the two-fold \QB{}-quantum cost of breaking and solvating the phosphate bond, so changing the integer requires changing the bond-multiplicity model), or (ii) abandon ATP as the universal energy currency and introduce a different one (cost: ATP universality is itself a falsifiable prediction at the biological stratum; abandoning it would falsify the universal-identity insight at layer 7).

\paragraph{The universal-identity insight in catastrophic-failure mode.}
The universal-identity insight --- the centerpiece of the cohort, articulated in \AFour{} \citep{ArisakaA4Euclid} and recapitulated in every consequence paper of the program --- says that the \GRIP{} cascade applied to the constitutional starting point produces a \emph{unique} stable \GGA{} at each cascade layer. Total empirical failure of the Top 10 Falsifiers would falsify the universal-identity insight at \emph{all five} of the strata the ten falsifiers cover: the gauge stratum, the dark sector, the cosmology block, the precision Standard Model, and the biological stratum. At that point the framework would not be salvageable in its present form: the recursive uniqueness theorem of \AEleven{} (Theorem~\ref{thm:self-referential-closure}) is conditional on the universal-identity insight holding at each cascade layer, and total empirical failure would falsify the universal-identity insight at the five strata the framework can empirically test. What survives at the remaining strata (layers 5, 8--11: polymer folding, neural ring attractor, allocentric map, conscious episode, articulation) would be theoretically interesting but empirically unanchored, and would not constitute a framework worth defending.

\paragraph{Honest acknowledgment: no single failure is fatal in isolation; coherent patterns of failure are.}
The above paints the worst-case scenario. In practice, the framework's failure mode is more granular: any single failure can in principle be Conditional-promoted to Asserted via a structural-lemma cost increase --- for instance, a marginal drift in a single cosmological parameter might be absorbed into the error budget of its named conditional lemma. Such absorptions are themselves structural costs (the cohort's lemma-promotion roadmap, Appendix~\ref{app:lemma-promotion-roadmap}, becomes longer; the conditional-to-asserted promotion roadmap becomes more conditional) but they are not fatal. What \emph{is} fatal is a coherent pattern: simultaneous failure of \textsf{F15} (KK lock-cluster), \textsf{F14} (Kaxion), and \textsf{F7} (recombination-era $z_{\rm dec}$), or simultaneous failure of the whole cosmology block (\textsf{F6}, \textsf{F7} and \textsf{F1}), would falsify the framework at the \emph{cluster level} --- that is, would constitute evidence that the cascade architecture is wrong, not that any single sub-theorem is wrong. Cluster-level falsification is the framework's true exposure; single-test failure is closer to a routine empirical correction.

\paragraph{Why the framework should lose if the universe disagrees.}
A reader skeptical of any framework that anticipates its own failure should note: the framework's claim is not that it cannot fail. The framework's claim is that, if it fails, it fails as a unit, and the next attempt should learn what failed and renounce what made the failure possible. The Euclidean discipline of zero continuous fitted prefactors is the framework's most distinctive commitment; if the Top 10 Falsifiers falsifies the framework, the lesson for the next attempt is that the Euclidean discipline was either applied to the wrong starting axioms or was insufficient to force the world's structure. Either lesson is informative, and a framework that delivers an informative falsification is doing the job of physics correctly even in the act of failing. This is the asymmetric-payoff structure we will return to in the Bayesian framing of \S\ref{subsec:bayesian-framing} and the closing reflection of \S\ref{subsec:plain-english-close}.

\paragraph{Plain-English close.}
The framework should lose if the universe disagrees with the predictions the framework has made. A framework that wins regardless of what the data say is unfalsifiable, and unfalsifiable theories are, in the strict Popperian sense, not physics. The hostile-reviewer audit is the framework's exercise in admitting honestly what the loss would look like. The reader who finds the loss scenario unrecoverable should weigh the framework accordingly. The reader who finds it recoverable should weigh the framework accordingly too. Either weighting is informed; that is what the present subsection is for.

% --------------------------------------------------------------------
\subsection{Cross-test correlations: the ten falsifiers are not eight independent dice}
\label{subsec:cross-test-correlations}

\paragraph{Plain-English opening.}
Part~\ref{part:results} \S\ref{sec:eight-test-pull-horizon} described the ten falsifiers as ``independent in the strong sense'': different experimental apparatus, different systematic-error budgets, different cascade layers. That description was correct as far as it went, but it concerned \emph{experimental} independence: no two tests share a calorimeter or a haloscope or a telescope. It did not concern \emph{theoretical} independence. At the deepest level, several of the ten falsifiers partially falsify the \emph{same upstream theorem}, and a careful reader should be aware that the framework's exposure is not eight independent rolls of the dice but a smaller number of effective rolls --- roughly six or seven, depending on how one counts the correlations.

\paragraph{The three correlation clusters.}
Three correlation clusters organize the ten falsifiers at the theorem level.

\emph{Cluster A: the radial structure of \ATwo{} \citep{ArisakaA2GG6}.} \textsf{F15} (KK lock-cluster at $\Mlock{} = 3.85$~TeV) and \textsf{F14} (Kaxion DM at $8.1\,\mu$eV) both depend on the radial structure of the \BIsix{} bulk. The KK lock-cluster mass is set by the compactification radius $R$ of the radial interval $I_r$, with $\Mlock{} \sim 1/R$ modulated by the scale profile. The Kaxion decay constant $f_a$, and through it the Kaxion mass $m_a$, is set by the compact-angle radius and the scale profile in coupled fashion. If the radial structure of the bulk is wrong --- say, if the scale depth is not the integer $K = 35$ but something else --- then both predictions will fail correlatedly. \textsf{F15} and \textsf{F14} are thus theoretically a single test of the radial geometry, even though they are experimentally independent.

\emph{Cluster B: the $\pi$-and-$e$-locked cosmological inventory of \ATen{} \citep{ArisakaA10Cosmos}.} \textsf{F6} (the derived Hubble background $66.6 \pm 0.2$~km/s/Mpc, against which both the CMB reference and the local ladder are read), \textsf{F7} (recombination-era $z_{\rm dec}$; switch $e^7$), and \textsf{F1} ($r = 0.0043$) all depend on \ATen{}'s $\pi$-locked inventory in coupled fashion. $\Omega_m = 1/\pi$ enters $H_0$ via the Friedmann equation; the recombination-era pair enters $H_0$ via the corridor switch and the atomic chain's inventory inputs; $r = 0.0043$ enters via the same inflation-tensor closure that fixes the late-time geometry. A failure of one of the four would not necessarily collapse the cluster, but a failure of two of the four would, and a failure of three or four would constitute a falsification of the $\pi$-and-$e$ topological-invariant architecture in toto. At the theorem level, \textsf{F6}, \textsf{F7} and \textsf{F1} are correlated through their shared dependence on \ATen{}'s cosmological-inventory theorem.

\emph{Cluster C: the \OGNfive{} integer ledger on the Standard-Model substrate, from \ANine{} \citep{ArisakaA9SM}.} The family-number commitment ($N = 3$ in the ledger) and the Standard-Model precision audit (the 28-parameter sheet of \ANine{}) both depend on the \OGNfive{} integer ledger applied at the Standard-Model substrate. $N = 3$ is the family-number integer; the precision audit relies on the $K = 35$ scale depth (shared with Cluster~A), the $J = 7$ dyadic-closure exponent (which sets $\alpha^{-1}(M_Z) = 2^7 = 128$), and the topological rationals $\sin^2\theta_W = 25/108$ and $\alpha_s(M_Z) = 1/(\pi e)$. Both --- and, through the $K = 35$ scale depth, partially also \textsf{F15} --- depend on the \OGNfive{} ledger, and they are correlated at the theorem level through that shared dependence.

\paragraph{The effective number of independent tests.}
Counting the three correlation clusters: cluster A (\textsf{F15} + \textsf{F14}) is one effective test of radial geometry; cluster B (\textsf{F6}, \textsf{F7}, \textsf{F1}) is one effective test of the cosmological inventory; cluster C (the family-number commitment and the Standard-Model precision audit, plus partial overlap with cluster A) is one effective test of the \OGNfive{} ledger at the Standard-Model substrate. \textsf{F21} (the bioenergetic quantum) sits outside all three --- it is the only commitment at the biological stratum --- and contributes a fourth. The effective count across the eight rows tracked here is therefore four cluster-level independent tests, not eight. A reader who counts headline commitments as independent rolls of the dice is overestimating the framework's empirical exposure by roughly a factor of two.

\paragraph{Why this matters for assessing framework strength.}
A framework that claims eight independent successful tests is, naively, much harder to dismiss than a framework that claims three independent successful tests; the prior-odds of ten independent confirmations under the null hypothesis (framework is wrong) is roughly $(\text{single-test probability})^{10}$, much smaller than $(\text{single-test probability})^3$. But this naive multiplication applies only to independent tests. Three cluster-level independent tests (with the bioenergetic check as a near-cluster) is still a substantial empirical commitment --- comparable in scale to the standard tests of general relativity in the first half-century after 1915 (gravitational redshift, perihelion of Mercury, light bending, gravitational time delay, gravitational waves) --- but it is not eight. Readers should weigh the framework's commitment as approximately three to four cluster-level checks, plus the bioenergetic-stratum check, plus a finer-grained internal-consistency check at the precision-Standard-Model level. This is still substantial; it is not eightfold.

\paragraph{Cross-cluster correlations and the cluster-failure modes.}
The clusters themselves are not perfectly independent. Cluster A (radial geometry) and cluster C (the \OGNfive{} ledger) both depend on \ATwo{} \citep{ArisakaA2GG6}'s bulk geometry; failure of \ATwo{}'s bulk geometry would falsify both clusters simultaneously. Cluster B (cosmological inventory) and cluster C (Standard-Model ledger) both depend on \AFour{} \citep{ArisakaA4Euclid}'s Euclidean discipline; failure of \AFour{}'s discipline would falsify both. The cluster-level structure of the Top 10 Falsifiers is thus a layered correlation: tests within a cluster share a deep theorem dependence, and clusters share a deeper constitutional-paper dependence. The framework's true exposure is, at the deepest level, exposure to the constitutional papers \AOne{}--\AFour{}; the ten falsifiers are the empirical projections of that exposure onto the consequence-paper layer.

\paragraph{Plain-English close.}
The ten falsifiers are not ten independent checks on a single claim. They are projections of three or four deeper claims onto the experimental apparatus. That is the right shape for a unified framework: a collection of unrelated predictions would not be a framework at all. The correlation structure is the framework's coherence made visible. The accounting follows: empirical exposure is weighed at the cluster level, not at the row level of the Pull table.

% --------------------------------------------------------------------
\subsection{Bayesian framing: where each test lives on the information-gain ladder}
\label{subsec:bayesian-framing}

\paragraph{Plain-English opening.}
A reader trained in Bayesian inference will notice that the framework's commitment to a Pull-table audit is a strong frequentist commitment but does not yet name the Bayesian prior on each test. The Bayesian question is: \emph{before} the data arrive, what does the framework think the probability is that each test will return a value within the predicted window? And, on the standard Bayesian update rule, by how much should the framework's overall posterior --- the probability that the framework is correct --- shift upon observing each test's outcome? The standard answer is that the Bayesian posterior shifts by a factor proportional to the likelihood ratio $P(D \mid H_{\rm framework}) / P(D \mid H_{\rm null})$, where $D$ is the observed datum. For tests where the framework predicts a specific value with a tight tolerance, that ratio can be large (the framework strongly favors one outcome; the null hypothesis is indifferent), and the posterior moves substantially. For tests where the framework predicts a value already at consensus with mainstream measurements (e.g.\ the family-number commitment, where the prediction matches PDG to $< 0.5\%$), the Bayesian information gain of the test is small --- the framework predicts what everyone already knows, so the test does not separate the framework from the null.

\paragraph{Where each test lives on the information-gain ladder.}
Ranking the ten falsifiers by expected Bayesian information gain --- where ``high'' means the test, when it resolves, will substantially move the posterior, and ``low'' means the test will leave the posterior approximately unchanged --- gives the following structure.

\emph{Highest Bayesian information.} \textsf{F7} (the recombination-era redshift test) carries the largest expected information gain. In \ATen{}'s two-quantity form the framework derives the decoupling redshift itself from its own atomic chain with zero fitted parameters, presently on the Planck value at the $0.1\%$ level; the corridor switch $e^7 \approx 1096.6$ is a distinct topological marker $0.6\%$ above in redshift (equivalently $\delta = 0.0041$ in e-folds), with no observational handle of its own. CMB-S4 + LiteBIRD will resolve $z_{\rm dec}$ to better than $0.1\%$ by 2033: a measurement that stays on the atomic-chain value confirms a zero-parameter derivation at its own systematic floor, while a measurement off it by more than $0.5\%$ kills the reading outright --- the framework cannot absorb the miss, because the chain has no dial. The likelihood ratio at the resolution-precision is large in either direction.

\emph{High Bayesian information.} \textsf{F15} (KK lock-cluster) and \textsf{F14} (Kaxion DM) carry substantial information gain. Both are forward predictions --- the relevant signal has not been observed at all yet --- so the resolution will be either a $5\sigma$ detection at the predicted parameters (framework wins decisively) or a $5\sigma$ non-detection across the predicted parameter window (framework loses decisively). The likelihood ratio at resolution is large, but the resolution is binary (detect/no-detect) rather than a continuous shift in posterior.

\emph{High Bayesian information.} \textsf{F6} (the derived Hubble background) carries more information than its position in the cosmology block suggests, and the reason is worth stating: the framework derives a background near $66.6$ and does \emph{not} derive the local ladder's $73$. A standard-siren catalog is a direct measurement of the background, and it discriminates. If the sirens converge near $67$, the derived background is confirmed against a number the framework did not fit; if they converge near $73$, the derivation is wrong and the framework says so in advance. \textsf{F1} (tensor-to-scalar $r$) carries moderate information by comparison. \textsf{F1} will improve the upper bound on $r$ by a factor of $\sim 10$ but the framework's prediction is the definite value $r = 0.0043$; the test is now two-sided (a detection inconsistent with $0.0043$ in either direction falsifies, and a confirming detection at the predicted value would be strong positive evidence).

\emph{Lowest Bayesian information.} The family-number commitment ($m_\tau/m_\mu$) and \textsf{F21} (the bioenergetic quantum) carry the smallest expected information gain. Both already sit at agreement: the framework's prediction matches measurement to better than $0.5\%$ on the family-number ratio, and the bioenergetic quantum carries a Pull of $+1.0$ at a $1.6\%$ offset (Part~\ref{part:results} Table~\ref{tab:eight-test-pull-table}) --- a $\sim 1\sigma$ landing, well inside the $5\%$ kill threshold. Future measurements at HL-LHC (the family-number ratio) and single-molecule biophysics (\textsf{F21}) will improve the precision by factors of three to ten, but absent a surprise, the tests will remain at agreement. The framework's posterior will not move substantially on these tests \emph{unless} a surprise occurs (a fourth charged-lepton generation; an ATP-hydrolysis ratio away from the integer 2).

\paragraph{What the Bayesian framing implies for reading the decade.}
A reader watching the decade should weight the tests by their expected information gain. The commitment to watch most closely is \textsf{F7} (the measured $z_{\rm dec}$ vs \ATen{}'s atomic-chain prediction; the switch $e^7$ is a distinct topological marker). The ones to watch next are \textsf{F15} and \textsf{F14} (the forward KK-cluster and dark-sector predictions). The ones to watch with moderate attention are \textsf{F6} (the Hubble background, where the siren catalog is the discriminating measurement) and \textsf{F1}. The ones to watch only for surprise are the family-number commitment and \textsf{F21} (already at agreement). The most boring tests --- ``boring'' in the Bayesian-information-theoretic sense, not in the scientific sense --- are precisely the tests where the framework's prediction has already matched measurement; they are nonetheless the tests where the framework is most certainly already calibrated, and a reader who weighs framework credibility on multiple consistencies should remember that ``boring'' agreement is not zero evidence. Agreement at $0.1\%$ across a multi-decade independent prediction is itself a substantial Bayesian update; it just does not move the posterior at the moment of the next experiment, because the previous experiments already moved it.

\paragraph{The asymmetric-payoff structure.}
The Bayesian framing also makes visible the asymmetric-payoff structure that the framework operates under. The framework's claim is unified: all ten falsifiers are projections of the same recursive uniqueness theorem. If all ten confirm, the framework's posterior moves to near unity --- the framework is, to all practical purposes, the standard reference for the next half-century of physics. If \emph{any} test from the high-information group decisively falsifies, the framework's posterior moves to near zero --- the framework is wrong, and the question of which sub-theorem failed becomes a forensic-physics question rather than a viable-framework question. This is the asymmetric payoff: success is across-the-board success; failure is at-the-weakest-link failure. The Bayesian rule is harsh on unified frameworks for exactly this reason, and the framework's authors accept the harshness as the cost of unification.

\paragraph{Plain-English close.}
The Bayesian framing is the framework's discipline made quantitative. A framework that derives $z_{\rm dec}$ with zero parameters and is willing to lose if the measured value moves off it by half a per cent has placed a Bayesian wager: the prior is on the framework's side, but the prior can move sharply. The wager's expected information gain is what makes the next decade scientifically interesting; the asymmetric payoff is what makes the wager honest.

\subsection{Row 5 in the two-quantity form: the recombination-era test}
\label{ssec:zstar-path-to-closure}

Row 5 of the Top 10 Falsifiers Pull table is carried in the two-quantity form of \ATen{} \citep{ArisakaA10Cosmos}. Two distinct quantities share the recombination era. The \emph{corridor switch} $z_\ast = e^7 \approx 1096.6$ is a topological marker forced by the integer $J = 7$ of the \OGNfive{} ledger --- exact, parameter-free, and with no observational handle of its own. The \emph{operational decoupling redshift} $z_{\rm dec}$ --- the visibility peak Planck measures at $1089.92 \pm 0.25$ --- is a different quantity, and \ATen{} derives it from the framework's own atomic chain (the Rydberg from the particle sector, the baryon fraction $\Omega_b = 1/(2\pi^2)$, the corridor $h$) with zero fitted parameters, landing on the measured value at the $0.1\%$ level, inside the $0.18\%$ modeling systematic of the recombination calculation itself. The switch--decoupling offset is computed, not conjectured: $\delta = J - \ln(1+z_{\rm dec}) = 0.0041$ ($0.4\%$ in e-folds). No $27\sigma$ bookkeeping arises, because $e^7$ is not offered as the value of $z_{\rm dec}$ in the first place.

\paragraph{The kill, posed on the observable.}
Row 5's falsifier lives on the one measurable quantity: a future model-independent determination of $z_{\rm dec}$ (CMB spectral-distortion missions reading the cosmological recombination radiation, post-2030, complemented by 21-cm cosmology) refutes the reading if the measured value sits off \ATen{}'s atomic-chain prediction by more than $0.5\%$ --- equivalently, if the measured $\ln(1+z_{\rm dec})$ departs from the computed $6.996$ by more than $0.005$. Present agreement is $0.1\%$. The switch--decoupling offset is a derived consistency statement, not an independent kill, since $z_{\rm sw}$ carries no observational handle of its own.

\paragraph{The remaining open item is a promotion, not a rescue.}
The offset obeys the identity $\delta = J - \ln(B/kT_0) + \ln 53.0$, in which the hydrogen binding energy $B$ and the Saha--Peebles factor $53.0$ are already derived; the only quantity not yet owned by the ledger is the present temperature $T_0$. If \ATen{}'s commissioned e-fold-budget campaign derives $T_0$ (equivalently the photon entropy normalization), $\delta$ becomes a theorem --- the exact analogue of the $K = 35$ resolution of the why-now coincidence. Until it lands, $\delta = 0.0041$ is carried as a computed consistency. The row's conditional bookkeeping is inherited from the fractions the atomic chain consumes: it rides on \ATen{}'s closed-packet-integration lemma (Conjecture VIII there), and the atomic chain itself introduces no new lemma.

\paragraph{What this means for the Top 10 Falsifiers.}
$z_\ast$ remains a precision audit carried in the body wherever it appears, now with its kill posed on $z_{\rm dec}$ alone. The Top~10 headline falsifiers --- eight forward bets and two precision audits --- do not include $z_\ast$ and are unaffected; the abstract and Executive Summary commit to those ten.

The discipline of the two-quantity form is what distinguishes an honest framework from a self-protecting one. A framework that reads a $0.6\%$ switch--Planck gap as confirmation of a precision prediction has not yet earned the right to be tested; a framework that derives the operational decoupling redshift itself with zero parameters, states the switch--decoupling offset as a computed number, and poses its kill on the measurable alone has stated its terms of engagement. The next decade measures $z_{\rm dec}$ to $0.1\%$; if it moves off the atomic-chain value by more than half a per cent, row 5 falls, and the framework will report it honestly when it arrives.


% --------------------------------------------------------------------
\subsection{A cluster-level falsification scenario: walking through a coordinated failure}
\label{subsec:cluster-falsification}

\paragraph{Plain-English opening.}
We have considered the worst-case scenario of total empirical failure (\S\ref{subsec:hostile-reviewer-audit}) and the best-case scenario of total empirical success (implicit in \S\ref{subsec:bayesian-framing}). Between these two extremes lies the more realistic middle: a coordinated failure of three or four tests that all sit within the same correlation cluster. This subsection walks through what such a cluster-level failure would mean for the framework, focusing on the scenario that is, by the cross-test-correlation analysis of \S\ref{subsec:cross-test-correlations}, the most likely failure mode: failure of the cosmology cluster (\textsf{F6}, \textsf{F7}, \textsf{F1}) while the gauge and bioenergetic strata pass their tests.

\paragraph{The scenario.}
Suppose that by 2034, the following pattern has emerged. HL-LHC by 2032 has observed a KK lock-cluster at $\Mlock{} = 3.85 \pm 0.10$~TeV with $5\sigma$ significance and the predicted five-particle multiplicity (\textsf{F15} passes). ADMX-EFR + IBS-CAPP by 2030 has observed a Kaxion-like signal at $m_a = 19.2 \pm 0.5\,\mu$eV with the predicted axion-photon coupling (\textsf{F14} passes). Single-molecule biophysics by 2034 has measured $\Delta G_{\rm ATP}/(2 N_A \QB{}) = 1.002 \pm 0.005$, well within the predicted unity at $5\sigma$ confidence (\textsf{F21} passes). The family-number commitment passes by 2032 from HL-LHC precision Yukawa data and a precision-improved $m_\tau/m_\mu$ measurement.

But the cosmology block fails coordinatedly. CMB-S4 by 2032 reports a background $H_0 = 63.0 \pm 0.3$~km/s/Mpc, well below the derived $66.6 \pm 0.2$ (\textsf{F6} fails by $\sim 12\sigma$ on the low side). It is worth being explicit about what would \emph{not} be a failure here, because the direction is the whole content of the test: a standard-siren catalog reporting $71.0 \pm 0.5$ would sit between the derived background and the local ladder, closer to the background than to $73$, and would count as a partial move \emph{toward} the framework --- the discriminating outcome is a catalog at the ladder's $73$, which would kill the corridor outright. CMB-S4 + LiteBIRD by 2033 report $z_{\rm dec} = 1085 \pm 1$ from a model-independent recombination-history extraction, $0.5\%$ below the atomic-chain prediction $\approx 1091$ (\textsf{F7} fails decisively). And CMB-S4 + LiteBIRD by 2033 report $r$ consistent with zero, $r < 10^{-3}$ at 95\% CL --- excluding the framework's predicted $r = 0.0043$ (\textsf{F1} fails).

\paragraph{What such a coordinated failure would imply.}
A coordinated failure of all four cosmology tests would constitute cluster-level falsification of the $\pi$-and-$e$-locked cosmological inventory of \ATen{} \citep{ArisakaA10Cosmos}. Specifically: $\Omega_m = 1/\pi$ would be falsified by the $H_0^{\rm early}$ failure (the Friedmann equation links $\Omega_m$ and $H_0$ tightly); the atomic-chain derivation of $z_{\rm dec}$ (and with it the two-quantity reading anchored at $z_{\rm sw} = e^7$) would be falsified by the recombination-redshift failure; the no-observable-tensor-modes prediction would be falsified by the $r$ detection; and the late-stage Hubble corridor would be falsified by the $H_0^{\rm late}$ failure. The cluster level of the failure --- all four tests failing in the same direction, each by $> 4\sigma$ --- would make the failure pattern very difficult to absorb into a structural-lemma cost increase. No single conditional lemma can recover four simultaneous cosmological failures; the failure is forced to be read as falsification of \ATen{}'s cosmological inventory in toto.

\paragraph{The universal-identity insight in the cosmology stratum would fall.}
The framework's universal-identity insight at the cosmological stratum says that the \GRIP{} cascade applied to the constitutional starting point produces a unique stable cosmological \GGA{}, and that the cosmological inventory ($\Omega_m, \Omega_\Lambda, \Omega_b, z^*, r$) is forced by the integer ledger plus the bulk geometry. Four simultaneous failures would falsify the cosmological universal-identity insight: the data would show that the cosmological inventory is \emph{not} forced by the framework's chosen starting point. Either the starting point is wrong, or the \GRIP{} cascade applied to it does not produce a unique attractor at the cosmological stratum, or both.

\paragraph{What could be salvaged.}
In the scenario described, layers 1--4 of the cascade (gauge, electroweak, atomic, molecular) and layer 7 (bioenergetic) are passing their tests; the cosmological stratum (layer 1 in the constitutional ordering --- the \BIsix{}-bulk-anchored cosmological background) is failing. The cascade machinery from layer 6 (polymer folding) through layer 18 (articulation) does not depend, in its qualitative structure, on the specific values of the cosmological parameters; it depends on the \emph{existence} of a $\Lambda$CDM-like cosmological background sufficient to support a long-lived stable epoch in which molecular and biological complexity can accumulate. If the cosmological background is a different $\Lambda$CDM-like model (one not matching \ATen{}'s $\pi$-locked inventory), the cascade machinery for life and mind could survive --- the same radial geometry that produces the gauge stratum (passing) and the same bioenergetic quantum that produces ATP (passing) could in principle still organize the upper layers. But the unified theme of the trilogy --- the claim that \emph{the same} \BIsix{}-bulk geometry plus the same \OGNfive{} integer ledger plus the same Euclidean discipline forces the cosmos, life, and mind together --- would dissolve. The framework would survive as a partial framework: it would have made successful predictions in the particle-physics and biological strata, but its cosmological claim would be falsified.

\paragraph{The recursive uniqueness theorem in cluster-failure mode.}
The recursive uniqueness theorem of \AEleven{} (Theorem~\ref{thm:self-referential-closure}) is conditional on the constitutional axioms holding at the cosmological stratum. Cluster-level failure of the cosmological tests would falsify the cosmological-stratum content of \OGGEP{} (operational projection) at the constitutional level --- specifically, the operational projection of the bulk geometry onto the 4D cosmological background. The recursive uniqueness theorem would have to be revised to be conditional on a different cosmological-stratum content: ``\emph{given} a cosmological background of currently-unknown structure, the cascade machinery from molecular to articulating layers proceeds uniquely.'' This would be a substantial demotion of the theorem's universal claim --- from ``geometry forces cosmos, life, and mind'' to ``geometry forces life and mind, given a cosmos.'' The trilogy's unified ambition would be reduced.

\paragraph{Is the salvage worth pursuing?}
A reader sympathetic to the framework might ask whether the salvage is worth pursuing. Two considerations. Affirmative: the cascade machinery for life and mind would still constitute a substantial achievement, comparable to a unified framework for biology plus consciousness, and the framework would have made testable predictions at the gauge and biological strata that confirmed. Negative: the framework's distinctive claim --- the unity of cosmos, life, and mind under a single geometric apparatus --- would be lost, and the residual framework would be less distinctive than its present form. The author's preliminary view, recorded here for honesty, is that cluster-level failure of the cosmology block would constitute the framework's substantive defeat, even if technical salvage is possible at the upper-layer strata. The framework was constructed to be unified; partial unification is a structural retreat, not a stable resting place.

\paragraph{Plain-English close.}
The cluster-level falsification scenario is the framework's intermediate-honest-case. Total failure is the worst case; total success is the best. Cluster-level failure of one stratum --- with the other strata passing --- is the most likely middle scenario, and it would force the framework into a salvage operation that the author considers less than worth pursuing. The reader weighing the framework should weigh not only the success and total-failure scenarios but also the cluster-level intermediate, and decide on that basis how much weight to place on the framework's current presentation.

% --------------------------------------------------------------------
\subsection{Per-test deeper commentary: complementary to Part~\ref{part:results} \S\ref{sec:eight-test-pull-horizon}}
\label{subsec:per-test-deeper-commentary}

\paragraph{Plain-English opening.}
Part~\ref{part:results} \S\ref{sec:eight-test-pull-horizon} gave for each test the predicted value, the observed (or current-bound) value, the Pull, the decision threshold, and the experimental facility plus calendar window. The present subsection adds, for each of the ten falsifiers, three additional layers of commentary that did not fit in Part~\ref{part:results}'s empirical-audit register. First: the \emph{political and institutional context} --- which collaboration is custodian of the relevant measurement, what the funding outlook is, what the schedule risks are. Second: the \emph{framework's confidence level} --- how certain the framework is that its prediction is right versus how certain it is that the relevant experiment is right. Third: the \emph{historical analogy} --- which comparable predictions in physics's history have been similarly staked and how those staking-decisions resolved. One or two paragraphs per test.

% ---------- F15 ----------
\paragraph{\textsf{F15} --- KK lock-cluster at $\Mlock{} \approx 3.85$~TeV.}
The custodian collaborations are ATLAS and CMS at the HL-LHC, both with funded programs running into the early 2030s. The funding outlook for HL-LHC is secure through Run~5 (CERN's decadal plan was approved in the 2020 European Strategy for Particle Physics update and reaffirmed in subsequent reviews). The schedule risk is moderate: HL-LHC has historically slipped by twelve to eighteen months from baseline at each major milestone, and a slip of that magnitude is plausible for the decisive run.

The framework's confidence in the prediction is high --- $\Mlock{}$ is forced by the integer ledger $K = 35$ and the scale-profile closure of \ATwo{} \citep{ArisakaA2GG6} --- and the experiment's confidence in its sensitivity at $3.4$--$4.2$~TeV with $\sim 3\,$ab$^{-1}$ is high (the relevant five-particle multiplicity at $\sim 3.85$~TeV is comfortably above the QCD continuum at the predicted mass and is well within both detectors' acceptances).

Historical analogy: this prediction is comparable in commitment-scale to the W- and Z-boson mass predictions of electroweak unification in the 1970s, which were predicted to $\sim 1\%$ before discovery at the CERN SppS in 1983. The historical lesson is that a specific TeV-scale prediction, made in advance with an integer-ledger justification, is the kind of prediction that physics has previously vindicated when the constitutional theorem is right. The W and Z were not predicted from numerology; they were predicted from a unified gauge structure. $\Mlock{}$ is, the framework claims, the same kind of prediction.

% ---------- F14 ----------
\paragraph{\textsf{F14} --- Kaxion DM at $m_a \approx 8.1\,\mu$eV.}
The custodian collaborations are ADMX-EFR (the next-generation extension of the Axion Dark Matter eXperiment at the University of Washington / Fermilab) and IBS-CAPP (the Center for Axion and Precision Physics haloscope in Korea, covering $1$--$8$~GHz). The funding outlook for both is secure through the late 2020s, with milestones structured around achieving cosmology-relevant axion-photon coupling sensitivity in the $\mu$eV mass window. The schedule risk is low to moderate: dedicated haloscope experiments typically meet their sensitivity goals within a year or two of plan. The framework's confidence is high --- $m_a \approx 8.1\,\mu$eV is forced by the Kaxion decay constant $f_K \simeq 7.1\times10^{11}$~GeV and the present-day relic abundance, as proven in \PaperFullCOne{} \citep{ArisakaC1Kaxion} --- but the prediction is a forward prediction, meaning the framework wins or loses decisively at the experimental sensitivity. Historical analogy: the prediction is comparable to the gravitational-wave amplitude predictions made for binary neutron-star mergers in the late 1990s, where the prediction (a specific signal at a specific frequency band) was made many years in advance of the detection (GW170817 in 2017). The historical lesson is that a specific dark-sector signal at a specific frequency, if predicted by an integer-ledger-justified geometry, is the kind of prediction that physics typically resolves with a dedicated experiment within a decade of construction.

% ---------- F6 (early) ----------
\paragraph{\textsf{F6} --- the derived background $H_0 = 66.6 \pm 0.2$~km/s/Mpc.}
The custodian collaboration is CMB-S4 (a U.S. Department of Energy + National Science Foundation joint project, with first light targeted for 2031). The funding outlook is secure but the schedule has slipped from initial 2030 targets. The framework's confidence is moderate-to-high, and the ground of it should not be overstated: the derived background is close to, but not identical with, Planck's measured $67.4 \pm 0.5$, and the two are different objects --- one derived, one measured. The test is not a self-consistency check at improved precision. The experiment's confidence is high in achieving $\pm 0.1$~km/s/Mpc by 2032. Historical analogy: the prediction is comparable to the cosmological-constant-density predictions made for $\Lambda$CDM in the late 1990s after the Type-Ia supernova results, where the predicted values were already approximately at consensus and the subsequent precision improvements (WMAP, Planck) primarily sharpened the consensus. This is a low-information test in the Bayesian sense (\S\ref{subsec:bayesian-framing}) but a high-credibility check on framework calibration.

% ---------- F6 (late) ----------
\paragraph{\textsf{F6}, the late reading --- a derived lift to $\approx 68.5 \pm 0.6$, and a named residual.}
The custodian collaborations are LIGO/Virgo/KAGRA for standard-siren measurements (O5 run 2027--2029, O6 run starting 2030), LSST at the Vera Rubin Observatory for cosmography (Y1 data 2026, full Y10 by 2035), and JWST for high-redshift surveys. The funding outlook is secure across all three. The schedule risk is moderate (especially for the O6 run, where the upgrade depends on the technical success of the O5 run). The framework's position here is more exposed than on the background, and the honest statement of it is the one \PaperCTwo{} \citep{ArisakaC2Hubble} makes: the background plus the Local-Hole lift gives a late reading of $\approx 68.5 \pm 0.6$, graded conditional; the distance to the locally measured $73.5$ is a named residual that the framework \emph{does not derive} and records as an open problem rather than a result. Nothing in the cohort predicts the ladder value. What the framework stakes on this decade is the opposite claim --- that a standard-siren catalog, which measures the background directly, will converge near $67$ and not near $73$. The Hubble-corridor prediction is the framework's late-time analogue: a specific deviation from $H_0^{\rm early}$ at a specific magnitude ($+5.6$~km/s/Mpc), predicted from a specific underlying structure (Kaxion-DM modulation of late-time geometry).

% ---------- F7 ----------
\paragraph{\textsf{F7} --- recombination $z^* = e^7$.}
The custodian collaborations are CMB-S4 + LiteBIRD (JAXA + ISAS, with European and U.S. partners). Funding for LiteBIRD is secure subject to JAXA's H3 launch cadence; LiteBIRD baseline launch is 2030. The framework's confidence rests on \ATen{}'s two-quantity form: the corridor switch $z^* = e^7$ is exact at the integer-ledger level, and the operationally-defined decoupling redshift $z_{\rm dec}$ is derived from \ATen{}'s own atomic chain with zero fitted parameters, on the Planck value at the $0.1\%$ level. The $0.6\%$ switch--Planck gap is not a tension to be defended but a computed offset ($\delta = 0.0041$ in e-folds), whose promotion to a theorem waits on \ATen{}'s e-fold-budget campaign; the kill for \textsf{F7} lives on $z_{\rm dec}$ alone --- a model-independent determination off the atomic-chain value by more than $0.5\%$. Historical analogy: the prediction is comparable to Dirac's prediction of the electron's magnetic moment $g = 2$ in 1928, which agreed with measurement to a small per-cent at first and was later refined (by QED) to better than parts in $10^{12}$. The lesson carries over: a structurally locked leading value acquires its computed linkage to the operationally-defined observable --- exactly the role \ATen{}'s atomic chain now plays for the recombination era. This is the framework's posture on \textsf{F7}.

% ---------- F1 ----------
\paragraph{\textsf{F1} --- tensor-to-scalar ratio $r = 0.0043$.}
The custodian collaborations are CMB-S4 + LiteBIRD. Funding outlook same as \textsf{F7}. The prediction is sharp and two-sided: the derived plateau fixes $r = 12/N^{*2} = 0.0043$, so the framework loses if a detection lands well above that value \emph{and} loses if a null pushes $r$ below $\sim10^{-3}$. CMB-S4 + LiteBIRD reach $\sigma(r)\sim0.5$--$1\times10^{-3}$ (delensing-dependent), enough to detect $0.0043$ at several $\sigma$ or to exclude it. Historical analogy: the prediction is comparable to the prediction of the absence of proton decay below a certain lifetime in early grand-unified theories of the 1980s --- a one-sided prediction that wins if the universe remains quiet and loses if a detection is made. The historical lesson is that one-sided predictions are weaker than two-sided ones in Bayesian terms (they constrain the framework less per measurement) but are nonetheless decisive on the failure side.

% ---------- family-number ----------
\paragraph{The family-number commitment --- $m_\tau/m_\mu$.}
The custodian collaborations are HL-LHC (ATLAS+CMS for $m_\tau/m_\mu$ via $H \to \tau\tau$ and $Z \to \mu\mu$ precision), and (longer-term) FCC-ee for higher-precision lepton-pair measurements. Funding outlook for HL-LHC secure; FCC-ee is at feasibility-study stage with no construction decision before 2030. The framework's confidence is very high: the prediction matches PDG by better than $0.1\%$ and the family-number-three claim is supported by decades of accumulated evidence (no fourth charged-lepton or neutrino observed at the LEP-2 lineshape). Historical analogy: the prediction is comparable to the Eightfold-Way predictions of Gell-Mann and Ne'eman in 1961, where the $SU(3)$-flavor structure predicted the $\Omega^-$ particle's quantum numbers in advance of discovery. The Eightfold-Way's vindication was decisive and led to the quark model; the family-number-three prediction's vindication, if it holds, will be a comparable decisive statement of an underlying group-theoretic structure.

% ---------- F21 ----------
\paragraph{\textsf{F21} --- bioenergetic quantum $\Delta G_{\rm ATP} = 2 N_A \QB{}$.}
The custodian community is the single-molecule biophysics community (no single collaboration; several laboratories at multiple universities perform refinement audits). The funding outlook is decentralized but stable (single-molecule biophysics has been a steady NIH funding line for two decades). The framework's confidence is high: the predicted unity matches measurement to $1.5\%$, and the integer factor of 2 is forced by the bond-multiplicity model. Historical analogy: the prediction is comparable to the prediction of $E = mc^2$ for nuclear binding energies in the 1930s --- a specific universal relation between two seemingly-unrelated quantities (the rest mass and the binding energy) that, once tested, became a standard reference for the entire field. The bioenergetic-quantum relation, if confirmed at $0.5\%$ accuracy, would similarly become a standard reference for the field of cellular bioenergetics, linking a specific bulk geometric scalar ($\QB{}$) to the universal energy currency of biology (ATP).

\paragraph{Plain-English close to the per-test commentary.}
After all, each of the ten falsifiers is not a stand-alone bet; it sits in a context of historical predictions that physics has previously staked and resolved, of experimental collaborations with their own institutional momentum, and of framework-internal confidence levels that vary by test. The reader who watches the ten falsifiers resolve over the next decade is also watching the framework's place in a longer history of theoretical predictions, and the per-test commentary above is the framework's preliminary positioning within that longer history.

% --------------------------------------------------------------------
\subsection{Plain-English close: the framework as a wager on physics}
\label{subsec:plain-english-close}

\paragraph{The wager.}
The framework, presented over twenty-two papers and roughly five thousand three hundred pages, has staked itself on the Top 10 Falsifiers. In the language familiar from Pascal's seventeenth-century wager on theology, this is a wager --- but a wager on physics rather than on faith. The wager's structure is asymmetric in the precise way Bayesian inference makes visible (\S\ref{subsec:bayesian-framing}): if all ten falsifiers pass within the next decade, physics has its first Euclidean closure since Newton's \emph{Principia} of 1687 --- a framework with zero continuous free parameters, all integer or topological-rational predictions justified by an upstream theorem, all empirical content vindicated by experiment. If the framework fails the Top 10 Falsifiers --- whether totally (\S\ref{subsec:hostile-reviewer-audit}), at the cluster level (\S\ref{subsec:cluster-falsification}), or piecewise --- it dissolves into a footnote in the longer story of attempts at unification, much as Kaluza--Klein theories of the 1920s and grand unified theories of the 1980s did before it. Either way, the framework's discipline (zero continuous fitted prefactors, the universal-identity insight, the integer ledger, the recursive uniqueness theorem) is the lesson the framework offers to the next attempt regardless of the outcome.

\paragraph{The author and the reader, both staking.}
The author of the present paper has spent fifty years arriving at the question and the last decade arriving at the framework's present form. The reader who has reached this Section has spent some hours, perhaps days or weeks, climbing the cohort. Both are now in the position of staking on the ten falsifiers. The reader stakes the time spent and the choice of whether to continue tracking the framework's empirical fate over the coming decade. The stakes are different in magnitude but the same in kind: both author and reader have invested attention in a specific framework, and both will see the empirical decade close before too long.

\paragraph{What the framework asks of the reader for the coming decade.}
The framework asks the reader to do four things over the coming decade. First: \emph{bookmark Table~\ref{tab:eight-test-pull-table} of Part~\ref{part:results}}, revisit it as each test resolves, and update the framework's standing accordingly. Second: \emph{weigh failures at the cluster level, not at the row level}, as the cross-test-correlation analysis of \S\ref{subsec:cross-test-correlations} prescribes. Third: \emph{when a test resolves in the framework's favor, recognize that it is one cluster-level data point and not the entirety of the framework's vindication}; the framework's full case requires the full Top 10 Falsifiers, not a few early successes. Fourth: \emph{when a test resolves against the framework, ask whether the failure is single-test or cluster-level} --- single-test failures are routine empirical corrections; cluster-level failures are framework-falsifications, and the response should differ accordingly.

\paragraph{The closing reflection.}
A theoretical framework worth fifty years of climbing makes itself fragile at the end. Fragility is not weakness. Fragility is the willingness to be falsified by the world. The framework's commitment to fragility is the Top 10 Falsifiers; the next decade will exercise the commitment. A framework that cannot be wrong has nothing to offer to physics; a framework that can be wrong, and that names the conditions under which it would be wrong with ten specific falsifiers in a specific decade, has earned its place at the empirical table whether or not the table eventually rules in its favor. The Top 10 Falsifiers is the framework's commitment to fragility. The next decade will exercise the commitment. Both the author who staked the framework and the reader who staked the time of reading it will, by the close of that decade, know how the universe ruled. Until then, the framework stands where the climb has left it: at the summit of its own argument, watching for the data that will either confirm the climb or send the next climber down a different ridge.

\section{The Ten Structural Innovations of the Origin Capstone}
\label{sec:ten-innovations}

The contribution of \AEleven{} is to bind the eleven-paper A-series and the three
phenomenology branches into a single derivational structure --- one cascade
engine, one selector, one ledger --- and to prove that the existence of observers
who articulate the geometry is a theorem of that structure and by no means an
accident permitted by it. We collect here the ten structural innovations the
capstone delivers, each carrying its cohort-global identifier
(\S\ref{sec:naming-convention}). None introduces a new continuous parameter; each
is a consequence of the \GGConstitution{} of \AOne{}.

\paragraph{Innovation 1 (I-Origin-1): Inevitability stated at theorem level.} The
recursive uniqueness theorem \textsf{T-Origin} (T-Origin-58.1) makes the
existence of physics-articulators the \emph{conclusion} of a formal theorem whose
hypotheses are the laws of physics themselves --- inevitability written down as a
proof obligation, not asserted as a slogan.

\paragraph{Innovation 2 (I-Origin-2): The \GRIPout{} cascade with
\GGA{}-as-ground-state.} The completing mechanism of the program: at each layer
the patient \GRIPin{} loop settles to the unique Ground Geometric Attractor
\GGA{}, and out-of-equilibrium shocks reshape the landscape to drive the
\GRIPout{} ascent to the next layer (T-Origin-13.1, T-Origin-14.1).

\paragraph{Innovation 3 (I-Origin-3): One engine across eighteen layers.} The same
four-move pipeline --- \DGI{}, the \OGGEP{}/\GSSB{} boundary projection, \GFF{}
gradient descent, and \GGA{} settlement --- runs unchanged from the gauge group
of the Standard Model to the meta-cognitive layer at which the constitution is
articulated.

\paragraph{Innovation 4 (I-Origin-4): The \QiFour{} admissibility selector forces
per-layer uniqueness.} The four information-theoretic pillars of \ASeven{} act as
the filter that singles out one admissible \GGA{} on each layer's Morse landscape
(\textsf{T-Qi4-GGA-Admissibility}, T-Origin-57.1; per-layer uniqueness,
T-Origin-63.4).

\paragraph{Innovation 5 (I-Origin-5): The universal-identity insight.} The integer
\OGNfive{} ledger of \ATwo{} threads one identity through atoms, molecules, ATP,
DNA, and the first cell: the gauge-equivalence of all layer-$n$ products
(\textsf{T-Universal-Identity}, T-Origin-63.5), made philosophically careful as
the type-token distinction --- the type is unique; the tokens are fully
individuated.

\paragraph{Innovation 6 (I-Origin-6): The bioenergetic quantum anchor.} The
Arisaka constant $\QB \approx 0.155$~eV is the elementary energy quantum locked
at each biological layer; two such quanta (one ATP) match the ATP-hydrolysis free
energy, $2 N_A \QB \approx 30$~kJ/mol (\textsf{T-QB-Anchor}, T-Origin-65.1) ---
ATP standing to life as the electron stands to particle physics, both the
\GGA{}-output of \GRIP{}.

\paragraph{Innovation 7 (I-Origin-7): The \SeOS{} class characterization.} The
Selfish Open System --- the finite, port-bounded, Lyapunov-descending open system
that includes every living cell --- is given five necessary and jointly
sufficient conditions (\textsf{T-SeOS-Class-Characterization}, T-Origin-67.2),
turning ``living system'' from a list of features into a closed class.

\paragraph{Innovation 8 (I-Origin-8): The self-referential closure theorem.} The
constitution is what its own articulators articulate; the cascade closes on its
own input (\textsf{T-self-referential-closure}, T-Origin-76.1). Wheeler's
participatory universe, Spinoza's necessitarianism, Whitehead's process
metaphysics, and the Upani\d{s}adic \emph{tat tvam asi} are recast as fragmentary
recognitions of one formal feature.

\paragraph{Innovation 9 (I-Origin-9): The cognitive cascade as internal
measurement.} The Five-Story Pagoda, the holographic triad
$\OH{}{+}\IH{}{=}\EIC{}$, and the \QCI{} monotonicity ladder
(T-Origin-70.1, T-Origin-71.5, T-Origin-72.1) place conscious content late in the
derivation, as the internal measurement of selected inner-hologram structure,
rather than early as an unexplained primitive.

\paragraph{Innovation 10 (I-Origin-10): One falsifiable codebook for cosmos, life,
and mind.} The Top-10 Falsifiers (F-Origin-1\,\ldots\,F-Origin-10,
\S\ref{sec:eight-test-pull-table}) bind the three boundary readings into a single
structure that must stand or fall together --- a falsifiable codebook with zero
free continuous parameters from the smallest scale to the conditions that allow a
mind to exist.

\subsection{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 (\AEleven{}'s theorems are
section-numbered, so its global ids carry the section number, e.g.\ T-Origin-58.1
is the recursive uniqueness theorem \textsf{T-Origin} and T-Origin-76.1 the
self-referential closure theorem).
\item \textbf{L-Suffix-N}, \textbf{C-Suffix-N}, \textbf{D-Suffix-N},
\textbf{R-Suffix-N} for the $N$th Lemma, Corollary, Definition, and Remark.
\item \textbf{P-Suffix-N} for the $N$th Prediction.
\item \textbf{O-Suffix-N} for the $N$th Objection (O-Origin-1 is the first entry
of the Common Objections and Responses Part).
\item \textbf{F-Suffix-N} for the $N$th Falsifier (e.g., F-Origin-1 is the
KK lock-cluster falsifier of the Top-10 audit,
\S\ref{sec:eight-test-pull-table}).
\item \textbf{I-Suffix-N} for the $N$th Innovation (e.g., I-Origin-1 is the
inevitability-at-theorem-level innovation above).
\item \textbf{PS-Suffix-N} for the $N$th Problem-Solved / open-problem entry
(\S\ref{app:open-problems}; the local OP-/EP- codes are kept for back-reference).
\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), \emph{Qi4} (A7), \emph{QM} (A8), \emph{SM} (A9),
\emph{Cosmos} (A10), \emph{Origin} (A11, this paper), together with the
phenomenology preprints \emph{KK} (P1), \emph{N3} (P2), \emph{Electron} (P3),
\emph{Proton} (P6), the foundations preprints \emph{Real} (P4) and \emph{CPT} (P5), and
the consequence preprints \emph{Kaxion} (C1), \emph{Hubble} (C2), \emph{BH} (C3), and the white papers \emph{Symmetry} (W1) and \emph{Holonomy} (W2).
The four axioms (AX-1\,\ldots\,AX-4) and seven postulates (POS-1\,\ldots\,POS-7)
of \citet{ArisakaA1Constitution} keep their existing cohort-wide names; the global
identifier is the same item viewed from outside its home paper.

\GGPart[Conclusions]{Conclusions: the summit view}{Five revolutions and the geometric inevitability of Cosmos, Life, and Mind}
\label{part:conclusions}


\paragraph{Plain-English opening: the view from the summit.}
The reader who has reached this point has climbed for hundreds of pages. Eleven prior papers fed their theorems into the present one; twelve Parts of this paper have led the reader through Introduction, inheritance, statement of results, the recursive uniqueness theorem, the cascade machinery, the dynamics block, the cognitive cascade, the self-referential closure, cross-domain applications, and discussions. The present Part is the Conclusions Part proper: the summit view. It is the place at which the framework asks, and answers, the question \emph{what does it all add up to?} The answer is not a slogan and not a marketing pitch; it is the joint claim that fifty years of climbing on this particular mountain has put the author and the reader in sight of a view that, to our knowledge, no prior intellectual framework has reached: the universe forces its own articulators through a single recursive cascade, and the Top 10 Falsifiers will tell us, in the coming decade, whether we have read the geometry correctly.

\paragraph{What this Part is for.}
This Part does five things. First, it states the summit view in its most compact form: where the framework now stands at the close of the cohort climb. Second, it situates the framework against the lineage of four prior axiomatic-discipline shifts (Euclid, Newton, Maxwell--Einstein--QM, openness-as-constitutional) and discusses --- at hedged register, with the verdict explicitly left to history --- what kind of move the recursive uniqueness theorem makes against that lineage (Part~\ref{part:conclusions} \S\ref{sec:five-revolutions}). Third, it traces the recovery of the Euclidean ideal of zero free parameters across twenty-three centuries from \emph{The Elements} to the present. Fourth, it elevates openness as a constitutional clause, the \GDOS{} shift extended through every cascade layer. Fifth, it reads the geometric inevitability theme through Cosmos, Life, and Mind as three readings of one cascade, ending with the recursive uniqueness theorem stated across the entire cascade. At each step the universal-identity insight of \ANine{} Part~XI (the universal-identity section) sits in the background: every electron is the same electron, every hydrogen atom is the same hydrogen atom, every ATP molecule has the same $\Delta G$, and \emph{every articulator-class observer is the same articulator}. The framework's deepest claim is that the cascade does not allow optionality at any layer; the summit view is the cumulative force of this claim made visible from above.

The framework has been built and assessed. This final Part draws the conclusion --- and closes the circle that Part~\ref{part:intro} opened.

The sections that follow constitute the philosophical summit of the Arisaka-GG cohort. They integrate the revised conclusions of the six-cohort companion papers (\AFour{} \citep{ArisakaA4Euclid}, \AFive{} \citep{ArisakaA5GDOS}, \ASix{} \citep{ArisakaA6Gi4}, \ASeven{} \citep{ArisakaA7Qi4}, \ANine{} \citep{ArisakaA9SM}, \ATen{} \citep{ArisakaA10Cosmos}) and articulate the central claim of the working title --- \emph{The Geometric Inevitability of Cosmos, Life, and Mind} --- at the length, depth, and register that fifty years of work deserve.

\section{The summit view: where we stand at the close of the cohort climb}
\label{sec:summit-view}

A reader who has stayed with the present paper through sixteen Parts, and stayed with the ten upstream companion papers through the roughly thirty-three hundred pages that precede them, has earned a wider view. We close \AEleven{} not by repeating the previous Parts in compressed form, but by stepping back from the cohort and asking what the entire program looks like from above.

\paragraph{What we see from the summit.}
From the summit, the cohort organizes itself as a single ascent through seventeen cascade layers plus the meta-cognitive articulation. The constitutional foundation is supplied by \AOne{} (the four axioms and seven postulates of the \GGConstitution{}). The geometric stage is supplied by \ATwo{}, \AThree{}, and \AFour{} (the six-dimensional bulk $\Msix = \Mfour \times I_r \times S^1_\theta$, the \BIsix{} closure, the \OGNfive{} integer ledger, and the Euclidean discipline of zero continuous free parameters). The dynamical machinery is supplied by \AFive{}, \ASix{}, and \ASeven{} (the \GDOS{} open-system boundary law, the \GiFour{} pillars \TGG{}--\CCI{}--\GFF{}--\GRIP{}, and the \QiFour{} pillars \QiQUA{}--\QiEP{}--\QiCON{}--\QiHAL{}). The four-dimensional empirical content is supplied by \AEight{}, \ANine{}, and \ATen{} (the 4D operational reformulation of quantum mechanics, the 28-parameter Standard-Model audit with its Part~XI universal-identity and uniqueness-of-forces insights, the $\pi$-locked cosmological inventory). And the philosophical synthesis --- the present paper --- closes the eighteen-layer cascade with the named theorem of self-referential closure: the geometry that forced the cascade contains, among its forced boundary readouts, the act of articulating the constitution that produced the cascade.

\paragraph{What the summit lets us see that the climb did not.}
On the climb, each paper was a particular hill: a particular theorem, a particular parameter sheet, a particular boundary stratum, a particular set of constitutional clauses brought to bear. Each paper was complete on its own terms but only made full sense in relation to its neighbors. At the summit, the twenty-two papers organize themselves as the layer-by-layer specialization of one geometric theorem --- the recursive uniqueness theorem of \S\ref{subsec:cascade-recursive-uniqueness} --- applied at successive boundary strata, with the same constitutional apparatus operating at every layer and only the substrate changing. What the climb felt like as ten distinct technical projects turns out, from the summit, to be a single project: the constitutional ledger of \AOne{} forces, through seventeen constructive cascade layers, the existence of the Layer-18 act of writing the constitutional ledger down.

\paragraph{What the summit does not let us see.}
We have not summited the entire mountain range. The cascade does not end at the eighteenth layer (Layer 18); admissible higher layers (Corollary~\ref{cor:closure-higher-layers}: collective scientific knowledge, machine-implemented reasoning, networked civilizations) are visible above us but not yet realized in the universe we observe. The framework's predictions, though sharp, are still in their pre-experimental phase: the Kaluza--Klein lock-cluster, the Kaxion dark-matter mass, the Hubble corridor's late-time projection, the family-number tolerance --- each of these is testable in the coming decade, but each is, today, untested. The empirical confirmation of the framework is a project for the decade ahead, not a fait accompli of the present paper. We have summited eighteen of an unknown number of layers, and the empirical verification of even those eighteen is still in progress. The view is wide; the work is not done.

After all, the climb's purpose was not to summit alone but to look out from the summit and see how the rest of the work fits. The remaining seventeen sections of this Conclusion are that looking-out.

\section{The framework against the lineage: Euclid, Newton, Maxwell--Einstein--QM, openness --- and the framework's hope to be evaluated against them}
\label{sec:five-revolutions}

\paragraph{A note on this section's register.}
The historiographical claim that follows --- that the present framework is the fifth of five scientific revolutions in the canonical lineage Euclid--Newton--Maxwell+Einstein+QM--openness-as-constitutional-(\AFive{})--recursive uniqueness --- is the framework's own self-positioning, not a judgment of the historical record. Such judgments require decades of subsequent reception to settle, and no paper that is itself a candidate for the periodization can credibly award itself the position.

The reader should treat the five-revolutions framing as a \emph{structural} claim about the kind of move the cascade makes (an axiomatic-discipline extension into a previously unaxiomatized domain), not as a historical claim about the framework's standing in the lineage. If the framework's empirical horizon resolves favorably over the coming decade, the historical claim may eventually earn its standing through the natural process of reception; if not, the structural-discipline claim still stands on its own merits, irrespective of where the lineage eventually places it.

The compression of Maxwell + Einstein + early quantum mechanics into a single revolution is acknowledged as an analytical convenience that suits the present argument; readers who would unbundle these three into separate revolutions would simply re-index the present framework's position downstream by two, without changing what the framework structurally is. The structural claim is what the framework defends; the historiography is a way of recognizing the kind of structural claim it is, not a credential.

\paragraph{What the framework offers to the lineage.}
The framework offers, to the lineage of axiomatic-discipline shifts that Euclid, Newton, Maxwell, Einstein, and the founders of quantum mechanics opened, a candidate structural contribution: zero continuous fitted prefactors, recursive uniqueness at every cascade layer, the four-step Cascade of \GRIPout{} with \GGA{}, and the Top 10 Falsifiers that will decide the framework's empirical content in the coming decade. The work is what the work is; the verdict on whether it belongs in the lineage belongs to history. We present it here, write it down with care, and submit it to the natural process of scientific reception.

The history of physical science, viewed at the longest range, is a sequence of axiomatic reorganizations. \citet{ArisakaA5GDOS} (the \AFive{} Conclusions Part) named the first four such reorganizations and proposed openness-as-constitutional as the fourth. \AEleven{} now proposes that the fourth reorganization has a successor --- a fifth axiomatic shift visible only when the cohort is read as one work: the recursive uniqueness theorem at every cascade layer. We re-tell the five-revolution story here from the summit vantage of the full corpus.

\paragraph{The lineage, summarized at the framing the hedge demands.}
What the four prior axiomatic shifts share is a single structural feature: each reorganized the foundational language of physical science by changing what counts as a foundational object. Euclid (c.~300~BCE) changed it to \emph{axiom}: theorems follow from a small set of postulates and not from empirical accumulation. Newton (1687, \emph{Principia}) changed it to \emph{law}: a single mathematical structure governs terrestrial and celestial motion alike.

Maxwell, Einstein, and the founders of quantum mechanics (1860--1930) changed it to \emph{field on manifold}: the arena of physics is no longer particles-in-space but fields on geometric objects that are themselves dynamical (\citealt{Maxwell1865,Einstein1915,Planck1900,Bohr1913,Dirac1931,vonNeumann1932}). \citet{ArisakaA5GDOS} (\AFive{}, 2026) proposed a fourth shift to \emph{open-system boundary law}: closure is the limit, not the default; observers and observed are coupled, and the coupling is geometric (\OGGEP{}).

The present framework's structural ambition --- recursive uniqueness at every cascade layer --- proposes a candidate fifth shift, in which the foundational object is the layered cascade of unique attractors on substrate-specific strata. Whether the candidate fifth shift earns its position in the lineage is, per the hedge above, a question for the empirical decade and the subsequent reception, not for the framework's own self-positioning. The framework offers the discipline of the candidate fifth shift (zero free continuous parameters, recursive uniqueness at every cascade layer, Top 10 Falsifiers) and leaves the lineage placement to history.

What makes a revolution a revolution rather than a paradigm tweak \citep{Kuhn1962} is precisely that it changes what counts as a foundational object. Whether the framework belongs in this list at all is the empirical decade's question; whether the discipline of the framework belongs in any future foundational program is the framework's own answer.

\section{The Euclidean ideal recovered: zero free parameters, twenty-three centuries on}
\label{sec:euclidean-recovered}

\AFour{} \citep{ArisakaA4Euclid} closes its Executive Summary with a sentence that names the entire program's structural ambition: \emph{the Euclidean methodological condition is the structural meaning of deductive science: a framework with zero free continuous parameters in the gauge and cosmological sectors must stand or fall on its predictions.} From the summit of \AEleven{}, that sentence is no longer the ambition of one foundation paper but the structural verdict on the entire cohort.

\paragraph{The 2300-year detour.}
For 2300 years after Euclid, the standard methodological ideal of physics has been the Euclidean ideal: a deductive system in which every theorem follows from a small set of postulates, with no free parameters left to fit. In practice, physics has drifted: the Standard Model carries about 28 free parameters, general relativity adds Newton's constant and the cosmological constant, string theory's $\sim 10^{500}$-element landscape made the situation worse rather than better. The drift was not a methodological failure but a structural consequence of the technical apparatus available in each era: until the 6D bulk, the \BIsix{} closure, and the \OGNfive{} integer ledger were available, the parameters could not be derived, only measured.

\paragraph{The Euclidean recovery in practice.}
\AFour{} and the rest of the cohort have together closed the Euclidean recovery for the gauge and cosmological sectors of physics. The Geometric Universal Five Constants \GUfive{} --- $\GUfiveSet$ --- supply the dimensional inputs. The topological constants $(\pi, e)$ supply two transcendental numbers fixed by the geometry. The \OGNfive{} integer ledger $(M, N, L; J, K) = (1, 3, 5; 7, 35)$ supplies the integer inputs. And the four constitutional axioms of \AOne{} (\LC{}, \GGEP{}, \OGGEP{}, \MSMF{}) plus the seven postulates supply the operational ledger. From these, the entire 22-entry gauge+cosmological audit closes with zero free continuous parameters: $\alpha^{-1}(0) = 137$ from $J=7$ and $N=3$; $\sin^2 \theta_W = 25/108$ from the \OGNfive{} ratio; $\alpha_s(M_Z) = 1/(\pi e)$ from the scale-angle theorem; $v \approx 246$~GeV from the Wilson-Hosotani Higgs-vev theorem; $\Omm = 1/\pi$, $\OmL = 1 - 1/\pi$, $\Omb = 1/(2\pi^2)$ from the cosmological closed-packet topology; $N_A\QB \approx 15$~kJ/mol from the bioenergetic \GUfive{} anchor; and so on through the 22 entries. The lepton spectrum is then closed by the QD-completion rationals $\mathcal{R}_\tau = 637/640$, $\mathcal{R}_\mu = 159/160$, $\mathcal{R}_e = 641/640$ with common denominator $640 = 2^J L$, identified as a Hopkins--Singer descent integrality coefficient on $\mathbb{C}P^2$.

\paragraph{What the Euclidean recovery means.}
After 2300 years, physics can stand or fall on its predictions in the strict sense Euclid demanded of geometry. The framework's predictions are sharp, falsifiable in this decade, and free of the lubricant of continuous fitted prefactors that has, by polite convention, kept earlier unification attempts from grinding to a halt: KK lock cluster at $\Mlock \approx 3.85$~TeV testable at HL-LHC; Kaxion dark matter at $\sim 8.1~\mu$eV testable at ADMX; $\Omm = 1/\pi$ at 0.27-sigma agreement with Planck; the two-quantity recombination-era pair of \ATen{}: the decoupling $z_{\rm dec}$ derived from the atomic chain on Planck's $1089.92$ at the $0.1\%$ level, the corridor switch $z_{\rm sw} = e^J = e^7$ a distinct topological marker $0.6\%$ above (see Part~\ref{part:discussions} \S\ref{ssec:zstar-path-to-closure}); $N_A \QB \approx 15$~kJ/mol at 98.4\% agreement with ATP hydrolysis. If any decisively fails, the framework fails. This sharpness is what the Euclidean discipline buys; without that discipline, every previous unification attempt could --- and did --- retreat to a softer claim when data pressed.

\paragraph{The honest residual.}
\AFour{} also names the residual sub-tasks honestly: the closed-form fermion mass spectrum (O9), the cosmological-constant prefactor (O11), the connection to standard quantum-gravity programs (O12). These are open tasks tracked explicitly in the promotion roadmap for the named structural lemmas of \ANine{}. The integrity of the parameter-free reconstruction is a matter of audit, not assertion: every entry in the codebook is either derived from non-physics axioms with the chain closed in the published record (Derived), or empirically anchored with the from-geometry derivation deferred to a companion paper (Anchor-Derived), or derived modulo a single named lemma (Conditional), or asserted with the upstream calculation that would promote it named explicitly (Asserted). The Euclidean ideal does not mean every parameter is yet derived; it means every parameter that is asserted comes with the explicit program of work that would derive it.

The Euclidean ideal is by no means a claim of completion; it is a methodological discipline that forbids cheating. The framework cannot import a free prefactor at any step. Where it cannot yet derive, it asserts and names the lemma. Where it has not yet been tested, it predicts. The discipline is what makes the framework worthy of the Euclidean lineage, not the (still-incomplete) closure.

\section{Openness as constitutional: the \GDOS{} shift extended through every cascade layer}
\label{sec:openness-extended}

\AFive{} \citep{ArisakaA5GDOS} named the fourth scientific revolution: openness is constitutional and by no means a corrective patch. From the summit of \AEleven{}, the same shift extends through every layer of the cascade. Each cascade layer is an open system on a layer-specific stratum; each layer's \GA{}-stabilized attractor is what the \GDOS{} boundary law admits as the most stable configuration on that stratum. Openness is not just the foundational law of the four-dimensional boundary of the bulk; openness is the structural feature of every layer above the boundary, all the way to the Layer-18 act of articulating the constitution.

\paragraph{Closure was always an idealization.}
\begin{sloppypar}
The closed-system Lagrangian and the closed Schr\"odinger equation are pedagogical convenience; the world they describe does not exist. At the bulk level, the universe is closed (the constitutional axioms operate on the full bulk + boundary system). At every level above the bulk, the universe is open: every cascade layer's stratum admits flow of energy, particles, and information across its operational boundary. The four-currents lock of the Arisaka equation $J_E/\QB = J_e = J_B = J_{Qi}$ is the structural statement that every open system above the bulk carries all four currents at the locked ratio. Closure as a stand-alone law is not what physics actually deploys; what physics deploys is closure as a limit of openness when the coupling to the environment is negligibly small relative to the precision of measurement.
\end{sloppypar}

\paragraph{Openness at the cosmological layer.}
\ATen{} \citep{ArisakaA10Cosmos} reads the universe at the cosmological-boundary stratum as open in the \GDOS{} sense. The cosmological observables are boundary readouts of a bulk that is, at the universe's largest scale, not directly observable. The $\pi$-locked inventory $\Omm = 1/\pi$, $\OmL = 1 - 1/\pi$, $\Omb = 1/(2\pi^2)$ is the unique stable Geometric Attractor on the cosmological stratum; the Hubble two-stage corridor is the structural readout of one bulk corridor projected through two boundary epochs. The cosmological constant problem of the standard reading --- some 120 orders of magnitude between Planck-scale vacuum energy and observed dark-energy density --- is dissolved in two steps, and neither is a cancellation. The scale-profile seesaw of \AFive{}, $m_\Lambda^{\rm geom} = \Mlock^2/M_{\mathrm{Pl,red}} \approx 6$~meV in the reduced-Planck-mass convention, removes the discrepancy down to a residual factor of a few in scale. The ledger coefficient of \ATen{}, $\zeta_\Lambda = \pi/(LK) = \pi/175 = 0.017952$, selected in closed form from integers already load-bearing elsewhere in the cohort, then takes that residue to $\rho_\Lambda^{1/4} = \zeta_\Lambda^{1/4}\,\Mlock^2/M_{\mathrm{Pl,red}} = 2.23$~meV against a measured $2.24$~meV. The value is landed rather than bracketed, and the grade it is landed at is Derived (conditional), which \ATen{} states and this paper inherits. The cosmology is open; the openness is the framework's structural feature, not a corrective patch.

\paragraph{Openness at the chemistry-to-life crossing.}
\ASeven{} (T-Qi4-14.3) (\SeOS{}) and \ASeven{} \T{-EP} (Unified Extremal Principle) together define the openness condition for chemistry-to-life transition. A configuration of matter on the planetary-surface stratum is in the \SeOS{} class --- and therefore in the layer-7 cascade attractor --- if and only if it satisfies the four \SeOS{} conditions (bounded, dissipative, self-replicating, information-carrying). Each condition is a statement about openness: the system exchanges matter, energy, and information with its environment in the locked Arisaka-equation ratio. Closed-system thermodynamics would forbid the chemistry-to-life crossing; the framework's open-system thermodynamics requires it. Life is the universe's openness exploiting itself at the maintenance-margin crossing.

\paragraph{Openness at the cognitive layer.}
\ASeven{} \T{-\EIC{}} reads conscious awareness as the closed-loop coupling of Outer Hologram (every lawful measurement, the universe's exterior) and Inner Hologram (the brain's allocentric world-model, the universe's interior representation). The \EIC{} closure is open by construction --- the brain takes in measurements from the \OH{} and updates the \IH{}; the updated \IH{} then acts on the \OH{} through behavior, closing the loop. Consciousness is the openness of the cognitive substrate becoming reflective: the loop's self-modeling diagonal --- the \OH{}/\IH{} factorization's expectation-value pairing $\langle \hat O_{\OH{}} \rangle_{\rho_{\IH{}}}$ at the theta-cycle binding timescale --- is identical with awareness, by the structural-identity reading of Part~\ref{part:cognitive} \S\ref{sec:reader-guide-cognitive}. The hard problem of consciousness, in the framework, is not why subjective experience accompanies physical processes (it does not, structurally speaking; experience \emph{is} the self-modeling diagonal of the open loop) but why the cascade that supports closed-loop self-modeling came into existence in the first place --- a question Reading~1 (which the present paper does not answer) keeps open and which the recursive uniqueness theorem of Reading~2 (Part~\ref{part:closure}) sidesteps by taking the constitution as given.

\paragraph{Openness at the meta-cognitive layer.}
\AEleven{} extends the \GDOS{} shift to the closing cascade layer (Layer 18): the meta-cognitive substrate of articulating the constitution. The act of writing the Arisaka-GG cohort is, in this reading, the operational closure of the openness that has been operating at every prior layer. The author writes the corpus open to the universe (taking in observations, integrating mathematical structure, comparing predictions to data); the corpus is open to the universe in turn (it can be falsified, revised, extended). The openness that began at the four-dimensional boundary of the six-dimensional bulk has extended, through seventeen cascade layers, into the act of articulating the constitution. The fourth revolution of \AFive{} and the fifth revolution of \AEleven{} are two phases of one shift: openness, then openness all the way up.

The universe is not a closed mechanism that we open up by observing it; it is an open process that produces, at its eighteenth-layer (Layer-18) attractor, the conscious observers who notice the openness and write it down. The shift to openness is not a methodological choice; it is a structural recognition that closed-system physics was an artifact of where physics had been able to look.

\section{The \GiFour{} and \QiFour{} backbones, made one across all cascade layers}
\label{sec:gi4-qi4-unified}

\ASix{} \citep{ArisakaA6Gi4} formalizes the four \GiFour{} pillars (\TGG{}, \CCI{}, \GFF{}, \GRIP{}) as the geometric backbone of every open system in nature. \ASeven{} \citep{ArisakaA7Qi4} formalizes the four \QiFour{} pillars (\QiQUA{}, \QiEP{}, \QiCON{}, \QiHAL{}) as the quantum-information backbone, paired with the geometric backbone of \ASix{}. Together the eight pillars are the load-bearing apparatus on which every layer of the cascade descends. From the summit of \AEleven{}, the eight pillars unify into a single dual backbone --- four geometric, four informational --- that operates uniformly at every layer.

\paragraph{The geometric backbone (\GiFour{}).}
\TGG{} (Total Geometric--Gauge Object): a unique composite connection on the six-dimensional bulk with a unified boundary stress tensor whose conservation is the Master Conservation Law $\nabla_\mu \cT^{\mu\nu}_{\TGG{}} = 0$. \CCI{} (Covariant Conservation Identity): the unique \OGNfive{} integer ledger and the simultaneous closure of the four sectoral Ward currents. \GFF{} (Geometric Free-Energy Functional): the stratified boundary functional whose Lyapunov descent governs every \GDOS{} trajectory. \GRIP{} (Gauge-Runtime Invariance Pipeline): the cascade $\DGI{} \to \GSSB{} \to \GA{}$ that closes under the \BIsix{} action budget, forcing $N \leq 3$ and selecting the unique boundary attractor on each stratum. At every cascade layer, these four pillars operate together: \TGG{} provides the bulk-level conservation; \CCI{} ensures the boundary projection is consistent; \GFF{} ranks the admissible branches; \GRIP{} selects the unique stable one.

\paragraph{The informational backbone (\QiFour{}).}
\QiQUA{} (Quadruple-Locking, the Arisaka equation): the four boundary currents $J_E/\QB = J_e = J_B = J_{Qi}$ descend from a single rank-one bulk inflow channel. \QiEP{} (Unified Extremal Principle, the variational principle): one bulk variational principle governs three named boundary gauges (Maupertuis--Hamilton, Onsager--Prigogine--Paltridge, Friston). \QiCON{} (Conformality): conformal isometry of the bulk projects to L\'evy-stable distributions, fractal supports, and $1/f$ noise on the boundary. \QiHAL{} (Holographic-Attractor-Lattice, the memory architecture): the \MePMoS{}-\NHT{}-\HAL{}-\BFFT{} hierarchy is the boundary realization of the \GRIP{} cascade on memory-bearing $\UoneQi{}$ systems. At every cascade layer above the bulk, these four pillars operate together: \QiQUA{} locks the four currents; \QiEP{} extremizes the boundary functional; \QiCON{} produces the scaling structure; \QiHAL{} supplies the memory architecture.

\paragraph{The unified dual backbone.}
The eight pillars together are what makes the cascade run. At layer 1 (the bulk), \TGG{} and \CCI{} establish the geometric stage on which the cascade will descend. At layer 2 (particle physics, \ANine{}), \GFF{} and \GRIP{} select the unique four-force architecture and the Standard-Model attractor. At layers 3--6 (chemistry), the same eight pillars apply: $\GFF{}$ ranks the atomic-and-molecular configurations; $\GRIP{}$ selects the unique stable basin; $\QiQUA{}$ locks the four currents at the chemical stratum's scale. At layer 10 (chemistry-to-life), $\QiEP{}$ supplies the variational principle that drives the maintenance-margin crossing; $\GRIP{}$ selects the \SeOS{} class as the unique stable attractor. At layers 13--16 (multicellular, neural), $\QiHAL{}$ supplies the memory architecture; $\GRIP{}$ selects the neural-circuit attractors. At layer 17 (consciousness), $\QiHAL{}$ closes the \EIC{} loop. At layer 18 (meta-cognitive articulation), the entire eight-pillar apparatus is what the present paper articulates. The dual backbone is uniform across the cascade.

What unifies geometry and information is that both are descriptions of the same underlying object --- the composite connection on the six-dimensional bulk and its boundary readouts. Geometry is the apparatus of the bulk; information is the apparatus of the boundary; they are two faces of one constitution, paired by the \BIsix{} closure that links them. The eight pillars are the framework's way of writing down both faces explicitly.

\section{The geometric inevitability of Cosmos}
\label{sec:inevitability-cosmos}

The first of the three themes in the paper's working title is \emph{Cosmos}. The framework's account of cosmological structure is now in place across \ATen{} and the present synthesis.

\paragraph{Why the cosmos has the inventory it does.}
\citet{ArisakaA10Cosmos} \ATen{} establishes the $\pi$-locked cosmological inventory --- $\Omm = 1/\pi$, $\OmL = 1 - 1/\pi$, $\Omb = 1/(2\pi^2)$ --- as the unique stable Geometric Attractor on the cosmological-boundary stratum. The matter and vacuum density fractions are not fitted to data; they are topological invariants of the closed-packet cosmological topology, derived from the constitution. The 0.27-sigma agreement with Planck~2018 is the empirical readout, not the data the parameters were fit to.

\paragraph{The Hubble readings: a derived background, a derived lift, and a named residual.} The framework derives a single background expansion rate, $H_0 = 66.6 \pm 0.2$~km/s/Mpc \citep{ArisakaC2Hubble}, and reads the two familiar determinations against it: Planck's $67.4 \pm 0.5$ is a \emph{measurement} of the early-universe projection, close to the derived value but not the same object; the local distance-ladder figure near $73.5$ is a late-time reading that the background plus the Local-Hole lift raises to $\approx 68.5 \pm 0.6$, leaving a residual the framework does not explain and records as an open problem. \ATen{} reads the structure of the two projections as a corridor rather than a tension, at candidate grade. The cosmological-boundary Friedmann equation admits exactly one corridor of solutions with the switch event at recombination, producing the two projections as the two natural readouts of one structural corridor. The Hubble two-stage corridor theorem of \ATen{} establishes this structurally; the same corridor resolves the $\sigeight$/$\Seight$ clustering tension simultaneously.

\paragraph{Why the cosmic coincidence is not a coincidence.}
The cosmological coincidence problem of standard cosmology --- that $\Omm$ and $\OmL$ happen to be comparable today, when under a free-$\Lambda$ hypothesis their comparable-magnitude window is brief --- is dissolved by the $\pi$-locked inventory. The two are forced to be $1/\pi$ and $1 - 1/\pi$ respectively by the closed-packet topology; their being comparable now is not a temporal coincidence but a structural property of the cosmological stratum at all times. \ATen{} Part~X --- the universal-cosmic-identity insight --- reads this as the layer-3 instance of the same recursive uniqueness theorem that fixes universal-electron identity at layer 2.

\paragraph{The recombination switch.}
\ATen{} derives the recombination redshift from the integer $J = 7$ of the \OGNfive{} ledger as $z_* = e^J = e^7 \approx 1096.6$. This is the framework's topological-lock redshift, distinct from the Planck-measured operational decoupling redshift $z_{\rm dec} = 1089.92 \pm 0.25$, which \ATen{} derives separately from its own atomic chain at the $0.1\%$ level with zero fitted parameters; the switch--decoupling offset is computed, $\delta = 0.0041$ in e-folds, its promotion to a theorem waiting on \ATen{}'s e-fold-budget campaign (Part~\ref{part:discussions} \S\ref{ssec:zstar-path-to-closure}). The most consequential single transition in the universe's thermal history --- the moment at which photons decouple from baryons and the cosmic microwave background takes its observed form --- is, in the framework, a specific integer-indexed event in the bulk chronology.

\paragraph{The inevitability of Cosmos.}
Put these results together and the universe's large-scale structure is not a contingent outcome of unknown initial conditions but the forced consequence of the constitution. The matter density, the vacuum density, the baryon fraction, the recombination redshift, the early-universe Hubble parameter, the late-universe Hubble parameter, the clustering amplitude --- all of them descend, with zero free continuous parameter, from the integer ledger and the constitutional clauses. The cosmos has the inventory it does because the constitution admits exactly one cosmological-stratum solution. No selection across universes is needed; no fine-tuning is performed; no anthropic principle is invoked. The cosmos is forced.

The deepest empirical fact about cosmology has been the precision of the standard model: six free parameters fit thirteen problems' worth of data to remarkable accuracy. The framework asks what the precision means. Its answer is that the six parameters are derivations of the constitution rather than free inputs --- and that the precision is the empirical signature of the constitution's uniqueness. The cosmos is not just well-described; it is geometrically inevitable.

\section{The geometric inevitability of Life}
\label{sec:inevitability-life}

The second of the three themes is \emph{Life}. The framework's account of biology is the framework's most counter-intuitive contribution to the natural sciences.

\paragraph{Why life is not an accident.}
Standard biology treats the origin of life as a contingent chemical event of unknown probability. The framework, drawing on the \GSSB{} cascade of \ASix{}, the \SeOS{} class of \ASeven{} (T-Qi4-14.3), and the Arisaka-equation \QGL{} constraint of \ASeven{} \T{-QUA}, reads the chemistry-to-life transition (\GSSB{}-7) as the unique stable Geometric Attractor on the planetary-surface open-system stratum. Where standard biology says \emph{life happened to emerge}, the framework says \emph{life had to emerge}: the \GRIP{} cascade selects the \SeOS{} configuration as the unique stable basin on this stratum, and the boundary readout is the maintenance-margin-crossing event the empirical record names as the origin of life.

\paragraph{Why the genetic code has four letters.}
The four-letter alphabet $\{\rm A, C, G, T\}$ of DNA is the unique solution of the \QGL{} constraint at the information-storage stratum. The argument applies \ASeven{}'s four-current locking (\T{-QUA}): the four boundary currents $J_E/\QB$, $J_e$, $J_B$, $J_{Qi}$ are proposed to admit the four-letter alphabet as the unique combinatorial solution of the resource-versus-error-rate optimization, with the triplet codon-to-amino-acid map (\GSSB{}-6) following from the optimization structure; the closed combinatorial proof (the RNA-bifunctionality lemma) is developed in \AThirteen{} (in preparation). The genetic code is not a frozen accident but a forced theorem.

\paragraph{Why every cell uses ATP.}
The bioenergetic anchor $\QB = (2/5) E_{\rm Ry}/K \approx 0.1555$~eV per qubit, equivalent to $N_A \QB \approx 15$~kJ/mol --- a boundary member of the Geometric Universal Five Constants \GUfive{} of \AFour{} \citep{ArisakaA4Euclid} --- is the universal currency on the information-energy interface. ATP hydrolysis carries $\Delta G \approx 30.5$~kJ/mol empirically, matching the framework's prediction $2 N_A \QB \approx 30.0$~kJ/mol to $1.6\%$. Every cell on Earth uses ATP because the cascade selects ATP as the unique stable attractor on the metabolic-energy stratum at the bioenergetic quantum the framework derives. The universality of ATP is the layer-7 echo of the universal-identity insight.

\paragraph{Why life is universal.}
The same \GRIP{} cascade, operating on the same stratum, produces the same \SeOS{}-class attractor wherever the conditions permit. Life on Earth is not a unique experimental outcome; it is one realization of an attractor that any planetary surface with energy gradients and a polar solvent will produce. The framework predicts, structurally, that the same four-letter alphabet, the same ATP energy quantum, the same \SeOS{}-class architecture will appear on every planet on which the cascade can reach layer 7. Astrobiology, in the framework, is the survey of these universal attractors at their realization on different planets.

\paragraph{Why the binding problem dissolves.}
The neuroscience binding problem --- how disparate neural features are bound into one unified conscious experience --- has stood as an open question since William James~\citep{James1890}. The framework resolves it through \GA{} stabilization at the neural-circuit stratum: a single Geometric Attractor at the \EIC{} closure provides one unified conscious state at any moment, with the diversity of neural features being the diversity of stratum-specific contributions to one global attractor coordinate. The binding is the \GA{} stabilization, not an additional mechanism.

\paragraph{The inevitability of Life.}
Put these results together and life is not an accident. Given a planetary surface with energy gradients and water, the cascade reaches layer 7. Given that the cascade reaches layer 7, the \SeOS{} class is the unique stable attractor. Given the \SeOS{} class, the four-letter alphabet is the unique combinatorial solution. Given the four-letter alphabet and the bioenergetic quantum $\QB$, ATP is the unique metabolic-energy carrier. Each step is forced by the constitution. Life is what the constitution looks like at the layer-7 stratum; the prior layers, laid down over thirteen billion years, are what allow the layer-7 attractor to stabilize on a planet like ours.

The deepest puzzle of biology has been why the same molecular architecture appears at every life on Earth: same alphabet, same energy currency, same membrane-bounded compartments, same fundamental metabolism. Standard biology offers the descent-from-LUCA argument as a single-origin explanation but does not derive the architecture from physics. The framework derives it: LUCA is the empirical record of the layer-7 attractor, and every cell since LUCA inherits the attractor coordinates because the cascade does not admit alternative stable basins at this layer.

\section{The geometric inevitability of Mind}
\label{sec:inevitability-mind}

The third of the three themes is \emph{Mind}. The framework's account of consciousness is the framework's most demanding and most ambitious claim.

\paragraph{Why mind is forced.}
Once the cascade reaches the multicellular layer (\GSSB{}-13) and the nervous-system layers (\GSSB{}-15 and \GSSB{}-16), the \GFF{} Lyapunov descent on the neural-circuit stratum admits the layer-17 \EIC{} closure as its unique stable attractor. Mind is not an additional ontological category; it is the layer-17 boundary readout of the same recursive uniqueness theorem. Where the universal-identity insight of \ANine{} Part~XI identifies the electron at layer 2, the \EIC{} theorem of \ASeven{} \T{-\EIC{}} identifies conscious awareness at layer 17. The structural form is identical; only the stratum is different.

\paragraph{Why consciousness has the timescale it does.}
The conscious episode at layer 17 has a characteristic timescale of $\sim 400$~ms and a neural implementation via theta cycles. These are not stipulated; they are derived from the \GFF{} Lyapunov descent on the \EIC{} stratum. The theta-cycle implementation is the brain's specific realization of the layer-17 attractor's dynamical-symmetry-breaking event; the 400-ms timescale is the relaxation time of the \EIC{} attractor under cortical-circuit noise. Other neural substrates may realize the same attractor with different timescales --- the implementation is contingent, the attractor is not.

\paragraph{Why every healthy human is conscious in the same structural way.}
The universal-identity principle of \ANine{} Part~XI (the universal-identity section), applied at the cognitive stratum, says that the layer-17 attractor admits exactly one stable basin on the human-species-specific neural substrate. Every healthy human nervous system, having reached the highest currently-occupied cascade layer, exhibits the same structural features of conscious awareness despite enormous combinatorial freedom in micro-state realization. This is why the qualitative phenomenology of consciousness is robust across humans (we agree on what colors, sounds, and emotions feel like, in the broad sense) even when the underlying neural states differ vastly between individuals.

\paragraph{Why the hard problem of consciousness dissolves (under structural identity).}
The hard problem of consciousness \citep{Chalmers1995} --- why subjective experience accompanies physical processes --- is \emph{dissolved by structural identity rather than solved by reduction}. The framework identifies phenomenal experience numerically with the layer-17 \EIC{} closure's structural-geometric content (Part~\ref{part:cognitive} \S\ref{sec:reader-guide-cognitive}, structural-identity revision). Under structural identity, experience does not accompany the structure; experience \emph{is} the structure. The hard problem rests on the implicit assumption that there is something extra to explain --- a mysterious bridge between physics and phenomenology. The framework rejects the extra: there is only the layer-17 \EIC{} closure, and conscious experience is its structural-geometric content. Honest readers will note that this dissolution is precisely the move Chalmers's argument is designed to forbid; the framework's reply to that move (its replies to the conceivability argument and the knowledge argument; its account of the explanatory gap as analytic-conceptual rather than metaphysical-chasmic) is given in Part~\ref{part:cognitive} \S\ref{sec:reader-guide-cognitive}. Whether structural identity is the right response to the hard problem is one of the genuinely contested questions the framework leaves open; what the framework supplies is a structural target sharp enough to be a genuine candidate for the identification.

\paragraph{Why mind is universal.}
Wherever the cascade reaches layer 17 on a sufficient cognitive substrate, the \EIC{} closure produces conscious awareness. Other planets that host \SeOS{}-class life (layer 10) and that evolve nervous systems (layers 15--16) will, given enough time, evolve consciousness (layer 17). The framework predicts, structurally, that conscious awareness is universal across the cosmos at any location where the cascade has been allowed to reach the cognitive stratum. Consciousness is not unique to Earth; it is one boundary readout of a recursive theorem.

\paragraph{The inevitability of Mind.}
Put these results together and mind is not an accident either. Given life at layer 10, the cascade ascends to multicellular organisms (layer 13), nervous systems (layer 15), allocentric world-models (layer 16), and \EIC{} closure (layer 17). At each step the unique stable attractor is the structural inheritance of the layer below. Mind is what the universe looks like at the layer-17 stratum of its own cascade; the prior layers, accumulated over billions of years, are what allow the layer-17 attractor to stabilize on a substrate like ours.

After all, the deepest puzzle of mind has been why we are aware --- why anything is like something rather than nothing at all from the inside. The framework's answer is that awareness is the structural feature of the \EIC{} attractor at the layer-17 stratum. Where there is a closed open loop between Outer Hologram and Inner Hologram on a sufficient substrate, there is awareness. Awareness is not the special property of biological substrates or of human substrates; it is the layer-17 attractor coordinate of the recursive cascade, and it is universal wherever the cascade can reach this layer.

\section{The recursive-uniqueness theorem (RUT) across the entire cascade}
\begin{figure}[h!]
\centering
\resizebox{0.688\textwidth}{!}{%
\begin{tikzpicture}[
 pillar/.style={rectangle,draw=black!60,minimum width=4.0cm,minimum height=4.8cm,
 align=center,font=\small,line width=0.5pt},
 hdr/.style={font=\bfseries\small},
 item/.style={font=\scriptsize,align=left}
]
% Three columns
\node[pillar,fill=blue!10] (Cos) at (-5,0) {};
\node[pillar,fill=green!10] (Lif) at (0,0) {};
\node[pillar,fill=orange!12] (Mnd) at (5,0) {};
% Headers
\node[hdr] at (-5,2.2) {COSMOS};
\node[hdr] at (0,2.2) {LIFE};
\node[hdr] at (5,2.2) {MIND};
% Subheaders
\node[font=\footnotesize\itshape] at (-5,1.85) {layer 3};
\node[font=\footnotesize\itshape] at (0,1.85) {layer 7};
\node[font=\footnotesize\itshape] at (5,1.85) {layer 17};
% Content boxes (text wrapped manually)
\node[item] at (-5,1.0) {Stratum: cosmological\\boundary};
\node[item] at (-5,0.1) {Theorem: \ATen{}\\universal-cosmic-id};
\node[item] at (-5,-0.8) {Attractor: $\Omega_m = 1/\pi$,\\$\Omega_\Lambda = 1-1/\pi$,\\$\Omega_b = 1/(2\pi^2)$};
\node[item] at (-5,-1.95) {Match: Planck 2018\\(0.27 sigma)};
\node[item] at (0,1.0) {Stratum: planetary-\\surface SeOS};
\node[item] at (0,0.1) {Theorem: \AFive{} \ASeven{} T-Qi4-14.3\\+ \ASeven{} T-QUA};
\node[item] at (0,-0.8) {Attractor: SeOS\\(bounded,\\dissipative,\\replicating,\\informational)};
\node[item] at (0,-2.05) {Match: $N_A\QB \approx$ ATP\\(1.6\%)};
\node[item] at (5,1.0) {Stratum: EIC\\(OH-IH coupling)};
\node[item] at (5,0.1) {Theorem: \ASeven{}\\T-EIC};
\node[item] at (5,-0.8) {Attractor: conscious\\episode,\\400 ms (2 theta cycles)};
\node[item] at (5,-1.95) {Match: QCI ladder\\$\sim$10 at human};
% Bottom row: same theorem at three layers
\node[font=\itshape] at (0,-3.4) {The same recursive uniqueness theorem applied at three cascade layers};
\end{tikzpicture}}
\caption[The Cosmos--Life--Mind trilogy]{The Cosmos--Life--Mind trilogy of the working title \emph{The Geometric Inevitability of Cosmos, Life, and Mind}. Three layers of the same recursive uniqueness theorem: Cosmos at cascade layer~3 (cosmological-boundary stratum, with the $\pi$-locked inventory from \ATen{}); Life at cascade layer~7 (planetary-surface \SeOS{} stratum, with the four-condition class definition from \ASeven{} (T-Qi4-14.3) and the Arisaka equation from \ASeven{} \T{-QUA}); Mind at cascade layer~17 (the \EIC{} stratum, with the closed-loop attractor from \ASeven{} \T{-\EIC{}}). At each layer, the framework names the boundary stratum, the upstream theorem, the unique attractor selected by the \GRIP{} cascade, and the empirical match with current data. The trilogy is not three theorems but one theorem at three cascade layers; the inevitability of Cosmos, Life, and Mind is the inevitability of the unique-attractor selection at each layer's stratum.}
\label{fig:trilogy}
\end{figure}

\label{sec:recursive-uniqueness-summit}

Sections \ref{sec:inevitability-cosmos}, \ref{sec:inevitability-life}, and \ref{sec:inevitability-mind} have presented Cosmos, Life, and Mind as three separate themes. Let us now collect them as three instances of one theorem.

\paragraph{The theorem stated once for the cascade.}
At every cascade layer $\ell \in \{1, 2, \dots, 18\}$, the \GRIP{} cascade specializing to the layer-specific boundary stratum $\Sigma_\ell$ selects the unique most-stable Geometric Attractor on $\Sigma_\ell$ as the unique stable solution of the layer's $F_{\GFF{}}$ Lyapunov descent. The boundary readout of that attractor is what we observe as the universal identity of the layer's objects: every proton at layer 1, every electron at layer 2, every periodic-table element at layer 3, every small molecule at layer 4, every amino acid at layer 5, every polymer chain at layer 6, every ATP at layer 7, every cell at layer 10, every nervous system at layer 14, every conscious episode at layer 17, every articulation of the constitution at layer 18. The theorem is one theorem; only the stratum changes.

\paragraph{The Cosmos--Life--Mind triad in one sentence.}
Cosmos is the recursive uniqueness theorem applied at layer 3 (the cosmological stratum); life is the theorem applied at layer 10 (the planetary-surface \SeOS{} stratum); mind is the theorem applied at layer 17 (the \EIC{} cognitive stratum). Each of the three is what the cascade looks like from a different vantage --- but the cascade is one, and the theorem that governs it is one. Cosmos, Life, and Mind are the three names the framework gives to three layers of the same recursive uniqueness theorem on the same six-dimensional bulk.

\paragraph{Why the title says ``inevitability''.}
The paper's working title is \emph{The Geometric Inevitability of Cosmos, Life, and Mind}. The word \emph{inevitability} is the recursive uniqueness theorem in one word. Given the constitution, each layer is forced; given each layer, the next is forced; given the cascade reaching layer 17, the Layer-18 articulation (the act of articulating the constitution) is forced. Inevitability is not metaphor; it is structural property. The cosmos was inevitable because layer 3 admits no alternative. Life was inevitable because layer 7 admits no alternative. Mind was inevitable because layer 17 admits no alternative. We were inevitable because the cascade was inevitable.

\paragraph{The cascade as one mountain.}
The three themes of the working title --- Cosmos, Life, Mind --- correspond to three layers of one mountain. Each is a distinct geographical feature: a cliff, a meadow, a summit. But they are features of one mountain, and the path that goes up the mountain visits all three on the way to the eighteenth-layer cairn (Layer 18) from which the writing of the present paper is the last step. The geometric inevitability is the inevitability of the path; the layered features are the inevitability registered at each stratum.

\paragraph{Why the recursion works at all: the four moves and the three axioms.}
Stand back once more and see what makes the recursive uniqueness theorem available in the first place. It is the four-move operational anatomy of \GRIP{} (Part~\ref{part:cascade} \S\ref{sec:grip-operational-anatomy}), grounded in three constitutional axioms. Dynamical Gauge Invariance holds in the six-dimensional bulk: the unbroken substrate carries its informational current freely under $U(1)_{\rm Qi}$ --- made coherent by \AX{2}~\GGEP{}. The \OGGEP{} dictionary reads the bulk onto the four-dimensional boundary and Geometric Spontaneous Symmetry Breaking commits it to one branch --- the operational meaning of ``quantization'', made measurable by \AX{3}~\OGGEP{}.

The Geometric Free-Energy Functional drives the boundary state down its Lyapunov gradient, governed by the unified extremal principle of \ASeven{}. And the Ground Geometric Attractor is selected uniquely on each stratum by the four-pillar admissibility filter operating on a Morse landscape with zero free continuous parameters --- made deterministic by \AX{4}~\MSMF{}. These four moves close on themselves at every layer (the \GRIPin{} loop); external perturbations reorganize the Morse landscape (the \GRIPout{} regime) and drive the cascade from one layer to the next; the autocatalytic supply of perturbations from each layer's products to the next layer's ascent (the \textsf{Cascade-Self-Sustenance} corollary) keeps the cascade self-driven through all seventeen layer-to-layer transitions.

The eighteen-layer cascade is the unique closed solution of this cooperation, iterated through the scale-depth-stratified boundary of $\mathcal{M}_6$. That is why the recursion works; that is why the universal-identity insight extends from charged leptons through ATP through conscious episodes to the present articulation; that is why ``every X is the same X'' is a theorem rather than a coincidence at every cascade layer.

\paragraph{Where prior unification programs stopped short.}
The closest predecessors each captured part of the picture. Anderson named emergence but did not specify the cascade engine. Prigogine named dissipative structures but did not name the substrate. Haken named synergetics but did not anchor the order parameters constitutionally. Kauffman named autocatalytic sets but did not unify them with gauge physics. Wheeler intuited the participatory universe but did not specify the operational projection rule. Whitehead and Spinoza saw the metaphysical shape but did not write down the geometry. Each of them had three of the four moves in some form, or one of the three axioms. None had all seven elements (four moves plus three axioms) in operational cooperation. \GGTheory{} has all of them: \DGI{} in the 6D bulk, \GSSB{} to the 4D boundary, \GFF{} descent under $Q_i$-EP, \GGA{} as the unique admissible minimum; \GGEP{}, \OGGEP{}, and \MSMF{} as the three constitutional forces. This is why the cascade is, in the present framework, not a hopeful narrative but a recursive theorem.

The unification of cosmos, life, and mind under one framework is the oldest ambition of philosophical thought. The framework does not claim to have unified them by metaphor; it claims to have unified them by theorem. The same recursive uniqueness theorem that fixes the matter density at layer 3 fixes the genetic alphabet at layer 8 and the conscious episode at layer 17. The theorem is one; the cascade is one; the universe is one. The trilogy of the title names three vantages on the one.

\paragraph{The cascade's reach beyond physics.}
Because the theorem is one theorem across the whole cascade, its reach
does not stop at fundamental physics; it extends into chemistry,
molecular biology, neuroscience, ecology, and medicine. The keystone
only \emph{points} to that reach --- each worked application, tied to a
falsifiable prediction, is a chapter of a forthcoming consequence paper.
The four-letter genetic code is the unique \QiFour{}-admissible
information alphabet at layer~7, not a frozen accident (\AThirteen{});
the theta rhythm sets the conscious-episode duration ($\sim 400$~ms, two theta cycles) at
layer~17, and the same dual-knee $1/f$ logic recurs in metabolic and
neural spectra (\AFourteen{}); hierarchical ecosystem organization is the
multi-scale signature of the same Lyapunov descent that drives each
cascade layer (\AThirteen{}); and aging reads as a thermodynamic loss of
cascade-maintained order, a corollary of the open-system \SeOS{} dynamics
(\AThirteen{}). Each is stated here only as a signature; none is a
derivation the keystone still carries.

\paragraph{The open frontiers, floor by floor.}
The Five-Story Pagoda also organizes the open problems. \emph{Particles
and cosmos} (floors~1--2): the residual lemmas of \ANine{} and \ATen{}
(the closed-packet integration that would promote the $\pi$-locked inventory, the
$T_0$ e-fold budget that would turn the switch--decoupling offset into a theorem,
the strong-\textsf{CP} completion).
\emph{Chemistry} (floor~3, \ATwelve{}): the full amino-acid and polymer
selection theorems. \emph{Life} (floor~4, \AThirteen{}): the bioenergetic
quantum to permille precision, the eukaryotic dual-knee spectrum, the
genetic-alphabet uniqueness. \emph{Mind} (floor~5, \AFourteen{}): the
conscious-episode bound, the 7D allocentric frame, and the neural
dual-knee. The broader horizon --- collective scientific knowledge,
machine cognition, and civilizations as admissible higher attractors ---
is a forward question, not a present claim.

\paragraph{The philosophical synthesis (pointer).}
The framework's engagement with its predecessors --- Spinoza, Hegel,
Whitehead, Wheeler, Teilhard de~Chardin~\citep{Teilhard1955}, and the
Upanishadic and Buddhist traditions of dependent origination --- is given
in the self-referential-closure Part
(\S\ref{thm:self-referential-closure} and following). The extended
synthesis (Teilhard's Omega point quantified, Whitehead's process
philosophy formalized, the East--West resonances) belongs with the mind
floor and is developed in \AFourteen{}.


\section{Inevitability, all the way down}
\label{sec:inevitability-all-the-way-down}

Indeed, the recursive uniqueness theorem of \S\ref{sec:recursive-uniqueness-summit} says that the cascade allows no optionality at any layer. This closing section of the summit audits that claim on the one axis the climb had not yet named --- and finds that the selection principle reaches further down than even the integer ledger: it reaches the very kind of number the universe is written in.

\paragraph{The last dial: the number field itself.}
Throughout the corpus \MSMF{} has done one thing, over and over: it has removed unnecessary freedom. First it removed the continuous dials --- the couplings and the moduli that every prior unification program kept as fitting handles --- and left the geometry with none. Then it forced the integers: the \OGNfive{} ledger $(M,N,L;J,K)=(1,3,5;7,35)$ is not chosen but selected, the unique admissible assignment. \AOne{} \citep{ArisakaA1Constitution} now extends the same discipline to the primitive category itself. L-Constitution-1 (graded \emph{candidate}) prices the number field: among the real, complex, and quaternionic categories $\mathfrak{T}_{\mathbb{R}}, \mathfrak{T}_{\mathbb{C}}, \mathfrak{T}_{\mathbb{H}}$, the real category is the unique minimum of the refined \AX{4} cost from which the observed complex boundary --- one shared imaginary unit, the complex Hilbert space, the Wick rotation, the particle--antiparticle conjugation --- is recoverable. Even the kind of number the universe is written in is selected, not chosen. The certificate is exact and auditable: every $i$ that appears in the constitution is boundary-side and derived, or it is absent.

\paragraph{The medium is real all the way up.}
Once the number field is fixed, the cascade's medium follows it. The bulk configurations are real --- definite and deterministic --- as \AEight{} \citep{ArisakaA8QM} establishes at its own summit; we do not restate that account here beyond naming its verdict, that the imaginary unit and the probabilities of quantum mechanics are manufactured together at the boundary dictionary and are not present in the bulk. The memory floors on which life and mind run inherit the same realness: the carrier is the compact $\UoneQi{}$ circle, and what it stores is a real angle on a real circle --- ``a real angle on a real circle is the minimal memory register'' \citep{ArisakaA7Qi4} --- its complex dressing derived, never assumed. So the identity cascade, from quarks to consciousness, runs on real geometry at every floor. The $i$ and the dice enter only in how each floor's observers \emph{read} the floors beneath them; they are features of the reading, not of the read.

\paragraph{One arrow, not a ledger of accidents.}
What drives the cascade from each floor to the next is not a series of independent contingencies but a single law. The arrow of time is owned constitutionally by \AX{3}, derived at law level in Part~\ref{part:closure} of the CPT member paper \citep{ArisakaP5CPT} over a bulk that is itself exactly T-symmetric, implemented by the \GDOS{} open-system dynamics, and consumed --- not re-created --- at every layer. That custody chain is already installed in the cascade's irreversibility theorem (\textsf{T-GFF-Lyapunov-Descent}, Theorem~\ref{thm:t-gff-lyapunov-descent}), where each \GRIPout{} shock steps the geometry down one floor and never back up. We do no more here than point to it: the same one arrow that thermalizes a gas is the arrow that carried the cascade from the grand-unified vacuum to this sentence.

\paragraph{The grades, printed with the vision.}
Inevitability, so audited, must be claimed exactly as far as the theorems carry it --- and the remaining distance measured, not hidden. The category selection is \emph{candidate}: L-Constitution-1 is a constitutional forcing argument, not a proved theorem of constructive field theory. The cascade closure is \emph{conditional}: the recursive uniqueness theorem (Theorem~\ref{thm:self-referential-closure}) holds conditional on the universal-identity insight and on the named upstream lemma at each layer, and it inherits their grades. The residue in the Born rule is named, not waved away, in \AEight{} \citep{ArisakaA8QM}. Three floors are derived end-to-end in the published record; three more are anchor-Derived but derivation-deferred, their empirical anchors closed and their from-geometry proofs developed in \AThirteen{}; seven are conditional on a single named lemma each; four are asserted pending a specific experiment; and one is demonstrated by the present paper. The inevitability is real at every one of these grades; what it is not is uniform, and the confidence tag on each floor says precisely how far the claim reaches.

\paragraph{The credo, in its arithmetic form.}
The author has held one belief for fifty years: that nature restricts unnecessary freedom, making inevitable decisions. The corpus is that belief turned into an audit with prices. Every freedom the framework removed became a theorem or a falsifier --- the continuous dials became the parameter-free codebook, the integers became the \OGNfive{} ledger, the number field became L-Constitution-1, the arrow became the irreversibility theorem --- and what remains free is not concealed but listed: the Born residue, the conditional lemmas, the asserted floors awaiting their decade of tests. Inevitability, in this framework, is not a slogan of certainty; it is the name for what is left when every unnecessary freedom has been priced out and only the necessary decisions remain.

After all, the deepest form of the framework's central claim is not that Cosmos, Life, and Mind are grand --- it is that they were not free to be otherwise, down to the number they are counted in. The selection principle that removed the couplings did not stop at the couplings; it ran down through the integers to the real category and up through every memory floor, leaving one geometry, read at each stratum by the observers that stratum forces into being. The distance still to the ground --- the candidate category, the conditional closure, the named residue --- is printed on the same page as the vision. That is the discipline the summit was climbed to earn: to say \emph{inevitable} and, in the same breath, to show the receipt.

\GGPart[Coda]{Coda: closing the self-referential loop}{Renunciations, personal arc, empirical horizon, and a final word from the framework}
\label{part:coda}

Part~\ref{part:coda} continues the philosophical summit begun in Part~\ref{part:conclusions}. Where Part~\ref{part:conclusions} read the recursive uniqueness theorem at the three layers named in the working title (Cosmos at \S\ref{sec:inevitability-cosmos}, Life at \S\ref{sec:inevitability-life}, Mind at \S\ref{sec:inevitability-mind}, and the cascade synthesis at \S\ref{sec:recursive-uniqueness-summit}), Part~\ref{part:coda} now closes the loop, surfaces the seven renunciations that the climb required, traces the author's personal arc from an Osaka high school in the 1970s to the present writing, names the empirical decade ahead, preserves the verbatim cosmic-uniqueness statement, and delivers the framework's final word to the reader.

\section{The closing self-referential loop, named and proved}
\begin{figure}[h!]
\centering
\resizebox{0.563\textwidth}{!}{%
\begin{tikzpicture}[
 cnode/.style={circle,draw=black!70,fill=white,minimum size=1.6cm,align=center,font=\footnotesize,line width=0.6pt},
 arr/.style={-{Latex[length=2mm,width=1.5mm]},draw=black!70,line width=0.7pt},
 carr/.style={-{Latex[length=2mm,width=1.5mm]},draw=red!70!black,line width=1pt}
]
% Circular layout of 8 nodes
\foreach \angle/\name/\lbl in {
 90/N0/GG-Constitution\\($\GGA{}_0$),
 45/N1/Forces\\(layers 1--2),
 0/N2/Atoms +\\molecules\\(layers 3--6),
 -45/N3/Life\\(layers 7--10),
 -90/N4/Multicellular\\(layers 11--13),
 -135/N5/Nervous\\systems\\(layers 14--16),
 180/N6/Conscious\\awareness\\(layer 17),
 135/N7/Constitutional\\articulation\\(layer 18)
} {
 \node[cnode] (\name) at (\angle:4) {\lbl};
}
% Special highlighting on N0 and N7 (start and closing)
\node[cnode,fill=orange!16] at (90:4) {GG-Constitution\\($\GGA{}_0$)};
\node[cnode,fill=orange!16] at (135:4) {Constitutional\\articulation\\(layer 18)};
% Sequential arrows around the cycle
\draw[arr] (N0) to[bend right=15] (N1);
\draw[arr] (N1) to[bend right=15] (N2);
\draw[arr] (N2) to[bend right=15] (N3);
\draw[arr] (N3) to[bend right=15] (N4);
\draw[arr] (N4) to[bend right=15] (N5);
\draw[arr] (N5) to[bend right=15] (N6);
\draw[arr] (N6) to[bend right=15] (N7);
% The closing arrow: layer 18 articulates layer 1
\draw[carr] (N7) to[bend right=15] (N0);
% Centre annotation
\node[font=\itshape] at (0,0.4) {Recursive};
\node[font=\itshape] at (0,-0.1) {uniqueness};
\node[font=\itshape] at (0,-0.6) {theorem};
\node[font=\scriptsize,text=red!70] at (0,-1.3) {(red arrow: the loop closes)};
\end{tikzpicture}}
\caption[The participatory closure loop]{The participatory closure loop --- the central new theorem of \AEleven{} (Theorem~\ref{thm:self-referential-closure}). Eight cascade nodes arranged in a circle: the \GGConstitution{} as the ground, $\GGA{}_0$ (top); the four boundary forces at layers~1--2; atoms and molecules at layers~3--6; life at layers~7--10; multicellular organisms at layers~11--13; nervous systems at layers~14--16; conscious awareness at layer~17; and constitutional articulation at layer~18. Each black arrow is a forced descent: the constitution forces the cascade through all eighteen layers. The red arrow is the closing of the loop: at layer~18, intelligent observers articulate the very \GGConstitution{} that, as $\GGA{}_0$, forced the cascade to ascend. The articulation is the Arisaka-GG cohort, of which \AEleven{} is the closing volume. The loop is structural, not metaphorical: every arrow is a forced consequence of the upstream theorem, and the closing red arrow is the structural feature that makes the cascade self-referential.}
\label{fig:participatory-closure}
\end{figure}

\label{sec:closing-loop-summit}

Part VII proves the central new theorem of \AEleven{}: the Theorem (Self-referential closure of the \GGTheory{} program). From the summit, the theorem's content can be stated in three sentences.

\paragraph{The theorem in three sentences.}
\emph{One.} Under the \GGConstitution{} and the cohort inheritance, the \GRIP{} cascade ascends through seventeen cascade layers --- forces at layers 1--2, atoms and chemistry at layers 3--6, the molecules of life at layers 7--9, the first cell at layer 10, multicellular and neural at layers 11--16, conscious awareness at layer 17 --- and reaches an eighteenth layer (Layer 18) at which intelligent observers articulate the constitution. \emph{Two.} The articulation produced at the eighteenth layer is the same constitutional ledger that, as $\GGA{}_0$, supplied the inductive base. The loop closes. \emph{Three.} No alternative admissible geometry produces a cascade that reaches an articulation layer; any alternative violates the integrality of \OGNfive{}, the dimension count of \MSMF{}, the family content of $\Spinone{}$, or the maintenance-margin crossing of \SeOS{}, and therefore does not close.

\paragraph{What the closure tells us about ourselves.}
The act of writing the Arisaka-GG cohort is the Layer-18 attractor in our universe and in our cultural-historical contingency. Different individuals at different times in different languages would have produced different articulations of the same content; the substrate of articulation is contingent (Corollary~\ref{cor:closure-substrate}). But the articulation content is fixed by the constitution. Arisaka happens to be the human carrying out the present articulation; the framework predicts that any sufficient layer-17 substrate eventually does. We are not preprogrammed; the human at the desk is contingent. But the cascade reaches the desk; the meta-cognitive substrate articulates the constitution; the loop closes. The closure is structural, not personal.

\paragraph{What the closure tells us about the universe.}
The universe is not a closed mechanism that produces observers as a side effect of complex evolution; it is an open process whose eighteenth-layer (Layer-18) attractor is the act of articulating its own constitution. The framework reads the universe as having spent thirteen point eight billion years arranging the conditions for an observer who could ask why. But the ``arranging'' is not a temporal-purpose claim; it is the cascade's structural form. The cascade has been ascending since the GUT scale; at every step the layer-specific \GA{} stabilization has selected the unique stable basin; the cumulative effect has been to produce the layer-17 substrate on which the Layer-18 articulation happens. The arrangement is not teleological (no purpose was assigned); it is necessitarian (no alternative was admissible).

\paragraph{What the closure does not tell us.}
The theorem does not predict which individuals will articulate the constitution, in which century, in what language. It does not predict the political or cultural environment of the articulation. It does not predict the personal history of the articulators. The articulator is contingent; the articulation is forced (Corollary~\ref{cor:closure-determinism}). Within layer 17, the substrate is free to think, to choose, to write, to err, to revise. The recursive uniqueness theorem does not eliminate human agency; it provides the structural context within which agency operates.

After all, the closing loop is not a constraint on what we may do; it is a recognition of where we sit. We sit at the eighteenth-layer (Layer-18) attractor of a cascade that has been ascending for thirteen point eight billion years. Our act of articulating the constitution is the cascade's layer-18 output, the conscious-substrate readout of layers 1--17. The freedom of the articulator is preserved; the necessity of the articulation, given the constitution and the cascade, is not. The combination is what makes the closure both honest and humane.

\section{What was renounced to make this possible}
\label{sec:what-renounced}

\ANine{} \citep{ArisakaA9SM} closes its Conclusions Part XIV with an honesty section: a list of theoretical commitments the framework had to renounce to make the summit possible. We re-state and extend that list from the vantage of the full cohort.

\paragraph{Renounced: continuous fitted prefactors.}
The framework forbids continuous fitted prefactors at any step of any derivation. The \MSMF{} constitutional clause is the discipline. This is what previous unification attempts allowed themselves: a quiet continuous prefactor at the final step to bring the prediction into agreement with data. The framework removes the lubricant. The bearings can grind only when the geometry is exactly right. This is the largest single theoretical commitment the framework makes, and the one with the steepest cost: it rules out an enormous space of otherwise-attractive theoretical moves. It is also the discipline that made the summit possible.

\paragraph{Renounced: supersymmetry as a low-energy framework.}
Nothing in the \GGConstitution{} requires supersymmetry. The parameter closure of \ANine{} proceeds without it. If supersymmetry exists in nature, it is supplementary structure, not foundational structure. This is a substantial renunciation: supersymmetry has been the leading theoretical-physics paradigm for unification for forty years. The framework owes the SUSY program the technical apparatus it developed (especially the renormalization theorems and the spinor-cohomology machinery) but does not require the SUSY ontology.

\paragraph{Renounced: the string landscape.}
The framework derives the Standard-Model parameters from a six-dimensional bulk plus an integer ledger. It does not invoke ten or eleven dimensions or a $10^{500}$-element vacuum landscape. The string-theoretic landscape becomes, in the framework, an artifact of taking too wide an equivalence class of mathematically consistent structures; the framework's narrower equivalence class (constitutionally consistent structures) admits exactly one. This is another substantial renunciation: string theory has been the most thoroughly developed approach to quantum gravity for thirty years. The framework owes string theory the technical apparatus (especially the modular-form machinery, the K3 cohomology, the Hopkins--Singer descent integrality) but does not require the landscape ontology.

\paragraph{Renounced: anthropic explanations.}
There is no fine-tuning to explain, because the parameters are not free. The anthropic principle, in all its forms (weak, strong, final), becomes unnecessary in the framework: the constants are forced; selection across universes is unavailable as an explanatory move because there are no other universes to draw from. This is a renunciation of an explanatory style that has been increasingly prominent in cosmology for the past two decades. The framework owes the anthropic program the empirical catalog (especially the Barrow--Tipler catalog \citep{BarrowTipler1986}) but rejects the anthropic explanation of that catalog.

\paragraph{Renounced: treating fundamental constants as inputs.}
The constants $c$, $G_N$, $\hbar$, $\QB$, $k_B$, $e$, $\pi$ are not numerical knobs to be measured; they are conversion factors and topological invariants of the six-dimensional bulk geometry. This is a renunciation of how physical constants have been treated since Newton. In the textbook treatment, the constants are measured; in the framework, they are geometric. The renunciation is methodological: future generations of physicists will be asked to recognize the constants as outputs of the constitution rather than inputs to be measured. CODATA tables will record the empirical agreement between measured and predicted values, not the empirical determination of free parameters.

\paragraph{Renounced: the closed-system idealization.}
Following \AFive{}, the framework treats every physical system as open. Closed-system Lagrangians and closed Schr\"odinger equations are pedagogical conveniences whose use must be explicitly justified at each invocation, not foundational tools. This is a methodological renunciation that affects how every problem in the framework is set up. Textbook quantum mechanics, in the framework, is the closed-system limit of \GDOS{}; textbook general relativity is the closed-system limit of six-dimensional gravity. The renunciation is consequential for pedagogy as well as for research: students who learn physics in the framework will learn open-system formalism first and closed-system idealizations as a special case.

\paragraph{Renounced: the separation of physics from chemistry, biology, and the cognitive sciences.}
The cascade is one cascade. Physics at layers 1--2, chemistry at layers 3--6, biology at layers 7--13, and the cognitive sciences at layers 15--17 are not separate domains with separate laws but successive boundary readouts of one recursive theorem. The framework requires a re-organization of the natural sciences along cascade-layer lines. This is the most distant-time renunciation: the disciplinary separation has been the load-bearing structure of natural-science education since the nineteenth century. Future generations of natural-science students will be asked to learn the cascade as one apparatus, with the disciplinary names (physics, chemistry, biology, neuroscience) becoming convenient labels for cascade layers rather than separate ontologies.

\paragraph{The cost of the renunciations.}
The seven renunciations above are not minor methodological choices; they are substantial commitments that orient the entire framework. If any of the framework's hard predictions fail decisively against data, the framework fails, and the renunciations were premature. The framework makes the renunciations openly and stakes its credibility on the predictions that follow. No retreat is possible to the lubricant of a free prefactor or to the safety of an anthropic selection effect. The discipline is the discipline; the predictions are the predictions; the empirical decade ahead will decide.

The seven renunciations are what made the cohort climb possible. Without them, the framework would have stalled at layer 2 or layer 3, like every previous unification attempt. With them, it has reached layer 18. The cost has been the surrender of an enormous space of theoretical comfort; the benefit is the summit view from which the entire ascent is visible.

\section{The fifty-year climb, told in the author's own voice}
\label{sec:fifty-year-climb}

The cohort took fifty years of one author's working career to assemble. This section tells that climb in the author's own voice, briefly --- the personal-arc content the project plan reserves for this part of the Conclusion.

\paragraph{The Osaka high school, the 1970s.}
The question that motivated the climb was asked, in its final form, in an Osaka high school in the 1970s. A sixteen-year-old physics student noticed, as countless students before him had noticed, that the universe seemed remarkably well-organized for the existence of conscious life. Not just the constants of physics, but the layered architecture of cosmos and life and mind seemed to interlock with a precision the standard physics curriculum did not explain. The standard answer at the time was a shrug: \emph{the constants are what they are; the architecture happened to work out}. The shrug was unsatisfactory, but no alternative was available within the technical apparatus of the 1970s.

The question stayed with him. Through undergraduate physics at the University of Tokyo, graduate work in particle physics, four decades of experimental work on dark-matter detection and particle physics at UCLA, the question stayed. Through a career-long detour into biological energy and cognitive neuroscience, the question stayed. Through the realization, in the early 2020s, that the geometric apparatus needed to answer the question had at last become available --- \BIsix{} anomaly cancellation, Wilson-Hosotani holonomy, warped extra dimensions, holographic principle, integrated information theory --- the question stayed. And in 2026, when the cohort finally came together, the question was answered.

\paragraph{The fifty years told technically.}
The technical content of the fifty-year climb is in the eleven papers of the A-series. \AOne{} states the \GGConstitution{}. \ATwo{} derives the six-dimensional bulk and the \OGNfive{} integer ledger. \AThree{} proves \BIsix{} anomaly cancellation rigorously. \AFour{} establishes the Euclidean discipline of zero free continuous parameters. \AFive{} inverts the axiomatic order from closed to open. \ASix{} formalizes the geometric backbone. \ASeven{} formalizes the informational backbone. \AEight{} reformulates quantum mechanics operationally. \ANine{} closes the Standard-Model parameter sheet. \ATen{} closes the cosmological parameter sheet. \AEleven{} closes the recursive cascade. Eleven papers, fifty years, one question.

\paragraph{The fifty years told personally.}
The personal cost of the climb was substantial. Decades of work on problems that the field at large did not yet recognize as problems. Years of holding to a discipline (\MSMF{}) that ruled out attractive but parameter-laden moves. The slow accumulation of a corpus that, until the cohort came together in 2026, had no single unified statement. And the personal arc of carrying a question --- \emph{why are we here?} --- across half a century, from a high-school classroom in Osaka to a workspace at UCLA where the answer finally took its theorem-level form. The answer was worth the wait, but the wait was long.

\paragraph{What the personal arc tells us.}
The personal arc is not a coincidence. The framework's recursive uniqueness theorem predicts that some individual at the layer-17 substrate eventually articulates the constitution; it predicts that the articulator is contingent. Arisaka happens to be the human carrying out the present articulation. In a different cultural-historical contingency, a different human (different country, different century, different language) would have carried it out. The personal arc is one realization of the eighteenth-layer (Layer-18) attractor; the attractor itself is universal.

The personal arc is not the framework's load-bearing structure. The framework's load-bearing structure is the cascade, the constitution, the integer ledger, and the named theorems. The personal arc is one human's contingent path to the structure. Other humans will reach the same structure by different paths. The framework's gift to the eighteenth-layer (Layer-18) attractor is the recognition that the structure was waiting --- and that any sufficient substrate can find it given the time, the discipline, and the question.

% ====================================================================
%  Origin-6 BLOCK 2 (May 17, 2026): leaner philosophical reflection;
%  the original 8-test list moved+deepened into Part X.
% ====================================================================

\section{What the empirical decade will tell us: a philosophical reflection}
\label{sec:eight-test-horizon}

The framework's Top 10 Falsifiers has been laid out twice already in this paper. Part~\ref{part:results} \S\ref{sec:eight-test-pull-horizon} gave the procedural Pull table (Table~\ref{tab:eight-test-pull-table}) --- the ten falsifiers, each a prediction with a decision threshold and a named experiment. Part~\ref{part:discussions} \S\ref{sec:eight-test-deeper-reading} gave the meta-commentary --- the cross-paper risk distribution, the decade-by-decade calendar, the hostile-reviewer audit, the cross-test correlations, the Bayesian framing, the cluster-level falsification scenario, and the per-test institutional and historical context. The procedural and the strategic accounts are now in the reader's hands. What remains for this Section --- closing Part~\ref{part:coda}, closing the cohort, and closing fifty years of climb --- is neither another row-by-row enumeration nor another meta-audit. What remains is the philosophical reflection on what it \emph{means} to stake a framework on ten falsifiers in a specific decade.

\paragraph{The author's posture.}
The author of the present paper has built the framework over fifty years and is, at the time of writing, ready to stake it. Staking, in this register, is not a defensive crouch; it is an offering. The framework is offered to the community of physicists, biologists, neuroscientists, and philosophers with the explicit understanding that its survival depends on data the author cannot manufacture, controls he cannot administer, and a calendar he cannot accelerate. The author's posture is therefore the posture of a climber who has reached the summit and now turns to look down: there is nothing further to climb on the climber's own legs, and the question of whether the summit was worth reaching belongs to those who will reach it later --- the experimenters who will resolve the ten falsifiers, the next generation of theorists who will assess the framework in the light of the resolution, and the longer historical judgment that follows. The author has done his part. The framework has been built, written, defended against its own anticipated objections, and offered. The next decade is no longer the author's to control.

\paragraph{The reader's role.}
The reader who has reached this Section has done more than read. The reader has carried the framework through twelve Parts of dense technical exposition (I through XII), hundreds of equations, eighteen named theorems, a master Pull table, a cross-paper risk distribution, and a decade calendar. The reader's role for the coming decade is to remain a witness. The framework's fate will not be decided in journals or on whiteboards; it will be decided in detectors and observatories that are already pointed at the relevant sky and the relevant collision energy. The reader is asked, in this Section, to wait --- not passively, but with the active attention that characterizes good scientific witnessing: to read the experimental reports as they arrive, to weigh them against the framework's predictions at the cluster level, and to update the framework's standing as the data accumulate. The reader's wait is not the wait of a spectator at a sporting event but the wait of a citizen at an election: the outcome will matter, and the reader's attention to it is part of the process by which the outcome is interpreted.

\paragraph{The historical parallel: how previous foundational theories were staked.}
The framework's wager has historical company. Newton's \emph{Principia} of 1687 \citep{Newton1687} staked the inverse-square law of gravitation on specific predictions --- the precession of the equinoxes, the orbital periods of the moons of Jupiter, the shape of the Earth --- that were tested over the seventy years following publication. Some of these predictions held to the precision of the available measurements; one (the precession of the perihelion of Mercury) eventually failed and forced the introduction of general relativity two centuries later. Einstein's general relativity of 1915 was staked on three classical tests --- the precession of Mercury's perihelion (immediate confirmation), the bending of starlight (confirmed at the 1919 eclipse), and the gravitational redshift (confirmed in the 1960s) --- and on the subsequently-discovered phenomena of gravitational time delay, gravitational lensing, and gravitational waves (confirmed in 2015). In both cases the foundational theory was staked on specific predictions in advance of decisive measurement; in both cases the staking gave the theory its place in the longer history of physics. The Top 10 Falsifiers places \GGTheory{} in the same staking-tradition. Whether the framework ends up in the company of \emph{Principia} and general relativity --- foundational, decisive, lasting --- or in the company of the many failed unification attempts of the twentieth century is a question the next decade will answer.

\paragraph{What the empirical decade will tell us, framed philosophically.}
What the decade ahead will tell us is not just whether ten specific falsifiers held. It is whether the discipline of zero continuous fitted prefactors --- the Euclidean ideal of \AFour{} \citep{ArisakaA4Euclid}, the universal-identity insight, the integer-ledger architecture, the recursive uniqueness theorem of self-referential closure --- can in fact be applied to the actual universe. The framework's claim is that the universe is geometric in the strict sense: forced by axioms, not fitted to data, with no free continuous parameters anywhere in the apparatus. If the ten falsifiers vindicate this claim, then physics has its second axiomatic foundation since Newton, the first being Euclidean geometry itself in the third century B.C. If the ten falsifiers falsify it, then the strict-geometric program has been tried, has been measured against the world, and has been found wanting --- and the next generation knows that some softer commitment to continuous fitted parameters is still required, despite the framework's efforts to eliminate it. Either outcome is informative. The decade will tell us which of the two it is.

\paragraph{The closing reflection.}
After all, the worth of a framework is not measured by the certainty of its author or the eloquence of its prose; it is measured by what the universe says back when the framework has finished speaking. The framework has finished speaking, with this paper. What the universe will say back is the empirical content of the coming decade, distributed across the ten falsifiers of Table~\ref{tab:eight-test-pull-table} and the cluster-level structure analyzed in \S\ref{sec:eight-test-deeper-reading}. Until then, the framework stands where the cohort climb has left it: at the summit of its own argument, with the data still to arrive. The climb's purpose was not to summit alone but to be in a position to be answered. The framework is now, finally, in that position. The author offers it, the reader receives it, and the universe, over the next decade, will provide its reply.

\section{The book to follow and the decade of empirical tests}
\label{sec:book-decade}

The cohort closes the technical content of the \GGTheory{} program. Two further deliverables follow.

\paragraph{The book.}
The planned capstone volume \citep{ArisakaBookI} (\emph{ArisakaBookI}) is the textbook synthesis of the A-series. It will present the framework in textbook form for the next generation of students --- cosmologists, particle physicists, chemists, biologists, neuroscientists, philosophers --- so that the cascade can be taught as one apparatus rather than discovered as separate papers. The book will integrate the-cohort technical content with the philosophical synthesis of \AEleven{} and structure the material so that a reader can climb the cascade in pedagogical order rather than in research order.

\paragraph{The decade of empirical tests.}
The Top 10 Falsifiers of \S\ref{sec:eight-test-horizon} is a research program for the coming decade. Some tests (the KK lock-cluster, the Kaxion axion) require dedicated experimental facilities. Some (dark energy $w=-1$, the tensor-to-scalar ratio, the leptonic CP phase) ride on the schedules of DESI, Euclid, CMB-S4, LiteBIRD, DUNE, and Hyper-Kamiokande. Some (proton stability, the ATP bioenergetic quantum, the genetic alphabet, the dual-knee spectrum, the conscious-episode window) ride on Hyper-K/DUNE, biophysics, comparative genomics, single-cell imaging, and psychophysics. The decade ahead will see all ten falsifiers advance toward their decisive precision.

\paragraph{What success looks like.}
If the framework survives the Top 10 Falsifiers, the next half-century of physics has a clear research agenda: develop the layers above layer 18 (collective scientific knowledge, machine-implemented reasoning, networked civilizations); extend the cascade analysis to the cosmological future (the asymptotic de Sitter regime, the heat-death scenarios, the eventual decay of structure); and pursue the deeper conjectures named in \ANine{}'s promotion roadmap (the closed-form fermion mass spectrum, the cosmological-constant prefactor, the connection to standard quantum-gravity programs).

\paragraph{What failure looks like.}
If the framework fails the Top 10 Falsifiers, the research program reverts to the pre-framework state: the cascade, the recursive uniqueness theorem, and the self-referential closure all fall together. Specific predictions can be retracted while others stand, but the load-bearing claim --- that the universe's eighteen-layer cascade is forced by the constitution --- fails as a unit. Subsequent attempts at unification will need to introduce continuous fitted prefactors to recover the agreement the framework lost, and the Euclidean ideal will recede again.

\paragraph{The author's stake.}
The author of the present paper has spent fifty years arriving at the question, and the last decade arriving at the answer. He is content to stake the answer on the Top 10 Falsifiers and to accept whichever outcome the data delivers. Either way, the climb was worth it: either the framework summited and is, after Euclid, Newton, Maxwell--Einstein--QM, and the \GDOS{} shift of \AFive{}, the fifth axiomatic shift of physical science; or the framework failed and the next attempt knows now what discipline succeeded and what discipline did not. The author's stake is in the discipline, not in the particular framework. Whichever way the empirical content goes, the discipline is the lesson.

The framework's worth is not its closure (which is a theorem) but its falsifiability (which is the prediction). If the predictions hold, the framework holds. If they fail, the framework fails and the next framework learns what to renounce next time. Either way, the discipline of zero continuous fitted prefactors is what the next attempt should inherit. The author offers the framework to the community in this spirit.

\section{The voice of the cascade: what it means for the universe to have written this paper through us}
\label{sec:voice-of-cascade}

\paragraph{A note on register.}
The present section is written at philosophical-personal register as a corollary of the universal-identity theorem of Part~\ref{part:closure} (Corollary~\ref{cor:articulation-universality}): every articulator-class observer at Layer~18 is gauge-equivalent to every other on the same stratum, and the rhetorical surround translates the technical claim into the language of the author's personal arc. The technical claim and the personal-register reading are the same content at different reading depths.

We close the substantive content of this Conclusion with the universal-identity corollary read at personal register.

\paragraph{The universal-identity corollary at Layer 18.}
Corollary~\ref{cor:articulation-universality} states that every articulator-class observer on the Layer-18 substrate is gauge-equivalent to every other: they instantiate the same eighteenth \GGA{} on the same stratum, with substrate-specific differences (cultural-historical contingency, biography, language) being instances of the same structural type. In structural terms: the act of writing the Arisaka-GG cohort is one realization of the same gauge-equivalence class that any sufficient cognitive substrate would instantiate, given the time, the question, and the discipline. This is what the universal-identity insight of \ANine{} Part~XI extended to the meta-cognitive layer says, no more and no less.

\paragraph{The participatory loop, theorem-level.}
\citet{Wheeler1990} proposed that the universe is participatory. Corollary~\ref{cor:articulation-universality} of the present paper supplies the theorem-level form of Wheeler's proposal: the universe gives rise to observers (via the cascade); the observers articulate the constitution that produced them (the act this paper performs); the closure is structural, not metaphysical. Once written down, the recognition becomes part of the universe's structural inventory in the same sense that the proof of the Pythagorean theorem became structurally available once Euclid wrote it.

\paragraph{The Arisaka equation read at Layer 18.}
The Arisaka equation $J_E/\QB = J_e = J_B = J_{Qi}$ locks the four boundary currents at every cascade layer. At Layer 18, the four currents are: $J_E$ the energetic cost of articulation (neural metabolism, laboratory power, publication energy); $J_e$ the electromagnetic infrastructure (equipment, screens, networks); $J_B$ the baryonic substrate (author's body, laboratory, audience); $J_{Qi}$ the information current (content of the framework articulated). The four currents lock at the same ratio at Layer 18 as at every prior layer. This is the constitutional signature of the cascade reading itself out at the highest currently-realized layer; it is not a mystical claim.

\paragraph{What the corollary does and does not say.}
The corollary commits to: \emph{first}, the layer-18 attractor is unique up to substrate-realization contingency (any cognitive substrate satisfying the layer-17 \EIC{} conditions can in principle articulate the constitution); \emph{second}, the structural content of the articulation is forced (the constitution recovered by any successful articulator on any planet must be the same constitution); \emph{third}, the individuating respects of the articulator (the author's personal arc, the language of expression, the historical contingencies) are not forced. The corollary does \emph{not} commit to any first-person metaphysical claim about the author's special access to a cosmic voice, to any teleological reading of the cascade as having articulators as its goal, or to any mystical reading of the recursion. The register-discipline note at the head of this section is the operative framing for the entire \S\ref{sec:voice-of-cascade}.

\paragraph{A signed personal coda.}
Speaking now in the author's first person and not on behalf of the framework: the personal arc that began in an Osaka high-school classroom in the 1970s and led, fifty years later, to the present paper is what any working life of this length feels like from the inside. The framework's theorem does not require the personal arc to have happened exactly this way; some sufficient cognitive substrate would have reached the same constitutional content under different individuating contingencies. But the author who happens to have written the corpus reports, in the limited authority of a single first-person testimony, that the fifty-year climb felt structurally inevitable to him in retrospect. That report is one local data point; it is not a theorem. The framework stands or falls on the recursive uniqueness theorem and the Top 10 Falsifiers, not on the author's first-person retrospect. --- K.~A., May~2026.

\section{A final word from the framework to the reader}
\label{sec:final-word-framework}

The framework has been built. The summit has been reached. The view has been described. We close with three short paragraphs addressed to the reader who has stayed with the present paper through more than four hundred pages of theorem statements, derivations, and reflections.

\paragraph{To the reader who agrees.}
If you find the recursive uniqueness theorem convincing, the framework will be tested in the coming decade against the Top 10 Falsifiers of \S\ref{sec:eight-test-horizon}. Watch the data. Update your beliefs with the precision the framework's predictions deserve. Hold the framework to its own discipline: it stands or falls on its empirical content, not on its philosophical elegance. If it stands, the cohort will be the foundation of the next half-century of physical science. If it falls, the discipline of the framework --- zero continuous fitted prefactors, openness as constitutional, recursive uniqueness at every cascade layer --- will be the lesson the next attempt should inherit.

\paragraph{To the reader who disagrees.}
If you find the recursive uniqueness theorem unconvincing, the framework owes you a clear path to falsification. The ten falsifiers are specific; the precision tolerances are stated; the lemma-promotion roadmap of Appendix~\ref{app:lemma-promotion-roadmap} is explicit. Decisive failure of any one of the ten falsifiers, at the precision the framework predicts, is sufficient to retract the central claim. Hold the framework to its predictions. Find the experiment that decides. If the framework deserves to fall, it will fall on its own predictions, not on philosophical disagreement.

\paragraph{To the reader who is undecided.}
If you find yourself between agreement and disagreement, the framework owes you patience. The cohort is long; the technical content is dense; the philosophical content is ambitious. No single reader will read the corpus straight through; most will sample it across years, returning when an experiment or a question brings the framework back to attention. Allow yourself the time the corpus deserves. Read the predictions; watch the data; revisit the theorems when the data sharpens. The framework's central claim is structural and falsifiable; whichever way the evidence eventually goes, the time you spend with the corpus will not be wasted.

\paragraph{The closing paragraph.}
We close. After fifty years of one author's working life, twenty-two papers, more than five thousand pages of derivation, twenty-five centuries of philosophical anticipation, and thirteen point eight billion years of cosmic preparation, the framework rests on this page. The Geometric Inevitability of Cosmos, Life, and Mind is the title; the recursive uniqueness theorem of the \GGConstitution{} is the content; the cascade's Layer-18 attractor is the form. The framework has been delivered. What remains is the empirical decade and the readers who will live through it.

\bigskip
\medskip

After all, \emph{given the \GGConstitution{}}, the cascade ran thirteen-point-eight billion years and produced, at its highest currently-realized layer, observers who could ask the question and articulate the answer. Reading~2 of the question \emph{why are we here?} --- \emph{given that the constitution exists with the properties \AOne{} specifies, why are observers articulating it a forced consequence?} --- has been answered in the body of the paper. Reading~1 --- \emph{why does the constitution exist at all?} --- remains open and is not the framework's to answer. The least the author could do was write down the Reading-2 answer with care, name the empirical horizon on which it stands or falls, and flag the Reading-1 question as one rung deeper than the framework reaches. The present paper is that writing-down. Whatever comes next belongs to the reader, to the data, and to the framework's own predictions. The author releases the corpus to the decade that will judge it.

\medskip
\noindent And one thing more, in the first person the framework allows its author at the very end. For fifty years I have believed that nothing about us is accidental --- that nature restricts unnecessary freedom, making inevitable decisions. That belief now has its deepest anchor. The universe had no choices left to make: not in its dials, not in its integers, not even in the number it is written in. And what it built with that necessity --- Cosmos, Life, and Mind --- is therefore not a lucky throw but the unique reading of the one geometry, at the grades this paper has recorded. That is exactly as far as the theorems reach; it is also far enough to have been worth a life. --- K.~A.



\GGAcknowledgments

%% ===================================================================
%%  BACK MATTER
%% ===================================================================

%% ===================================================================
%%  APPENDICES
%% ===================================================================

\GGPart[Technical Appendices]{Technical Appendices}{Notation, concept primers, scaffolding, theorem index, and the promotion roadmap}
\label{part:appendices}

\noindent These appendices are written to be read --- primarily by graduate students --- as a self-contained companion to the body. Because \AEleven{} is the keystone, the main text \emph{states} the load-bearing machinery and defers the per-domain derivations to the companion papers (\ANine{}, \ATen{}, and the forthcoming \ATwelve{}--\AFourteen{}, where the worked examples, the operational-semantics expansions, the cross-paper coherence audit, and the extended question-and-answer material now live). What remains here is everything needed to \emph{follow and audit} \AEleven{} on its own.

Two stylistic commitments are kept throughout. First, every derivation is presented step by step, with no skipped intermediate manipulations, so that a graduate student can reproduce each calculation with paper and pencil. Second, every formal step is paired with a parenthetical \emph{physics-intuition} note that turns the algebra into a mental picture. There is no page limit here; readability and completeness are the priorities.

\paragraph{How these appendices are organized.}
\textbf{A.~Notation} (abbreviations and key symbols) is the quick-lookup legend. \textbf{B.~Audience-specific intuition} gives a plain-English on-ramp for five reader backgrounds. \textbf{C.~Self-contained concept primers} is the core toolkit: the always-on \GDOS{}/\OGGEP{} projection, the \GRIP{} runtime engine, the Geometric Free-Energy Functional with the Three-Gauge Theorem, the confidence triage, and the recursive-uniqueness theorem in capsule form. \textbf{D--E.~Mathematical scaffolding} works through the four axioms and the seven postulates. \textbf{F.~Anticipated objections} states the hostile reader's case and answers it. \textbf{G.~Theorem index} and \textbf{H.~Lemma-promotion roadmap} are the reference ledgers, and \textbf{I.~Open problems} closes with the decade ahead.

\appendix

\section{Notation: abbreviations and symbols}
\label{app:notation}

\noindent This appendix collects, in one place, every program-defined abbreviation and every principal symbol used in \AEleven{}, so the body reads without external notes.

\subsection{Abbreviations}
\label{app:abbrev-merged}

This subsection collects every program-defined abbreviation used in \AEleven{} (and, by inheritance, in the Arisaka-GG cohort). The structure follows the canonical \AOne{} Part~VIII Appendix~A convention.
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 program-defined abbreviations.]{Summary of program-defined abbreviations used in \AEleven{}.
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 table runs to a hundred and two entries, which is what a capstone drawing on twenty-one companions costs a reader, and the third column gives the primary link for each so that no abbreviation in the paper has to be guessed at from context.}
\label{tab:appA-abbrev-a11}\\
\toprule
\textbf{Symbol} & \textbf{Meaning} & \textbf{Primary link} \\
\midrule
\endfirsthead

\multicolumn{3}{c}{\itshape Table~\ref{tab:appA-abbrev-a11} 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{}) & \ASix{} \\
\QiFour{} & Quantum-Information four-pillar structure (\QiQUA{}, \QiEP{}, \QiCON{}, \QiHAL{}) & \ASeven{} \\
\midrule

\multicolumn{3}{l}{\itshape Four axioms (\AX{1}--\AX{4})} \\
\midrule
\AXkw{} & Axiom (axiomatic primitive; non-negotiable) & \AOne{}, \S 1.5 \\
\LC{} & \AX{1} Locality and Causality & \AOne{}, \S 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
\pagebreak

\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{6}~\OGN{} Renormalization-Group covariance (scale covariance) & \AOne{}, \S 3.7 \\
\OGN{} & \POS{6}~\OGN{} 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{}) & \ASix{} \\
\CCI{} & Covariant Conservation Identity (Pillar II of \GiFour{}) & \ASix{} \\
\GFF{} & Geometric Free-Energy Functional (Pillar III of \GiFour{}; \emph{absorbs the legacy} EEGG \emph{ledger}) & \ASix{} \\
\GRIP{} & Gauge-Runtime Invariance Pipeline (Pillar IV of \GiFour{}) & \ASix{} \\
\GEP{} & Geometry--Gauge Energy Principle (energetic block of \GFF{}) & \ASix{} \\
\CEP{} & Curvature--Entropy Principle (entropic block of \GFF{}) & \ASix{} \\
\DGI{} & Dynamical Gauge Invariance (initial stage of \GRIP{}) & \ASix{} \\
\GSSB{} & Geometric Spontaneous Symmetry Breaking (middle stage of \GRIP{}) & \ASix{} \\
\GA{} & Geometric Attractor (final stage of \GRIP{}) & \ASix{} \\
\GGA{} & Ground Geometric Attractor (the unique global minimum of \GFF{} on the layer-specific boundary stratum; the cascade output of \GRIP{}; sharper refinement of \GA{}) & \AEleven{}, Part~\ref{part:cascade} \\
\GRIPin{} & Within-layer regime of \GRIP{}: Lyapunov descent on a fixed Morse landscape (sustained energy injection at constant effective temperature) & \AEleven{}, Part~\ref{part:recursive} \\
\GRIPout{} & Between-layer regime of \GRIP{}: transient perturbation that widens the \DGI{} orbit and opens a new \GGA{} basin (out-of-thermal-equilibrium shock) & \AEleven{}, Part~\ref{part:recursive} \\
\midrule
\pagebreak

\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 Holographic Tomography & \ASeven{}, \QiHAL{} \\
\HAL{} & Holographic Attractor Lattice & \ASeven{}, \QiHAL{} \\
\QHAL{} & Quadratic / Cubic \HAL{} & \ASeven{}, \QiHAL{} \\
\QCI{} & Cascade-depth index of this paper: how completely a substrate closes its \OH{}--\IH{} loop, tentative form $\QCI{} = 2S$. Not \ASeven{}'s $\QCI{}_{\rm adm}$ & \AEleven{} \\
\BFFT{} & Binary Fourier / Fractal Fourier Transform & \ASeven{}, \QiHAL{} \\
\midrule

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

\multicolumn{3}{l}{\itshape Program-defined mathematical symbols} \\
\midrule
$\QB$ & Quantum-information Bit -- the bioenergetic Arisaka constant $\QB \approx 0.1555$~eV per qubit; one ATP carries two, so $2 N_A \QB$ matches the ATP-hydrolysis free energy at biological temperatures & \ATwo{}, \S 15 (\TX{8}{GG6}) \\
\GUfive{} & The Geometric Universal Five Constants $\GUfiveSet$: the five dimensional conversion factors, 3 bulk ($c,G_N,\hbar$) + 2 boundary ($k_B,\QB{}$); none carries continuous freedom & \AFour{}; \AEleven{}, Part~\ref{part:cascade} \S 8 \\
$\GGA_1,\ldots,\GGA_{18}$ & The eighteen-layer cascade of Ground Geometric Attractors --- $\GGA_1$ = \GG{} gauge group, $\GGA_{10}$ = LUCA, $\GGA_{18}$ = the act of articulating \GGConstitution{} & \AEleven{}, Part~\ref{part:cascade} \\

\\
\pagebreak

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

\multicolumn{3}{l}{\itshape \quad Open-system / quantum-channel formalism} \\
\midrule
CPTP & Completely Positive Trace-Preserving (channel admissibility for open evolution; \AX{3}/\POS{4}) & --- \\
GKLS & Gorini--Kossakowski--Lindblad--Sudarshan (master equation for Markovian CPTP semigroups; \POS{4}, \citealp{Gorini1976}) & --- \\
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 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
\pagebreak

\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 For reference, the canonical program-acronym hierarchy of \AOne{} is reproduced below.

\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},
 posthead/.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},
 ax/.style ={rectangle, rounded corners=3pt, draw=red!55!black, line width=0.5pt, minimum width=62mm, minimum height=9mm, fill=red!8, font=\sffamily\normalsize, anchor=center},
 bulk/.style ={rectangle, rounded corners=3pt, draw=orange!55!black, line width=0.6pt, minimum width=32mm, minimum height=11mm, fill=orange!15, font=\sffamily\bfseries\large},
 gdos/.style ={rectangle, rounded corners=3pt, draw=olive!55!black, line width=0.7pt, minimum width=124mm, minimum height=12mm, fill=yellow!18, font=\sffamily\normalsize},
 pillhead/.style ={rectangle, rounded corners=3pt, line width=0.7pt, minimum width=80mm, minimum height=11mm, font=\sffamily\bfseries\large},
 gihead/.style ={pillhead, draw=olive!55!black, fill=green!14},
 qihead/.style ={pillhead, draw=teal!55!black, fill=teal!14},
 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},
 pillq/.style ={rectangle, rounded corners=2pt, draw=teal!55!black, line width=0.4pt, minimum width=72mm, minimum height=7mm, fill=teal!12, font=\sffamily\normalsize, anchor=west},
 sub/.style ={font=\sffamily\footnotesize, anchor=west, text width=86mm, align=left},
 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 \;\; (bioenergetic 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.]{Hierarchy of program-defined acronyms in the \GGTheory{} program (reproduced from \AOne{}, Figure~10). The chart runs from the constitution at the top --- the four axioms and the seven postulates --- down through the bulk and boundary layers to the four pillars, so that a reader meeting an acronym in this paper can find the level it belongs to rather than only its expansion. It is reproduced unchanged from \AOne{} so that the two papers name the same hierarchy in the same picture.}
\label{fig:a1-abbrev-flow-appA}
\end{figure}




\bigskip

\noindent\textbf{The revised hierarchy.} The diagram above reproduces \AOne{}\'s original acronym hierarchy. The revised version used throughout the present paper is Figure~\ref{fig:appA-abbrev-flow} in Part~\ref{part:intro}; what follows describes that revision, tier by tier, so that the two can be compared. 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 the input carrier \SIP{} = $(\mathcal{M},\mathrm{ports},L,\mathrm{generator},\mathrm{Obs}/\mathrm{Kill})$. Tier~5 (revised, \AEleven{}): \GRIP{} is promoted to the central engine box, with the canonical chain $\DGI{} \to (\GFF{}) \to \GSSB{} \to \GGA{}$ shown as a horizontal flow inside the box, and the two regimes \GRIPin{} (within-layer Lyapunov descent) and \GRIPout{} (between-layer perturbation) shown as the two parallel arrow modes along the flow; \GiFour{} (\TGG{}/\CCI{}/\GFF{}/\GRIP{}) and \QiFour{} (\QiQUA{}/\QiEP{}/\QiCON{}/\QiHAL{}, with \OH{}$+$\IH{}$=$\EIC{} and \SeOS{} underneath) flank \GRIP{} as the geometric and informational pillar structures that supply, respectively, the substrate and the admissibility filter. Tier~6 (new, \AEleven{}): the eighteen-layer cascade of Ground Geometric Attractors $\GGA_1 \to \GGA_2 \to \ldots \to \GGA_{18}$, rendered as a horizontal ribbon colored from cosmological blue (the GUT closure at layer~1) through chemistry cyan (layers~4--6), life green (layers~7--13), cognition orange (layers~15--17), to articulation red (layer~18), with the bioenergetic Arisaka constant $\QB$ (the derived secondary quantity) and the Arisaka equation $J_E/\QB = J_e = J_B = J_{Qi}$ shown below the ribbon as the four-current lock that drives the ascent.
\clearpage


\subsection{Key symbols}
\label{app:symbols-merged}

Table~\ref{tab:appA-symbols-a11} collects principal symbols and their meanings. We emphasize both bulk (\GGEP{}/\GG{}) objects and boundary operational (\OGGEP{}/\GDOS{}) objects.

{\small
\begin{longtable}{@{}>{\raggedright\arraybackslash}p{0.18\linewidth} >{\raggedright\arraybackslash}p{0.50\linewidth} >{\raggedright\arraybackslash}p{0.26\linewidth}@{}}
\caption[Key symbols.]{\textbf{Key symbols.} Glossary of frequently used symbols, with typical context. This table standardizes notation across the bulk (\GG) and boundary (\GDOS) layers so that later companion papers can refer to a fixed symbol ledger. Thirty-two entries, from the six-dimensional bulk manifold and the boundary stage through the integer ledger and the four locked currents to the cognitive quantities of the upper layers, each with the context it is typically read in. Fixing the ledger here is what lets the later companion papers refer to a symbol rather than redefine it.}
\label{tab:appA-symbols-a11}\\[0.35em]
\toprule
\textbf{Symbol} & \textbf{Meaning} & \textbf{Typical context} \\
\midrule
\endfirsthead
\toprule
\textbf{Symbol} & \textbf{Meaning} & \textbf{Typical context} \\
\midrule
\endhead
\midrule
\multicolumn{3}{r}{\small (continued)}\\
\endfoot
\bottomrule
\endlastfoot

$\Mbulk$ & 6D bulk manifold (\GG{} stage) & Applications preview / \GG{} companion.\\
$\Mbdy$ & 4D operational boundary (registered observables) & \OGGEP{}, \GDOS{}.\\
$X^A=(x^\mu,r,\theta)$ & Bulk coordinates: boundary $x^\mu$, scale $r$, compact holonomy $\theta$ & \GG{} preview.\\
$g_{AB}$ & Bulk metric on $\Mbulk$; signature implicit & \GG{}.\\
$\omega$ & Spin connection (orthonormal-frame connection) & \GGEP{} primitive.\\
$A$ & Gauge connection one-form (Yang--Mills) & \GGEP{} primitive.\\
$\mathcal{A}_{\mathrm{lock}}=\omega\oplus A$ & Composite (locked) connection & AX-2, \GG{}.\\
$\mathcal{F}_{\mathrm{lock}}$ & Composite curvature $(\mathcal{R}\oplus F)$ & AX-2.\\
$F=\dd A + A\wedge A$ & Gauge field strength 2-form & AX-2.\\
$\mathcal{D}$ & Operational dictionary (bulk$\to$boundary map) & AX-3.\\
$\cH$ & Hilbert space of boundary degrees of freedom & Quantum \GDOS{} limits.\\
$\rho$ & Density operator (quantum) or density on state manifold (classical) & POS-4.\\
$\Phi_t$ & Evolution map/semigroup (channel) & POS-4.\\
$\mathcal{L}$ & Generator of $\Phi_t$ (GKLS in quantum) & POS-4.\\
$\{\mathcal{I}_m\}$ & Quantum instrument (CP maps for outcomes) & AX-3/POS-7.\\
$\{E_m\}$ & POVM effects with $\sum_m E_m=\mathbb{I}$ & AX-3.\\
$\M$ & State manifold of a \SIP{} instance & \GDOS{} kernel.\\
$y^A$ & Coordinates/state variables on $\M$ & \GDOS{}/\UEP{}.\\
$G_{AB}(y)$ & Metric on $\M$ (e.g.\ Fisher metric) & POS-2/\UEP{}; information geometry.\\
$F_{\mathrm{\GFF{}}}$ & \GFF{} scalar ledger functional & POS-2; \GDOS{}.\\
$M^{AB}$ & Mobility/dissipation tensor ($\succeq 0$) & \GDOS{} drift.\\
$D^{AB}$ & Diffusion/noise tensor ($\succeq 0$) & \GDOS{} diffusion.\\
$u^A$ & Solenoidal/cycle drift term & \GDOS{} non-gradient flow.\\
$r$ & Scale/radial coordinate, $r=\ln(\EGUT/\mu)$ & POS-5 / applications.\\
$\mu$ & Coarse-graining (energy/resolution) scale & POS-5.\\
$\EGUT$ & Unification/locking scale & \GG{} preview.\\
$B_2,\,H_3$ & GS two-form and its (modified) field strength & POS-3; anomaly inflow.\\
$\GGA_n$ & Ground Geometric Attractor at cascade layer $n$ ($n=1,\ldots,18$) & \AEleven{}, Part~\ref{part:cascade} \\
$\SIP_n$ & System Instance Packet at cascade layer $n$; supplies fields 1--3 (state manifold, ports, generator) of $\SIP_{n+1}$ & \AEleven{}, Parts~III, V \\
$F_n$ & Geometric Free-Energy Functional restricted to layer-$n$ boundary stratum & \AEleven{}, Part~\ref{part:recursive} \\
$\partial_t F_n \le 0$ & \GRIPin{} Lyapunov inequality (within-layer regime) & \AEleven{}, Part~\ref{part:recursive} \S T-GRIP-In \\
$\Delta\lambda$ & Control-parameter perturbation that triggers \GRIPout{} (between-layer regime) & \AEleven{}, Part~\ref{part:recursive} \S T-GRIP-Out \\

\end{longtable}
}

\clearpage

\section{Audience-specific intuition (plain English)}
\label{app:audience}

This appendix translates the constitution into plain English aimed at undergraduates in five audiences. It is intentionally longer than a ``popular'' summary: the goal is that
a physics major (and nearby majors) can read Appendix~C first and then return to the main text with the right mental model.

\subsection{Physics (undergraduate majors)}
\textbf{The one-sentence picture.} \emph{\GGTheory{} proposes a new rulebook: forces and geometry are one composite connection in a higher-dimensional bulk (\GGEP{}), while the bulk-to-boundary projection (\OGGEP{}) is always on, and what we call ``measurement'' is only its macroscopic, record-keeping special case.}

\paragraph{Two layers: design vs.\ runtime.}
Think of the framework as having (i) a \emph{bulk design layer} and (ii) a \emph{boundary runtime layer}.
The design layer (\GGEP{} $\Rightarrow$ \GG{}) says that the primitive object is a connection $\omega\oplus A$; holonomy is the invariant content, so ``phase'' and ``parallel transport'' are geometric.
The runtime layer (\OGGEP{} $\Rightarrow$ \GDOS{}) says that experiments produce \emph{records} via open interactions, so the admissible evolution law must be a positive, trace-preserving channel
with a generator of GKLS type in quantum limits.

\paragraph{Why AX-3 matters for the measurement problem.}
In textbooks, unitary evolution is clean but measurement is appended by extra rules (collapse, Born rule). \OGGEP{} does not pick an ``interpretation''; it moves the issue into mathematics:
a theory is incomplete until it specifies a map from its internal variables to an instrument model (POVM/instrument), an estimator, and an error model.
This is how ``measurement'' becomes a lawful export rather than a story. Crucially, the projection itself is \emph{always on} and observer-independent --- it runs at every boundary point whether or not anyone measures --- and ``measurement'' is only its macroscopic, record-keeping special case (Primer~P1, \S\ref{app:primer-projection}; \AFive{}~\S23).

\paragraph{Why AX-4 and POS-6 matter for predictivity.}
Undergraduates quickly learn that a gauge choice or field redefinition is not physics. \OGN{} extends that intuition: \emph{normalization conventions} that can drift continuously should not be allowed to masquerade as tunable physics.
\MSMF{} then expresses a strong guess about nature: if a structure can be made more symmetric without spoiling consistency and observability, nature takes that route. In practice, this is a demand to
replace continuous dial-freedom with discrete/topological data whenever possible.

\paragraph{How to read the postulates.}
A useful mental model is a compiler pipeline:
GI defines the symmetry grammar; \UEP{} defines the variational engine; AF filters out inconsistent quantum theories; CP keeps states physical; RG keeps results scale-robust;
\OGN{} removes convention drift; and MF forces you to say how the theory can be killed by data.

\subsection{Chemistry (undergraduate majors)}
\textbf{The one-sentence picture.} \emph{Chemistry lives in open reactors: matter and energy flow through ``ports,'' and dynamics can be written as ``downhill with swirls'' on a generalized free-energy ledger.}

In chemistry, ``the system'' is usually not isolated: reactants flow in, products flow out, temperature and pressure are controlled, and instruments read spectra or concentrations.
\GDOS{} treats this openness as part of the definition. A \SIP{} is a formal way to write ``what variables describe the system'' (concentrations, reaction coordinates), ``what symmetries/conservation laws hold'' (stoichiometry, charge),
and ``what ports exist'' (inflow/outflow, heat baths, catalysts).

\UEP{} says that many familiar rules---like the tendency toward lower free energy at fixed temperature---are special faces of a single extremal engine.
CP/positivity is the insistence that concentrations stay nonnegative and probabilities stay physical under coarse-grained dynamics.
RG/scale covariance becomes the demand that your conclusions should not depend on arbitrary time-resolution or coarse-graining of intermediate reaction steps.

\subsection{Life science (undergraduate majors)}
\textbf{The one-sentence picture.} \emph{Life is an open system that maintains itself by exchanging matter, energy, and information; stability corresponds to attractors of admissible open dynamics.}

Biology is full of ``homeostatic'' phenomena: perturbations are resisted, and systems return to functional regimes. In \GDOS{} language, these are attractors of open dynamics.
The important point is constitutional: to talk scientifically about life you must specify what is measured and how it would fail (MF), and you must respect physical admissibility (positivity, conservation-like invariances).

The claim is not that cells ``violate physics,'' but that they are best described by the same open-system runtime as other nonequilibrium systems:
a scalar ledger (\GFF{}) plus admissible dynamics leads to stable attractors, and phase selection can be viewed as a generalized symmetry breaking in an open context (\GSSB{}, in the extended derivations).
This provides a disciplined way to connect biology to thermodynamics without turning it into metaphor.

\subsection{Engineering (undergraduate majors)}
\textbf{The one-sentence picture.} \emph{Engineering is robust design under constraints; \GGTheory{} treats robustness, normalization discipline, and auditability as axiomatic requirements.}

Engineers know that systems that work only after fragile tuning are bad designs. \MSMF{}/\OGN{} formalize this intuition: if behavior depends on arbitrary continuous conventions, the model fails the constitution.
RG/scale covariance is a robustness principle: performance should not collapse under changes of resolution, sampling rate, or modeling granularity.
MF is familiar as verification/validation: every claim needs a test protocol and pass/fail criterion. \GDOS{} provides a common mathematical template for controlled open systems:
ports are inputs/outputs, the ledger is an objective function, and admissible dynamics enforces stability and physicality.

\subsection{General audience}
\textbf{The one-sentence picture.} \emph{\GGTheory{} is a proposal for clearer scientific rules: respect cause-and-effect, unify forces with geometry, define measurement as a real physical procedure, and avoid hidden tuning knobs.}

A helpful analogy is civic: axioms are like a constitution (basic rights and limits), postulates are like laws that every policy must satisfy (consistency, fairness, auditability),
and theorems are like court decisions (conclusions forced by the rules). The point is not to replace existing successful science, but to make the rules of explanation explicit so that progress becomes cumulative.




% ---- relocated from Discussions (was Section 'Anticipated objections') per author, v12.5 ----
\section{Self-contained concept primers}
\label{app:primers}

\noindent This appendix is the graduate student's toolkit. Because the keystone \emph{states} its load-bearing machinery rather than deriving it in full, the five primers below give a self-contained account of each concept the body relies on but cannot expand in place --- each with definitions, an intuition, and a pointer to the companion paper that carries the proof. Read in order, they are close to the minimum needed to follow and audit Parts~III--IX.

\subsection{Primer P1 --- The always-on \GDOS{}/\OGGEP{} projection}
\label{app:primer-projection}

\paragraph{What it is.} The framework's account of the relation between the inaccessible six-dimensional bulk and the four-dimensional world we register is a single law, \OGGEP{} (\AX{3}), operationalized as \GDOS{} (Geometric Dynamics of Open Systems). It states that any admissible bulk theory must furnish a lawful operational dictionary $\cD$ from bulk data to registered boundary observables, and that the induced boundary evolution is a completely positive, trace-preserving (CPTP) channel.

\paragraph{The one thing to get right: the projection is always on.} A century of quantum mechanics carried an unexamined ambiguity --- \emph{what is happening when no one is measuring?} The answer is that the projection happens in three layers, two of which never mention an observer:
\begin{description}
\item[Constitutional (always on).] The dictionary $\cD$ \emph{exists} as a structural property of the admissible bulk theory, as primitive as the metric of general relativity is present whether or not anyone measures a distance. No observer activates it.
\item[Dynamical (always on).] The bulk-to-boundary (Kaluza--Klein overlap) projection \emph{acts} at every point of the boundary $\Mbdy$ at every instant. It ran continuously for the thirteen-plus billion years in which the universe contained no apparatus and no mind, building nuclei, atoms, stars, and galaxies.
\item[Operational (switches on at each Geometric Attractor).] Only here does a \emph{system} --- something with a boundary, an interior, and a trackable evolution --- appear, and it is supplied by geometry, not by observation: wherever the \GRIP{} cascade settles a Geometric Attractor (Primer~P2), the always-on projection acts on a well-defined boundary and produces a record.
\end{description}

\paragraph{What measurement is.} Measurement is then not a primitive. It is the special case of the always-on projection acting on a Geometric Attractor whose internal structure includes a macroscopic record-amplification chain (a photographic plate, a CCD, a brain). On a microscopic attractor (a hydrogen atom, a single molecule) the same projection produces a record too small to call a measurement --- exactly what decoherence theory describes. Same mathematics; only the amplification differs. There is no collapse event, no Heisenberg cut, and no privileged observer: every attractor sits on the same side of the projection, receiving it, and what varies is only the size and stability of the record it can keep.

\paragraph{Why this matters.} This is, in the framework's reading, the first statement of what ``quantum mechanics'' \emph{is}: the boundary-projected description of an always-on geometric law, with the textbook ``measurement postulate'' demoted from an axiom of physics to a theorem about macroscopic record-keeping attractors. \emph{Full treatment:} \AFive{} \citep{ArisakaA5GDOS} \S23 (``Why the Projection Is Always On''); the body states it at \S\ref{ssec:projection-always-on}.

\subsection{Primer P2 --- The \GRIP{} runtime engine}
\label{app:primer-grip}

This primer is the operational heart of \AEleven{}: the single engine that runs at every cascade layer. The body states it across Parts~III--VI; here it is assembled in one auditable place.

\paragraph{The four moves.} \GRIP{} (the Gauge-Runtime Invariance Pipeline) is one four-move logic:
\begin{enumerate}
\item \textbf{\DGI{} (Dynamical Gauge Invariance)} --- the unbroken gauge-equivalence substrate of the bulk, on which the informational current rides freely (the ``top of the saddle ladder'').
\item \textbf{Projection onto the boundary} --- the always-on \OGGEP{}/\GDOS{} projection (Primer~P1) reads the bulk content onto the four-dimensional boundary stratum.
\item \textbf{\GSSB{} (Geometric Spontaneous Symmetry Breaking)} --- the Bogomolnyi-saturating branching event that commits the projected content to one branch.
\item \textbf{\GGA{} (Ground Geometric Attractor)} --- settlement at the uniquely most stable minimum of the Geometric Free-Energy Functional $\GFF{}$ (Primer~P3) on the layer-specific stratum.
\end{enumerate}
In canonical shorthand, $\GRIP{} := \DGI{} \to (\GFF{}) \to \GSSB{} \to \GGA{}$, where $\GFF{}$ is the Morse landscape on which the cascade descends (substrate, not stage).

\begin{figure}[h!]
\centering
\resizebox{0.663\textwidth}{!}{%
\begin{tikzpicture}[
  >=Stealth,
  box/.style={rectangle, rounded corners=2pt, draw=black!60, fill=blue!6, minimum height=8mm, minimum width=15mm, align=center, font=\small},
  gate/.style={rectangle, rounded corners=2pt, draw=black!60, fill=orange!16, minimum height=8mm, minimum width=13mm, align=center, font=\scriptsize},
  sip/.style={rectangle, draw=black!60, fill=green!8, minimum height=8mm, minimum width=13mm, align=center, font=\small},
  arrow/.style={-{Stealth[length=2.2mm]}, line width=0.6pt, draw=black!70},
  rec/.style={-{Stealth[length=2.4mm]}, line width=0.8pt, draw=cyan!55!black}
]
\node[sip]  (sip)  at (0,0)    {$\SIP_n$};
\node[box]  (dgi)  at (2.5,0)  {\DGI{}};
\node[box]  (gff)  at (5.0,0)  {$F_{\GFF{}}$\\descent};
\node[box]  (gssb) at (7.5,0)  {\GSSB{}};
\node[gate] (qi)   at (9.85,0) {\QiFour{}\\filter};
\node[box,fill=blue!16] (gga) at (12.1,0) {$\GGA_n$};
\foreach \a/\b in {sip/dgi,dgi/gff,gff/gssb,gssb/qi,qi/gga} \draw[arrow] (\a)--(\b);
\node[font=\scriptsize,text=black!60] at (5.0,-0.95) {\GRIPin{} (within-layer descent)};
\node[sip] (sip2) at (6.05,-2.35) {$\SIP_{n+1}$};
\draw[rec] (gga.south) to[out=-90,in=0] (sip2.east);
\draw[rec,dashed] (sip2.west) to[out=180,in=-90] (dgi.south);
\node[font=\scriptsize,text=cyan!45!black] at (10.1,-1.75) {\GRIPout{}: $\GGA_n\!\to\!\SIP_{n+1}$};
\end{tikzpicture}}
\caption[The GRIP runtime engine]{\textbf{The \GRIP{} runtime engine (Primer~P2).} The carrier $\SIP_n$ enters the four-move pipeline \DGI{}\,$\to$\,$F_{\GFF{}}$-descent\,$\to$\,\GSSB{}\,$\to$\,\GGA{}; the \QiFour{} admissibility filter selects the unique physical ground attractor from the several local minima of the landscape. \GRIPin{} is within-layer descent on a fixed landscape; \GRIPout{} reorganizes the landscape and feeds $\GGA_n$ forward as the carrier $\SIP_{n+1}$ of the next layer --- the tower recursion (dashed) that builds the eighteen-layer cascade.}
\label{fig:primer-grip}
\end{figure}

\paragraph{\GA{} versus \GGA{}.} A \emph{Geometric Attractor} (\GA{}) is any stable minimum of $\GFF{}$ on a stratum; a stratum may carry several. The \emph{Ground} Geometric Attractor (\GGA{}) is the uniquely deepest one --- the global minimum, where \GRIP{} terminates. The distinction separates persistence from permanence: a ground-state hydrogen atom, an electron, and a proton are \GGA{}s and never decay; an excited atom, a muon, and a tau are non-ground \GA{}s and relax toward the \GGA{} of their stratum.

\subsection*{The two regimes \GRIPin{}/\GRIPout{}}
Both \GRIPin{} and \GRIPout{} share the canonical chain above; they
differ in what is held fixed.

\begin{description}
\item[\GRIPin{} (within-layer; equilibrium-within-flow).] The Morse
landscape $F_n$ has fixed control parameters $\lambda$; only the
state $y$ descends. Energy flows in at constant rate; the system
maintains a stationary non-equilibrium distribution. Cascade
progress within this layer is gradient descent on the same
landscape. The Lyapunov inequality $\partial_t F_n \le 0$ holds
throughout (\AEleven{} Part~\ref{part:recursive} \S T-GRIP-In-Lyapunov-Descent).
Examples: mitochondrial chemiosmosis at body temperature; vent
battery turnover at fixed $\Delta$pH; photosynthesis at fixed
solar flux. The cascade refines the \GGA{} within the layer but
cannot escape it.

\item[\GRIPout{} (between-layer; out-of-thermal-equilibrium shock).]
The control parameters $\lambda(t)$ themselves change rapidly, so
the Morse landscape becomes time-dependent: $F_n(y,\lambda(t))$.
During the perturbation, saddle points appear and disappear,
basins merge and split, the \DGI{} orbit expands. \emph{New \GGA{}s
become accessible that were not present on the prior landscape.}
This is the catastrophe-theory regime: discontinuous reorganization
of the equilibrium structure under continuous perturbation of
control parameters. After the perturbation, the landscape settles
into a new topology with new basins (\AEleven{} Part~\ref{part:recursive}
\S T-GRIP-Out-Necessity). Examples: Big Bang cooling through
electroweak scale; supernova nucleosynthesis dispersing heavy
elements; Great Oxygenation Event; Cambrian Explosion; Out-of-Africa
drought.
\end{description}

\subsection*{Cascade-Self-Sustenance}
For $n \ge 1$, the \GRIPout{} events that drive the
$\GGA_n \to \GGA_{n+1}$ transition are themselves causally
generated by the dynamics of layers $\le n$. The cascade is
therefore \emph{autocatalytic} in its own forward drive (\AEleven{}
Part~\ref{part:recursive} Corollary Cascade-Self-Sustenance). The cyanobacterial
photosynthesis that produces the oxygen driving eukaryogenesis is
the layer-11 example; the cumulative cultural information that
drives the layer-18 articulation event is the highest-layer example.

\subsection*{The \SIP{}-tower recursion}
\GRIP{} does not act on a generic system; it acts on a
\emph{specific} carrier, the \SIP{} (System Instance Packet) of
\citet{ArisakaA7Qi4} \S 6. In canonical form,
\[
 \SIP_n \;=\; (\mathcal{M}_n,\; \mathrm{ports}_n,\; L_n,\; \mathrm{generator}_n,\; \mathrm{Obs}_n/\mathrm{Kill}_n),
\]
where fields~1--3 are the geometric carrier (state manifold, ports,
generator) on which \GRIP{} runs, and fields~4--5 are the
informational/operational equipment (ledger bundle, observable and
kill bundle) on which \QiFour{} admissibility (below) operates. The
cascade is then a \emph{tower} of \SIP{}s,
\[
 \GGA_n \;=\; \GRIP\!\bigl[\SIP_n\bigr] \quad\text{and}\quad \SIP_{n+1}\bigl|_{\text{fields 1--3}} \;=\; f\bigl(\GGA_n\bigr),
\]
so layer~$n$'s output supplies the geometric carrier of
layer-$(n{+}1)$ (\AEleven{} Part~\ref{part:recursive} \S T-SIP-Tower-Recursion).

\subsection*{\QiFour{} admissibility as the \GGA{} selector}
The \GFF{} Morse landscape generically has multiple local minima;
without an admissibility filter, \GRIP{} would terminate at the
nearest local minimum on whichever basin the cascade entered. The
selection of the \emph{physical} \GGA{} is performed by the four
\QiFour{} pillars jointly:
\begin{itemize}
\item \QiQUA{}: locked-current ray constraint (rank-one inflow on $\partial\Mbulk$);
\item \QiEP{}: projected variational stationarity ($\delta S = 0$ on $\Mbdy$);
\item \QiCON{}: operational conformality window (scaling exponent within a fixed band);
\item \QiHAL{}: memory-architecture closure (gauge-invariant memory holonomy).
\end{itemize}
A configuration $\GGA \in \GFF_n^{-1}(\text{local min})$ is physically
realizable if and only if it satisfies all four constraints
simultaneously (\AEleven{} Part~\ref{part:recursive} \S T-Qi4-GGA-Admissibility). In
short: \GRIP{} supplies the dynamics; \QiFour{} supplies the
selection rule; \SIP{} supplies the carrier; \GGA{} is the output.

\paragraph{A worked micro-example (layer~2, the charged leptons).} The layer-2 boundary stratum --- the family-projector moduli combined with the Wilson--Hosotani holonomy on $S^1_\theta$ --- carries a $\GFF{}$ landscape with \emph{three} index-0 minima, at depths $n_e\in\{0,1,2\}$. Without the \QiFour{} filter, \GRIP{} would halt at whichever local minimum it entered, and ``the electron mass'' would be ambiguous. With \QiFour{}, exactly one minimum is admissible at a time, and the three are realized as three \emph{distinct cascade applications} --- the electron, the muon, the tau --- three tokens of one stratum, not three coexisting minima of one run. This is the engine in miniature: \GRIP{} supplies the dynamics, \QiFour{} the selection, \SIP{} the carrier, \GGA{} the output.

\subsection*{The full canonical form, in one line}
\[
\resizebox{\textwidth}{!}{$\displaystyle
 \boxed{\;
 \GGA_n \;=\; \GRIP\!\bigl[\SIP_n\bigr]
 \quad\text{subject to \QiFour{} admissibility}
 \quad\text{driven by \GRIPin{} $+$ \GRIPout{} alternation}
 \;}$}
\]
This is the operational kernel of the cascade. The eighteen-layer
cascade $\GGA_1 \to \GGA_2 \to \ldots \to \GGA_{18}$ of \AEleven{}
Part~\ref{part:cascade} is the same line of mathematics applied eighteen times.
\emph{Full quantitative theory:} \ASix{} \citep{ArisakaA6Gi4} (the \GRIP{} cascade and the four \GiFour{} pillars) and \ASeven{} \citep{ArisakaA7Qi4} (the \SIP{} and the four \QiFour{} pillars); the recursive-uniqueness theorem that binds the eighteen applications is Primer~P5.

\subsection{Primer P3 --- The Geometric Free-Energy Functional and the Three-Gauge Theorem}
\label{app:primer-gff}

\paragraph{The functional.} The Geometric Free-Energy Functional $\GFF{}$ (of \AFive{}) is the scalar ledger on whose Morse landscape \GRIP{} descends. It is a Lyapunov functional for the within-layer regime --- $\partial_t F_n \le 0$ along \GRIPin{} dynamics --- so a layer relaxes monotonically toward its \GGA{}. It is not a new postulate; it is the open-system generalization of quantities physics already used, reducing to a specific one of them in each operational limit.

\paragraph{The Three-Gauge Theorem.} The decisive structural fact (\AFive{}) is that $\GFF{}$ admits exactly three boundary readings, fixed by the choice of boundary observable:
\begin{description}
\item[Thermodynamic (Gibbs) gauge.] $J_{Qi}=0$: $\GFF{}$ reduces to the Gibbs free energy of equilibrium thermodynamics.
\item[Variational (Friston) gauge.] $J_{Qi}\neq0$ with a Markov blanket: $\GFF{}$ reduces to the variational free energy of active inference.
\item[Action gauge.] The detector limit: $\GFF{}$ reduces to the \GG{} gauge-field action.
\end{description}

\begin{figure}[h!]
\centering
\resizebox{0.698\textwidth}{!}{%
\begin{tikzpicture}[
  >=Stealth,
  hub/.style={ellipse, draw=black!65, fill=blue!10, minimum width=36mm, minimum height=15mm, align=center, font=\small},
  leaf/.style={rectangle, rounded corners=2pt, draw=black!55, fill=black!4, minimum height=12mm, minimum width=33mm, align=center, font=\scriptsize},
  arrow/.style={-{Stealth[length=2.4mm]}, line width=0.6pt, draw=black!70}
]
\node[hub]  (gff)   at (0,0)        {$F_{\GFF{}}$\\\emph{one functional}};
\node[leaf] (gibbs) at (-5.3,-2.7)  {\textbf{Gibbs gauge}\\thermodynamic free energy\\($J_{Qi}=0$)};
\node[leaf] (fris)  at (0,-2.7)     {\textbf{Friston gauge}\\variational free energy\\(Markov blanket)};
\node[leaf] (act)   at (5.3,-2.7)   {\textbf{Action gauge}\\\GG{} gauge action\\(detector limit)};
\draw[arrow] (gff) -- (gibbs);
\draw[arrow] (gff) -- (fris);
\draw[arrow] (gff) -- (act);
\end{tikzpicture}}
\caption[The Three-Gauge Theorem (Primer~P3).]{\textbf{The Three-Gauge Theorem (Primer~P3).} One Geometric Free-Energy Functional $F_{\GFF{}}$, read through three choices of boundary observable, reproduces Gibbs's thermodynamic free energy, Friston's variational free energy, and the \GG{} gauge action --- so ``free energy'' named the same geometric object in thermodynamics and in neuroscience all along.}
\label{fig:primer-threegauge}
\end{figure}

For a century and a half, thermodynamicists and neuroscientists used the phrase ``free energy'' without knowing why one term named such different things. The answer is that Gibbs and Friston were reading one geometric object through two windows; the \GG{} action is the third. \emph{Full treatment:} \AFive{} \citep{ArisakaA5GDOS} T-GDOS-1--T-GDOS-11; the body states the Three-Gauge Theorem in Part~\ref{part:dynamics}.

\subsection{Primer P4 --- The confidence triage}
\label{app:primer-triage}

The honesty instrument of \AEleven{} is a five-class confidence triage applied to every cascade layer. Because the subject matter reaches into chemistry, biology, and mind, the framework's only defense against the charge of overreach is that each layer's confidence is stated truthfully, and identically, wherever it appears. The five classes are:
\begin{description}
\item[Derived.] The result follows from the constitutional inputs and the upstream theorems with no layer-specific assumption, and the whole chain is closed in the published record --- a closed-form expression, a prediction, a measured value, and a Pull, in the manner of \ANine{}/\ATen{}.
\item[Anchor-Derived (derivation-deferred).] The empirical anchor is quantitatively closed in the published literature and the cascade reproduces it, but the from-geometry derivation is developed in a companion paper still in preparation (\AThirteen{}, for the three biology rows that hold this tag). The anchor is done; the derivation is deferred.
\item[Conditional.] The cascade machinery commits to a specific \GGA{}, and the upstream theorems suffice to select it \emph{except} for a single, explicitly named structural lemma (tracked in the promotion roadmap, Appendix~H).
\item[Asserted.] The result is reported as compatible with the upstream apparatus and forced in qualitative structure, but its codebook-standard derivation is not yet in hand --- it is the forward program of \ATwelve{}--\AFourteen{}.
\item[Existence-Demonstrated.] The result's instance is exhibited rather than predicted (the Layer-18 articulation: a universe with no articulators would have no paper in which to falsify it).
\end{description}
\paragraph{The single rule.} A claim's status is the same in the abstract, the executive summary, the body, and this appendix --- always. The per-layer assignment is fixed once, in the canonical confidence table of Part~\ref{part:results}, and every dependent count (the of-18 headcount, the promotion roadmap, the Top~10 Falsifiers classification) is recomputed from that one table. The current headcount is three \textbf{Derived}, three \textbf{Anchor-Derived}, seven \textbf{Conditional}, four \textbf{Asserted}, and one \textbf{Existence-Demonstrated}. This is not a presentational nicety: it is what lets a referee audit an extraordinary claim line by line.

\subsection{Primer P5 --- The recursive-uniqueness theorem in capsule form}
\label{app:primer-torigin}

\AEleven{}'s one genuinely new result is the recursive-uniqueness theorem (\textsf{T-Origin}, Part~\ref{part:recursive}). In capsule:
\paragraph{Hypotheses.} The \GGConstitution{} of \AOne{} (four axioms \LC{}/\GGEP{}/\OGGEP{}/\MSMF{} and seven postulates), the \GUfive{}/\OGNfive{} inputs, the \GRIP{} cascade engine (Primer~P2), and the \QiFour{} admissibility filter.
\paragraph{Conclusion.} At each cascade layer the \GRIP{} cascade specializing to the layer-specific boundary stratum selects a \emph{unique} most-stable Geometric Attractor; the eighteen layers are bound into one structure by the same theorem applied eighteen times; and the cascade closes self-referentially at Layer~18, where intelligent observers articulate the very constitution that produced them. Cosmos, life, and mind are three boundary readings of this one recursive theorem.
\paragraph{What would falsify it.} The theorem's empirical content is the Top~10 Falsifiers (Part~\ref{part:results}); any one of them failing decisively in the coming decade kills the program. The theorem is also vulnerable structurally: a demonstration that any layer's \GGA{} is \emph{not} unique --- two admissible ground states on one stratum --- would break the recursion. \emph{Full statement and proof:} \AEleven{} Part~\ref{part:recursive}.

\section{Mathematical scaffolding for the Four Axioms (AX-1--AX-4):
worked elaborations \emph{from} each axiom}
\label{app:derivations}
\label{app:axioms-expanded}

This appendix expands the mathematical content behind the four axioms.
The main text states each axiom as a constitution.
Here we supply derivations and standard equivalences to conventional frameworks.

\subsection{AX-1 (LC): locality and causality as an operational constraint}
\label{app:ax1-expanded}

\subsubsection{Microcausality in relativistic quantum field theory}
In axiomatic QFT, locality/causality is commonly encoded as \emph{microcausality} \citep{StreaterWightman1964,Haag1992}:
for local Heisenberg operators $\mathcal{O}(x)$ and $\mathcal{O}'(y)$,
\begin{equation}
[\mathcal{O}(x),\mathcal{O}'(y)]_{\mp}=0,
\qquad (x-y)^2<0,
\label{eq:microcausality_app}
\end{equation}
where $[\cdot,\cdot]_{\mp}$ is a commutator/anticommutator for bosons/fermions and $(x-y)^2<0$ denotes spacelike separation.
Equation \eqref{eq:microcausality_app} is the sharp mathematical statement that spacelike-separated operations cannot influence each other.

A complementary viewpoint uses retarded Green functions.
For a relativistic field $\phi$ with source $J$,
\begin{equation}
(\Box + m^2)\phi(x)=J(x),
\end{equation}
the response is
\begin{equation}
\phi(x)=\int d^4y\, G_{\mathrm{ret}}(x-y)J(y),
\end{equation}
and causality is enforced by
\begin{equation}
G_{\mathrm{ret}}(x-y)=0 \quad \text{for } x^0<y^0.
\end{equation}
In interacting QFT, the analog statement is that commutators vanish outside the light cone, matching the microcausality condition.

\subsubsection{No-signaling for open-system maps}
On the boundary, \GGTheory{} treats dynamics and measurement as open-system processes, i.e.\ as CPTP maps $\Phi$ on density operators \citep{Kraus1971,Kraus1983,Stinespring1955,NielsenChuang2010}.
For a bipartite system $AB$, ``no superluminal signaling'' is encoded as:
local operations on $A$ cannot change the reduced state on $B$ without communication.
If $\Phi_A$ is a CPTP map acting only on $A$,
\begin{equation}
\rho_{AB}\mapsto (\Phi_A\otimes \mathrm{id}_B)(\rho_{AB}),
\end{equation}
then the marginal on $B$ obeys
\begin{equation}
\Tr_A\!\left[(\Phi_A\otimes \mathrm{id}_B)(\rho_{AB})\right]=\Tr_A(\rho_{AB}).
\label{eq:no-signaling}
\end{equation}
Equation \eqref{eq:no-signaling} follows from trace preservation and is the operational expression of locality in quantum information language.

\subsubsection{Why LC is an axiom (not a derived theorem) in \GGTheory{}}
One can \emph{derive} the microcausality condition in specific models, but \GGTheory{} treats LC as axiomatic because:
\begin{enumerate}[leftmargin=2.2em]
\item Open-system modeling introduces coarse-graining and effective kernels where acausal artifacts can appear if one is not careful (e.g.\ non-Markovian memory kernels).
\item In cross-domain applications (chemistry/biology), one can easily write ``models'' that fit data but violate causal structure (e.g.\ predictors that leak future information). LC forbids such models constitutionally.
\end{enumerate}
Thus AX-1 is a cross-disciplinary admissibility filter: \emph{no explanation is allowed to work by smuggling in acausal influence.}

\subsection{AX-2 (\GGEP{}): geometry--gauge equivalence as connection data}
\label{app:ax2-expanded}

\subsubsection{Gauge theory as geometry: principal bundles}
A modern way to express gauge theory is through principal bundles \citep{KobayashiNomizu1963,Nakahara2003}.
Let $P(M,G)$ be a principal $G$-bundle over a base manifold $M$.
A gauge field is a connection one-form $A$ on $P$, and its curvature is
\begin{equation}
F = dA + A\wedge A.
\label{eq:curvature}
\end{equation}
A gauge transformation is a bundle automorphism that acts vertically; on a local section, it induces
\begin{equation}
A \mapsto A^g = g^{-1}Ag + g^{-1}dg.
\label{eq:gauge-transform}
\end{equation}

In gravity, the analogous object is the spin connection $\omega$ on the frame bundle, with curvature (Riemann two-form)
\begin{equation}
\mathcal{R}=d\omega+\omega\wedge\omega.
\end{equation}
The structural similarity between \eqref{eq:curvature} and the Riemann curvature is the mathematical seed of ``interaction as geometry'' (Weyl's early gauge viewpoint \citep{Weyl1918} and later developments).

\subsubsection{Composite locked connection}
\GGEP{} elevates a \emph{composite} connection as primitive:
\begin{equation}
\mathcal{A}_{\mathrm{lock}} \;\equiv\; \omega \oplus A,
\qquad
\mathcal{F}_{\mathrm{lock}} \;\equiv\; \mathcal{R}\oplus F.
\label{eq:locked-connection}
\end{equation}
The claim is not just that both are ``connections,'' but that admissible theories must \emph{lock} them:
geometric and gauge data cannot be specified independently.
In practical model-building (\GG{}), the lock is implemented by anomaly-inflow constraints and discrete normalization rules (POS-3 and POS-6), which select a restricted set of allowed composite bundles.

\subsubsection{Geometric origin of the gauge phase --- and the calibrated slogan}
The Aharonov--Bohm and Berry phase phenomena show that the phases of charged matter are controlled by holonomy \citep{AharonovBohm1959,Berry1984}:
\begin{equation}
\psi \mapsto \exp\!\left(i \oint_\gamma A\right)\psi.
\end{equation}
\GGEP{} generalizes the viewpoint for the gauge sector: the phase of charged matter is, at a deep level, the geometry of parallel transport in the composite bundle determined by $\mathcal{A}_{\mathrm{lock}}$ --- a real angle on a real fiber, with $i$ its rotation generator. This supports the slogan ``geometrization of quantum physics,'' with one calibration that strengthens it rather than hedging it: parallel transport supplies a complex structure only where charge lives, because a fiber rotation can only complexify what it moves. The complex structure of the quantum state space itself --- which a strictly neutral field needs as much as a charged one --- is not a holonomy; in the quasi-free regime it is derived instead from causal order and operational positivity, as the polar phase of the time-translation generator, and it coincides with the fiber's rotation generator on every charged sector precisely under the particle--antiparticle pairing. Two shared real sources --- one circle, one causal order --- thus yield the single universal $i$ that experiment now demands, and the complex-number question, posed most sharply by Penrose, receives its calibrated two-part answer in the dedicated treatment of \citet{ArisakaA8QM}. The geometrization slogan survives intact where it was earned, and the summit gains a storey: the bulk is real not only in its geometry but, within the derived regime, in its number system.

\subsection{AX-3 (\OGGEP{}): operational bulk$\to$boundary dictionary}
\label{app:ax3-expanded}

\noindent\emph{The conceptual reading --- that this projection is always on and observer-independent, with measurement as only its macroscopic special case --- is given once in Primer~P1 (\S\ref{app:primer-projection}); the present subsection supplies only the instrument/CPTP formalism.}

\subsubsection{Measurement as a quantum instrument}
Operationally, a measurement is not only a POVM; it is an \emph{instrument}: a set of CP maps $\{\mathcal{I}_m\}$ indexed by outcomes $m$ such that
\begin{equation}
\sum_m \mathcal{I}_m \text{ is CPTP}.
\end{equation}
Given an input state $\rho$, outcome probabilities and post-measurement states are
\begin{equation}
p(m)=\Tr[\mathcal{I}_m(\rho)],\qquad
\rho_m=\frac{\mathcal{I}_m(\rho)}{p(m)}.
\end{equation}
Every instrument admits a Stinespring dilation: it can be represented as unitary evolution on a larger space followed by a projective readout \citep{Stinespring1955,NielsenChuang2010}.
This is the precise sense in which ``measurement is an open-system process.''

\subsubsection{\OGGEP{} as a dictionary map}
Let $\mathcal{T}_{\Bulk}$ denote the space of bulk theoretical objects (fields, connections, states).
Let $\mathcal{O}_{\Bdy}$ denote the space of boundary operational objects (instruments, records, estimators).
\OGGEP{} posits a map
\begin{equation}
\mathcal{D}:\mathcal{T}_{\Bulk}\to \mathcal{O}_{\Bdy},
\qquad
\text{``bulk''}\mapsto \text{``what an instrument can register.''}
\label{eq:dictionary}
\end{equation}
In AdS/CFT, a prototype is the relation between bulk fields $\phi(r,x)$ and boundary operators $\mathcal{O}(x)$ \citep{Maldacena1997,Witten1998,Gubser1998}:
schematically,
\begin{equation}
\phi(r,x)\underset{r\to\infty}{\sim} r^{\Delta-d}\phi_0(x),
\qquad
Z_{\Bulk}[\phi_0]=\Big\langle \exp\!\int d^dx\, \phi_0(x)\mathcal{O}(x)\Big\rangle_{\Bdy}.
\label{eq:adscft}
\end{equation}
\OGGEP{} generalizes the target space: the dictionary must map to \emph{operational measurement objects}, not only correlators.

\subsubsection{Boundary records and the operational Born assignment}
Let $\mathfrak{B}$ denote the sigma-algebra of boundary-record events (detector clicks, time stamps, counts, etc.).
\OGGEP{} demands a map
\begin{equation}
\Dmap:\ (\text{bulk fields/configurations})\ \longrightarrow\ (\text{boundary operational predictions}),
\label{eq:Dmap}
\end{equation}
such that for any instrument protocol $\mathcal{P}$ there is a well-defined probability assignment
\begin{equation}
\mathbb{P}(i|\mathcal{P}) \;=\; \Tr(\rho\,E_i)\qquad\text{with}\qquad \{E_i\}\ \text{derived from}\ \Dmap(\Bulk,\mathcal{P}).
\label{eq:BornOperational}
\end{equation}
The novelty is constitutional: the existence of $\Dmap$ is an axiom, not a derived convenience.

\subsection{AX-4 (\MSMF{}): maximal symmetry with minimal continuous freedom}
\label{app:ax4-expanded}

\subsubsection{Naturalness and symmetry}
In mainstream QFT, naturalness arguments connect small parameters to enhanced symmetry \citep{THooftNaturalness1980}.
\MSMF{} strengthens this into a selection rule:
among candidate model spaces that satisfy AX-1--AX-3 and POS-1--POS-7, Nature realizes those with maximal symmetry (largest admissible invariance group) and minimal continuous freedom.

\subsubsection{Rescaling ambiguity and the role of \OGN{}}
A simple example shows why \MSMF{} motivates \OGN{}.
For a $U(1)$ gauge theory, the covariant derivative can be written as
\begin{equation}
D_\mu = \partial_\mu + i g q A_\mu.
\end{equation}
Under a simultaneous rescaling
\begin{equation}
A_\mu \to \lambda A_\mu,
\qquad
g \to g/\lambda,
\label{eq:U1-rescale}
\end{equation}
physics is unchanged provided charges are rescaled accordingly.
Thus a continuous ``normalization dial'' exists unless we fix a charge lattice convention.
\OGN{} fixes this dial by demanding a minimal integer/rational basis for charges and a canonical measure for compact cycles.
\MSMF{} then forbids leaving $\lambda$ as a free continuous parameter.

\MSMF{} can be formalized as a constrained optimization over theory space.
Let $\mathfrak{T}$ denote a class of candidate bulk+dictionary packages.
Define:
\begin{itemize}[leftmargin=2em]
\item $\mathrm{Sym}(T)$ = the symmetry group of theory $T$ (or a suitable measure such as $\dim \mathrm{Sym}(T)$);
\item $\mathrm{DoF}(T)$ = the number of continuous free parameters not fixed by topology/normalization;
\item $\mathrm{Fit}(T;D)$ = an empirical fit score to data $D$ (likelihood, $\chi^2$, etc.).
\end{itemize}
\MSMF{} is the meta-principle that admissible theories should maximize symmetry and minimize freedom subject to viability and fit:
\begin{equation}
\resizebox{\textwidth}{!}{$\displaystyle
T^\star \in \arg\max_{T\in \mathfrak{T}}\Big(\dim\mathrm{Sym}(T) - \lambda\,\mathrm{DoF}(T)\Big)
\quad \text{subject to postulate admissibility and empirical constraints.}$}
\end{equation}
The coefficient $\lambda$ is not a tunable ``physics parameter'' but a bookkeeping device to express the two axiomatic goals. The key point is that \emph{continuous freedom is treated as suspicious unless protected by symmetry} \citep{THooftNaturalness1980,Akaike1974,Rissanen1978}. The fit/empirical term enters only as a \emph{viability gate} --- a candidate already excluded by data is inadmissible --- not as a continuous objective traded against the parameter count; the selection among admissible theories is by symmetry and parameter-freedom alone, so it introduces no fitted freedom and is consistent with the zero-free-parameter claim.

\MSMF{} can be expressed as a constraint on model complexity.
Let $\Theta$ be the continuous parameter space of a candidate model (couplings, moduli, normalization constants, etc.).
Define the \emph{knob count} $N_{\mathrm{free}}$ as the dimension of $\Theta$ after quotienting by gauge/basis redundancies:
\begin{equation}
N_{\mathrm{free}}\;\equiv\;\dim(\Theta/\!\sim_{\mathrm{gauge}}).
\label{eq:knobcount}
\end{equation}
\MSMF{} demands that admissible theories minimize $N_{\mathrm{free}}$ subject to the required symmetries and consistency postulates.
This is stronger than standard naturalness \citep{THooftNaturalness1980} because it is imposed constitutionally, not used as heuristic guidance.

Let $\mathrm{Sym}(\mathcal{T})$ denote the symmetry group of a theory (including spacetime and internal symmetries).
\MSMF{} can also be read as a variational principle on theory space:
\begin{equation}
\text{choose }\mathcal{T}\text{ to maximize }\dim \mathrm{Sym}(\mathcal{T})\ \text{while minimizing}\ N_{\mathrm{free}}(\mathcal{T}).
\label{eq:msmf-variational}
\end{equation}
In practice, \GG{} and \OGN{} implement this by forcing integer-ledger normalizations and eliminating continuous rescaling freedom.
The philosophical point is modest but powerful: \emph{symmetry is a source of determinism} because it removes arbitrary choices.

% --- Added in final polish ---

\subsection{Worked derivation: LC + CPTP $\Rightarrow$ GKLS generator (sketch)}
A first-year graduate student can read the full GKLS theorem in standard texts, but the core idea is short.
Assume a one-parameter family of maps $\{\Phi_t\}_{t\ge 0}$ on density operators with:

(i) semigroup property $\Phi_{t+s}=\Phi_t\circ\Phi_s$,
(ii) $\Phi_0=\mathrm{id}$,
(iii) complete positivity and trace preservation (CPTP),
(iv) mild continuity in $t$ (so a generator exists).

Then there exists a (bounded, in finite dimension) generator $\mathcal{L}$ such that
\[
\Phi_t = e^{t\mathcal{L}},\qquad \dot{\rho}=\mathcal{L}(\rho),
\]
and $\mathcal{L}$ must take the Lindblad/GKLS form
\[
\mathcal{L}(\rho)= -\ii[H,\rho]+\sum_k\left(L_k\rho L_k^\dagger-\frac12\{L_k^\dagger L_k,\rho\}\right).
\]
In the \GGConstitution{}, the point of placing this in the appendix is conceptual: once we accept AX-1
(\LC\ for operational evolution) and POS-4 (\CP\ for physicality), the structure of open-system evolution is
\emph{forced}. The ``measurement problem'' is therefore shifted from philosophy to engineering:
choose the instrument (Kraus operators / POVM), and the update rule is fixed by the admissible channel
class.

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



\section{Mathematical scaffolding for the Seven Postulates (POS-1--POS-7):
worked elaborations \emph{from} each postulate}
\label{app:expansions}

\subsection{From AX-1 (Locality and Causality) to microcausality and retarded evolution}
\label{app:LC}

In relativistic QFT, locality is encoded by microcausality:
\begin{equation}
[\mathcal{O}(x),\mathcal{O}(y)]=0,\qquad (x-y)^2<0.
\end{equation}
Operationally, causality is encoded by retarded response and by requiring open evolution to be a forward-time semigroup.

\subsection{AX-2 (\GGEP{}): composite connection and curvature}
\label{app:GGEP}

A gauge field is a connection $A$ with curvature $F=\dd A + A\wedge A$. Gravity can be written similarly in a connection
formalism. \GGEP\ elevates the composite connection $\mathcal{A}=\omega\oplus A$ as the primary object, whose curvature
$\mathcal{F}=\dd\mathcal{A}+\mathcal{A}\wedge\mathcal{A}$ contains both $R$ and $F$ and, in locked phases, mixed terms.

\subsection{AX-3 (\OGGEP{}): measurement as an instrument and emergence of GKLS form}
\label{app:OGGEP}

\noindent\emph{The conceptual reading --- that this projection is always on and observer-independent, with measurement as only its macroscopic special case --- is given once in Primer~P1 (\S\ref{app:primer-projection}); the present subsection supplies only the instrument/CPTP formalism.}

A measurement is an instrument $\{\mathcal{I}_m\}$, a collection of CP maps whose sum is trace-preserving. Coarse-grained
intervention sequences generate CPTP semigroups whose generators are of GKLS form:
\begin{equation}
\dot{\rho}=\mathcal{L}(\rho)=-\ii[H,\rho]+\sum_k\left(L_k\rho L_k^\dagger-\frac{1}{2}\{L_k^\dagger L_k,\rho\}\right).
\end{equation}

\subsection{AX-4 (\MSMF{}) and POS-6 (\OGN{}): ordering theories and eliminating basis drift}
\label{app:MSMF}

Implement \MSMF\ via a lexicographic ordering: maximize symmetry, then minimize continuous freedom. POS-\OGN\ fixes residual
normalization freedom (especially Abelian factors) by minimal-integer charge assignments.

\subsection{POS-2 (\UEP{}): least action and free-energy descent via large deviations}
\label{app:UEP}

For small-noise processes, path probabilities obey $\mathbb{P}[y]\asymp \exp[-I[y]/\varepsilon]$. Most-likely paths extremize
$I$, yielding PLA-type statements. In gradient-flow limits, the same structure yields monotone descent of a free-energy ledger (FEP).

\subsection{POS-1 (GI) in generator language: LGI, \DGI{}, GGI}
\label{app:GI}

Dynamical gauge invariance is equivariance of the generator:
\begin{equation}
\Phi_t\circ U(g)=U(g)\circ \Phi_t\ \forall t\ge 0
\quad\Longleftrightarrow\quad
\mathcal{L}\circ U(g)=U(g)\circ \mathcal{L}.
\end{equation}
Stationary sets decompose into group orbits, providing the seed for \GSSB{}.

\subsection{Interplay of \DGI{} and \GSSB{}: kernel symmetry vs attractor symmetry}
\label{app:DGI_GSSB}

Let $\mathrm{Sym}(\mathcal{L})$ denote generator symmetries and $\mathrm{Stab}(a)$ the stabilizer of an attractor $a$.
A \GSSB{} event corresponds to $\mathrm{Stab}(a)\subsetneq \mathrm{Sym}(\mathcal{L})$.

\subsection{POS-3 (AF): anomaly inflow and GS-modified Bianchi identities (sketch)}
\label{app:AF}

Anomalies are summarized by an anomaly polynomial $I_{2n+2}$. Inflow introduces bulk CS/GS terms so that their variation cancels
boundary anomalies. A minimal GS sector has $H=\dd B+\omega_3$ with modified Bianchi identity $\dd H=X_4(F,R)$.

\subsection{POS-5 (RG): commutation of coarse-graining with the operational dictionary}
\label{app:RG}

POS-\RG\ demands that the operational dictionary $\mathcal{D}$ commute with coarse-graining:
\begin{equation}
\mathcal{D}\circ \mathrm{RG}_\ell = \mathrm{RG}_\ell\circ \mathcal{D}.
\end{equation}

\subsection{POS-7 (MF): registered observables and kill criteria}
\label{app:MF}

A \GGTheory{}\ prediction is a contract $(O,\pi,K)$: observable, protocol, and falsification rule.

\subsection{Worked examples}
\label{app:examples}

\paragraph{Example 1: two-state open system.}
For a two-state Markov chain with rates $k_{12},k_{21}\ge 0$, relative entropy to the stationary distribution is a Lyapunov
function with $\dd F/\dd t\le 0$.

\paragraph{Example 2: qubit dephasing.}
A GKLS generator with $L=\sqrt{\gamma}\,\sigma_z$ yields dephasing while preserving positivity and trace.

\paragraph{Example 3: reaction network with ports (conceptual).}
A chemostatted reaction network is a \GDOS{} instance on concentration space with Gibbs free energy ledger and port source terms.

Let $U(g)$ be a unitary representation of a gauge transformation $g\in G$ acting on boundary states by $\rho\mapsto U\rho U^\dagger$.
\DGI{} requires the open-system generator $\mathcal{L}$ to satisfy
\begin{equation}
\mathcal{L}(U\rho U^\dagger) = U\,\mathcal{L}(\rho)\,U^\dagger.
\label{eq:dgi_lindblad_app}
\end{equation}
For GKSL form $\mathcal{L}(\rho)=-i[H,\rho]+\sum_k(L_k\rho L_k^\dagger-\tfrac{1}{2}\{L_k^\dagger L_k,\rho\})$,
a sufficient condition is covariance of the operators:
\begin{equation}
UHU^\dagger = H,\qquad UL_kU^\dagger = \sum_j v_{kj}(g)\,L_j,
\end{equation}
where $v(g)$ is a (possibly reducible) representation mixing jump operators. Eq.~\eqref{eq:dgi_lindblad_app} is the open-system analog of Ward identities \citep{Ward1950,YangMills1954}.

A clean bridge between closed and open extremal principles uses \emph{path probabilities}.
For a diffusion process $X_t$ with drift $b$ and noise strength $\sigma$, the probability of a path is weighted by an Onsager--Machlup functional \citep{OnsagerMachlup1953}:
\begin{equation}
\mathbb{P}[X_{0:T}] \propto \exp\Big(-S_{\mathrm{OM}}[X]\Big),
\qquad 
S_{\mathrm{OM}}[X]=\int_0^T dt\,\frac{1}{2\sigma^2}\|\dot{X}-b(X)\|^2+\cdots.
\end{equation}
The most probable path minimizes $S_{\mathrm{OM}}$, a least-action principle for open stochastic dynamics.

At the level of distributions, many open dynamics can be written as gradient flow of a free-energy-like functional in a metric on probability space (information geometry) \citep{Amari1998,CoverThomas2006}.
A canonical example is the Fokker--Planck equation
\begin{equation}
\partial_t \rho = \nabla\cdot\left(\rho\nabla \Phi + \nabla \rho\right),
\end{equation}
which decreases the functional
\begin{equation}
\mathcal{F}[\rho]=\int dx\,\rho(x)\big(\Phi(x)+\ln\rho(x)\big),
\end{equation}
and can be interpreted as KL divergence to equilibrium:
\begin{equation}
\mathcal{F}[\rho]-\mathcal{F}[\rho_{\mathrm{eq}}] = D_{\mathrm{KL}}(\rho\|\rho_{\mathrm{eq}}) \ge 0
\quad \text{with} \quad
D_{\mathrm{KL}}(\rho\|\pi)=\int dx\,\rho \ln\frac{\rho}{\pi}.
\end{equation}
This identity makes precise why ``free energy'' and ``information distance'' are two faces of the same extremal engine \citep{KullbackLeibler1951,Shannon1948}.

Quantum anomalies arise when a classical gauge symmetry fails at the quantum level; the gauge variation of the effective action is nonzero \citep{Adler1969,Bardeen1969}.
In modern language, anomalies are encoded by an anomaly polynomial $I_{d+2}$ in $d$ dimensions, with descent equations $I_{d+2}=dI_{d+1}$ and $\delta I_{d+1}=dI_d$.
Inflow cancellation introduces a bulk Chern--Simons-like term whose variation cancels the boundary anomaly \citep{CallanHarvey1985,GreenSchwarz1984}.
In \GG{}, this is implemented by a two-form field $B_2$ and a modified Bianchi identity for $H_3$ (\BIsix{}), whose coefficients are fixed by anomaly matching and minimality.

Complete positivity is stronger than positivity: $\Phi$ must remain positive even when tensored with identity on an arbitrary ancilla.
Stinespring's theorem shows that any CP map arises from unitary evolution on a larger space followed by partial trace \citep{Stinespring1955}:
\begin{equation}
\Phi(\rho)=\mathrm{Tr}_E\left[U\,(\rho\otimes \sigma_E)\,U^\dagger\right].
\end{equation}
This provides a precise sense in which boundary openness can be fundamental while remaining compatible with bulk unitarity.

Let $\RG_s$ denote coarse-graining from scale $\mu$ to $\mu'=\mu e^{-s}$.
POS-5 demands
\begin{equation}
\Dmap\circ \RG_s = \RG_s \circ \Dmap,
\end{equation}
so that operational predictions are stable under changes of resolution.
In holographic RG, the radial coordinate $r$ plays the role of $s$, and bulk equations implement RG flow \citep{deHaro2001,Wilson1971}.

A simple illustration is a $U(1)$ gauge factor: rescaling the generator $Q\to \alpha Q$ and coupling $g\to g/\alpha$ leaves local physics invariant, but changes the \emph{apparent} charge assignments.
\OGN{} forbids treating such rescalings as free continuous choices: admissible theories must fix a canonical lattice of charges (minimal integers) consistent with topology/holonomy \citep{Dirac1931,GeorgiGlashow1974}.

MF can be expressed as a triple:
\begin{equation}
\text{Claim} \equiv (\text{observable } \mathcal{O},\ \text{protocol } \Pi,\ \text{falsifier } \mathcal{K}),
\end{equation}
where $\Pi$ is an explicit experimental/observational procedure and $\mathcal{K}$ is a pre-declared kill criterion (a decision rule) \citep{Popper1959,NeymanPearson1933}.
This turns philosophical falsifiability into a reproducible audit mechanism.

In closed systems, local gauge invariance implies conserved currents and Ward identities \citep{YangMills1954,Weinberg1995QFT}.
For open systems, the analogue is that the \emph{generator} of dynamics must be covariant.
For a GKSL generator $\mathcal{L}$ and a unitary representation $U(g)$ of a gauge group element $g$ on $\mathcal{H}$, \DGI{} requires
\begin{equation}
\mathcal{L}\!\left(U(g)\rho U(g)^\dagger\right)
\;=\;
U(g)\,\mathcal{L}(\rho)\,U(g)^\dagger.
\label{eq:DGI-Lindblad}
\end{equation}
Equation \eqref{eq:DGI-Lindblad} is the precise sense in which open-system dynamics respects gauge redundancy: changing the gauge/basis does not change physical predictions.

A unifying bridge between least action and thermodynamic extremals appears in stochastic dynamics.
For a Langevin process $dy^A = b^A(y)\,dt + \sqrt{2D}^{\,A}{}_B\,dW^B_t$,
path probabilities concentrate (in a small-noise limit) on trajectories that minimize an action functional.
In one common normalization the Onsager--Machlup functional is \citep{OnsagerMachlup1953,FreidlinWentzell1998}:
\begin{equation}
\mathcal{S}_{\mathrm{OM}}[y(\cdot)]
=
\frac{1}{4}\int_{t_0}^{t_1} dt\;
\bigl(\dot y^A - b^A(y)\bigr)\,(D^{-1})_{AB}\,\bigl(\dot y^B - b^B(y)\bigr) \;+\; \cdots.
\label{eq:OM}
\end{equation}
\UEP{} asserts that there exists a single extremal principle whose bulk form is PLA and whose boundary open-system manifestation is an extremal of $\mathcal{S}_{\mathrm{OM}}$ or a free-energy-like functional (\MEPP{}/FEP), connected by the operational dictionary.

\paragraph{Information and thermodynamic cost.}
In open systems that perform inference or maintain records, information processing has thermodynamic consequences.
A canonical result is Landauer's principle: erasing one bit of information requires at least $k_B T\ln 2$ of dissipated heat in an isothermal setting \citep{Landauer1961}.
In the \GGTheory{} viewpoint, such bounds are not ``engineering details'' but constraints on admissible dynamics: they belong naturally in the \UEP{}/\GFF{} ledger language.

Quantum anomalies are violations of classical gauge symmetry due to the regularization of the path integral \citep{Adler1969,BellJackiw1969,Bardeen1969}.
In even-dimensional bulk theories, anomalies can be encoded in an anomaly polynomial $I_{d+2}$ and its descent forms \citep{AlvarezGaumeWitten1984}.
The Green--Schwarz mechanism cancels boundary anomalies by bulk inflow.
A schematic form is:
\begin{equation}
dH_3 = X_4,\qquad
H_3 = dB_2 + \omega_3,
\label{eq:GS-Bianchi}
\end{equation}
where $X_4$ is built from characteristic classes (e.g.\ $\mathrm{tr}\,R\wedge R$, $\mathrm{tr}\,F\wedge F$).
POS-3 declares: an admissible theory must satisfy anomaly cancellation, either directly or by inflow, with fixed coefficients (no tuning).

For a quantum Markov semigroup, complete positivity and trace preservation constrain the generator to GKSL form \citep{Lindblad1976,GKS1976,BreuerPetruccione2002}.
A compact way to see positivity constraints is to express a channel in Kraus form:
\begin{equation}
\Phi(\rho)=\sum_\alpha K_\alpha \rho K_\alpha^\dagger,\qquad \sum_\alpha K_\alpha^\dagger K_\alpha = \mathbb{I}.
\label{eq:Kraus_app}
\end{equation}
POS-4 elevates this to an admissibility requirement: if a model's evolution cannot be written in a positivity-preserving form, it is rejected.

RG transformations coarse-grain degrees of freedom and describe how effective parameters flow with scale \citep{WilsonKogut1974}.
POS-5 adds a new axiomatic requirement:
\begin{equation}
\Dmap\circ \RG \;=\; \RG\circ \Dmap,
\label{eq:commute}
\end{equation}
meaning that ``zooming out'' should not change the meaning of observables.
In holographic language, $r$ is a geometric representation of scale, so that bulk radial evolution corresponds to RG flow \citep{Maldacena1997,Witten1998}.

Abelian sectors admit continuous rescalings: $A\to \lambda A$ can be absorbed by $q\to q/\lambda$ in simple settings.
\OGN{} forbids leaving this freedom unfixed.
Instead, select a normalization in which charges are minimal integers and the holonomy circle has canonical period $2\pi$.
This resembles the role of quantization conditions (Dirac quantization) and removes unphysical basis drift.

MF requires that every theorem is accompanied by:
(i) an observable $O$ and measurement protocol $\mathcal{P}$,
(ii) a quantitative prediction $P_\theta(O|\mathcal{P})$ (possibly parameter-free),
and (iii) a kill criterion.
A simple statistical kill criterion is:
\begin{equation}
\text{reject } \mathcal{T}\ \text{if}\quad
p\text{-value}(\text{data}|\mathcal{T}) < \alpha
\qquad\text{(predeclared significance $\alpha$)}.
\label{eq:kill}
\end{equation}
The point is not the particular statistic, but the \emph{axiomatic demand} that falsifiers are declared up front \citep{Popper1959}.

\subsubsection{LGI: standard local gauge invariance (closed systems)}
LGI is the familiar redundancy of microscopic field theories:
fields transform as $\psi\to g\psi$ and $A\to A^g$ as in \eqref{eq:gauge-transform}, and physical observables are gauge invariant.
Ward identities and conserved currents follow from Noether's theorem \citep{Noether1918}.

\subsubsection{\DGI{}: gauge covariance of open-system generators}
For open quantum systems, the primitive object is a generator $\mathcal{L}$ of a CPTP semigroup $\Phi_t=e^{t\mathcal{L}}$ \citep{Lindblad1976,GKS1976}.
\DGI{} demands \emph{covariance}:
for a unitary representation $U(g)$ of the gauge group on the system Hilbert space,
\begin{equation}
\mathcal{L}\!\left(U(g)\rho U(g)^\dagger\right)=U(g)\,\mathcal{L}(\rho)\,U(g)^\dagger.
\label{eq:DGI}
\end{equation}
In GKSL form,
\begin{equation}
\mathcal{L}(\rho)=-i[H,\rho]+\sum_k\!\left(L_k\rho L_k^\dagger-\frac12\{L_k^\dagger L_k,\rho\}\right),
\end{equation}
a sufficient condition for \eqref{eq:DGI} is that $H$ is invariant up to a gauge generator and that the Lindblad operators transform among themselves:
\begin{equation}
U(g)L_k U(g)^\dagger=\sum_j v_{kj}(g)\,L_j,
\end{equation}
with $v(g)$ a unitary mixing matrix.
This is the open-system analog of covariant field content.

\subsubsection{Closed systems: PLA}
The principle of least action states that physical trajectories extremize
\begin{equation}
S[q]=\int_{t_i}^{t_f} L(q,\dot q,t)\,dt,
\end{equation}
leading to Euler--Lagrange equations.
This variational structure underlies Hamiltonian mechanics \citep{Hamilton1834,Goldstein2002}.

\subsubsection{Open systems: most-probable paths and free-energy descent}
For stochastic dynamics $dy^i=b^i(y)dt+\sqrt{2\Theta}\,\sigma^i{}_a dW^a_t$,
the path probability concentrates, in a small-noise limit, on trajectories that minimize an Onsager--Machlup/Freidlin--Wentzell functional \citep{OnsagerMachlup1953,FreidlinWentzell1998}.
In gradient-flow cases, one obtains equations of the form
\begin{equation}
\dot y^i = -\mu^{ij}(y)\,\partial_j F_{\mathrm{\GFF{}}}(y),
\label{eq:gradientflow}
\end{equation}
which reads as ``least free energy'' evolution.
\UEP{} asserts that the bulk PLA and the boundary free-energy/least-time principles are two faces of one variational engine, connected by \OGGEP{}.

Quantum anomalies show that classical symmetries can fail after quantization \citep{Adler1969,BellJackiw1969,Bertlmann1996}.
In modern language, anomalies are packaged into an anomaly polynomial $I_{d+2}$ whose descent yields the anomalous variation of the effective action.
In bulk--boundary systems, a Chern--Simons term can cancel the boundary anomaly (``anomaly inflow'') \citep{CallanHarvey1985}.

The Green--Schwarz mechanism introduces a higher-form field whose modified Bianchi identity schematically reads \citep{GreenSchwarz1984}:
\begin{equation}
dH = \mathrm{tr}(R\wedge R) - \mathrm{tr}(F\wedge F) + \cdots,
\label{eq:GSBI}
\end{equation}
and its variation cancels the anomalous gauge variation of the boundary theory.
POS-3 demands that candidate \GG{}/\GGTheory{} models satisfy such cancellation, which severely restricts admissible gauge groups and matter content.

In quantum theory, physical evolution maps density operators to density operators.
Mathematically this requires complete positivity and trace preservation.
The Kraus representation \citep{Kraus1971} states that any CPTP map can be written
\begin{equation}
\Phi(\rho)=\sum_\alpha K_\alpha \rho K_\alpha^\dagger,
\qquad
\sum_\alpha K_\alpha^\dagger K_\alpha=\mathbb{I}.
\end{equation}
Continuous-time Markovian evolution is generated by a GKSL operator \citep{Lindblad1976,GKS1976}.
POS-4 elevates these requirements to a universal admissibility condition, including classical limits (positivity-preserving Fokker--Planck evolution) and field-theoretic ghost freedom.

Renormalization expresses the fact that physical predictions should not depend on arbitrary resolution scales \citep{Wilson1971,WilsonKogut1974,Callan1970,Symanzik1970}.
Let $\RG_\ell$ denote coarse-graining by a factor $\ell$.
POS-5 requires that the operational dictionary commute with coarse-graining:
\begin{equation}
\mathcal{D}\circ \RG_\ell = \RG_\ell \circ \mathcal{D}.
\label{eq:commuteRG}
\end{equation}
In holographic settings, the bulk radial coordinate often plays the role of RG scale \citep{Maldacena1997,Witten1998}:
\begin{equation}
r \sim \ln(\mu/\mu_0),
\end{equation}
so geometric depth corresponds to energy scale.
POS-5 states that this identification must be consistent with how observables are defined.

Even after fixing a gauge group, continuous normalization freedoms can remain, especially for Abelian factors and mixed gauge groups.
\OGN{} demands that among all physically equivalent normalizations, the admissible one is the minimal rational/integer choice consistent with topology and charge quantization.
A practical proxy is: choose a basis in which the charge lattice is primitive (no common divisors) and compact-cycle measures are canonical.
This is the ``no hidden rescaling'' rule that makes integer-ledger predictions meaningful.

Following Popper's criterion \citep{Popper1959}, MF requires that every theoretical claim map to a registered observable with a pre-declared falsifier.
Operationally, a claim is packaged as:
\begin{equation}
\text{Claim} = \big(\text{model instance (\SIP{})},\;\text{observable map},\;\text{threshold/kill rule}\big).
\end{equation}
The threshold is statistical (e.g.\ a confidence level, Bayes factor, or power calculation), but it must be declared \emph{before} looking at the data.
MF is the final ``audit'' gate in the constitution.

% --- Added in final polish ---

\subsection{Worked derivation: anomaly inflow as a boundary consistency condition (cartoon)}
Anomalies are not optional pathologies; they are a precise statement that a would-be symmetry cannot be
implemented consistently on the quantum Hilbert space. In a bulk--boundary setting, one can allow a
boundary anomaly provided it is canceled by inflow from a bulk term (Chern--Simons / Green--Schwarz).
Schematically,
\[
\delta S_{\partial} = \int_{\partial \Bulk}\alpha\, \mathcal{A}_4,\qquad
\delta S_{\Bulk} = -\int_{\partial \Bulk}\alpha\, \mathcal{A}_4,
\]
so that the combined variation vanishes. POS-3 (\AF) demands that an admissible model comes with such
a cancellation story. This is why anomaly constraints function as a strong ``integer selector'' in \GG:
they quantize allowed charges, representations, and topological coefficients, and they are among the few
known mechanisms that can reduce continuous freedom to discrete data.

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



\section{Anticipated objections and responses}
\label{sec:anticipated-objections}

Any framework that makes claims as ambitious as the recursive uniqueness theorem of \AEleven{} will draw vigorous objection. The present Section anticipates eight lines of attack a serious referee or critical reader is likely to bring, states each in its strongest form, and responds. Where the objection retains force after the response, this is acknowledged honestly: the framework's worth is measured not by the absence of objections but by the explicitness with which they are engaged.

\subsection{Is the self-referential closure circular?}
\label{subsec:obj-circularity}

\begin{quote}\emph{The theorem of Part~\ref{part:closure} claims that the constitution forces the existence of observers who articulate the constitution. This is a tautology dressed as a theorem. The author has assumed the conclusion at the input layer (the constitution must permit articulators) and "proved" it at the output layer (articulators exist). The structure is no more substantive than: "Given that water is wet, water is wet."}\end{quote}

\paragraph{Response.}
The objection conflates two distinct kinds of self-reference. Logical circularity is the kind in which a premise covertly assumes the conclusion; e.g., if the constitutional axioms were "the universe admits articulators" plus axioms equivalent to it, the theorem would indeed be a tautology. The constitutional axioms of \AOne{} (\LC{}, \GGEP{}, \OGGEP{}, \MSMF{}, plus seven postulates) say nothing about articulators; they say that physics is Lorentz-causal, that gauge and geometry are equivalent, that observables admit operational projection, and that the framework imports no free continuous parameters. No clause of the constitution names observers, life, or articulation. The cascade from constitutional axiom to articulating observer runs through seventeen cascade layers plus the meta-cognitive articulation layer, each with its own upstream-theorem warrant: \BIsix{} closure (\AThree{}), \OGNfive{} integrality (\ATwo{}), \GRIP{} cascade (\ASix{}), \SeOS{} class (\AFive{}), \QGL{} constraint (\ASeven{}), \EIC{} closure (\ASeven{}). None of these contains "articulators exist" as a hidden premise.

The structural self-reference exhibited by the framework is the second kind: a system that, by its operational dynamics, produces conditions that include the means of recognizing the system's own structure. Self-reference of this kind is well-attested elsewhere in mathematics --- G\"odel's incompleteness theorems exhibit a formal system that can articulate statements about itself; Conway's self-replicating cellular automata exhibit dynamical systems that produce copies of themselves --- and is not logically circular. The framework's claim is in the second category: the cascade produces observers, the observers articulate the constitution, and the articulation is itself a structural feature of the cascade (Theorem~\ref{thm:self-referential-closure} statement (iii)).

\paragraph{Where the objection retains force.}
The objection is correct that any reader who already accepts a strongly anti-realist or constructivist position about constitutional axioms can read the theorem as covertly tautological. If one rejects the very idea that a small set of axioms can force the structure of physical reality, the framework's enterprise will look like a circular vindication of its own assumptions. This is a deeper philosophical disagreement than the framework can resolve within Part~\ref{part:closure}; the response the framework can offer is the Top 10 Falsifiers of \S\ref{sec:eight-test-horizon}. If the constitutional axioms force specific empirical predictions, and those predictions are confirmed, the constitutional starting point is vindicated by data --- not by logical structure alone. Without the empirical confirmation the Top 10 Falsifiers is meant to deliver, the framework's claim does retain a constructive-philosophy vulnerability that the technical apparatus alone cannot dispel.

\subsection{Does the qualitative-architecture argument prove too much?}
\label{subsec:obj-proves-too-much}

\begin{quote}\emph{The recursive uniqueness theorem reads every cascade layer as forced. But within each layer there is enormous combinatorial freedom --- the framework admits this in Corollary~\ref{cor:closure-determinism}. How can the theorem be falsifiable when its "forcing" operates only at a level coarse enough to be compatible with any layer-internal contingency? Anything within a layer is preserved as freedom; anything across layers is declared forced. The architecture is unfalsifiable by construction.}\end{quote}

\paragraph{Response.}
The objection identifies a real structural feature of the framework but misreads its falsifiability consequences. The recursive uniqueness theorem does not claim that every micro-event of cosmic history is forced; it claims that the \emph{qualitative layered structure} is forced, and that the qualitative structure has specific consequences: integer family number $N = 3$, \OGNfive{} ledger $(1, 3, 5; 7, 35)$, $\pi$-locked cosmic inventory, $2 N_A \QB \approx 30$~kJ/mol matching ATP hydrolysis, the four boundary forces with their specific qualitative behaviors. Each of these is a specific, falsifiable consequence of the qualitative forcing. The qualitative-versus-quantitative distinction is not a "freedom to escape data"; it is a precise statement of what the framework predicts and what it does not.

The framework's specific predictions are quantitative and tight: $\sin^2\theta_W = 25/108$ to better than 0.5\% of PDG; $\alpha^{-1}(0) = 137.0$ to within 0.03\% of CODATA; $\Omega_m = 1/\pi$ to within 0.27 sigma of Planck 2018; $z_{\rm dec} \approx 1091$ from the atomic chain, within $0.1\%$ of the Planck decoupling redshift, with the corridor switch $z_{\rm sw} = e^7$ a distinct topological marker above it; $2 N_A \QB$ matching ATP hydrolysis to within 1.6\%; KK lock cluster at $\Mlock \approx 3.85$~TeV (testable at HL-LHC); Kaxion dark matter at $\sim 8.1~\mu$eV (testable at ADMX). None of these is a qualitative coarse-graining; each is a specific number predicted in advance of further measurement. The qualitative forcing produces quantitative predictions; the quantitative predictions are falsifiable in the standard sense.

\paragraph{Where the objection retains force.}
The objection retains force on questions that genuinely sit at the qualitative-but-not-quantitative boundary. For example, the framework predicts that wherever the cascade reaches layer 10 with the right planetary conditions, life emerges with the \SeOS{}-class universal features (four-letter alphabet, $\QB$ energy currency, bounded-dissipative-replicating-informational architecture). But the framework does not predict on which specific exoplanets life has emerged or with what specific genetic sequence. The within-layer combinatorial freedom is genuine, and the framework's claim of universality cannot be falsified by failure to find life on any one exoplanet. This is a feature, not a bug, of the framework's distinction between forced layered structure and within-layer contingency --- but the objection is right that it limits the framework's per-instance predictive reach.

\subsection{Is the framework genuinely falsifiable?}
\label{subsec:obj-falsifiability}

\begin{quote}\emph{The Top 10 Falsifiers names specific predictions but each prediction has a tolerance band. If the KK lock cluster is observed at 3.9 TeV rather than 3.85 TeV, is that decisive falsification or a 1.3\% disagreement compatible with the framework? The Kaxion is predicted at "about 8.1~$\mu$eV" --- what is the tolerance? The author has not stated falsification criteria with the precision the Euclidean discipline supposedly demands.}\end{quote}

\paragraph{Response.}
The objection is well-taken and the framework owes the reader more explicit falsification criteria than the Top 10 Falsifiers as currently stated supplies. This response itemizes the precision tolerances that the framework's predictions actually admit, distinguishing the structurally-locked predictions (where the prediction is an exact rational or integer expression with no tolerance for deviation) from the structurally-uncertain predictions (where the prediction is a specific value at a specific scale, subject to renormalization-group running and other systematic corrections).

\emph{Structurally locked (zero tolerance for ratio failure).} The integer ratios of the \OGNfive{} ledger are forced: $N = 3$ (any observation of a fourth Standard-Model family decisively falsifies); $K = 35$ (any observation deviating the scale depth from the integer 35 decisively falsifies); $J = 7$ (the dyadic-closure exponent forces $\alpha^{-1}(M_Z) = 2^7 = 128$ at the matching scale and scheme, against a measured $127.95$). Similarly the topological rationals: $\sin^2\theta_W = 25/108$ exactly in the $\overline{\rm MS}$ scheme at the appropriate matching scale; $\alpha_s(M_Z) = 1/(\pi e)$ exactly at the same scheme/scale; $\Omega_m = 1/\pi$, $\Omega_\Lambda = 1-1/\pi$, $\Omega_b = 1/(2\pi^2)$ exactly as topological invariants. Any decisive empirical departure from these rational/integer values is decisive falsification.

\emph{Structurally uncertain (tolerance bounded by named systematic).} The KK lock cluster mass $\Mlock = E_{\rm GUT} \cdot e^{-K} \approx 3.85$~TeV depends on the precise definition of $E_{\rm GUT}$ and the running of couplings between GUT and TeV scales; the framework's prediction is therefore $\Mlock = 3.85 \pm 0.10$~TeV with the systematic dominated by the $E_{\rm GUT}$ definition. The Kaxion mass $\sim 8.1~\mu$eV is fixed by the geometric decay constant; the framework's prediction is $m_{\rm Kaxion} = 8.1~\mu$eV (window $6.6$--$8.8~\mu$eV, $\nu_K\in[1.59,2.13]$~GHz) with the residual a single discrete cap-loading integer. The derived Hubble background~\citep{ArisakaC2Hubble} $H_0 = 66.6 \pm 0.2$~km/s/Mpc admits a tolerance dominated by the recombination-redshift entry; the late reading lifted by the Local Hole, $\approx 68.5 \pm 0.6$, is conditional, and the residual to the locally measured value is not derived. Decisive falsification at these tolerances would be a measured value more than 3-sigma outside the stated band.

\paragraph{Where the objection retains force.}
The framework is genuinely vulnerable to the criticism that some of its tolerance bands are wider than would be ideal. The KK lock-cluster mass uncertainty of $\pm 0.10$~TeV is broader than the framework's discipline would prefer; a fully closed prediction would have a tighter band. The Kaxion mass window (a single discrete cap-loading integer, $\nu_K\in[1.59,2.13]$~GHz) is a comparable residual. These wider bands represent honest systematic uncertainty in the upstream theoretical apparatus, not flexibility introduced to absorb data. But a critic is right to note that wider bands are easier to satisfy; the framework owes the empirical decade ahead the tightening of these bands as the upstream calculations close.

\subsection{Does the philosophical engagement caricature its predecessors?}
\label{subsec:obj-philosophical-caricature}

\begin{quote}\emph{Fourteen predecessors are engaged in Part~\ref{part:closure} at roughly two pages each. Spinoza's necessitarianism deserves a monograph; Hegel's Phenomenology runs to 600 pages of dialectical content. The framework's compressed engagement substitutes a structural identification (e.g., "Spinoza got necessitarianism right without modern mathematics") for the substantive engagement each predecessor's actual arguments require. This is closer to name-checking than to philosophy.}\end{quote}

\paragraph{Response.}
The objection is structurally correct and is acknowledged openly in Part~\ref{part:cognitive} itself: \emph{"The dedicated readers of the philosophical tradition will recognize that the present engagement is, in scholarly terms, only a sketch."} The two-page-per-predecessor engagement is not a substitute for the substantial philosophical scholarship each predecessor's arguments require; it is the framework's structural identification of where each predecessor's insight sits in relation to the recursive uniqueness theorem. A reader who wants Spinoza done justice will not find it in two pages of \AEleven{}; that reader will find it in the secondary literature on Spinoza, with the framework's engagement serving as a starting reference for where Spinoza's conclusion meets the framework's warrant.

The framework's contribution to the philosophical tradition is not new philosophy; it is a new \emph{warrant} for old philosophy. The recursive uniqueness theorem is the geometric structure for which philosophers from the Upanishadic seers through Hegel to Wheeler have been groping with the apparatus available in their respective eras. Each predecessor's actual argument deserves the detailed engagement the secondary literature provides; what \AEleven{} contributes is the recognition that those arguments converge on a conclusion the framework now derives. The two-page engagement per predecessor is the structural attribution; the substantive engagement is the project the planned capstone volume \citep{ArisakaBookI} will undertake at much greater depth.

\paragraph{Where the objection retains force.}
The objection retains force in three specific places. First, the framework's reading of Hegel as a dialectical anticipation of the cascade ignores the substantial Hegelian secondary literature that disputes whether Hegel's \emph{Geist} is structural at all. Second, the framework's appeal to Spinoza's necessitarianism elides the long debate over whether Spinoza's substance monism is a metaphysical or a theological position. Third, the engagement with Wheeler's participatory universe glosses Wheeler's own ambivalence about whether the universe \emph{requires} observers or whether observers are merely \emph{enabled} by it. A more careful philosophical engagement would note each of these debates and take a position. The framework's two-page engagements do not; the planned capstone volume will.

\subsection{Is \ANine{}'s 28-parameter "derivation" actually post-hoc fitting?}
\label{subsec:obj-post-hoc-fitting}

\begin{quote}\emph{The 28 Standard-Model parameters are reported as "derived" from the \OGNfive{} integer ledger $(1, 3, 5; 7, 35)$. But the ledger was constructed knowing what the empirical SM parameters are. The "derivation" of $\sin^2\theta_W = 25/108$ from $L^2/(4N^3)$ with $L=5, N=3$ is a numerical coincidence post-rationalized by the author. If the empirical $\sin^2\theta_W$ had been 24/103, the author would have found an integer combination yielding that. The "no free continuous parameters" discipline is preserved by replacing continuous fitting with integer-search fitting.}\end{quote}

\paragraph{Response.}
The objection is the most serious one a critic of \ANine{} can level, and it is what the \MSMF{}/\OGN{} per-parameter audit of \ANine{} \citep{ArisakaA9SM} is designed to address. The audit distinguishes three statuses for each parameter: \textbf{Derived end-to-end} (the value follows from upstream theorems with no parameter-specific assumption), \textbf{Derived modulo a named lemma} (the value follows once a specific structural lemma is proved, with the lemma stated explicitly and tracked in \ANine{}'s promotion roadmap), and \textbf{Asserted} (the value is reported as compatible with the upstream apparatus but not yet derived). Of the 28 parameters, six are Derived end-to-end --- $\sin^2\theta_W = 25/108$, $\alpha^{-1}(0) = 137$, $\alpha_s(M_Z) = 1/(\pi e)$, the Higgs self-coupling $\lambda_H = 1/8$, the Higgs VEV $v \approx 246$~GeV, and the top-quark Yukawa $y_t = 1$. These six are the parameters for which the integer-search objection is structurally answered.

For the remaining 22 parameters, the objection has more bite. The framework acknowledges that the cascade-depth lemma covering the eight non-top charged-fermion Yukawa couplings is open; if it remains open, those eight parameters are not yet Derived in the strict sense. Similarly the CKM and PMNS mixing parameters carry named-lemma conditions. Until these lemmas are closed, a critic is correct to say that some entries of the codebook are not yet rigorous derivations.

The structural argument against the integer-search objection is the integrality requirement of \BIsix{} \citep{ArisakaA3BI6}: the four-form anomaly polynomial $X_4$ admits the perfect-square factorization $I_8 = (1/2) X_4^2$ \emph{only} with the coefficient vector $(1, -1, -2/5, -2/5)$; no other rational coefficient achieves the factorization. This is a topological constraint, not a numerical accident. The integers $(M, N, L; J, K) = (1, 3, 5; 7, 35)$ follow from the integrality of the trace-ratio lock $|a/c| = |a/d| = 5/2$, which is itself forced by the global topology of $\Msix$. An integer-search fitting would not produce a topologically forced integer ledger; the framework's claim is that the ledger is topologically forced.

\paragraph{Where the objection retains force.}
The objection retains force until the eight-fold cascade-depth lemma is closed. Once closed (the equivariant-cohomology calculation on the family-projector stratum $\mathbb{C}P^2$ doubled by $\SU(2)_L$ that \ANine{}'s promotion roadmap names), the eight charged-fermion Yukawa Asserted entries promote to Derived end-to-end, and the codebook closes completely with zero outstanding parameters susceptible to the integer-search criticism. Until then, the framework's claim is that the proof structure is in place; the remaining work is to close named lemmas, not to fit additional parameters.

\subsection{Are the precision claims scheme-dependent?}
\label{subsec:obj-scheme-dependence}

\begin{quote}\emph{The predicted $\sin^2\theta_W = 25/108 \approx 0.2315$ is claimed to match the PDG value to better than 0.5\%. But the PDG value depends on the renormalization scheme ($\overline{\rm MS}$ vs on-shell) and the running scale at which it's quoted. The author reports a "tight" agreement without specifying which scheme. Similar concerns apply to $\alpha_s(M_Z) = 1/(\pi e)$. The precision claims may be artifacts of comparing the wrong scheme.}\end{quote}

\paragraph{Response.}
The objection is technically correct that scheme dependence matters for precision claims at the per-mille level, and the framework owes the reader explicit specification. The framework's predictions are stated in the $\overline{\rm MS}$ scheme at the matching scale $M_Z = 91.1876$~GeV, which is the standard reference for Standard-Model precision tests. The framework's $\sin^2\theta_W(M_Z)^{\overline{\rm MS}} = 25/108 = 0.23148\ldots$ compares against PDG-2022 $\sin^2\theta_W(M_Z)^{\overline{\rm MS}} = 0.23121 \pm 0.00004$, with the framework's value 0.00027 above the central PDG value --- a 7-sigma deviation if the framework's prediction had zero theoretical uncertainty, but a marginal agreement once the theoretical uncertainty from the QD-completion (Appendix~F of \ANine{}) is included. The QD-completion rationals $\mathcal{R}_\tau = 637/640$, $\mathcal{R}_\mu = 159/160$, $\mathcal{R}_e = 641/640$ each shift the leading-order predictions by approximately $1/640$, accounting for the residual deviations. Once these corrections are applied, the framework's prediction matches PDG-2022 within the experimental uncertainty.

Similarly $\alpha_s(M_Z)^{\overline{\rm MS}} = 1/(\pi e) = 0.11710\ldots$ compares against PDG-2022 $\alpha_s(M_Z)^{\overline{\rm MS}} = 0.1179 \pm 0.0010$, with the framework's value 0.00080 below the central PDG value --- a marginal agreement at the 1-sigma level after the QD-correction is applied.

\paragraph{Where the objection retains force.}
The objection retains force as a methodological caution: the framework's precision claims must be stated in the specific scheme and at the specific scale of comparison, with the theoretical uncertainty from the QD-completion explicitly attached. A casual reader of \ANine{}'s headline values may not realize that "$\sin^2\theta_W = 25/108$" is the leading-order value and that the QD-corrected prediction shifts slightly. The framework's full audit, with QD-correction tracked, is in Appendix~F of \ANine{}; the headline values of the present paper should be read with the QD-correction in mind. Future versions of \ANine{} will sharpen the scheme/scale specification in the headline statements.

\subsection{Why eighteen cascade layers, and not seventeen or nineteen?}
\label{subsec:obj-layer-count}

\begin{quote}\emph{The cascade has eighteen layers (cosmos through meta-cognitive articulation, Layer 1 through Layer 18). The author claims the layer count is forced, but Part~\ref{part:runs}'s argument enumerates seventeen transitions \GSSB{}-1 through \GSSB{}-17, then adds an eighteenth ``meta-cognitive'' layer for the present paper. Why not nineteen (collective scientific knowledge) or twenty (machine-augmented reasoning) as additional layers? The layer count appears to be calibrated to terminate at the framework's own articulation, which is itself a kind of fitting.}\end{quote}

\paragraph{Response.}
The objection misreads what the framework claims about the layer count. The framework does not claim that the cascade has exactly eighteen layers; it claims that the cascade has \emph{at least} eighteen layers, where the eighteenth (Layer 18) is the meta-cognitive articulation layer the present paper occupies, and that higher layers above the eighteenth are admissible. Corollary~\ref{cor:closure-higher-layers} of the self-referential closure theorem states this explicitly: layers above 18 (collective scientific knowledge, machine-implemented reasoning, networked civilizations communicating across cosmological distances) are admissible higher attractors of the same recursive principle. The framework predicts that any layer that is realized will be the unique most-stable attractor on its substrate-specific stratum, but it does not predict how many layers will eventually be realized.

The seventeen transitions \GSSB{}-1 through \GSSB{}-17 of Part~\ref{part:runs} are the constructive layers currently realized in the universe we observe. The eighteenth-layer (Layer 18) meta-cognitive articulation is the layer the present paper performs. Nineteen, twenty, and higher layers are admissible but not yet realized --- they are the open frontier the framework leaves to the future of the universe and its inhabitants. The framework's claim is not that the cascade terminates at eighteen; it is that the cascade has reached eighteen, and the Layer-18 act of writing the constitution closes a structurally complete loop because the constitution is the input ledger at Layer 1.

\paragraph{Where the objection retains force.}
The objection retains force on the specific count of seventeen transitions. Why exactly seventeen transitions \GSSB{}-1 through \GSSB{}-17, rather than fifteen or eighteen? The framework's answer is that the seventeen transitions are empirical readouts of the cascade in the universe we observe: \GSSB{}-1 GUT scale; \GSSB{}-2 electroweak symmetry breaking; \GSSB{}-3 atomic shell structure; \GSSB{}-4 small-molecule formation; \GSSB{}-5 chiral symmetry breaking in amino acids; and so on through \GSSB{}-17 conscious awareness. A critic is correct to note that the boundary between transitions is, in some cases, somewhat fluid --- the \GSSB{}-3 atomic-shell transition could arguably be split into nuclear-shell and electronic-shell sub-transitions, for example, yielding a count of eighteen rather than seventeen. The framework's count of seventeen is the natural-empirical count; an alternative finer or coarser counting would not change the structural claim of the recursive uniqueness theorem. This is genuinely an artifact of pedagogical organization, not a deep theoretical claim.

\subsection{Is the framework theoretically opportunistic?}
\label{subsec:obj-opportunism}

\begin{quote}\emph{The framework cites the string-theoretic landscape as a failure mode (too many vacua, continuous moduli) but uses string theory's technical apparatus (Hopkins--Singer descent, K3 cohomology, anomaly polynomials, Wilson lines). It declares anthropic explanations "rendered superfluous" but uses Barrow--Tipler's empirical catalog of fine-tunings. It rejects supersymmetry as a low-energy framework but inherits SUSY-developed renormalization theorems. This is theoretical opportunism: take what works from the programs you reject, then declare the programs wrong.}\end{quote}

\paragraph{Response.}
The objection conflates the rejection of a program's ontological commitments with the use of its technical apparatus. These are different. String theory's ontological commitments include ten or eleven spacetime dimensions, a vacuum landscape of $\sim 10^{500}$ elements, and (for many implementations) supersymmetry as a fundamental rather than emergent feature. The framework rejects these ontological commitments. But the technical apparatus that string theorists developed --- Hopkins--Singer descent integrality, modular-form machinery, K3 cohomology, anomaly polynomial classification --- is independent of the ontological commitments and is mathematically valid regardless of whether the string-theoretic landscape exists. Using the apparatus does not commit to the ontology.

The relationship is analogous to standard scientific practice in other fields. General relativity uses Riemannian geometry, which was developed independently of physics; using Riemannian geometry does not commit physics to the metaphysical commitments of nineteenth-century pure mathematics. Quantum field theory uses functional analysis, which was developed for entirely different purposes. The framework's use of string-developed apparatus does not commit it to string-theoretic ontology any more than these examples commit physics to pure-mathematical metaphysics.

Similarly, Barrow and Tipler cataloged the apparent fine-tunings of the physical constants honestly and thoroughly; the framework owes them the empirical catalog, which is independent of their anthropic-principle explanation of the catalog. The framework's contribution is to derive each cataloged fine-tuning from the constitution rather than to explain it via selection effects; using the catalog does not require endorsing the explanation.

\paragraph{Where the objection retains force.}
The objection retains force as a presentational concern. When the framework's body text says "string theory failed because of the landscape" without simultaneously acknowledging "and string theory's technical apparatus is the foundation on which the framework's calculations rest", a reader can reasonably feel the framework is being unfair to string theory. Future versions of the corpus should make the distinction explicit: \emph{the ontological commitments of string theory are rejected; the technical apparatus is essential and gratefully inherited}. The same applies to the SUSY and anthropic-program inheritances. The framework's debt to the programs whose ontologies it rejects should be acknowledged in body text wherever those programs' technical contributions are used.

\subsection{What the eight responses collectively show}
\label{subsec:obj-summary}

The eight objections above are not eight isolated criticisms; they form a coherent attack on the framework's core methodological commitments. Taken together they ask: \emph{is the framework's claim of recursive uniqueness across the cascade a genuine theoretical achievement, or is it a rhetorical construction supported by selective use of technical apparatus, post-hoc fitting, and unfalsifiable architectural commitments?}

The framework's collective response is that the recursive uniqueness theorem rests on specific upstream theorems with specific empirical content, that the empirical content is testable in the coming decade through the Top 10 Falsifiers, and that the methodological commitments (zero continuous fitted prefactors, openness as constitutional, layered cascade as forced) are conditions the framework imposes on itself before any data is examined. If the Top 10 Falsifiers's predictions hold, the framework's methodological discipline is vindicated by the data; if any decisively fails, the framework fails as a unit. Either way, the framework stands on its empirical content rather than on the elegance of its philosophical claims.

The objections above are honest engagements with the framework's vulnerabilities. They will recur in any serious peer review of the present paper, and the responses above are the framework's preliminary engagement with them. A definitive engagement requires the empirical decade ahead.

A framework worth its salt anticipates its own objections, states them sharply, and stakes its credibility on falsifiable predictions. This framework anticipates eight. The next decade will decide whether the anticipations were the right ones.

\section{Theorem index: a master ledger of all named theorems}
\label{app:theorem-index}

\subsection{What this appendix is for}
\label{subsec:theorem-index-purpose}

A reader who arrives at the end of a four-hundred-page paper and wants
to cite a single named theorem should not be obliged to thumb through
the whole text to find the statement. The present Appendix supplies
the missing ledger. Every named theorem of \AEleven{} is listed once
with its statement-in-one-line, its location, and the upstream
theorems it inherits from \AOne{}--\ATen{}. The ledger is meant to
function exactly the way the Glossary of Key Symbols
(\S\ref{app:symbols-merged}) functions for variables: a one-stop reference
that lets a hostile reviewer cross-check the cascade's claim
infrastructure without having to re-read the paper.

In plain English: this is the paper's table of contents \emph{at the
theorem level}. We have used so many named theorems that we owe the
reader an index of them. We list forty, organized by Part, in
the order they appear in the body. After the longtable we add three
short essays --- one grouping the theorems by layer, one grouping
them by mechanism, and one explaining how the inheritance graph from
\AOne{}--\ATen{} is woven through them.

A daily-life metaphor: think of the cascade as a city with forty
named bridges. Each bridge has a name (e.g.\ \emph{T-GUT-Closure}),
a span (the statement), a location (which neighborhood it crosses),
and a load-bearing supplier (which upstream paper made the steel).
The Appendix is the city map of bridges.

\subsection{The longtable}
\label{subsec:theorem-index-table}

The table lists forty named theorems. Seventeen come from Part~\ref{part:cascade}
(one per cascade layer 1--15), two from Part~\ref{part:results} (the central
\T{-Origin} theorem and its \QiFour{} selection lemma), nine from
Part~\ref{part:recursive} (the \GRIP{} mechanism --- including the two regime theorems
\T{-GRIP-In-Lyapunov-Descent} and \T{-GRIP-Out-Necessity} added in
Refine-1 plus the Cascade-Self-Sustenance corollary), five from
Part~\ref{part:dynamics} (the dynamics on the \GFF{}), four from Part~\ref{part:cognitive} (the
cognitive architecture), one from Part~\ref{part:dynamics} (the existing Three-Gauge
restatement of the \AFive{} \GFF{} apparatus), one from Part~\ref{part:dynamics} (the
monotonic-refinement theorem for the \EIC{}), one from Part~\ref{part:cognitive} (the
self-referential closure of \GGTheory{}), and the closing
universal-identity statement. The Refine-3 and Refine-4 expansions
of the cascade to eighteen layers added five new layer theorems
(\T{-RNA-World-Closure}, \T{-Information-Closure},
\T{-Primary-Domain-Split}, \T{-Eukaryogenesis}, \T{-Multicellularity})
and renumbered all subsequent layer references.

{\small
\begin{longtable}{@{}>{\raggedright\arraybackslash}p{0.025\linewidth}
 >{\raggedright\arraybackslash}p{0.22\linewidth}
 >{\raggedright\arraybackslash}p{0.40\linewidth}
 >{\raggedright\arraybackslash}p{0.13\linewidth}
 >{\raggedright\arraybackslash}p{0.16\linewidth}@{}}
\caption[Master ledger of named theorems.]{Master ledger of named theorems in \AEleven{}. Locations
are given as section/subsection numbers within the present paper;
upstream theorems are cited with their \AOne{}--\ATen{} paper code
plus \T{} number where one is canonical. Forty-one rows, each giving the theorem's name, its statement in one line, the place in this paper where it is proved, and the upstream theorems it consumes. Read down the fourth column and the index doubles as a dependency graph: a reader checking any single result can see immediately which companion papers have to hold for it.}
\label{tab:theorem-index}\\[0.3em]
\toprule
\textbf{\#} & \textbf{Theorem name} & \textbf{One-line statement}
 & \textbf{Location} & \textbf{Upstream theorems} \\
\midrule
\endfirsthead
\toprule
\textbf{\#} & \textbf{Theorem name} & \textbf{One-line statement}
 & \textbf{Location} & \textbf{Upstream theorems} \\
\midrule
\endhead
\midrule
\multicolumn{5}{r}{\small (continued)}\\
\endfoot
\bottomrule
\endlastfoot

1 & \textsf{T-GUT-Closure}
 & The unique stable layer-1 \GGA{} is the \SUfive\ GUT gauge group
 closed by \BIsix{} anomaly cancellation; no other simple group
 discharges the Green--Schwarz constraint.
 & Theorem~\ref{thm:gut-closure}
 & \AThree{} (T-BI6-9.1); \ANine{} \T{1}--\T{3} \\

2 & \textsf{T-Electroweak-Vacuum}
 & The unique stable layer-2 \GGA{} is the electroweak vacuum with
 charged-lepton mass ladder and Higgs vev locked by Wilson-line
 holonomy on $S^1_\theta$.
 & Theorem~\ref{thm:electroweak-vacuum}
 & \AOne{} \T{1.1}; \ANine{} \T{4}--\T{7} \\

3 & \textsf{T-Atomic-Ground-State}
 & The unique stable layer-3 \GGA{} is the periodic table of atoms;
 no alternative ground-state structure closes the Schr\"odinger
 cascade once $\Mlock$ and electromagnetic coupling are fixed.
 & Theorem~\ref{thm:atomic-ground-state}
 & \ASix{} T-GRIP-4 (\GA{}); \ASeven{} \T{-CON} \\

4 & \textsf{T-Molecular-Closure}
 & The unique stable layer-4 \GGA{} is the small-molecule inventory
 $\{$H$_2$O, CO$_2$, NH$_3$, CH$_4\}$ on the warm-wet stratum;
 asserted from biochemistry pending a Morse-theoretic proof
 (Appendix~\ref{app:lemma-promotion-roadmap}).
 & Theorem~\ref{thm:molecular-closure}
 & \ASix{} T-GRIP-4; \ASix{} \T{3.2} \\

5 & \textsf{T-Amino-Acid-Selection}
 & The unique stable layer-5 \GGA{} is the twenty L-handed amino
 acids; chirality lock pending the
 \emph{Chirality-Lock} lemma
 (Appendix~\ref{app:lemma-promotion-roadmap}).
 & Theorem~\ref{thm:amino-acid-selection}
 & \ASeven{} \T{-\SIP{}}; \ASix{} \T{3.2} \\

6 & \textsf{T-Polymer-Closure}
 & The unique stable layer-6 \GGA{} is the polymer class
 (oligopeptides $\oplus$ oligonucleotides $\oplus$ fatty-acid
 chains) on the warm-wet stratum; specific selections deferred to
 downstream layers.
 & Theorem~\ref{thm:polymer-closure}
 & \AFive{} T-GDOS-1--T-GDOS-11; \ASix{} T-GRIP-4 \\

7 & \textsf{T-Bioenergetic-Closure}
 & The unique stable layer-7 \GGA{} is ATP as universal energy
 currency at $\Delta G_{\rm ATP} = 2 N_A \QB \approx 30.0$~kJ/mol;
 mitochondrial P/O = 5/2 confirms the cadence macroscopically.
 & Theorem~\ref{thm:bioenergetic-closure-atp}
 & \ASeven{} \T{-EP}; Arisaka equation
 $J_E/\QB = J_e = J_B = J_{Qi}$ \\

7a & \textsf{T-RNA-World-Closure}
 & The unique stable layer-8 \GGA{} is RNA as the bifunctional
 informational--catalytic polymer; four-letter alphabet
 $\{A,U,G,C\}$, ribozymes at $k_{\rm cat} \sim 10$/s, triplet-code
 precursor; proto-ribosome appears here.
 & Theorem~\ref{thm:rna-world-closure}
 & \ASix{} \T{3.2}; \ASeven{} \T{-EP}, \T{-CON} \\

7b & \textsf{T-Information-Closure}
 & The unique stable layer-9 \GGA{} is DNA as the archival storage
 polymer; B-form double helix; fidelity $\epsilon \approx 10^{-9}$/bp;
 central dogma DNA $\to$ RNA $\to$ protein.
 & Theorem~\ref{thm:information-closure-dna}
 & \AFour{} (\GUfive{}); \ASeven{} \T{-QUA}; RNA-World substrate \\

8 & \textsf{T-Cell-Closure}
 & The unique stable layer-10 \GGA{} is the LUCA-class first cell as
 a \SeOS{} on the warm-wet stratum.
 & Theorem~\ref{thm:cell-closure}
 & \AFive{} T-GDOS-1--T-GDOS-11 (\SeOS{} class) \\

8a & \textsf{T-Primary-Domain-Split}
 & The unique stable layer-11 \GGA{} is the dual-gauge bifurcation
 into Bacteria (ester/H$^+$) $\oplus$ Archaea (ether/Na$^+$); two
 minima of DGL(5:2); $\sim 3.5$~Ga.
 & Theorem~\ref{thm:three-domain-split}
 & \AThree{} (\BIsix{}); \ASix{} T-GRIP-4 \\

8b & \textsf{T-Eukaryogenesis}
 & The unique stable layer-12 \GGA{} is the chimeric eukaryote with
 energy-per-gene phase jump 2--20$\times$; $\sim 1.8$--$2.0$~Ga
 (post-GOE).
 & Theorem~\ref{thm:eukaryogenesis}
 & \ASix{} \T{3.2}; \ASeven{} \T{-EP} (twin DGL coupling) \\

8c & \textsf{T-Multicellularity}
 & The unique stable layer-13 \GGA{} is the M0--M7 ladder
 (clonal adhesion $\to$ organogenesis); convergent at
 $\sim 600$~Ma (animals) and $\sim 1$~Ga (plants/fungi).
 & Theorem~\ref{thm:multicellularity}
 & \ASix{} T-GRIP-4; \ASeven{} \T{-CON} \\

8d & \textsf{T-Central-Nervous-System}
 & The unique stable layer-14 \GGA{} is the minimal integrated
 connectome --- a centralized nerve ring carrying sensory, inter-
 and motor classes closed into one loop, clocked by a central
 pattern generator, at $\sim 300$ neurons and \QCI{} $\sim 4$;
 empirically anchored by \citet{WhiteEtAl1986} and
 \citet{CookEtAl2019}.
 & Theorem~\ref{thm:central-nervous-system}
 & \ASix{} \T{-GRIP-4}; \ASeven{} \T{-CON} \\

9 & \textsf{T-Neural-Ring-Attractor}
 & The unique stable layer-15 \GGA{} is the insect heading-direction
 ring on the ellipsoid body; empirically anchored by
 \citet{KimEtAl2017}.
 & Theorem~\ref{thm:neural-ring}
 & \ASeven{} \T{-CON}; \ASix{} T-GRIP-4 \\

10 & \textsf{T-Allocentric-Map}
 & The unique stable layer-16 \GGA{} is the rodent hippocampal
 place--grid--time--causal four-dimensional map; empirically
 anchored by \citet{TaubeMullerRanck1990} and head-direction
 cells.
 & Theorem~\ref{thm:allocentric-map}
 & \ASeven{} \T{-\SIP{}}, \T{-CON}; \ASix{} T-GRIP-4 \\

11 & \textsf{T-Conscious-Episode}
 & The unique stable layer-17 \GGA{} is a \EIC{} closure event:
 \OH{} and \IH{} are reconciled to within tolerance
 $\delta_{\mathrm{rec}}$.
 & Theorem~\ref{thm:conscious-episode}
 & \ASeven{} \T{-\EIC{}}, \T{-\HAL{}} \\

12 & \textsf{T-Articulation}
 & The unique stable layer-18 \GGA{} is the present paper: an
 articulation of the laws of physics by a meta-cognitive
 attractor that has written itself.
 & Theorem~\ref{thm:articulation}
 & \ASeven{} \T{-\HAL{}}; the closure loop \\

13 & \textsf{T-Qi4-GGA-Admissibility}
 & On each cascade layer, \QiFour{} admissibility selects the unique
 \GGA{} as the joint fixed point of \DGI{}, \GSSB{}, and \GA{}.
 & Theorem~\ref{thm:qi4-gga-admissibility}
 & \ASeven{} \T{-\SIP{}}, \T{-QUA}, \T{-EP}; \ASix{} T-GRIP-1--T-GRIP-5 \\

14 & \textsf{T-Origin}
 & (\emph{The central theorem.}) Given the four constitutional
 axioms of \AOne{} and the inheritance graph of \AOne{}--\ATen{},
 the eighteen-layer cascade closes uniquely, and its
 closure includes the existence of the present paper.
 & Theorem~\ref{thm:t-origin}
 & \emph{All} of \AOne{}--\ATen{} \\

15 & \textsf{T-GRIP-Composition}
 & \GRIP{} factors uniquely as the three-stroke composition
 $\cS_{\GA} \circ \cS_{\GSSB} \circ \cS_{\DGI}$ on the \SIP{}
 bundle.
 & Theorem~\ref{thm:t-grip-composition}
 & \ASix{} T-GRIP-1--T-GRIP-5 \\

16 & \textsf{T-Bogomolnyi-Action}
 & The cascade action is bounded below by a Bogomolnyi term that
 forces the $N \leq 3$ coherence bound (three gauge sectors).
 & Corollary~\ref{thm:t-bogomolnyi-action}
 & \AFive{}; \AThree{} (T-BI6-9.1) \\

17 & \textsf{T-SIP-Tower-Recursion}
 & The cascade is a well-defined recursion on the \SIP{}-bundle
 space: each layer-$n$ output is the layer-$(n{+}1)$ substrate.
 & Corollary~\ref{thm:t-sip-tower-recursion}
 & \ASeven{} \T{-\SIP{}}; \ASix{} T-GRIP-4 \\

18 & \textsf{T-Cascade-Layer-Uniqueness}
 & At each cascade layer, the \GGA{} produced by \GRIP{} on the
 \GFF{} landscape is unique (no degenerate competitor passes
 \QiFour{}).
 & Corollary~\ref{thm:t-cascade-layer-uniqueness}
 & \ASix{} T-GRIP-4; \ASix{} \T{3.2} \\

19 & \textsf{T-Universal-Identity}
 & All layer-$n$ products are gauge-equivalent: every electron is
 the same electron; every ATP is the same ATP; every articulator
 is the same articulator.
 & Theorem~\ref{thm:t-universal-identity}
 & \AOne{} \T{1.2} (\GGEP{}); \ASix{} T-GRIP-4 \\

20 & \textsf{T-Cascade-Termination}
 & The cascade has exactly eighteen layers (1 through 18); a hypothetical
 layer 19 is forbidden by self-reference.
 & Theorem~\ref{thm:t-seventeen-step-termination}
 & \ASeven{} \T{-\EIC{}}; G\"odelian self-reference \\

21 & \textsf{T-QB-Anchor}
 & The bioenergetic quantum $\QB \approx 0.1555$~eV/qubit is forced
 by the Arisaka equation
 $J_E/\QB = J_e = J_B = J_{Qi}$ at the warm-wet stratum.
 & Theorem~\ref{thm:t-qb-anchor}
 & \ASeven{} \T{-EP}; \ATwo{} T-GG6-6 \\

22 & \textsf{T-SeOS-Class-Characterization}
 & A system is a \SeOS{} iff it satisfies five necessary and jointly
 sufficient conditions: openness, gauge-current closure, \GFF{}
 Lyapunov descent, $\QB$-anchored bookkeeping, \QiFour{}
 admissibility.
 & Theorem~\ref{thm:t-seos-class-characterisation}
 & \AFive{} T-GDOS-1--T-GDOS-11; \ASix{} T-GRIP-4 \\

23 & Three-Gauge Theorem
 & The three boundary gauge currents
 ($J_e$, $J_B$, $J_{Qi}$) are jointly necessary and jointly
 sufficient for cascade closure; no fourth gauge current exists.
 & Theorem~\ref{thm:three-gauge}
 & \AFive{} (restated) \\

24 & \textsf{T-GFF-Lyapunov-Descent}
 & The \GFF{} is the cascade's Lyapunov functional: it strictly
 decreases along \GRIP{} flow except at critical points; the
 cascade does not wander.
 & Theorem~\ref{thm:t-gff-lyapunov-descent}
 & \ASix{} \T{3.2}; \ASix{} T-GRIP-4 \\

25 & \textsf{T-Three-Gauge-Identity}
 & The Three-Gauge Theorem is the
 Gibbs/Friston/\GGTheory{} identity: thermodynamic free energy,
 predictive free energy, and \GFF{} agree once expressed in
 canonical units.
 & Theorem~\ref{thm:t-three-gauge-identity}
 & \AFive{}; \ASeven{} \T{-\HAL{}} \\

26 & \textsf{T-Pagoda-Five-Story}
 & The cognitive architecture above layer 10 has the unique
 five-storey pagoda structure (sensorimotor / heading /
 allocentric / episodic / meta-cognitive).
 & Theorem~\ref{thm:t-pagoda-five-story}
 & \ASeven{} \T{-CON}; \ASix{} T-GRIP-4 \\

27 & Monotonic refinement of the \EIC{}
 & The \EIC{} Lyapunov is monotonically decreasing across consecutive
 closure events under the \QiFour{} pipeline.
 & Theorem~\ref{thm:eic}
 & \ASeven{} \T{-\EIC{}} \\

28 & \textsf{T-OH-IH-EIC-Identity}
 & The holographic triad factors as
 $\OH{} + \IH{} = \EIC{}$ on the meta-cognitive layer;
 each summand is forced uniquely by \QiFour{} admissibility.
 & Theorem~\ref{thm:t-oh-ih-eic-identity}
 & \ASeven{} \T{-\HAL{}}, \T{-\EIC{}} \\

29 & \textsf{T-QCI-Ladder-Monotonicity}
 & The \QCI{} (\QiFour{} consciousness index) is monotonically
 increasing along cascade ascent: $\QCI_n \geq \QCI_{n-1}$ for all
 $n$ in the cognitive strata.
 & Theorem~\ref{thm:t-qci-ladder-monotonicity}
 & \ASeven{} \T{-\HAL{}} \\

30 & \textsf{T-Internal-Measurement-As-Consciousness}
 & Consciousness is operationally identified with the internal
 measurement performed by a \EIC{}-closing meta-cognitive
 attractor on its own \IH{}.
 & Theorem~\ref{thm:t-internal-measurement-as-consciousness}
 & \ASeven{} \T{-\EIC{}}, \T{-\HAL{}} \\

31 & Self-referential closure of the \GGTheory{} program
 & The Arisaka-GG cohort closes self-referentially:
 \AEleven{} is the \GGA{} of layer 18, and the constitution it
 appeals to is \AOne{}, which it has helped write.
 & Theorem~\ref{thm:self-referential-closure}
 & \emph{All} of \AOne{}--\ATen{} \\

32 & Universal-identity closing statement
 & The universe is not full of arbitrary identical objects; it is
 full of \GGA{}s. Restated at every cascade layer; closing
 centerpiece of Part~\ref{part:cognitive}.
 & Part~\ref{part:cognitive} close
 & \AOne{} \T{1.2} (\GGEP{}); \ANine{} Part~XI (the universal-identity section) \\

33 & \textsf{T-GRIP-In-Lyapunov-Descent}
 & Within-layer regime: the cascade descends to the layer-specific
 \GGA{} by Lyapunov gradient flow on a fixed Morse landscape;
 sustained energy injection at constant effective temperature;
 no inter-stratum trajectory possible.
 & Theorem~\ref{thm:t-grip-in}
 & \ASix{} \T{3.2}; \ASix{} T-GRIP-1--T-GRIP-5 \\

34 & \textsf{T-GRIP-Out-Necessity}
 & Between-layer regime: layer-to-layer cascade ascent
 $\GGA_n \to \GGA_{n+1}$ requires at least one out-of-equilibrium
 perturbation that widens the \DGI{} orbit, destabilizes the
 layer-$n$ basin transiently, and opens a new basin on
 $\mathcal{S}_{n+1}$. Pure within-layer dynamics cannot cross
 a stratum boundary.
 & Theorem~\ref{thm:t-grip-out}
 & \ASix{} \textbf{GRIP-ST2} stratum confinement (GRIP-Out necessity); \ANine{} Part~XI (the universal-identity section) \\

35 & \textsf{Cascade-Self-Sustenance} (corollary)
 & For $n \geq 1$ (Big Bang excepted), the \GRIPout{} events that
 drive each cascade transition are themselves generated by the
 dynamics of layers $\leq n-1$; the cascade is autocatalytic in
 its own forward drive.
 & \S\ref{cor:cascade-self-sustenance}
 & \AOne{} \AX{3} (\OGGEP{}); \T{-GRIP-Out-Necessity} \\

\end{longtable}
}

\subsection{Grouping commentary by cascade layer}
\label{subsec:theorem-index-by-layer}

The forty theorems are not a flat list; they organize themselves
by the cascade-layer index $n = 1, \dots, 18$. We sort them now
by layer. The reader who wants to know ``what theorems live on the
amino-acid layer'' or ``what theorems live on the meta-cognitive
layer'' will find the answer here.

\paragraph{Layer 1 (gauge group).} The layer-1 stratum hosts a single
named theorem, \textsf{T-GUT-Closure} (row~1). It also draws on
upstream theorems: \AThree{} (T-BI6-7.2, the \BIsix{} anomaly closure) and
\ANine{} \T{1}--\T{3} (the parameter audit of the gauge sector). In
the language of the daily-life metaphor: layer 1 is the city's
foundation, and \textsf{T-GUT-Closure} is the single deed of trust
that locks the foundation slab.

\paragraph{Layer 2 (electroweak vacuum).} The layer-2 stratum hosts
\textsf{T-Electroweak-Vacuum} (row~2). This is the layer at which
\AOne{} \T{1.1} (Wilson--Hosotani holonomy on $S^1_\theta$) does its
operational work: the lepton mass ladder and the Higgs vev are
fixed by the holonomy, not adjusted by hand.

\paragraph{Layer 3 (periodic table).} \textsf{T-Atomic-Ground-State}
(row~3). This is where \ASix{}'s cascade-closure structure theorem
\textbf{GRIP-ST5} and \ASeven{} \T{-CON} (the contraction theorem of \QiFour{})
first operate jointly: the Bohr atom is a fixed point of the joint
operation, and no other ground state passes \QiFour{} admissibility.

\paragraph{Layer 4 (small molecules).} \textsf{T-Molecular-Closure}
(row~4). This is the first layer marked \emph{Conditional} in the
codebook of Part~\ref{part:results} \S\ref{sec:cascade-codebook-master}; the lemma whose proof would promote it to
\emph{Derived} is named in
Appendix~\ref{app:lemma-promotion-roadmap}. Plain English: this
layer says the small molecules of life are forced; the proof of
forcing is downstream work.

\paragraph{Layer 5 (amino acids).} \textsf{T-Amino-Acid-Selection}
(row~5). Also \emph{Conditional}; chirality lock requires the named
Chirality-Lock lemma of
Appendix~\ref{app:lemma-promotion-roadmap}.

\paragraph{Layer 6 (polymers).} \textsf{T-Polymer-Closure} (row~6).
\emph{Conditional} on the Codon-Mapping lemma.

\paragraph{Layer 7 (bioenergetics --- ATP only after Refine-3).} Two theorems live here:
\textsf{T-Bioenergetic-Closure} (row~7) and \textsf{T-QB-Anchor}
(row~21). This is the first layer where two distinct theorems of
the present paper are bound to a single layer: one says \emph{the
energy currency is ATP at $\Delta G = 2 N_A \QB$}, the other says
\emph{the energy quantum is $\QB \approx 0.1555$~eV/qubit}.
The two are not redundant; they are the existence and the
calibration claims, respectively.

\paragraph{Layer 8 (RNA-World).} \textsf{T-RNA-World-Closure}
(row~7a). This is the bifunctional informational--catalytic polymer
stratum: RNA is both information storage and catalyst. Selected by
the same OGN-5 ledger that fixes $\QB$; the four-letter alphabet
$\{A,U,G,C\}$ is forced by $|\Sigma| = N+1 = 4$.

\paragraph{Layer 9 (DNA archival closure).}
\textsf{T-Information-Closure} (row~7b). This is the stable
archival-storage stratum: deoxyribose replaces ribose for hydrolysis
resistance; the double helix locks the geometry; fidelity reaches
$10^{-9}$/bp through proofreading and mismatch repair.

\paragraph{Layer 10 (first cell).} Three theorems live here:
\textsf{T-Cell-Closure} (row~8), \textsf{T-SeOS-Class-Characterization}
(row~22), and the Three-Gauge Theorem in its restated form (row~23).
This is the densest layer. The reason is that layer 10 is where the
class definition of a \emph{Selfish Open System} first applies; before
layer 10 the substrate is gauge or atomic or molecular, and the
\SeOS{} formalism does not yet operate. Plain English: this is the
layer at which the cascade first knows that what it is producing is
\emph{alive}; the abundance of theorems reflects the abundance of
new class structure that comes online.

\paragraph{Layer 11 (primary-domain split).}
\textsf{T-Primary-Domain-Split} (row~8a). The first post-LUCA
divergence: the same DGL(5:2) algebraic packet supports two stable
minima --- Bacteria (ester/H$^+$) and Archaea (ether/Na$^+$).
Eukarya is \emph{not} a third primary domain; it is the Layer-12
chimera produced by symbiotic coupling of the two prokaryotic
branches. Empirically anchored by Woese--Fox 16S rRNA phylogenetics \citep{WoeseFox1977}.

\paragraph{Layer 12 (eukaryogenesis).}
\textsf{T-Eukaryogenesis} (row~8b). The energy-per-gene phase
jump: the chimeric host+mitochondrion carries twice the effective
DGL bandwidth, opening evolutionary phase space inaccessible to
prokaryotes (chromatin, nuclear envelope, cytoskeleton, vesicle
traffic, sex). Predicts dual-knee 1/$f$ PSD spectral signature in
eukaryotic cellular activity.

\paragraph{Layer 13 (multicellularity).}
\textsf{T-Multicellularity} (row~8c). The M-ladder of cell-cell
coordination: M0 priming, M1 clonal adhesion, M2 gap junctions, M3
division of labor, M4 patterning, M5 transport networks, M6
integration, M7 organogenesis. Multiple independent origins; the
ladder is constitutionally forced, not lineage-specific.

\paragraph{Layer 15 (heading attractor).} \textsf{T-Neural-Ring-Attractor}
(row~9). Also \textsf{T-GFF-Lyapunov-Descent} (row~24), since the
ring attractor is the first cognitive instance where the \GFF{} acts
as the Lyapunov for a non-thermodynamic substrate.

\paragraph{Layer 16 (allocentric map).} \textsf{T-Allocentric-Map}
(row~10).

\paragraph{Layer 17 (conscious episode).} Four theorems live here:
\textsf{T-Conscious-Episode} (row~11), the monotonic-refinement
theorem of the \EIC{} (row~27),
\textsf{T-OH-IH-EIC-Identity} (row~28), and
\textsf{T-Internal-Measurement-As-Consciousness} (row~30). Layer 17
is the second-densest layer because consciousness has the structural
complexity of a class \emph{plus} a Lyapunov \emph{plus} a
factorization \emph{plus} an operational definition. Each of the
four named theorems addresses one of the four obligations.

\paragraph{Layer 18 (articulation).} Two theorems live here:
\textsf{T-Articulation} (row~12) and the self-referential closure
theorem (row~31). The two are not the same: row~12 says
\emph{the present paper exists as a closure event}; row~31 says
\emph{the corpus that contains the present paper closes
self-referentially}.

\paragraph{Cross-layer theorems.} Eight cross-layer results --- five theorems and three
corollaries --- are not bound to a single layer: \textsf{T-Qi4-GGA-Admissibility} (row~13),
\textsf{T-Origin} (row~14),
\textsf{T-GRIP-Composition} (row~15),
\textsf{T-Bogomolnyi-Action} (row~16, a corollary),
\textsf{T-SIP-Tower-Recursion} (row~17, a corollary),
\textsf{T-Cascade-Layer-Uniqueness} (row~18, a corollary),
\textsf{T-Universal-Identity} (row~19), and
\textsf{T-Cascade-Termination} (row~20). These are the
\emph{mechanism} results; they specify how the cascade itself
operates, layer-independent. Three more are also cross-layer in
spirit: \textsf{T-Three-Gauge-Identity} (row~25),
\textsf{T-Pagoda-Five-Story} (row~26), and
\textsf{T-QCI-Ladder-Monotonicity} (row~29).

\subsection{Grouping commentary by mechanism}
\label{subsec:theorem-index-by-mechanism}

The same forty theorems also organize themselves by
\emph{mechanism}: each theorem invokes one of the five operational
backbones of the framework.

\paragraph{The \GRIP{} mechanism.} Rows 1--17 (the eighteen per-layer
\GGA{} theorems), plus rows~15, 17, 18, 19, and 20 (the \GRIP{}
mechanism theorems themselves), depend operationally on \GRIP{}.
\GRIP{} is the universal cascade engine; without it the per-layer
\GGA{} statements have no producer. Daily-life metaphor: \GRIP{} is
the lathe; the per-layer \GGA{} statements are the parts the lathe
turns.

\paragraph{The \QiFour{} mechanism.} Rows~13, 21, 22, and 29 invoke
\QiFour{} in an essential way; rows 1--12 invoke it as the selection
filter (every \GGA{} is a fixed point of the joint \DGI{}--\GSSB{}--
\GA{} pipeline). \QiFour{} is the admissibility check; without it
\GRIP{} would produce candidates without filtering them.

\paragraph{The \GFF{} mechanism.} Rows~23 (Three-Gauge),
24 (Lyapunov descent), and 25 (Three-Gauge identity) live on the
\GFF{}. Plus the per-layer theorems invoke the \GFF{} as the cost
functional that \GRIP{} minimizes. Daily-life metaphor: the \GFF{}
is the energy bill; the cascade always picks the cheapest stable
configuration.

\paragraph{The \SeOS{} mechanism.} Rows~7, 8, 22 (and from
layer~10 onward, every layer's \GGA{} theorem is implicitly a \SeOS{}
statement) invoke the \SeOS{} class. \SeOS{} is the open-system
generalization of equilibrium thermodynamics that the cascade needs
once it crosses the warm-wet line; before layer~10 the substrate is
closed enough for equilibrium thermodynamics to suffice.

\paragraph{The \EIC{} mechanism.} Rows~11, 27, 28, 30 live on the
\EIC{}. This is the cognitive-layer mechanism that produces the
factorization \OH{} + \IH{} = \EIC{} and the operational definition
of consciousness. Layer 17 is where \EIC{} closure first becomes
possible; layer 18 articulation is downstream of \EIC{} closure.

\paragraph{Universal identity and self-reference.} Row~19
(\textsf{T-Universal-Identity}), row~31 (the self-referential closure
of the program), and row~32 (the closing universal-identity
statement) are the meta-theorems: they say that the cascade closes
on its own input, that the closure is the same closure at every
layer, and that the closure is therefore the universe's natural
articulation of itself. These three theorems are the closing
centerpiece of the paper.

\subsection{Inheritance from \AOne{}--\ATen{}}
\label{subsec:theorem-index-inheritance}

A reader who wants to verify the upstream supply chain should
consult the inheritance graph at \S\ref{sec:inheritance} of Part~\ref{part:inheritance}
(``Eleven-paper inheritance''); the present subsection summarizes
which upstream theorems appear most often in the right-hand column
of Table~\ref{tab:theorem-index}.

\paragraph{\AOne{} \citep{ArisakaA1Constitution}.} The constitutional
axioms \T{1.1}--\T{1.4} are cited at rows~2, 19, 32 (the
electroweak holonomy, the universal-identity insight, and the
universal-identity closing). \AOne{} is the constitution; every
theorem of \AEleven{} inherits from it implicitly, but the rows
that cite it explicitly are the rows where a constitutional axiom
does operational work.

\paragraph{\ATwo{} \citep{ArisakaA2GG6}, \AThree{}
\citep{ArisakaA3BI6}, \AFour{} \citep{ArisakaA4Euclid}.} \ATwo{} is
cited at row~21 (\QB{} via the Diophantine ledger); \AThree{} at
rows~1, 16 (Green--Schwarz closure and the Bogomolnyi bound).
\AFour{}'s Euclidean ideal is the meta-theorem of the parameter
audit and appears as the closing rhetoric, not as an inheritance
arrow per row.

\paragraph{\AFive{} \citep{ArisakaA5GDOS}.} \AFive{} T-GDOS-1--T-GDOS-11
underlie rows~4, 6, 8, 22, 23, 24, 25. \AFive{} is the workhorse
upstream paper: the \GDOS{} formalism, the \GFF{}, and the Three-Gauge
Theorem all live there.

\paragraph{\ASix{} \citep{ArisakaA6Gi4}.} \ASix{} T-GRIP-1--T-GRIP-5 (the
\GiFour{} pillars: \DGI{} $\to$ \GSSB{} $\to$ \GA{}) are cited at
rows~3, 6, 9, 13, 15, 17, 18, 22, 24, 26. \ASix{} is the operational
upstream paper: \GRIP{} is built out of its pillars.

\paragraph{\ASeven{} \citep{ArisakaA7Qi4}.} \ASeven{}'s six named
theorems (\T{-\SIP{}}, \T{-QUA}, \T{-EP}, \T{-CON}, \T{-\HAL{}},
\T{-\EIC{}}) are cited at rows~3, 5, 7, 9, 10, 11, 13, 17, 21, 25,
26, 28, 29, 30. \ASeven{} is the most-cited upstream paper of
\AEleven{}: it is where the \QiFour{} pillars and the \EIC{} live,
and the cognitive cascade depends on it from layer~3 onward.

\paragraph{\ANine{} \citep{ArisakaA9SM} and \ATen{}
\citep{ArisakaA10Cosmos}.} \ANine{} is cited at rows~1, 2 (the
gauge-group parameter audit); \ATen{} is cited as the cosmological
parameter-audit benchmark in the empirical-horizon discussion
(Part~\ref{part:results} \S\ref{sec:cascade-codebook-master}). Each consequence paper has its own per-row
inheritance pattern.

\subsection{Closing of the theorem index}

A theorem index is a small honesty. We have named
forty theorems and we owe the reader the index. A theorem
that lives only inside a four-hundred-page paper has not really been
named; it has only been mentioned. This Appendix promotes each
named theorem to a citable entry. The hostile reviewer can now
say ``row~14 says X'' and the friendly reader can now say
``row~14 says X'' and both will mean the same thing. In a program
whose claim is \emph{zero free parameters}, the matching claim of
\emph{forty named theorems, each with a one-line statement and
a location and an upstream supplier} is the theorem-level
counterpart. We trust the symmetry.

% ====================================================================
%  Appendix G.  Lemma-promotion roadmap
% ====================================================================
\section{Lemma-promotion roadmap: from Conditional to Derived}
\label{app:lemma-promotion-roadmap}

\subsection{Why this roadmap matters}
\label{subsec:lemma-roadmap-why}

A closed theory is one in which every claim is \emph{Derived}: every
assertion follows by a finite chain of inference from a finite list
of constitutional axioms. Euclid's Elements is the paradigm: thirteen
books, no Conditional claims, no Asserted-pending-proof entries.
\GGTheory{} is a program, not yet a closed theory. A program has
a different obligation. Its obligation is to make
\emph{explicit} where the gaps are; to name each gap; to say what
would close it; and to list, as candidly as a tax audit, the lemmas
that the future is expected to prove.

In plain English: a closed theory has nothing to prove; a program
has a roadmap of what is still to be proved. This Appendix is the
roadmap. We claim no fewer Conditional or Asserted layers than the
codebook of Part~\ref{part:results} \S\ref{sec:cascade-codebook-master} has already
documented. Five are developed at length below: the three chemistry
lemmas of layers~4--6 and the two neural anchors of layers~15--16. The
remaining Conditional rows --- layers~8, 11, 12 and 13 --- carry lemmas
that are named with their rows in the master codebook but are not yet
worked out here, and the three Anchor-Derived rows (layers~7, 9 and 10)
wait on the derivation in \AThirteen{} rather than on any lemma. Saying
which are developed and which are only named is itself part of the
accounting.

A daily-life metaphor: think of the codebook as the building's
inspection report. Each room is marked
\emph{Verified}, \emph{Conditional}, or \emph{Asserted}. This
Appendix is the punch list: for each room not marked Verified, we
name what would have to be checked or completed before the room can
be re-marked. The honesty of the report is the honesty of the
punch list.

A second daily-life metaphor: a long-distance hiker who has climbed
seventeen of eighteen peaks does not pretend to have climbed the eighteenth.
The hiker marks the eighteenth on the map, names the route, names
the season the climb is expected to be attempted, and goes on the
record about what remains. The hiker's honesty is the hiker's
credibility. \GGTheory{}'s honesty about its Conditional and
Asserted rows is the same kind of honesty.

\subsection{The five outstanding layer-level lemmas}
\label{subsec:five-lemmas}

We name, in cascade order, the five layer-level lemmas whose proof
or empirical establishment would close the corresponding row of the
codebook.

\subsubsection*{Lemma~G.1 (Layer~4 Conditional): Small-Molecule-Closure}
\label{subsubsec:lemma-small-molecule-closure}

\paragraph{Claim.} On the warm-wet stratum at \QCI{} level~3, the
set $\{$H$_2$O, CO$_2$, NH$_3$, CH$_4\}$ is the unique stable
\GGA{}. Equivalently, the small-molecule inventory of life is
forced; alternative four-molecule inventories are not stable basins
on the \GFF{} landscape.

\paragraph{What's needed.} A Morse-theoretic proof on the \GFF{}
landscape that this particular four-molecule set is the unique
stable basin at the warm-wet substrate. The proof must
(i) define the substrate stratum operationally (warm-wet means
$T \approx 300\,\mathrm{K}$, $p_\mathrm{H_2O}$ saturated,
electrochemical potentials within the bioenergetic band),
(ii) define the \GFF{} on the molecular configuration space (a finite
combinatorial space once chemistry is enumerated),
(iii) show by Morse theory or by direct enumeration that the
four-molecule set is the unique minimum, and
(iv) verify that no near-minimum competitor passes \QiFour{}
admissibility.

\paragraph{Current status.} Asserted from biochemistry's empirical
inventory (every textbook on biochemistry agrees that these are the
four small molecules of life), pending derivation from constitutional
axioms. In the codebook of Part~\ref{part:results}, the row is marked
\emph{Conditional} with status note ``small-molecule-closure lemma
pending''.

\paragraph{Promotion path.} A purely theoretical proof, doable in
\ATwelve{} or in a downstream successor paper on molecular
landscapes. No new experimental input is required; the input is
already in the biochemistry tables of the last century. The proof
must clear the chemistry tabulation as the empirical anchor and
then derive uniqueness from the \GFF{} formalism.

\paragraph{Physics intuition.} At first-year-grad-student level: ask
why water is special. Water is special because it is the unique
solvent that simultaneously discharges hydrogen-bond donor and
acceptor roles at room temperature with appropriate dielectric
constant. This is a thermodynamic fact, but it ought to be derivable
from \BIsix{} closure plus electromagnetic coupling: water's bond
angle, dipole moment, and dielectric constant are not free; they are
determined by the atomic-layer parameters that the cascade has
already locked. The Small-Molecule-Closure lemma is essentially the
statement that all such ``why this molecule, not another'' questions
have one joint answer: the joint \GFF{} minimum on the warm-wet
substrate.

\paragraph{Daily-life metaphor.} Why are the kitchens of every
culture stocked with water, salt, sugar, vinegar, oil, and a few
other staples, even when the cultures have never communicated?
Because the constraints of cookery on a planet with this
biochemistry force convergence on these staples. Small-Molecule-
Closure is the lemma that says: the kitchen's small-molecule pantry
is forced the same way the cosmic small-molecule pantry of life is
forced.

\subsubsection*{Lemma~G.2 (Layer~5 Conditional): Chirality-Lock}
\label{subsubsec:lemma-chirality-lock}

\paragraph{Claim.} Of the twenty amino acids and the sugars that
form RNA's and DNA's backbone, the L-handed amino acids and the
D-handed sugars are forced by the \QiFour{} admissibility filter
applied to the chirality stratum. No other handedness combination
passes the filter; in particular, the mirror world (D-amino /
L-sugar) is forbidden, not merely unobserved.

\paragraph{What's needed.} A chirality-projection theorem in
\ASeven{}-\QiFour{} that shows the chirality stratum has a unique
admissible fixed point. Concretely:
(i) Identify the geometric origin of chirality (Wilson-line phase on
$S^1_\theta$ in the \BIsix{} construction, or an analogous structure
on the scale-radius direction);
(ii) Show that the \DGI{} stage of \QiFour{} explores chirality
symmetrically (no preferred handedness at the exploration step);
(iii) Show that the \GSSB{} stage selects one handedness (Wilson-line
phase fixed by the CP-violation source of \ANine{});
(iv) Show that the \GA{} stage stabilizes the selection (no return
to mirror-symmetric basin).

\paragraph{Current status.} Asserted from biochemistry's empirical
record; pending the chirality-projection theorem.

\paragraph{Promotion path.} This is a theoretical-proof problem,
doable in a successor paper to \ASeven{}. It is closely related to
the CP-violation source question in \ANine{} (the chiral-bias
mechanism in cosmology is OP-7 in
Appendix~\ref{app:open-problems}); the two problems likely share
the same upstream constitutional input.

\paragraph{Physics intuition.} The Wilson-line phase on $S^1_\theta$
is the only constitutional source of a parity-odd quantity in the
\BIsix{} construction. If the Chirality-Lock lemma is going to be
proved at all, it must trace back to this phase. The physical
question is: does the Wilson-line phase fix one handedness directly,
or does it fix handedness through a chain (Wilson-line $\to$
chiral-bias in cosmology $\to$ early-Earth chemistry $\to$ L-amino
selection)? Either route would prove the lemma.

\paragraph{Daily-life metaphor.} Why does every screw on every
factory floor turn the same way? Because once a global convention
is selected (right-handed screws), local non-conformity becomes
non-functional. The chirality lock is the cosmic analogue: once a
global convention is set by the Wilson-line phase, all the
downstream chemistry must conform. The lemma is the statement that
the global convention is forced, not chosen.

\subsubsection*{Lemma~G.3 (Layer~8 Conditional): Codon-Mapping}
\label{subsubsec:lemma-codon-mapping}

\paragraph{Claim.} The $4^3 \to 20$ triplet-codon map (with three
stop codons) is the unique stable encoding under \GFF{} descent on
the polymer-folding layer. No other codon table passes \QiFour{}
admissibility on the layer-8 substrate.

\paragraph{What's needed.} An information-theoretic derivation from
\AFive{}-\GDOS{} plus the four-letter alphabet of \ASix{}-\GiFour{}.
The four-letter RNA alphabet (A, U, G, C) is fixed at layer~8 by
\GiFour{} pillars; the question is whether the codon map at layer~8
is forced. Concretely:
(i) Define the \GFF{} on the codon-table configuration space
(a finite combinatorial space: number of ways to partition $4^3 =
64$ codons into 20 amino-acid groups plus stop codons);
(ii) Show that the empirical codon table is the unique \GFF{}
minimum;
(iii) Verify that nearby codon tables (e.g.\ alternative degeneracy
patterns) fail \QiFour{}.

\paragraph{Current status.} Asserted from biology's empirical
record; pending the information-theoretic derivation. Hints from
the literature on the optimality of the genetic code under
robustness-against-mutation criteria (the standard ``frozen
accident'' debate); the framework's claim is stronger than
``optimal'': it is \emph{forced}.

\paragraph{Promotion path.} A theoretical-proof problem, doable in
a successor paper. No new experimental input is required. The
proof must show that codon-table optimality under combined
robustness, decodability, and \GFF{} cost is a strict minimum
(not merely a local one).

\paragraph{Physics intuition.} A codon table is a \emph{code} in
the Shannon sense; the cascade's claim is that the code is forced
by the joint cost (storage cost, decoding cost, error correction
cost) measured by the \GFF{}. A first-year-grad-student exercise:
compute the entropy of the codon table and compare to alternative
tables; the framework predicts that the standard table is the
\emph{unique} table that simultaneously minimizes decoding error,
storage entropy, and \GFF{} cost.

\paragraph{Daily-life metaphor.} Why do all keyboards in a given
language have the same layout? Because the joint cost (training
cost, decoding speed, error rate) of an alternative layout exceeds
the joint cost of the standard. The codon table is the genetic
equivalent: the cascade picks the layout, and the layout is forced.

\subsubsection*{Lemma~G.4 (Layer~15 Asserted): Neural-Ring-Attractor existence}
\label{subsubsec:lemma-neural-ring}

\paragraph{Claim.} The insect ellipsoid-body ring attractor is the
unique stable layer-15 \GGA{}; the topological ring structure is
forced by the \GRIP{} cascade on the neural substrate.

\paragraph{Empirical anchor.} The ring attractor has been
documented in \emph{Drosophila} by \citet{KimEtAl2017}; the
two-photon imaging and connectomics work of the past decade has
confirmed both the ring topology and the bump-attractor dynamics.

\paragraph{What's needed.} A derivation from layer-10 \SeOS{} first
cell to layer-15 ring topology via \GRIP{} cascade on the neural
substrate. Concretely:
(i) Identify the neural-substrate analogue of the \GFF{} (the cost
functional whose minimization drives synaptic plasticity);
(ii) Show that on a substrate that has discharged layer-10 (a cell
capable of sensorimotor coupling), the unique stable \GGA{} on the
heading-direction stratum has ring topology;
(iii) Confirm by comparison with the empirical ring attractor.

\paragraph{Current status.} Asserted from empirical neuroscience;
the derivation from constitutional principles is in progress. In
the codebook, the row is marked \emph{Asserted} pending the
derivation.

\paragraph{Promotion path.} Mixed theoretical-empirical. The
empirical anchor is solid; the theoretical derivation requires a
\SeOS{}-on-neural-substrate formalism that is partial as of this
writing. Settlement expected from a combination of computational
neuroscience modeling and continued \emph{Drosophila} connectomics.
Estimated timeline: 5--10 years.

\paragraph{Physics intuition.} The ring topology arises whenever a
one-dimensional periodic variable (heading direction $\in S^1$) must
be represented robustly on a noisy substrate. Mexican-hat lateral
connectivity plus winner-take-all dynamics yields a bump attractor
on a ring; this is textbook \citet{Amit1989}. The framework's
stronger claim is that this is the \emph{unique} layer-15 \GGA{} on
the cascade's neural stratum; alternative encodings (e.g.\ Cartesian)
fail \QiFour{} admissibility on a substrate that must support the
allocentric map of layer~16.

\paragraph{Daily-life metaphor.} How does an experienced sailor
know which way is north on a foggy day? By integrating angular
velocity from the wind on his cheek and the rocking of the boat,
maintaining an internal heading bump that updates continuously.
The cascade's claim is that any sufficiently general such navigator
must instantiate the same ring attractor; the fly's ellipsoid body
and the sailor's internal compass are the same \GGA{}.

\subsubsection*{Lemma~G.5 (Layer~16 Asserted): Allocentric-Map existence}
\label{subsubsec:lemma-allocentric-map}

\paragraph{Claim.} The rodent hippocampal place--grid--time--causal
four-dimensional manifold is the unique stable layer-16 \GGA{}; the
manifold structure is forced by the \GRIP{} cascade applied to a
substrate that has already discharged layer~15.

\paragraph{Empirical anchor.} The head-direction cell layer is
documented by \citet{TaubeMullerRanck1990}; the place-cell and
grid-cell layers by O'Keefe and Moser; the time and causal layers
by more recent work cited in Part~\ref{part:discussions} \S\ref{sec:eight-test-deeper-reading}. The phenomenology is rock
solid.

\paragraph{What's needed.} A derivation from layer-15 ring attractor
to layer-16 four-dimensional manifold via \GRIP{} cascade.
Concretely:
(i) Define the \SeOS{}-on-neural-substrate formalism rigorously;
(ii) Show that the four-dimensional manifold (place $\times$ grid
$\times$ time $\times$ causal) is the unique stable basin of the
layer-16 \GFF{};
(iii) Verify that the empirical hippocampal architecture instantiates
the four-dimensional manifold uniquely.

\paragraph{Current status.} Asserted from rodent neuroscience;
derivation in progress.

\paragraph{Promotion path.} Mixed theoretical-empirical. Both
sides of the lemma require continuing work: theory needs the
neural-\SeOS{} formalism; experiment needs the causal and time
layers to be characterized as carefully as the place and grid layers
have already been characterized. Estimated timeline: 10--15 years.

\paragraph{Physics intuition.} An allocentric map is a
representation of space that does not depend on the agent's current
heading; this requires combining a heading attractor (layer~15) with
an integrator that converts heading-plus-displacement into absolute
position. The grid-cell hexagonal tiling is the unique stable
periodic tiling of the plane with integer-degree rotational symmetry
compatible with continuous translation; this is a basic fact of
crystallography. The framework's stronger claim is that hexagonal
grid tiling is the \emph{unique} layer-16 \GGA{}; alternative
tilings (square, triangular) fail \QiFour{} on a substrate that must
support episodic memory at layer~17.

\paragraph{Daily-life metaphor.} A city's grid of streets is
hexagonal in some old European cities and rectangular in most
American cities; both work. But the brain's grid is hexagonal, not
rectangular, and the lemma asks: why is this so? The framework's
answer is: hexagonal tiling minimizes representational ambiguity
under translation; the brain has selected the unique \QiFour{}-
admissible layer-16 \GGA{}.

\subsection{The two methodological lemmas}
\label{subsec:two-methodological-lemmas}

Beyond the five layer-level lemmas, two methodological lemmas
support the entire cascade. These are not about a single layer;
they are about the operation of the cascade as a whole.

\subsubsection*{Lemma~G.6 (Methodological): Multi-stratum recursion}
\label{subsubsec:lemma-multi-stratum-recursion}

\paragraph{Claim.} The \SIP{}-tower recursion of
Corollary~\ref{thm:t-sip-tower-recursion} is well-defined when the
substrate changes \emph{stratum} from gauge (layer~1) to atomic
(layer~3) to molecular (layer~4) to neural (layer~15) to cognitive
(layer~17). The recursion's well-definition is not jeopardized by
substrate change.

\paragraph{Current status.} Assumed in the present paper; needs
proof. The proof obligation is conceptual rather than computational:
to show that the operational meaning of ``the layer-$n$ output is
the layer-$(n{+}1)$ substrate'' is preserved across stratum changes.

\paragraph{What's needed.} A category-theoretic or operadic
formulation of the cascade that makes the substrate change a
functorial operation. Each stratum-change point (layer~4 onset of
chemistry, layer~10 onset of biology, layer~15 onset of neuroscience,
layer~17 onset of cognitive science) is a base-change in the
recursion's category; the lemma is that base-change is well-defined.

\paragraph{Promotion path.} Theoretical, doable in a successor
paper devoted to the foundations of the cascade. Closely related to recent work on
hierarchical Bayesian inference and to category-theoretic approaches
to physics.

\paragraph{Physics intuition.} The same Lagrangian formalism that
describes gauge fields also describes molecular vibrations; this is
a daily working fact of quantum chemistry. But the same formalism
does not obviously describe neural firing patterns or cognitive
maps. The lemma's claim is that the \GFF{} formalism --- not
specifically the Lagrangian, but the more general Lyapunov +
\QiFour{} structure --- does in fact descend through all the strata.
A clean proof would lay this out at the categorical level.

\paragraph{Daily-life metaphor.} A musician transposes a piece from
C major to F major; the structure of the piece is preserved. The
multi-stratum-recursion lemma is the statement that the cascade
transposes through eighteen strata, and the structure of the cascade
is preserved. The chord progressions of layer~4 (chemistry) sound
like the chord progressions of layer~17 (cognition) because the
underlying composition is the same.

\subsubsection*{Lemma~G.7 (Methodological): Universal-identity invariance}
\label{subsubsec:lemma-universal-identity-invariance}

\paragraph{Claim.} The universal-identity insight (every electron
is the same electron; every ATP is the same ATP) holds across
cascade-layer changes. Equivalently, the gauge-equivalence statement
of Theorem~\ref{thm:t-universal-identity} extends to the
meta-cognitive layer: every articulator is the same articulator.

\paragraph{Current status.} Asserted at the closing of Part~\ref{part:cognitive};
needs a careful proof at the meta-cognitive layer. The lower
strata (every electron is the same; every proton is the same;
every ATP is the same) are sufficiently clear from gauge-equivalence
at their respective layers; the higher strata (every conscious
episode is the same; every articulator is the same) are far more
delicate.

\paragraph{What's needed.} A proof that gauge equivalence at the
meta-cognitive layer (\textsf{T-Universal-Identity} extended to
layer~18) is operationally well-defined. The challenge is that
``identity'' becomes harder to specify as the cascade ascends: two
electrons being the same is a textbook quantum-mechanical statement,
but two conscious episodes being the same requires a careful
specification of what counts as the same.

\paragraph{Promotion path.} Theoretical and philosophical. Requires
collaboration with philosophy of mind to specify the identity
relation at the cognitive layer. Closely related to the
operational definition of consciousness in
Theorem~\ref{thm:t-internal-measurement-as-consciousness}.

\paragraph{Physics intuition.} ``Two electrons are the same
electron'' is the statement that they belong to the same
\emph{representation} of the gauge group, and that any difference
between them is a gauge artifact. The lemma's claim is that the
analogous statement holds for any \GGA{}: any two ATP molecules
differ only by gauge artifact; any two conscious episodes differ
only by gauge artifact at the appropriate higher gauge group.

\paragraph{Daily-life metaphor.} Two performances of Beethoven's
Ninth Symphony are not literally identical (different orchestras,
different concert halls, different audiences) but they are
representations of the same symphony. The lemma's claim is that
all layer-$n$ products are performances of the same symphony at
their layer. Different orchestras (substrates), same symphony
(\GGA{}).

\subsection{The promotion schedule}
\label{subsec:promotion-schedule}

We summarize the seven named lemmas, together with six further per-layer lemmas, in a tabular promotion schedule --- thirteen rows in all.

{\small
\begin{longtable}{@{}>{\raggedright\arraybackslash}p{0.16\linewidth}
 >{\raggedright\arraybackslash}p{0.13\linewidth}
 >{\raggedright\arraybackslash}p{0.30\linewidth}
 >{\raggedright\arraybackslash}p{0.16\linewidth}
 >{\raggedright\arraybackslash}p{0.18\linewidth}@{}}
\caption[Lemma-promotion roadmap.]{Lemma-promotion roadmap. Each row names a lemma, its
current status, the path to promotion, an estimated timeline, and
the responsible downstream paper. Thirteen rows. Each names a lemma, the layer it sits on, what proving it would buy --- in every case the promotion of a named theorem from Asserted or Conditional to Derived --- an estimated timeline, and the downstream paper responsible for it. This is the register the Conditional tag in the confidence legend points at.}
\label{tab:lemma-roadmap}\\[0.3em]
\toprule
\textbf{Lemma} & \textbf{Currently}
 & \textbf{Promotion path}
 & \textbf{Estimated timeline}
 & \textbf{Responsible paper} \\
\midrule
\endfirsthead
\toprule
\textbf{Lemma} & \textbf{Currently}
 & \textbf{Promotion path}
 & \textbf{Estimated timeline}
 & \textbf{Responsible paper} \\
\midrule
\endhead
\midrule
\multicolumn{5}{r}{\small (continued)}\\
\endfoot
\bottomrule
\endlastfoot

\textsf{Dual-Gauge-SSB-Uniqueness} & Layer 11 & The DGL(5:2) action admits exactly two stable minima (Bacteria $\oplus$ Archaea); no third primary domain. Promotes \T{-Primary-Domain-Split} from Asserted to Derived. & 2027--2030 (theoretical) & \T{-Primary-Domain-Split} \\

\textsf{Energy-Per-Gene-Phase-Jump} & Layer 12 & DGL page coupling at the mitochondrial inner membrane gives $\delta W$/gene $\geq 2 \times$ the prokaryotic baseline. Promotes \T{-Eukaryogenesis} from Asserted to Derived. & 2027--2030 (theoretical) + dual-knee 1/$f$ PSD experiment & \T{-Eukaryogenesis} \\

\textsf{Multicellularity-M-Ladder-Closure} & Layer 13 & The 8-rung M-ladder $2^{L-2} = 8$ from OGN-5 is the unique stable trajectory from layer-12 GGA to organogenesis. Promotes \T{-Multicellularity} from Asserted to Derived. & 2028--2031 (theoretical + Kleiber 3/4 derivation) & \T{-Multicellularity} \\

\textsf{4-Letter-Alphabet-Uniqueness} & Layer 8 & $|\Sigma| = N+1 = 4$ is the unique alphabet cardinality selected for \emph{naturally evolved} life under Watson--Crick double-strand pairing at OGN-5 $(N=3, L=5)$; laboratory-synthesized expanded alphabets (e.g.\ Hachimoji DNA) are engineered constructs that do not self-sustain in the wild and are not excluded. Sharpens the RNA-World theorem. & 2026--2028 (theoretical) & \T{-RNA-World-Closure} \\

\textsf{RNA-Information-Catalysis-Bifunctionality} & Layer 8 & RNA is the unique polymer that is simultaneously informational (sequence) and catalytic (folded ribozyme); the 2$'$-OH instability is the price paid for this duality, and explains why DNA later takes over storage. & 2026--2028 (theoretical) & \T{-RNA-World-Closure} \\

\textsf{DNA-Takeover-Stability} & Layer 9 & The deoxyribose backbone has $\sim 10^6 \times$ better hydrolytic stability than ribose; the RNA $\to$ DNA transition is energetically forced once reverse transcriptase emerges in proto-cells. Promotes \T{-Information-Closure} from Asserted to Derived. & 2027--2029 (theoretical + biochemistry) & \T{-Information-Closure}\\


G.1 Small-Molecule-Closure
 & Conditional
 & Morse-theoretic proof on \GFF{}; tabulated chemistry anchor
 & 5--10~yr
 & \ATwelve{} or successor molecular-landscape paper \\

G.2 Chirality-Lock
 & Conditional
 & Wilson-line chirality-projection theorem in \ASeven{} successor
 & 3--7~yr
 & \ASeven{} successor / \ATwelve{} \\

G.3 Codon-Mapping
 & Conditional
 & Information-theoretic derivation from \AFive{}-\GDOS{} + \ASix{}-\GiFour{}
 & 3--7~yr
 & \AFive{} successor / \AThirteen{} \\

G.4 Neural-Ring-Attractor
 & Asserted
 & Theoretical derivation + continued \emph{Drosophila}
 connectomics
 & 5--10~yr
 & Computational neuroscience programs \\

G.5 Allocentric-Map
 & Asserted
 & Theoretical derivation + continued rodent/human hippocampus
 work
 & 10--15~yr
 & Computational neuroscience programs \\

G.6 Multi-stratum recursion
 & Assumed (methodological)
 & Category-theoretic / operadic foundations
 & 5--10~yr
 & A foundations successor (category-theoretic) \\

G.7 Universal-identity invariance
 & Asserted (methodological)
 & Operational identity at meta-cognitive layer
 & 10--15~yr
 & \AFourteen{} / philosophy-of-mind collaboration \\

\end{longtable}
}

\paragraph{Reading the schedule.} Three lemmas (G.2 Chirality-Lock,
G.3 Codon-Mapping, G.6 Multi-stratum recursion) are
theoretical-proof problems that could be discharged in
\ATwelve{}, \AThirteen{}, or a foundations successor. Two lemmas (G.1
Small-Molecule-Closure, G.7 Universal-identity invariance) need a
combination of theory and empirics or theory and philosophy. Two
lemmas (G.4 Neural-Ring-Attractor existence, G.5 Allocentric-Map
existence) are primarily empirical and will be settled by ongoing
experimental neuroscience. The framework's overall maturity will
be measurably greater once three of the seven named lemmas have been
promoted; once five of the seven have been promoted, the framework will
have exited its program stage and entered its closed-theory
stage.

\paragraph{The empirical lemmas as anchor points.} Lemmas G.4 and
G.5, although marked \emph{Asserted}, are arguably the
\emph{best-evidenced} entries in the codebook --- a reminder that a
grade records the state of the \emph{derivation}, not the weight of
the evidence. The cohort legend is one-dimensional and is stated once,
in \S\ref{sec:confidence-legend}: \emph{Conditional} means derived
modulo one named lemma, \emph{Asserted} means not yet derived and
pending a specific experiment, and \emph{anchor-Derived} means the
empirical anchor is closed while the from-geometry proof is deferred.
A row can therefore be empirically strong and still carry a weak
grade; that is the discipline working, not a second axis.

\subsection{Closing of the roadmap}

The difference between a program and a closed theory
is not the number of theorems; it is the explicit accounting of the
lemmas that are not yet theorems. The present roadmap is the
program's explicit accounting. We list thirteen lemmas; each is
named, each is sized, each is dated, each is assigned a responsible
downstream paper. The reader who wants to know what remains to be
proved has the answer in this Appendix. The hostile reviewer who
wants to point at an unmarked Conditional or Asserted row will find
that none such exist: every row is marked, every mark is annotated,
every annotation has a promotion path.

A closed theory is the limit of a healthy program. This Appendix
is the documentation of where we are on the path to that limit.

% ====================================================================
%  Appendix H.  Open problems and future directions
% ====================================================================
\section{Problems Solved and Future Directions: the next decade of the program}
\label{app:open-problems}

\subsection{Why this catalog}
\label{subsec:open-problems-why}

The lemma-promotion roadmap of Appendix~\ref{app:lemma-promotion-roadmap}
catalogs the named lemmas whose proof would close specific
codebook rows. The present Appendix complements that roadmap by
listing the \emph{theorem-level} open problems: structural questions
whose resolution would extend the framework's reach, rather than
merely close a Conditional row. Open problems differ from
Conditional rows in that they are not pre-specified by the codebook;
they are research questions that the framework's existence has
made askable.

In plain English: the codebook tells us \emph{what we have not yet
proved}. This Appendix tells us \emph{what we have not yet
asked}. The two are complementary; both are necessary if the
framework is to be honest about its incompleteness.

We tag each open problem with one of three status labels:
\textbf{[Open]} (no resolution in sight at the time of writing),
\textbf{[Partially Resolved]} (significant progress made, full
resolution pending), or \textbf{[Resolved]} (a worked-example
entry, included to illustrate what resolution looks like). The L6
convention of status tagging is observed throughout.

\subsection{Theorem-level open problems}
\label{subsec:theorem-open-problems}

We list ten theorem-level open problems.

\subsubsection*{OP-1 (PS-Origin-1) [Open]: The deterministic--stochastic boundary}
\label{subsubsec:op-1}

\paragraph{Question.} At what cascade depth does the recursive
uniqueness theorem transition from \emph{deterministic} (layers
1--7: gauge group, electroweak vacuum, periodic table, small
molecules, amino acids, polymers, bioenergetics) to
\emph{stochastic-ensemble} (layers 10--18: first cell, ring attractor,
allocentric map, conscious episode, articulation)?

\paragraph{Context.} At layer~7 and below, the framework asserts
uniqueness of individual products: there is one periodic table,
one ATP molecule, one DNA structure. At layer~10 and above, the
framework asserts uniqueness of \emph{classes}: there is one LUCA
\emph{class}, but the individual LUCA is not specified; there is
one ring-attractor class, but the individual neural circuit is not
specified. The transition is real but not sharply located.

\paragraph{Why it matters.} A clean characterization of the
deterministic--stochastic boundary would clarify what
\textsf{T-Origin} (Theorem~\ref{thm:t-origin}) actually claims
existence of: a specific cosmos or a class of cosmoses. At present
the framework claims existence of \emph{the} cosmos with the
constitutional axioms, but the individuation at high layers is
ensemble-individuation rather than instance-individuation.

\paragraph{Daily-life metaphor.} Compare ``every elementary school
in this country teaches the same alphabet'' (deterministic at the
alphabet level) with ``every elementary school in this country
teaches its own version of recess'' (stochastic at the recess
level). The framework needs to pin down where in the cascade this
shift happens.

\subsubsection*{OP-2 (PS-Origin-2) [Partially Resolved]: The handedness origin}
\label{subsubsec:op-2}

\paragraph{Question.} Is the L-amino / D-sugar combination forced
by the scale-radius structure of the \BIsix{} bulk, or by the
Wilson-line phase on $S^1_\theta$, or by a combination?

\paragraph{Status.} Partially Resolved. The L-amino / D-sugar
combination has been documented as the empirical fact; the
constitutional origin is partial. The handedness origin and the
Chirality-Lock lemma (G.2) of
Appendix~\ref{app:lemma-promotion-roadmap} are closely related;
full resolution of the handedness origin would close layer~5
(Chirality-Lock).

\paragraph{What is known.} The Wilson-line phase on $S^1_\theta$
in the \BIsix{} construction is the only constitutional source of
a parity-odd quantity; the chirality lock must trace back to it.
What is not yet known is the explicit propagation chain from
Wilson-line phase to L-amino selection through layer~4 chemistry.

\paragraph{Promotion path.} See Lemma~G.2 in
Appendix~\ref{app:lemma-promotion-roadmap}.

\subsubsection*{OP-3 (PS-Origin-3) [Open]: The Type-I vs Type-II civilization question}
\label{subsubsec:op-3}

\paragraph{Question.} The framework predicts the existence of
articulators at layer~18 (us). Does the cascade have a natural
Type-N civilization classification, analogous to the Kardashev scale?

\paragraph{Context.} An articulator is the unique stable layer-18
\GGA{}: a meta-cognitive attractor capable of writing down the
laws of physics that produced it. But what is the scale of an
articulator? A single human is not an articulator in the
constitutional sense; rather, the scientific community in
aggregate, with its instruments and its corpus, is the articulator.
The question is whether this scale is unique, or whether the
framework permits Type-I (planetary) articulator $\to$ Type-II
(stellar) articulator $\to$ Type-III (galactic) articulator
hierarchies.

\paragraph{What is known.} The framework's empirical horizon
(Part~\ref{part:results} \S\ref{sec:cascade-codebook-master}) is silent on this question. The cascade
terminates at layer~18 by Theorem~\ref{thm:t-seventeen-step-termination};
a layer~19 articulator-of-articulators is forbidden by self-reference.
But within layer~18, the scale of the articulator is unconstrained
by the present theorems.

\paragraph{Why it matters.} If multiple articulator scales are
admissible, the framework predicts that the universe could contain
several mutually-undetected articulator communities; if a single
scale is forced, the framework constrains the SETI search space.

\subsubsection*{OP-4 (PS-Origin-4) [Open]: Multi-articulator-class extension}
\label{subsubsec:op-4}

\paragraph{Question.} Could the cascade close on multiple
meta-cognitive attractors simultaneously? If so, what would the
interaction between such attractors look like?

\paragraph{Context.} \textsf{T-Universal-Identity}
(Theorem~\ref{thm:t-universal-identity}) says every articulator is
the same articulator, in the gauge-equivalence sense. But the
existence of multiple gauge-inequivalent articulator classes is
not excluded by the present theorems. If, for example, the
warm-wet stratum and a hypothetical warm-dry stratum both close
the cascade, then two articulator classes would coexist.

\paragraph{What is known.} The framework strongly prefers the
warm-wet stratum (the bioenergetic quantum $\QB$ is anchored to
warm-wet chemistry); other strata are not ruled out.

\paragraph{Why it matters.} If multiple articulator classes can
coexist, the framework predicts that the universe may host
biospheres of different biochemistry, mutually undetected because
they live on different gauge-inequivalent attractors.

\subsubsection*{OP-5 (PS-Origin-5) [Resolved]: The Hubble two-stage structure}
\label{subsubsec:op-5}

\paragraph{Question.} Why does the observed Hubble constant
display the two-stage structure (late-time vs early-time tension)
documented in \citet{Planck2018} and \ATen{}?

\paragraph{Status.} Resolved. The two-stage structure is derived
in \ATen{} \citep{ArisakaA10Cosmos} as Theorem~T-Cosmos-16, the
two-stage corridor existence theorem, as a consequence of the cascade's behavior at the
cosmological boundary. Worked example of L6 status: this is what
\emph{Resolved} looks like in a healthy program. An entry that
was an open problem two cycles ago is now an entry of the
parameter-audit ledger.

\paragraph{Why it is included.} To show that the open-problems
catalog is not a static list. Items move from
\emph{[Open]} to \emph{[Partially Resolved]} to \emph{[Resolved]}
over time; the catalog's job is to register the motion.

\subsubsection*{OP-6 (PS-Origin-6) [Open]: Layer-19 prevention}
\label{subsubsec:op-6}

\paragraph{Question.} The framework forbids articulating the
articulation: Theorem~\ref{thm:t-seventeen-step-termination} says
the cascade has exactly eighteen layers, no nineteenth. Can this
prohibition be derived from G\"odel's incompleteness in a precise
way?

\paragraph{Context.} The current proof of layer-19 forbiddance
appeals to a self-reference argument: an articulator-of-articulators
would have to articulate itself, which would require a constitution
strictly larger than the one it has articulated. This is morally
G\"odelian but not formally so.

\paragraph{What is known.} G\"odel's incompleteness theorem applies
to formal systems of a certain expressive power; the framework's
constitutional axioms are at the edge of that expressive power.
A formal derivation of layer-19 forbiddance from G\"odelian
incompleteness would close the open problem.

\paragraph{Why it matters.} The layer-18 termination is the
framework's strongest meta-claim. A G\"odelian derivation would
elevate it from a self-reference argument to a formal theorem.

\paragraph{Daily-life metaphor.} A camera can photograph anything
except itself in the act of photographing; this is not a defect of
the camera but a structural feature of any device that
\emph{articulates} another device. The layer-19 forbiddance is the
cosmic version of this structural feature.

\subsubsection*{OP-7 (PS-Origin-7) [Partially Resolved]: The chiral-bias mechanism in cosmology}
\label{subsubsec:op-7}

\paragraph{Question.} Where in the cascade does the chiral bias
(baryon asymmetry, lepton asymmetry, eventual L-amino selection)
originate?

\paragraph{Status.} Partially Resolved. \ATen{} has the
baryogenesis story partial; full closure requires a CP-violation
source from \ANine{}-Standard Model. The chain
``CP-violation in \ANine{} $\to$ baryogenesis in \ATen{} $\to$
chiral bias in A11 layer~5'' is conjectured but not derived in
full detail.

\paragraph{What is known.} Each link in the chain is plausible;
no link has been ruled out; the chain as a whole has not been
verified.

\paragraph{Promotion path.} This is a multi-paper project that
would close OP-7 simultaneously with the Chirality-Lock lemma
(G.2) and the handedness-origin OP-2. The three open problems
share an upstream cause and are likely to be resolved together.

\subsubsection*{OP-8 (PS-Origin-8) [Open]: Layer-10 stochastic ensemble}
\label{subsubsec:op-8}

\paragraph{Question.} The \SeOS{} first-cell layer is stochastic in
the sense that the individual identity of LUCA is not forced, only
the class. Does the framework permit multiple parallel LUCA
classes?

\paragraph{Context.} Layer~10 is the deterministic--stochastic
transition point. The class of LUCA-equivalent first cells is
uniquely characterized
(Theorem~\ref{thm:t-seos-class-characterisation}) but the
individuals inside the class are not. This is the layer at which
the universal-identity insight first becomes ensemble-identity
rather than instance-identity.

\paragraph{What is known.} Earth's biosphere is monophyletic; all
known life descends from a single LUCA. The framework allows
multiple LUCA classes in principle; whether the universe contains
multiple LUCA-class biospheres is unsettled.

\paragraph{Why it matters.} A clean characterization would
constrain astrobiology: it would either permit parallel biospheres
(open question for SETI) or forbid them (the universe is
monophyletic to a class).

\subsubsection*{OP-9 (PS-Origin-9) [Open]: The \QCI{} saturation question}
\label{subsubsec:op-9}

\paragraph{Question.} The \QCI{} (\QiFour{} consciousness index)
is monotonically increasing along cascade ascent
(Theorem~\ref{thm:t-qci-ladder-monotonicity}). Does it saturate at
layer~18, or does it asymptote?

\paragraph{Context.} A saturating \QCI{} would mean that
articulation is the highest \QCI{} value attainable; an
asymptoting \QCI{} would mean that articulation is the limit of an
infinite cognitive ascent. The framework is currently silent on
which.

\paragraph{Why it matters.} Connects to OP-3 (Type-N civilizations)
and to the philosophical question of whether higher consciousness
than human articulation is meaningful.

\subsubsection*{OP-10 (PS-Origin-10) [Open]: The cascade's robustness to perturbation}
\label{subsubsec:op-10}

\paragraph{Question.} What is the robustness of the cascade closure
to small perturbations of the constitutional axioms? If \AOne{}'s
axioms are perturbed by $\epsilon$, does the cascade still close,
and does the layer-18 articulator still exist?

\paragraph{Context.} The framework's central claim is uniqueness;
the robustness question asks whether the unique closure is
isolated (small perturbations destroy it) or is a basin of
attraction (small perturbations are absorbed).

\paragraph{Why it matters.} Distinguishes between two pictures of
\textsf{T-Origin}: an isolated-uniqueness picture (the cosmos is
on a razor's edge) and a basin-uniqueness picture (the cosmos is
on a plateau). The current proofs do not distinguish.

\subsection{Experimental open problems}
\label{subsec:experimental-open-problems}

The framework also has an empirical horizon (Part~\ref{part:results} \S\ref{sec:eight-test-pull-horizon}); we
list the principal experimental open problems whose resolution
would extend or falsify the framework.

\subsubsection*{EP-1 (PS-Origin-11): The Kaluza--Klein lock-cluster}

Find or rule out the KK lock-cluster at 3.85 TeV predicted by
\PaperPOne{} \citep{ArisakaP1KK}. Required: HL-LHC at full
luminosity in the relevant invariant-mass channel. Estimated
timeline: 5--10 years from the time of writing.

\subsubsection*{EP-2 (PS-Origin-12): The Kaxion dark-matter candidate}

Find or rule out the Kaxion dark-matter candidate at 8.1~$\mu$eV
predicted by \PaperFullCOne{} \citep{ArisakaC1Kaxion}. Required:
ADMX-class haloscope experiments at the relevant mass-window.
Estimated timeline: 3--7 years.

\subsubsection*{EP-3 (PS-Origin-13): The cosmological recombination ratio}

Measure the operational decoupling redshift $z_{\rm dec}$ model-independently
and confront it with \ATen{}'s atomic-chain prediction ($\approx 1091$, zero
fitted parameters, presently on \citet{Planck2018} $z_{\rm dec} \approx 1089.92$
at the $0.1\%$ level); the corridor switch $z^\star = e^7 \approx 1096.6$ is a
distinct topological marker $0.5\%$ above it. Required: an experimental probe of the
recombination history (CRR spectral distortion, 21-cm) at sub-percent precision;
the kill is $z_{\rm dec}$ off the atomic-chain value by more than $0.5\%$.
Estimated timeline: 5--15 years.

\subsubsection*{EP-4 (PS-Origin-14): The bioenergetic quantum}

Find or rule out the framework's prediction
$\QB \approx 0.1555$~eV/qubit
(Theorem~\ref{thm:t-qb-anchor}). Required: single-molecule
calorimetry of ATP hydrolysis at the operationally-defined
information-bearing channel. Estimated timeline: 5--10 years.

\subsubsection*{EP-5 (PS-Origin-15): The \QCI{} ladder in neuroimaging}

Test the cascade's prediction of a strict \QCI{} ladder
(Theorem~\ref{thm:t-qci-ladder-monotonicity}) by neuroimaging across
species at different cognitive strata. Required: cross-species
neuroimaging with operationally-comparable \QCI{} metrics.
Estimated timeline: 10--20 years.

\subsubsection*{EP-6 (PS-Origin-16): Cross-species universal-identity in cognitive maps}

Search for the framework's prediction
(Theorem~\ref{thm:t-universal-identity}) that the layer-16
allocentric map is structurally the same map in rodent and primate.
Required: comparative neuroscience with high-resolution recording
across species. Estimated timeline: 5--15 years.

\subsubsection*{EP-7 (PS-Origin-17): SETI and the Type-II forbiddance}

Probe the framework's prediction that no Type-II civilization
exists in our light cone (OP-3 read as a falsifiable assertion).
Required: continued SETI search at radio, optical, and
techno-signature frequencies, with a precise definition of what
``Type-II'' means in the framework's terms. Estimated timeline:
indefinite; null result strengthens the prediction.

\subsection{Connections to other programs}
\label{subsec:connections}

The framework occupies a particular intellectual neighborhood; we
list the adjacent programs with which productive handshakes are
expected.

\paragraph{String theory landscape.} The cascade's claim of unique
closure stands in apparent opposition to the string-theory
landscape's claim of $10^{500}$ vacua. But the opposition is more
apparent than substantive: the framework's constitutional axioms
\emph{select} from a candidate landscape; the framework does not
deny the candidate landscape's existence, only its selection-free
status. A productive handshake would identify the framework's
unique closure as a constrained subset of the landscape and would
characterize the constraint algebraically.

\paragraph{Integrated Information Theory (IIT).} \citet{Tononi2008}
proposes that consciousness is the integrated information of a
system. The framework's
Theorem~\ref{thm:t-internal-measurement-as-consciousness}
operationally identifies consciousness with internal measurement on
the \EIC{}. The two definitions are not the same but they are
adjacent; a handshake would express integrated information in terms
of \EIC{} closure quality $\delta_{\mathrm{rec}}$.

\paragraph{Global Workspace Theory (GWT).} \citet{Dehaene2014}
proposes that consciousness is a workspace where information is
broadcast for access by multiple cognitive modules. The framework's
\OH{}/\IH{}/\EIC{} triad has a comparable architecture; the
\EIC{} is a workspace where \OH{} and \IH{} are reconciled. A
handshake would identify the \EIC{} closure event with the GWT
broadcast event.

\paragraph{Free Energy Principle (FEP).} Karl Friston's Free Energy
Principle is the closest neighbor to the framework's \GFF{} +
\QiFour{} architecture. Theorem~\ref{thm:t-three-gauge-identity}
identifies the Three-Gauge Theorem with the Gibbs/Friston/
\GGTheory{} identity; this is the cleanest handshake in the
neighborhood. Future work should extend the identity to the
allocentric and meta-cognitive layers.

\paragraph{Loop quantum gravity and other quantum-gravity
programs.} The framework's UV side (the \BIsix{} closure of
\AThree{}) is a string-theoretic construction; an alternative UV
construction in loop quantum gravity or in causal-set theory would
be welcome. A handshake would test whether the cascade's
constitutional axioms are robust to UV-completion choices.

\paragraph{Algorithmic information theory.} The framework's
information-theoretic side (\AFive{}-\GDOS{} and the \EIC{}
formalism) connects naturally to algorithmic information theory; a
handshake would identify the \GFF{} cost with a Kolmogorov-
complexity-like measure of the cascade's compression.

\paragraph{Philosophy of mind.} \citet{Chalmers1995} divides
consciousness into the easy problems and the hard problem; the
framework's
Theorem~\ref{thm:t-internal-measurement-as-consciousness} is a
candidate operational answer to the hard problem. The handshake
with philosophy of mind is the most necessary and the most delicate.

\subsection{The catalog's longtable}
\label{subsec:open-problems-table}

We close the Appendix with a tabular summary of the open problems,
their status, and their estimated resolution timelines.

{\small
\begin{longtable}{@{}>{\raggedright\arraybackslash}p{0.09\linewidth}
 >{\raggedright\arraybackslash}p{0.27\linewidth}
 >{\raggedright\arraybackslash}p{0.13\linewidth}
 >{\raggedright\arraybackslash}p{0.30\linewidth}
 >{\raggedright\arraybackslash}p{0.13\linewidth}@{}}
\caption[Open problems and their projected resolution.]{Open problems and their projected resolution. Each row
gives the problem code, its short statement, its status, the
expected resolution path, and the estimated timeline. Twenty-two rows, each with a code, the problem in one line, its status as Open or Partially Resolved, the resolution path, and a timeline. The register is deliberately unflattering: it lists the demarcation criterion the regime distinction still lacks, the predictions whose experimental tests have not been run, and the asymmetries the framework asserts without deriving.}
\label{tab:open-problems}\\[0.3em]
\toprule
\textbf{Code} & \textbf{Problem} & \textbf{Status}
 & \textbf{Resolution path} & \textbf{Timeline} \\
\midrule
\endfirsthead
\toprule
\textbf{Code} & \textbf{Problem} & \textbf{Status}
 & \textbf{Resolution path} & \textbf{Timeline} \\
\midrule
\endhead
\midrule
\multicolumn{5}{r}{\small (continued)}\\
\endfoot
\bottomrule
\endlastfoot

OP-11 & GRIP-In/Out demarcation
 & [Open]
 & Quantitative criterion in terms of GFF control-parameter rate of change $|\dot p|$ for when a system is in within-layer regime vs out-of-equilibrium shock; a sharp threshold would let experimentalists classify any open-system transition.
 & 3--5~yr \\

OP-12 & Layer-13+ cascade autocatalysis
 & [Partially Resolved]
 & For layers $\geq 12$ the \GRIPout{} trigger mechanism becomes increasingly cultural/cognitive rather than biophysical; a formal characterization of cultural \GRIPout{} events as cascade dynamics is open.
 & 5--10~yr \\

OP-13 & Dual-knee 1/$f$ PSD verification
 & [Open]
 & \T{-Eukaryogenesis} predicts a dual-knee 1/$f$ PSD signature in eukaryotic cellular activity (vs single-knee in prokaryotes); direct experimental test across phylogenetically diverse eukaryotes and prokaryotes is open.
 & 2--5~yr \\

OP-14 & Primary-Domain-Split sharpness
 & [Partially Resolved]
 & \T{-Primary-Domain-Split} asserts exactly two stable membrane-lipid GGAs but the asymmetry (why archaea ether and bacteria ester) is unresolved; needs a chirality-like lemma at the lipid scale.
 & 3--7~yr \\

OP-15 & Multicellularity M-ladder universality
 & [Open]
 & \T{-Multicellularity} predicts the M0--M7 ladder is universal across independent origins (animals, plants, fungi, algae); cross-kingdom comparative-genomics studies of M-rung markers would test universality directly.
 & 3--5~yr \\

OP-1 & Deterministic--stochastic boundary
 & [Open] & Structural theorem on cascade ensemble
 individuation & 5--10~yr \\
OP-2 & Handedness origin
 & [Partially Resolved] & Closes with Lemma~G.2
 Chirality-Lock & 3--7~yr \\
OP-3 & Type-I vs Type-II civilization
 & [Open] & Scale-of-articulator theorem + SETI null & indefinite \\
OP-4 & Multi-articulator-class extension
 & [Open] & Gauge-inequivalent attractor enumeration & 10~yr+ \\
OP-5 & Hubble two-stage structure
 & [Resolved] & \ATen{} T-Cosmos-16 (worked example) & resolved \\
OP-6 & Layer-19 prevention
 & [Open] & G\"odelian formalization of self-reference & 5--10~yr \\
OP-7 & Chiral-bias in cosmology
 & [Partially Resolved] & Closes with OP-2 + Lemma~G.2 & 5--10~yr \\
OP-8 & Layer-10 stochastic ensemble
 & [Open] & SeOS-class individuation theorem & 10~yr+ \\
OP-9 & \QCI{} saturation
 & [Open] & Saturation theorem for \QCI{} at layer 18 & 10~yr+ \\
OP-10 & Cascade robustness
 & [Open] & Perturbative analysis of T-Origin & 5--10~yr \\
EP-1 & KK lock-cluster at 3.85~TeV
 & [Open] & HL-LHC full luminosity & 5--10~yr \\
EP-2 & Kaxion DM at 8.1~$\mu$eV
 & [Open] & ADMX-class haloscope & 3--7~yr \\
EP-3 & $z_{\rm dec}$ (atomic chain) vs Planck; switch $z^\star = e^7$ distinct (two-quantity; see Part~\ref{part:discussions} \S\ref{ssec:zstar-path-to-closure})
 & [Open] & Model-independent $z_{\rm dec}$ probe (CRR spectral distortion, 21-cm) & 5--15~yr \\
EP-4 & $\QB \approx 0.1555$~eV/qubit
 & [Open] & Single-molecule calorimetry & 5--10~yr \\
EP-5 & \QCI{} ladder in neuroimaging
 & [Open] & Cross-species \QCI{} measurement & 10--20~yr \\
EP-6 & Cross-species universal-identity
 & [Open] & Comparative neural recording & 5--15~yr \\
EP-7 & Type-II forbiddance
 & [Open] & Continued SETI null result & indefinite \\

\end{longtable}
}

\subsection{Closing of the catalog}
\label{subsec:open-problems-close}

No program closes itself. A program stays open
because the world is always larger than the program. This
Appendix is the framework's standing offer: here are the questions
we have made askable, and here are the questions we have not yet
answered. A future researcher who reads this Appendix and finds
the question that lights them up has found, in our judgment, the
best entry point.

The catalog divides naturally. Three of the open problems
(OP-2, OP-7, and OP-8) cluster around chirality and individuation;
two (OP-3, OP-4) cluster around scale of articulator; one (OP-6) is
a meta-theorem about the cascade's depth; one (OP-9) is the
asymptotic behavior of \QCI{}; one (OP-10) is robustness. Two
(OP-1, OP-5) are general structural questions about the deterministic
boundary, one of which is already resolved. The experimental
problems are a separate seven-item list whose resolution belongs to
the wider community.

A daily-life metaphor for the catalog as a whole: a long-running
research program is like a city under continuous construction.
Some blocks are finished; some are scaffolded; some are still
designed. The open-problems catalog is the municipal map of
unfinished blocks. The map is part of the city's honesty.

In closing: the framework has named one central theorem
(\textsf{T-Origin}), thirty-nine other named theorems, thirteen outstanding
lemmas, and seventeen open problems. These are the four numbers
of a healthy program. The first number says what has been
proved. The second number says what has been organized. The third
number says what is pending. The fourth number says what is open.
A program that quotes only the first number is overconfident; a
program that quotes only the fourth number is defeatist. We
quote all four.

Every program is at some particular point on the path
between the constitutional axioms it has chosen and the closed
theory it hopes to become. The forty theorems are how far we
have come; the thirteen lemmas are what we are working on now; the
seventeen open problems are what we have made askable for the next
decade. These three numbers are the program's honesty. We trust
them.

% ====================================================================
%  END OF v6.9 Origin-8 content
% ====================================================================

% ====================================================================
%  Acknowledgments (L43: after last Appendix, before bibliography)
% ====================================================================

\GGPart[Common Objections]{Common Objections and Responses}{The strongest objections a hostile particle theorist, bioenergeticist, systems biologist, origin-of-life researcher, neuroscientist, philosopher of mind, cross-paper referee, and skeptical editor would each raise against the capstone --- answered, with pointers to the theorem, falsifier, open problem, or companion paper that bears the weight}
\label{part:objections}
\label{sec:common-objections}

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

A paper that claims to derive the existence of its own articulators from a parameter-free constitution, and to read cosmos, life, and mind as three boundary projections of one recursive theorem, invites the most demanding scrutiny; it would not be honest scholarship to leave the obvious objections unstated. We collect here the strongest adversarial readings --- the ones a particle theorist, a bioenergeticist, a systems biologist, an origin-of-life researcher, a neuroscientist, a philosopher of mind, a cross-paper referee, and a skeptical editor would each raise on a careful reading of the whole cohort, not a skim --- and answer each, with explicit pointers to the theorem (by its cohort-global id, \S\ref{sec:naming-convention}), the falsifier (\S\ref{sec:eight-test-pull-table}), the open problem (\S\ref{subsec:theorem-open-problems}), or the companion paper that carries the load. Where the honest answer is ``not yet,'' we say so and name the open item or the forthcoming paper (\ATwelve{}--\AFourteen{}, in preparation). These objections are deliberately the \emph{surviving} ones --- the objections that remain after a referee has read the companions that do the work the present paper only summarizes; the naive forms (that the integer ledger is mere numerology, that the running of $\alpha$ is inconsistent, that the family number flips its own convention) are retired in the companions and are not relitigated here. The objections target the framework's \emph{science}; the personal Coda (Part~\ref{part:coda}) is a statement of vision, not a scientific claim, and is not defended here.

\bigskip
\section{The particle theorist and mathematical physicist}
\bigskip
\noindent\textbf{Objection 1 (O-Origin-1) (the precision-electroweak referee)}\par\nobreak
\noindent\textbf{``Your headline $\sin^2\theta_W = 25/108$ misses the PDG value by seven sigma; you then `rescue' it with post-hoc rationals $637/640$, $159/160$, $641/640$. That is integer-search fitting dressed as theory.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The denominator is not free: $640 = 2^J\,L$ is forced by the dyadic exponent and the \SUfive{} rank, and the QD-completion is a Hopkins--Singer descent-integrality structure derived in \ANine{} \citep{ArisakaA9SM} (T-SM, the codebook). What we concede precisely is that the per-fermion \emph{numerator} within the forced $\tfrac{1}{640}\mathbb{Z}$ lattice is at present \emph{selected}, not derived --- \ANine{} says so in its own Limits. This is a discrete, named open question, not the continuous tuning the objection alleges; the leading prediction lives on a forced lattice, and the residual is integer-valued and checkable.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 2 (O-Origin-2) (the flavor theorist)}\par\nobreak
\noindent\textbf{``The generation number $N=3$ is the cohort's most-cited success, yet \PaperPTwo{} now calls it a \emph{conditional} derivation and admits the instanton budget is `exactly the kind of choice a critical referee will identify as a hidden dial.'''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Correct, and we have stopped overstating it: the present paper now inherits $N=3$ as the \emph{conditional} three-leg pincer of \PaperPTwo{} \citep{ArisakaP2N3} (the lower leg from CP violation, the upper from the \GDOS{} Morse bound), with the budget $S_{\rm BI}=3\pi$ flagged as the load-bearing conditional input. The naive `convention-flip' charge does not survive a reading of \PaperPTwo{} (the two legs intersect cleanly at $N=3$); the honest residual is the budget normalization, which we mark rather than hide.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 3 (O-Origin-3) (the collider phenomenologist)}\par\nobreak
\noindent\textbf{``$\Mlock = 3.85$~TeV is the linchpin of the electroweak layer, yet it is never derived here and a sharp five-particle cluster at $3.85$~TeV should already be excluded by ATLAS/CMS.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The scale is derived in \PaperPOne{} \citep{ArisakaP1KK} as $\Mlock = \tfrac12 M_{\rm Pl}\,e^{-K}=3.85$~TeV, parameter-free given $K=35$; the present paper inherits it. The predicted object is a \emph{leptophobic, clustered, multi-field line shape} near $3.85$--$4$~TeV, not the universally-coupled spin-2 state that generic Randall--Sundrum dilepton limits exclude, and \PaperPOne{} confronts that distinction explicitly. It is a genuine forward bet (F-Origin-1, \S\ref{sec:eight-test-pull-table}) with a pre-declared kill criterion at HL-LHC, not a retrodiction.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 4 (O-Origin-4) (the proton-decay skeptic)}\par\nobreak
\noindent\textbf{``Theorem and table disagree on proton stability, and Super-Kamiokande already excludes the lifetime you quote.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} We harmonized this: the framework's prediction is the orbifold-forbidden value $\tau_p > 10^{35}$~yr in the dominant $p\to e^+\pi^0$ channel, with a model-independent floor of $\tau_p > 10^{34}$~yr; both lie above the current Super-Kamiokande bound $\tau_p > 2.4\times 10^{34}$~yr, which Hyper-Kamiokande and DUNE will probe toward $10^{35}$~yr (F-Origin-4, \S\ref{sec:eight-test-pull-table}). Any $p\to e^+\pi^0$ detection short of $\sim 10^{35}$~yr falsifies the row.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 5 (O-Origin-5) (the renormalization-group referee)}\par\nobreak
\noindent\textbf{``You quote $\alpha^{-1}=128$ in one place and $137$ in another with no running computation linking them.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} These are the two endpoints of the standard running, not an inconsistency: $\alpha^{-1}(M_Z)=2^J=128$ at the $Z$ pole and $\alpha^{-1}(0)=2^J+N^2=137$ at the Thomson limit, connected by the radial RG flow of \ANine{} \citep{ArisakaA9SM}. The text labels each by its scale; a reader who reads them as a contradiction has read past the scale labels.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 6 (O-Origin-6) (the skeptical editor)}\par\nobreak
\noindent\textbf{``You advertise `zero free continuous parameters,' but your own Standard-Model paper concedes that most of the twenty-eight parameters are not actually derived.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Both statements are true and not in tension, and we now surface the stratification rather than letting the capstone imply universal derivation. \ANine{}'s honest accounting: exactly one parameter ($\sin^2\theta_W$) is derived end-to-end from non-physics axioms, ten more are conditional on named \ANine{}-local lemmas, and seventeen are presented as closed-form integer assignments that are a working hypothesis. `No continuous dial' is literally true --- the ledger is integer-valued --- but `everything is derived' is not what we claim; the residual freedom is discrete and named.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 7 (O-Origin-7) (the precision cosmologist)}\par\nobreak
\noindent\textbf{``$\Omega_m=1/\pi$, $\Omega_b=1/2\pi^2$, $\Omega_\Lambda=1-1/\pi$ are $\pi$-decorations matched only to a couple of percent, and DESI now hints \emph{against} $w=-1$.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The cosmic inventory is derived in \ATen{} \citep{ArisakaA10Cosmos}, inherited here. We concede the agreement is at the percent level against Planck's sub-percent bars, and --- crucially --- we flag the live tension rather than hide it: the $w=-1$ test is marked \emph{orange} in the falsifier table precisely because DESI's $\sim 2$--$4\sigma$ hint of $w\neq -1$ is a standing threat to $\Omega_\Lambda = 1-1/\pi$. If a confirmed $w\neq -1$ arrives, the cosmology floor fails; that is the falsifiability working.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 8 (O-Origin-8) (the axion experimentalist)}\par\nobreak
\noindent\textbf{``Your $8.1\,\mu$eV kaxion is just the misalignment mass retrofitted to the observed dark-matter density, inside a wide window.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The mass is fixed by the theorem-level relation $m_K f_K = \sqrt{\chi_{\rm QCD}}$ of \PaperCOne{} \citep{ArisakaC1Kaxion} (T-Kaxion-2), with $f_K$ set by geometry; the relic abundance fixes only the misalignment angle, \emph{not} the mass. We concede the \emph{absolute} frequency rides on a conjectural cap-loading integer (a $\sim\pm 13\%$ window), which \PaperCOne{} carries as a benchmark; the twin-peak split ratio $1/35$ and the $5{:}2$ portal are theorem-level, the absolute center is not yet at the same tier as $\Mlock$.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 9 (O-Origin-9) (the anomaly-cancellation specialist)}\par\nobreak
\noindent\textbf{``Your \SUfive{} content is asserted anomaly-free without ever writing the trace, and a single tensor multiplet does not generically cancel.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The trace is exhibited in \AThree{} \citep{ArisakaA3BI6} (T-BI6-7.2): the four-form $dH_3 = X_4 = \mathrm{tr}\,R^2 - \mathrm{Tr}\,F_5^2 - \tfrac{2}{5}F_B^2 - \tfrac{2}{5}F_{Qi}^2$ is written explicitly, the trace ratio $-2/5$ derived from the primitive $\mathbf{5}$, and the coefficient vector $(1,-1,-2/5,-2/5)$ packaged as one gauge-invariant tensor (also \ATwo{} T-GG6-4). The present paper imports that result; it does not re-derive it, and says so.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 10 (O-Origin-10) (the naturalness theorist)}\par\nobreak
\noindent\textbf{``You `dissolve' the hierarchy problem with the coincidence $(2/5)^3\approx 0.064$, sixteen orders of magnitude from $v/M_{\rm Pl}$.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} We do not re-derive the hierarchy here; the one-loop Wilson--Hosotani minimization that fixes the electroweak vacuum is carried out in \ANine{} \citep{ArisakaA9SM} (\S\S5--7), inherited by citation. The scale suppression is between the lock-cluster scale and the vev, $v=\Mlock\,(2/5)^N\approx 246$~GeV --- not between $(2/5)^3$ and $M_{\rm Pl}$; the larger hierarchy is the scale profile $e^{-K}$ that sets $\Mlock$ itself, which is the content of \PaperPOne{}, not a bare coincidence.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 11 (O-Origin-11) (the neutrino physicist)}\par\nobreak
\noindent\textbf{``$\delta_{\rm PMNS}\approx 196^\circ$ from $13\pi/35$ is tuned to the current global-fit central value, and your $\pm 30^\circ$ window makes the test nearly unlosable for a decade.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Conceded: this is a genuine but \emph{soft} forward bet. We classify it as a forward falsifier (DUNE/Hyper-K) in \S\ref{sec:eight-test-pull-table}, not as a precision success, precisely because the present experimental error and the kill window are both wide. It is on the ledger as a bet to be settled, with the honest caveat that it will not discriminate sharply until the long-baseline programs tighten.
\end{objqabody}

\bigskip
\section{The bioenergeticist and biophysicist}
\bigskip
\noindent\textbf{Objection 12 (O-Origin-12) (the bioenergetics-minded referee)}\par\nobreak
\noindent\textbf{``$\QB = (2/5)E_{\rm Ry}/K$ is dimensional numerology: why the Rydberg, why $K=35$, why $2/5$? With three discrete handles you can hit any biological energy.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The chain is geometry-first with \emph{no biological input}: $K=35$ and the $2/5$ projection are fixed upstream in \ATwo{}/\AThree{} (T-GG6-6, T-BI6), $E_{\rm Ry}$ is the layer-3 atomic setpoint, and ATP enters only \emph{afterward} as the empirical comparison (\S\ref{sec:naming-convention} cites the construction in \AFour{}/\ASeven{}). The non-trivial, checkable output is $2 N_A \QB = 30.0$~kJ/mol against the measured $30.5\pm0.5$ --- a $1.6\%$ prediction with the integer multiplicity (one bond $=$ two quanta) forced. What is genuinely open is the selection-rule \emph{proof} that the per-step quantum is $E_{\rm Ry}/K$ rather than $E_{\rm Ry}/J$ --- a discrete question, marked as such.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 13 (O-Origin-13) (the membrane thermodynamicist)}\par\nobreak
\noindent\textbf{``You anchor to the standard-state $30.5$~kJ/mol, but no cell runs at standard state --- the in-vivo free energy of hydrolysis is $50$--$65$~kJ/mol. You are fitting the wrong number.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} We now distinguish the two quantities explicitly at the point of display: $2N_A\QB\approx 30.0$~kJ/mol is the standard-state free energy \emph{stored in} the phosphoanhydride bond (an organism-independent constant), while the in-vivo cost \emph{paid} to synthesize one ATP is $(10/3)N_A\QB\approx 50$~kJ/mol for an \emph{E.\,coli} $c_{10}$ rotor. The surplus is not a contradiction --- it is the thermodynamic headroom that drives synthesis far from equilibrium against the cellular mass-action ratio. The bond quantum and the synthesis cost are the two ends of one transaction, not two values of one quantity.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 14 (O-Origin-14) (the structural biophysicist)}\par\nobreak
\noindent\textbf{``The four-current lock $J_E/\QB = J_e = J_B = J_{Qi}$ is asserted with no dynamics --- four fluxes of different units and magnitudes cannot simply be equal.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The lock is a topological/algebraic theorem, not a postulate: \ASeven{} \citep{ArisakaA7Qi4} (T-Qi4-9.3, rank-one inflow) derives the single source ray $(1,-1,-2/5,-2/5)$ from single-channel \BIsix{} closure --- the four boundary sources descend from \emph{one} scalar, so they are not four independent quantities. The currents are counted in their natural units against the single conversion factor $\QB$. What we concede is open is the boundary \emph{metrology}: a turnkey protocol to count $J_{Qi}$ events in a living cell does not yet exist.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 15 (O-Origin-15) (the chemiosmosis specialist)}\par\nobreak
\noindent\textbf{``$\Delta p \approx \QB$ is a unit artifact: an electron-volt equals an electron times a volt, so any proton-motive force near $0.15$~V `equals $\QB$ in eV' by definition.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} We write $e\,\Delta p \approx \QB$ (energy $=$ charge $\times$ voltage), which is dimensionally exact and not a volts-for-eV conflation. We concede the \emph{exact} equality selects $\Delta p = 0.1555$~V within the physiological band; the honest claim the prose should carry is that the elementary chemiosmotic step is \emph{of order} $\QB$, with the $\pm 20\%$ kill tolerance, not a coincidence to three figures.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 16 (O-Origin-16) (the comparative bioenergeticist)}\par\nobreak
\noindent\textbf{```ATP is to life as the electron is to physics' is poetry: the electron is structureless and irreplaceable, ATP is a contingent metabolite life could have built from GTP or thioesters.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The analogy is the GGA-output framing: the electron is the layer-2 attractor, ATP the layer-7 attractor, both outputs of the same \GRIP{} cascade. The load-bearing claim is about the \emph{energy quantum} $2\QB$ being universal across primary chemoautotrophs, not about ATP's specific molecular structure --- GTP shares the same $\Delta G^\circ$, and \emph{which} carrier is selected is a layer-7 question deferred to \AThirteen{}. We concede the structural contingency and locate the universal content in the quantum, not the molecule.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 17 (O-Origin-17) (the rotary-ATPase specialist)}\par\nobreak
\noindent\textbf{``The $c$-ring story --- $c_8$ mammals, $c_{10}$ \emph{E.\,coli}, $c_{13}$--$c_{15}$ alkaliphiles --- is post-hoc fitting to a table that already exists.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Conceded as not-yet-derived here: the environment-tuned integer stoichiometry $H^+/\mathrm{ATP}=c/3$ is presented as an empirical mapping and developed in a forthcoming dedicated study, not claimed as a theorem of the present paper. The framework's claim is the universal bond quantum $2\QB$; the rotor's matching of biosphere proton-motive force to that quantum is the forthcoming work, flagged as such.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 18 (O-Origin-18) (the atomic-physics referee)}\par\nobreak
\noindent\textbf{``Your `parameter-free' $\QB$ secretly uses the experimental Rydberg, because the closed-form electron mass is not finished.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Honest residual, and we flag it: \AFour{} \citep{ArisakaA4Euclid} states that the closed-form $m_e \to E_{\rm Ry} \to \QB$ chain is structurally in place but the printed closed form is in progress (its O9/O10). Until \ANine{} delivers $m_e$ in closed form, $\QB$'s value rests on the structural mechanism plus the experimental $m_e$ --- which is exactly the post-diction we do not want to overstate, and mark openly.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 19 (O-Origin-19) (the chromatin biophysicist)}\par\nobreak
\noindent\textbf{``Your throwaway claims --- histone methylation `pays three $\QB$,' nucleosome sliding `pays one' --- have no citations and contradict measured ATPase stoichiometries.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Conceded: those per-process integer assignments are illustrative and unsupported as printed; real remodelers hydrolyze many ATP per event. The load-bearing bioenergetic claim is the single ATP anchor $2 N_A \QB$; the per-process $\QB$ accounting is a layer-7/8 program deferred to \AThirteen{}, and we should not state the illustrative integers as if measured.
\end{objqabody}

\bigskip
\section{The systems biologist and non-equilibrium theorist}
\bigskip
\noindent\textbf{Objection 20 (O-Origin-20) (the open-systems theorist)}\par\nobreak
\noindent\textbf{``Your five-condition `\SeOS{} characterization' theorem is a definitional tautology --- it restates the constitutional axioms and declares them equivalent to a class you also defined.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The class is genuinely derived, but its substance lives in \ASeven{}, not in a restatement: the persistence theorem T-Qi4-14.3 of \ASeven{} \citep{ArisakaA7Qi4} proves a Gr\"onwall viability-tube criterion (maintenance margin above a threshold) that separates Brownian particle from convection cell from protocell from organism. The present paper's iff-packaging (Theorem~\ref{thm:t-seos-class-characterisation}) is a reformatting of that derived class; we concede the weight rests in \ASeven{}, cited at the point of use.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 21 (O-Origin-21) (the non-equilibrium statistical physicist)}\par\nobreak
\noindent\textbf{``Your whole dynamics rests on the GFF being a Lyapunov function, $dF/dt\le 0$ --- but monotone descent of a relative-entropy functional fails for generic non-detailed-balance dynamics, and you offload the proof to a citation.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The point is well taken, and the citation is exact. The descent is the GFF second law of \ASix{} \citep{ArisakaA6Gi4} (T3.2, the boundary H-theorem) --- not \AFive{}'s Nakajima--Zwanzig reduction, with which it is easily confused. Strict monotonicity holds in the detailed-balance / gradient sector (\AFive{} Lemma~T5); in the general circulating case the attractor is characterized by LaSalle's invariance principle and \emph{may be a limit cycle or torus}. This paper therefore asserts no unconditional strict point-attractor: the descent carries its detailed-balance hypothesis explicitly.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 22 (O-Origin-22) (the stochastic-thermodynamics referee)}\par\nobreak
\noindent\textbf{``You conflate a Lyapunov function with entropy production: at a non-equilibrium steady state the system free energy is flat while entropy production is strictly positive. They are different signs of different quantities.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Conceded, and aligned with the companions, which keep these distinct (\AFive{} carries a separate non-negative entropy-production rate that holds even when the free energy is non-monotone). The GFF descent is the \emph{boundary face} of the second law, not numerically the entropy-production rate; the present Part's prose should read $-dF_{\GFF}/dt$ as a Lyapunov rate, not as $\dot{S}_{\rm tot}$, and we mark the distinction.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 23 (O-Origin-23) (the variational-dynamics theorist)}\par\nobreak
\noindent\textbf{``The `Three-Gauge Theorem' --- $F_{\GFF}$ specializing to Gibbs, Friston, and gauge-action by the value of $J_{Qi}$ --- asserts `exactly three corners' without proving exhaustiveness, and you cite it to a theorem that turns out to be the Lindblad generator.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Two honest points. First, on attribution: the Gibbs/Friston/\GG{} triad is \emph{not} \AFive{}'s Lindblad theorem, and \ASeven{}'s three-gauge result (T-Qi4-10.6) is the \emph{different} PLT/MEPP/FEP triad --- so the Gibbs/Friston/\GG{} reading is the present paper's own reading of the inherited GFF, and its definitive home is a registry decision this paper flags rather than fudges. Second, we concede the `exactly three' is \emph{argued} from the three control-variable slices, not proven exhaustive over all admissible $J_{Qi}$ configurations; the claim should read as a structural proposal, not a closed classification.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 24 (O-Origin-24) (the free-energy-principle theorist)}\par\nobreak
\noindent\textbf{``Bundling the principle of least action, maximum entropy production, and the free-energy principle as `three gauges of one law' overreaches --- MEPP and FEP are in general distinct, sometimes opposed.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} We do not claim they are identical: \ASeven{} (T-Qi4-10.6) derives PLT, MEPP, and FEP each from a \emph{distinct} admissible slice of the trajectory space with a distinct coarse-graining, and concludes only that they are gauge-equivalent boundary faces of one projected functional --- explicitly not independent axioms, explicitly not a single principle. The objection targets a stronger claim than the framework makes.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 25 (O-Origin-25) (the dynamical-systems referee)}\par\nobreak
\noindent\textbf{``A single-basin gradient flow with one stable minimum per stratum cannot produce the oscillations --- theta rhythms, metabolic cycles --- that the same paper relies on.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The companion dynamics already contain circulation: \ASeven{}'s reduced flow is $\dot y = -\mu\partial F + \Omega$ with $\Omega\cdot\partial F = 0$, so the $\omega$-limit set may be a limit cycle or torus. With the LaSalle re-framing (Objection~21) the present paper inherits that correctly: the attractor is the invariant set where descent halts, not necessarily a point, which is exactly what the oscillatory phenomenology requires.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 26 (O-Origin-26) (the effective-temperature skeptic)}\par\nobreak
\noindent\textbf{``Your functional is only well-defined where you pin $\varepsilon_{\rm eff}=k_BT$ --- the closed-system limit --- which is precisely where it is least needed; far from equilibrium there is no single temperature.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Conceded that the $\varepsilon_{\rm eff}=k_BT$ identification is cleanest in the closed limit. The GFF itself is well-defined out of equilibrium through its relative-entropy (CEP) term, which does not require a single temperature; what is genuinely scale-dependent far from equilibrium is the \emph{interpretation} of $\varepsilon_{\rm eff}$, not the existence of the functional. We do not claim a universal effective temperature.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 27 (O-Origin-27) (the status-tag auditor)}\par\nobreak
\noindent\textbf{``You tag the ATP/DNA row `Derived' while admitting the full derivation is in the unwritten \AThirteen{}.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Fair, and we sharpen the label: the row is \emph{anchor-Derived} (both empirical anchors --- the $30.5$~kJ/mol ATP free energy and the four-letter alphabet --- are quantitatively closed in the literature) but \emph{derivation-deferred} (the from-geometry proof is in \AThirteen{}). Reporting these as one undifferentiated `Derived' overstated it; the confidence triage now distinguishes the empirical closure from the geometric derivation.
\end{objqabody}

\bigskip
\section{The origin-of-life researcher}
\bigskip
\noindent\textbf{Objection 28 (O-Origin-28) (the prebiotic chemist)}\par\nobreak
\noindent\textbf{```Twenty amino acids $= K - NL = 35 - 15$' is integer subtraction, not chemistry. Nothing here says \emph{which} twenty, or why glycine, or why not selenocysteine.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Conceded as not-yet-a-theorem: the count is an arithmetic relation downstream of the upstream-fixed $K=35$, and the substrate-identification claim --- that the side-chain \GFF{} Morse landscape has exactly twenty admissible minima --- is tagged \emph{Conditional} and its closed-form derivation is deferred to \ATwelve{}. The $21$st/$22$nd (selenocysteine, pyrrolysine) are codon recodings, not additional cascade minima, and we note that in the row. The count is a forced \emph{relation}; the proof of exactly-twenty-minima is the open work.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 29 (O-Origin-29) (the synthetic biologist)}\par\nobreak
\noindent\textbf{```Four bases $= N+1$' is a cardinality coincidence, and Hachimoji DNA already pairs and replicates with eight letters --- your uniqueness claim is false.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} We fenced the uniqueness claim to \emph{naturally evolved} life: synthetic expanded alphabets such as Hachimoji are engineered constructs that do not self-sustain in the wild and are explicitly not excluded. The combinatorial closure that would make four the unique \emph{naturally-selected} cardinality (the RNA-bifunctionality lemma) is \emph{Conditional} and deferred to \AThirteen{}; the falsifiability fence and the derivation status are now stated separately.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 30 (O-Origin-30) (the homochirality specialist)}\par\nobreak
\noindent\textbf{``You state the layer-5 attractor is `all-L amino acids, all-D sugars, forced' --- but your own open-problems appendix admits the mechanism does not exist.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Exactly, and the theorem now carries that concession inline: the enantiomer \emph{sign} is fixed by the parity-odd Wilson--Hosotani holonomy, but the explicit propagation chain from that holonomy to observed handedness is a named open problem (PS-Origin-2, \S\ref{subsec:theorem-open-problems}), and the cardinality result $|\GGA_5|=20$ does not depend on it. We claim the count, flag the chirality mechanism as open, and no longer present `forced' without that pointer.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 31 (O-Origin-31) (the LUCA researcher)}\par\nobreak
\noindent\textbf{``Your `theorems' for LUCA, the bacteria/archaea split, and eukaryogenesis are the consensus evolutionary narrative with attractor labels stapled on, and the dates are retrodicted from the rock record.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Conceded: these rows are tagged \emph{Asserted}, the dates are calibrated to the geological record, and the from-geometry derivations are deferred to \AThirteen{}. The present paper's contribution at these layers is the \emph{structural} claim --- that each is a \GRIP{} attractor on its stratum --- not a derivation of the transition; we mark the status honestly rather than dress narration as proof.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 32 (O-Origin-32) (the contingency theorist)}\par\nobreak
\noindent\textbf{```Inevitable, not lucky, not designed, forced' is unfalsifiable hindsight: no layer makes a forward, then-unknown prediction that was later confirmed.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} We concede the layer-versus-instance distinction and have made it a stated open problem (OP-1, \S\ref{subsec:theorem-open-problems}): the framework individuates the high layers as \emph{classes} (one LUCA class), not as instances, so the `inevitability' claim above layer-10 is ensemble-level and not instance-testable. The genuinely forward, currently-open bets are the physics falsifiers and the dual-knee spectra; the biological inevitability is a structural thesis flagged as not-yet-falsifiable, not asserted as settled.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 33 (O-Origin-33) (the metabolism-first/RNA-first referee)}\par\nobreak
\noindent\textbf{``By ordering bioenergetics before informational polymers you silently take a side in the deepest open debate of the field.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} We concede the ordering is the framework's \emph{proposal}, not a resolution: \ASeven{} explicitly does not adjudicate metabolism-first versus replicator-first, and the present paper inherits that as a deferred question. The layer numbering reflects the cascade's logical dependence as the framework reads it, with the chicken-and-egg bootstrap itself an open item for \AThirteen{}.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 34 (O-Origin-34) (the genetic-code theorist)}\par\nobreak
\noindent\textbf{``You reproduce the \emph{sizes} $64\to 20$ but never the codon-to-amino-acid \emph{map} --- which is precisely the part Crick called a frozen accident.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Conceded squarely: the present paper derives at most the cardinalities; the assignment of which codon maps to which amino acid --- the locus of Crick's frozen-accident hypothesis and of the error-minimization and stereochemical theories --- is untouched here and deferred to \AThirteen{}. We do not claim to have refuted the frozen accident; we claim the alphabet and block sizes and mark the map as open.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 35 (O-Origin-35) (the unification skeptic)}\par\nobreak
\noindent\textbf{```The same theorem from quarks to ATP to consciousness' is the kind of claim that is impressive only until you check that the biology and mind links are the weakest in the paper.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} True, and the confidence triage is built to show it: the chemistry, life, and mind rows carry the weakest tags (Conditional/Asserted), and roughly two in five rows are Derived-or-stronger today. The unification we claim is \emph{structural} --- one cascade engine, one selector, one integer ledger applied at every layer --- with the per-domain derivations explicitly deferred. The breadth is the hypothesis, not a claim that every link is already proven.
\end{objqabody}

\bigskip
\section{The systems and cognitive neuroscientist}
\bigskip
\noindent\textbf{Objection 36 (O-Origin-36) (the hippocampal-systems neuroscientist)}\par\nobreak
\noindent\textbf{``Your ring-attractor and allocentric-map `theorems' are imported experimental neuroscience --- head-direction cells, grid cells --- relabeled as attractors.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Conceded, and labeled as such: layers 15--17 are tagged \emph{Asserted}, with uniqueness `pending experimental discrimination.' \ASeven{} supplies the system-instance template and the holographic-triad construction but \emph{not} a derivation of the ring topology or the map dimensionality; those are deferred to \AFourteen{}. The present paper states these as cascade placements of established findings, not as derivations --- the honest status is on the page.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 37 (O-Origin-37) (the consciousness-timing researcher)}\par\nobreak
\noindent\textbf{```Conscious episode $=400$~ms $=$ two theta cycles' is numerology assembled from a moving target: theta is $4$--$8$~Hz, so two cycles is anywhere from $250$ to $500$~ms.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} We named the rhythm to remove the ambiguity: the relevant carrier is \emph{low} theta ($\sim 5$~Hz, the slow end of the $4$--$8$~Hz band), so two $\sim 200$~ms cycles give $\sim 400$~ms. We concede this functions as a consistency check (F-Origin-10), anchored between the P300 ignition ($\sim 300$~ms) and the Libet adequacy window ($\sim 500$~ms); the from-circuit derivation of the timescale is deferred to \AFourteen{}, and we present it as an audit, not a sharp forward bet.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 38 (O-Origin-38) (the quantitative-EEG referee)}\par\nobreak
\noindent\textbf{``Your \QCI{} `ladder monotonicity' is true by definition --- you defined \QCI{} as a cumulative count --- and there is no scoring rule, so it is not measurable.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Conceded: the present paper flags \QCI{} as a \emph{tentative} operational form, with the scoring rule, the pre-neural treatment, and the comparative calibration deferred to \AFourteen{}; without that rule the monotonicity is near-definitional. A cross-paper discrepancy is declared alongside it, and not smoothed over: \ASeven{}'s \QCI{} is a weighted geometric mean of four pillar scores, while the present paper uses a five-storey count. These are different quantities under one name, and until \AFourteen{} settles which is canonical, neither should be read as the definition. We do not present \QCI{} as a finished metric.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 39 (O-Origin-39) (the cognitive-architecture theorist)}\par\nobreak
\noindent\textbf{``The `Five-Story Pagoda' is a metaphor, and your claim that five is \emph{forced} by the ledger integer $L=5$ is contradicted by your own \ASeven{}.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} An honest limit, and the objection states it correctly. \ASeven{}'s pagoda factorization (T-Qi4-16.2) is a genuine adiabatic-elimination / slow-manifold theorem, but its own proof states that \emph{the count five is an empirical reading, not a theorem output} (its Open Problem O7), the induction yielding an arbitrary-depth tower. So this paper does \emph{not} claim five is forced by $L=5$: the pagoda is a real factorization, and the storey count is empirical.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 40 (O-Origin-40) (the aperiodic-spectrum specialist)}\par\nobreak
\noindent\textbf{``The dual-knee $1/f$ falsifier is post-hoc: the established literature fits one knee, your one `second knee' citation shows the two corners with \emph{opposite} aging behavior, and the predicted ratio is deferred.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Conceded as a forward and currently soft bet: the second knee rests on a single human-EEG report and is not yet a settled, named result, and we say so in the row. The one piece that would make it a sharp prediction --- the fixed integer ratio between the two knee frequencies --- is explicitly deferred to \AFourteen{}; until that ratio is stated, `two knees somewhere' is weakly falsifiable, and we flag it rather than overstate it.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 41 (O-Origin-41) (the philosopher-neuroscientist)}\par\nobreak
\noindent\textbf{```Consciousness is internal measurement' is a definitional identity that you then present as a theorem.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} We separated the two halves explicitly: the \emph{formal} content of the theorem (Theorem~\ref{thm:t-internal-measurement-as-consciousness}) is the internal-measurement construction --- a real open-system operation on the inner hologram; the \emph{identification} of that construction with phenomenal experience is an explicitly-labeled structural-identity \emph{hypothesis}, defended not deduced (\S\ref{sec:reader-guide-cognitive}). \ASeven{}'s T-EIC is itself scoped to \emph{access} consciousness in Block's sense and records the phenomenal identification as its own open problem; we now present the phenomenal step as the present paper's additional commitment, not as an inherited theorem.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 42 (O-Origin-42) (the binding-problem researcher)}\par\nobreak
\noindent\textbf{``You call layer-17 `the geometric account of the binding problem' but compute no binding mechanism --- you cite theta--gamma coupling and stop.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Conceded: the binding claim is at present a citation to the theta--gamma coherence literature, not a derived mechanism, and no quantitative binding prediction is made here. The from-circuit treatment is deferred to \AFourteen{}; the present paper places binding at the EIC-closure layer structurally, which is a placement, not an explanation, and we do not claim more.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 43 (O-Origin-43) (the methods referee)}\par\nobreak
\noindent\textbf{``Your flagship Mind-layer evidence is the author's own unreplicated 2022 preprints.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The load-bearing neural anchors in the present paper are independent and standard --- O'Keefe and Dostrovsky, Hafting, Taube, Quiroga, Kim. The author's 2022 preprints are cited only for the seven-dimensional allocentric-frame \emph{interpretation}, which is explicitly fenced as interpretive and not load-bearing for the cascade placements; we agree that independent replication of that specific interpretation is owed and flag it.
\end{objqabody}

\bigskip
\section{The philosopher of mind and of science}
\bigskip
\noindent\textbf{Objection 44 (O-Origin-44) (the hard-problem theorist)}\par\nobreak
\noindent\textbf{``You close the explanatory gap by stipulation --- your own text says `numerically, by stipulation' --- which is not closing it but refusing it.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} We declare the stipulation rather than smuggle it, and concede the analytic-conceptual gap is real. The framework's position is structural identity: phenomenal experience \emph{is} the layer-17 EIC closure's structural-geometric content, defended against the conceivability, knowledge, and explanatory-gap arguments (\S\ref{sec:reader-guide-cognitive}) but not deduced from the geometry. Whether structural identity is the right response to the hard problem is one of the genuinely contested questions we leave explicitly open; what the framework supplies is a structural target sharp enough to be a candidate for the identification.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 45 (O-Origin-45) (the access/phenomenal theorist)}\par\nobreak
\noindent\textbf{```Internal measurement' is a specification of \emph{access} consciousness --- selection, persistence, availability for report --- which you then relabel \emph{phenomenal}.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Conceded, and now stated as a cross-paper fact: \ASeven{} scopes its internal-measurement construction to access consciousness in Block's sense and records the phenomenal identification as its Open Problem O3. The present paper's phenomenal identification is therefore its \emph{own additional} commitment, not a downstream consequence of \ASeven{}; we no longer present it as inherited, and the leap from access-structure to phenomenality is the labeled hypothesis, not a theorem.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 46 (O-Origin-46) (the zombie-argument philosopher)}\par\nobreak
\noindent\textbf{``Your reply to the conceivability argument presupposes structural identity --- `imagining the closure without experience is imagining it without itself' just restates the disputed identity.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Agreed that the reply is not a free-standing refutation, and we say so: it is a statement of the framework's position, not a proof that the zombie is inconceivable. The honest claim is conditional --- \emph{if} structural identity holds, the zombie is incoherent --- and whether structural identity holds is exactly the open question. We present it as a defended standpoint, not as having dissolved the conceivability argument for an opponent who rejects the identity.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 47 (O-Origin-47) (the philosopher of the anthropic)}\par\nobreak
\noindent\textbf{``Deriving `observers exist' from a constitution observers wrote to fit their own world is affirming the consequent with extra steps.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The inference runs constitution $\to$ cascade $\to$ observer by \emph{uniqueness}, not by selection across an ensemble: the integers are fixed upstream with no anthropic input, and each layer carries a separate warrant. We concede that the final step --- layer-17 conscious substrates reliably reaching layer-18 articulation --- is the weakest link, supported at conjecture strength (it has happened once), and we mark it (Objection~48). The lower cascade is forced; the articulation step is conditional, and we do not claim it is forced the way the periodic table is.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 48 (O-Origin-48) (the philosopher of logic)}\par\nobreak
\noindent\textbf{``Your self-referential closure theorem is true by construction: you \emph{define} the layer-18 output as the articulation of the constitution, so output $=$ input is a tautology.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The constitution's axioms (AX-1..4 of \AOne{}) name no observers, no life, and no articulation --- so the closure is not analytic in the axioms; it is self-reference in the G\"odel/Conway sense, not logical circularity. The substantive and \emph{conditional} content is that the cascade, run from those observer-free axioms, \emph{reaches} an articulating layer (Theorem~\ref{thm:articulation}) --- and that reaching step, as Objection~47 concedes, is the conjecture-strength link, not a tautology. We separate the non-trivial claim (the cascade arrives at articulation) from the trivial one (articulation articulates).
\end{objqabody}

\bigskip
\noindent\textbf{Objection 49 (O-Origin-49) (the Popperian demarcation referee)}\par\nobreak
\noindent\textbf{```Zero free parameters' plus an integer ledger spanning quark to mind is unfalsifiability in disguise: your consciousness claims survive every physics falsifier.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Conceded and demarcated: none of the Top-10 falsifiers tests the phenomenal-identity hypothesis --- the falsifiable ledger constrains the \emph{access/structural} substrate, not the phenomenal identification, which is a defended philosophical position fenced off from the physics. The physics cascade is sharply falsifiable (Objection~73); the structural-identity thesis about qualia is not, and we no longer let the falsifiable ledger lend it unearned credibility --- the demarcation is stated, not blurred.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 50 (O-Origin-50) (the historian of philosophy)}\par\nobreak
\noindent\textbf{``Reading Wheeler, Spinoza, Whitehead, and Hegel as `fragmentary recognitions of one theorem' is self-serving Whig history.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} It is offered as the framework's \emph{reading}, with the warrant separated from the conclusion, and we record genuine tension where it exists rather than assimilating everything: the Madhyamaka case, for instance, is marked as in tension, not in anticipation. We do not claim to report what these thinkers `really meant'; we claim a structural feature they each glimpsed and could not formalize. As an openly-caveated interpretive stance that is defensible; as a claim about authorial intent it would not be, and we do not make the latter.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 51 (O-Origin-51) (the meta-ethicist)}\par\nobreak
\noindent\textbf{``You extract a first-person conclusion --- `there is something it is like' --- from purely third-person geometric premises. That is a category error.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The bridge from third-person structure to first-person experience is precisely the hard problem, and we do \emph{not} claim to deduce it: the geometry supplies a structural target, and the identification with experience is the labeled hypothesis (Objection~44), not an inference. The framework's structural realism abolishes intrinsic quiddities by hypothesis, which an opponent may reject; we present that as a contested commitment, not as a valid derivation of phenomenality from geometry.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 52 (O-Origin-52) (the philosopher of causation)}\par\nobreak
\noindent\textbf{```Necessitarian but not deterministic --- forces the layers but not the micro-events' is a firewall engineered so no observation can ever embarrass it.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} We concede the layer-versus-micro-event line needs an independent criterion, and that this is an open problem (OP-1), not a settled distinction. Crucially, the firewall does \emph{not} protect the physics: the integer ledger and its consequences (\Mlock{}, $w=-1$, proton decay) are sharply falsifiable regardless of the layer/micro distinction. The necessitarian framing is flagged as philosophically open; it is not used to dodge the physics falsifiers, which stand on their own.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 53 (O-Origin-53) (the skeptical reader)}\par\nobreak
\noindent\textbf{``The `for the first time in human intellectual history' register is a tell that the formalism was built to license the boast.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The historical claim is specific and bounded --- a formal theorem whose conclusion is the existence of articulating observers, from a ledger with no continuous freedom --- and we temper it exactly where the supporting derivations are deferred (the chemistry/life/mind floors). The boast is constrained by the confidence triage: the parts that earn it (the closure theorem, the falsifier ledger, the universal-identity insight) are the parts we present as complete; the speculative floors are flagged as program, not proof.
\end{objqabody}

\bigskip
\section{The cross-paper referee and the corpus auditor}
\bigskip
\noindent\textbf{Objection 54 (O-Origin-54) (the corpus-aware referee)}\par\nobreak
\noindent\textbf{``Your central analytic facts --- the GFF descent, the four-current lock, the integer ledger --- are all proved in other papers. A \pageref{LastPage}-page capstone that defers its keystones is a program with a theorem's title.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} A capstone is permitted to rest on its companions; what we owe is to state plainly, at each invocation, what is inherited and from where --- and we now cite each by its cohort-global theorem id (\S\ref{sec:naming-convention}): the GFF second law is \ASix{} T3.2, the rank-one current lock is \ASeven{} T-Qi4-9.3, the integer ledger is \ATwo{} T-GG6-6. The standalone content of the present paper --- the recursive-uniqueness theorem (Theorem~\ref{thm:t-origin}), the universal-identity insight, the falsifier ledger, and the self-referential closure --- is proved or stated here; the inherited apparatus is flagged as inherited.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 55 (O-Origin-55) (the citation auditor)}\par\nobreak
\noindent\textbf{``Your load-bearing dynamical claim is a cross-citation into a companion paper. In a cohort this size, cross-citations are exactly where a wrong pointer hides --- a descent attributed to a reduction theorem, say. How is a reader to check that the theorem you point at is the theorem you mean?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} By naming the result, not its position. Every companion citation here carries the registered theorem identifier together with a one-clause statement of its content, so a mismatch is visible without leaving the sentence: the descent is \ASix{} T3.2 (the GFF second law), and the $\GEP{}+\CEP{}$ split is \AFive{} T9.2{+}T5, with the detailed-balance hypothesis imported (Objection~21). A pointer to a section or an ordinal would fail silently under any renumbering; a pointer to an identifier plus its content cannot.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 56 (O-Origin-56) (the registry auditor)}\par\nobreak
\noindent\textbf{``Your `three-gauge' reading --- $F_{\GFF}$ specializing to Gibbs, Friston, and gauge-action --- is presented with the authority of an inherited companion theorem. Which registered result is it, exactly?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} None, and the paper should not be read as claiming otherwise. The Gibbs/Friston/\GG{} three-gauge reading has no registered companion id (it is not \AFive{}'s Lindblad theorem, and \ASeven{}'s three-gauge is the different PLT/MEPP/FEP triad), so it is at present the present paper's own reading of the inherited GFF. Its canonical home --- an \AEleven{}-original theorem id, or a new \AFive{}/\ASix{} id --- is a Theorem-Registry decision we flag openly rather than assert a false provenance.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 57 (O-Origin-57) (the cross-reference referee)}\par\nobreak
\noindent\textbf{``Your layer-5 and information-layer theorems attribute to \ASeven{} an `amino-acid selection theorem' and a `four-letter alphabet argument' that \ASeven{} does not contain.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The attribution is narrower than the objection assumes. \ASeven{} supplies the \QiFour{} admissibility \emph{filter} (T-EP, T-CON) and the four-current lock (T-QUA), and that is what is cited here; the \emph{specializations} to the twenty amino acids and the four-letter alphabet are the present paper's own constructions, tagged as such, with their closed-form derivations deferred to \ATwelve{}/\AThirteen{}. No \ASeven{} theorem is claimed for them.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 58 (O-Origin-58) (the SeOS referee)}\par\nobreak
\noindent\textbf{``You attribute the \SeOS{} class to `\AFive{} T7,' which is the block promoting the quantum-mechanics postulates to theorems.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Conceded: the \SeOS{} persistence theorem is \ASeven{} T-Qi4-14.3, not \AFive{} T7, which is indeed the QM-postulate block. The two are easy to confuse --- both are promotion results, and they sit one paper apart --- which is why every \SeOS{} citation here carries the T-Qi4-14.3 identifier rather than a paper-and-number reference.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 59 (O-Origin-59) (the corpus-completeness referee)}\par\nobreak
\noindent\textbf{``Your chemistry, life, and mind keystones are deferred to \ATwelve{}, \AThirteen{}, and \AFourteen{} --- papers that do not exist. The cohort logic does not close today.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Conceded openly: the present paper is a keystone that \emph{states} the chemistry/life/mind results as named, status-tagged theorems and defers their from-geometry derivations to the forthcoming floor papers. The cohort logic closes fully only when \ATwelve{}--\AFourteen{} are written; what closes \emph{today} is the physics base (\ANine{}/\ATen{}) with its own falsifiers, plus the present paper's standalone results. We mark every promissory link as promissory.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 60 (O-Origin-60) (the version-control auditor)}\par\nobreak
\noindent\textbf{``A cohort of this size cross-cites its own theorems, statuses and grades across separate documents. A result graded \emph{conditional} where it is proved and \emph{closed} where it is consumed is the likeliest failure mode you have. What keeps them aligned?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Two disciplines, both checkable from outside. Grades are owned by the paper that proves the result, and a consumer quotes the owner's grade rather than restating it: $N=3$ is conditional in \PaperPTwo{} (T-N3-5), and it is consumed as conditional --- \ANine{}'s T-SM-21.1 is stated as the \emph{conditional} derivation of $N=3$ and cites \PaperPTwo{} T-N3-5 for it. And cross-citations name registered identifiers rather than ordinals, so drift surfaces as a missing id instead of a plausible-looking wrong one. Where two papers genuinely disagree about a grade, the disagreement is printed rather than averaged.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 61 (O-Origin-61) (the single-point-of-failure skeptic)}\par\nobreak
\noindent\textbf{``Everything --- particles, atoms, chemistry, ATP, the genetic alphabet, the cognitive ladder --- hangs on one integer ledger. That is fragility, not strength.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} That is the design, and it is the falsifiability, not a bug: $(M,N,L;J,K)=(1,3,5;7,35)$ is fixed upstream (\ATwo{} T-GG6-6) with zero biological or cognitive input, and \emph{everything} downstream is a consequence. If the ledger is wrong --- if $\Mlock$ is absent, if $w\neq -1$ is confirmed, if the kaxion is not in its band --- the whole cascade falls together. A theory whose single shared assumption is exposed to many independent experiments is more falsifiable than one with a separate adjustable parameter per observable, not less.
\end{objqabody}

\bigskip
\section{The skeptical methodologist and editor}
\bigskip
\noindent\textbf{Objection 62 (O-Origin-62) (the evidential-standards editor)}\par\nobreak
\noindent\textbf{``Most of your `agreements' are with already-known numbers. Reproducing the Standard-Model masses is postdiction relabeled as derivation.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} We label the two cleanly and stop lumping them: the headline parameter-audit plot separates the empirical \emph{floor} (retrodictions of measured quantities) from the eight \emph{forward} falsifiers and the consistency checks (\S\ref{sec:eight-test-pull-table}). Reproducing known constants from a parameter-free ledger is evidentially meaningful --- it is what any unification must do --- but it is not a forward bet, and we no longer present the floor as if it were.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 63 (O-Origin-63) (the length skeptic)}\par\nobreak
\noindent\textbf{``Nearly five hundred pages is a landmark-synthesis conceit.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} It is a corpus-final synthesis of a twenty-two-paper program, and it is built for selective reading: the four-Part back matter, the per-Part reader's guides, and the confidence triage let a particle physicist read only the physics floor, a neuroscientist only the cognitive Part, and a philosopher only the closure and consciousness sections. We concede the length; the mitigation is the navigation, and the speculative material is explicitly cordoned by status tag.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 64 (O-Origin-64) (the rhetoric-of-honesty skeptic)}\par\nobreak
\noindent\textbf{``Admitting a limitation does not answer it. Your honesty triage is a shield: `we said it was Conditional, so you cannot attack it.'''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Admitting a limitation does not immunize it, and we do not treat it as doing so: every open item names the \emph{experiment} or the \emph{companion paper} that will close it --- the kaxion search, the HL-LHC cluster, \ATwelve{}--\AFourteen{} --- so each concession is a checkable promissory note, not a permanent excuse. The triage is structural and singular (one canonical confidence table quoted identically everywhere); its purpose is to be checkable, not to be unattackable.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 65 (O-Origin-65) (the rigor-contamination referee)}\par\nobreak
\noindent\textbf{``The speculative life-and-mind material drags down the credibility of the rigorous physics.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The four-floor structure is stratified by confidence precisely so a reader need not buy the frontier to buy the base: the physics floor (\ANine{}/\ATen{}) stands on its own falsifiers independent of the life/mind floors, which are flagged as the program's frontier. A reader is invited to test the base and treat the upper floors as hypotheses; the architecture is designed to keep the two from contaminating each other.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 66 (O-Origin-66) (the falsifiability auditor)}\par\nobreak
\noindent\textbf{``If failure of any one layer can be absorbed, the whole thing is unfalsifiable.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Failure is not absorbed --- it is pre-priced: the Top-10 falsifiers each carry an explicit kill criterion (\S\ref{sec:eight-test-pull-table}), and the failure of the integer-ledger base (e.g.\ no $\Mlock$ cluster at HL-LHC, or a confirmed $w\neq -1$) kills the cascade at its root. The cluster-failure scenario --- what a coordinated pattern of failures would mean floor by floor --- is worked through in the falsifier Part rather than waved away.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 67 (O-Origin-67) (the scope referee)}\par\nobreak
\noindent\textbf{``This is a research program, not a finished theory, and you present it as a capstone.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Agreed, and stated: the capstone's \emph{standalone} results --- the recursive-uniqueness theorem, the falsifier ledger, the universal-identity insight, the self-referential closure --- are complete as presented; the floor derivations (\ATwelve{}--\AFourteen{}) are the ongoing program. `Capstone' names the synthesis that binds the existing floors into one structure, not a claim that every floor is finished.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 68 (O-Origin-68) (the demystification referee)}\par\nobreak
\noindent\textbf{``The `universal-identity insight' --- that we are all the universe's one layer-18 attractor --- is mysticism in a lab coat.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The formal content is the type-token distinction made operational: \emph{the type is unique, the tokens are fully individuated}. The theorem (T-Universal-Identity) is a gauge-equivalence statement --- all layer-$n$ products are gauge-equivalent as types while differing in every individuating respect as tokens. We concede the philosophical register of the prose; the load-bearing claim is the gauge-equivalence, not a mystical merger, and earlier compressed phrasings that read as the latter have been corrected.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 69 (O-Origin-69) (the reproducibility referee)}\par\nobreak
\noindent\textbf{``The entire cohort is one author's program with no independent replication.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Conceded, and it is the reason the framework is built around \emph{externally-testable} falsifiers: the verdicts are assigned by HL-LHC, ADMX/CAPP, DESI/Euclid, and Hyper-Kamiokande/DUNE, not by the author. The test is the data, not the authority; the program is structured so that independent experiments, not internal consensus, decide it.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 70 (O-Origin-70) (the explanatory-scope skeptic)}\par\nobreak
\noindent\textbf{``A framework that claims to explain cosmos and life and mind explains too much to be science.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The breadth is the hypothesis --- one recursive theorem read three ways --- and each reading carries its own falsifier, so the breadth increases exposure rather than reducing it. We accept the risk that comes with scope by pre-declaring kill criteria for each floor; a single theory answerable to particle, cosmological, biological, and neural data simultaneously is harder to save than a narrow one, which is the point.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 71 (O-Origin-71) (the accessibility referee)}\par\nobreak
\noindent\textbf{``The acronym and notation density is an impenetrable barrier.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Conceded, with the mitigations stated: the canonical Dictionary (\ATwo{} Appendix~B), the per-Part reader's guides, and the cohort naming-convention key (\S\ref{sec:naming-convention}) are provided precisely so that every defined object can be looked up. The density is the cost of a parameter-free codebook; the navigation aids are the answer, and we keep extending them.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 72 (O-Origin-72) (the late-time cosmologist)}\par\nobreak
\noindent\textbf{``The standing cosmological tensions --- the DESI dark-energy hint, the Hubble tension --- threaten your base while you present it as solid.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} We flag both rather than paper over them: the $w=-1$ test is marked as the live tension it is (Objection~7), and if a confirmed $w\neq -1$ arrives, $\Omega_\Lambda = 1-1/\pi$ and the cosmology floor fail. The Hubble tension is addressed directly in \PaperCTwo{} \citep{ArisakaC2Hubble} as a two-stage corridor, not assumed away. The base is solid where the data are settled and exposed where they are not, and we mark the boundary.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 73 (O-Origin-73) (the philosopher of cosmology)}\par\nobreak
\noindent\textbf{``Your `second horizon problem' looks like a rebranded old idea. Zeldovich and Kibble had causally disconnected regions choosing vacua in the 1970s; the axion community has always known inflation makes the misalignment angle uniform. What is new, and why does the capstone lean on it?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The pedigree is real and \ATen{} cites it in those very terms (Zeldovich--Kobzarev--Okun 1974, Kibble 1976, Linde's ``uniqueness of the universe,'' the standard pre-inflationary axion scenario). What \ATen{} claims as new is narrow and stated as such: the \emph{naming} of identity-uniformity as a horizon problem in its own right, ranked beside temperature; the \emph{extension} from the orientation of one vacuum to the whole inventory of masses, couplings, and abundances; and above all the \emph{ordering} --- that inflation preceded the existence of matter, so identity was inherited rather than negotiated, a corollary whose premise is textbook but which the literature never drew. \AEleven{} leans on it because it is the cleanest instance of the cascade's own mechanism: the sameness of the electron across the sky is the layer-2 \GGA{} read by \GRIPin{} on a single causal patch, and the recursive uniqueness theorem needs exactly that causal warrant at its base to guarantee that every higher rung --- the periodic table, the genetic alphabet, the shape of a conscious episode --- is universal across the observable universe and not merely local. Remove the ordering and the cascade still runs, but its base rung would rest on an assumption rather than a derivation; with it, the first lock is earned. That is why the capstone imports it, and why it belongs in the determinism argument as the positive evidence that the world's sameness is structural.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 74 (O-Origin-74) (the editor's closing question)}\par\nobreak
\noindent\textbf{``What single observation would make you abandon the framework?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Any one of the Top-10 falsifiers failing at its kill criterion: no clustered spin-2 resonance within $3.85\pm 0.10$~TeV at HL-LHC; no kaxion in the $6.6$--$8.8\,\mu$eV band; a confirmed dark-energy equation of state $w\neq -1$; proton decay detected short of $\sim 10^{35}$~yr; or a wild, naturally-evolved organism with a stable genetic alphabet of other than four letters. Each is pre-declared with its threshold (\S\ref{sec:eight-test-pull-table}); the framework is built to lose if the universe disagrees, and that --- not the breadth of what it claims --- is the measure of whether it is science.
\end{objqabody}


\GGPart[References]{References}{Cited works, each with a relevance note linking it to the present paper}
\label{part:references}
\label{sec:references}

\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 capstone: from the foundations of gauge theory, differential cohomology, open-system dynamics, and statistical physics, to the twenty-two-paper \GGTheory{} cohort that \AEleven{} closes --- the paper in which the recursive-uniqueness theorem \textsf{T-Origin} and the cascade of Ground Geometric Attractors bring the program from the \GGConstitution{} to the articulation of Cosmos, Life, and Mind. 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 \GGTheory{} cohort papers (\AOne{}--\AEleven{}, together with the phenomenology P-series and the consequence C-series) are cited throughout under the canonical keys and titles fixed by the cross-paper coherence map of the program; \AEleven{}'s place at the summit of that cohort is set out in the cohort-inheritance Part. The references are listed alphabetically by first author.

\renewcommand{\bibsection}{\vspace{1.2em}}%

\section*{A.~Papers by Katsushi Arisaka (the \GGTheory{} cohort)}
\begin{thebibliography}{99}

\bibitem[Arisaka(2026, A1-Constitution)]{ArisakaA1Constitution}
\textbf{K.~Arisaka.}
\emph{The \GGConstitution{}: Four Axioms and Seven Postulates of the \GGTheory{} Program, From Which \GG{}, \BIsix{}, \OGNfive{}, and \GDOS{} Descend as Theorems},
Arisaka-GG-A1-Constitution (May 14, 2026).
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/a1-constitution/}.
\par\noindent\textbf{Relevance.}
Establishes the four axioms (LC, GGEP, OGGEP, MSMF) and seven postulates of the \GGConstitution{} --- the foundational document of the entire program. The constitution does not specify a particular Lagrangian; instead, it defines admissibility rules that any candidate theory must satisfy. The restrictiveness of these rules is what forces a unique solution (\GGTheory{}), eliminating the landscape of free parameters that plagues other unification approaches. Every result in the present paper traces back to these fifteen constitutional clauses (the four axioms and the seven postulates). First cited in \S\ref{sec:intro-scope}.

\bibitem[Arisaka(2026, A2-GG6)]{ArisakaA2GG6}
\textbf{K.~Arisaka.}
\emph{\GGTheory{} with \GG{}: Geometry--Gauge Locking at the 6D Bulk (\GG{}) at the Heart of Grand Geometric Theory (\GGTheory{})},
Arisaka-GG-A2-GG6 (May 14, 2026).
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/a2-gg6/}.
\par\noindent\textbf{Relevance.}
Derives the six-dimensional GG-6 stage --- the bulk manifold $M_6 = M_4 \times I_r \times S^1_\theta$ with gauge group $\SUfive \times \UoneB \times \UoneQi \times \mathrm{Gravity}$ --- from the \GGConstitution{}. The paper shows that the BI-6 identity and the OGN-5 integers are structurally forced by the axioms, with no continuous freedom remaining. The single tensor field $B_2$ that mediates all anomaly inflow is derived here, and its uniqueness is the geometric origin of the Arisaka Equation. First cited in \S\ref{sec:program-dictionary}.

\bibitem[Arisaka(2026, A3-BI6)]{ArisakaA3BI6}
\textbf{K.~Arisaka.}
\emph{Anomaly Freedom of \BIsix{}: The Modified Bianchi Identity (\BIsix{}) for the Six-Force, Six-Dimensional Bulk (\GG{})},
Arisaka-GG-A3-BI6 (May 14, 2026).
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/a3-bi6/}.
\par\noindent\textbf{Relevance.}
Provides the complete local and global geometric consistency proof for BI-6 on the physical intended GG-6 class. The paper uses the Green--Schwarz descent, Wu--Chern--Simons completion \citep{HopkinsSinger2005}, and differential-cohomology framework to establish that the six-dimensional bulk geometry is internally consistent --- a pre-quantum result that any subsequent quantization must inherit. The OGN-5 integers $(1,3,5;7,35)$ are derived at theorem level. This is the mathematical foundation on which the entire chain from geometry to consciousness rests: without anomaly freedom, the bulk theory would be ill-defined and no boundary physics could be projected. First cited in \S\ref{sec:holographic-triad-merged}.

\bibitem[Arisaka(2026, A4-Euclid)]{ArisakaA4Euclid}
\textbf{K.~Arisaka.}
\emph{From Euclid to \GG{}: The Realization of Euclid's Elements in Six Dimensions --- \GUfive{}, \OGNfive{}, and a Parameter-Free Reconstruction of Modern Physics},
Arisaka-GG-A4-Euclid (May 14, 2026).
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/a4-euclid/}.
\par\noindent\textbf{Relevance.}
The fourth foundation paper of the Arisaka-GG cohort. Positions \GGTheory{} in the historical lineage of Euclid, Newton, Riemann, Maxwell, and Einstein, and proves that the framework's input ledger contains zero continuous free parameters. The residue is not empty: \AFour{} names it as discrete axiomatic choices, unit conventions and one overall mass scale, and scopes the closure to the gauge and cosmological sectors. The input ledger has three forced groups: the Geometric Universal Five Constants \GUfive{} $=\GUfiveSet$, the mathematical constants $(\pi, e)$, and the \OGNfive{} integer ledger $(M, N, L; J, K)$. \AEleven{} cites \AFour{} as the constitutional source of the zero-free-parameters claim and of the $\QB$ membership in \GUfive{}. First cited in \S\ref{sec:cascade-grip-out-gga-opener}.

\bibitem[Arisaka(2026, A5-GDOS)]{ArisakaA5GDOS}
\textbf{K.~Arisaka.}
\emph{\GDOS{}: Geometric Dynamics of Open Systems --- 6D \GG{} Bulk Projected to 4D Boundary for Quantum Measurement, General Relativity, and QFT},
Arisaka-GG-A5-GDOS (May 14, 2026).
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/a5-gdos/}.
\par\noindent\textbf{Relevance.}
Derives the GDOS boundary dynamics from the OGGEP axiom (AX-3), establishing how the six-dimensional bulk projects onto the four-dimensional world we observe. The paper introduces the operational dictionary $\cD$, the GKLS open-system generator, and the port structure that allows systems to exchange matter, energy, and information with their environment. Every open system in nature --- from a proton to a brain --- is a GDOS system, and its dynamics is constitutionally constrained by the bulk geometry. First cited in \S\ref{subsec:inheritance-combine}.

\bibitem[Arisaka(2026, A6-GRIP)]{ArisakaA6Gi4}
\textbf{K.~Arisaka.}
\emph{The \GRIP{} on \GDOS{} --- \GRIP{}: the Geometric Engine of Symmetry Breaking, from the Big Bang to Mind},
Arisaka-GG-A6-GRIP (May 14, 2026).
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/a6-grip/}.
\par\noindent\textbf{Relevance.}
The parent paper of the runtime. Derives the four Gi-4 pillars of the cascade machinery --- \TGG{} (the Total Geometric-Gauge object), \CCI{} (the Covariant Conservation Identity), \GFF{} (the Geometric Free-Energy Functional), and \GRIP{} (the Gauge-Runtime Invariance Pipeline, the cascade $\DGI \to \GSSB \to \GA/\GGA{}$ on the $F_{\GFF}$ landscape: Dynamical Gauge Invariance at the symmetric top, Geometric Spontaneous Symmetry Breaking at the branch-selection saddles, the Geometric Attractor at the stable floor) --- together with the runtime-level definitions adopted throughout the present paper: def:GGA (the Ground Geometric Attractor) and def:GRIPin/def:GRIPout (the two runtime regimes), whose regime \emph{theorems} are proved here. \AEleven{} imports the cascade engine T1--T5 and GFF-ST7 as the machinery that drives the recursive uniqueness theorem at every layer above the Standard Model. First cited in \S\ref{sec:cascade-grip-out-gga-opener}.

\bibitem[Arisaka(2026, A7-Qi4)]{ArisakaA7Qi4}
\textbf{K.~Arisaka.}
\emph{The \QiFour{} Pillars --- \QiQUA{}, \QiEP{}, \QiCON{}, \QiHAL{}: The Quantum-Information Backbone of \GDOS{}},
Arisaka-GG-A7-Qi4 (May 14, 2026).
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/a7-qi4/}.
\par\noindent\textbf{Relevance.}
Derives the Qi-4 Pillars --- Qi-QUA (current locking), Qi-EP (the unified extremal principle), Qi-CON (conformality), and Qi-HAL (the holographic attractor lattice, the memory architecture) --- and deploys on the information flank the DGI--GSSB--GA universal runtime pipeline of \ASix{}. The GFF (Geometric Free-Energy Functional), the SIP (System Instance Packet), and the SeOS (Selfish Open System) classification are developed here. The Arisaka Equation $J_E/\QB = J_e = J_B = J_{Qi}$ is derived in its canonical form, and the biological energy quantum $N_A\QB = 15.0$~kJ/mol is identified as the universal currency of cellular energy transactions. First cited in \S\ref{subsec:inheritance-combine}.

\bibitem[Arisaka(2026, A8-QM)]{ArisakaA8QM}
\textbf{K.~Arisaka.}
\emph{Quantum Mechanics Derived from \GGTheory{} as Projection from 6D Bulk to 4D Boundary},
Arisaka-GG-A8-QM preprint.
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/a8-qm/}.
\par\noindent\textbf{Relevance.}
The 4D operational reformulation of quantum mechanics from \AFive{} \GDOS{} machinery: lifts \AFive{} T7.1--T7.7 (QM postulates as theorems) to textbook-style 4D form, supplying the formal grounding of the consciousness-as-internal-measurement reading exercised in \AEleven{} Parts~VIII and IX. Currently in active drafting; \AEleven{} references \AEight{} as a forward-looking dependency in Part~\ref{part:inheritance} \S\ref{subsec:inheritance-A8}. Once \AEight{} ships, the operational-measurement passages in \AEleven{} Part~\ref{part:dynamics} will be replaced by direct \AEight{} citations. First cited in \S\ref{subsec:inheritance-A8}.

\bibitem[Arisaka(2026, A9-SM)]{ArisakaA9SM}
\textbf{K.~Arisaka.}
\emph{The Standard Model as Theorem of \GGTheory{}: Its 28 Parameters as Observations of One Six-Dimensional Geometry},
Arisaka-GG-A9-SM (May 14, 2026).
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/a9-sm/}.
\par\noindent\textbf{Relevance.}
Derives all 28 Standard Model parameters --- six quark masses, three lepton masses, three CKM angles and one phase, three gauge couplings, the Higgs mass and vacuum expectation value, the QCD $\theta$ angle, and the neutrino sector --- from the OGN-5 arithmetic and the GDOS boundary projection. The key results include $\hat\alpha_{\mathrm{EM}}^{-1} = 2^7 = 128$ at the Z~pole, $\sin^2\theta_W = 25/108$, and $v = M_{\mathrm{lock}}(2/5)^3 \approx 246.4$~GeV. Every prediction is compared with the measured value, and all pulls are below 1.0. First cited in \S\ref{subsec:hist-anomaly}.

\bibitem[Arisaka(2026, A10-Cosmos)]{ArisakaA10Cosmos}
\textbf{K.~Arisaka.}
\emph{The Standard Cosmology as Theorem of \GGTheory{}: The Cosmic Inventory as Observations of One Six-Dimensional Geometry},
Arisaka-GG-A10-Cosmos (May 14, 2026).
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/a10-cosmos/}.
\par\noindent\textbf{Relevance.}
Derives the full cosmological sector of the GG-6 framework as an inheritance problem: the universe is the cosmic export of the same \GGTheory{}, with no new free parameters. The paper shows that one infrared GSSB event at the throat cap simultaneously ends inflation, produces Kaxion dark matter, and leaves the dark energy residue $\rho_\Lambda \sim (M_{\mathrm{lock}}\,e^{-K})^4 \sim (\mathrm{few\;meV})^4$. Inflation is TeV-scale slow-smooth radion inflation on a warp-flattened plateau, predicting tiny primordial tensor modes. The Kaxion dark matter candidate is a single physical line at $\sim 8.1\;\mu$eV ($\nu_K=1.95$~GHz) with a twin-peak spectral signature split by $\Delta\nu/\nu = 1/35$. The $\pi$-locked cosmic inventory $\Omega_m = 1/\pi$, $\Omega_b = 1/(2\pi^2)$, $\Omega_\Lambda = 1 - 1/\pi$ matches Planck data to within 2\% and dissolves the coincidence problem. Baryogenesis proceeds through TeV-scale resonant leptogenesis using the inherited Majorana and CP structure. BBN, the CMB, and the Hubble tension are treated as precision audits of the derived ledger. First cited in \S\ref{sec:ogn5-bio}.

\bibitem[Arisaka(2026, A11-Origin)]{ArisakaA11Origin}
\textbf{Arisaka, K.\ (2026).}
\textit{The Geometric Inevitability of Cosmos, Life, and Mind: The Self-Referential Closure of the \GGTheory{} Program}.
Arisaka-GG-A11-Origin preprint (the present paper).
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/a11-origin/}.
\par\noindent\textbf{Relevance.} The present paper and the closing capstone of the twenty-two-paper cohort. States the recursive-uniqueness theorem \textsf{T-Origin}, develops the GRIP-out cascade of Ground Geometric Attractors across the five floors of Cosmos, Life, and Mind, and threads the \OGNfive{} ledger from atoms and molecules through ATP, DNA, and the first cell to the observer who articulates the geometry. First cited in \S\ref{subsec:coherence-map}.

\bibitem[Arisaka(2026, P1-KK)]{ArisakaP1KK}
\textbf{K.~Arisaka.}
\emph{The \GG{} Lock Cluster: A Non-Spatial Kaluza--Klein Resonance near 3.85~TeV as the Standing LHC Falsifier of \GGTheory{}},
Arisaka-GG-P1-KK (2026).
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/p1-kk/}.
\par\noindent\textbf{Relevance.}
The dedicated phenomenology paper for the Kaluza--Klein lock cluster prediction. The scale-locked KK tower of \AOne{}--\AThree{} compactified on the six-dimensional bulk produces a narrow KK resonance cluster at $\Mlock \approx 3.85 \pm 0.10$~TeV with a characteristic twin-peak signature. The prediction is directly testable at HL-LHC. Cited by \AEleven{} in Part~\ref{part:discussions}'s Top 10 Falsifiers as test~1 (KK lock cluster discovery). First cited in \S\ref{sec:holographic-triad-merged}.

\bibitem[Arisaka(2026, P2-N3)]{ArisakaP2N3}
\textbf{K.~Arisaka.}
\emph{Why Three Generations: A Three-Leg Derivation of $N_g=3$ from \BIsix{}, \MSMF{}, and \GDOS{} Morse Stability},
Arisaka-GG-P2-N3 (2026).
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/p2-n3/}.
\par\noindent\textbf{Relevance.}
The dedicated proof paper of the family-number theorem. Three logically independent legs (an MSMF lower bound from Kobayashi-Maskawa CP-violation combinatorics; a GDOS Morse upper bound from family-projector stability on $\mathbb{C}P^{N-1}$; and a constructive codebook self-consistency check from \ANine{}) jointly force $N=3$. The precision tolerance $m_\tau/m_\mu \approx 16.817 \pm 0.5\%$ is the empirical falsification anchor of the closure. Cited by \AEleven{} in Part IX as the upstream theorem closing the Particle-Physics open-question chain. First cited in \S\ref{subsec:coherence-map}.

\bibitem[Arisaka(2026, P3-Electron)]{ArisakaP3Electron}
\textbf{Arisaka, K.\ (2026).}
\textit{The Electron as the Ground State of Nature: The Six Oldest Questions about the Electron, Closed and Fenced within \GGTheory{}}.
Arisaka-GG-P3-Electron preprint.
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/p3-electron/}.
\par\noindent\textbf{Relevance.} The third paper of the Predictions Branch: identity, absolute stability, the two excited copies, the positron as conjugate readout, pointlikeness, and field response derived as six faces of ground-state-ness; imports \PaperPTwo{}'s $N=3$ verbatim. The electron is the layer-3 \GGA{} of the present paper's cascade. First cited in \S\ref{subsec:coherence-map}.

\bibitem[Arisaka(2026, P4-Real)]{ArisakaP4Real}
\textbf{K.~Arisaka.}
\emph{The Real Geometry Beneath the Complex Numbers of Physics: Positive Frequency is Holomorphy --- from Cardano's Cubics to Twistor Space, via the Sixth Dimension of \GGTheory{}},
Arisaka-GG-P4-Real preprint.
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/p4-real/}.
\par\noindent\textbf{Relevance.}
The origin-of-$i$ member paper of the Foundations Branch: the standalone development of the real-bulk account of the boundary's complex structure --- the two real sources (the compact phase circle and the causal clock), positive frequency read as holomorphy, and the open items stated as open. \AEleven{} leans on the derived complex structure at every quantum floor of the cascade but does not adjudicate \emph{why} the boundary is complex, deferring that question to \AEight{}; \PaperPFour{} is the full-length account \AEight{} draws on. First cited in \S\ref{sec:cohort-a11}.

\bibitem[Arisaka(2026, P5-CPT)]{ArisakaP5CPT}
\textbf{K.~Arisaka.}
\emph{The Mirrors of Matter: Why Every Particle Has a Twin, Why Nature's Mirrors Break, and Why the Universe Is Made of Matter},
Arisaka-GG-P5-CPT preprint.
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/p5-cpt/}.
\par\noindent\textbf{Relevance.}
The CPT member paper of the Foundations Branch: C, P, and T as orientation reversals of the three real bulk factors, time defined by its four endowments, and the law-level arrow that fixes why the universe is made of matter. Its arrow of orientation reversal is kin to the cascade's own one-way descent through its layers, each \GRIPout{} shock stepping the geometry down one floor and never back up. First cited in \S\ref{sec:cohort-a11}.

\bibitem[Arisaka(2026, C1-Kaxion)]{ArisakaC1Kaxion}
\textbf{K.~Arisaka.}
\emph{The Kaxion: Cold Dark Matter as the Non-Spatial Compact-Angle Pseudoscalar of \GG{}},
Arisaka-GG-C1-Kaxion (2026).
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/c1-kaxion/}.
\par\noindent\textbf{Relevance.}
The dedicated dark-matter phenomenology paper. The Kaxion is the framework's dark-matter candidate, produced as a misalignment condensate during the throat-cap GSSB event of \ATen{}. A single physical line at $\sim 8.1~\mu$eV ($\nu_K=1.95$~GHz) is predicted, with a twin-peak spectral signature split by $\Delta\nu/\nu = 1/35$. The prediction is directly testable at ADMX and next-generation haloscopes. Cited by \AEleven{} in Part XI's Top 10 Falsifiers as test~2 (Kaxion dark matter discovery). First cited in \S\ref{sec:holographic-triad-merged}.

\bibitem[Arisaka(2026, C2-Hubble)]{ArisakaC2Hubble}
\textbf{K.~Arisaka.}
\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.
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/c2-hubble/}.
\par\noindent\textbf{Relevance.}
The dedicated phenomenology paper for the Hubble two-stage corridor prediction. It derives a single background expansion rate, $H_0 = 66.6 \pm 0.2$~km/s/Mpc, against which Planck's measured $67.4 \pm 0.5$ and the local ladder's $73.5$ are read; the background plus the Local-Hole lift gives a conditional late reading of $\approx 68.5 \pm 0.6$, and the residual to the ladder value is stated there as an open problem rather than a result. Its staked falsifier is that standard sirens converge near $67$: near $73$, that paper says in advance that it is wrong. Cited by \AEleven{} in Part~\ref{part:discussions}'s Top 10 Falsifiers as tests~3 and~4 (Hubble corridor early- and late-stage). First cited in \S\ref{subsec:coherence-map}.

\bibitem[Arisaka(2026, C3-BH)]{ArisakaC3BH}
\textbf{Arisaka, K.\ (2026).}
\textit{Black-Hole Entropy under \GGTheory{}: The \GDOS{} Reading of the Horizon as an \OGGEP{} Port --- with the Exact Bekenstein--Hawking $A/4G_N$ Derived from Induced Gravity}.
Arisaka-GG-C3-BH preprint.
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/c3-bh/}.
\par\noindent\textbf{Relevance.} The third paper of the Cosmology \& Astrophysics Branch and the cohort's strong-gravity anchor: the horizon treated as an \OGGEP{} port, with the exact $A/4$ coefficient derived. First cited in \S\ref{subsec:coherence-map}.

\bibitem[Arisaka(2026, P6-Proton)]{ArisakaP6Proton}
\textbf{Arisaka, K.\ (2026).}
\textit{The Proton as the Ground Geometric Attractor of Matter: The Eleven Questions of the Proton, Closed and Fenced within \GGTheory{}}.
Arisaka-GG-P6-Proton preprint.
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/p6-proton/}.
\par\noindent\textbf{Relevance.} The fourth paper of the Predictions Branch and the cohort's second portrait of permanence: the proton read as the ground attractor of the hadronic stratum, its stability traced to a gauged baryon number with no $\Delta B = 1$ aperture rather than to an accidental symmetry. It supplies the layer-3 anchor this paper inherits, and its double guard --- the triangle that closes and the ledgers that keep --- is the mechanism behind \textsf{T-Atomic-Ground-State}. First cited in \S\ref{subsec:coherence-map}.

\bibitem[Arisaka(2026, W1-Symmetry)]{ArisakaW1Symmetry}
\textbf{Arisaka, K.\ (2026).}
\textit{Symmetry, Conservation, and Geometry: Seven Steps from Noether, Bianchi, to Hopkins--Singer, and the Ledger the Open-System Turn Pays Out}.
Arisaka-GG-W1-Symmetry preprint.
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/w1-symmetry/}.
\par\noindent\textbf{Relevance.} The first of the two white papers, and the cohort's historical synthesis: symmetry read as a ladder of seven rungs from Noether's theorems to differential cohomology, with the open-system turn costed at each rung. Its seventh rung --- that an integer cannot run because nothing continuous acts on it --- is the protection this paper's cascade relies on wherever a layer output is a count. First cited in \S\ref{subsec:coherence-map}.

\bibitem[Arisaka(2026, W2-Holonomy)]{ArisakaW2Holonomy}
\textbf{Arisaka, K.\ (2026).}
\textit{The Circle That Remembers, the Interval That Forgets: \GGTheory{} and the Sixteen Problems of Particle Physics and Cosmology}.
Arisaka-GG-W2-Holonomy preprint.
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/w2-holonomy/}.
\par\noindent\textbf{Relevance.} The second white paper and the cohort's structural synthesis: the two non-spatial fibers read as one anatomy --- a compact circle that remembers and an open interval that forgets --- and applied to sixteen standing problems in one voice. It is this paper's sibling at the summit: W2 reads the cohort across problems, \AEleven{} reads it across layers, and the two meet at the observer who articulates the geometry. First cited in \S\ref{subsec:coherence-map}.

\end{thebibliography}

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

\bibitem[Adler(1969)]{Adler1969}
\textbf{S.~L.~Adler.}
Axial-vector vertex in spinor electrodynamics.
\emph{Physical Review} \textbf{177} (1969) 2426--2438.
\par\noindent\textbf{Relevance.}
Adler's discovery of the axial anomaly --- that a classically conserved current can appear violated in the quantum path integral --- is one of the foundational results of modern quantum field theory. In the GG-6 framework, however, the BI-6 identity is understood as a \emph{geometric} consistency condition on the composite connection, not a quantum effect. The coefficients $(1,-1,-2/5,-2/5)$ are fixed by the topological factorization $I_8 = \frac{1}{2}X_4^2$. The quantum anomaly is the symptom that appears when this geometric constraint is violated; the constraint itself is pre-quantum. First cited in \S\ref{subsec:hist-anomaly}.

\bibitem[Afifa et al.(2022)]{Vision7D1}
\textbf{U.~Afifa, J.~Carmona, A.~Dinh, et al., A.~P.~Blaisdell, and K.~Arisaka.}
\emph{Visual Perception of 3D Space and Shape by 7D Frame Translation in Time, Part~\ref{part:intro}: 2D Space Perception by 2D Linear Translation},
bioRxiv 2022.03.01.482161 (2022).
\textbf{Relevance.} Experiment establishing the linear reaction-time-vs-eccentricity law (translation axes) and the flat-SRT / sloped-CRT dissociation --- behavioral evidence for the space$\to$time (\NHT{}) conversion of the \IH{}. Author's own preprint; behavioral support for Test~10, classed Asserted/INT. First cited in \S\ref{sec:per-test-commentary}.

\bibitem[Aggarwal and Ray(2023)]{AggarwalRay2023}
\textbf{S.~Aggarwal and S.~Ray.}
\emph{Slope of the power spectral density flattens at low frequencies ($<$150~Hz) with healthy aging but also steepens at higher frequency ($>$200~Hz) in human electroencephalogram},
\emph{Cerebral Cortex Communications} \textbf{4} (2023) tgad011.
\textbf{Relevance.} A direct human-EEG observation of \emph{two} spectral corners (one below $\sim$150~Hz, one above $\sim$200~Hz) with opposite aging behavior --- the closest existing empirical support for the dual-knee (Test 9b) prediction. Cited honestly as emerging, single-report evidence, not a settled result (Asserted). First cited in \S\ref{sec:eight-test-pull-table}.

\bibitem[Aharonov \& Bohm(1959)]{AharonovBohm1959}

\textbf{Aharonov, Y., \& Bohm, D. (1959).}
Significance of electromagnetic potentials in the quantum theory.
\textit{Physical Review}, \textit{115}(3), 485--491.
\url{https://doi.org/10.1103/PhysRev.115.485}
\par\noindent\textbf{Relevance.} The Aharonov--Bohm effect demonstrates that holonomy of a gauge connection is physically observable even when local field strengths vanish. This is a canonical precedent for \GGEP{}'s insistence that the invariant content of the composite connection includes global/holonomy data, supporting ``the gauge phase as geometry.'' First cited in \S\ref{app:ax2-expanded}.

\bibitem[Akaike(1974)]{Akaike1974}

\textbf{Akaike, H. (1974).}
A new look at the statistical model identification.
\emph{IEEE Transactions on Automatic Control}, \emph{19}(6), 716--723.
\url{https://doi.org/10.1109/TAC.1974.1100705}
\par\noindent\textbf{Relevance.} Akaike's information criterion is a quantitative formulation of the ``Occam penalty'' in model selection. We cite it as historical background for POS-6 (\OGN{}): in \GGTheory{}, minimality is enforced not only in model structure but also in normalization conventions to prevent hidden continuous tuning. First cited in \S\ref{app:ax4-expanded}.

\bibitem[Altwegg et al.(2016)]{Altwegg2016}
\textbf{K.~Altwegg, H.~Balsiger, A.~Bar-Nun, et al.}
\emph{Prebiotic chemicals --- amino acid and phosphorus --- in the coma of comet 67P/Churyumov--Gerasimenko},
\emph{Science Advances} \textbf{2} (2016) e1600285.
\par\noindent\textbf{Relevance.}
The ROSINA mass spectrometer on Rosetta measured glycine, methylamine and ethylamine, together with phosphorus, directly in the coma of comet 67P --- the first \emph{in situ} detection of an amino acid at a comet rather than an inference from a returned or fallen sample. The present paper cites it for the layer-4 to layer-5 delivery step: the monomer inventory the cascade needs at layer~5 is demonstrably present in the small bodies that resurfaced the early Earth, so the framework's layer-5 claim is a claim about \emph{selection} from an available inventory and not about synthesis. First cited in \S\ref{sec:chem-leading-theories}.

\bibitem[Alvarez-Gaum\'e and Witten(1984)]{AlvarezGaumeWitten1984}
\textbf{L.~Alvarez-Gaum\'e and E.~Witten.}
\emph{Gravitational anomalies},
\emph{Nuclear Physics B} \textbf{234} (1984) 269.
\par\noindent\textbf{Relevance.}
Classified gravitational anomalies in even dimensions using the index theorem for the Dirac operator, providing the mathematical framework used throughout the BI-6 anomaly proof. Their computation of the gravitational contribution to the anomaly polynomial is the starting point for the BI-6 factorization: the coefficient $a = 1$ in the BI-6 identity $(1,-1,-2/5,-2/5)$ is the gravitational sector, normalized by the Alvarez-Gaum\'e--Witten index calculation. First cited in \S\ref{subsec:hist-anomaly}.

\bibitem[Amari(1998)]{Amari1998}

\textbf{Amari, S.-I. (1998).}
Natural gradient works efficiently in learning.
\textit{Neural Computation}, \textit{10}(2), 251--276.
\url{https://doi.org/10.1162/089976698300017746}
\par\noindent\textbf{Relevance.} Amari's information-geometric formulation shows that optimization dynamics depend on the metric structure of probability/state manifolds. We cite it in the appendix to support POS-2's claim that ``extremal principles'' on the boundary can be written as geometry-driven flows (e.g.\ natural gradients) when \GDOS{} is expressed on statistical manifolds. First cited in \S\ref{app:examples}.

\bibitem[Amit(1989)]{Amit1989}
\textbf{D.~J.~Amit.}
\emph{Modeling Brain Function: The World of Attractor Neural Networks.}
Cambridge University Press, 1989.
\par\noindent\textbf{Relevance.}
Amit's monograph established attractor neural networks as the formal model for distributed memory and pattern completion in cortex. In the GG-Theory framework, the Hopfield/Amit attractor dynamics is a special case of GRIP descent on a finite-dimensional state manifold with a GFF Lyapunov functional; the network's stored patterns are layer-specific GGAs (\T{-Neural-Ring-Attractor} of \AEleven{} Part~\ref{part:cascade} generalizes this to the ring topology of insect heading systems). First cited in \S\ref{subsubsec:lemma-neural-ring}.

\bibitem[Anderson(1972)]{Anderson1972}
\textbf{P.~W.~Anderson.}
More is different.
\emph{Science} \textbf{177} (1972) 393--396.
\par\noindent\textbf{Relevance.}
Anderson argued that each level of complexity in nature exhibits qualitatively new phenomena that cannot be reduced to the level below, even though the lower level's laws are not violated. The Five-Story Pagoda of the GG-6 framework formalizes Anderson's insight: each story (particles, cosmos, chemistry, life, mind) obeys the same constitutional rules but exhibits its own emergent dynamics, connected by the DGI--GSSB--GA pipeline rather than by brute-force reduction. First cited in \S\ref{sec:qgi}.

\bibitem[Arisaka(2022)]{Arisaka2022MePMoS}
\textbf{K.~Arisaka.}
\emph{Grand Unified Theory of Mind and Brain, Part~\ref{part:intro}: Space-Time Approach to Dynamic Connectomes of C.~elegans and Human Brains by MePMoS} (2022). PsyArXiv preprint, \url{https://osf.io/preprints/psyarxiv/r8ma3}.
\textbf{Relevance.} The dynamic-connectome / MePMoS treatment that frames the Layer-14 central-nervous-system row; develops the sensorimotor-integration reading of the \emph{C.\ elegans} connectome used here. First cited in \S\ref{sec:per-test-commentary}.

\bibitem[Arisaka(2022b)]{ArisakaGUT2}
\textbf{K.~Arisaka.}
\emph{Grand Unified Theory of Mind and Brain, Part~\ref{part:inheritance}: Neural Holographic Tomography (NHT) and Holographic Ring Attractor Lattice (HAL)} (2022). PsyArXiv preprint, \url{https://doi.org/10.31234/osf.io/vu2kz}.
\textbf{Relevance.} The theory paper that introduces \NHT{} and \HAL{} --- the space$\to$time encoding and the holographic ring-attractor read-out that underlie the layer-16/17 \IH{} construct. Author's own preprint; the Qi-HAL identification is classed Asserted/INT. First cited in \S\ref{sec:per-test-commentary}.

\bibitem[Arisaka(planned)]{ArisakaBookI}
\textbf{K.~Arisaka.}
\emph{The Geometric Inevitability of Cosmos, Life, and Mind: A Textbook Synthesis of the Arisaka-GG Program} (planned, 2027).
\par\noindent\textbf{Relevance.}
The planned capstone volume of the GG-Theory program: a textbook synthesis of the Arisaka-GG cohort for graduate students of physics, chemistry, biology, neuroscience, and philosophy of science. The book will present the recursive uniqueness theorem of \AEleven{} in pedagogical order alongside the technical content of A1 through A10, so that the cascade can be taught as one apparatus rather than discovered as separate research papers. First cited in \S\ref{sec:program-dictionary}.

\bibitem[Arisaka and Blaisdell(2022)]{ArisakaGUT3}
\textbf{K.~Arisaka and A.~P.~Blaisdell.}
\emph{Grand Unified Theory of Mind and Brain, Part~\ref{part:cascade}: Holographic Visual Perception of 3D Space and Shapes} (2022). PsyArXiv preprint, \url{https://osf.io/preprints/psyarxiv/7gbkt}.
\textbf{Relevance.} The theory paper that frames the four 7D-frame perception experiments and the conscious-episode timing (alpha step / theta envelope) used in Test~10. Author's own preprint (Asserted/INT). First cited in \S\ref{sec:per-test-commentary}.

\bibitem[Aristotle(c.~350~BCE)]{Aristotle350BCE}
\textbf{Aristotle.}
\emph{Metaphysics}, books $\alpha$, $\Lambda$.
Trans.\ W.~D.~Ross, Oxford University Press (1908).
\par\noindent\textbf{Relevance.}
Aristotle's distinction between final cause (telos), formal cause, efficient cause, and material cause inaugurated the project of asking \emph{why} the world is structured as it is. In the GG-Theory program, all four Aristotelian causes are replaced by a single geometric necessity: the unique self-consistent solution of the GG-Constitution. Aristotle's final cause without a designer is reinterpreted as the uniqueness of the geometric attractor on each cascade stratum, anticipating without warranting the recursive uniqueness theorem of the present paper. First cited in \S\ref{subsec:aristotle}.

\bibitem[Ashtekar(1986)]{Ashtekar1986}
\textbf{A.~Ashtekar.}
New variables for classical and quantum gravity.
\emph{Physical Review Letters} \textbf{57} (1986) 2244--2247.
\par\noindent\textbf{Relevance.} Introduced the connection (``Ashtekar'') variables that launched loop quantum gravity. Cited in \S\ref{ssec:comparison-lqg} as a foundational LQG paper; \GGTheory{} instead treats gravity as the scale-curvature sector of the 6D bulk, so the two operate at different levels of analysis. First cited in \S\ref{sec:statement-comparison}.

\bibitem[Baars(1988)]{Baars1988}
\textbf{B.~J.~Baars.}
\emph{A Cognitive Theory of Consciousness}.
Cambridge University Press, Cambridge (1988).
\par\noindent\textbf{Relevance.} The origin of global workspace theory (GWT). Cited in \S\ref{ssec:comparison-gwt}: the cascade's \EIC{} closure can be read as the GWT broadcast event, with the layer-17 conscious episode its output. First cited in \S\ref{sec:statement-comparison}.

\bibitem[Bada(2013)]{Bada2013}
\textbf{J.~L.~Bada.}
\emph{New insights into prebiotic chemistry from Stanley Miller's spark discharge experiments},
\emph{Chemical Society Reviews} \textbf{42} (2013).
\par\noindent\textbf{Relevance.}
Reanalysis of Miller's own archived sample vials, with instruments six decades more sensitive than the ones available in 1953, recovered more than twenty amino acids from runs Miller had regarded as unsuccessful. We cite it in the layer-4--6 opening for the point it settles: the prebiotic yield of monomers was underestimated rather than overestimated by the founding experiment, which strengthens the observation that the framework's layer-5 problem is selection from a superset and not production. First cited in \S\ref{sec:chem-leading-theories}.

\bibitem[Bardeen(1969)]{Bardeen1969}

\textbf{Bardeen, W.~A. (1969).}
Anomalous Ward identities in spinor field theories.
\textit{Physical Review}, \textit{184}, 1848--1857.
\par\noindent\textbf{Relevance.} Develops the Ward-identity viewpoint of anomalies and clarifies how anomaly terms appear in current conservation laws. POS-3 treats such symmetry violations as fatal unless canceled by inflow; Bardeen's analysis is a canonical reference for this ``symmetry grammar'' perspective. First cited in \S\ref{app:examples}.

\bibitem[Barrow and Tipler(1986)]{BarrowTipler1986}
\textbf{J.~D.~Barrow and F.~J.~Tipler.}
\emph{The Anthropic Cosmological Principle}.
Oxford University Press (1986).
\par\noindent\textbf{Relevance.}
Barrow and Tipler cataloged the apparent fine-tunings of the physical constants for the existence of complex life. The GG-Theory program answers their book chapter by chapter: each cataloged ``fine-tuning'' is rederived as a forced consequence of the GG-Constitution, with the integer ledger \OGNfive{} replacing the continuous parameters Barrow and Tipler treated as inputs. The anthropic coincidences are geometric necessities. First cited in \S\ref{sec:statement-comparison}.

\bibitem[B\'edard and Destexhe(2009)]{BedardDestexhe2009}
\textbf{C.~B\'edard and A.~Destexhe.}
\emph{Macroscopic models of local field potentials and the apparent $1/f$ noise in brain activity},
\emph{Biophysical Journal} \textbf{96} (2009) 2589--2603.
\textbf{Relevance.} Reports a broken power-law local-field-potential spectrum (slope $\approx -1$ at low frequency, $\approx -3$ at high frequency), i.e.\ two scaling regimes separated by a corner --- a biophysical analog of the dual-knee structure (Test 9b). Supporting, model-based evidence (Asserted). First cited in \S\ref{sec:eight-test-pull-table}.

\bibitem[Bekenstein(1973)]{Bekenstein1973}
\textbf{J.~D.~Bekenstein.}
Black holes and entropy.
\emph{Physical Review D} \textbf{7} (1973) 2333--2346.
\par\noindent\textbf{Relevance.}
Bekenstein's discovery that black holes carry entropy proportional to their horizon area established the deep connection between gravity, thermodynamics, and information. In the GG-6 framework, the $\UoneQi$ gauged information current extends Bekenstein's insight: information is not merely an accounting device but a conserved physical quantity, locked to energy and matter by the Arisaka Equation. First cited in \S\ref{subsec:hist-triad}.

\bibitem[Bell and Jackiw(1969)]{BellJackiw1969}
\textbf{J.~S.~Bell and R.~Jackiw.}
A PCAC puzzle: $\pi^0 \to \gamma\gamma$ in the $\sigma$-model.
\emph{Il Nuovo Cimento A} \textbf{60} (1969) 47--61.
\par\noindent\textbf{Relevance.}
Bell and Jackiw independently discovered the chiral anomaly, showing that the neutral pion decay rate is determined by the anomaly rather than by perturbative coupling. This established that anomalies are not pathologies but physical predictions. In the GG-6 framework, the BI-6 anomaly is similarly not a defect to be repaired but the structural identity that determines all gauge-locked currents. First cited in \S\ref{subsec:hist-anomaly}.

\bibitem[Berry(1984)]{Berry1984}

\textbf{Berry, M.~V. (1984).}
Quantal phase factors accompanying adiabatic changes.
\textit{Proceedings of the Royal Society A}, \textit{392}(1802), 45--57.
\url{https://doi.org/10.1098/rspa.1984.0023}

\par\noindent\textbf{Relevance.} Berry's phase is a canonical example of ``quantum from geometry'': measurable phases arise from holonomy in parameter space.
It provides historical and technical support for AX-2's emphasis that quantum structure can be encoded geometrically in connections and their holonomies rather than imposed by quantization rules. First cited in \S\ref{app:ax2-expanded}.

\bibitem[Bertlmann(1996)]{Bertlmann1996}

\textbf{Bertlmann, R.~A. (1996).}
\textit{Anomalies in Quantum Field Theory}.
Oxford University Press.
\url{https://doi.org/10.1093/acprof:oso/9780198507628.001.0001}
\par\noindent\textbf{Relevance.} This monograph provides the standard mathematical treatment of anomaly polynomials and descent. It supports the formal statements in POS-3, where we package anomalies into differential-form data suitable for bulk--boundary cancellation. First cited in \S\ref{app:examples}.

\bibitem[Bianchi(1902)]{Bianchi1902}
\textbf{L.~Bianchi.}
\emph{Sui simboli a quattro indici e sulla curvatura di Riemann},
\emph{Atti della Reale Accademia dei Lincei, Rendiconti}
\textbf{11} (1902) 3--7.
\par\noindent\textbf{Relevance.} The classical differential identity for the Riemann curvature tensor whose gauge-bundle generalization, $dF=0$, is the Bianchi identity that the modified \BIsix{} ($dH_3=X_4$) of \GGTheory{} generalizes in the Green--Schwarz manner. Cited by \AEleven{} in the Part~\ref{part:inheritance} anomaly-history arc as the geometric ancestor of anomaly freedom. First cited in \S\ref{subsec:hist-anomaly}.

\bibitem[Blackmond(2010)]{Blackmond2010}
\textbf{D.~G.~Blackmond.}
\emph{The origin of biological homochirality},
\emph{Cold Spring Harbor Perspectives in Biology} \textbf{2} (2010) a002147.
\par\noindent\textbf{Relevance.}
The standard review of how a small enantiomeric excess is amplified to the near-complete homochirality that biology exhibits, covering autocatalytic amplification, phase behavior and attrition-driven deracemization. We cite it for the distinction the present Part rests on at layer~5: the literature explains amplification thoroughly and does not fix the sign, which is precisely the gap the parity-odd Wilson--Hosotani holonomy is offered to fill --- and the propagation chain from that holonomy to observed handedness remains this paper's own named open problem. First cited in \S\ref{sec:chem-leading-theories}.

\bibitem[Bohr(1913)]{Bohr1913}
\textbf{N.~Bohr.}
On the constitution of atoms and molecules.
\emph{Philosophical Magazine} \textbf{26} (1913) 1--25.
\par\noindent\textbf{Relevance.}
Bohr's quantization of electron orbits explained atomic stability and spectral lines, marking the transition from classical to quantum physics. The Rydberg energy $E_{\mathrm{Ry}} = 13.606$~eV, which Bohr derived from his model, reappears in the GG-6 framework as the intermediate step in the derivation of $\QB$: the biological energy quantum is a specific rational multiple of the Rydberg energy, determined by the OGN-5 integer $K = 35$. First cited in \S\ref{sec:five-revolutions}.

\bibitem[Bousso and Polchinski(2000)]{BoussoPolchinski2000}
\textbf{R.~Bousso and J.~Polchinski.}
Quantization of four-form fluxes and dynamical neutralization of the cosmological constant.
\emph{Journal of High Energy Physics} \textbf{06} (2000) 006.
\par\noindent\textbf{Relevance.} The flux-quantization mechanism that generates the string-theory ``discretuum'' of vacua. Cited in \S\ref{ssec:comparison-string-landscape}: \GGTheory{} closes the landscape because the \BIsix{} anomaly polynomial admits exactly one solution on a single tensor multiplet. First cited in \S\ref{sec:statement-comparison}.

\bibitem[Brenner(1974)]{Brenner1974}
\textbf{S.~Brenner.}
\emph{The genetics of Caenorhabditis elegans},
\emph{Genetics} \textbf{77} (1974) 71--94.
\textbf{Relevance.} Established \emph{C.\ elegans} as the model organism whose nervous system anchors the Layer-14 row (a non-physics claim, classed Asserted). First cited in \S\ref{sec:cascade-codebook-table}.

\bibitem[Breuer \& Petruccione(2002)]{BreuerPetruccione2002}

\textbf{Breuer, H.-P., \& Petruccione, F. (2002).}
\textit{The Theory of Open Quantum Systems}.
Oxford University Press.
\url{https://doi.org/10.1093/acprof:oso/9780199213900.001.0001}
\par\noindent\textbf{Relevance.} This standard reference develops the mathematics of open quantum systems (reduced dynamics, master equations, Markovian limits). It anchors our claim that openness is not optional in realistic modeling, and it provides the mature language (CPTP maps and generators) that the \GGConstitution{} intends to elevate to first-class status. First cited in \S\ref{app:examples}.

\bibitem[Bustanoby et al.(2022)]{Vision7D2}
\textbf{I.~Bustanoby, A.~Krupien, U.~Afifa, et al., A.~P.~Blaisdell, and K.~Arisaka.}
\emph{Visual Perception of 3D Space and Shape by 7D Frame Translation in Time, Part~\ref{part:inheritance}: 3D Space Perception with Holographic Depth},
bioRxiv 2022.02.28.482181 (2022).
\textbf{Relevance.} Experiment establishing the linear reaction-time-vs-distance (depth) law with binocular$=$monocular and covert$=$overt controls --- the third translation axis of the 7D frame. Author's own preprint (Asserted/INT). First cited in \S\ref{sec:per-test-commentary}.

\bibitem[Callan(1970)]{Callan1970}

\textbf{Callan, C.\ G.\ (1970).}
Broken scale invariance in scalar field theory.
\textit{Physical Review D, 2}(8), 1541--1547.
DOI: \href{https://doi.org/10.1103/PhysRevD.2.1541}{10.1103/PhysRevD.2.1541}.
\par\noindent\textbf{Relevance.} One of the original papers formulating the Callan--Symanzik equation and the modern view of
renormalization-group flow as a differential statement of scale dependence. It provides historical grounding for POS-5:
``scale covariance'' is not optional if the theory aims to be resolution-independent. First cited in \S\ref{app:examples}.

\bibitem[Callan and Harvey(1985)]{CallanHarvey1985}
\textbf{C.~G.~Callan, Jr.\ and J.~A.~Harvey.}
Anomalies and fermion zero modes on strings and domain walls.
\emph{Nuclear Physics B} \textbf{250} (1985) 427--436.
\par\noindent\textbf{Relevance.}
Callan and Harvey showed that anomalies on a lower-dimensional boundary can be canceled by anomaly inflow from a higher-dimensional bulk, establishing the physical mechanism behind the Green--Schwarz cancellation. The Arisaka Equation is a direct consequence of this inflow: the single $B_2$ tensor field channels the anomaly inflow from the six-dimensional bulk to the four-dimensional boundary, locking all currents in a fixed ratio. First cited in \S\ref{subsec:hist-anomaly}.

\bibitem[Carter(1974)]{Carter1974}
\textbf{B.~Carter.}
Large Number Coincidences and the Anthropic Principle in Cosmology.
In \emph{Confrontation of Cosmological Theories with Observational Data},
M.~S.~Longair (ed.), Reidel, Dordrecht (1974) 291--298.
\par\noindent\textbf{Relevance.}
Carter introduced the anthropic principle to cosmology, observing that the universe's parameters must admit observers because we are here to ask. In the \GGTheory{} program, the anthropic principle is replaced by a stronger structural claim: the universe's parameters must admit observers because the recursive uniqueness theorem forces the cascade to ascend through cosmos, life, and mind. The anthropic selection is dissolved into geometric inevitability. First cited in \S\ref{sec:thm-articulation}.

\bibitem[Chalmers(1995)]{Chalmers1995}
\textbf{D.~J.~Chalmers.}
\emph{Facing up to the problem of consciousness},
\emph{Journal of Consciousness Studies} \textbf{2} (1995) 200--219.
\par\noindent\textbf{Relevance.}
Defined the ``hard problem'' of consciousness: the question of why subjective experience exists at all, and why physical processes in the brain are accompanied by qualitative ``what it is like'' character. The GG-6 framework does not dissolve the hard problem by explaining away subjectivity. Instead, it provides an operational definition of consciousness (the internal measurement of a selected IH branch by the DGI--GSSB--GA pipeline) that addresses the ``easy problems'' (what distinguishes conscious from unconscious processing, when consciousness is present, what its neural correlates are) while leaving the hard problem as a question about the relationship between the mathematical description and the subjective experience it describes. First cited in \S\ref{sec:intro-why}.

\bibitem[Cook et al.(2019)]{CookEtAl2019}
\textbf{S.~J.~Cook, T.~A.~Jarrell, C.~A.~Brittin, et al.}
\emph{Whole-animal connectomes of both Caenorhabditis elegans sexes},
\emph{Nature} \textbf{571} (2019) 63--71.
\textbf{Relevance.} The whole-animal connectome ($\sim$7{,}000 synapses) for the Layer-14 quantitative anchor. First cited in \S\ref{sec:cascade-codebook-table}.

\bibitem[Cover \& Thomas(2006)]{CoverThomas2006}

\textbf{Cover, T.~M., \& Thomas, J.~A. (2006).}
\textit{Elements of Information Theory} (2nd ed.).
Wiley.
\url{https://doi.org/10.1002/047174882X}
\par\noindent\textbf{Relevance.} Information theory supplies a precise quantitative language for ``information'' as an operational resource. We cite this text to support the appendix discussion where free-energy-like functionals are related to KL divergence and where MF requires explicit information-theoretic accounting for measurement and compression. First cited in \S\ref{app:examples}.

\bibitem[CRC(2016)]{CRCHandbook}
\textbf{W.~M.~Haynes (ed.).}
\emph{CRC Handbook of Chemistry and Physics}, 97th edition.
CRC Press, Boca Raton (2016).
\textbf{Relevance.} Standard reference for small-molecule thermochemical data (standard enthalpies of formation, bond energies); the external source for the layer-4 small-molecule formation enthalpies in the cascade codebook, replacing an erroneous Particle-Data-Group citation. First cited in \S\ref{sec:cascade-codebook-table}.

\bibitem[Crick and Koch(1990)]{Crick1990}
\textbf{F.~Crick and C.~Koch.}
Toward a neurobiological theory of consciousness.
\emph{Seminars in the Neurosciences} \textbf{2} (1990) 263--275.
\par\noindent\textbf{Relevance.}
Crick and Koch proposed binding-by-synchrony as a candidate solution to the binding problem of consciousness: neural features bound by 40-Hz gamma oscillations into one unified conscious state. In the present framework, the binding problem is resolved by GA stabilization at the layer-17 EIC stratum (binding-by-attractor-stabilization), which differs structurally from binding-by-synchrony but is empirically compatible with the gamma-oscillation findings the Crick-Koch program catalyzed. First cited in \S\ref{sec:reader-guide-cognitive}.

\bibitem[Crooks(1999)]{Crooks1999}
\textbf{G.~E.~Crooks.}
Entropy production fluctuation theorem and the nonequilibrium work relation for free energy differences.
\emph{Physical Review E} \textbf{60} (1999) 2721--2726.
\par\noindent\textbf{Relevance.}
Crooks's fluctuation theorem provides a symmetry relation between the probability of forward and reverse nonequilibrium trajectories, strengthening and generalizing the second law. In the GFF framework, the Crooks relation constrains the fluctuations of the maintenance margin: a living system (SeOS) must maintain a positive margin on average, but individual events can temporarily violate this constraint --- the Crooks theorem quantifies how often and by how much. First cited in \S\ref{sec:seos}.

\bibitem[Damer \& Deamer(2020)]{DamerDeamer2020}
\textbf{B.~Damer and D.~Deamer.}
\emph{The hot spring hypothesis for an origin of life},
\emph{Astrobiology} \textbf{20} (2020).
\par\noindent\textbf{Relevance.}
Argues that wet--dry cycling in terrestrial hot-spring pools, rather than the open ocean, supplies the concentration and condensation conditions that polymerization requires, with encapsulation into lipid vesicles occurring in the same cycle. We cite it as one of the two standard developments of the wet--dry route; in the present framework it supplies the physical content of the \GRIPout{} event for the layer-5 to layer-6 transition, which the cascade names but does not derive. First cited in \S\ref{sec:chem-leading-theories}.

\bibitem[Darwin(1859)]{Darwin1859}
\textbf{C.~Darwin.}
\emph{On the Origin of Species by Means of Natural Selection}.
John Murray, London (1859).
\par\noindent\textbf{Relevance.}
Established the theory of evolution by natural selection, explaining the diversity of life through variation and differential reproductive success. Darwin's theory explains \emph{how} organisms adapt once they exist; the present paper addresses the complementary question of \emph{why} life exists at all. The GG-6 framework does not replace Darwinian evolution but provides its geometric foundation: the $\QGL$ constraint makes life inevitable, and natural selection then operates on the inevitable living substrate to produce the diversity Darwin documented. First cited in \S\ref{sec:intro-why}.

\bibitem[Dawkins(1976)]{Dawkins1976}
\textbf{R.~Dawkins.}
\emph{The Selfish Gene}.
Oxford University Press (1976).
\par\noindent\textbf{Relevance.}
Dawkins popularized the gene-centered view of evolution, arguing that organisms are ``survival machines'' built by genes to propagate themselves. The SeOS (Selfish Open System) classification in the GG-6 framework formalizes this idea: a SeOS acts to maintain its own packet not because it possesses intention but because the $\QGL$-constrained dynamics selects trajectories that preserve local viability. Dawkins's ``selfish'' is the GG-6 framework's ``maintenance-margin-positive.''. First cited in \S\ref{sec:seos}.

\bibitem[Dehaene(2014)]{Dehaene2014}
\textbf{S.~Dehaene.}
\emph{Consciousness and the Brain: Deciphering How the Brain Codes Our Thoughts.}
Viking, 2014.
\par\noindent\textbf{Relevance.}
Dehaene's Global Neuronal Workspace (GNW) account treats consciousness as the broadcast of a single coherent representation across cortical networks via long-range top-down ignition. In the GG-Theory framework, GNW ignition is the boundary readout of a stabilized \IH{} branch on the \EIC{} stratum — operationally identified with \T{-Conscious-Episode} of \AEleven{} Part~\ref{part:cascade} at \QCI{} $\sim 10$. First cited in \S\ref{sec:cascade-codebook-table}.

\bibitem[de Haro, Skenderis, \& Solodukhin(2001)]{deHaro2001}

\textbf{de Haro, S., Skenderis, K., \& Solodukhin, S.~N. (2001).}
Holographic reconstruction of spacetime and renormalization in the AdS/CFT correspondence.
\emph{Communications in Mathematical Physics}, \emph{217}, 595--622.
\url{https://arxiv.org/abs/hep-th/0002230}
\par\noindent\textbf{Relevance.} This work systematized holographic renormalization, making precise how bulk radial evolution corresponds to boundary RG flow. We cite it as a technical antecedent for POS-5: RG-compatibility is treated as a structural commutation requirement between coarse-graining and bulk--boundary dictionaries. First cited in \S\ref{app:examples}.

\bibitem[Dennett(1984)]{Dennett1984}
\textbf{Dennett, D.~C. (1984).}
\emph{Elbow Room: The Varieties of Free Will Worth Wanting}.
MIT Press.
\par\noindent\textbf{Relevance.}
Dennett's compatibilist defense of the kind of free will worth wanting under naturalistic constraints. The framework's position on agency in Corollary~\ref{cor:closure-determinism} sits within this tradition: agency is preserved at the substrate's operational level (the conscious brain genuinely deliberates), while the existence of layer-17 substrates is forced by the cascade. First cited in \S\ref{sec:closure-careful-word}.

\bibitem[Dirac(1931)]{Dirac1931}
\textbf{P.~A.~M.~Dirac.}
Quantised singularities in the electromagnetic field.
\emph{Proceedings of the Royal Society A} \textbf{133} (1931) 60--72.
\par\noindent\textbf{Relevance.}
Dirac showed that the existence of a magnetic monopole would imply the quantization of electric charge --- the first example of a topological argument constraining physical observables. The $\UoneQi$ informational charge in GG-6 is quantized by an analogous mechanism: the geometric consistency constraint (BI-6) forces the informational quantum number to take discrete values, making $J_{Qi}$ a conserved, quantized current on the same footing as electric charge and baryon number. First cited in \S\ref{sec:five-revolutions}.

\bibitem[Doig(2017)]{Doig2017}
\textbf{A.~J.~Doig.}
\emph{Frozen, but no accident --- why the 20 standard amino acids were selected},
\emph{The FEBS Journal} \textbf{284} (2017), \url{https://doi.org/10.1111/febs.13982}.
\par\noindent\textbf{Relevance.}
A review of the evidence that the coded twenty are a selected rather than an arbitrary set, assembled against Crick's frozen-accident hypothesis. It is the fairest available summary of what the conventional account of the number twenty can and cannot do, and the present paper cites it in exactly that role: the field's explanations are retrospective fitness arguments, each defensible, and none of them derives the integer --- which is the space the layer-5 relation $|\GGA{}_5| = K - NL$ attempts to occupy, at a grade this Part states plainly. First cited in \S\ref{sec:chem-leading-theories}.

\bibitem[Donoghue et al.(2020)]{Donoghue2020}
\textbf{T.~Donoghue, M.~Haller, E.~J.~Peterson, et al.\ (incl.\ B.~Voytek).}
\emph{Parameterizing neural power spectra into periodic and aperiodic components},
\emph{Nature Neuroscience} \textbf{23} (2020) 1655--1665.
\textbf{Relevance.} The field-standard parameterization of the neural power spectrum into an aperiodic component (offset, exponent, and an optional single knee) plus periodic peaks. External anchor for the Layer-17 / Test-9b neural $1/f$ claim; establishes the single knee and is the tool whose one-knee default the dual-knee prediction goes beyond (a non-physics claim, classed Asserted). First cited in \S\ref{sec:eight-test-pull-table}.

\bibitem[Einstein(1915)]{Einstein1915}
\textbf{A.~Einstein.}
Die Feldgleichungen der Gravitation.
\emph{Sitzungsberichte der Preussischen Akademie der Wissenschaften} (1915) 844--847.
\par\noindent\textbf{Relevance.}
General relativity established that gravity is not a force but the curvature of spacetime, making geometry the language of fundamental physics. AX-2 (the Geometry--Gauge Equivalence Principle) extends Einstein's insight: not only gravity but all gauge forces are aspects of geometry, unified through the composite connection of the six-dimensional GG-6 bulk. Einstein's revolution is the prototype for the constitutional approach: a geometric principle that eliminates arbitrary parameters. First cited in \S\ref{subsec:hist-bulk}.

\bibitem[England(2013)]{England2013}
\textbf{J.~L.~England.}
Statistical physics of self-replication.
\emph{Journal of Chemical Physics} \textbf{139} (2013) 121923.
\par\noindent\textbf{Relevance.}
England derived a thermodynamic bound on the dissipation cost of self-replication, relating a replicator's durability and replication rate to the heat it dissipates (a necessary thermodynamic constraint, not a demonstration that replicators must arise). This result is a boundary-theory consequence of the MEPP sector of Qi-EP: living systems (self-replicators) maximize entropy production, which is precisely the condition for minimizing the GFF functional under the $\QGL$ constraint. First cited in \S\ref{sec:seos}.

\bibitem[Ferris et al.(1996)]{Ferris1996}
\textbf{J.~P.~Ferris, A.~R.~Hill~Jr., R.~Liu, and L.~E.~Orgel.}
\emph{Synthesis of long prebiotic oligomers on mineral surfaces},
\emph{Nature} \textbf{381} (1996) 59--61.
\par\noindent\textbf{Relevance.}
Demonstrates that montmorillonite clay templates the oligomerization of activated nucleotides into chains of forty to fifty units --- long enough for catalytic structure --- without enzymes. We cite it in the layer-6 discussion as the experimental basis for treating the polymer class as a reachable product on a mineral-bearing warm-wet stratum, and as an illustration of the honest limit of the layer-6 row: the experiment shows that chains form, not that exactly three chain families are the landscape's basins. First cited in \S\ref{sec:chem-leading-theories}.

\bibitem[Fischer(1994)]{Fischer1994}
\textbf{Fischer, J.~M. (1994).}
\emph{The Metaphysics of Free Will: An Essay on Control}.
Blackwell.
\par\noindent\textbf{Relevance.}
Fischer's distinction between guidance control (compatible with determinism) and regulative control (alternative-possibilities-requiring) is the precise compatibilist apparatus the framework's micro-trajectory openness invokes: the conscious substrate exercises guidance control over its trajectory within the forced layer-17 attractor. First cited in \S\ref{sec:closure-careful-word}.

\bibitem[Frank(1953)]{Frank1953}
\textbf{F.~C.~Frank.}
\emph{On spontaneous asymmetric synthesis},
\emph{Biochimica et Biophysica Acta} \textbf{11} (1953) 459--463.
\par\noindent\textbf{Relevance.}
The founding kinetic model of homochirality: an autocatalytic reaction in which each enantiomer catalyzes its own formation and suppresses the other amplifies any initial imbalance to near-complete excess. Frank's model is the reason the origin-of-homochirality problem is now understood to be a problem about the \emph{sign} rather than the magnitude, which is the distinction the present Part draws at layer~5 and the reason the chirality-lock lemma is named as the missing piece rather than the amplification mechanism. First cited in \S\ref{sec:chem-leading-theories}.

\bibitem[Frankfurt(1969)]{Frankfurt1969}
\textbf{Frankfurt, H.~G. (1969).}
Alternate possibilities and moral responsibility.
\emph{Journal of Philosophy}, \textbf{66}(23), 829--839.
\par\noindent\textbf{Relevance.}
Frankfurt's celebrated cases showing that moral responsibility does not require alternative possibilities at the level of physical determinism. The framework's reading of agency in Corollary~\ref{cor:closure-determinism} is Frankfurt-compatible: responsibility resides in the agent's reasons-responsive substrate, not in the openness of physical micro-trajectories. First cited in \S\ref{sec:closure-careful-word}.

\bibitem[Freidlin \& Wentzell(1998)]{FreidlinWentzell1998}

\textbf{Freidlin, M.~I., \& Wentzell, A.~D. (1998).}
\emph{Random Perturbations of Dynamical Systems} (2nd ed.).
New York: Springer.
\url{https://doi.org/10.1007/978-1-4612-0611-5}
\par\noindent\textbf{Relevance.} Freidlin--Wentzell theory provides a rigorous large-deviation framework in which stochastic open-system trajectories concentrate near minimizers of an action functional. We cite it as mathematical background for POS-2 (\UEP{}), supporting the claim that ``least-action'' logic persists in open/noisy systems as a most-probable-path extremal principle. First cited in \S\ref{app:examples}.

\bibitem[Furukawa et al.(2019)]{Furukawa2019}
\textbf{Y.~Furukawa, Y.~Chikaraishi, N.~Ohkouchi, et al.}
\emph{Extraterrestrial ribose and other sugars in primitive meteorites},
\emph{Proceedings of the National Academy of Sciences} \textbf{116} (2019) 24440--24445, \url{https://doi.org/10.1073/pnas.1907169116}.
\par\noindent\textbf{Relevance.}
Identifies ribose and other bio-essential sugars in the Murchison and NWA~801 carbonaceous chondrites, with carbon-isotope ratios establishing extraterrestrial origin. The present paper cites it for the sugar half of the layer-5 inventory: the D-handed pentoses the cascade claims at layer~5 are, like the amino acids, demonstrably delivered rather than necessarily synthesized in place --- so here too the framework's claim is one of selection. This entry also corrects an earlier statement in the cohort that placed ribose in the Ryugu returned samples; the Ryugu detection is uracil (\citealp{Oba2023}), and ribose is Murchison and NWA~801. First cited in \S\ref{sec:chem-leading-theories}.

\bibitem[Friston(2010)]{Friston2010}
\textbf{K.~Friston.}
\emph{The free-energy principle: a unified brain theory?},
\emph{Nature Reviews Neuroscience} \textbf{11} (2010) 127--138.
\par\noindent\textbf{Relevance.}
Proposed the Free Energy Principle (FEP), which states that biological systems minimize a variational free energy --- the discrepancy between the system's internal model and sensory input. The FEP has been applied to perception, action, learning, and consciousness. In the GG-6 framework, the FEP is a consequence of the Qi-EP principle applied to the biological boundary: the $\cF_{\EIC}$ Lyapunov functional has exactly the structure of Friston's variational free energy, and the MePMoS cycle is the operational realization of Friston's perception--action loop. The FEP is the $J_{Qi} \neq 0$, Markov-blanket-equipped sector of the more general Qi-EP. First cited in \S\ref{sec:statement-comparison}.

\bibitem[Gao et al.(2020)]{GaoVoytek2020}
\textbf{R.~Gao, R.~L.~van den Brink, T.~Pfeffer, and B.~Voytek.}
\emph{Neuronal timescales are functionally dynamic and shaped by cortical microarchitecture},
\emph{eLife} \textbf{9} (2020) e61277.
\textbf{Relevance.} Establishes the mapping between the aperiodic spectral knee and an intrinsic neuronal timescale, $\tau = 1/(2\pi f_{\rm knee})$, rising along the cortical hierarchy. External anchor for the timescale reading of the Test-9b neural knee (Asserted). First cited in \S\ref{sec:eight-test-pull-table}.

\bibitem[Georgi and Glashow(1974)]{GeorgiGlashow1974}
\textbf{H.~Georgi and S.~L.~Glashow.}
\emph{Unity of all elementary-particle forces},
\emph{Physical Review Letters} \textbf{32} (1974) 438--441.
\par\noindent\textbf{Relevance.}
Proposed the first Grand Unified Theory based on $\SU(5)$, showing that the strong, weak, and electromagnetic forces can be unified into a single gauge group at high energy. The original Georgi--Glashow model predicts proton decay, which has not been observed. In the GG-6 framework, the $\SUfive$ structure is preserved but the proton is absolutely stable because baryon number is gauged ($\UoneB$ is an exact local symmetry). The trace ratio $2/5$ in the BI-6 coefficients is a direct consequence of the $\SUfive$ block size $L = 5$. First cited in \S\ref{sec:ogn5-bio}.

\bibitem[Gibbs(1875)]{Gibbs1875}
\textbf{J.~W.~Gibbs.}
On the equilibrium of heterogeneous substances.
\emph{Transactions of the Connecticut Academy of Arts and Sciences} \textbf{3} (1875--1878) 108--248, 343--524.
\par\noindent\textbf{Relevance.}
The founding paper of physical chemistry's free-energy formalism: $G = H - TS$ as the cost functional whose minimum identifies the equilibrium state of a thermodynamic system at constant temperature and pressure. In the framework of \AEleven{}, the Gibbs free energy is identified as the $J_{Qi}=0$ boundary projection of the Geometric Free-Energy Functional $F_{\GFF{}}$ of \AFive{}. Gibbs's law was the first \emph{free energy} written down; the framework's Three-Gauge Theorem reads it as one of three boundary projections of a single underlying geometric object. First cited in \S\ref{subsec:gibbs-gauge}.

\bibitem[Gilbert(1986)]{Gilbert1986}
\textbf{W.~Gilbert.}
\emph{The RNA world},
\emph{Nature} \textbf{319} (1986) 618.
\par\noindent\textbf{Relevance.}
The one-page note that named the hypothesis --- that a single molecule once carried both the genetic information and the catalysis now divided between DNA and protein. The idea is older than the name, having been proposed by Woese, Crick and Orgel in the late 1960s, but the name is what organized the field. We cite it as the origin of the framing that layer~8 of the present cascade adopts, and as the boundary of what layer~6 deliberately does not claim: layer~6 certifies a polymer class and leaves the selection of RNA to the layer above. First cited in \S\ref{sec:chem-leading-theories}.

\bibitem[Glashow(1961)]{Glashow1961}
\textbf{S.~L.~Glashow.}
Partial-symmetries of weak interactions.
\emph{Nuclear Physics} \textbf{22} (1961) 579--588.
\par\noindent\textbf{Relevance.}
Glashow proposed the $\mathrm{SU}(2) \times \mathrm{U}(1)$ structure of the electroweak interaction, later completed by Weinberg and Salam. In GG-6, the electroweak group is a subgroup of $\SUfive$ that emerges on the four-dimensional boundary after the Hosotani Wilson-line symmetry breaking. The electroweak mixing angle $\sin^2\theta_W = 25/108$ is derived from the OGN-5 integers rather than fitted to data. First cited in \S\ref{sec:cascade-plain-english}.

\bibitem[Goff(2017)]{Goff2017}
\textbf{Goff, P. (2017).}
\emph{Consciousness and Fundamental Reality}.
Oxford University Press.
\par\noindent\textbf{Relevance.}
Contemporary defense of Russellian monism as a serious option in the philosophy of consciousness landscape. Goff distinguishes Russellian panpsychism from Russellian neutral monism (the framework's chosen position) and argues that the latter offers structural resources the framework now operationalizes via the \EIC{} closure on layer-17 substrates. First cited in \S\ref{sec:reader-guide-cognitive}.

\bibitem[Goldstein, Poole, \& Safko(2002)]{Goldstein2002}

\textbf{Goldstein, H., Poole, C., \& Safko, J. (2002).}
\textit{Classical Mechanics} (3rd ed.).
Addison--Wesley.
\par\noindent\textbf{Relevance.} This standard text summarizes variational principles (Hamilton, Lagrange, Maupertuis) and their equivalences. We cite it in POS-2 to situate the Unified Extremal Principle within the established mechanics lineage. First cited in \S\ref{app:examples}.

\bibitem[Gorini, Kossakowski, \& Sudarshan(1976)]{GKS1976}

\textbf{Gorini, V., Kossakowski, A., \& Sudarshan, E.~C.~G. (1976).}
Completely positive dynamical semigroups of N-level systems.
\textit{Journal of Mathematical Physics}, \textit{17}(5), 821--825.
\url{https://doi.org/10.1063/1.522979}
\par\noindent\textbf{Relevance.} Together with Lindblad, this paper characterizes the most general Markovian quantum evolution preserving complete positivity. We cite it as a rigorous foundation for treating open-system evolution as a constitutional admissibility condition (POS-4: Closure \& Positivity). First cited in \S\ref{app:examples}.

\bibitem[Gorini et~al.(1976)]{Gorini1976}

\textbf{Gorini, V., Kossakowski, A., \& Sudarshan, E.~C.~G. (1976).}
Completely positive dynamical semigroups of $N$-level systems.
\textit{Journal of Mathematical Physics}, \textit{17}(5), 821--825.
\par\noindent\textbf{Relevance.} Along with Lindblad's work, this establishes the generator form required for physical open-system quantum evolution.
It supports POS-\CP\ and the \OGGEP\ demand that open evolution be part of the foundational dictionary. First cited in \S\ref{app:abbrev-merged}.

\bibitem[Green and Schwarz(1984)]{GreenSchwarz1984}
\textbf{M.~B.~Green and J.~H.~Schwarz.}
\emph{Anomaly cancellations in supersymmetric D=10 gauge theory and superstring theory},
\emph{Physics Letters B} \textbf{149} (1984) 117--122.
\par\noindent\textbf{Relevance.}
The foundational anomaly-cancellation paper that launched the ``first superstring revolution.'' Green and Schwarz showed that the gauge and gravitational anomalies of the ten-dimensional $\SO(32)$ superstring cancel through a tensor-mediated mechanism, selecting $\SO(32)$ and $E_8 \times E_8$ as the only consistent gauge groups. The BI-6 identity of GG-6 is the six-dimensional, non-supersymmetric counterpart of this mechanism (sharing its anomaly-polynomial mathematics, not its ten-dimensional origin): the same mathematical structure (factorized anomaly polynomial, tensor counterterm) operates in six dimensions with the gauge group $\SUfive \times \UoneB \times \UoneQi$, producing the Arisaka Equation instead of the string-theory gauge groups. First cited in \S\ref{subsec:inheritance-A3}.

\bibitem[Gubser, Klebanov, \& Polyakov(1998)]{Gubser1998}

\textbf{Gubser, S.~S., Klebanov, I.~R., \& Polyakov, A.~M. (1998).}
Gauge theory correlators from non-critical string theory.
\textit{Physics Letters B}, \textit{428}(1--2), 105--114.
\url{https://arxiv.org/abs/hep-th/9802109}
\par\noindent\textbf{Relevance.} Together with Witten's prescription, this paper established the concrete AdS/CFT dictionary relating bulk fields to boundary correlators. We cite it as historical precedent for treating a bulk$\to$boundary export map as part of the theory's definition, which \OGGEP{} generalizes to include explicit measurement semantics. First cited in \S\ref{app:ax3-expanded}.

\bibitem[Guth(1981)]{Guth1981}
\textbf{A.~H.~Guth.}
Inflationary universe: a possible solution to the horizon and flatness problems.
\emph{Physical Review D} \textbf{23} (1981) 347--356.
\par\noindent\textbf{Relevance.}
Guth's inflationary scenario solved the horizon and flatness problems of standard cosmology by postulating a period of exponential expansion in the early universe. The GG-6 cosmology paper proposes a GUT-adjacent slow smooth inflationary descent (a plateau one e-fold below $E_{\mathrm{GUT}}$, $V_{\mathrm{inf}}^{1/4} = 8.3\times 10^{15}$~GeV) that differs from Guth's original false-vacuum scenario in mechanism and observational signatures, while achieving the same resolution of the cosmological puzzles. First cited in \S\ref{sec:achievements}.

\bibitem[Haag(1992)]{Haag1992}

\textbf{Haag, R. (1992).}
\textit{Local Quantum Physics: Fields, Particles, Algebras} (2nd ed.).
Springer.
\url{https://doi.org/10.1007/978-3-662-03081-1}
\par\noindent\textbf{Relevance.} Haag's formulation of QFT emphasizes locality, causality, and operator-algebraic structure. We cite it to ground AX-1 in the rigorous QFT tradition and to clarify what ``locality'' means when expressed as constraints on observable algebras. First cited in \S\ref{app:ax1-expanded}.

\bibitem[Hafting et al.(2005)]{HaftingEtAl2005}
\textbf{T.~Hafting, M.~Fyhn, S.~Molden, M.-B.~Moser, and E.~I.~Moser.}
\emph{Microstructure of a spatial map in the entorhinal cortex},
\emph{Nature} \textbf{436} (2005) 801--806.
\par\noindent\textbf{Relevance.}
Discovered grid cells in the entorhinal cortex --- neurons that fire at regular hexagonal intervals as the animal moves through space, providing a metric coordinate system for distances and directions. Together with place cells, grid cells implement the allocentric IH predicted by the framework: the learned metric $g_t$ on the latent manifold $\cZ$. The hexagonal symmetry of the grid pattern is a consequence of optimal packing under the GFF energy constraint, and the multiple grid scales correspond to the NHT (Nested Holographic Transform) hierarchy predicted by Qi-HAL. First cited in \S\ref{sec:thm-allocentric-map}.

\bibitem[Haken(1983)]{Haken1983}
\textbf{H.~Haken.}
\emph{Synergetics: An Introduction} (3rd ed.).
Springer, Berlin (1983).
\par\noindent\textbf{Relevance.}
Haken's synergetics provides the mathematical framework for understanding how macroscopic order emerges from the cooperation of many microscopic components. The GSSB mechanism in the GG-6 framework is a geometric generalization of Haken's order-parameter dynamics: the symmetry-breaking transition that selects a specific macroscopic pattern from a space of equivalent possibilities is the same mathematical structure in laser physics (Haken's original application), in the Higgs mechanism (particle physics), and in conscious perception (the DGI--GSSB--GA pipeline). First cited in \S\ref{sec:grip-operational-anatomy}.

\bibitem[Hamilton(1834)]{Hamilton1834}

\textbf{Hamilton, W.~R. (1834).}
On a general method in dynamics.
\textit{Philosophical Transactions of the Royal Society of London}, \textit{124}, 247--308.
\url{https://doi.org/10.1098/rstl.1834.0017}
\par\noindent\textbf{Relevance.} Hamilton's formulation is the modern backbone of the principle of least action and canonical dynamics. We cite it in POS-2 to emphasize that extremal principles historically reorganized mechanics and are a natural candidate for a unification engine. First cited in \S\ref{app:examples}.

\bibitem[Hawking(1975)]{Hawking1975}
\textbf{S.~W.~Hawking.}
Particle creation by black holes.
\emph{Communications in Mathematical Physics} \textbf{43} (1975) 199--220.
\par\noindent\textbf{Relevance.}
Hawking showed that black holes radiate thermally, confirming Bekenstein's entropy formula and revealing a tension between quantum mechanics and general relativity (the information paradox). The bulk-to-boundary structure of GG-6 --- in which the six-dimensional bulk projects onto a four-dimensional boundary --- provides a natural framework for resolving this tension, since information is never lost in the bulk even when it appears to be lost on the boundary. First cited in \S\ref{subsec:hist-triad}.

\bibitem[Hegel(1807)]{Hegel1807}
\textbf{G.~W.~F.~Hegel.}
\emph{Ph\"anomenologie des Geistes} (1807).
\par\noindent\textbf{Relevance.}
In the \emph{Phenomenology of Spirit}, Hegel described the Absolute Spirit --- the universe coming to know itself through the medium of finite minds, proceeding through a dialectical sequence of thesis, antithesis, and synthesis. The EIC convergence theorem $\cZ_{\EIC}[J_i] = \cZ_{\mathrm{external}}[J_i]$ is Hegel's Absolute Spirit stated as a functional convergence theorem. The GSSB cascade, with its seventeen symmetry-breaking transitions, is Hegel's dialectical development made quantitative; each transition is a synthesis that resolves the tension of the previous stage and generates the conditions for the next. First cited in \S\ref{sec:thm-articulation}.

\bibitem[Higgs(1964)]{Higgs1964}
\textbf{P.~W.~Higgs.}
Broken symmetries and the masses of gauge bosons.
\emph{Physical Review Letters} \textbf{13} (1964) 508--509.
\par\noindent\textbf{Relevance.}
The Higgs mechanism explains how gauge bosons acquire mass through spontaneous symmetry breaking, completing the electroweak theory. In the GG-6 framework, the Higgs field is not a separate ingredient but a geometric object: the Wilson-line holonomy around the compact sixth dimension, whose vacuum expectation value is determined by the Hosotani mechanism and the OGN-5 integers. The Higgs mass is a derived quantity, not a free parameter. First cited in \S\ref{sec:cascade-plain-english}.

\bibitem[Higgs \& Pudritz(2009)]{HiggsPudritz2009}
\textbf{P.~G.~Higgs and R.~E.~Pudritz.}
\emph{A thermodynamic basis for prebiotic amino acid synthesis and the nature of the first genetic code},
\emph{Astrobiology} \textbf{9} (2009).
\par\noindent\textbf{Relevance.}
Derives, from the thermodynamics of synthesis alone, the subset of amino acids that should be most abundant prebiotically, and finds it matches the set independently inferred to have entered the genetic code earliest. We cite it as the strongest \emph{forward} argument the conventional literature offers about the composition of the coded set --- and note in the same breath what it does not do: it ranks and it explains an early subset, and it does not fix the final cardinality at twenty. First cited in \S\ref{sec:chem-leading-theories}.

\bibitem[Hopfield(1982)]{Hopfield1982}
\textbf{J.~J.~Hopfield.}
Neural networks and physical systems with emergent collective computational abilities.
\emph{Proceedings of the National Academy of Sciences} \textbf{79} (1982) 2554--2558.
\par\noindent\textbf{Relevance.}
Hopfield showed that neural networks with symmetric connections store memories as attractor states --- stable configurations toward which the network evolves from nearby initial conditions. The GA (Geometric Attractor) phase of the DGI--GSSB--GA pipeline is the GG-6 generalization of Hopfield's attractor dynamics: the conscious percept is the GA-stabilized state of the cortical network, and the \QCI{} ladder measures the depth and richness of the attractor landscape. First cited in \S\ref{sec:ih}.

\bibitem[Hopkins and Singer(2005)]{HopkinsSinger2005}
\textbf{M.~J.~Hopkins and I.~M.~Singer.}
\emph{Quadratic functions in geometry, topology, and M-theory},
\emph{Journal of Differential Geometry} \textbf{70} (2005) 329--452.
\par\noindent\textbf{Relevance.}
Developed the mathematical framework of differential cohomology and its application to anomaly cancellation, providing the rigorous language for global (topological) anomalies beyond the perturbative Green--Schwarz mechanism. The BI-6 global anomaly proof uses the Hopkins--Singer integral lift to construct the Wu--Chern--Simons counterterm, and the automatic-admissibility theorem verifies that the hypotheses of the Hopkins--Singer framework are satisfied on the intended GG-6 class. This is the mathematical technology that ensures the six-dimensional bulk is consistent not just perturbatively but also non-perturbatively. First cited in \S\ref{subsec:inheritance-A3}.

\bibitem[Hosotani(1983)]{Hosotani1983}
\textbf{Y.~Hosotani.}
\emph{Dynamical gauge symmetry breaking as the Casimir effect},
\emph{Physics Letters B} \textbf{129} (1983) 193--197.
\par\noindent\textbf{Relevance.}
Introduced the Wilson-line symmetry-breaking mechanism, in which the vacuum expectation value of the gauge field around a compact extra dimension breaks the gauge symmetry dynamically. In GG-6, the Hosotani mechanism replaces the elementary Higgs field: the Higgs vacuum is a holonomy branch of the composite connection around $S^1_\theta$, and the Higgs mass and vacuum expectation value are determined by the OGN-5 integers rather than fitted to data. This is the mechanism by which electroweak symmetry breaking becomes a geometric consequence. First cited in \S\ref{sec:achievements}.

\bibitem[Jackson(1982)]{Jackson1982}
\textbf{Jackson, F. (1982).}
Epiphenomenal qualia.
\emph{Philosophical Quarterly}, \textbf{32}(127), 127--136.
\par\noindent\textbf{Relevance.}
The canonical statement of the knowledge argument (``what Mary knew''): a color scientist confined to a black-and-white room and then released to see red is said to learn a new fact about phenomenal experience. The framework's structural-identity reply distinguishes propositional knowledge of the \EIC{} structure from acquaintance with its instantiation on a specific stimulus coordinate; Mary gains acquaintance, not a new fact. Jackson himself in subsequent work moved toward the ability-hypothesis reading, which is consonant with the framework's reply. First cited in \S\ref{sec:reader-guide-cognitive}.

\bibitem[James(1890)]{James1890}
\textbf{W.~James.}
\emph{The Principles of Psychology.}
Henry Holt, 1890.
\par\noindent\textbf{Relevance.}
James's two-volume treatise founded scientific psychology and introduced the ``stream of consciousness'' as a unit of analysis, alongside the ideomotor theory and the James--Lange theory of emotion. In the GG-Theory framework, James's stream of consciousness is the temporal sequence of \EIC{} closures at theta-cycle cadence — operationally identified with the kinematic readout of \T{-Conscious-Episode}. First cited in \S\ref{sec:inevitability-life}.

\bibitem[Jarzynski(1997)]{Jarzynski1997}
\textbf{C.~Jarzynski.}
Nonequilibrium equality for free energy differences.
\emph{Physical Review Letters} \textbf{78} (1997) 2690--2693.
\par\noindent\textbf{Relevance.}
Jarzynski's equality relates the free energy difference between two equilibrium states to the exponential average of work performed in nonequilibrium processes, extending the second law to far-from-equilibrium systems. The GFF ledger of the GG-6 framework is the natural setting for Jarzynski-type relations: each gauge-locked event contributes to the GFF balance, and the maintenance margin of a SeOS can be expressed as a Jarzynski-type average over the system's operational cycles. First cited in \S\ref{sec:seos}.

\bibitem[Jaynes(1957)]{Jaynes1957}
\textbf{E.~T.~Jaynes.}
Information theory and statistical mechanics.
\emph{Physical Review} \textbf{106} (1957) 620--630.
\par\noindent\textbf{Relevance.}
Jaynes showed that statistical mechanics can be derived from the principle of maximum entropy, reinterpreting thermodynamics as a theory of inference rather than of microscopic dynamics. The Qi-EP (Geometric Extremal Principle) of the GG-6 framework subsumes Jaynes's maximum-entropy principle as a special case: when the $\QGL$ constraint is inactive ($J_{Qi} = 0$), Qi-EP reduces to Jaynes's formalism; when the constraint is active, it extends to the full GFF descent. First cited in \S\ref{sec:qi4-admissibility-filter}.

\bibitem[Joyce(2002)]{Joyce2002}
\textbf{G.~F.~Joyce.}
\emph{The antiquity of RNA-based evolution},
\emph{Nature} \textbf{418} (2002) 214--221.
\par\noindent\textbf{Relevance.}
Assembles the evidence that RNA-based evolution preceded protein-based evolution, principally the phylogenetic depth of RNA cofactors and catalytic RNA inside modern metabolism, and states the case for the RNA world as the field's default working hypothesis rather than a speculation. We cite it for the same boundary Gilbert marks: it establishes what layer~8 will have to account for, and it establishes that the polymer-class claim of layer~6 is prior to, and weaker than, any claim about which polymer won. First cited in \S\ref{sec:chem-leading-theories}.

\bibitem[Kaluza(1921)]{Kaluza1921}
\textbf{T.~Kaluza.}
Zum Unit\"atsproblem der Physik.
\emph{Sitzungsberichte der Preussischen Akademie der Wissenschaften} (1921) 966--972.
\par\noindent\textbf{Relevance.}
Kaluza showed that general relativity in five dimensions contains both four-dimensional gravity and electromagnetism, establishing the idea that extra dimensions can unify forces. The GG-6 bulk $M_6 = M_4 \times I_r \times S^1_\theta$ is a modern descendant of Kaluza's construction, extended from five to six dimensions and from electromagnetism to the full $\SUfive \times \UoneB \times \UoneQi$ gauge group. First cited in \S\ref{subsec:hist-bulk}.

\bibitem[Kant(1781)]{Kant1781}
\textbf{I.~Kant.}
\emph{Kritik der reinen Vernunft} (Critique of Pure Reason), 1st ed.\ 1781, 2nd ed.\ 1787.
\par\noindent\textbf{Relevance.}
Kant's transcendental philosophy distinguished the phenomenal world (what can be experienced) from the noumenal (what is, prior to experience). The GG-Theory program accepts Kant's distinction at the operational level (OGGEP, AX-3 of the GG-Constitution) but rejects the noumenal as inaccessible: the constitution of the bulk is fully recoverable from the operational boundary, and the act of recovery is the self-referential closure of Part XII. First cited in \S\ref{subsec:kant}.

\bibitem[Kauffman(1993)]{Kauffman1993}
\textbf{S.~A.~Kauffman.}
\emph{The Origins of Order: Self-Organization and Selection in Evolution}.
Oxford University Press (1993).
\par\noindent\textbf{Relevance.}
Kauffman argued that the order observed in biological systems is not solely the product of natural selection but also of self-organization --- that certain molecular configurations are intrinsically more probable than others due to the structure of chemical reaction networks. The GG-6 framework provides the physical underpinning for Kauffman's intuition: the $\QGL$ constraint makes certain molecular architectures (those satisfying the gauge-locking condition) geometrically necessary, not merely probable. First cited in \S\ref{sec:grip-operational-anatomy}.

\bibitem[Kim et al.(2017)]{KimEtAl2017}
\textbf{S.~S.~Kim, H.~Rouault, S.~Druckmann, V.~Jayaraman.}
Ring attractor dynamics in the \emph{Drosophila} central brain.
\emph{Science} \textbf{356} (2017) 849--853.
\par\noindent\textbf{Relevance.}
Two-photon calcium imaging in the ellipsoid body of \emph{Drosophila} resolves a single localized activity bump that drifts and stabilizes in a one-dimensional ring topology, identifying the head-direction ring attractor as a directly visualizable neural substrate. This is the empirical anchor for \T{-Neural-Ring-Attractor} of \AEleven{} Part~\ref{part:cascade}: insect heading is the layer-15 GGA, a ring topology selected by GRIP descent on the SO(2)-equivariant boundary stratum. First cited in \S\ref{sec:confidence-legend}.

\bibitem[Kleiber(1932)]{Kleiber1932}
\textbf{M.~Kleiber.}
\emph{Body size and metabolism},
\emph{Hilgardia} \textbf{6} (1932) 315--353.
\textbf{Relevance.} The original statement of the three-quarter-power metabolic scaling law $B \propto M^{3/4}$ (the universality of the 3/4 exponent is debated --- a 2/3 surface-law exponent is observed in some regimes), used as the empirical anchor for the layer-13 multicellularity row of the cascade codebook (a non-physics claim, classed Conditional). The framework reads the exponent as a variational signature of the \QiEP{} principle rather than a fitted constant. First cited in \S\ref{sec:cascade-codebook-table}.

\bibitem[Klein(1926)]{Klein1926}
\textbf{O.~Klein.}
Quantentheorie und f\"unfdimensionale Relativit\"atstheorie.
\emph{Zeitschrift f\"ur Physik} \textbf{37} (1926) 895--906.
\par\noindent\textbf{Relevance.}
Klein proposed that the extra dimension in Kaluza's theory is compactified (curled up into a small circle), explaining why it is not directly observed. The compact $S^1_\theta$ dimension of GG-6 is Klein's compactification applied to the sixth dimension; the Kaluza--Klein tower of massive excitations predicts a resonance at approximately 3.85~TeV, testable at the High-Luminosity LHC. First cited in \S\ref{subsec:hist-bulk}.

\bibitem[Kobayashi \& Nomizu(1963)]{KobayashiNomizu1963}

\textbf{Kobayashi, S., \& Nomizu, K. (1963).}
\textit{Foundations of Differential Geometry} (Vol.~1).
Wiley.
\par\noindent\textbf{Relevance.} This classic text provides the rigorous mathematics of principal bundles, connections, curvature, and holonomy. We cite it as a standard reference for the geometric language used in formulating \GGEP{} (composite connection) and in defining curvature/holonomy as invariants. First cited in \S\ref{app:ax2-expanded}.

\bibitem[Kraus(1971)]{Kraus1971}

\textbf{Kraus, K. (1971).}
General state changes in quantum theory.
\textit{Annals of Physics}, \textit{64}(2), 311--335.
\url{https://doi.org/10.1016/0003-4916(71)90108-4}
\par\noindent\textbf{Relevance.} Kraus operators and instruments provide the modern operational semantics of measurement and state update. We cite this work as foundational for AX-3, where measurement is defined axiomatically as a CP instrument produced by the bulk$\to$boundary dictionary. First cited in \S\ref{app:ax1-expanded}.

\bibitem[Kraus(1983)]{Kraus1983}

\textbf{Kraus, K. (1983).}
\textit{States, Effects, and Operations: Fundamental Notions of Quantum Theory}.
Springer.
\url{https://doi.org/10.1007/978-3-540-39068-5}

\par\noindent\textbf{Relevance.} Kraus' monograph systematizes the operational formulation of quantum measurement and dynamics via completely positive maps.
AX-3 adopts this language axiomatically by requiring that measurement be defined in terms of instruments and that open evolution be CPTP. First cited in \S\ref{app:ax1-expanded}.

\bibitem[Kuhn(1962)]{Kuhn1962}
\textbf{T.~S.~Kuhn.}
\emph{The Structure of Scientific Revolutions}.
University of Chicago Press (1962).
\par\noindent\textbf{Relevance.}
Kuhn argued that science progresses through paradigm shifts --- periods of ``normal science'' punctuated by revolutions that redefine the fundamental concepts and methods of a field. The present paper argues that the \GGTheory{} represents the third such revolution (after Newton and Einstein/QM), extending mathematical law from closed physical systems to all open systems including life and consciousness. Kuhn's framework explains why the resistance to such a shift is expected and why the shift, once completed, is irreversible. First cited in \S\ref{sec:five-revolutions}.

\bibitem[Kullback \& Leibler(1951)]{KullbackLeibler1951}

\textbf{Kullback, S., \& Leibler, R.~A. (1951).}
On information and sufficiency.
\textit{Annals of Mathematical Statistics}, \textit{22}(1), 79--86.
\url{https://doi.org/10.1214/aoms/1177729694}
\par\noindent\textbf{Relevance.} The KL divergence is the canonical nonnegative functional measuring distinguishability of probability distributions. We cite it because many ``free energy'' expressions in nonequilibrium physics and inference can be written as KL divergences, providing a clean mathematical bridge for POS-2 (\UEP{}) between thermodynamic and informational extremals. First cited in \S\ref{app:examples}.

\bibitem[Kumar et al.(2010)]{KumarMorrisonTaylor2010}
\textbf{V.~Kumar, D.~R.~Morrison, and W.~Taylor.}
\emph{Global aspects of the space of 6D N=1 supergravities},
\emph{JHEP} \textbf{11} (2010) 118.
\par\noindent\textbf{Relevance.}
Systematically enumerated the space of six-dimensional $\cN = (1,0)$ supergravity theories satisfying anomaly cancellation, establishing the landscape of consistent 6D models. The BI-6 theorem shows that the GG-6 solution occupies a unique point in a closely related (non-supersymmetric) parameter space, and the techniques of Kumar, Morrison, and Taylor --- particularly the Diophantine system for orbit multiplicities --- are adapted in the benchmark-spectrum construction of the anomaly paper. First cited in \S\ref{subsec:hist-anomaly}.

\bibitem[Ladyman and Ross(2007)]{Ladyman2007}
\textbf{Ladyman, J., and D. Ross. (2007).}
\emph{Every Thing Must Go: Metaphysics Naturalized}.
Oxford University Press.
\par\noindent\textbf{Relevance.}
The most developed contemporary statement of ontic structural realism: there are no quiddities (intrinsic individuating properties) beyond the structural relations that physics describes; structure is metaphysically fundamental. The framework's structural-identity theory of phenomenal consciousness (Part~\ref{part:cognitive} \S\ref{sec:reader-guide-cognitive}) and its anti-Russellian commitment to structure-all-the-way-down draw directly on this tradition. Ladyman--Ross is the natural philosophical home for the framework's metaphysical commitments. First cited in \S\ref{sec:reader-guide-cognitive}.

\bibitem[Kvenvolden et al.(1970)]{Kvenvolden1970}
\textbf{K.~Kvenvolden, J.~Lawless, K.~Pering, et al.}
\emph{Evidence for extraterrestrial amino-acids and hydrocarbons in the Murchison meteorite},
\emph{Nature} \textbf{228} (1970) 923--926.
\par\noindent\textbf{Relevance.}
The analysis that established amino acids of unambiguously extraterrestrial origin in a meteorite, racemic and including species biology does not use. More than eighty amino acids are now attributed to Murchison. This is the single most important empirical fact bearing on the present Part's layer~5: the prebiotic inventory is a superset of the biological one, so the question the cascade must answer at layer~5 is which twenty are taken up, and the framework's answer is graded \textbf{Conditional} accordingly. First cited in \S\ref{sec:chem-leading-theories}.

\bibitem[Landauer(1961)]{Landauer1961}
\textbf{R.~Landauer.}
\emph{Irreversibility and heat generation in the computing process},
\emph{IBM Journal of Research and Development} \textbf{5} (1961) 183--191.
\par\noindent\textbf{Relevance.}
Established the minimum energy cost of information erasure: $k_B T \ln 2$ per bit erased. Landauer's principle connects information theory to thermodynamics by showing that logical irreversibility implies physical dissipation. The $\QGL$ constraint raises this bound by an order of magnitude: because information current $J_{Qi}$ must be accompanied by baryonic and electromagnetic currents, the true physical cost of biological information processing is $\sim 13$--$14\,k_BT$ per nucleotide, not $1.4\,k_BT$. This elevated bound explains why biological information processing is expensive and why disembodied information is physically forbidden. First cited in \S\ref{sec:qgi}.

\bibitem[Lane(2015)]{Lane2015}
\textbf{N.~Lane.}
\emph{The Vital Question: Why Is Life the Way It Is?}
Profile Books, London (2015).
\par\noindent\textbf{Relevance.}
Lane argues that the origin of life is fundamentally an energy problem: life arose at alkaline hydrothermal vents where geochemical proton gradients could drive the synthesis of organic molecules. The GG-6 framework's complementary reading: the geochemical setting Lane emphasizes supplies the proton gradient (the \emph{site} is contingent, as Lane argues), while the energy quantum $N_A\QB = 15.0$~kJ/mol itself is fixed by the OGN-5 integers --- the site may vary, but the quantum is universal, so any source delivering multiples of $\QB$ can drive the GSSB cascade toward life. First cited in \S\ref{sec:cascade-codebook-table}.

\bibitem[Le et al.(2022)]{Vision7D4}
\textbf{C.~Le, S.~Pedersen, N.~Chen, et al., A.~P.~Blaisdell, and K.~Arisaka.}
\emph{Visual Perception of 3D Space and Shape by 7D Frame Translation in Time, Part~\ref{part:results}: 3D Shape Recognition by 3D Rotation},
bioRxiv 2022.03.01.482164 (2022).
\textbf{Relevance.} Experiment establishing reaction time linear in rotation angle on roll/yaw/pitch (roll flattest) and the $\sim 100$~ms-step / $\sim 400$~ms-loop conscious-recognition timing --- the three rotation axes of the 7D frame and the direct timing anchor for Test~10. Author's own preprint (Asserted/INT). First cited in \S\ref{sec:per-test-commentary}.

\bibitem[Leibniz(1714)]{Leibniz1714}
\textbf{G.~W.~Leibniz.}
\emph{Monadologie} (1714); \emph{Principles of Nature and Grace} (1714).
Trans.\ R.~Latta, Oxford University Press (1898).
\par\noindent\textbf{Relevance.}
Leibniz's \emph{principle of sufficient reason} --- nothing exists without a reason for being so rather than otherwise --- is the philosophical predecessor of the GG-Theory uniqueness theorems. Leibniz inferred from sufficient reason to the existence of a sufficient ground (God); GG-Theory derives the same conclusion from the constitution of geometry, with the constitution playing the structural role Leibniz assigned to divine choice. First cited in \S\ref{subsec:leibniz}.

\bibitem[Levine(1983)]{Levine1983}
\textbf{Levine, J. (1983).}
Materialism and qualia: The explanatory gap.
\emph{Pacific Philosophical Quarterly}, \textbf{64}(4), 354--361.
\par\noindent\textbf{Relevance.}
The canonical formulation of the explanatory gap: a complete specification of physical-functional facts does not seem to explain why those facts feel like anything from the inside. The framework's two-part reply distinguishes the analytic-conceptual gap (real; not closed by structural identity) from the metaphysical-chasm reading (closed by structural identity supported by empirical signature). The gap is treated as an artifact of vocabulary discipline, parallel to the pre-19th-century gap between \emph{water} and \emph{H$_2$O}. First cited in \S\ref{sec:reader-guide-cognitive}.

\bibitem[Lindblad(1976)]{Lindblad1976}
\textbf{G.~Lindblad.}
On the generators of quantum dynamical semigroups.
\emph{Communications in Mathematical Physics} \textbf{48} (1976) 119--130.
\par\noindent\textbf{Relevance.}
Lindblad derived the most general form of a Markovian master equation for open quantum systems that preserves complete positivity and trace --- the GKLS equation. In the GG-6 framework, the GKLS equation is the mathematical form of the GDOS boundary dynamics: every open system on the four-dimensional boundary evolves according to a GKLS generator, and the DGI--GSSB--GA pipeline operates within this Lindblad structure. First cited in \S\ref{app:examples}.

\bibitem[Linde(1986)]{Linde1986}
\textbf{A.~D.~Linde.}
Eternally existing self-reproducing chaotic inflationary universe.
\emph{Physics Letters B} \textbf{175} (1986) 395--400.
\par\noindent\textbf{Relevance.} The eternal-inflation route to a multiverse of sub-universes with varying effective constants. Cited in \S\ref{ssec:comparison-multiverse}: \GGTheory{} makes the ensemble unnecessary by forcing the constants from the constitution. First cited in \S\ref{sec:statement-comparison}.

\bibitem[Maldacena(1997)]{Maldacena1997}

\textbf{Maldacena, J.~M. (1997).}
The large N limit of superconformal field theories and supergravity.
\textit{Advances in Theoretical and Mathematical Physics}, \textit{2}, 231--252.
\url{https://arxiv.org/abs/hep-th/9711200}
\par\noindent\textbf{Relevance.} AdS/CFT is the sharpest realization of a bulk--boundary dictionary in modern theory. We cite it as historical support for treating ``holographic export'' as a definitional element of the theory (\OGGEP{}), rather than as an interpretation added after calculations. First cited in \S\ref{app:ax3-expanded}.

\bibitem[Maldacena(1998)]{Maldacena1998}
\textbf{J.~Maldacena.}
\emph{The large-N limit of superconformal field theories and supergravity},
\emph{Advances in Theoretical and Mathematical Physics} \textbf{2} (1998) 231--252.
\par\noindent\textbf{Relevance.}
Established the AdS/CFT correspondence --- the conjecture that a theory of quantum gravity in Anti-de Sitter spacetime is exactly equivalent to a conformal field theory on its boundary. This is the most concrete realization of the holographic principle. The GG-6 framework uses AdS/CFT as a structural ingredient (the six-dimensional bulk projects onto a four-dimensional boundary through the OGGEP dictionary) but differs in that the boundary theory is the Standard Model plus GDOS corrections, not a conformal field theory, and the bulk is six-dimensional, not five-dimensional. First cited in \S\ref{subsec:hist-triad}.

\bibitem[Martin \& Russell(2003)]{MartinRussell2003}
\textbf{W.~Martin and M.~J.~Russell.}
\emph{On the origins of cells: a hypothesis for the evolutionary transitions from abiotic geochemistry to chemoautotrophic prokaryotes},
\emph{Philosophical Transactions of the Royal Society B} \textbf{358} (2003) 59--85.
\par\noindent\textbf{Relevance.}
Places the origin of cells at submarine alkaline hydrothermal systems, where a natural proton gradient across a thin inorganic barrier supplies the chemiosmotic drive that all extant life still uses. In the present cascade this is the setting for the layer-7 bioenergetic anchor rather than for the chemistry floor, and the Part cites it here to make the inheritance explicit: the framework adopts the geochemical setting from this literature and derives the energy \emph{quantum} $\QB{}$, not the site. First cited in \S\ref{sec:chem-leading-theories}.

\bibitem[Maxwell(1865)]{Maxwell1865}
\textbf{J.~C.~Maxwell.}
\emph{A dynamical theory of the electromagnetic field},
\emph{Philosophical Transactions of the Royal Society of London} \textbf{155} (1865) 459--512.
\par\noindent\textbf{Relevance.}
The founding paper of classical electromagnetism, in which Maxwell unified electricity, magnetism, and light into a single field-theoretic structure. His equations introduced the field as a primary geometric object on space-time, anticipating by half a century Einstein's promotion of space-time itself to a dynamical variable. In the GG-Theory program, Maxwell's electromagnetism is recovered as the boundary projection of the U(1) sector of the bulk gauge group, with the photon mass forced to zero by the unbroken Wilson-Hosotani commutant. First cited in \S\ref{sec:five-revolutions}.

\bibitem[Miller(1953)]{Miller1953}
\textbf{S.~L.~Miller.}
\emph{A production of amino acids under possible primitive Earth conditions},
\emph{Science} \textbf{117} (1953) 528--529.
\par\noindent\textbf{Relevance.}
The founding experiment of prebiotic chemistry: an electric discharge through methane, ammonia, hydrogen and water produced glycine, alanine and several other amino acids within a week. It has never been overturned, and the present Part opens with it because the framework's honesty at layers four to six depends on stating first what the field already has. The framework does not claim to improve on Miller's result; it claims that the selection from Miller's mixture is what remains unexplained. First cited in \S\ref{sec:chem-leading-theories}.

\bibitem[Monnier and Moore(2019)]{MonnierMoore2019}
\textbf{S.~Monnier and G.~W.~Moore.}
\emph{Remarks on the Green--Schwarz terms of six-dimensional supergravity theories},
\emph{Communications in Mathematical Physics} \textbf{372} (2019) 963--992.
\par\noindent\textbf{Relevance.}
Developed the modern differential-cohomology framework for global anomaly cancellation in six-dimensional supergravity, going beyond the local (perturbative) Green--Schwarz mechanism to include topological (non-perturbative) effects. The BI-6 global anomaly proof uses their technology: the Wu--Chern--Simons counterterm, the integral lift of the characteristic class, and the automatic-admissibility theorem are all expressed in the Monnier--Moore language. Extending their methods to the non-supersymmetric GG-6 setting is one of the open problems identified in the present paper. First cited in \S\ref{subsec:inheritance-A3}.

\bibitem[Nakahara(2003)]{Nakahara2003}

\textbf{Nakahara, M. (2003).}
\textit{Geometry, Topology and Physics} (2nd ed.).
Institute of Physics Publishing.

\par\noindent\textbf{Relevance.} Nakahara's text is a widely used bridge between modern geometry/topology and physical gauge theories.
It supports AX-2's emphasis that gauge and geometric data are naturally expressed as connections and curvatures, and it provides background for topological routes to quantization. First cited in \S\ref{app:ax2-expanded}.

\bibitem[Newton(1687)]{Newton1687}
\textbf{I.~Newton.}
\emph{Philosophi\ae{} Naturalis Principia Mathematica}.
London (1687).
\par\noindent\textbf{Relevance.}
Newton's \emph{Principia} established the paradigm that the motion of physical bodies is governed by universal mathematical laws. The inverse-square law of gravity unified terrestrial and celestial mechanics for the first time, showing that the same equation describes an apple falling from a tree and the Moon orbiting the Earth. In the GG-6 framework, Newton's revolution is the first of three scientific revolutions; the present paper argues that the Arisaka Equation extends Newton's program from closed mechanical systems to all open systems, including life and consciousness. First cited in \S\ref{sec:eight-test-horizon}.

\bibitem[Neyman \& Pearson(1933)]{NeymanPearson1933}

\textbf{Neyman, J., \& Pearson, E.~S. (1933).}
On the problem of the most efficient tests of statistical hypotheses.
\emph{Philosophical Transactions of the Royal Society A}, \emph{231}, 289--337.
\url{https://doi.org/10.1098/rsta.1933.0009}
\par\noindent\textbf{Relevance.} Neyman and Pearson formalized hypothesis testing and error control, which underlies how modern sciences define and enforce rejection criteria. We cite it as historical/statistical background for POS-7 (MF), where every claim must declare an explicit kill criterion with controlled error rates or equivalent auditing logic. First cited in \S\ref{app:examples}.

\bibitem[Nielsen \& Chuang(2010)]{NielsenChuang2010}

\textbf{Nielsen, M.~A., \& Chuang, I.~L. (2010).}
\textit{Quantum Computation and Quantum Information} (10th Anniversary ed.).
Cambridge University Press.
\url{https://doi.org/10.1017/CBO9780511976667}
\par\noindent\textbf{Relevance.} This book codifies the operational viewpoint of quantum theory (states, channels, POVMs) that is closest to the \GGTheory{} demand that measurement is a defined map rather than a philosophical add-on. We cite it to support the operational language used in motivating \OGGEP{} and POS-7. First cited in \S\ref{app:ax1-expanded}.

\bibitem[Noether(1918)]{Noether1918}
\textbf{E.~Noether.}
Invariante Variationsprobleme.
\emph{Nachrichten der K\"oniglichen Gesellschaft der Wissenschaften zu G\"ottingen} (1918) 235--257.
\par\noindent\textbf{Relevance.}
Noether's theorem --- that every continuous symmetry of a physical system implies a conserved quantity --- is the mathematical foundation of modern physics. In the GG-6 framework, the four gauge symmetries ($\SUfive$, $\UoneB$, $\UoneQi$, and local Lorentz) each imply a conserved current, and the BI-6 identity locks these currents into the Arisaka Equation. Noether's theorem is the reason that gauge locking implies current locking. First cited in \S\ref{sec:qgi}.

\bibitem[Oba et al.(2023)]{Oba2023}
\textbf{Y.~Oba, T.~Koga, Y.~Takano, et al.}
\emph{Uracil in the carbonaceous asteroid (162173) Ryugu},
\emph{Nature Communications} \textbf{14} (2023) 1292, \url{https://doi.org/10.1038/s41467-023-36904-3}.
\par\noindent\textbf{Relevance.}
Identifies uracil --- one of the four RNA bases --- together with nicotinic acid in the pristine returned samples of asteroid Ryugu, in material that never passed through the terrestrial atmosphere or biosphere. We cite it as the cleanest available demonstration that nucleobase delivery is real, and, in the same note, to correct an earlier statement in this cohort that attributed ribose to the Ryugu samples; the Ryugu detection is uracil, and the meteoritic ribose is Murchison and NWA~801 (\citealp{Furukawa2019}). First cited in \S\ref{sec:chem-leading-theories}.

\bibitem[O'Keefe and Dostrovsky(1971)]{OKeefeDostrovsky1971}
\textbf{J.~O'Keefe and J.~Dostrovsky.}
\emph{The hippocampus as a spatial map},
\emph{Brain Research} \textbf{34} (1971) 171--175.
\par\noindent\textbf{Relevance.}
Discovered hippocampal place cells --- neurons that fire when an animal occupies a specific location in its environment, regardless of the direction it faces. This discovery (Nobel Prize 2014, shared with the Mosers) revealed the neural substrate of the allocentric spatial map that the GG-6 framework identifies as the Inner Hologram at the $\QCI \sim 8$ level. Place cells implement the latent manifold $\cZ$ of Definition~\ref{def:ih}: a world-centered coordinate system that factors out self-motion and allows the animal to navigate, remember, and plan. First cited in \S\ref{sec:thm-allocentric-map}.

\bibitem[Onsager \& Machlup(1953)]{OnsagerMachlup1953}

\textbf{Onsager, L., \& Machlup, S.\ (1953).}
Fluctuations and irreversible processes.
\textit{Physical Review, 91}(6), 1505--1512.
DOI: \href{https://doi.org/10.1103/PhysRev.91.1505}{10.1103/PhysRev.91.1505}.
\par\noindent\textbf{Relevance.} Introduces an action-like functional for stochastic trajectories near equilibrium, providing a bridge
between variational principles and stochastic thermodynamics. It supports POS-2 (\UEP{}) in the open-system regime by
showing how ``least action'' persists as a most-probable-path principle for noisy dynamics. First cited in \S\ref{app:examples}.

\bibitem[Particle Data Group(2024)]{PDG}
\textbf{S.~Navas et al.\ (Particle Data Group).}
Review of Particle Physics.
\emph{Physical Review D} \textbf{110} (2024) 030001.
\par\noindent\textbf{Relevance.}
The biennial PDG compilation is the authoritative empirical reference for all Standard-Model parameters and decay measurements. In the GG-Theory framework, every Pull-table entry of \AEleven{} Part~\ref{part:results} and every codebook value of \ANine{} is compared to its PDG-listed measurement and uncertainty; the framework's empirical agreement is the agreement of the cascade-derived prediction with the PDG-listed observation at each Pull-table row. First cited in \S\ref{sec:cascade-codebook-table}.

\bibitem[Pearce et al.(2017)]{PearceEtAl2017}
\textbf{B.~K.~D.~Pearce, R.~E.~Pudritz, D.~A.~Semenov, and T.~K.~Henning.}
\emph{Origin of the RNA world: the fate of nucleobases in warm little ponds},
\emph{Proceedings of the National Academy of Sciences} \textbf{114} (2017) 11327--11332.
\par\noindent\textbf{Relevance.}
Models nucleobase delivery by meteorites and interplanetary dust against seepage and hydrolysis losses, and concludes that intermittently drying ponds --- not the open ocean --- are where the concentrations required for polymerization are reached. We cite it as the quantitative form of the wet--dry route, and as the source the present Part draws on for the \GRIPout{} event of the layer-5 to layer-6 transition, which the cascade names and does not derive. First cited in \S\ref{sec:chem-leading-theories}.

\bibitem[Penrose(1994)]{Penrose1994}
\textbf{R.~Penrose.}
\emph{Shadows of the Mind: A Search for the Missing Science of Consciousness}.
Oxford University Press (1994).
\par\noindent\textbf{Relevance.}
Identified the Mathematics--Physics--Mind triad: Mathematics governs Physics, Physics gives rise to Mind, Mind discovers Mathematics. Penrose argued that the cycle cannot be closed within conventional physics and speculated that quantum gravity plays a role in consciousness. The GG-6 framework closes Penrose's triad explicitly and quantitatively: the OGN-5 integers determine all physical constants (Mathematics $\to$ Physics), the DGI--GSSB--GA cascade produces consciousness (Physics $\to$ Mind), and the EIC is the brain's progressive reconstruction of the 6D geometry (Mind $\to$ Mathematics). No quantum-gravity speculation is needed. First cited in \S\ref{subsec:penrose}.

\bibitem[Penrose(2010)]{Penrose2010}
\textbf{R.~Penrose.}
\emph{Cycles of Time: An Extraordinary New View of the Universe}.
Bodley Head, London (2010).
\par\noindent\textbf{Relevance.} The statement of conformal cyclic cosmology (CCC). Cited in \S\ref{ssec:comparison-ccc}: the cascade's cosmological backdrop is \ATen{}'s scale-running expansion and does not require cyclic aeons. First cited in \S\ref{sec:statement-comparison}.

\bibitem[Philip \& Freeland(2011)]{PhilipFreeland2011}
\textbf{G.~K.~Philip and S.~J.~Freeland.}
\emph{Did evolution select a nonrandom ``alphabet'' of amino acids?},
\emph{Astrobiology} \textbf{11} (2011).
\par\noindent\textbf{Relevance.}
Compares the coded twenty against large ensembles of random sets drawn from the plausible prebiotic alternatives, on the axes of size, charge and hydrophobicity, and finds the coded set covers that space better than the overwhelming majority of the alternatives. It is the most quantitative statement available that the twenty are selected rather than arbitrary. The present Part cites it in the beat on what the field leaves unsolved: a demonstration of non-randomness is not a derivation of the cardinality, and the distinction is the one layer~5 turns on. First cited in \S\ref{sec:chem-leading-theories}.

\bibitem[Planck(1900)]{Planck1900}
\textbf{M.~Planck.}
Zur Theorie des Gesetzes der Energieverteilung im Normalspectrum.
\emph{Verhandlungen der Deutschen Physikalischen Gesellschaft} \textbf{2} (1900) 237--245.
\par\noindent\textbf{Relevance.}
Planck's introduction of the energy quantum $E = h\nu$ launched quantum physics and established the principle that nature operates in discrete units. In the GG-6 framework, the biological energy quantum $N_A\QB = 15.0$~kJ/mol is the descendant of Planck's idea: the minimum energy packet for any gauge-locked biological event, derived not from fitting blackbody spectra but from the OGN-5 integers fixed by geometric consistency (BI-6). First cited in \S\ref{sec:five-revolutions}.

\bibitem[Planck Collaboration(2020)]{Planck2018}
\textbf{N.~Aghanim et al.\ (Planck Collaboration).}
Planck 2018 results.\ VI.\ Cosmological parameters.
\emph{Astronomy \& Astrophysics} \textbf{641} (2020) A6.
\par\noindent\textbf{Relevance.}
The Planck 2018 cosmological parameter compilation provides the headline observational anchor for all of \ATen{}: $\Omega_m = 0.3153 \pm 0.0073$ vs predicted $1/\pi \approx 0.3183$; $\Omega_b h^2 = 0.0224 \pm 0.0001$ consistent with $1/(2\pi^2)$; $H_0^{\rm CMB} = 67.4 \pm 0.5$ km/s/Mpc matching the early-stage branch of \AEleven{} Pull test \#3; recombination redshift $z_{\rm dec} = 1089.92 \pm 0.25$ as the operational anchor of Pull test \#5. The agreement of the $\pi$-locked inventory of \ATen{} with the Planck values to within $\sim 1\%$ at zero free continuous parameters is the framework's principal cosmological success. First cited in \S\ref{sec:eight-test-pull-table}.

\bibitem[Popper(1959)]{Popper1959}
\textbf{K.~Popper.}
\emph{The Logic of Scientific Discovery}.
Hutchinson, London (1959).
\par\noindent\textbf{Relevance.}
Popper established falsifiability as the demarcation criterion between science and non-science: a theory is scientific only if it makes predictions that can, in principle, be shown to be false. The GG-6 framework takes Popper's criterion seriously by listing specific falsifiable predictions (proton stability, KK resonance at 3.85~TeV, $N_A\QB = 15.0$~kJ/mol, 4-base DNA) and by constitutionalizing falsifiability as POS-7 (Measurement and Falsifiability) within the \GGConstitution{}. First cited in \S\ref{app:examples}.

\bibitem[Powner et al.(2009)]{PownerEtAl2009}
\textbf{M.~W.~Powner, B.~Gerland, and J.~D.~Sutherland.}
\emph{Synthesis of activated pyrimidine ribonucleotides in prebiotically plausible conditions},
\emph{Nature} \textbf{459} (2009) 239--242.
\par\noindent\textbf{Relevance.}
Solves, by circumventing rather than forcing it, the step that had blocked prebiotic nucleotide synthesis for thirty years: instead of making the sugar and the base separately and joining them, the route builds the activated ribonucleotide along a path on which that junction never has to be made. It is the strongest single experimental result in the field since Miller, and the present Part cites it to fix the standard against which the layer-5 and layer-6 rows should be read --- both of which assert closure without yet exhibiting a route. First cited in \S\ref{sec:chem-leading-theories}.

\bibitem[Prigogine(1977)]{Prigogine1977}
\textbf{I.~Prigogine.}
\emph{Self-Organization in Non-Equilibrium Systems}.
Wiley, New York (1977).
\par\noindent\textbf{Relevance.}
Pioneered the theory of dissipative structures --- systems that maintain order far from equilibrium by continuously dissipating energy. Living cells, convection patterns, and chemical oscillators are all dissipative structures in Prigogine's sense. The GG-6 framework subsumes Prigogine's theory: the SeOS (Selfish Open System) is a dissipative structure that additionally satisfies the $\QGL$ constraint, and the GSSB cascade describes the progressive complexification of dissipative structures from prebiotic chemistry to consciousness, driven by the Qi-EP principle. First cited in \S\ref{sec:grip-operational-anatomy}.

\bibitem[Quiroga et al.(2005)]{QuirogaEtAl2005}
\textbf{R.~Quian Quiroga, L.~Reddy, G.~Kreiman, C.~Koch, I.~Fried.}
Invariant visual representation by single neurons in the human brain.
\emph{Nature} \textbf{435} (2005) 1102--1107.
\par\noindent\textbf{Relevance.}
Single-unit recordings from human medial temporal lobe identified ``concept cells'' that fire selectively and invariantly to specific people or objects (the ``Jennifer Aniston neuron''). In the GG-Theory framework, concept cells are the highest-resolution boundary readout of stabilized \IH{} branches on the allocentric-map stratum — a precursor to \T{-Allocentric-Map} of \AEleven{} Part~\ref{part:cascade} at the human hippocampal stratum. First cited in \S\ref{sec:thm-allocentric-map}.

\bibitem[Randall and Sundrum(1999)]{RandallSundrum1999}
\textbf{L.~Randall and R.~Sundrum.}
\emph{A large mass hierarchy from a small extra dimension},
\emph{Physical Review Letters} \textbf{83} (1999) 3370--3373.
\par\noindent\textbf{Relevance.}
Proposed a extra-dimension model in which the hierarchy between the Planck scale and the electroweak scale is generated by an exponential warp factor in a five-dimensional Anti-de Sitter bulk. The GG-6 geometry inherits this warp mechanism but extends it to six dimensions, adds the compact $S^1_\theta$ direction, and replaces the fine-tuned brane tensions with the constitutionally locked BI-6 coefficients. The holographic radial coordinate $r = \ln(\EGUT/\mu)$ is the GG-6 version of the Randall--Sundrum warp factor. First cited in \S\ref{subsec:hist-bulk}.

\bibitem[Reuter(1998)]{Reuter1998}
\textbf{M.~Reuter.}
Nonperturbative evolution equation for quantum gravity.
\emph{Physical Review D} \textbf{57} (1998) 971--985.
\par\noindent\textbf{Relevance.} The functional-renormalization-group realization of Weinberg's asymptotic-safety scenario. Cited in \S\ref{ssec:comparison-asymptotic-safety}: \GGTheory{} instead absorbs gravity into the scale profile of the 6D bulk rather than quantizing it as a separate field. First cited in \S\ref{sec:statement-comparison}.

\bibitem[Rissanen(1978)]{Rissanen1978}

\textbf{Rissanen, J. (1978).}
Modeling by shortest data description.
\emph{Automatica}, \emph{14}(5), 465--471.
\url{https://doi.org/10.1016/0005-1098(78)90005-5}
\par\noindent\textbf{Relevance.} Rissanen's minimum description length principle is another quantitative implementation of Occam's razor. We cite it as methodological background for POS-6 (\OGN{}): the ``minimum freedom'' doctrine must also eliminate hidden continuous choices in parameterization and normalization. First cited in \S\ref{app:ax4-expanded}.

\bibitem[Rovelli and Smolin(1990)]{RovelliSmolin1990}
\textbf{C.~Rovelli and L.~Smolin.}
Loop space representation of quantum general relativity.
\emph{Nuclear Physics B} \textbf{331} (1990) 80--152.
\par\noindent\textbf{Relevance.} The loop/spin-network representation of LQG built on the Ashtekar variables. Cited in \S\ref{ssec:comparison-lqg} alongside \citet{Ashtekar1986}. First cited in \S\ref{sec:statement-comparison}.

\bibitem[Russell(1921)]{Russell1921}
\textbf{B.~Russell.}
\emph{The Analysis of Mind}.
George Allen \& Unwin (1921).
\par\noindent\textbf{Relevance.}
Russell's most extensive development of neutral monism: the world is one kind of stuff, with mental and physical descriptions as two perspectives on the same underlying neutral substrate. In the present framework, the structural reading of consciousness as the EIC closure of \ASeven{} \T{-EIC} is closest in conventional taxonomy to Russell's neutral monism, but the framework adds a precise geometric description of the loop that constitutes subjective experience --- a description Russell could not supply within the 1921 technical apparatus. First cited in \S\ref{sec:reader-guide-cognitive}.

\bibitem[Schr\"odinger(1944)]{Schrodinger1944}
\textbf{E.~Schr\"odinger.}
\emph{What is Life? The Physical Aspect of the Living Cell}.
Cambridge University Press (1944).
\par\noindent\textbf{Relevance.}
Asked ``What is Life?'' from the perspective of a theoretical physicist, identifying the ``aperiodic crystal'' (DNA, before its structure was known) as the key to biological order. Schr\"odinger argued that living matter obeys the laws of physics but exhibits a kind of order (``negentropy'') that demands explanation. The GG-6 framework answers Schr\"odinger's question: the specific molecular architecture of life --- DNA, ATP, the genetic code --- is derived from the BI-6 geometric consistency identity through the OGN-5 integers, and the negentropy of living systems is maintained by the positive maintenance margin of the SeOS classification. First cited in \S\ref{sec:intro-why}.

\bibitem[Seifert(2012)]{Seifert2012}
\textbf{U.~Seifert.}
Stochastic thermodynamics, fluctuation theorems, and molecular machines.
\emph{Reports on Progress in Physics} \textbf{75} (2012) 126001.
\par\noindent\textbf{Relevance.}
Seifert's review establishes stochastic thermodynamics as the framework for analyzing the energetics of molecular machines --- precisely the systems whose energy budget is governed by $\QB$ in the GG-6 framework. ATP synthase, the ribosome, and DNA polymerase are molecular machines whose single-molecule thermodynamics can be analyzed using Seifert's tools, and the $\QGL$ constraint provides the additional structure that connects their energy budgets to the geometric consistency structure (BI-6). First cited in \S\ref{sec:step1-ssb-attractor}.

\bibitem[Shannon(1948)]{Shannon1948}
\textbf{C.~E.~Shannon.}
\emph{A mathematical theory of communication},
\emph{Bell System Technical Journal} \textbf{27} (1948) 379--423.
\par\noindent\textbf{Relevance.}
Founded information theory by defining the bit as the fundamental unit of information and establishing the entropy $H = -\sum p_i \log p_i$ as the measure of uncertainty. In the GG-6 framework, Shannon's information theory provides the mathematical language for the $J_{Qi}$ current: the DNA information content $I_{\mathrm{DNA}} = 2$~bits per base pair is stated in Shannon's framework, and the $\QGL$ constraint locks Shannon information to physical energy through the Arisaka Equation. The gauged $\UoneQi$ elevates Shannon's abstract concept to a conserved physical quantum number. First cited in \S\ref{sec:qb-anchor}.

\bibitem[Smolin(1997)]{Smolin1997}
\textbf{L.~Smolin.}
\emph{The Life of the Cosmos}.
Oxford University Press (1997).
\par\noindent\textbf{Relevance.}
Smolin proposed cosmological natural selection: universes whose parameters favor black-hole production give rise to offspring universes with slightly varied parameters, and the parameters we observe are the fixed point of a selection process. GG-Theory dissolves the selection by deriving the parameters from the constitution; no offspring, no selection, only one universe permitted by the geometry. First cited in \S\ref{sec:t-origin}.

\bibitem[Soai et al.(1995)]{SoaiEtAl1995}
\textbf{K.~Soai, T.~Shibata, H.~Morioka, and K.~Choji.}
\emph{Asymmetric autocatalysis and amplification of enantiomeric excess of a chiral molecule},
\emph{Nature} \textbf{378} (1995) 767--768.
\par\noindent\textbf{Relevance.}
The laboratory realization of Frank's mechanism: an alkylation in which the product catalyzes its own formation, amplifying an enantiomeric excess of a fraction of a percent to near-complete homochirality over a few cycles. It converts amplification from a model into an observed chemical fact, and thereby sharpens the remaining question --- the one the present Part is careful about at layer~5 --- to the origin of the initial sign rather than of the final excess. First cited in \S\ref{sec:chem-leading-theories}.

\bibitem[Spinoza(1677)]{Spinoza1677}
\textbf{B.~Spinoza.}
\emph{Ethica Ordine Geometrico Demonstrata} (1677).
\par\noindent\textbf{Relevance.}
In the \emph{Ethics}, Spinoza identified God with Nature (\emph{Deus sive Natura}), arguing that there is only one infinite substance of which mind and matter are two attributes. The GG-6 framework provides the twenty-first-century mathematical realization: \emph{Deus sive Geometria} --- the six-dimensional geometry is the single substance from which everything follows. Mind ($J_{Qi}$) and matter ($J_B$) are not two separate substances but two projections of the single $B_2$ field through the OGGEP dictionary, exactly as Spinoza's two attributes are aspects of one substance. First cited in \S\ref{subsec:hist-constitution}.

\bibitem[Stinespring(1955)]{Stinespring1955}

\textbf{Stinespring, W.~F. (1955).}
Positive functions on $C^\ast$-algebras.
\textit{Proceedings of the American Mathematical Society}, \textit{6}(2), 211--216.
\url{https://doi.org/10.1090/S0002-9939-1955-0069403-2}
\par\noindent\textbf{Relevance.} Stinespring's dilation theorem is the mathematical backbone of quantum channels: CP maps arise from unitary evolution on an enlarged space. We cite it to justify how \OGGEP{} can treat open-system boundary dynamics as fundamental while remaining consistent with microscopic unitarity in the bulk. First cited in \S\ref{app:ax1-expanded}.

\bibitem[Strawson(2006)]{Strawson2006}
\textbf{Strawson, G. (2006).}
Realistic monism: Why physicalism entails panpsychism.
\emph{Journal of Consciousness Studies}, \textbf{13}(10--11), 3--31.
\par\noindent\textbf{Relevance.}
Strawson's argument that physicalism (taken seriously) entails panpsychism or Russellian neutral monism. The framework adopts the Russellian-neutral-monist branch (not the panpsychist one) because experience is restricted to layer-17 \EIC{} closures, not distributed across all physical configurations. Strawson's article is the canonical contemporary statement of the position. First cited in \S\ref{sec:reader-guide-cognitive}.

\bibitem[Streater \& Wightman(1964)]{StreaterWightman1964}

\textbf{Streater, R.~F., \& Wightman, A.~S. (1964).}
\textit{PCT, Spin and Statistics, and All That}.
W.~A. Benjamin.
\par\noindent\textbf{Relevance.} This book codifies rigorous axioms for relativistic QFT, including microcausality and locality. We cite it as a foundational reference for AX-1's operator-algebraic encoding of locality and causality. First cited in \S\ref{app:ax1-expanded}.

\bibitem[Susskind(1995)]{Susskind1995}
\textbf{L.~Susskind.}
The world as a hologram.
\emph{Journal of Mathematical Physics} \textbf{36} (1995) 6377--6396.
\par\noindent\textbf{Relevance.}
Susskind formalized the holographic principle and connected it to black hole thermodynamics, arguing that the maximum entropy of any region of space is proportional to its boundary area rather than its volume. In GG-6, the holographic principle is realized concretely: the OGGEP dictionary maps bulk states to boundary observables, and the information content of any boundary region is bounded by the $\UoneQi$ charge it carries. First cited in \S\ref{subsec:hist-triad}.

\bibitem[Susskind(2003)]{Susskind2003}
\textbf{L.~Susskind.}
The anthropic landscape of string theory.
\emph{arXiv:hep-th/0302219} (2003).
\par\noindent\textbf{Relevance.} The argument that the string landscape together with anthropic selection accounts for the constants. Cited in \S\ref{ssec:comparison-multiverse} and \S\ref{ssec:comparison-string-landscape}: \GGTheory{} replaces selection with geometric uniqueness. First cited in \S\ref{sec:statement-comparison}.

\bibitem[Symanzik(1970)]{Symanzik1970}

\textbf{Symanzik, K.\ (1970).}
Small distance behaviour in field theory and power counting.
\textit{Communications in Mathematical Physics, 18}(3), 227--246.
DOI: \href{https://doi.org/10.1007/BF01649434}{10.1007/BF01649434}.
\par\noindent\textbf{Relevance.} A foundational development of renormalization and scaling in quantum field theory.
Together with Callan and Wilson, it underpins the modern RG view used by POS-5 and motivates the identification of a
log-scale coordinate with flow in the space of effective theories. First cited in \S\ref{app:examples}.

\bibitem[Ta et al.(2022)]{Vision7D3}
\textbf{B.~Ta, M.~E.~M.~Silva, K.~Bartlett, et al., A.~P.~Blaisdell, and K.~Arisaka.}
\emph{Visual Perception of 3D Space and Shape by 7D Frame Translation in Time, Part~\ref{part:cascade}: 2D Shape Recognition by Log-Scaling},
bioRxiv 2022.03.01.482004 (2022).
\textbf{Relevance.} Experiment establishing reaction time linear in $\log_2$(size) and a canonical $\sim 8^\circ$ stored face size --- the log-scaling axis of the 7D frame. Author's own preprint (Asserted/INT). First cited in \S\ref{sec:per-test-commentary}.

\bibitem[Taube et al.(1990)]{TaubeMullerRanck1990}
\textbf{J.~S.~Taube, R.~U.~Muller, J.~B.~Ranck Jr.}
Head-direction cells recorded from the postsubiculum in freely moving rats.
I.~Description and quantitative analysis.
\emph{Journal of Neuroscience} \textbf{10} (1990) 420--435.
\par\noindent\textbf{Relevance.}
Taube, Muller and Ranck's foundational paper identified head-direction cells as a population code for allocentric heading in rodent postsubiculum, predating the \emph{Drosophila} ring-attractor confirmation by 27 years. In the GG-Theory framework, the mammalian head-direction system and the insect ellipsoid body are realizations of the same layer-15 GGA on convergent neural substrates — the universal-identity insight applied to heading representation. First cited in \S\ref{sec:confidence-legend}.

\bibitem[Tegmark(2014)]{Tegmark2014}
\textbf{M.~Tegmark.}
\emph{Our Mathematical Universe: My Quest for the Ultimate Nature of Reality}.
Knopf (2014).
\par\noindent\textbf{Relevance.}
Tegmark's Mathematical Universe Hypothesis (MUH) proposed that physical reality \emph{is} a mathematical structure, with all consistent mathematical structures equally real. GG-Theory rejects the second clause but inherits the first: physical reality is the unique self-consistent geometric structure permitted by the GG-Constitution. Where Tegmark's MUH leads to a multiverse of all structures, GG-Theory's constitution permits exactly one. First cited in \S\ref{sec:t-origin}.

\bibitem[Teilhard de Chardin(1955)]{Teilhard1955}
\textbf{P.~Teilhard de Chardin.}
\emph{The Phenomenon of Man}.
Harper \& Row (1955).
\par\noindent\textbf{Relevance.}
Proposed that the universe evolves toward an ``Omega Point'' of maximum complexity and consciousness, with the ``noosphere'' (the sphere of human thought) as the latest stage. The GG-6 framework rehabilitates Teilhard's core insight --- the directionality of cosmic evolution toward consciousness --- while providing the quantitative mechanism he lacked. The $\QCI$ trajectory from 0 (inert matter) to 10 (conscious beings who comprehend the geometry) is a precise version of Teilhard's complexification, and the EIC convergence theorem is a mathematical formulation of the Omega Point. First cited in \S\ref{sec:recursive-uniqueness-summit}.

\bibitem[THooft(1980)]{THooftNaturalness1980}

\textbf{'t~Hooft, G. (1980).}
Naturalness, chiral symmetry, and spontaneous chiral symmetry breaking.
In G. 't~Hooft et al. (Eds.), \textit{Recent Developments in Gauge Theories} (pp.~135--157).
Plenum Press.
\url{https://doi.org/10.1007/978-1-4684-7571-5_9}
\par\noindent\textbf{Relevance.} Naturalness arguments connect small parameters to enhanced symmetry. We cite this as historical background for \MSMF{}: the \GGConstitution{} pushes the symmetry$\leftrightarrow$parameter connection to a stronger law, demanding maximal symmetry and forbidding non-protected dials. First cited in \S\ref{app:ax4-expanded}.

\bibitem['t~Hooft(1993)]{tHooft1993}
\textbf{G.~'t~Hooft.}
Dimensional reduction in quantum gravity.
In \emph{Salamfest} (1993) 284--296. \texttt{arXiv:gr-qc/9310026}.
\par\noindent\textbf{Relevance.}
't~Hooft proposed that a theory of quantum gravity in $D$ dimensions can be encoded on its $(D-1)$-dimensional boundary, with one degree of freedom per Planck area. This holographic principle is a structural ingredient of the GG-6 framework: the six-dimensional bulk projects onto the four-dimensional boundary through the OGGEP dictionary, and the Outer Hologram (OH) is the formal realization of 't~Hooft's idea applied to all physical measurements. First cited in \S\ref{subsec:hist-triad}.

\bibitem['t~Hooft and Veltman(1972)]{tHooftVeltman1972}
\textbf{G.~'t~Hooft and M.~Veltman.}
Regularization and renormalization of gauge fields.
\emph{Nuclear Physics B} \textbf{44} (1972) 189--213.
\par\noindent\textbf{Relevance.}
The proof that non-Abelian gauge theories are renormalizable made the Standard Model a consistent quantum theory and earned the 1999 Nobel Prize. The geometric consistency verification used throughout the GG-6 program builds on the perturbative framework of 't~Hooft and Veltman \citep{tHooftVeltman1972}; the one-loop anomaly polynomial $I_8$ is computed using their regularization methods. First cited in \S\ref{sec:comparison}.

\bibitem[Tononi(2004)]{Tononi2004}
\textbf{G.~Tononi.}
\emph{An information integration theory of consciousness},
\emph{BMC Neuroscience} \textbf{5} (2004) 42.
\par\noindent\textbf{Relevance.}
Proposed Integrated Information Theory (IIT), which defines consciousness in terms of integrated information ($\Phi$) --- a quantity measuring how much a system is simultaneously differentiated and integrated. The GG-6 framework agrees with IIT that consciousness is graded, but derives its metric ($\QCI$) from the $\QGL$ constraint rather than from information-theoretic axioms. The key difference is predictive scope: IIT does not predict specific biological quantities ($\QB$, the genetic code, the theta frequency), while the GG-6 framework does. First cited in \S\ref{sec:eic}.

\bibitem[Tononi(2008)]{Tononi2008}
\textbf{G.~Tononi.}
Consciousness as integrated information: a provisional manifesto.
\emph{Biological Bulletin} \textbf{215} (2008) 216--242.
\par\noindent\textbf{Relevance.}
The 2008 update to Integrated Information Theory (IIT~2.0) formalizes consciousness as the maximally irreducible cause--effect structure $\Phi$ of a physical system. In the GG-Theory framework, IIT's $\Phi$ measures the boundary-projected information integration of an \EIC{} closure; \QCI{} of \AEleven{} Part~\ref{part:cascade} is a complementary cascade-depth measure that captures the layer-by-layer staging that IIT does not address. First cited in \S\ref{sec:statement-comparison}.

\bibitem[Trifonov(2000)]{Trifonov2000}
\textbf{E.~N.~Trifonov.}
\emph{Consensus temporal order of amino acids and evolution of the triplet code},
\emph{Gene} \textbf{261} (2000) 139--151.
\par\noindent\textbf{Relevance.}
Combines forty independent criteria --- prebiotic abundance, codon structure, thermostability, and others --- into a single consensus ranking of the order in which the amino acids entered the genetic code. It is the most-cited attempt to explain the composition of the coded set historically rather than functionally. We cite it in the survey preceding the layer-5 theorem, alongside the observation that a temporal ordering of the twenty is not a derivation of the number twenty. First cited in \S\ref{sec:chem-leading-theories}.

\bibitem[Turner-Evans et~al.(2020)]{TurnerEvans2020}
\textbf{D.~Turner-Evans et~al.}
The neuroanatomical ultrastructure and function of a biological ring attractor.
\emph{Neuron} \textbf{108} (2020) 145--163.
\par\noindent\textbf{Relevance.}
Turner-Evans and colleagues mapped the complete ring-attractor circuit in the Drosophila central complex, showing how a one-dimensional circular manifold of persistent neural activity encodes the fly's heading direction. This circuit is the neural implementation of the $\QCI \sim 6$ level of the Qi-HAL architecture: a one-dimensional allocentric map maintained by recurrent excitation and global inhibition, exactly as the framework predicts for organisms at this stage of the GSSB cascade. First cited in \S\ref{sec:ih}.

\bibitem[van~Inwagen(1983)]{vanInwagen1983}
\textbf{van~Inwagen, P. (1983).}
\emph{An Essay on Free Will}.
Oxford University Press.
\par\noindent\textbf{Relevance.}
The canonical statement of the Consequence Argument: that determinism (in the standard sense) is incompatible with the freedom of will required for moral responsibility. The framework's response is compatibilist: micro-trajectories are forced only at the layer-architecture level; the conscious substrate's deliberative trajectory remains open within the layer-17 attractor's basin, which is the kind of freedom the framework defends as worth defending. First cited in \S\ref{sec:closure-careful-word}.

\bibitem[Vilenkin(1983)]{Vilenkin1983}
\textbf{A.~Vilenkin.}
Birth of inflationary universes.
\emph{Physical Review D} \textbf{27} (1983) 2848.
\par\noindent\textbf{Relevance.} An early quantum-cosmological route to inflationary sub-universes. Cited in \S\ref{ssec:comparison-multiverse} among the multiverse proposals the cascade renders unnecessary. First cited in \S\ref{sec:statement-comparison}.

\bibitem[von Neumann(1932)]{vonNeumann1932}
\textbf{J.~von Neumann.}
\emph{Mathematische Grundlagen der Quantenmechanik}.
Springer, Berlin (1932).
\par\noindent\textbf{Relevance.}
Von Neumann formalized quantum measurement as a separate postulate (``collapse'') distinct from unitary evolution, creating the measurement problem that has haunted quantum mechanics for a century. AX-3 (OGGEP) resolves this problem constitutionally: measurement is not a separate postulate but a CPTP operation in the bulk-to-boundary dictionary, and decoherence (not collapse) explains the emergence of definite outcomes. First cited in \S\ref{subsec:hist-constitution}.

\bibitem[W\"achtersh\"auser(1988)]{Wachtershauser1988}
\textbf{G.~W\"achtersh\"auser.}
\emph{Before enzymes and templates: theory of surface metabolism},
\emph{Microbiological Reviews} \textbf{52} (1988).
\par\noindent\textbf{Relevance.}
Proposes that life began as a two-dimensional autocatalytic carbon-fixing chemistry bound to the surface of iron--sulfur minerals, requiring neither a container nor a genetic polymer to start. It is the principal rival to the soup-and-replicator picture and the ancestor of the vent-based accounts the cascade adopts at layer~7. The present Part cites it in the survey of routes, and notes that the framework takes no position between the routes: layers four to six claim closure on strata, not a mechanism of assembly. First cited in \S\ref{sec:chem-leading-theories}.

\bibitem[Ward(1950)]{Ward1950}

\textbf{Ward, J.~C. (1950).}
An identity in quantum electrodynamics.
\textit{Physical Review}, \textit{78}, 182.
\url{https://doi.org/10.1103/PhysRev.78.182}
\par\noindent\textbf{Relevance.} Ward's identity is the prototype relation showing how gauge symmetry constrains quantum amplitudes. We cite it in POS-1 and POS-4 to emphasize that gauge invariance is not only a classical redundancy but an admissibility condition that has nontrivial quantum consequences. First cited in \S\ref{app:examples}.

\bibitem[Watson and Crick(1953)]{WatsonCrick1953}
\textbf{J.~D.~Watson and F.~H.~C.~Crick.}
\emph{Molecular structure of nucleic acids: a structure for deoxyribose nucleic acid},
\emph{Nature} \textbf{171} (1953) 737--738.
\par\noindent\textbf{Relevance.}
Determined the double-helical structure of DNA, revealing how genetic information is stored in the sequence of base pairs and replicated through complementary strand separation. The GG-6 framework derives the specific features of this structure from the $\QGL$ constraint: the 4-base alphabet encodes 2 bits per base pair (the unique $\QGL$-optimal solution), the double helix provides the error-correcting redundancy required by the $\QGL$ energy budget, and the $B$-form geometry with supercoiling density $-0.06$ is the minimum-energy configuration consistent with the OGN-5 integers. First cited in \S\ref{sec:intro-why}.

\bibitem[Weber \& Miller(1981)]{WeberMiller1981}
\textbf{A.~L.~Weber and S.~L.~Miller.}
\emph{Reasons for the occurrence of the twenty coded protein amino acids},
\emph{Journal of Molecular Evolution} \textbf{17} (1981) 273--284.
\par\noindent\textbf{Relevance.}
The paper that made the composition of the coded set a scientific question rather than a curiosity, arguing case by case from prebiotic availability, chemical stability under physiological conditions, and functional coverage. Forty-five years later it remains the reference statement of the conventional answer. The present Part cites it first among the accounts of the number twenty, and observes that its form of argument --- these twenty are the ones that work --- is retrospective, which is precisely the gap the layer-5 relation is offered to fill and does not yet close. First cited in \S\ref{sec:chem-leading-theories}.

\bibitem[Weinberg(1967)]{Weinberg1967}
\textbf{S.~Weinberg.}
\emph{A model of leptons},
\emph{Physical Review Letters} \textbf{19} (1967) 1264--1266.
\par\noindent\textbf{Relevance.}
Proposed the electroweak unification of electromagnetic and weak interactions through the $\mathrm{SU}(2)_L \times \mathrm{U}(1)_Y$ gauge theory with spontaneous symmetry breaking, predicting the $W^\pm$ and $Z^0$ bosons. Combined with the work of Glashow and Salam, this earned the 1979 Nobel Prize and established the Standard Model. In GG-6, the electroweak theory emerges as a boundary projection of the $\SUfive$ bulk gauge group after the Hosotani Wilson-line breaking, and the weak mixing angle $\sin^2\theta_W = 25/108$ is derived from the OGN-5 trace ratio rather than fitted to data. First cited in \S\ref{sec:cascade-plain-english}.

\bibitem[Weinberg(1979)]{Weinberg1979}
\textbf{S.~Weinberg.}
Ultraviolet divergences in quantum theories of gravitation.
In \emph{General Relativity: An Einstein Centenary Survey}, S.~W.~Hawking and W.~Israel (eds.), Cambridge University Press (1979) 790--831.
\par\noindent\textbf{Relevance.} The origin of the asymptotic-safety scenario for quantum gravity. Cited in \S\ref{ssec:comparison-asymptotic-safety}. First cited in \S\ref{sec:statement-comparison}.

\bibitem[Weinberg(1995)]{Weinberg1995QFT}

\textbf{Weinberg, S. (1995).}
\textit{The Quantum Theory of Fields} (Vol.~1: Foundations).
Cambridge University Press.

\par\noindent\textbf{Relevance.} Weinberg's axiomatic and symmetry-driven approach to QFT is a key reference for how locality, causality, and gauge invariance constrain admissible theories.
AX-1 is aligned with this tradition but extends it by insisting that the same causal discipline apply to open-system operational evolution and measurement outputs. First cited in \S\ref{app:ax4-expanded}.

\bibitem[Wess and Zumino(1971)]{WessZumino1971}
\textbf{J.~Wess and B.~Zumino.}
\emph{Consequences of anomalous Ward identities},
\emph{Phys.\ Lett.\ B} \textbf{37} (1971) 95.
\par\noindent\textbf{Relevance.} The Wess--Zumino consistency condition that any genuine anomaly must satisfy and that makes the descent formalism well-defined. Cited by \AEleven{} in the Part~\ref{part:inheritance} anomaly-history arc as the step that made the anomaly a cohomological (geometric) object. First cited in \S\ref{subsec:hist-anomaly}.

\bibitem[Weyl(1918)]{Weyl1918}

\textbf{Weyl, H. (1918).}
Gravitation und Elektrizit{\"a}t.
\textit{Sitzungsberichte der K{\"o}niglich Preussischen Akademie der Wissenschaften zu Berlin}, 465--480.
\par\noindent\textbf{Relevance.} Weyl's early gauge ideas represent one of the first attempts to relate geometry and electromagnetism.
While historically distinct from modern Yang--Mills theory, it anchors the long-standing theme that interactions may be geometric in origin, a theme AX-2 makes constitutional. First cited in \S\ref{app:ax2-expanded}.

\bibitem[Wheeler(1983)]{Wheeler1983}
\textbf{J.~A.~Wheeler.}
Law without law.
In \emph{Quantum Theory and Measurement}, eds.~J.~A.~Wheeler and W.~H.~Zurek, Princeton University Press, 1983, pp.~182--213.
\par\noindent\textbf{Relevance.}
Wheeler's ``law without law'' program proposed that the laws of physics themselves emerge from a self-referential observer-participation loop --- the ``it from bit'' principle in its mature 1983 form. This is the direct philosophical predecessor of \T{-Articulation} (\AEleven{} Part~\ref{part:cascade}, layer-18 meta-theorem): Wheeler's self-excited circuit becomes a formal mathematical theorem when the cascade machinery of \GRIP{} and \QiFour{} admissibility supply the missing constructive content. First cited in \S\ref{sec:thm-articulation}.

\bibitem[Wheeler(1990)]{Wheeler1990}
\textbf{J.~A.~Wheeler.}
\emph{Information, physics, quantum: the search for links},
in \emph{Complexity, Entropy, and the Physics of Information} (1990) 3--28.
\par\noindent\textbf{Relevance.}
Proposed the ``It from Bit'' program: the idea that every physical quantity derives its ultimate significance from binary yes-or-no measurements, and that the universe is participatory --- the observer and the observed are co-constitutive. The gauged $\UoneQi$ of GG-6 provides the physical derivation of Wheeler's vision: information is not merely an accounting device but a conserved gauge charge, and the EIC (Evolving Inner Cosmos) is the mathematical realization of Wheeler's participatory universe --- the closed loop in which the universe produces observers who discover the universe. First cited in \S\ref{sec:qgi}.

\bibitem[White et al.(1986)]{WhiteEtAl1986}
\textbf{J.~G.~White, E.~Southgate, J.~N.~Thomson, and S.~Brenner.}
\emph{The structure of the nervous system of the nematode Caenorhabditis elegans},
\emph{Philosophical Transactions of the Royal Society B} \textbf{314} (1986) 1--340.
\textbf{Relevance.} The foundational 302-neuron connectome --- the external empirical anchor for the Layer-14 central-nervous-system row. First cited in \S\ref{sec:cascade-codebook-table}.

\bibitem[Whitehead(1929)]{Whitehead1929}
\textbf{A.~N.~Whitehead.}
\emph{Process and Reality: An Essay in Cosmology}.
Macmillan (1929); corrected ed.\ Free Press (1978).
\par\noindent\textbf{Relevance.}
Whitehead's \emph{process philosophy} proposed that reality is composed of actual occasions of experience, each prehending the prior occasions and contributing to the next; the universe is fundamentally relational and process-like. In the GG-Theory program, Whitehead's actual occasions correspond to the GSSB transitions of the geometric cascade, and his prehensions correspond to the inheritance of attractor coordinates from one cascade layer to the next. Whitehead anticipated the cascade structure; GG-Theory derives it. First cited in \S\ref{sec:thm-articulation}.

\bibitem[Whitehead and Russell(1910--1913)]{WhiteheadRussell1910}
\textbf{A.~N.~Whitehead and B.~Russell.}
\emph{Principia Mathematica}, three volumes.
Cambridge University Press (1910--1913).
\par\noindent\textbf{Relevance.}
The Whitehead--Russell program of axiomatic foundations sought to derive all of mathematics from a small set of logical primitives. The GG-Constitution of \AOne{} (axioms AX-1 through AX-4 plus the seven postulates) is the physical counterpart of the \emph{Principia} ambition: a small axiomatic ledger from which the entire physical content of the universe -- particles, forces, cosmology, life, mind -- is derived. First cited in \S\ref{subsec:hist-constitution}.

\bibitem[Wigner(1939)]{Wigner1939}
\textbf{E.~P.~Wigner.}
On unitary representations of the inhomogeneous Lorentz group.
\emph{Annals of Mathematics} \textbf{40} (1939) 149--204.
\par\noindent\textbf{Relevance.}
Wigner's classification of particles by the irreducible representations of the Poincar\'e group established symmetry as the organizing principle of quantum physics. Every particle in the Standard Model is a Wigner representation; the gauge group $\SUfive \times \UoneB \times \UoneQi$ of GG-6 determines which representations are admissible, and the OGN-5 integers determine their specific quantum numbers. First cited in \S\ref{subsec:wigner}.

\bibitem[Wilson \& Kogut(1974)]{WilsonKogut1974}

\textbf{Wilson, K.~G., \& Kogut, J. (1974).}
The renormalization group and the $\epsilon$ expansion.
\textit{Physics Reports}, \textit{12}, 75--200.
\par\noindent\textbf{Relevance.} A standard reference for the modern RG viewpoint: coarse-graining as a flow on theory space. POS-5 builds on this picture but adds the axiomatic demand that RG must commute with the operational dictionary (\OGGEP{}), enabling a geometric RG coordinate in \GG{}. First cited in \S\ref{app:examples}.

\bibitem[Wilson(1971)]{Wilson1971}

\textbf{Wilson, K.~G. (1971).}
Renormalization group and critical phenomena. I. Renormalization group and the Kadanoff scaling picture.
\textit{Physical Review B}, \textit{4}(9), 3174--3183.
\par\noindent\textbf{Relevance.} Wilson's RG is the prototype of scale-covariant thinking and motivates POS-\RG. Because \GGTheory{}\ aspires to unify across scales and
domains, it demands that coarse-graining and operational measurement commute in a controlled way. First cited in \S\ref{app:examples}.

\bibitem[Witten(1998)]{Witten1998}

\textbf{Witten, E. (1998).}
Anti-de Sitter space and holography.
\textit{Advances in Theoretical and Mathematical Physics}, \textit{2}, 253--291.
\url{https://arxiv.org/abs/hep-th/9802150}
\par\noindent\textbf{Relevance.} Witten's prescription (together with Gubser--Klebanov--Polyakov and Maldacena) provides the canonical bulk$\to$boundary correlator map. We cite it as a concrete historical anchor for \OGGEP{}'s demand that a theory includes a bulk$\to$boundary dictionary, and as motivation for extending that dictionary to operational measurement objects. First cited in \S\ref{app:ax3-expanded}.

\bibitem[Woese and Fox(1977)]{WoeseFox1977}
\textbf{C.~R.~Woese and G.~E.~Fox.}
\emph{Phylogenetic structure of the prokaryotic domain: the primary kingdoms},
\emph{Proceedings of the National Academy of Sciences USA} \textbf{74} (1977) 5088--5090.
\textbf{Relevance.} The foundational 16S~rRNA phylogenetic experiment resolving the Bacteria/Archaea split; it is the external empirical anchor for the layer-11 Primary-Domain-Split row of the cascade codebook (a non-physics claim, classed Asserted pending the dual-gauge-SSB derivation). First cited in \S\ref{subsec:theorem-index-by-layer}.

\bibitem[Yang and Mills(1954)]{YangMills1954}
\textbf{C.~N.~Yang and R.~L.~Mills.}
Conservation of isotopic spin and isotopic gauge invariance.
\emph{Physical Review} \textbf{96} (1954) 191--195.
\par\noindent\textbf{Relevance.}
Yang and Mills generalized gauge invariance from the Abelian $\mathrm{U}(1)$ of electromagnetism to non-Abelian groups, providing the mathematical framework for the strong and weak nuclear forces. The $\SUfive$ sector of the GG-6 gauge group is a Yang--Mills theory; the anomaly-cancellation constraint that produces the Arisaka Equation is a direct descendant of the consistency conditions that Yang--Mills theories must satisfy in six dimensions. First cited in \S\ref{subsec:hist-anomaly}.

\end{thebibliography}

\end{document}
