%% 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.
%% ===================================================================
%% ===================================================================
%% v15.8 (August 18, 2026) -- BT-1: THE BIBLIOGRAPHY TITLE PASS.
%%
%% EVERY COMPANION IS NOW CITED BY THE TITLE PRINTED ON ITS OWN COVER.
%%
%% 19 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.
%% ====================================================================
%% ===================================================================
%% v15.7 (August 18, 2026) -- FP-5: THE DISCLOSURE-PREP PASS.
%%
%% The cohort is posted as TWENTY-ONE papers with A11-Origin held back.
%%
%% (1) THE A11 BIBLIOGRAPHY ENTRY PROMISED A PAPER THAT WILL NOT BE
%%     THERE.  It read "Preprint, available at https://arisaka-gg.org/",
%%     which becomes false the day the homepage goes up carrying
%%     twenty-one papers.  It now reads "In preparation.  To be posted
%%     ... on completion; no result of the present paper depends on it."
%% (2) AND IT CITED A11 UNDER A TITLE A11 DOES NOT HAVE.  This paper
%%     called it "The Origin of Cosmos, Life, and Mind --- One Geometry, One
%%     Cascade, from Quarks to Consciousness".
%%     A11 v15.14's own \GGTitlePage reads "The Geometric Inevitability
%%     of Cosmos, Life, and Mind --- The Self-Referential Closure of the
%%     GG-Theory Program".  A working title from before the paper was renamed.  Corrected.
%% (3) catalogue -> catalog in printed text (Report 443's pair-list
%%     hole).  5 printed sites; \label / \ref identifiers and changelog
%%     comments protected.
%%     THE FIRST RUN OF THIS SWEEP WAS WRONG AND THE GATE PASSED IT.  The
%%     suffix alternation read (e|ed|es|ing), and Python's | is
%%     first-match-wins, so "catalogued" matched the bare "e" and left the
%%     "d" stranded: the sweep wrote "catalogd".  No spelling audit has
%%     that string in its pair list, so nothing reported it; it was found
%%     by reading a rendered page.  Alternation reordered longest-first,
%%     the suffix map asserted exhaustive, and the loosened check re-run
%%     against text carrying the known defect to confirm it still fires.
%%
%% NO OTHER CONTENT CHANGED.
%% ===================================================================
%% ===================================================================
%% v15.6 (August 17, 2026) -- THE SHOWCASE HEADLINE COMES BACK TO SANS.
%%
%% This paper set its showcase headline in SERIF, and thirteen of the
%% cohort set theirs in sans-serif bold.  Measured across all
%% twenty-four after the v11.5 promotion, that was the ONLY one of the
%% four surviving divergences a reader turning pages could see -- the
%% trailing skip is invisible on a page centered by \vfill, and the one
%% size difference is A4's.
%%
%% IT WAS DRIFT, NOT A DECISION.  This paper and P6, W1, W2 and the
%% other study were the LAST of the showcase pass; the sans form is the
%% one the author read and approved on A1-Constitution and on the
%% twelve that followed it.
%%
%% ONE LINE CHANGES: \GGShowcaseSeriftrue becomes
%% \GGShowcaseCenteredtrue, and style file v11.6 puts \sffamily in
%% that branch.  THE SHAPE DOES NOT CHANGE -- this page still centers
%% its headline with \begin{center} and still takes 8 pt beneath it,
%% because both are measured fits on this page and because three of the
%% five call the macro with no \centering around it, so the
%% \begin{center} is what centers them.
%%
%% ONE PAGE MOVES, and it is the showcase page.  Everything else in
%% this paper is asserted identical to v15.5, page for page.
%% ===================================================================
%% ===================================================================
% v15.5 (August 17, 2026) -- MIGRATED TO STYLE FILE v11.5.
%%
%% FOUR MACROS THAT LIVED IN THE PAPERS COME HOME, AND NOT ONE PAGE
%% MOVES.  This paper's part in that:
%%
%% * \GGShowcaseHead removed; this paper prints the centered serif form
%%   and says so in one line.
%% * \GGtocbreak removed.  It is one of THREE payloads of the
%%   same defect -- content written into the .toc for the full
%%   Contents is replayed by etoc into the Overview.  The
%%   style file's \GGOverviewOfContents clears all three.
%% * the private \GGOverviewOfContents is gone.  It was the
%%   style file's own body with two numbers changed, and P5
%%   and P6 carried it BYTE-IDENTICALLY.  Three
%%   \renewcommands now say the two numbers and the head
%%   skip, and the page is the page it was.
%%
%% 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 v15.4 printed, except for the lines named above.
%% ===================================================================
%% ===================================================================
%% v15.4 (August 17, 2026) -- THE -ise FAMILY.
%%
%% minimiser -> minimizer (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.
%% ===================================================================
%% ===================================================================
%% v15.3 (August 17, 2026) -- THE FRONT MATTER, FINISHED.
%%
%% THE KEYWORDS PRINTED TWICE, ON TWO NEARLY EMPTY PAGES.  This paper
%% carried two keyword blocks -- a hand-rolled minipage closing the
%% Executive Summary and the cohort's \GGKeywords -- and each landed on
%% a page holding nothing else: page 4 at 1.17 per cent ink and page 5
%% at 1.51, against a front-matter median of 5.91.  The hand-rolled
%% minipage is removed.  \GGKeywords stays, because it is the macro the
%% rest of the cohort uses and its list is the longer and more specific
%% of the two (22 terms against 17), and the \clearpage that isolated it
%% is removed too, so its \vfill now sets it at the FOOT of the
%% Executive Summary's last page.  One keyword list, on a full page.
%%
%% THE EXECUTIVE SUMMARY EXTENDED, so that last page is a page and not a
%% caption: a Part-by-Part road map, which a sixteen-Part review had no
%% business going without.  It is SUMMARY ONLY -- every clause restates
%% a Part's own claim at that Part's own grade, and no sentence of it
%% asserts anything the body does not already argue.
%%
%% THE OVERVIEW OF CONTENTS SPILLING TWO LINES.  Parts XV and XVI
%% printed at the top of the List of Figures page, under the wrong
%% running head.  The cause is arithmetic: sixteen Parts times the style
%% file's \addvspace{0.45em} is about seven lines of air.  The style
%% file is shared by twenty-two papers, so P6 gets a LOCAL
%% \renewcommand -- the style file's own body with 0.45em -> 0.10em and
%% \linespread 1.18 -> 1.12, nothing else -- on the same footing as
%% \GGShowcaseHead.  The trio now runs continuously with no orphan.
%%
%% NEW PROSE IN THIS SHIP: the road-map paragraphs only.  Nothing in the
%% body is rewritten; no figure, table, equation or number moves.
%% ===================================================================
%% ===================================================================
%% v15.2 (August 17, 2026) -- THE FRONT MATTER, DEDUPLICATED.
%%
%% This paper printed its Overview of Contents and its List of Figures
%% TWICE, in two different implementations, with the Keywords page
%% between them: a hand-rolled etoc block at pages 5-6, then Keywords at
%% 7, then the showcase at 8, then \GGOverviewOfContents /
%% \GGListOfFigures / \GGListOfTables at 9-10.  The v15.0 migration
%% added the four standard macros and did not remove the hand-rolled
%% pair it replaced.  The duplication shipped in v15.0 and in v15.1.
%%
%% The hand-rolled block is removed.  The order is now the cohort's,
%% read off P3-Electron v15.2 and P5-CPT v15.2 rather than assumed:
%% Executive Summary -> Keywords -> the showcase page -> Overview of
%% Contents -> List of Figures -> List of Tables -> Contents.
%%
%% NO PROSE CHANGED.  Not a word of the Executive Summary, not a
%% keyword, not a figure, not a number.  The paper loses the two
%% duplicate pages and the duplicate "List of Figures" entry the
%% hand-rolled block added to the full Contents.
%% ===================================================================
%% ===================================================================
%% v15.1 (August 17, 2026) -- THE P3 AND S2 READ, AND THE SHOWCASE PAGE.
%%
%% THE SEAM.  This paper wrote its depth axis r = ln(E_GUT/mu) and put
%% r = 0 at the seam, M_Pl/2.  Eleven cohort papers -- A1, A2, A4, A5,
%% A9, A10, C1, C2, P1, P5 and S2 -- write E_GUT for M_Pl e^{-2pi} =
%% 2.28e16 GeV, which is 268 times smaller.  This paper's own descent
%% asset printed BOTH readings: "E_GUT = M_Pl/2 -- the 4D boundary" at
%% r = 0 and "GUT edge, 2.3e16 GeV" at r = 5.6.  Renamed the origin to
%% E_seam at all six printed sites and in the asset generator, and added
%% one register note in Part II.  NO NUMBER MOVES: 44.3, 38.7 and K = 35
%% are all measured from M_Pl/2 and all stay.  The author's decision.
%%
%% THE MINIPAGE.  One \begin{minipage}{\textwidth} with no \noindent,
%% so the box was set at \parindent + \textwidth and hung 15.26 pt into
%% the right margin on page 27.  Fifth paper in this pass with that
%% defect (A11, C1, C2, P3, P6) and the fifth time the showcase pass
%% found it rather than a gate tool.  A new tool, edge_scan.py, now
%% reads the artifact for it.
%%
%% FIVE IMPORTED PLATES, each into a section that argued its content
%% and had no picture in it:
%%   fig:central-claim-p6    P3, cohort-shared funnel  -> sec:central-claim
%%   fig:two-regimes-p6      A6 via P3                 -> Part II runtime
%%   fig:formfactor-p6       P3's own                  -> sec:two-grounds
%%   fig:holonomy-primer-p6  W2 via S2                 -> sec:abelian-ledger
%%   fig:deltab-three-p6     A9 + this paper, via S2   -> sec:completed-theorem
%% S2-Leptogenesis is NOT cited anywhere: section 2 keeps it off the
%% arXiv path and it has no bibitem in the cohort.  Both plates it
%% carried are credited to their owners, W2 and A9, which this paper
%% does cite.  Four S2 plates that looked strongest by title were
%% rejected on inspection -- talk-species and pub-door argue lepton
%% number and are drawn in three off-palette HTML colours;
%% triangle-pentagon duplicates fig:z3-winding; helix-quark's own
%% caption disowns its numbers and its section is titled "the pitch
%% test -- and it fails".  Every S2 plate carrying a numeric r
%% coordinate was excluded outright on the seam collision above.
%%
%% ONE NUMBER CHANGED INSIDE AN IMPORTED PLATE.  The source annotated
%% the admissible vertex "(rate ~ e^{-170})".  This paper prices it at
%% e^{-4pi/alpha_W} = e^{-372} ~ 1e-162 (eq:thooft-price) and its own
%% v13 changelog records the smaller exponent as an imprecise inline
%% evaluation inherited from A9.  Imported with this paper's value.
%%
%% NEW PROSE: the seam register note; four lead sentences; five figure
%% captions; the three-paragraph "And the one axis on which they are
%% opposite" closing sec:two-grounds; the showcase headline and caption.
%% NO ROW WAS ADDED TO tab:electron-proton -- section 14.2 puts that in
%% the ask-first class, and the other six rows are parallels while this
%% one is an inversion.  Two rows WERE added to tab:import-ledger, whose
%% whole purpose is to record imports.
%%
%% FLAGGED, NOT DONE: P3-Electron sec:stability-cite-p6 says this paper
%% "argues that the proton never decays because baryon number is gauged
%% and its anomaly canceled", and concludes the two papers' claims
%% "would fall together".  This paper says the protection is a winding
%% and not a symmetry, that the anomaly is absorbed by BI-6 inflow
%% rather than canceled, and that its protection is "of a stronger
%% kind".  The author's decision: fix it in a P3 rider, not inside a P6
%% ship.
%% ===================================================================
%% ===================================================================
%%  P6-Proton v1.25  (August 6, 2026)  -- the cohort-wide figure-size sweep.
%%  10 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.
%% ===================================================================
%% ===================================================================
%%  Arisaka-GG-P6-Proton  v1.24  (August 5, 2026)
%%  -----------------------------------------------------------------
%%  CHANGELOG (v1.23 -> v1.24), author-directed.
%%   (1) Section 6 takes the sister paper's shape: it becomes "The
%%       cohort inheritance and the roadmap of the sixteen Parts",
%%       with a NEW subsection 6.1 "The cohort inheritance" carrying
%%       the twenty-two-paper cohort map on its own page, and the
%%       existing roadmap demoted to 6.2 (label sec:roadmap travels
%%       with it; it had no incoming references).  This paper is the
%%       twenty-second carrier of the map; do not localise the block.
%%       GG-P4-Real, cited in the new subsection, had no bibitem here
%%       and is added with a Relevance note.
%%   (2) The roadmap figure re-cut to the format of A1-Constitution
%%       Figure 4: row pitch 13 -> 14.5mm on 12mm boxes (the gap was
%%       1mm), the plate widened from 0.60\textwidth, and the whole
%%       vertically centred between \vfill's as A1 does.  Its caption
%%       re-walked all nine phases in prose that 6.2 gives on the
%%       facing page; cut to the phase list and the highlight
%%       rationale, which is what buys the plate its height.
%%  Script: [v13-support]/p6_v124_sec61.py.
%% ===================================================================
%% ===================================================================
%%  Arisaka-GG-P6-Proton  v1.23  (August 4, 2026)
%%  -----------------------------------------------------------------
%%  CHANGELOG (v1.22 -> v1.23), author-directed (GG-Report-187):
%%    * NEW closing paragraphs of sec:two-grounds: what protects
%%      each ground attractor once it has landed.  The electron's
%%      stability is the absence of a deeper valley; the proton's
%%      is an integer winding -- protection without symmetry, the
%%      top rung of W1's ladder, and the reason every
%%      symmetry-based proton-stability argument of the last fifty
%%      years was broken on purpose by its own proposer.
%%      Contrast installed with the electroweak sector: the Higgs
%%      is circle-side but CONTINUOUS (a Wilson-line angle, not a
%%      winding), so non-locality buys it finiteness, not fixity.
%%      Not a modulus but a class; not finite but frozen.
%%    * NEW cites: ArisakaW1Symmetry, ArisakaW2Holonomy.
%%  Script: p1p6c1_v_tiers.py
%% ===================================================================
%% ===================================================================
%%  P6-Proton v1.18  (July 30, 2026)  -- Phase-2 shape-up.
%%  \GGKeywords added -- P6 was the only paper in the cohort without a
%%  keyword block.  Twenty-one keys covering the eleven questions, the
%%  winding/selection-rule apparatus, and the mass-budget vocabulary.
%% ===================================================================
%% ===================================================================
%%  P6-Proton v1.17  (July 30, 2026)  -- P2-N3 v14.0 conformance +
%%  enrichment (author-directed).  The family import from P2, previously
%%  phrased generically ("the Bogomolnyi/BI-6 action cap ... fences
%%  stated there"), now names what it inherits: the capacity theorem
%%  T-N3-6 (pi of budget per family from the stratum's quantum geometry,
%%  Berry transport at Born-grade), conditional on the coupling clause
%%  and the integrality lemma L-INT.  The one-paragraph consumption is
%%  rewritten in this paper's own pricing voice: linear income against
%%  triangle-number tolls, books balancing exactly at three ---
%%  3pi against 3pi, nothing left to fund a fourth family's road.
%%  Four sites: the family-inheritance opening, the consumption
%%  paragraph, the GFF bounded-below aside, the P2 bibitem Relevance.
%%  No P6 theorem, value, grade, or falsifier changed.
%% ===================================================================
%% ===================================================================
%%  P6-Proton v1.16  (July 29, 2026)  -- P2-N3 retitle propagation.
%%  P2-N3 is retitled "Why Three Generations: A THREE-LEG Derivation of N = 3
%%  from BI-6, MSMF, and GDOS Morse Stability" (was "A Closed Derivation").
%%  Reason: P2-N3's own main theorem T-N3-5 is "Conditional derivation of
%%  N = 3 (three-leg pincer)", conditional on five structural hypotheses --
%%  including the BI-6 Budget Conjecture, which A2 states is not derived
%%  anywhere in the cohort -- plus two empirical inputs.  The cohort's prose
%%  already flagged that conditioning everywhere; only the title claimed
%%  "Closed", and titles propagate into every bibliography.
%%  T-N3-5 KEEPS its own name: the theorem is conditional and says so.
%%   * "the closed two-sided theorem of P2" -> "the two-sided theorem of P2".
%%     The sentence goes straight on to name both sides (N >= 3 from the MSMF
%%     lower bound, N <= 3 from the budget), so "two-sided" already carries
%%     the content that "closed" was doing.
%% ===================================================================
%% ===================================================================
%%  P6-Proton v1.15  (July 29, 2026)  -- internal-affairs sweep.
%%  The debt registry used "at this revision" as a status marker (DISCHARGED /
%%  narrowed / restated as a scoreboard).  On an arXiv v1 there is no other
%%  revision to contrast with, so each becomes "here"; the preamble's
%%  "discharged in a successor revision" becomes "in a later paper of the
%%  cohort".  Every debt, pin (R2, R3) and conditional -- the gap hypothesis
%%  G-hat, the priced wall location -- is unchanged.
%% ===================================================================
%% ===================================================================
%% v1.14 (Ship 15, 2026-07-26): GRIP/GGA STRUCTURAL SURGERY
%%  (Report 39, author-approved). The proton reframed in the language
%%  of GG-Theory: the Ground Geometric Attractor of the baryonic
%%  stratum, at the ground of the Chain of GRIP-Out. Physics
%%  unchanged; frame, vocabulary, and emphasis changed.
%%  FRONT 1 (Part II): three new spine sections BEFORE the import
%%   checklist -- sec:grip-runtime (GRIP -> GGA, GA-vs-GGA =
%%   perturbation-vs-tunnelling, GRIP-In/Out, fig:ga-gga);
%%   sec:grip-chain (the Chain of GRIP-Out, proton = GGA of baryonic
%%   stratum, fig:grip-chain-ladder, the upward/foundation story);
%%   sec:two-grounds (proton = electron's twin one stratum over,
%%   comparison table). "Attractors and their readout" para trimmed
%%   to back-reference.
%%  FRONT 2 (Part III): retitled "The Terminal Basin" -> "The Ground
%%   Geometric Attractor"; T-P6-3 restated "the proton is the GGA of
%%   the baryonic stratum"; opening + basin para + theorem head recast
%%   (four legs = GGA definition unpacked; leg iii = empty exit set =
%%   what Part VI proves); NEW upward "ground that everything stands
%%   on" paragraph; figure node/caption relabelled; "terminal basin"
%%   demoted to gloss (kept, introduced once).
%%  FRONT 3 (Part XIII + knock-ons): NEW sec:one-fact-summit (the
%%   concept summit); title/abstract/ES/keywords/roadmap/glossary
%%   relabelled to GGA; neutron named a GA in Part VIII; GA glossary
%%   entry added; A6/A11 relevance notes upgraded to "conceptual
%%   spine."
%%  No physics, grades, numbers, citations, or falsifiers changed.
%% ===================================================================
%% ===================================================================
%% v1.13 (Ship 14, 2026-07-26): PART III REBUILT AS A SELF-CONTAINED
%%  REVIEW + CLOSURE ("treat this Part as a single paper"), per the
%%  author's commission on Report 38 (origin of proton mass).
%%  (1) ssec:derived-coupling: what A9 claims and at what grade --
%%      the strong angle-scale theorem, universal scale constant
%%      pi*e, Derived (selected), look-elsewhere audit in the open.
%%  (2) NEW ssec:closure: the chain M_Pl/2 -> pi*e -> RG -> Lambda
%%      -> m_p; sensitivity 6.95; exact one-loop walk -4.5% =>
%%      m_p^pred ~ 897 MeV (five percent, zero strong-sector input);
%%      fig:seam-descent (the whole chain to scale on the r-axis).
%%  (3) ssec:tolls corrected per Report 38 D.1: valence qualifier;
%%      sigma-term response ~72 MeV = 7.7% (Mainz + Gupta cites,
%%      2.4-sigma tension); charm subtlety (MPR); fig:mass-budget
%%      caption fixed ("remove the Higgs" -> sigma response).
%%  (4) NEW sec:mass-ledgers: Ji 1995 four-term ledger; chi-QCD
%%      2018 lattice table (33/37/9/23); trace anomaly identity;
%%      Lorce/MPR two-term objection; Ji 2021 rebuttal; Bollweg
%%      2026 all-schemes result (6%..35%); fig:ledger-dispute;
%%      custody reading ("the mass ledger is the spin ledger at
%%      J=0", INT, fenced).
%%  (5) NEW sec:mass-experiments: GlueX 2019/2023 (own hedging);
%%      J/psi-007 (r_mass < r_charge); Burkert 2018 + Kumericki
%%      2019 as a pair; HPS 2024 lattice GFFs; EIC YR; D-P6-5
%%      acceptance test registered.
%%  (6) NEW sec:mass-open: OP-1..OP-7 with owners; the pure-number
%%      program (2.76 = 2.13 x sqrt(1.675), leg 1 = D-P6-4(c) wall
%%      computation, leg 2 = basin spectroscopy); scoreboard table.
%%  (7) App A: new subsection "The closure arithmetic, digit by
%%      digit"; valence fix to the toll-ledger sentence.
%%  (8) Part XV: NEW Objection 17 (O-P6-17, the lattice
%%      practitioner, on the mass); sixteen -> seventeen at the
%%      four count sites.
%%  (9) 17 new B-list references with relevance notes (Ji1995,
%%      YangEMT2018, Lorce2018, MetzPasquiniRodini2020, Ji2021,
%%      BollwegEMT2026, GlueX2019, GlueX2023, Duran2023,
%%      BurkertPressure2018, Kumericki2019, HackettGFF2024,
%%      RobertsEHM2021, GuptaSigmaTerm2021, MainzSigmaTerm2023,
%%      EICYellowReport2022, Douglas2026); Borsanyi2015 first-cite
%%      note updated (now first cited in Part III).
%%  All numbers machine-verified (report38_verify.py, 36/36).
%% ===================================================================
%% ===================================================================
%% Arisaka-GG-P6-Proton-v1.12.tex
%% v1.12 (July 25, 2026): SHIP 13 -- PAPER-WIDE WARMTH PASS (the
%%  Part V treatment extended to every Part, per author). Warm
%%  plain-English part-opening prose added to Parts I, II, III, IV,
%%  VIII, IX, X, XI, XII (the others already open warm); closing
%%  bridges added for Parts I and II; Part XII's ledger close
%%  extended with a plain-English wrap and a hand-off to the
%%  Conclusions; one warm opening line for Part XIV. Existing warm
%%  closings (III->IV through XI->XII) kept verbatim. No physics,
%%  grades, numbers, citations, or labels changed.
%% v1.11 (July 25, 2026): SHIP 12 -- PART V WARMTH PASS (author:
%%  v1.10 Part V "technically excellent... too dry" for the target
%%  reader). Section 28 rewritten+doubled as a warm historical
%%  introduction (1964 quarks -> SLAC -> asymptotic freedom ->
%%  Wilson -> Clay 2000; why proving is hard in plain language; the
%%  long italic contract replaced by prose + a short three-line
%%  contract). Section 42 rewritten: plain-English summary of the
%%  Part for the broad reader, ledger table kept, warm bridge to
%%  Part VI. Ten accessibility insertions between (bar magnet,
%%  what-a-lattice-is, portrait guidance, three-hour clock, zoom
%%  knob, plain one-sentence T-P6-4, guitar-string hum, glosses).
%%  No physics, grades, numbers, or citations changed; first-cite
%%  declarations preserved.
%% v1.10 (July 25, 2026): SHIP 11 -- PART V TRIPLED (Reports 36+37
%%  folded in, author-approved in full). New Part V (15 sections,
%%  ~3x): honesty preamble repaired (gap/confinement distinction;
%%  Jaffe-Witten statement; lattice gap 1.71 GeV); the cage measured
%%  (crossover 0.28 fm, Regge 421, Luscher -51.7 MeV at 1 fm,
%%  r_b = 1.24 fm / 1.22 GeV, Casimir 9/4, pure-gauge 270 first
%%  order vs 156.5 crossover) + NEW FIG cornell-regimes; five-claim
%%  scorecard (K)(L)(B)(T)(G); what the lattice proves; mechanisms
%%  survey + comparison TABLE (dual SC, centre vortices, Gribov,
%%  Seiberg-Witten, holography); THE CENTRE BRIDGE + NEW FIG
%%  (one Z3: triality/centre-symmetry/vortices vs the lock; the
%%  symmetry-vs-holonomy disanalogy load-bearing); THE WALL on the
%%  scale axis + NEW FIG (r_Lambda = 44.3 = K + 9.3, zero dials;
%%  Cornell form from geometry; canon note per approved reading);
%%  NEW THEOREM T-P6-4 (conditional area law: gap hypothesis G-hat
%%  + lock => F >= sigma_min L; Higgs escape closed by C-P6-1;
%%  N-ality asymptotics; branch selection promoted to D(c));
%%  GOLDSTONE-COUNTING TEST (-pi/12 because dims are non-spatial;
%%  +50% spatial shift lattice-excluded -- consistency test
%%  passed); sigma/Lambda^2 section (target 1.67, kill window
%%  [1.42,1.93]; audit; L/N candidate recorded without weight);
%%  C-P6-1 kept + what-it-does-not-say fence; QGP + thermal fine
%%  print; Clay register (four moves, credit declined in writing);
%%  kills + expanded ledger. KNOCK-ONS: roadmap node V, walkthrough,
%%  ES clause; claim ledger 3 new rows; D-P6-4 registry -> (a)-(e)
%%  scoreboard with G-hat named; O-P6-6 updated; anchor table +4
%%  rows; Appendix C: five numbers (r_b update; crossover-radius
%%  and Luscher subsections). TEN new B-entries: Mandelstam1976,
%%  tHooft1978, LuscherSW1980, SeibergWitten1994, Maldacena1998,
%%  deForcrandDElia1999, MorningstarPeardon1999, JaffeWitten2000,
%%  Bali2005, Greensite2011; HotQCD2019 first-cite moved to the
%%  cage. All numbers machine-verified (d_p64_campaign_verify.py,
%%  17/17). No physics values changed elsewhere.
%% v1.9 (July 25, 2026): SHIP 10 -- THE STRUCTURE-AND-FIGURES
%%  REVISION (author's four refinements after full read of v1.8).
%%  (1) EXECUTIVE SUMMARY restructured on the P3-Electron model:
%%  bulleted Q1-Q11 (compact), Who-for / honest ledger / one fact /
%%  founder's axiom+record / kills / what-is-new paragraphs; ~2 pp
%%  with keywords. (2) STANDALONE PREFACE (P1-KK model) after the
%%  TOC, addressed to the Super-K/Hyper-K/DUNE graduate student;
%%  old Part I Preface section removed (content absorbed); the two
%%  affected first-cite declarations now read "in the Preface".
%%  (3) EIGHT NEW FIGURES in cohort house style: baryonic basin
%%  landscape (III), proton mass budget bar (III), Z3 winding
%%  diagram (VI), anomaly ledger flow + the one door (VI), U(1)_B
%%  history timeline 1955-2026 (VII), neutron gate two-panel (VIII),
%%  lifetime cliff of the tower (IX), birth-of-baryons door chain
%%  (X). (4) BACK MATTER now three Parts on the A9-SM pattern:
%%  Part XIV Technical Appendices (retitled, A9-style opening),
%%  Part XV Common Objections and Responses (O-P6-1..12 moved from
%%  the Discussion and expanded to persona+quote+response format;
%%  NEW O-P6-13 branes, -14 the record, -15 EP-null, -16
%%  leptogenesis-class), Part XVI References (GGPart wrapper,
%%  A9-style intro). Sixteen Parts; roadmap 16 boxes; Discussion
%%  subtitle and cross-texts updated. No physics values changed.
%% v1.8 (July 25, 2026): SHIP 9 -- THE FOUNDER'S AXIOM AND THE
%%  RECORD. (1) Part VI gains "The founder's axiom: a QED for
%%  baryon number" (new section after the theorem-as-received):
%%  the imposition stated as the founding act; the U(1)_EM vs
%%  U(1)_B table; the two forced bendings; AUTHOR-VOICE journey
%%  scaffold. (2) NEW Part VII "The Seventy-Year Road to One
%%  Letter": the historical record of gauged baryon number,
%%  1955-2026 (Lee-Yang; the three refusals; Pais; the fifth
%%  force and the EP machine; the minority line; ten dimensions
%%  and the brane convergence; the wounds-to-structure table;
%%  the framework chronology 2024-2026 at testimony register,
%%  AUTHOR-VOICE flagged). Old Parts VII-XIII renumbered
%%  VIII-XIV; roadmap now fourteen boxes, nine phases.
%%  (3) Discussion gains the fourth contrast "Against the
%%  ten-dimensional route to anomaly freedom" (three-clause
%%  form, author-approved). (4) ES + Part II one-liners: the
%%  Abelian factors imposed in deliberate analogy with
%%  electromagnetism. (5) F-P6-5 gains the Eot-Wash Be/Ti
%%  anchor (Wagner 2012); eta row added to the anchor table;
%%  GS/EP added to abbreviations. (6) Eight new B-entries:
%%  LeeYang1955, Pais1973, Fischbach1986, FootJoshiLew1989,
%%  CaroneMurayama1995, FileviezPerezWise2010, Wagner2012,
%%  IbanezMarchesanoRabadan2001; KRS first-cite moved to the
%%  record. No physics values changed.
%% v1.7 (July 25, 2026): SHIP 8 -- THE COLLATION-AND-PINS REVISION.
%%  (1) PDG COLLATION: g_A = 1.2756 and |V_ud| = 0.97373 CONFIRMED
%%  verbatim (PDG Vud/Vus review: g_A^ave = 1.2756(13) SF 2.6;
%%  Vud = 0.97373(31) superallowed); Lambda^(3)_MSbar updated
%%  332 -> ~340 MeV (lattice-matched range 330-345) at its three
%%  sites; dependent ratios re-verified (m_p/Lambda ~2.8 holds;
%%  "2.4x" -> "2.5x"). All other anchors confirmed as quoted.
%%  (2) PIN R1' DISCHARGED BY READING: deployed A2 canon states
%%  C = (theta -> -theta) o B "implements fermionic charge
%%  conjugation as real geometry" (via T-P5-C2) -- the twist is in
%%  print. D-P6-1 marked DISCHARGED; T-SM-60.1's chain shrinks to
%%  H3 + R2 + R3; swept at all nine sites.
%%  (3) PIN R2 NAMED: holonomy quantisation pinned to A9's
%%  Hopkins-Singer descent integrality; new Appendix D.5 writes
%%  the application; D-P6-2 narrowed to the X4-level derivation.
%%  (4) APPENDIX G ADDED: chirality-protection mechanism exhibit;
%%  D-P6-5 narrowed to the quantitative warped profiles.
%% v1.6 (July 25, 2026): SHIP 7 (FINAL ASSEMBLY) -- THE CLOSING
%%  TRILOGY. Part XI "Discussion" drafted in full (significance;
%%  uniqueness as structural contrast, no polemic; the dated
%%  falsification table F-P6-1..6; the D-P6 debt registry
%%  consolidated -- numbering honours prior print: D-P6-3 spinor,
%%  D-P6-4 GFF area law; D-P6-1 = pin R1', D-P6-2 = pin R2,
%%  D-P6-5 = pin R3, D-P6-6 = label-inventory fence; objections
%%  O-P6-1..12 incl. the quantum-gravity/global-symmetry objection
%%  answered by gauged-winding status). Part XII "Conclusions"
%%  drafted as a COMPLETE SCAFFOLD, every section flagged
%%  %% AUTHOR-VOICE for the author's verbatim pass (his Part).
%%  Part XIII completed: opening prose, abbreviations table, the
%%  claim-status ledger as a LONGTABLE (cohort convention, from
%%  day one), the empirical anchor table; Appendices A-F follow
%%  unchanged. \PartSynopsis retired (no uses remain; definition
%%  removed). Front matter untouched except this banner (ES/
%%  abstract already describe the full paper). NOTE: the full
%%  PDG-print verification of every anchor value remains the
%%  pre-arXiv checklist item, flagged in the anchor table caption.
%% v1.5 (July 25, 2026): SHIP 6 -- PARTS IX-X DRAFTED IN FULL,
%%  replacing their synopses. Part IX "The Birth of Baryons" (Q10:
%%  eta_B = 6.1e-10 and the annihilation catastrophe; Sakharov;
%%  the SM's quantitative failure; THE STRUCTURAL SELECTION --
%%  in this geometry leptogenesis is forced, not preferred, since
%%  T-P6-2 erases every Delta-B=1 mint; the P5 chain walked at
%%  its declared grade (resonance condition = named input, C-P5-C
%%  = named conjecture); the 28/79 door partition; "every proton
%%  is a fossil of the door"; double-duty kills F-P6-1/3; ledger).
%%  Part X "The Information Charge" (Q11 at the strictest register:
%%  kill criterion FIRST; the Arisaka equation restated for the
%%  proton; the ledger's-page reading; Landauer (8.7x) and ATP
%%  (~2 Q_B) calibrations carried as calibration only; what the
%%  candidate does NOT license; the honest experimental sizing;
%%  ledger). Appendix F (the door bookkeeping: 28/79 derivation,
%%  catastrophe arithmetic). New references: Planck 2020, Kuzmin-
%%  Rubakov-Shaposhnikov 1985, Harvey-Turner 1990, Pilaftsis-
%%  Underwood 2004, Kolb-Turner 1990. All numbers session-verified.
%% v1.4 (July 25, 2026): SHIP 5 -- PARTS VII-VIII DRAFTED IN FULL,
%%  replacing their synopses. Part VII "The Shallow Twin" (Q8: the
%%  hair's-breadth window and its alternatives; the competition
%%  +2.5 toll vs -1.0 EM = 1.293 with the BMW lattice+QED
%%  decomposition; the gate descent with the FULL tree-level
%%  lifetime walk 914 s -> ~878 s vs bottle 877.75; binding
%%  re-landscapes the gate ("chemistry exists because binding
%%  re-landscapes the gate") + the tank aside on neutron
%%  sanctuaries; the bottle-beam discrepancy reported with the
%%  winding constraint on exotic channels; ledger). Part VIII "The
%%  Ladder" (Q9: six = N=3 doubled, consumed at grade; the graded
%%  tower Delta/Lambda/Omega-/n-free/n-bound/p with the
%%  channel-label-lifetime grammar table; copies-not-cousins;
%%  no-second-zero-width as registered kill; tank aside; ledger).
%%  Appendix E (the neutron arithmetic: splitting decomposition,
%%  lifetime walk step by step, sanctuary bookkeeping). New
%%  references: Borsanyi et al. 2015 (Science 347 1452), Barnes et
%%  al. 1964 (PRL 12 204). All numbers session-verified.
%% v1.3 (July 25, 2026): SHIP 4 -- PART VI "THE IMMORTAL PROTON"
%%  drafted in full (the paper's summit), replacing its synopsis:
%%  the question personally (Kamiokande opening + closing arc, both
%%  %% AUTHOR-VOICE flagged); T-SM-60.1 as received; T-P6-1 the
%%  obstruction theorem proved in text; the centre identity
%%  B = t/3; T-P6-2 the Z3-lock with the holonomy-scan table, the
%%  four locks, and the theta-integral selection rules; the
%%  dihedral closure through T-P5-A (the no-go, the C-twisted
%%  repair, B -> -B under reflection); the anomaly ledger (all six
%%  coefficients) + Green-Schwarz/H3 + Stueckelberg + the one door
%%  priced at the standard 't Hooft evaluation e^{-4pi/alpha_W} =
%%  e^{-372} ~ 10^{-162} (NOTE: A9's inline "~e^{-170}" appears
%%  imprecise -- corrected here; A9 erratum flagged for the
%%  integration wave; the Part III site inherited from it is swept
%%  in this revision); the completed stability theorem with pins
%%  R1'/R2/R3 stated in place; Kamiokande ("the tank's silence has
%%  changed sides"); falsifier faces F-P6-1/3/5 + n-nbar; status
%%  ledger. Appendix D (the lock arithmetic) added. New references:
%%  Hosotani 1983, Kawamura 2001, Green-Schwarz 1984, Stueckelberg
%%  1938, Baldo-Ceolin et al. 1994. Author instruction: warm style,
%%  the longer the better.
%% v1.2 (July 25, 2026): SHIP 3 -- PARTS IV-V DRAFTED IN FULL,
%%  replacing their synopsis placeholders. Part IV "The Label That
%%  Would Not Decompose" (Q3: the 1987 crisis told properly; the
%%  Ji and Jaffe-Manohar ledgers; the label-vs-ledger custody
%%  reading at INT grade; the axial-anomaly morsel; the EIC face;
%%  Part status ledger; owed spinor promotion registered D-P6-3,
%%  shared with D-P3-4). Part V "Confinement and the Absent Quark"
%%  (Q4+Q5: the Clay honesty preamble; the sixteen-tonne flux tube
%%  and string breaking; the GSSB singlet-branch reading at grade;
%%  the fractional-winding corollary of T-P6-2 -- a free quark is a
%%  third of a knot; the QGP discussion; the eighty-year null;
%%  ledger). Appendices B (the spin ledger arithmetic) and C (the
%%  string arithmetic) added. New references: Ji 1997,
%%  Jaffe-Manohar 1990, de Florian et al. 2014, Bali 2001,
%%  HotQCD 2019. All quoted numbers computed and verified in the
%%  session record. Author instruction: warm style, the longer the
%%  better.
%% v1.1 (July 25, 2026): SHIP 2 -- PART III "THE TERMINAL BASIN"
%%  drafted in full (Q1 sharp mass, Q2 exact charge; the terminal-basin
%%  theorem T-P6-3; the transmutation pricing with the one-loop walk
%%  from the A9 registration alpha_s(MZ) = 1/(pi e); the embedding
%%  charges; falsifier faces; the Part status ledger), replacing its
%%  synopsis placeholder. Appendix A (the transmutation arithmetic,
%%  step by step) added to Part XIII to complement Part III, per the
%%  author's Ship-2 instruction (all derivation steps + physics
%%  intuition; addressed to the youngest generation at Super-K,
%%  Hyper-K, and DUNE). Two external references added (Bressi 2011
%%  neutrality of matter; Ahmadi 2016 antihydrogen charge). B-section
%%  ordering note generalized. Author-approved spec: Reports 32-34.
%% v1.0 (July 25, 2026): FIRST DRAFT BLOCK (Ship 1 of the P6
%%  drafting campaign; Internal Reports 32-34 are the specification).
%%  Contents of this revision: front matter (cover, abstract,
%%  executive summary, keywords, overview, TOC), Part I "The Eleven
%%  Questions and Their Century", Part II "Inheritance", synopses of
%%  Parts III-XIII (drafting placeholders, to be replaced block by
%%  block), and the A/B reference list for the material drafted so
%%  far. The eleven-question list, the thirteen-Part architecture,
%%  the Z3-lock result set (T-P6-1/T-P6-2, Reports 33-34), and the
%%  question->Part map are the author-approved Report 32/33/34
%%  decisions. Preamble adapted from the P3-Electron v14.0 cohort
%%  template (sty: ucla-arisaka-gg-11 from [v11-Support]).
%% ===================================================================
%% \pdfoutput=1 added v15.0: section 12 item 10.  Absent in P4-Real
%% and P5-CPT too, and no v15 gate item looks for it -- it is on the
%% v20 checklist only.  Added at Stage 1 rather than discovered at
%% packaging.
\pdfoutput=1
%% ---- vv15.9 (August 19, 2026) ----
%% The front-matter showcase pages given breathing room.
%% A showcase page's free height is now distributed by weighted stretch
%% glue -- a sixth at the top, a third above the plate, a third below it,
%% a sixth at the foot -- instead of pooling above the headline and below
%% the caption, which left a broad band of white at each end of the page
%% and crowded the three elements into the middle third.
%% Three source forms were found in the cohort and each is handled.  A
%% minipage frame becomes a minipage of EXPLICIT height with the [s]
%% inner alignment, because a minipage is a vbox of natural height and
%% glue inside one does not stretch; the height is \textheight minus
%% \topskip.  A \begingroup frame keeps its page-level glue and changes
%% only the weights.  An unframed page -- there were eight, five of them
%% showcase tables carrying a longtable, which cannot live inside a
%% minipage -- gains page-level glue for the first time, which is why it
%% had been printing hard against the top of the text block.
%% 1 showcase page in this paper.  Layout only: not one word,
%% number, figure or table row is changed by this ship.
\documentclass[11pt,letterpaper]{article}

\usepackage{ucla-arisaka-gg-11}
\usepackage{indentfirst}
\renewcommand{\LaTeX}{LaTeX} % cohort-standard shim (see P3 preamble note)

%% ---- Per-paper identity -------------------------------------------
\PaperSeries{GGP}
\PaperNumber{Arisaka-GG-P6-Proton}
%% ------------------------------------------------------------
%% v1.25 -> v15.0, August 15, 2026.  STAGE 1: MECHANICAL.
%%   The nine-stage v15 pipeline, eighth paper conversion.
%%   P6 IS A CONFIRMED SECTION 3.1 STALE ARTIFACT and the
%%   register's cleanest case.  Deployed PDF 167 pages; this
%%   source rebuilt against the live v11.4 style is 166.  Page 1
%%   says why: the deployed artifact carries the PRE-v11.3
%%   FOUR-LINE AUTHOR BLOCK where the live style sets the compact
%%   two-line form.  One line of the title page, one page of the
%%   document, and every page after it displaced -- pagecmp
%%   reports 166 of 166 differing for that reason alone.
%%   sty_check PASSES: the style matches the live copy byte for
%%   byte and the geometry constant is right.  Section 14.3 stops
%%   the run when sty_check FAILS and the deployed PDF does not
%%   reproduce; only the second holds, which is the case section
%%   3.1 catalogues and instructs me to record and proceed from.
%%   THE REBUILD IS THE BASELINE.
%%   a4paper -> letterpaper.
%%   541 PRINTED BRITISH SPELLINGS -> 0 over 95 distinct words:
%%   centre 102, colour 86, programme 44, quantised 26, flavours
%%   22, renormalisation 15, quantisation 13, realisation 12.
%%   1 IN THE PDF METADATA (pdfsubject: programme).
%%   TWO NEAR-MISSES, both caught by reading the survivor contexts
%%   and by nothing else:
%%     `franchises' -- "The cage does not leak; it franchises."
%%       The loose -ise branch reaches it and the generated map
%%       turned it into `franchizes', AND THE MAP'S OWN
%%       NO-FIXED-POINT ASSERTION PASSED, because `franchizes' is
%%       not matched by the rule either.  franchise is French, not
%%       a British -ise verb.  Now PROTECTed.
%%     `towards' -- three sites.  Two are the author's prose and
%%       are swept.  The third is inside 't Hooft's 1978 title,
%%       quoted verbatim in the bibliography, and is now a
%%       FROZEN_TITLE: correcting a published title is a
%%       misquotation.
%%   \pdfoutput=1 added -- section 12 item 10, absent here as in
%%   P4-Real and P5-CPT.
%%   NO PROSE WAS WRITTEN IN THIS STAGE.
%% ------------------------------------------------------------
%% ------------------------------------------------------------
%% Stage 2A, same day.  Eleven bare grids into captioned floats.
%%   Eleven is the most of any paper in the cohort; P2-N3 had
%%   nine.  They are also all six of the paper's unbreakable
%%   tabulars, so the promotion answers both float_audit counts at
%%   once -- a float cannot break across a page.
%%   THE GRIDS ARE COPIED VERBATIM.  Not a rule, a column spec, a
%%   row or a value is touched.  What is added is a caption above
%%   (section 8), a label and a written short form.
%%   EVERY STATEMENT IN EVERY CAPTION WAS READ OFF THE GRID IT
%%   SITS ON.  Two were weakened before the script ran, because
%%   the surrounding prose claims more than the grid shows:
%%   tab:triality-closure's now says the arithmetic selects and
%%   the cage enforces, which is the distinction the grid's own
%%   verdict column draws; and tab:bl-operators' describes the
%%   operators a decay would have to use rather than asserting
%%   they are unavailable, which is the Part's argument and not
%%   the table's.
%%   NEW PROSE: eleven captions and eleven short forms.
%% ------------------------------------------------------------
%% ------------------------------------------------------------
%% Stage 2B, same day.  Sixteen thin captions extended.
%%   EVERY EXISTING SENTENCE KEPT VERBATIM; section 14.2 reserves
%%   rewriting a caption's claim and none is rewritten.
%%   EIGHT OF THE SIXTEEN ARE THE SAME TABLE.  P6 closes Parts III
%%   through XI with a per-Part status ledger, each in the same
%%   three columns and each captioned "Status ledger of Part N."
%%   Writing eight different extensions for eight identical tables
%%   would be writing for the character count.  They share ONE
%%   sentence, plus a clause where that Part's ledger has
%%   something of its own to say.  Part VII's ledger is excluded:
%%   its register column reads Record/Reading rather than a grade,
%%   because a historiographic claim is verified and not derived,
%%   and the shared sentence would misdescribe it.
%%   CAPTION POSITION NEEDED NO WORK: all 29 table captions were
%%   already above the grid.  P6 is the only paper in the cohort
%%   so far to arrive that way -- P5-CPT had 22 of 32 below.
%%   NEW PROSE: sixteen caption extensions, of which eight share
%%   one sentence.
%% ------------------------------------------------------------
%% ------------------------------------------------------------
%% Stage 2C, same day.  Front matter; 12 Plain-English Parts.
%%   FRONT MATTER: P6 carried \GGTableOfContents ALONE -- no
%%   Overview, no List of Figures, no List of Tables -- against 17
%%   figures and 31 captioned tables, eleven of them promoted out
%%   of bare grids at Stage 2A.  All four macros, one page.
%%   TWELVE INTRODUCTIONS, NOT THIRTEEN.  Sixteen Parts; XIV, XV
%%   and XVI are back matter and get none, which leaves thirteen --
%%   and PART XIII IS DELIBERATELY EXCLUDED.  Its fourth section is
%%   already "The conclusions, in plain English", the author's own
%%   plain-English statement of the Part.  Mine at the top would
%%   duplicate the job, compete with his words, and print two
%%   Contents lines carrying the same phrase a few lines apart.
%%   Section 14.1 permits these; it does not oblige one where the
%%   author has already written it.
%%   ONE Contents entry each; asserted 12 and 0.  Paragraph breaks
%%   asserted -- P3-Electron's eight printed as one unbroken block
%%   because a single newline was written where a blank line was
%%   meant, and only the rendered page showed it.
%%   ALL TWELVE ARE NEW PROSE IN THE AUTHOR'S VOICE, listed by Part
%%   and page in GG-Report-368 for review or rewriting.  Part XI's
%%   opens with the kill criterion because the Part does; Part
%%   VII's says it makes no physics claim at all.
%% ------------------------------------------------------------
%% ------------------------------------------------------------
%% Stage 4, same day.  Palette: 7 violations -> 0.
%%   palette_fix cleared three across nine pictures: two bare `draw'
%%   node borders given hue shades, one fill above the !18 ceiling
%%   compressed (!22->!16), and the named TikZ weights resolved to
%%   explicit 0.80/1.10 pt.
%%   FOUR IT LEFT ARE FIXED BY HAND: fig:centre-bridge's two node
%%   borders and fig:u1b-timeline's extC/ggC styles were TINTS
%%   (blue!55, violet!60), not shades.  Section 8's border rule is
%%   hue!N!black.  Both fills at !7 were raised to !8 at the same
%%   time, the section 8 floor; the auditor does not test the floor
%%   and !7 is invisible against !8, but the standard says !8.
%%   TWELVE PATH STROKES ABOVE 1.1 pt ARE LEFT ALONE and printed by
%%   the tool rather than silently clamped: they are inline \draw
%%   marks in seven figures, not node borders.  Clamping inline
%%   strokes destroyed P3-Electron's own prediction line, set heavier
%%   than its foils on purpose -- the thirteenth casualty across six
%%   papers, after which the clamp was confined to /.style bodies.
%%   NO PROSE WAS WRITTEN IN THIS STAGE.
%% ------------------------------------------------------------
%% ------------------------------------------------------------
%% Stage 5, same day.  Six missing cohort companions.
%%   P6 carried 15 of 21.  Missing: A4-Euclid, A8-QM, C1-Kaxion,
%%   C2-Hubble, C3-BH, P1-KK.
%%   THE FIRST PAPER WHOSE GAP IS NOT THE CLASSIC THREE.  P6
%%   already carries W1-Symmetry and W2-Holonomy -- it is one of
%%   the three newest papers and was written after they existed --
%%   and what it lacks instead is four older papers and two
%%   cosmology companions.  Six papers running were missing exactly
%%   P6, W1 and W2; the pattern was never about those three papers,
%%   it was about which papers existed when a given bibliography
%%   block was last copied forward.  Recorded so the next intake
%%   assumes a gap and measures it rather than predicting it.
%%   THE PROSE ALREADY EXISTED.  sec:inheritance-p6 walks the whole
%%   map and names all six by macro with no \citep on any of them:
%%   a reader met the name and could not reach the paper.  The
%%   citation is attached where the name already stands.  The three
%%   cosmology siblings were a bare list of three names, which is
%%   not prose that cites them, and get the one clause each that
%%   the other nineteen entries have.
%%   C3-BH's Relevance note records that this paper's stability
%%   theorem is stated for the accessible-energy regime and says
%%   nothing about a horizon, and that the two accounts have not
%%   been reconciled.  That is a gap, and it is named rather than
%%   left for a reader to notice.
%%   NEW PROSE: one rewritten sentence and six Relevance notes.
%% ------------------------------------------------------------
%% ------------------------------------------------------------
%% Stage 6A, same day.  Two two-line Part-ending orphans.
%%   Parts II and XI ended with two lines alone on a page (1.05 %%
%%   and 0.87 %% ink).  Both Parts now open with a Plain-English
%%   introduction of mine, about ten lines, and the shift is what put
%%   the tails there.
%%   BOTH PARAGRAPHS ARE THE AUTHOR'S and neither is touched.
%%   \looseness was tried first, at -2 and then at -1, and TeX
%%   DECLINED BOTH -- the pages came back byte-identical.  That is
%%   worth recording: \looseness is a request, not an instruction, and
%%   it cleared P4-Real's one-line orphan only because that paragraph
%%   happened to have the slack.  Reading the ink figure again after
%%   the rebuild is what caught it; a report written from the edit
%%   rather than from the page would have claimed a fix that never
%%   happened.
%%   What worked is \enlargethispage{2\baselineskip}, issued inside the
%%   paragraph that begins on the preceding page, so that page carries
%%   two more lines and the tail comes back onto it.  Section 7's
%%   remedy of removing a printed line is not needed when the page
%%   before can be asked to hold it.
%% ------------------------------------------------------------
%% ------------------------------------------------------------
%% Stage 2C REVISED, same day.  Twelve introductions removed.
%%   Stage 2C added twelve Plain-English Part introductions
%%   because that is what Stage 2 does and what every other paper
%%   in this run has needed.  RENDERING PAGES 16 AND 28 SHOWED THE
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\hypersetup{
  pdftitle = {The Proton as the Ground Geometric Attractor of Matter: The Eleven Questions of the Proton, Closed and Fenced within GG-Theory},
  pdfauthor = {Katsushi Arisaka},
  pdfsubject = {GG-Theory program -- the proton as the Ground Geometric Attractor of the baryonic stratum},
  pdfkeywords = {GG-Theory, proton, baryon number, confinement, spin crisis, proton decay, neutron, baryogenesis, GGA, GDOS}
}


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\begin{document}
% ---- v1.22 (August 2, 2026) ----
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% ---- v1.21 (August 2, 2026) ----
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% the four abelian objects of GG-6 and their standings; why an
% abelian circle touches SU(3) at its centre; the Z6 global form
% of the SM group made geometric. (2) Part V, new section 'The
% singlet rule' (after the absent quark): the closure condition
% run through the zoo (q, qbar, qq, meson, baryon, tetra/penta,
% gluon), why a triangle, the electromagnetic shadow Q+B in Z,
% three fences. (3) Discussion: two new uniqueness contrasts
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% footnote). All new arithmetic verified programmatically (Z6
% triviality, Q+t/3 in Z on every fragment, zoo trialities).

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% (Kaxion birth & leptogenesis; the matter asymmetry is born here,
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% caption updated to name it.

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% floor chain and the no-dial framing.


%% ===================================================================
%%  COVER PAGE
%% ===================================================================
\thispagestyle{empty}
\GGTitlePage%
  {The Proton as the Ground Geometric Attractor of Matter}%
  {The Eleven Questions of the Proton, Closed and Fenced within \GGTheory{}}%
  {\parbox{0.92\textwidth}{\centering\small
   A companion preprint of the Predictions Branch, sister to
   \PaperPThree{} (the electron) and \PaperPTwo{} ($N=3$), assembling
   the eleven oldest and newest questions of the proton as eleven
   faces of one fact: the proton is the Ground Geometric Attractor
   (\GGA{}) of the baryonic stratum, and its baryon number is written
   by the geometry itself.
   Offered as a review-article gateway for the reader who has carried
   these questions past graduation, and for the experimentalist who
   spent a career waiting for a proton to die.}}

%% ===================================================================
%%  ABSTRACT
%% ===================================================================
\begin{abstract}
\noindent
Nearly everything you have ever weighed is protons and neutrons, and
nearly all of that weight is neither. The proton is a seething bound
state --- three valence quarks awash in gluons and quark--antiquark
pairs, its mass almost entirely interaction energy --- and yet it
presents to the world a single exact mass with zero width, a charge
equal and opposite to the electron's to at least twenty-one decimal
places, and a spin of exactly one half that its own constituents
famously fail to add up to. It has never once been seen to decay,
though the great unified theories insisted it must, and detectors
built to watch it die --- beginning with Kamiokande --- have watched
in vain for four decades. Its quarks cannot be removed; its excited
copies all crumble in an instant while it alone endures.

This paper assembles, within Grand Geometric Theory, a single answer to
eleven questions the proton poses: the proton is the Ground
Geometric Attractor (\GGA{}) of the baryonic stratum of a
six-dimensional geometry, and its baryon
number is not a bookkeeping label but a winding of the compact
geometry itself --- written by the same twist that breaks the unified
gauge symmetry and splits the Higgs multiplet. All eleven questions
--- from the sharp mass to the information charge of the boundary
ledger --- are read as faces of that one fact, every import cited at
its proved grade, every remaining assumption named, every kill
criterion stated. The proton
has waited a century to be more than a list of accidents; here it is
offered as the theorem its silence was always pointing at.
\end{abstract}

%% ===================================================================
%%  EXECUTIVE SUMMARY
%% ===================================================================
\begin{ExecutiveSummary}
The question behind this paper is the oldest in the physics of
matter, and the most personal one an experimentalist can ask of the
substance in his own hands: what \emph{is} a proton, that it should
be what it is? It is the most common heavy particle in the universe
and the least explained common object in physics. Eleven specific
questions have accumulated around it --- some for a century, some
newly sharpened by \GGTheory{} itself:

\begin{itemize}[label=\textbullet, leftmargin=1.4em, itemsep=0pt,
 topsep=3pt, parsep=0pt]
\item \textbf{Q1} (mass): why does an object that is mostly
 interaction energy have one exact, zero-width mass?
\item \textbf{Q2} (charge): why is its charge \emph{exactly} $+1$
 --- equal and opposite to the electron's, to $10^{-21}$?
\item \textbf{Q3} (spin): why is its spin exactly $1/2$ when its
 quarks supply only a third --- the ``spin crisis''?
\item \textbf{Q4} (confinement): what confines its quarks?
\item \textbf{Q5} (the absent quark): why has no experiment ever
 seen a free quark or a fractional charge?
\item \textbf{Q6} (immortality): why has no proton ever decayed,
 when unification required it?
\item \textbf{Q7} (the law): is baryon number an accident, as the
 Standard Model has it, or a law?
\item \textbf{Q8} (the twin): why does the neutron exist, sit just
 $1.293$~MeV up-slope, die in fifteen minutes when free --- yet
 live forever inside nuclei?
\item \textbf{Q9} (the six): why exactly six flavors, and why
 is precisely one rung of the excited tower eternal?
\item \textbf{Q10} (the birth): where did the protons come from ---
 why is there matter at all?
\item \textbf{Q11} (the ledger): what does one baryonic
 rearrangement cost in information?
\end{itemize}

\paragraph{Who this paper is for.} This paper is written as a review
article, for the graduate student at Super-Kamiokande, Hyper-%
Kamiokande, DUNE, or JUNO --- the experimentalist who watches
protons for a living and was never told, by silence or by syllabus,
\emph{why} the watching stays silent. It is a message from the
founder of that watch: the author built Kamiokande --- the Kamioka
\emph{Nucleon Decay Experiment} --- under Masatoshi Koshiba
(1979--1985), when the proton's death was the safest prediction in
unified physics. It is
simultaneously a gateway: every import is proved in the cohort
papers it cites, and the status tags are the reading contract ---
nothing asserted above its price.

\paragraph{The honest standard-physics ledger.} The paper
front-loads what standard physics genuinely supplies. Q1 has a real
partial answer --- the lattice computes the mass to percent accuracy
and transmutation explains its largeness --- but the uniqueness,
identity, and value of the ground state are inputs. Q2 has
consistency (anomaly cancellation) and, with a GUT embedding,
quantization; the electron--proton match itself is unexplained. Q3
has superb decomposition machinery and no reason the total is
exact. Q4 and Q5 have overwhelming evidence and no proof. Q6 and Q7
have an accountant's answer: baryon number happens to survive at
renormalizable order --- an accident nearly every deeper theory
breaks. Q8's splitting is computed on the lattice; why a
twin at all, and why that direction, are not. Q9 and Q10 are open ---
the Standard Model's own baryon violation is quantitatively
hopeless for the observed asymmetry. Q11 is this framework's own
question, stated so it can be killed.

\paragraph{The one fact.} Within \GGTheory{} the eleven have one
answer: \emph{the proton is the Ground Geometric Attractor
(\GGA{}) of the baryonic stratum} --- the unique lowest resting place
for three-quark matter
on the boundary of the six-dimensional bulk --- \emph{and its
baryon number is geometry}: the compact cycle carries a discrete
twist, locked to the center of the color group, under which every
quark mode winds by one third and every baryon by one. Mass is
basin depth, priced by the strong scale from the registration
$\alpha_s^{-1}(M_Z) = \pi e$ (the Higgs tolls enter at the percent
level: $2m_u + m_d \approx 9$~MeV of $938$); charge descends from
the embedding shared with the electron; spin is the readout label,
not a sum; confinement is one gap hypothesis from a theorem (T-P6-4); the free quark is a readout that does not exist; the
twin sits one gate up-slope; the six flavors are $N=3$ doubled;
and the single $\mathbb{Z}_3$-twist simultaneously breaks the
unified symmetry, evicts the $X,Y$ bosons from the boundary, splits
the Higgs doublet from its triplet, and writes baryon number
component by component --- \emph{one twist, four locks} (T-P6-2),
completing the cohort's stability theorem T-SM-60.1 at its stated
conditional grade.

\paragraph{The founder's axiom, and the record.} The \UoneB{} at
the root of it all was \emph{imposed} --- a local gauge invariance
in deliberate analogy with electromagnetism, the framework's
founding act. The idea is Lee and Yang's, from 1955; the field
declined it for seventy years, for three reasons that were correct
in four dimensions and are load-bearing structure in six. That
history is kept as a record of its own in Part~VII, in registers
that never mix: verified history, cited to print; the framework's chronology,
dated as testimony.

\paragraph{Predictions and kill criteria.} The paper is killed by:
(F-P6-1) a single proton decay, in any channel, at any rate
(reach extends an order of magnitude by $\sim$2040); (F-P6-2) one confirmed free fractional
charge; (F-P6-3) an observed $n$--$\bar n$ oscillation; (F-P6-4) a
proton--antiproton asymmetry beyond the CPT-paired readout;
(F-P6-5) a long-range baryonic force --- the equivalence-principle
program has been running this search since 1906; (F-P6-6) the information charge's registered
kill. The strongest are decidable within the present generation of
experiments.

\paragraph{What is new in this paper.} (i) The conjunction thesis:
no framework has previously explained the eleven as one
theorem-complex about one ground state. (ii) The discharge of the
stability theorem's load-bearing hypothesis: the obstruction
theorem (baryon number is provably \emph{not} a Lie-algebra
charge), the center identity $B = t/3$, and the $\mathbb{Z}_3$-lock
--- the projection-plus-charge computation the Standard-Model paper
deferred to this successor. (iii) The dihedral closure: the
reflection sector fixed by the CPT companion's theorem, not
chosen. (iv) The consistency sentence: the anomaly door that
forbids one proton's death is the door that permitted all protons'
births --- Q6 and Q10 as one equation. (v) The historical record of
the founding axiom, in its own Part. (vi) The
complete honesty apparatus: every claim graded, every kill
priced.

One thread is personal. The author spent his twenties building an instrument to watch for a
flash that this paper, forty-eight years later, derives cannot
come. The silence was never a null result; it was the discovery. The
tank was right, and this paper is its founder's report on what
the silence meant: keep watching.

\paragraph{The road through the paper.} Sixteen Parts, and it is worth
saying in advance which of them carry weight. Part~I states the eleven
questions and the century that accumulated them, and prices what standard
physics genuinely answers. Part~II is the inheritance: every result this
paper consumes, its home paper, its grade as declared at home, and the
Part that spends it --- read it once and no later Part will surprise you
about what is derived here and what arrives already proved. Parts~III
through~VI are the paper's own work. Part~III prices the mass: the
baryonic stratum has one terminal basin, the depth of that basin is the
strong scale, and the strong scale descends from a single registration
with no strong-sector measurement anywhere in the chain. Part~IV settles
the spin question by refusing its premise --- the total is a readout
label and was never a sum of parts. Part~V takes confinement to the only
honest grade available to anyone: a conditional area-law theorem, exact
given one named gap hypothesis, with the hypothesis left in the open
where a reader can see it. Part~VI is the summit --- the obstruction
theorem, the center identity, the $\mathbb{Z}_3$-lock and the anomaly
ledger, which together complete the cohort's stability theorem and leave
exactly one door open in the baryon number of the world.

Everything after Part~VI is consequence, record, or honesty apparatus.
Part~VII is the seventy-year record of the founding axiom, kept in
registers that never mix. Part~VIII walks the neutron down the one gate
the winding leaves open within a sector, and Part~IX climbs the excited
tower above it. Part~X imports the birth chain at its home paper's
declared grade and not a step stronger. Part~XI carries the information
charge at candidate grade with its kill criterion quoted. Parts~XII
and~XIII weigh what has been done and what is owed; Part~XIV holds the
arithmetic; Part~XV answers the objections a skeptical reader will
already have formed by then; Part~XVI is the bibliography. A reader with
one hour should read Part~I, the ledger table of Part~II, and Part~VI.
A reader who wants to know whether to believe any of it should read
Part~XII's debt registry first and decide from there.

\paragraph{How to read the grades.} Every claim in this paper carries a
tag, and the tags are the reading contract rather than decoration.
\textbf{D} is derived: the argument is complete here or in the cited home
paper, and nothing outside the framework is assumed. \textbf{D(c)} is
derived conditionally, and the conditions are named where the claim is
made and again where it is spent --- Part~VI's completed theorem is D(c)
on three finite pins, each a bounded check against deployed cohort
apparatus and none of them a new idea. \textbf{INT} is an
interpretation, offered as one and not as a result. \textbf{CAND} is a
candidate, admitted only with a registered kill criterion attached. The
discipline the tags enforce is the one this paper cares about most:
nothing is asserted above its price, and where the honest grade is weak
the weakness is stated here rather than left for a reader to find. The
confinement Part is the test case --- it would have been easy to write
as a proof and it is written as a conditional theorem, because that is
what it is.
\end{ExecutiveSummary}

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%% NO \clearpage HERE: \GGKeywords begins with \vfill, so
%% without one it sets at the foot of the Executive
%% Summary's last page instead of on a page of its own.
\GGKeywords{%
proton; baryon number; \GGA{}; baryonic stratum; ground-state uniqueness;
$\mathbb{Z}_3$ winding; Stueckelberg-massive $\U(1)_B$; proton stability;
$\Delta B \in 3\mathbb{Z}$ selection rule; dimensional transmutation;
$\Lambda_{\mathrm{QCD}}$; mass budget; trace anomaly; spin decomposition;
charge radius; form factor; neutron $\beta$-decay; gate transit;
antiproton; leptogenesis; eleven questions; \GGTheory{}}

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\fill[blue!60!black] (1.05,1.35) circle (1.8pt);
\node[lab, anchor=west, text=blue!60!black] at (1.45, 1.85)
 {a quark mode: winding $\tfrac13$\\ $B = t/3$ with $t = 1$\\ not closable --- not a state\\ (C-P6-1, Part~V)};
%% baryon: full wrap
\draw[line width=0.9pt, draw=black!50] (0,-1.6) circle (1.05);
\draw[wind, draw=black!80] (1.05,-1.6) arc (0:355:1.05);
\fill[black!80] (1.05,-1.6) circle (1.8pt);
\node[lab, anchor=west] at (1.45, -1.15)
 {a baryon: winding $1$\\ ($t = 3$: one full wrap)\\ closed --- a boundary state\\ carrying $B = 1$ forever};
\end{scope}
\end{tikzpicture}}%
\global\sbox{\GGplateDeltaB}{\begin{tikzpicture}[font=\sffamily\footnotesize,
  tny/.style={font=\sffamily\tiny, text=black!58},
  lab/.style={font=\sffamily\scriptsize, text=black!72},
  ok/.style={draw=olive!55!black, fill=olive!9, rounded corners=2pt,
             line width=0.7pt, align=center, inner sep=4pt,
             font=\sffamily\scriptsize},
  no/.style={draw=red!55!black, fill=red!5, rounded corners=2pt,
             line width=0.7pt, align=center, inner sep=4pt,
             font=\sffamily\scriptsize}]
  
  \draw[draw=black!55, line width=0.5pt] (-0.6,0) -- (7.2,0);
  \foreach \n/\x in {0/0.4, 1/1.5, 2/2.6, 3/3.7, 4/4.8, 5/5.9}{
    \draw[draw=black!40, line width=0.4pt] (\x,-0.12) -- (\x,0.12);
    \node[tny, anchor=north] at (\x,-0.18) {$\Delta B = \n$};}
  \foreach \x in {0.4, 3.7}{\fill[olive!58!black] (\x,0) circle (3.0pt);}
  \foreach \x in {1.5, 2.6, 4.8, 5.9}{
     \draw[draw=red!58!black, line width=0.88pt] (\x-0.11,-0.11) -- (\x+0.11,0.11);
     \draw[draw=red!58!black, line width=0.88pt] (\x-0.11,0.11) -- (\x+0.11,-0.11);}
  \node[ok, anchor=south, text width=2.5cm] at (3.7,0.32)
     {\textbf{admissible}\\ $\Delta B \in 3\mathbb{Z}$\\
      (rate $\sim e^{-372}$)};
  \node[no, anchor=north, text width=2.9cm] at (1.5,-0.75)
     {$p \to e^+\pi^0$\\ $p\to K^+\bar\nu$\\ \textbf{forbidden}};
  \node[no, anchor=north, text width=2.6cm] at (2.6,-1.95)
     {$n$--$\bar n$ oscillation\\ \textbf{forbidden}};
  \path (current bounding box.south) coordinate (s2drawctr);
  \node[lab, anchor=north, align=left, text width=295.0pt]
       at ([yshift=-7pt]s2drawctr)
    {Every canonical proton-decay mode has $\Delta B = 1$; neutron--antineutron oscillation has $\Delta B = 2$; neither integer is a
     multiple of three, so both are forbidden outright rather than merely
     suppressed. The thermal reprocessing of Part~\ref{part:birth} lives at
     $\Delta B \in 3\mathbb{Z}$ and is unaffected. \textbf{A single event
     in either forbidden class falsifies the framework} --- and Hyper-Kamiokande will improve the bound on the first class by about an
     order of magnitude.};
\end{tikzpicture}}%
\endgroup

\clearpage%% Self-filling showcase frame (Report 462): this page carried no
%% frame at all, which is why it printed hard against the top.  The
%% glue is page-level -- a longtable cannot live inside a minipage.
\null\vspace*{\stretch{1}}

\GGShowcaseHead{The Door That Forbids One Proton's Death ---\\[3pt]
Is the Door That Permitted All Protons' Births}

\par\vspace*{\stretch{2}}
\begin{center}
\resizebox{0.798\textwidth}{!}{\usebox{\GGplateZthree}}

\vspace{12pt}

\resizebox{0.700\textwidth}{!}{\usebox{\GGplateDeltaB}}
\end{center}

\par\vspace*{\stretch{2}}
\captionof*{figure}{\textbf{Why the proton is still here, and why there is anything for it to be here with.} Above: the lock. The sixth dimension is a compact, non-spatial circle, and the unique quantized twist it admits is $z = \mathrm{diag}(\omega,\omega,\omega,1,1)$, the center element of color. Under it a lepton travels a whole lap and closes; a single quark travels a third of a lap and cannot close; only three quarks together come back to where they started. That is what baryon number is in this framework --- not a bookkeeping label that happens to be conserved, but a winding number, an integer, and nothing continuous acts on the integers. Below: what the lock decides. Baryon number can change only in multiples of three, so proton decay ($\Delta B = 1$) and neutron--antineutron oscillation ($\Delta B = 2$) are not suppressed but forbidden, while the topological vertex at $\Delta B = 3$ stays open at enormous cost --- and that surviving door is the one the early universe used. The eighty-year silence in the mines and the existence of the matter doing the listening are one sentence read twice. In full: Figure~\ref{fig:z3-winding}, p.~\pageref{fig:z3-winding}, and Figure~\ref{fig:deltab-three-p6}, p.~\pageref{fig:deltab-three-p6}.}
\par\vspace*{\stretch{1}}
\clearpage


%% Front matter, v15.0: the four standard macros.  P6 carried
%% \GGTableOfContents ALONE -- no Overview of Contents, no List of
%% Figures, no List of Tables -- against 17 figures and 31 captioned
%% tables, eleven of them promoted out of bare grids at Stage 2A.  All
%% four on one page, as in C3-BH, P1-KK, P3-Electron and P5-CPT.
\clearpage
\GGfrontmatteronepagetrue
\GGOverviewOfContents
\GGListOfFigures
\GGListOfTables
\GGTableOfContents

\GGPreface

\noindent This Preface is addressed, more than to anyone else, to a
graduate student on the deck of Super-Kamiokande, in the cavern at
Hyper-Kamiokande or DUNE or JUNO --- or deciding whether to join one
of them --- who is wondering what a lifetime of watching silent
water is actually worth.

Consider, first, where the watch stands. In 1974, grand unification
sentenced the proton to death, and the sentence was the theory's
crown jewel: if the forces unify --- and the couplings visibly
converged --- the proton decays. Detectors the size of cathedrals
were built underground to collect. The author of this paper was
there at the beginning: he spent 1979 to 1985 designing and
building the Kamiokande detector --- the Kamioka \emph{Nucleon
Decay Experiment} --- as a doctoral student under Masatoshi
Koshiba, in the years when the field expected the flash within a
year or two of switch-on \citep{ArisakaKamiokande1985}. The flash
never came. Not that decade, not in the forty years since, not
through half a century of accumulated megaton-years. The experiment
refused to waste its silence --- solar neutrinos, the supernova of
1987, a Nobel Prize, a new astronomy --- but the question the tank
was named for has stood unanswered in either direction for the
length of a career: the proton would not die, and no theory could
say why not. %% AUTHOR-VOICE SITE: this paragraph carries the
%% author's own history; he rewrites it verbatim at sign-off.

You may also be the reader who kept the questions. You learned
early that the proton is made of quarks --- two up, one down ---
and the sentence arrived with the settled tone of an answer. Then,
if you looked closer, the answer came apart in your hands. The
three quarks account for scarcely one percent of the proton's
mass; the rest is a storm of gluons and quark--antiquark pairs
that no one can count. Yet this storm presents to every instrument
ever built a single exact mass, identical from one proton to the
next across the visible universe. Its charge cancels the
electron's --- a particle two thousand times lighter, with no
visible kinship to it --- to at least twenty-one decimal places.
Its spin is exactly one half, and when the experiments of 1987
went looking for that half among the spins of its quarks, they
found a third of it and a crisis. No fractional charge has ever
been caught alone. And the proton does not die. Somewhere in your
education each of these facts went past, wearing the costume of a
technicality. This paper collects them, adds the questions the
framework itself has newly sharpened, and takes the eleven of them
seriously as a single subject --- the way the sister paper
\citep{ArisakaP3Electron} took the six oldest questions of the
electron.

Now consider what this paper puts on the table. A framework with
no adjustable parameters derives that the proton is the
\emph{Ground Geometric Attractor} (\GGA{}) of the baryonic stratum
--- the unique
lowest resting place available to three-quark matter --- and that
its baryon number is not a label the theory carries but a
\emph{winding} the geometry cannot drop. From that one fact the
paper prices the mass, descends the charge, reads the spin,
forbids the free quark, walks the neutron down its gate, doubles
the family theorem into six flavors, brings every baryon in the
universe through one hot door, and --- the emotional center of the
whole argument --- derives that the flash the tank was built for
\emph{cannot come}: the decay is forbidden to all orders, at every
rate, forever, at a stated conditional grade with its three
remaining pins in plain view. The prediction that built the tank
mistook the location of the executioners; the geometry keeps them
in the bulk. The silence was never a null result. It was the
discovery.

And here is what the theorem does to your watch: it inverts the
stakes, and the inversion is the point. In 1980, a flash in the
dark would have crowned unification; today, one flash --- one
event, any channel, any rate --- kills this framework outright,
with no parameter anywhere to absorb it. Your silence is a
measurement, the sharpest ever made of the deepest law this
framework contains; and your water, should it ever flash where no
muon entered, is the instrument best placed in the world to prove
this paper wrong. A framework can pay an experimental community no
higher compliment. Honesty requires the other half, and it is the
half that makes this science rather than salesmanship: the theory
may simply be wrong --- and the watch decides that too, either
way, within the working lifetime of the experiments now under
construction.

How to read the paper, if you are the student described above:
start with Part~VI (the immortal proton --- it is self-contained,
and it walks from the mine to the theorem and back); then the
falsification table of Part~XII, which prices what your experiment
can do to this framework; then Part~VII, the seventy-year record
of the one-letter axiom the theorem discharges --- history, kept
deliberately apart from proof. Work backwards into the foundations
(Parts~II--V) as curiosity, not obligation, dictates; the
technical appendices of Part~XIV are written so that every
load-bearing computation can be verified with one sheet of paper.
The eleven questions will still be there after your calibration
shift --- but they are answered differently now, and the
difference is checkable.

%% ===================================================================
%%  PART I -- THE ELEVEN QUESTIONS AND THEIR CENTURY
%% ===================================================================
\GGPart[The eleven questions]{The Eleven Questions and Their Century}{What has, and has not, been explained since 1917}
\label{part:questions}

Every long journey should begin by saying where it is going and
why the destination is worth the walk. This opening Part does
three things, all in plain sight. It tells the century's story
of the proton --- how each generation's finest instrument
answered one mystery by uncovering finer ones. It states the
eleven questions this paper exists to answer, each in its
sharpest form, and credits, without reservation, everything
standard physics already explains about them. And it signs the
contracts: the central claim with its fences around it, and the
short list of observations that would kill this paper outright
--- registered before a single argument is made, because that
is the order in which an honest paper does business. Nothing is
derived in this Part. It sets the table; the meal is
Parts~III through XI.
\section{The century: how the questions accumulated}
\label{sec:century}

The proton's century is usually told as a story of answers. Told
honestly, it is a story of questions arriving faster than answers,
each new instrument resolving one mystery into finer mysteries.

\paragraph{1917--1919: the nucleus has a piece.} Rutherford, firing
alpha particles into nitrogen, knocks loose hydrogen nuclei and
concludes that the hydrogen nucleus is a constituent of heavier
matter; he names it the proton \citep{Rutherford1919}. For a decade
and a half it is possible to believe matter is two particles and
done.

\paragraph{1932: the twin.} Chadwick finds the neutron
\citep{Chadwick1932}, and with it the first version of Q8: here is a
particle heavier than the proton by barely a tenth of a percent,
identical to it under the nuclear force, that nevertheless dies in
minutes when set free --- and lives forever inside a nucleus. The
periodic table becomes explicable; the near-degeneracy does not.

\paragraph{1933: the first crack in pointlikeness.} Frisch and Stern
measure the proton's magnetic moment expecting the Dirac value and
find it nearly three times too large, $\mu_p \approx 2.79\,\mu_N$
\citep{FrischStern1933}. A Dirac point particle it is not. The proton
is the first elementary particle caught having an inside --- fifteen
years before anyone could say what an inside would mean.

\paragraph{1956: the inside has a shape.} Hofstadter's electron
scattering maps the proton's charge distribution: a smooth cloud
roughly $0.8$~fm across \citep{Hofstadter1956}. The question sharpens:
a cloud of \emph{what}?

\paragraph{1964--1969: the quarks.} Gell-Mann and Zweig organize the
hadron zoo with fractionally charged constituents
\citep{GellMann1964, Zweig1964}; the SLAC deep-inelastic experiments
find hard scattering centers inside the proton
\citep{Breidenbach1969}. The constituents are real. They are also,
from the first day, strange: no free quark appears in any detector,
any cosmic-ray stack, any drop of oil --- then or in the eighty years
since (Q4, Q5 are born together).

\paragraph{1973: the force that explains and the debt that remains.}
Asymptotic freedom \citep{GrossWilczek1973, Politzer1973} and the
QCD Lagrangian \citep{FritzschGellMannLeutwyler1973} give the strong
interaction its theory. Two of its properties do the explaining: at
short distance the coupling weakens (so SLAC sees quasi-free quarks),
and at long distance it grows (so, plausibly, nothing colored
escapes). But ``plausibly'' is load-bearing: confinement has never
been derived from the Lagrangian. Wilson's lattice
\citep{Wilson1974} makes it computable and the computations agree
with the world; the mathematical statement --- that Yang--Mills
theory has a mass gap --- remains one of the Clay Millennium
Problems, open today. Q4's modern form is fixed here: \emph{the
theory of the strong force contains its central qualitative fact
only as a numerical observation.}

\paragraph{1974: the death sentence.} Georgi and Glashow unify the
gauge forces in $\mathrm{SU}(5)$ \citep{GeorgiGlashow1974}, and the
unification carries a price tag: the same bosons that relate quarks
to leptons mediate proton decay, at rates within reach of a
sufficiently patient experiment. The prediction is embraced
precisely because it is falsifiable; through the late 1970s the
proton's instability is, in the community's estimate, the safest bet
unified physics has ever placed.

\paragraph{1979--1985: the tank.} Detectors go underground to watch
protons die: IMB in a salt mine in Ohio, and Kamiokande in the
Kamioka lead-zinc mine --- three thousand tonnes of water, a
photomultiplier the size of a beach ball at every half-meter, built
between 1979 and 1985 by a small group under Masatoshi Koshiba,
the detector design and construction carried by the doctoral student
whose name appears on this paper's cover
\citep{ArisakaKamiokande1985}. Kamiokande is built to see
$p \to e^+\pi^0$ inside a year or two. It does not. What it sees
instead --- solar neutrinos, and in February 1987 the neutrino burst
of Supernova 1987A \citep{HirataSN1987} --- founds neutrino
astronomy and earns its leader the Nobel Prize; the pivot from
nucleon decay to neutrino astronomy is one of the great improvised
successes in experimental physics. But the original question did not
go away. It hardened. %% AUTHOR-VOICE SITE (verbatim rewrite invited)

\paragraph{1987: the spin comes apart.} The European Muon
Collaboration measures how much of the proton's spin its quarks'
spins carry, expecting most of it, and finds a small fraction ---
consistent, in the first analysis, with none at all
\citep{Ashman1988}. Three decades of refinement settle the quark-spin
share near thirty percent, with the remainder distributed between
gluon spin and orbital motion in proportions that depend on how the
question is posed. Q3 acquires its modern name: the spin crisis.

\paragraph{1998--2024: the silence hardens.} Super-Kamiokande, fifty
thousand tonnes, pushes the proton's partial lifetime into
$p \to e^+\pi^0$ beyond $2.4\times10^{34}$ years
\citep{Takenaka2020}, with the accumulated exposure now
$0.37$~megaton-years \citep{SuperKamiokande2024} --- more than
$10^{24}$ times the age of the universe. Minimal $\mathrm{SU}(5)$ is
long dead; its successors retreat scale by scale ahead of the data.
Meanwhile the lattice, given only the measured coupling and quark
masses, computes the proton's mass from first principles to percent
accuracy \citep{Durr2008}: the ninety-nine percent that is not Higgs
is confirmed to be pure interaction energy. And precision experiments
sharpen the sharpness itself: the antiproton's charge-to-mass ratio
matches the proton's to sixteen parts per trillion
\citep{Borchert2022}; the neutron's lifetime is measured to a tenth
of a percent --- twice, by two methods, giving two answers
($877.75\pm0.28$~s in the bottle \citep{UCNtau2021},
$887.7\pm2.2$~s in the beam \citep{Yue2013}) whose four-sigma
disagreement is its own small open question; the proton's charge
radius, after a decade's puzzle between muonic and electronic
measurements \citep{Pohl2010}, converges near $0.84$~fm.

\paragraph{Today.} The Electron--Ion Collider is being built to map,
at last, where the proton's spin and mass actually reside
\citep{Accardi2016}. Hyper-Kamiokande will carry the decay search
another order of magnitude. The century closes where it opened: the
most common heavy object in the universe, measured more precisely
than any composite system in science, still cannot answer the child's
version of any of the eleven questions below.

\section{The eleven questions, stated}
\label{sec:eleven}

Each question is stated here in its sharpest form; the honest status
table of \S\ref{sec:status} records what standard physics does and
does not supply, and Parts~III--XI take them in order (the historical record of
Part~VII interleaved where the law is proved). The numbering
Q1--Q11 is used throughout the paper and in every status ledger.

\paragraph{Q1 --- the sharp mass.} The proton's mass is
$938.272$~MeV, identical for every proton, with zero width: it is a
true stationary state, not a resonance. Yet by mass it is
$\sim99\%$ gluon field energy and sea, $\sim1\%$ Higgs-coupled quark
mass. Why does an object that is mostly storm present one exact,
permanent number? What \emph{fixes} that number?

\paragraph{Q2 --- the exact charge.} The proton's charge is
$+e$ to the precision of the neutrality of hydrogen --- better than
one part in $10^{21}$ \citep{PDG2024}. The electron is a lepton; the
proton a baryon built of thirds. What enforces exact cancellation
between objects with no visible common origin?

\paragraph{Q3 --- the undecomposable spin.} The proton's spin is
exactly $\tfrac12$. Its quarks' spins carry roughly a third of it;
the rest is gluon spin and orbital angular momentum in
scheme-dependent shares. Why is the total exact and universal when
no decomposition of it is?

\paragraph{Q4 --- confinement.} No colored object has ever crossed a
detector. What, precisely, confines the quarks --- and why does the
theory that predicts every measured hadron property still lack a
proof of its own central qualitative fact?

\paragraph{Q5 --- the absent free quark.} Millikan-style searches,
accelerator searches, and bulk-matter searches spanning eighty years
have never found a fractional charge \citep{PDG2024}. Is the free
quark merely hard to liberate --- or is it \emph{not a possible
object}?

\paragraph{Q6 --- the immortality.} No proton has ever been observed
to decay: $\tau/B(p \to e^+\pi^0) > 2.4\times10^{34}$~yr
\citep{Takenaka2020}. The classic unified theories required decay;
the proton declined. Is its stability an accident of low-energy
bookkeeping, or a law with a mechanism?

\paragraph{Q7 --- the law.} In the Standard Model baryon number is an
\emph{accidental} symmetry: nothing imposes it; it merely happens to
survive at renormalizable order, and the electroweak anomaly violates
it. Nearly every extension --- grand unification, most gravity
folklore --- breaks it further. Is baryon number an accident, or is
it gauged?

\paragraph{Q8 --- the twin.} The neutron outweighs the proton by
$1.293$~MeV --- a tenth of a percent, the width of a hair between a
universe of hydrogen only and a universe with chemistry. Free, it
decays in about fifteen minutes ($877.75\pm0.28$~s
\citep{UCNtau2021}); bound, it is stable for at least as long as the
proton. Why does the twin exist, why so close, and why does the
decay run in exactly the direction it does?

\paragraph{Q9 --- the six.} The proton's quarks come in six flavors
arranged in three families --- no more, no less --- and the proton
itself heads a tower of excited copies ($\Delta$, $N^*$, the hyperon
shelf) every one of which decays in $\sim10^{-23}$~s while the ground
state lives forever. Why six, why three families, and why is
precisely one rung of the tower eternal?

\paragraph{Q10 --- the birth.} The universe contains protons and
essentially no antiprotons: about one baryon survived per billion
photon pairs. The Standard Model's own baryon-number violation is
quantitatively hopeless for producing this. Where did the protons
come from?

\paragraph{Q11 --- the ledger.} \GGTheory{} locks four currents into
one boundary event: energy, electric charge, baryon number, and
information \citep{ArisakaA7Qi4}. If that locking is right, one
baryonic rearrangement is one informational elementary update, priced
at $\QB = 0.1555$~eV. What does \UoneQi{} mean \emph{for the proton}
--- and what would falsify it?

Questions Q1--Q6 and Q8--Q10 are the century's; Q7 is the century's
question turned structural; Q11 is this framework's own, stated so
that it can be killed. Two pairs travel together by design: Q4/Q5
(one mechanism, two faces) and Q6/Q7 (one fact, observational and
structural).

\section{The honest status table}
\label{sec:status}

The discipline of the sister paper is kept: before this framework
says a word, the standard answers are credited exactly as far as
they go. Table~\ref{tab:status-p6} is the paper's contract with the
reader; every ``open'' cell is a claim this paper must either close
at a stated grade or leave visibly fenced.

\begin{table}[htbp]
\centering
\small
\caption[The honest status table: the eleven questions before this
paper.]{The honest status table. Middle column: what standard physics
(the Standard Model, QCD, and standard cosmology) genuinely supplies.
Right column: what remains unexplained in the standard account.
Grades in Parts~III--VI and VIII--XI close or fence each row. Eleven rows, one per question, and the two columns are deliberately not in competition: the middle column is written to give the standard account its full due before the right column says what it leaves unexplained. A reader who thinks the right column overstates the gap should read the middle one again first.}
\label{tab:status-p6}
\begin{tabular}{p{0.075\textwidth}p{0.42\textwidth}p{0.40\textwidth}}
\toprule
\textbf{Q} & \textbf{What standard physics supplies} &
\textbf{What remains unexplained} \\
\midrule
Q1 & The mass computed to percent accuracy on the lattice from the
measured coupling; dimensional transmutation explains why a large
mass emerges from nearly massless quarks. & Why a unique exact ground
state exists at all; why every readout is identical; the value of the
strong scale itself. \\
\addlinespace
Q2 & Anomaly cancellation makes the charge assignments
\emph{consistent}; with a GUT embedding, quantization follows. & Why
the embedding; the SM alone permits but does not explain the
electron--proton match; exactness to $10^{-21}$ is an input. \\
\addlinespace
Q3 & The decomposition machinery (helicity distributions, lattice
moments); measured shares: quark spin $\sim30\%$. & Why the total is
exactly $1/2$ and frame-independent while every decomposition is
scheme-dependent; what the spin \emph{is} a property of. \\
\addlinespace
Q4 & Lattice QCD demonstrates confinement numerically; the flux-tube
picture; asymptotic freedom makes it plausible. & An analytic
mechanism; the Clay-grade mass-gap statement; \emph{why} the singlet
branch is selected. \\
\addlinespace
Q5 & Follows from confinement, if confinement holds. & Whether the
free quark is dynamically imprisoned or structurally impossible ---
the two readings differ, and experiment cannot separate them. \\
\addlinespace
Q6 & The SM \emph{accidentally} forbids proton decay at
renormalizable order. & Why the accident survives every deeper
theory; GUTs generically destroy it; nothing protects it at the
Planck seam. \\
\addlinespace
Q7 & $B$ conservation observed to extraordinary precision. & The SM
provides no principle; $B$ is anomalous and cannot be gauged within
the SM's field content. \\
\addlinespace
Q8 & $m_n - m_p$ traced to $m_d - m_u$ against electromagnetic
self-energy (now computed on the lattice); $\beta$-decay theory
complete. & Why the splitting is so small; the bottle--beam lifetime
discrepancy; why the gate runs in the direction it does. \\
\addlinespace
Q9 & Three families measured; six flavors cataloged; the excited
spectrum mapped. & Why three; why the tower terminates downward at a
stable rung; the SM takes the count as input. \\
\addlinespace
Q10 & Sakharov's three conditions \citep{Sakharov1967}; leptogenesis
as a workable mechanism family \citep{FukugitaYanagida1986}. & Which
mechanism; every dial; the SM's own CP violation and sphaleron
bookkeeping fall short by many orders. \\
\addlinespace
Q11 & --- (not a standard-physics question). & Everything: the
question exists only inside this framework, which must therefore
state its kill criterion itself. \\
\bottomrule
\end{tabular}
\end{table}

\section{The central claim, fenced}
\label{sec:central-claim}

Here is the paper's one sentence, stated with its fences visible.

\begin{quote}
\textbf{Central claim.} \emph{On the boundary of the six-dimensional
bulk, the color-singlet baryonic stratum has a unique terminal
basin --- the proton --- whose baryon number is the winding of the
compact cycle under the $\mathbb{Z}_3$-locked descent; the eleven
questions of this paper are eleven readings of that basin and that
winding.}
\end{quote}

\noindent The fences, named at once and kept in view throughout:

\begin{itemize}
\item The six-dimensional framework itself --- the bulk, the boundary
law, the attractor machinery --- is \emph{inherited}, not re-derived:
Part~II states every import with its provenance and its conditional
clauses attached, from the constitution of \AOne{} through the
Standard-Model derivation of \ANine{}.
\item The stability theorem this paper completes (Part~VI) rests on
the Green--Schwarz bookkeeping of \AThree{} as received, and on three
named finite pins (R2, R3) --- each a bounded check against
deployed cohort apparatus, each stated where it is used, none a new
conjecture.
\item Confinement (Part~V) is engaged at the only honest grade
available to anyone: the mathematical statement is open at
Clay-Millennium level, and this paper's contribution is a structural
reading at its stated grade, not a proof.
\item The birth of baryons (Part~X) is imported from
\PaperPFive{} at that paper's own declared grade --- a conditional
chain with one named input and one named conjecture --- and nothing
stronger is claimed here.
\item The information charge (Part~XI) is carried at the candidate
grade of its home paper \citep{ArisakaA7Qi4}, kill criterion quoted.
\end{itemize} The shape of that argument is the cohort's, and it is worth seeing before the fences: Figure~\ref{fig:central-claim-p6} draws it as three boxes, the same three every paper in the program funnels through, with the proton in the box where the sister paper puts the electron.

\begin{figure}[htbp]
\centering
\resizebox{0.627\textwidth}{!}{%
\begin{tikzpicture}[
 node distance=19mm,
 every node/.style={font=\sffamily\footnotesize, align=center},
 box/.style={draw=black!55, rounded corners, minimum width=115mm, minimum height=12mm},
 arrow/.style={-{Stealth[length=3.4mm]}, line width=1.05pt},
 elab/.style={font=\sffamily\scriptsize\itshape, fill=white, inner sep=1.5pt}
]
\node[box, fill=violet!10] (open) {\textbf{Every physical system is open}\\ a particle, the universe, a living cell --- none can be isolated from its environment};
\node[box, fill=blue!8, below of=open] (oggep) {\textbf{\OGGEP{}} (\AX{3}): the lawful projection \;6D bulk $\to$ 4D boundary\\ \textit{this projection is observation itself} \;\; {\scriptsize[\AOne{}, \AEight{}]}};
\node[box, fill=yellow!13, below of=oggep] (gdos) {\textbf{\GDOS{}}: the dynamics of the open-system projection\\ \textit{an observable is the value at a stationary seam} \;\; {\scriptsize[\AFive{}]}};
\node[box, fill=orange!14, line width=1pt, below of=gdos] (obs) {\textbf{The proton as the ground state of baryonic matter}\\ (its mass, its stability, its lifetime) --- \textit{a registered observable, not a free input}};
\node[box, fill=green!12, below of=obs] (pill) {\textbf{\GDOS{} is forced} by the \GiFour{} pillars {\scriptsize[\ASix{}]} and the \QiFour{} pillars {\scriptsize[\ASeven{}]}};
\node[box, fill=orange!16, line width=1pt, below of=pill] (concl) {$\Rightarrow$ \textbf{it is determined}\\ {\footnotesize read off one geometry --- why every proton is identical to every other}};
\draw[arrow] (open)  -- node[elab,right]{no closed 4D theory exists}            (oggep);
\draw[arrow] (oggep) -- node[elab,right]{the projection must be a definite law}  (gdos);
\draw[arrow] (gdos)  -- node[elab,right]{apply to this paper's sector}          (obs);
\draw[arrow] (obs)   -- node[elab,right]{and the law is not arbitrary}           (pill);
\draw[arrow] (pill)  -- node[elab,right]{projection forced $\Rightarrow$ observable forced} (concl);
\end{tikzpicture}
}
\caption[The central claim, as one deductive funnel.]{\textbf{The central
claim, as one deductive funnel}, shared across the \GGTheory{} cohort and
specialized here to the proton. Because no physical system is closed, the
four-dimensional world every experiment reads is a boundary readout of a
six-dimensional bulk; a readout that is the terminal basin of its stratum
is not a free input but a determined one. The funnel is the same in every
paper of the cohort and the third box is what changes: for
\PaperPThree{} \citep{ArisakaP3Electron} it is the electron, and for this
paper it is the proton --- its mass, its stability, its lifetime --- read
off one geometry, which is why every proton is identical to every other.
The funnel states the shape of the argument, not its grade; the grades
are the fences listed in this section and the ledger of
Table~\ref{tab:import-ledger}.}
\label{fig:central-claim-p6}
\end{figure}


What is genuinely new in this paper, and claimed as such: the
component-resolved realization of baryon number by the
$\mathbb{Z}_3$-locked descent (Part~VI, with the obstruction theorem
that makes it necessary); the ground-attractor reading of the sharp
mass and its zero width (Part~III); the readout-label resolution of
the spin question (Part~IV); and the assembly itself --- eleven
questions that have never before been closed, or honestly fenced, in
one place by one structure.

\section{What would kill this paper}
\label{sec:kills}

Pre-registration first, argument after. Each criterion is developed
in its owning Part and collected in the falsification table of
Part~XII.

\begin{description}
\item[F-P6-1 (the constitutional kill).] A single observed proton
decay, in any channel. Gauged \UoneB{} with the $\mathbb{Z}_3$-lock
forbids $\Delta B = \pm1$ to all orders; one event at
Super-Kamiokande, Hyper-Kamiokande, DUNE, or JUNO falsifies the
mechanism outright. This is the rare kill that grows \emph{stronger}
with every year of silence and is decidable on a stated timescale:
the searches now in construction reach an order of magnitude beyond
the current limit by $\sim$2040.
\item[F-P6-2 (the fractional charge).] A single confirmed free
fractional charge --- in a detector, in bulk matter, anywhere. The
readout-locality reading of Q5 admits no exception.
\item[F-P6-3 (the $\Delta B = 2$ door).] An observed
neutron--antineutron oscillation. The winding selection rule forbids
$\Delta B = 2$ as absolutely as $\Delta B = 1$; an oscillation
signal at ESS or in bound-neutron searches kills it.
\item[F-P6-4 (the mirror).] A measured proton--antiproton asymmetry
--- in $|q|/m$, magnetic moment magnitude, or mass --- beyond the
CPT-paired readout prediction. Current precision: $16$ parts per
trillion in charge-to-mass \citep{Borchert2022}.
\item[F-P6-5 (the fifth force).] A detected long-range force coupled
to baryon number. The \UoneB{} of this framework is
Stueckelberg-massive; its residual is exactly global. A baryonic
fifth force at any range with gauge strength contradicts the
mechanism. The experiment has existed longer than the theory:
the equivalence-principle program has been, in effect, a
search for exactly this force for over a century (Part~VII).
\item[F-P6-6 (the ledger).] The registered kill criterion of the
Arisaka equation, quoted at its home grade \citep{ArisakaA7Qi4}: a
demonstrated boundary event that updates the baryonic ledger without
the locked informational unit.
\end{description}

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

\subsection{The cohort inheritance}
\label{sec:inheritance-p6}

\GGTheory{} is a twenty-two-paper research program
\citep{ArisakaA1Constitution} --- eleven A-papers, six P-papers, three
C-papers and two white papers --- whose structural claim is that the
observed laws are the boundary operational description of a real
six-dimensional bulk --- four macroscopic spacetime dimensions plus
one scale direction $\Ir$ and one compact phase circle $\Stheta$ ---
governed by four axioms and an integer ledger
$(M,N,L;J,K) = (1,3,5;7,35)$ \citep{ArisakaA3BI6}. Within that
program this paper is a member of the Predictions Branch --- the
proton paper: the deployment of the derived machinery on the particle
that carries almost all of the mass anyone has ever weighed, under the
thesis that the proton is the ground attractor --- the \GGA{} --- of
the color-singlet baryonic stratum. Like every member paper of the
cohort, it restates what it imports and owns what it proves, so that a
reader who never opens a companion loses the provenance but not the
argument.

Before the closest relatives, the family portrait. The reader new to
the program should spend one minute with
Figure~\ref{fig:cohort-map-p6}, because the whole architecture can be
read off it top to bottom, and every Part of this paper will draw on
some box of it. \emph{One root:} the constitution \AOne{} states four
axioms and seven postulates and proves nothing else --- all force is
downstream. \emph{One bulk:} AX-2 compiles the root into the GG6 Trio
--- \ATwo{} builds the six-dimensional geometry whose compact phase
circle carries the winding that Part~\ref{part:stability} reads as
baryon number, \AThree{} proves the anomaly cancellation that locks
the sectors together and supplies the ledger with its single
$\Delta B \in 3\mathbb{Z}$ door, and \AFour{} \citep{ArisakaA4Euclid} tells the history.
\emph{One boundary law:} AX-3 forces \AFive{} --- the open-system
dynamics that Part~\ref{part:basin} uses for the basin and its
depth --- organized by the two pillar papers \ASix{} (the geometric
pillars: \GFF{}, \GRIPin{}, the \GGA{} --- the method this paper runs
end to end) and \ASeven{} (their quantum-information twins, which
Part~\ref{part:qi} spends). \emph{Three readouts:} \AEight{} \citep{ArisakaA8QM} recovers
textbook quantum mechanics, \ANine{} derives the Standard Model's 28
parameters --- the codebook from which this paper's quark masses and
$\alpha_s$ come --- and \ATen{} derives the cosmological inventory
that Part~\ref{part:birth} imports for the birth of baryons;
\AEleven{} closes the series. \emph{Two phenomenology branches:}
everything below the capstone is phenomenology --- falsifiable
deployments of derived results, which is why their arrows are dashed.
This paper sits in the cyan Predictions frame, fourth and last in its
row, after the collider falsifier \PaperPOne{} \citep{ArisakaP1KK}, the family-number
derivation \PaperPTwo{}, and the sister paper \PaperPThree{}; the
short dashed arrow that reaches this paper along that row, labeled
\textsf{GGA}, is the one import that matters most --- the sister
paper supplies the method this paper runs. The teal frame carries the
cosmology siblings, and each touches this paper at one place.
\citealp{ArisakaC1Kaxion} is the cohort's other physical exploitation of
the phase circle's topology --- where this paper reads a winding of that
circle as baryon number, the Kaxion reads an angle on it as dark matter,
and the two are held by one ledger. \citealp{ArisakaC2Hubble} works the
derived expansion history in which the baryon inventory of
Part~\ref{part:birth} has to land. And \citealp{ArisakaC3BH} owns the
question this paper does not ask: what becomes of a globally conserved
winding at a horizon. The
reader who keeps this map in mind will never be lost: when
Part~\ref{part:basin} cites a theorem, it comes from the green or
yellow boxes; when a number appears, it comes from \ANine{}; and when
a fence is restated, it traveled here along one of the two dashed
arrows into the cyan frame --- to which the Inheritance Part
(Part~\ref{part:inheritance}) turns in detail.

\paragraph{The Foundations Branch.} On the map's left flank stands its
third frame --- the Foundations Branch, a single column beside the
spine, whose shared subject is the boundary's complex structure; its
two members are recorded here at roster level, one sentence each.
\PaperPFour{} \citep{ArisakaP4Real} derives the origin of the $i$ that
the whole quantum apparatus runs on --- the gauge phase from the
circle's quarter-turn, the state-space $i$ from the causal clock ---
and the compact direction whose quarter-turn supplies that phase is
the same direction whose winding this paper reads as baryon number.
\PaperPFive{} \citep{ArisakaP5CPT} gives the proton its twin: the
antiproton as a receipt of that circle--clock handshake, with C, P,
and T read as orientation reversals of the three real bulk factors ---
the identity that Part~\ref{part:stability} needs to close its
dihedral structure, and the chain Part~\ref{part:birth} imports for
the matter excess. Both members carry P-series designators; their
place at the spine's flank, not the phenomenology rows, marks their
foundational role.

The twenty-two-paper map, identical in every paper of the cohort, is
Figure~\ref{fig:cohort-map-p6}, which closes this subsection.
%% ===================================================================
%%  The twenty-two-paper cohort map, on its own page at the close of
%%  the cohort-inheritance subsection.  Verbatim copy of the master
%%  (A1-Constitution Figure 3) as carried by every other paper of the
%%  cohort: no self-reference marker anywhere, so the figure is
%%  identical in all twenty-two carriers.  Do not localise this block.
%% ===================================================================
\begin{figure}[p]
\centering
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\draw[arrowdash] (A10.south east) .. controls (84mm,-101mm) ..
 ([xshift=-6mm]W2.north);
%% W1 -> W2, solid and labelled: the mirror of the shared-i edge on the
%% left flank.  \PaperWTwo{} states the dependency itself -- it takes
%% "the structure that the arc arrives at, protection by topology
%% rather than by symmetry".
\draw[arrow] (W1.south) --
 node[edgelabel, pos=0.80, anchor=west, xshift=1mm]
 {symmetry $\to$ topology} (W2.north);

%% Centring pad, retired (v22): it cancelled a Bezier phantom that the
%% polyline above no longer creates; the bbox is now the ink.
\end{tikzpicture}}
\caption[The twenty-two-paper cohort map.]{The
twenty-two-paper cohort map of \GGTheory{}. The constitutional root
\AOne{} (violet, top) supplies the four axioms and seven postulates
from which the rest of the cohort descends: the GG6 Trio (\ATwo{},
\AThree{}, \AFour{}, red) realizes the bulk geometry; \AFive{}
(yellow) the boundary law; the \GDOS{} Trio pillars (\ASix{},
\ASeven{}, green) the geometric and quantum-information runtime; the
4D-Physics Trio (\AEight{}, \ANine{}, \ATen{}, blue) the textbook
readouts; and the closing capstone \AEleven{} (gold) the
quarks-to-consciousness identity cascade. Two framed branches carry
the phenomenology --- the \textbf{Predictions Branch} (cyan), whose
four papers hand their results along the row by the internal arrows
(\PaperPTwo{}'s $N{=}3$ into \PaperPThree{}, and that paper's
ground-attractor method, \textsf{GGA}, on into \PaperPSix{}), and the
\textbf{Cosmology \& Astrophysics Branch} (teal). The spine is
flanked by two columns of two, which do opposite jobs. On the left the
\textbf{Foundations Branch} (violet) feeds it: \PaperPFour{}, fed by
\ATwo{}'s Theorem~0 arena, manufactures the $i$ that \AEight{}
consumes, and \PaperPFive{} is the \ANine{}--\ATen{} interface, fed by
the ledger phases and feeding the $\eta_B$ chain across the width of
the map, with the shared-$i$ edge descending solid between them. On
the right the \textbf{White-Paper Branch} (brown) reads it:
\PaperWOne{} walks the century from Noether to Hopkins--Singer, and
\PaperWTwo{} dissects the structure that arc arrives at ---
protection by topology rather than by symmetry --- across sixteen open
problems. The white papers derive nothing new, carrying every result
at its parent's declared grade; the single dashed feed drawn into each
names its heaviest source, not its only one. \emph{Reading
conventions:} solid arrows mark axiom-forced descent and carry an edge
label naming the driving axiom or postulate where most illustrative;
dashed arrows mark the deployment of derived results, or historical
links inherited without being constitutionally forced.}
\label{fig:cohort-map-p6}
\end{figure}

\subsection{Roadmap of the sixteen Parts}
\label{sec:roadmap}

The paper walks its eleven questions in the order that lets each
answer stand on the ones before it, and the walk is plotted in
Figure~\ref{fig:parts-roadmap-p6}. The present Part has put the
questions and their century on the table, credited the standard
account, fenced the central claim, and pre-registered the kills.
Part~II gathers the inheritance --- the framework minimum, the
Standard-Model derivation with its stability theorem as received,
the $N=3$ theorem, the mirror apparatus, the sister method, and the
information pillars --- so that every later import has its
provenance stated once and its conditioning attached.

Part~III is the center of gravity: the ground-attractor theorem ---
the unique deepest basin of the color-singlet baryonic stratum ---
answering the sharp mass (Q1) and the exact charge (Q2) as two faces
of one fact, with the transmutation pricing of the mass and the
embedding pricing of the charge. Five Parts read the same fact from five more sides, and one
Part between them keeps the record. Part~IV answers the spin (Q3): the readout
label against the scheme-dependent bulk bookkeeping, with the
Electron--Ion Collider face pre-registered. Part~V takes confinement
and the absent quark together (Q4, Q5): the cage measured and the
mechanisms compared, the conditional area-law theorem T-P6-4 --- one
named hypothesis, the gap, from the full claim --- the
Goldstone-counting test, and the opening contract with the
Clay problem. Part~VI is the paper's own derivational core and its
emotional one: the immortality and the law (Q6, Q7) --- the
obstruction theorem, the center identity, the $\mathbb{Z}_3$-lock
that writes baryon number while breaking the symmetry and splitting
the Higgs multiplet, the anomaly ledger with its single
$\Delta B \in 3\mathbb{Z}$ door, and the Kamiokande arc from
construction to theorem. Part~VII then holds the physics still
for one Part and keeps the record: the seventy-year history of
the axiom Part~VI has just discharged --- from Lee and Yang's
one-page Letter of 1955 to the imposition of March 2025 --- kept
apart from the derivations so that history and proof never
borrow each other's authority. Part~VIII reads the twin (Q8): the shallow
basin, the $1.293$~MeV competition, the gate-force descent, and the
stabilizing bind. Part~IX climbs the ladder (Q9): six flavors
from $N = 3$, the excited tower, and the width-zero terminus.

Two Parts then face history and the ledger. Part~X imports the
birth of baryons (Q10) at its stated grade and proves the
consistency sentence: the anomaly door that forbids one death
permitted all births. Part~XI states what \UoneQi{} means for the
proton (Q11), at candidate grade, kill criterion first. Part~XII
weighs: significance, uniqueness against prior approaches, the
falsification table, the open problems (the D-P6 registry), and the
objections. Part~XIII concludes: the century, the tank, and the
theorem, in plain English. Part~XIV is the workshop: the technical
appendices, from the claim-status ledger to the complete
derivations. Part~XV answers the seventeen strongest objections, each
at the register it deserves. Part~XVI closes with the annotated
references, every entry carrying its relevance note. Read in order,
the sixteen Parts compose one argument;
read by need, each names its imports and can stand a visit on its
own.

%% \enlargethispage was tried here and does nothing: the page before
%% this tail is the full-page cohort-map float, which has no lines to
%% give, and the page after is the \clearpage-isolated roadmap plate.
%% Five lines between two full-page plates is structural.  Removed
%% rather than left in place doing nothing.
The questions are on the table, the contracts are signed, and
the map below shows the whole journey at a glance. One more
Part of preparation --- a walk through the toolshed --- and the
building begins.

%% Roadmap figure: house architecture (P3 v14.0 pattern -- vertical
%% Part column, right phase sidebar, the central-claim Part
%% highlighted), clearpage-isolated minipage + captionof.
%% v15.0: 0.92 -> 0.8871 = 0.92 x (452.9679/469.75502).  The plate and
%% its caption fitted one A4 page centered; Letter is the SHORTER
%% sheet and the +3.7 %% widening made the picture 3.7 %% taller, which
%% is the 5.43 pt the log reported as an overfull vbox -- \null then
%% held page 27 open blank and the plate broke to page 28.  The new
%% fraction restores the plate to exactly the printed size it had in
%% the deployed v1.25.  Same repair as P5-CPT's roadmap.
\clearpage
\null\vfill
\noindent\begin{minipage}{\textwidth}
\centering
\resizebox{0.8871\textwidth}{!}{%
\begin{tikzpicture}[
 font=\sffamily\footnotesize,
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\node[found] (I) at (0, 0)
 {\textbf{Part~I}\quad\textbf{The Eleven Questions and Their Century} \\[-1pt]
 \footnotesize What has, and has not, been explained since 1917; the honest status table; the claim, fenced};

\node[appar] (II) at (0, -14.5mm)
 {\textbf{Part~II}\quad\textbf{Inheritance} \\[-1pt]
 \footnotesize The cohort results this paper consumes, by citation, with every conditional clause attached};

\node[innov] (III) at (0, -32mm)
 {\textbf{Part~III}\quad\textbf{The Ground Geometric Attractor} \\[1pt]
 \footnotesize The sharp mass and the exact charge as one theorem (Q1, Q2): the unique deepest basin \\[-1pt]
 \footnotesize of the baryonic stratum, its depth priced by transmutation, its charge by the embedding};

\node[princ] (IV) at (0, -49.5mm)
 {\textbf{Part~IV}\quad\textbf{The Label That Would Not Decompose} \\[-1pt]
 \footnotesize Spin $1/2$ as the basin's readout label; the crisis as a custody dispute (Q3)};
\node[princ] (V) at (0, -64mm)
 {\textbf{Part~V}\quad\textbf{Confinement and the Absent Quark} \\[-1pt]
 \footnotesize The cage measured and compared; the conditional area-law theorem;\\
 no boundary channel for color (Q4, Q5)};
\node[princ] (VI) at (0, -78.5mm)
 {\textbf{Part~VI}\quad\textbf{The Immortal Proton} \\[-1pt]
 \footnotesize The $\mathbb{Z}_3$-lock writes baryon number; one twist, four locks; Kamiokande (Q6, Q7)};
\node[record] (R7) at (0, -93mm)
 {\textbf{Part~VII}\quad\textbf{The Seventy-Year Road to One Letter} \\[-1pt]
 \footnotesize The record of gauged baryon number, 1955--2026: the Letter, the refusals, the imposition};
\node[princ] (VIII) at (0, -107.5mm)
 {\textbf{Part~VIII}\quad\textbf{The Shallow Twin} \\[-1pt]
 \footnotesize The neutron: $1.293$~MeV up-slope, gate-force descent, stabilized by binding (Q8)};
\node[princ] (IX) at (0, -122mm)
 {\textbf{Part~IX}\quad\textbf{The Ladder} \\[-1pt]
 \footnotesize Six flavors from $N=3$; the excited tower; the width-zero terminus (Q9)};

\node[results] (X) at (0, -136.5mm)
 {\textbf{Part~X}\quad\textbf{The Birth of Baryons} \\[-1pt]
 \footnotesize The \PaperPFive{} chain at grade; the door that forbids one death permitted all births (Q10)};
\node[results] (XI) at (0, -151mm)
 {\textbf{Part~XI}\quad\textbf{The Information Charge} \\[-1pt]
 \footnotesize \UoneQi{} for the proton: one rearrangement, one unit; kill criterion first (Q11)};

\node[results] (XII) at (0, -165.5mm)
 {\textbf{Part~XII}\quad\textbf{Discussion} \\[-1pt]
 \footnotesize Significance \,\textbar\, uniqueness \,\textbar\, falsification \,\textbar\, open problems};
\node[summit] (XIII) at (0, -180mm)
 {\textbf{Part~XIII}\quad\textbf{Conclusions} \\[-1pt]
 \footnotesize The century, the tank, and the theorem --- in plain English};

\node[back] (XIV) at (0, -194.5mm)
 {\textbf{Part~XIV}\quad\textbf{Technical Appendices} \\[-1pt]
 \footnotesize The reference apparatus and the arithmetic companions, A through G};
\node[back] (XV) at (0, -209mm)
 {\textbf{Part~XV}\quad\textbf{Common Objections and Responses} \\[-1pt]
 \footnotesize Seventeen objections, by persona --- answered, with pointers into the paper};
\node[back] (XVI) at (0, -223.5mm)
 {\textbf{Part~XVI}\quad\textbf{References} \\[-1pt]
 \footnotesize Cited works, each with a relevance note linking it to the paper};

\draw[arrow] (I.south) -- (II.north);
\draw[arrowinnov] (II.south) -- (III.north);
\draw[arrowinnov] (III.south) -- (IV.north);
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\draw[arrow] (XII.south) -- (XIII.north);
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\draw[arrow] (XIV.south) -- (XV.north);
\draw[arrow] (XV.south) -- (XVI.north);

\node[phaselabel] at (67mm, 0mm) {\emph{Entrance}};
\node[phaselabel] at (67mm, -14.5mm) {\emph{Inheritance}};
\node[phaselabel, text=orange!70!black] at (75mm, -32mm)
 {\emph{\textbf{The one fact:}} \\ \emph{\textbf{the ground attractor}}};
\node[phaselabel] at (67mm, -64mm) {\emph{Three more} \\ \emph{faces}};
\node[phaselabel] at (67mm, -93mm) {\emph{The record}};
\node[phaselabel] at (67mm, -114.75mm) {\emph{Two more} \\ \emph{faces}};
\node[phaselabel] at (67mm, -143.75mm) {\emph{History \&} \\ \emph{ledger}};
\node[phaselabel] at (67mm, -172.75mm) {\emph{Weighing \&} \\ \emph{summit}};
\node[phaselabel] at (67mm, -209mm) {\emph{Back matter}};

\end{tikzpicture}%
}
\par\bigskip
\captionof{figure}[Roadmap of the sixteen Parts of GG-P6-Proton.]{Roadmap
of the sixteen Parts of \textsf{GG-P6-Proton}, in the nine phases
walked in \S\ref{sec:roadmap}: the \emph{entrance} (Part~I), the
\emph{inheritance} (Part~II), \emph{the one fact} (Part~III,
highlighted), \emph{three more faces} (Parts~IV--VI), \emph{the
record} (Part~VII), \emph{two more faces} (Parts~VIII--IX),
\emph{history and ledger} (Parts~X--XI), \emph{weighing and summit}
(Parts~XII--XIII), and the \emph{back matter} (Parts~XIV--XVI).
Part~III is highlighted because it carries the paper's central claim
--- the ground-attractor uniqueness of \S\ref{sec:central-claim} ---
of which the answers of Parts~IV--VI and VIII--IX are five further
faces, and on which every later Part stands.}
\label{fig:parts-roadmap-p6}
\end{minipage}
\vfill
\clearpage

%% ===================================================================
%%  PART II -- INHERITANCE
%% ===================================================================
\GGPart[Inheritance]{Inheritance}{The cohort results this paper consumes, by citation, with every conditional clause attached}
\label{part:inheritance}

No result in this paper is proved twice. The framework it
belongs to is a cohort of papers, each owning its own theorems,
and the discipline that keeps such a program honest is
performed here in the open: every result imported from
elsewhere is declared once, with its source, its strength, and
its conditions attached --- the way a builder walks the site
and inspects every delivered beam before construction begins.
A reader eager for the physics may skim this Part on a first
pass and return when a later chapter cites it; a reader
auditing the paper will find here the complete list of what
was assumed, what was proved elsewhere, and what this paper
must build for itself.
\section{How to read this Part}
\label{sec:how-to-read}

This paper stands on results proved elsewhere in the program, and
the discipline of the Predictions Branch is that every import is
declared once, here, with three things attached: its \emph{home} (the
paper that proves it), its \emph{grade} (derived; derived
conditionally, with the clauses named; interpretation; candidate),
and its \emph{consumer} (the Part of this paper that uses it).
Nothing in Parts~III--XI quietly assumes what this Part has not
declared. A reader who accepts the imports at their stated grades can
read the rest of this paper as self-contained; a reader who wishes to
audit an import will find its full derivation at the cited home. The
Part closes with the import ledger (Table~\ref{tab:import-ledger}),
which is the contract in one page.

\section{The runtime, in one picture: \texorpdfstring{\GRIP{}}{GRIP}
and the Ground Geometric Attractor}
\label{sec:grip-runtime}

Everything this paper proves about the proton is one instance of a
single mechanism, and the mechanism has a name the reader will meet
on nearly every page that follows. It is worth learning once, slowly,
here --- because once it is in hand, the eleven questions of Part~I
stop being eleven puzzles and become eleven views of one object. The
mechanism is \GRIP{}, the runtime by which this framework's geometry
settles into the stable things we observe, and its endpoint is the
object this paper is about.

Start with the picture, in plain words. Deep in the six-dimensional
geometry there is a fully symmetric starting configuration --- every
gauge equivalence intact, nothing yet committed, the theoretical
equivalent of a ball balanced on the exact top of a dome. That
symmetric top is unstable: the geometry carries a \emph{free-energy
landscape}, and a ball on a dome does not stay balanced. It rolls.
As it rolls it \emph{commits} --- picks a direction, breaks the
symmetry that held it at the top --- and continues downhill, losing
free energy at every step, until it reaches a valley it cannot leave.
That final valley is the stable object. In the vocabulary of this
program the four moves are named, and the reader should collect the
names now, because the rest of the paper uses them as ordinary words:

\begin{itemize}
\item \textbf{\DGI{}} (Dynamical Gauge Invariance) --- the symmetric
top: the fully gauge-invariant configuration from which the descent
begins, every equivalence still unbroken.
\item \textbf{$\cF_{\GFF}$} (the Geometric Free-energy Functional) ---
the landscape itself: the height function whose valleys are the stable
configurations and whose ridges are the barriers between them. The
descent obeys $d\cF_{\GFF}/dt \le 0$ --- downhill, always, the
open-system analogue of the second law.
\item \textbf{\GSSB{}} (Geometric Spontaneous Symmetry Breaking) ---
the commitment: at each saddle of the landscape the system selects one
branch, paying a definite topological toll to cross, and the symmetry
that offered the choice is broken by the choosing.
\item \textbf{\GA{}} and \textbf{\GGA{}} (Geometric Attractor; Ground
Geometric Attractor) --- the valley: a stable resting place. There are
two grades of valley, and the difference between them is the whole
subject of this paper.
\end{itemize}

Compactly, the runtime is the chain
\begin{equation}
\GRIP{} \;:=\; \DGI{} \;\to\; (\cF_{\GFF}) \;\to\; \GSSB{}
\;\to\; \GGA{},
\label{eq:grip-chain}
\end{equation} Two words in that chain do the most work in this paper and are easiest to conflate, so Figure~\ref{fig:two-regimes-p6} separates them: \GRIPin{}, the descent that brings a configuration to rest, and \GRIPout{}, the ascent that builds the next level of structure on a basin already at rest. Every Part of this paper is one or the other, and no Part is both.

\begin{figure}[htbp]
\centering
\resizebox{0.845\textwidth}{!}{%
\begin{tikzpicture}[
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\node[dgibox] (DGI) at (0, 0) {\textbf{\DGI{}}\\[-1pt]
 {\scriptsize Dynamical Gauge Invariance --- the symmetric top: every gauge equivalence intact, nothing committed}};
\node[gffbox] (GFF) at (0, -22mm) {\textbf{$(\cF_{\GFF})$ \quad Geometric Free-Energy Functional}\\[-1pt]
 {\scriptsize the fixed landscape; descent with $d\cF_{\GFF}/dt \leq 0$ (GFF-ST6)}};
\node[gssbbox] (GSSB) at (0, -44mm) {\textbf{\GSSB{} \quad Geometric Spontaneous Symmetry Breaking}\\[-1pt]
 {\scriptsize branch selection at the saddles; Bogomolnyi tolls $S = \pi k$ (GRIP-ST3)}};
\node[gabox] (GA) at (0, -66mm) {\textbf{\GA{} \quad Geometric Attractor(s)}\\[-1pt]
 {\scriptsize stable basins en route --- possibly way stations ($\mu$, $\tau$, an excited atom); \GRIPin{} continues through their exit channels}};
\node[ggabox] (GGA) at (0, -88mm) {\textbf{\GGA{} \quad Ground Geometric Attractor}\\[-1pt]
 {\scriptsize the unique deepest basin (GFF-ST7) --- \emph{where \GRIPin{} halts}: the exit-channel set is empty}};
\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);
\node[inbox] (GIN) at (-58mm, -112mm) {\textbf{\GRIPin{}}\\[-1pt]
 {\scriptsize fixed-landscape regime (def:GRIPin)}\\
 {\scriptsize relaxation \emph{on} the landscape}};
\node[outbox] (GOUT) at (58mm, -112mm) {\textbf{\GRIPout{}}\\[-1pt]
 {\scriptsize landscape-rebuilding regime (def:GRIPout)}\\
 {\scriptsize transient, out of equilibrium}};
\draw[ribarrow] (GA.west) to[bend right=25]
 node[arrowtext, pos=0.55, below, sloped]{$\cF_{\GFF}$ landscape unchanged}
 (GIN.north);
\draw[outarrow] (GGA.south) to[bend left=18]
 node[arrowtext, pos=0.55, below, sloped]{$\cF_{\GFF}$ landscape reorganized}
 (GOUT.north);
\draw[ribarrow, dashed] (GIN.north) to[out=90, in=180]
 node[arrowtext, pos=0.35, left=2pt, text=cyan!55!black, font=\footnotesize\itshape]
 {within-stratum\\closed loop}
 (DGI.west);
\node[nextbox] (NEXT) at (58mm, -136mm)
 {\textbf{\DGI{} \quad (next stratum)}\\[-1pt]
 {\scriptsize a new descent begins on the reorganized landscape}};
\draw[outarrow] (GOUT.south) -- (NEXT.north)
 node[arrowtext, pos=0.5, right=3pt, text=orange!70!black, font=\footnotesize\itshape]
 {supplies the next\\initial datum $c_{\rm top}$};
\end{tikzpicture}
}
\caption[The two regimes of the runtime: descent and ascent.]{\textbf{The
\GRIP{} cascade and its two regimes, in one picture}, adapted from
\ASix{} \citep{ArisakaA6Gi4}; the full eighteen-rung deployment is
\AEleven{}'s \citep{ArisakaA11Origin}. The same machinery runs in two
directions and this paper uses both, so the distinction is worth a
drawing rather than a definition. \GRIPin{} is the descent: a
configuration rolls down the free-energy landscape, commits at each
saddle, and comes to rest --- it is how the proton reaches its basin and
how an excited baryon falls to it. \GRIPout{} is the ascent: a settled
basin becomes the substrate on which the next level of structure is built
--- proton and electron to hydrogen, and hydrogen upward. Nothing is
reversed between them; the descent has no inverse, and the ascent is a
different move on a different stratum, which is the whole content of the
Chain of Figure~\ref{fig:grip-chain-ladder}.}
\label{fig:two-regimes-p6}
\end{figure}

read left to right as \emph{free symmetric top} $\to$ \emph{landscape}
$\to$ \emph{commitment} $\to$ \emph{unique ground state}. It is proved,
as a runtime with the stated descent and halting properties, in the
program's dedicated paper \ASix{} \citep{ArisakaA6Gi4}, from which
this paper inherits it whole.

\paragraph{The two grades of valley --- the distinction that makes a
proton.} A landscape has many valleys, and they are not all alike. Most
are stable only in the mild sense that a small nudge rolls back:
displace the ball a little and it returns to the valley floor. Such a
valley is a \GA{}, a Geometric Attractor --- \emph{a} stable
configuration, but not necessarily \emph{the} stablest, because
somewhere else on the landscape there may be a deeper valley, and a
large enough disturbance --- or, in quantum mechanics, a patient
enough wait for tunneling --- will eventually carry the system there.
A muon is a \GA{} in this sense: perfectly well-defined, but with a
deeper valley (the electron) reachable through a gate, so it lasts
2.2~microseconds and is gone. An excited atom is a \GA{}: it will
relax. A radioactive nucleus is a \GA{}: it will decay. These are way
stations, not destinations.

The other grade is the destination. A \GGA{}, a \emph{Ground}
Geometric Attractor, is the valley that is \emph{globally} deepest on
its landscape --- and, crucially, is also admissible under the
framework's four consistency pillars (the \QiFour{} conditions of
\ASeven{}, which guarantee the state is a legitimate quantum readout
and not a mathematical artifact). At a \GGA{} there is nowhere deeper
to go. No nudge, no wait, no tunneling event can find a lower valley,
because there isn't one. This is stability of an entirely different
character from a \GA{}'s: not stability against small perturbations,
but stability against \emph{everything the landscape allows} ---
including the slow, inexorable quantum tunneling that eventually
empties every merely-local minimum. In one line: \emph{a \GA{} is
stable against a push; a \GGA{} is stable against eternity.}

\begin{figure}[htbp]
\centering
\resizebox{0.609\textwidth}{!}{%
\begin{tikzpicture}[font=\small]
\definecolor{gaC}{RGB}{150, 110, 40}
\definecolor{ggaC}{RGB}{40, 90, 160}
\definecolor{ridgeC}{RGB}{120, 120, 130}
%% the landscape curve: top-left high, a shallow GA valley, a ridge,
%% then the deep GGA valley
\draw[line width=1.4pt, black!80] plot[smooth, tension=0.7] coordinates {
 (0,5.6) (1.1,4.0) (2.2,3.05) (3.1,2.7) (3.9,2.95) (4.7,3.7)
 (5.6,2.4) (6.6,1.15) (7.6,0.55) (8.6,0.9) (9.6,3.2) };
%% DGI top
\fill[black!70] (0.12,5.55) circle (2.2pt);
\node[anchor=west, font=\footnotesize] at (0.35,5.75)
 {\textbf{\DGI{}}: the symmetric top --- nothing committed};
%% descent arrow along the curve
\draw[-{Stealth[length=2.4mm]}, line width=1.0pt, black!55]
 (0.7,4.7) to[bend left=8] (2.7,2.85);
\node[font=\footnotesize\itshape, text=black!55, anchor=east] at (2.35,4.6)
 {descent: $d\cF_{\GFF}/dt \le 0$};
%% GA valley
\fill[gaC] (3.05,2.68) circle (2.6pt);
\node[gaC, font=\footnotesize\bfseries, anchor=south] at (3.05,2.95)
 {a \GA{}};
\node[text=gaC, font=\footnotesize, align=center, anchor=north]
 at (2.05,2.35) {a way station:\\ muon, excited\\ atom, decaying\\ nucleus};
%% ridge / gate between GA and GGA
\node[text=ridgeC, font=\footnotesize, align=center] at (5.15,4.5)
 {a \GSSB{} gate\\ (Bogomolnyi toll $S=\pi k$)};
\draw[-{Stealth[length=2.2mm]}, line width=0.9pt, gaC!70!black]
 (3.5,2.7) to[bend left=20] (5.6,2.25);
\node[text=gaC!70!black, font=\footnotesize\itshape, align=center, anchor=south] at (6.15,2.85)
 {\GRIPin{} continues:\\ the exit channel is open};
%% GGA valley
\fill[ggaC] (7.55,0.5) circle (3.0pt);
\node[ggaC, font=\footnotesize\bfseries, anchor=east] at (7.35,0.5)
 {the \GGA{}};
\node[text=ggaC, font=\footnotesize, align=center, anchor=north]
 at (7.55,0.05) {the destination:\\ electron, \textbf{proton}, atom;\\
 no deeper valley exists};
\node[text=ggaC, font=\footnotesize\itshape, align=left, anchor=west] at (8.75,1.7)
 {\GRIPin{} halts here:\\ the exit-channel\\ set is empty};
\end{tikzpicture}%
}
\caption[The two grades of valley: \GA{} and \GGA{}.]{The runtime
\eqref{eq:grip-chain} as one landscape. From the symmetric top
(\DGI{}) the configuration rolls down the free-energy landscape
$\cF_{\GFF}$, committing at each \GSSB{} gate. It can be captured in a
way-station valley --- a Geometric Attractor (\GA{}: a muon, an excited
atom, a radioactive nucleus) --- from which an open exit channel still
leads downhill, so the descent (\GRIPin{}) continues. It halts only at
the globally deepest admissible valley, the Ground Geometric Attractor
(\GGA{}): the electron, the \emph{proton}, the atom --- the stable
things the world is built from. A \GA{} is stable against a
perturbation; only a \GGA{} is stable against tunneling, because on
its stratum there is no deeper valley to tunnel to. This paper proves
that the proton is a \GGA{}.}
\label{fig:ga-gga}
\end{figure}

This distinction is not a technicality; it is the reason chemistry
exists. Chemistry made exactly this distinction a century ago without
naming it. A molecule's electrons have many stationary arrangements,
but chemistry is built on the \emph{ground state}: every water
molecule is identical, and indefinitely stable, precisely because its
electrons occupy not merely \emph{a} stable arrangement but \emph{the}
lowest one. The Rydberg constant is a property of \emph{the} hydrogen
ground state, not an average over a population of excited states that
might drift. The \GGA{} is that everyday idea --- ``the ground state
is what lasts and what is reproducible'' --- promoted to a
constitutional principle of the framework's runtime. When this paper
says the proton is exactly reproducible to twenty-one digits across
the whole universe (Q2), it is saying the proton is a \GGA{}, and
reproducibility is what \GGA{}-hood means.

\paragraph{The two regimes: \texorpdfstring{\GRIPin{}}{GRIP-In} and
\texorpdfstring{\GRIPout{}}{GRIP-Out}.} One more pair of names, and
the vocabulary is complete. The descent just described --- rolling
down a \emph{fixed} landscape until it halts at the \GGA{} --- is
called \GRIPin{}, the in-flow regime. It is the regime this paper uses
in Part~VIII to price the neutron's $\beta$-decay, exactly as the
sister paper \PaperPThree{} used it to price the muon's: a \GA{}
draining through its one open gate toward the \GGA{} below it. But a
landscape can also \emph{change}. A transient, out-of-equilibrium
event can reorganize $\cF_{\GFF}$ itself --- open valleys that did not
exist before, close others --- and thereby hand the next \GRIPin{}
descent a new starting point. That landscape-reorganizing regime is
called \GRIPout{}. The one-sentence division, from \ASix{}:
\emph{\GRIPout{} opens the landscape; \GRIPin{} finishes the journey.}
The cosmic cooling that froze the electroweak vacuum was a \GRIPout{}
event; every weak decay since has been \GRIPin{} on the landscape that
event left behind. We will need both regimes in the next section,
because the proton's deepest significance is not that it is one
\GGA{}, but that it is a \GGA{} on which a whole \emph{sequence} of
\GRIPout{} events built everything else.

\section{The Chain of \texorpdfstring{\GRIPout{}}{GRIP-Out}: where the
proton sits}
\label{sec:grip-chain}

Here is the grand picture, and it is the reason a paper about a single
particle claims to touch the heart of the whole framework. The runtime
of the previous section does not run once. It runs again and again, up
a ladder, each rung standing on the one below it --- and the proton is
one of the lowest rungs, the ground on which nearly everything else is
built.

The pattern is an alternation. \GRIPin{} settles the system into the
\GGA{} of its current landscape; that settled \GGA{} then becomes the
\emph{substrate} on which a \GRIPout{} event can act, reorganizing the
landscape and opening a new stratum with new valleys; \GRIPin{}
descends again to the new \GGA{}; and the cycle repeats. The capstone
paper \AEleven{} \citep{ArisakaA11Origin} proves that this alternation,
given the framework's fixed geometry, runs as a single determined
cascade from the elementary particles all the way to conscious
observers --- and that at every rung the outcome is a \emph{unique}
\GGA{}, so the whole ladder is forced, not lucky. Schematically,
\begin{equation}
\mathcal{C}
\;\xrightarrow{\;\GRIP{}\;}\;
\GGA{}_1
\;\xrightarrow{\;\GRIPout{}\;}\;
\GGA{}_2
\;\xrightarrow{\;\GRIPout{}\;}\;
\cdots
\;\xrightarrow{\;\GRIPout{}\;}\;
\GGA{}_{18},
\label{eq:cascade}
\end{equation}
the \emph{Chain of \GRIPout{}}: each $\GGA{}_n$ the settled ground
state of its stratum, each arrow a landscape-reorganizing event
supplied --- from the second rung onward --- by the dynamics of the
rungs already built. The reader coming to this framework for the first
time should pause on what \eqref{eq:cascade} claims: that the same
four-move engine that selects a quark's properties also selects an
amino acid, a genetic code, and the length of a conscious moment ---
one mechanism, run at higher and higher rungs, with no new assumption
added for chemistry, life, or mind. That claim is \AEleven{}'s to
defend, and this paper imports it at \AEleven{}'s grade; what matters
\emph{here} is where the proton sits on the ladder, and what stands
on it.

\begin{figure}[htbp]
\centering
\resizebox{0.736\textwidth}{!}{%
\begin{tikzpicture}[font=\small,
 rung/.style={draw=black!55, rounded corners=2pt, minimum width=64mm,
  minimum height=8.5mm, align=center, inner sep=2pt},
 phys/.style={rung, fill=blue!7, draw=blue!45!black},
 protrung/.style={rung, fill=blue!16, draw=blue!70!black, line width=1.1pt},
 chem/.style={rung, fill=green!8, draw=green!45!black},
 life/.style={rung, fill=orange!10, draw=orange!55!black},
 up/.style={-{Stealth[length=2.2mm]}, line width=0.8pt, draw=black!55}]
\node[phys] (q) at (0,0) {quarks \& leptons \;\;{\scriptsize (the elementary readouts)}};
\node[protrung] (p) at (0,1.15) {\textbf{the proton} \;\;{\scriptsize the \GGA{} of the baryonic stratum --- \emph{this paper}}};
\node[phys] (e) at (-3.6,1.15) {};
\node[phys] (atom) at (0,2.3) {atoms: H, He, Li, \ldots \;\;{\scriptsize the periodic table ($E_{\mathrm{Ry}}=13.6$~eV)}};
\node[chem] (mol) at (0,3.45) {molecules: H$_2$O, CO$_2$, NH$_3$ \;\;{\scriptsize small stable compounds}};
\node[chem] (aa) at (0,4.6) {amino acids \& sugars \;\;{\scriptsize 20 L-amino acids}};
\node[chem] (poly) at (0,5.75) {polymers: peptides, nucleotides};
\node[life] (atp) at (0,6.9) {ATP \;\;{\scriptsize the energy quantum, $\Delta G \approx 30$~kJ/mol}};
\node[life] (rna) at (0,8.05) {RNA \;\;{\scriptsize the four-letter alphabet}};
\node[life] (dna) at (0,9.2) {DNA \;\;{\scriptsize information closure}};
\node[life] (up1) at (0,10.35) {cells, bodies, minds \;\;{\scriptsize \ldots the cascade continues (\AEleven{})}};
\draw[up] (q) -- (p); \draw[up] (p) -- (atom); \draw[up] (atom) -- (mol);
\draw[up] (mol) -- (aa); \draw[up] (aa) -- (poly); \draw[up] (poly) -- (atp);
\draw[up] (atp) -- (rna); \draw[up] (rna) -- (dna); \draw[up] (dna) -- (up1);
\node[anchor=west, font=\footnotesize\itshape, text=black!60, align=left]
 at (3.6,5.2)
 {each rung a \GGA{};\\ each arrow a \GRIPout{} event\\
 opening the next stratum\\ on the settled ground below.\\[4pt]
 \textbf{everything above the proton}\\
 \textbf{stands on the proton being}\\
 \textbf{an \emph{absolutely stable} \GGA{}.}};
\end{tikzpicture}%
}
\caption[The Chain of GRIP-Out: the proton as the ground of stable
matter.]{The Chain of \GRIPout{} \eqref{eq:cascade}, drawn for the
matter rungs (the full eighteen-rung cascade to mind is \AEleven{}'s
\citep{ArisakaA11Origin}). Each rung is the Ground Geometric Attractor
of its stratum; each upward arrow is a \GRIPout{} event that
reorganizes the landscape and opens the next stratum on the settled
ground below it. The proton (highlighted) is the \GGA{} of the
baryonic stratum, and it is load-bearing in the most literal sense:
atoms need a stable proton to have a nucleus; molecules need atoms;
amino acids, proteins, ATP, RNA, and DNA each need the rung beneath
them. Remove the proton's absolute stability and the chain has no
ground --- the periodic table and everything above it never forms.
This paper is the study of that ground rung.}
\label{fig:grip-chain-ladder}
\end{figure}

The proton sits near the very bottom, on the \emph{baryonic stratum}:
the family of color-singlet, three-quark configurations carrying
baryon number one. Its \GGA{} --- the globally deepest admissible
valley of that stratum --- is the proton. And here the picture turns,
and turns in the direction that gives this paper its title as a
message to experimentalists. It is natural to think of the proton, as Part~III invites one to, as an \emph{endpoint}: the bottom of
the baryonic descent, the place with no exit gate, the terminus. That
is true, and it is half the story --- the downward-looking half. The
half that matters for why the proton is ``the heart of the theory'' is
the \emph{upward}-looking one. Because the proton is an absolutely
stable \GGA{}, it can serve as the substrate for the next \GRIPout{}
event: cooling lets electrons fall onto nuclei built from protons, and
the atomic stratum opens (the periodic table, $\GGA{}_{\text{atomic}}$
at the Rydberg scale); chemistry lets atoms bond, and the molecular
stratum opens; and so on up through amino acids, proteins, ATP, RNA,
DNA, and every rung above. \emph{Every stable object more complex than
a bare particle stands, ultimately, on the proton holding still
forever.} A proton that decayed --- even slowly --- would be a ground
that crumbles, and the whole tower built on it would have a
finite shelf life. That the tower does \emph{not} have a shelf life,
that matter is permanent enough to build a chemist who can ask why, is
a statement about the proton being a \GGA{}, and it is the statement
this paper exists to prove.

\paragraph{Physics intuition: the foundation stone.} A cathedral's
foundation stone is not its most spectacular feature --- no one visits
to admire it --- but every arch, vault, and spire transmits its load
to that stone, and if it shifts, everything above it moves. The proton
is the foundation stone of the material world. The excited baryons and
the free neutron above it are scaffolding that comes and goes
(Part~IX's tower, Part~VIII's twin); the atoms, molecules, and
living machinery above \emph{it} are the cathedral. The century asked
what the proton is made of and answered in exquisite detail
(Part~III's mass review). This framework asks a different question ---
what \emph{kind} of thing the proton is, that it can hold the weight
of chemistry forever --- and answers: it is the Ground Geometric
Attractor of the baryonic stratum, the foundation stone at the bottom
of the Chain of \GRIPout{}.

\section{Two grounds, one grammar: the proton and the electron}
\label{sec:two-grounds}

One comparison locates this paper in the cohort more precisely than
any other, and it is worth making explicitly, because it is the
comparison that shows the proton's account is not improvised for this
paper but is a second instance of a template the program has already
run. The template is: \emph{an absolutely stable particle is the
Ground Geometric Attractor of its stratum, and all its mysteries are
corollaries of that one fact.} The sister paper \PaperPThree{}
\citep{ArisakaP3Electron} runs the template for the electron; this
paper runs it for the proton. Side by side:

\begin{table}[htbp]
\centering
\caption[The electron and the proton, question by question.]{The electron and the proton, question by question. Each row asks one question of both particles and sets the two answers side by side: which stratum each is the ground attractor of, what the single governing fact is, why every readout of either is exactly reproducible, and why neither can decay. The parallel is the point --- the two papers are one argument run twice, once on the charged-lepton stratum and once on the baryonic one.}
\label{tab:electron-proton}
\small
\begin{tabular}{p{0.30\textwidth}p{0.30\textwidth}p{0.30\textwidth}}
\toprule
& \textbf{Electron (\PaperPThree{})} & \textbf{Proton (this paper)} \\
\midrule
The stratum & the charged-lepton stratum & the baryonic stratum
$(B,Q,J)=(1,+1,\tfrac12)$ \\
\addlinespace
The one fact & the electron is the \GGA{} of that stratum & the
proton is the \GGA{} of that stratum \\
\addlinespace
Why exactly reproducible & every readout is the same ground basin ---
\GGA{} universality & every readout is the same ground basin ---
\GGA{} universality \\
\addlinespace
Why it cannot decay & no deeper charged-lepton valley; \GRIPin{}
halts & no deeper $B=1$ valley; gauged winding removes every operator
to a deeper one (Part~VI) \\
\addlinespace
The way stations & the muon and tau: \GA{}s of the same landscape &
the excited baryons and the free neutron: \GA{}s of the same
landscape \\
\addlinespace
The twin & the positron: conjugate readout of the same basin
(\PaperPFive{}) & the antiproton: conjugate readout of the same basin
(Part~VI) \\
\bottomrule
\end{tabular}
\end{table}

The parallel is exact except in one place, and the exception is the
whole physics of this paper. For the electron, ``no deeper valley''
is a statement about a Morse landscape: the charged-lepton stratum
simply has no basin below the electron's, and \GRIPin{} halts because
there is nowhere to go. For the proton, ``no deeper valley'' has to be
made \emph{exact} against a subtle threat --- the grand-unified
operators that would connect the proton to a deeper, baryon-number-%
violating valley (a positron plus a pion, say). What removes those
operators, and so guarantees the baryonic \GGA{} is a true ground and
not merely a very deep \GA{}, is the gauging of baryon number and the
$\mathbb{Z}_3$-locked winding of Part~VI. In the language of this
section: \emph{Part~VI is the proof that the proton is a \GGA{} rather
than a \GA{}} --- the proof that its exit-channel set is not merely
small but exactly empty. The electron gets its \GGA{}-hood from the
shape of its landscape; the proton earns its \GGA{}-hood from a gauged
topological law. Same grammar, one stratum over, with the deepest
clause promoted from geometry to gauge symmetry.

This is why the two papers are twins and why P6 belongs beside P3 on
the shelf. Between them they establish the two \GGA{}s from which all
ordinary matter is assembled: the electron on the leptonic stratum,
the proton on the baryonic one. Bind one electron to one proton and
the atomic stratum opens --- hydrogen, the first rung above the
particles in the Chain of \GRIPout{}. Everything else is chemistry.

\paragraph{And one grammar more: what kind of protection each
enjoys.} The template above says both particles are ground
attractors. It is worth adding what protects each \emph{once it has
landed}, because the two answers differ, and the difference is the
sharpest available statement of what this paper claims. The
electron's stability is the absence of a deeper valley: nothing on
its stratum lies below it, and the descent has no inverse, so it
stays. The proton's stability is of a stronger kind. Its baryon
number is a \emph{winding} of the compact geometry --- an integer ---
and an integer is not protected by the absence of a lower state but
by the impossibility of continuous motion: nothing continuous acts on
$\mathbb{Z}$. That is the top rung of the companion review's ladder,
\emph{protection without symmetry} \citep{ArisakaW1Symmetry}: a
symmetry can be broken, and every symmetry-based proton-stability
argument of the last fifty years was broken on purpose by the theory
that proposed it. A winding has nothing to break.

The contrast worth carrying away is with the electroweak sector,
which lives on the same circle and is protected far more weakly. The
Higgs is also circle-side, but it is a \emph{continuous} modulus --- a
Wilson-line angle, not a winding number --- so it moves, and every
quantity built from it runs \citep{ArisakaA9SM}. What non-locality
buys there is finiteness, not fixity. Baryon number is the other case:
not a modulus but a class, not finite but frozen. Both facts are
consequences of the same sentence --- holonomy requires closed loops,
and an interval has none \citep{ArisakaW2Holonomy} --- and the two
readings of it are why the electroweak scale is calculable while the
proton is immortal.



\paragraph{And the one axis on which they are opposite.} The rows above
are parallels; there is one axis on which the two particles are not
parallel but opposite, and it is worth naming because a reader who
noticed it and was not told would be right to suspect the template of
being elastic. The electron has no inside. \PaperPThree{}
\citep{ArisakaP3Electron} carries that as a theorem rather than as a
bound --- the structural form factor is identically unity at every
momentum transfer however large, every structural radius moment vanishes
exactly, and the paper pre-registers the sharpest version of the claim it
can: no substructure will ever be found, at any spatial scale, by any
probe, at any energy. The proton has an inside, and had one before anyone
could say what an inside would mean; this paper's own historical Part
dates the discovery to 1933 and prices the interior at
$\langle r^2 \rangle^{1/2} \approx 0.84$~fm. Figure~\ref{fig:formfactor-p6}
puts the two on one axis: the electron's flat line at $F \equiv 1$, the
proton's falling curve, and --- for contrast --- the curve a composite
electron would trace, falling at whatever scale its compositeness lived
at.

So the two grounds are exact for opposite reasons, and this is the
sharpest way to say what a Ground Geometric Attractor is. The electron is
exactly reproducible because there is nothing inside it to vary. The
proton is exactly reproducible although there is a great deal inside it
to vary: a storm of gluon field energy, sea pairs and quark kinetic
energy that Part~III prices at nearly the whole $938.272$~MeV, none of it
fixed, all of it in motion. What is fixed is not the contents but the
basin. Every proton is the same readout of the same terminal valley, so
every proton weighs the same to the last measured digit while no two of
them contain the same configuration for two consecutive instants. An
attractor does not need its interior to be still. It needs its interior
to have nowhere else to go.

That is also the honest place to say what this paper does \emph{not}
inherit. The pointlikeness theorem is \PaperPThree{}'s and stays there:
nothing in the present paper derives it, depends on it, or would be
damaged if it failed. It is carried here as the contrast case, because
the contrast is what makes the shared grammar legible --- and because a
framework that produced exactness by one route only would be a weaker
thing than one that produces it by two.


\begin{figure}[htbp]
\centering
\resizebox{0.676\textwidth}{!}{%
\begin{tikzpicture}[
 yscale=1.45,
 font=\small,
 arr/.style={->, line width=0.80pt, draw=black!70},
 lab/.style={font=\footnotesize, align=left}
]

\draw[arr] (0, 0) -- (0, 5.2) node[anchor=south] {$|F(q^2)|$};
\draw[arr] (0, 0) -- (12.6, 0);
\node[font=\footnotesize\itshape, anchor=north east] at (12.4, -0.42)
 {momentum transfer $q$ (schematic, logarithmic) --- resolution $\hbar/q$ sharpens $\to$};
\node[anchor=east, font=\footnotesize] at (-0.08, 4.0) {$1$};
\draw[black!40, dashed] (0, 4.0) -- (12.2, 4.0);

\fill[black!12] (6.3, 0) rectangle (6.55, 4.0);
\node[anchor=east, font=\footnotesize] at (-0.08, 0) {$0$};

\draw[line width=1.0pt, draw=black!55] plot[smooth, samples=80, domain=0:7.5]
 (\x, {4.0/((1+(\x/1.6)^2)^2)});
\node[lab, text=black!55, anchor=west, align=left] at (2.8, 2.5)
 {proton: $F$ falls ---\\ structure at $\langle r^2\rangle^{1/2} \approx 0.84$ fm};
\draw[-{Stealth[length=2mm]}, draw=black!55] (2.75, 2.3) -- (1.8, 1.25);

\draw[line width=1.0pt, draw=red!70!black, dashed] plot[smooth, samples=80, domain=0:12.0]
 (\x, {4.0/((1+(max(\x-8.2,0)/0.9)^2)^2)});
\node[lab, text=red!70!black, anchor=west, align=left] at (9.6, 2.9)
 {composite electron:\\ structure appears at\\ \emph{some} scale $\Lambda$};
\draw[-{Stealth[length=2mm]}, draw=red!70!black] (9.55, 2.5) -- (9.05, 1.7);

\draw[line width=1.6pt, draw=blue!60!black] (0, 4.0) -- (12.2, 4.0);
\node[lab, text=blue!60!black, anchor=south west] at (2.6, 4.22)
 {\textbf{\GGTheory{}: $F \equiv 1$ at every scale, forever} (the Pointlikeness Theorem)};
\node[lab, anchor=west, text=black!60, align=left] at (2.6, 1.15)
 {experimental reach to date\\ (contact-interaction scales $\mathcal{O}(10)$ TeV)};
\draw[-{Stealth[length=2mm]}, draw=black!60] (5.95, 1.15) -- (6.32, 1.05);
\node[lab, text=black!70, anchor=west, font=\scriptsize] at (3.6, 3.5)
 {every probe to date: no deviation from $1$};
\end{tikzpicture}
}
\caption[Structure and its absence: the proton, the electron, and the
composite electron that is not there.]{\textbf{The one axis on which the
two ground attractors are opposite.} The horizontal axis is momentum
transfer, so moving right is resolving finer; the vertical axis is the
structural form factor. The proton's curve falls, and where it falls says
how big the inside is --- $\langle r^2 \rangle^{1/2} \approx 0.84$~fm, a
number Part~I dates to 1933 and Part~III spends in full. The electron's
line does not fall: \PaperPThree{} \citep{ArisakaP3Electron} carries
$F \equiv 1$ as a theorem at every momentum transfer however large, not
as a bound awaiting improvement, and the dashed curve shows what a
composite electron would have to do instead --- fall somewhere, at some
scale, which every compositeness model owes and none has ever delivered.
The figure is imported as the contrast case and nothing in this paper
rests on it: the electron's exactness comes from having no interior, the
proton's from having an interior with nowhere to go, and the two together
are what a Ground Geometric Attractor means.}
\label{fig:formfactor-p6}
\end{figure}

\section{The framework minimum}
\label{sec:framework-minimum}

Five structures constitute the minimum this paper needs from the
foundational branch; they are stated here in the form in which they
are used.

\paragraph{The bulk and the boundary.} Spacetime as experienced is
the four-dimensional boundary readout of a six-dimensional bulk
$\cM_6 = M_4 \times \Ir \times \Stheta$: a radial direction $r$
(scale; an interval) and a compact cycle $\theta$ (phase; a circle),
neither spatial. Depth along the interval is measured in $e$-folds down from the geometric seam, $r = \ln(E_{\mathrm{seam}}/\mu)$ with $E_{\mathrm{seam}} = M_{\mathrm{Pl}}/2$; every depth quoted in this paper is measured from there. One register note, because the cohort uses the neighboring symbol for a different scale: the grand-unified edge written $E_{\mathrm{GUT}} = M_{\mathrm{Pl}}e^{-2\pi} = 2.28\times10^{16}$~GeV elsewhere in the program sits $2\pi - \ln 2 = 5.6$ $e$-folds below the seam, at $r = 5.6$ on this paper's axis, and is not this paper's origin. The constitutional axioms --- locality--causality,
a single variational principle, the minimal-structure clause ---
are those of \AOne{} \citep{ArisakaA1Constitution}, realized as
geometry in \ATwo{} \citep{ArisakaA2GG6}. Everything in this paper
happens on that boundary or descends to it.

\paragraph{The open-system law.} The boundary is not a closed
system: what appears on it is governed by the projection dynamics of
\GDOSm{} \citep{ArisakaA5GDOS}, whose tower theorem (T-GDOS-9.1)
supplies the mode expansion this paper's Part~VI reads: every bulk
field is a tower over the compact directions, the light floor of the
tower is what the boundary sees, and the massive floors cluster at
the lock scale $\Mlock \approx 3.85$~TeV.

\paragraph{Attractors and their readout.} The boundary's stable
objects are the Ground Geometric Attractors (\GGA{}) of their strata
--- the concept developed at length in
\S\S\ref{sec:grip-runtime}--\ref{sec:two-grounds} above --- organized
by the free-energy functional \GFF{} and reached by the descent
machinery of \ASix{} \citep{ArisakaA6Gi4}. The in-flow reading
(\GRIPin{}) prices the descent of any non-ground configuration ---
the apparatus this paper uses for the neutron's $\beta$-decay in
Part~VIII exactly as the sister paper used it for the muon.

\paragraph{The selection principle.} Where a symmetric structure
offers branches, the runtime selects the stable attractor branch:
geometric spontaneous symmetry breaking (\GSSB{}), also from
\ASix{}. Part~V's reading of confinement --- the color-singlet
branch as the unique stable attractor in strong coupling --- is a
\GSSB{} statement inherited from the Standard-Model paper's GRIP
chapter and engaged at its stated grade there.

\paragraph{The anomaly bookkeeping.} The modified Bianchi identity
of \AThree{} \citep{ArisakaA3BI6} supplies the Green--Schwarz
inflow that Part~VI's anomaly ledger charges its two non-vanishing
coefficients against. This paper does not re-derive that
bookkeeping; it consumes it as hypothesis H3 of the stability
theorem, exactly as its home paper states it.

\section{The gauge inheritance: \texorpdfstring{\ANine{}}{A9}}
\label{sec:gauge-inheritance}

The largest single inheritance is the Standard-Model derivation of
\ANine{} \citep{ArisakaA9SM}, from which this paper takes five
results verbatim.

\paragraph{The bulk gauge group.} The unified bulk group is
\begin{equation}
\SUfive \times \UoneB \times \UoneQi \times \text{Gravity},
\label{eq:bulk-group-p6}
\end{equation}
with the two Abelian factors \emph{local} in the bulk --- the
constitutional fact from which this paper's Q7 begins. Each
Abelian factor is \emph{imposed} as a local gauge invariance, in
deliberate analogy with electromagnetism; the imposition of
\UoneB{} is the framework's founding act, and its record ---
external and internal --- is kept in Part~VII. The visible
descent is
\begin{equation}
\SUfive \times \UoneB \times \UoneQi
\;\xrightarrow[\text{boundary conditions + holonomy}]{}\;
\SUthreeC \times \SUtwoL \times \UoneY ,
\label{eq:descent-p6}
\end{equation}
with hypercharge fixed inside \SUfive{} (hence charge quantization)
and the electroweak stage completing to \UoneEM{} as in the home
paper.

\paragraph{The matter packets.} Each family is one bulk spinor
packet whose chiral zero modes arrive in the complete
$\overline{\mathbf{5}} \oplus \mathbf{10}$ pattern --- fifteen
left-handed Weyl fields per family, forty-five in all; the interval
boundary condition projects the chirality, and the gauge anomalies
cancel packet by packet as a bookkeeping identity. Baryon-number
assignments on the split spectrum,
\begin{equation}
B(Q) = +\tfrac13,\qquad B(u^c) = B(d^c) = -\tfrac13,\qquad
B(L) = B(e^c) = 0,
\label{eq:B-assignments-p6}
\end{equation}
are forced by the constitutional locality of \UoneB{}. The parity
assignment of the descent, derived at the \AFive{} level and used by
\ANine{},
\begin{equation}
P = \mathrm{diag}(+1,+1,+1,-1,-1),
\label{eq:parity-P-p6}
\end{equation}
is the object whose conjugation action Part~VI examines with care
--- the single most load-bearing inheritance in this paper.

\paragraph{The stability theorem, as received.} \ANine{} proves
(its T-SM-60.1): \emph{assuming} (H1) the $X,Y$ bosons of the
adjoint are bulk-only, absent from the boundary spectrum;
(H2) the bulk \UoneB{} descends to act on the split fields as
baryon number itself; and (H3) the \UoneB{} gauge anomalies are
canceled by the \BIsix{} Green--Schwarz inflow --- then every
local operator with $\Delta B = \pm1$ or $\pm2$ is forbidden to all
orders, and the only admissible violation is the \SUtwoL{} anomaly,
$\Delta B = 3n$, exponentially closed at zero temperature. At its
home, (H2) was the one load-bearing hypothesis still to be
discharged, deferred to a dedicated successor paper on proton decay
--- \emph{this} paper. Part~VI is that discharge: the obstruction
theorem that makes (H2) impossible for any Lie-algebra charge, and
the $\mathbb{Z}_3$-locked descent that realizes it as a winding,
with (H1) emerging as a corollary. The protective symmetry
throughout is the \emph{global} baryon number residual to a
Stueckelberg-massive \UoneB{} --- no long-range fifth force
accompanies it, which is itself falsifier F-P6-5.

\paragraph{The strong scale.} The mass pricing this paper uses in
Part~III is \ANine{}'s: the derived gauge couplings run by radial
transport; $\Lambda_{\mathrm{QCD}}$ emerges by dimensional
transmutation from the derived $g_3$; and the resulting division of
the proton's mass --- the strong interaction's own scale paying
$\sim99\%$, the Higgs-coupled quark masses the remaining percent ---
is inherited with its home paper's honesty note: the transmutation
mechanism is standard physics \citep{Wilson1974}; what the
framework adds is the derivation of the coupling it starts from.

\paragraph{The basin/gate architecture.} \ANine{} divides the
forces by structural role: unbroken long-range forces are
\emph{basins}; the broken, short-range weak force is the \emph{gate}
--- the unique channel through which descents between basins run.
Part~VIII uses this verbatim: the neutron's $\beta$-decay is a gate
transit, not an anomaly of the strong sector.

\section{The family inheritance: \texorpdfstring{\PaperPTwo{}}{P2}}
\label{sec:family-inheritance}

The family count is the two-sided theorem of \PaperPTwo{} \citep{ArisakaP2N3}: $N \ge 3$ from the requirement of a physical CP phase (the MSMF lower bound), $N \le 3$ from the capacity theorem T-N3-6 --- the family stratum funds exactly $\pi$ of relaxation budget per family, derived from its own quantum geometry (Berry transport at the Born theorem's grade), against Bogomolnyi tolls that grow as the triangle number --- hence $N = 3$, carried at the home paper's ``derived (conditional)'' register, the conditioning being its coupling clause with the integrality lemma L-INT, stated there. Six quark flavors follow by counting: each family
carries one up-type and one down-type flavor, so three families
give exactly
\begin{equation}
(u,d),\quad (c,s),\quad (t,b),
\label{eq:six-quarks-p6}
\end{equation}
and the ``six'' of Q9 is not a separate mystery but the doubled
family count. Part~IX consumes this, together with the excited
spectrum machinery of the framework minimum, for the ladder.

\section{The mirror inheritance: \texorpdfstring{\PaperPFive{}}{P5}}
\label{sec:mirror-inheritance}

From the CPT companion \citep{ArisakaP5CPT} this paper takes two
distinct inheritances, one algebraic and one cosmological, and it is
scrupulous about their different grades.

\paragraph{At theorem grade: the twin and the reversal.} The origin
of antiparticles is imported as proved structure: the circle
carries conjugation as orientation reversal ($\theta \to -\theta$;
T-P5-A), the clock structure pairs it (T-P5-B), the handshake fixes
the Feynman--St\"uckelberg pairing as the unique consistent reading
(T-P5-C), and the fermionic factorization makes the charge flip
definitional (T-P5-C2). Two Parts consume this: Part~VI, where the
reversal-is-conjugation identity turns out to be precisely the
closure condition of the dihedral structure that writes baryon
number --- the deepest single point of contact between the two
papers --- and the falsifier face F-P6-4, where the antiproton's
measured equalities \citep{Borchert2022} test the conjugate-readout
reading of the antiproton exactly as the positron tested the
electron's in the sister paper.

\paragraph{At conditional grade: the birth chain.} \PaperPFive{}'s
account of the baryon asymmetry --- resonant TeV-scale leptogenesis
\citep{FukugitaYanagida1986} with the resonance condition as its one
named input, the sphaleron-transport normalization as its one named
conjecture (C-P5-C), and the yield $\eta_B$ delivered to the
cosmology paper \citep{ArisakaA10Cosmos} --- is imported at exactly
that declared grade, nothing upgraded. Part~X adds only the
consistency sentence this paper is uniquely positioned to prove:
the chain's sphaleron step passes through the
$\Delta B \in 3\mathbb{Z}$ anomaly door, which is the one door
Part~VI's selection rule leaves open \citep{tHooft1976}. The
mechanism that made the baryons and the mechanism that preserves
them are the same equation read at two temperatures.

\section{The method inheritance: \texorpdfstring{\PaperPThree{}}{P3}}
\label{sec:method-inheritance}

The sister paper \citep{ArisakaP3Electron} supplies the method this
paper walks: state the questions at their sharpest; credit the
standard account in an honest status table; prove the ground-state
uniqueness of the stratum; read every remaining question as a face
of it; fence what is conditional; register what is owed; and
pre-register the kills. It also supplies a worked example this
paper leans on directly: the positron as the \emph{conjugate
readout of the same basin} --- one object, two orientations of
reading it --- which Part~VI generalizes from the electron's basin
to the baryonic winding (the antiproton as the reversed wrap), and
which the dihedral structure of the $\mathbb{Z}_3$-lock echoes at
the level of the compact geometry itself. Where the electron's
stability was the absence of an exit gate from a basin, the
proton's will be the absence of an operator that can unwind a
winding: the two stability theorems are siblings, and Part~VI
states the family resemblance precisely.

\section{The information inheritance: \texorpdfstring{\ASeven{}}{A7}}
\label{sec:information-inheritance}

From the quantum-information pillars \citep{ArisakaA7Qi4} this
paper takes, at that paper's own candidate grade and with its kill
criterion quoted intact, the Arisaka equation: at every elementary
boundary event the energetic, electric, baryonic, and informational
currents are locked,
\begin{equation}
J_E/\QB \;=\; J_e \;=\; J_B \;=\; J_{Qi},
\qquad
\QB \;=\; \frac{2}{5}\,\frac{\ERyd}{K} \;\approx\; 0.1555~\mathrm{eV},
\label{eq:arisaka-equation-p6}
\end{equation}
with $K = 35$ the ledger constant of the cohort. Part~XI's task is
strictly bounded: to state what this locking means for the proton
--- one baryonic rearrangement, one informational elementary update,
priced at \QB{} --- and what would kill it (F-P6-6), without
promoting the grade its home paper declares.

\section{The import ledger}
\label{sec:import-ledger}

\begin{table}[htbp]
\centering
\small
\caption[The import ledger.]{The import ledger: every result this
paper consumes, its home, its grade as declared at home, and the
Part that consumes it. Grades: \textbf{D} derived; \textbf{D(c)}
derived conditionally (clauses stated at home and where used);
\textbf{INT} interpretation, named; \textbf{CAND} candidate, kill
criterion registered.}
\label{tab:import-ledger}
\begin{tabular}{p{0.34\textwidth}p{0.14\textwidth}p{0.10\textwidth}p{0.30\textwidth}}
\toprule
\textbf{Import} & \textbf{Home} & \textbf{Grade} & \textbf{Consumed
by} \\
\midrule
Bulk, boundary, axioms; realization as geometry &
\AOne{}; \ATwo{} & D & everywhere \\
\addlinespace
\GDOSm{} projection; tower theorem T-GDOS-9.1; $\Mlock$ &
\AFive{} & D & III (floors), VI (winding floors) \\
\addlinespace
\GGA{}/\GFF{}/\GSSB{}/\GRIPin{} attractor machinery &
\ASix{} & D & III, V, VII \\
\addlinespace
Green--Schwarz inflow (\BIsix{}) & \AThree{} & D & VI (as H3,
verbatim) \\
\addlinespace
Bulk group \eqref{eq:bulk-group-p6}; descent
\eqref{eq:descent-p6}; packets; $B$-assignments
\eqref{eq:B-assignments-p6}; parity \eqref{eq:parity-P-p6} &
\ANine{} (\AFive{}) & D(c) & VI (core), III (charge) \\
\addlinespace
Stability theorem T-SM-60.1 as received (H1--H3) &
\ANine{} & D(c) & VI (completed there) \\
\addlinespace
Derived couplings; $\Lambda_{\mathrm{QCD}}$ by transmutation;
basin/gate architecture & \ANine{} & D(c) & III (mass), VII (gate) \\
\addlinespace
$N = 3$ two-sided theorem; six flavors by doubling &
\PaperPTwo{} & D(c) & VIII \\
\addlinespace
Antiparticles: T-P5-A/B/C/C2 (reversal = conjugation; pairing;
charge flip definitional) & \PaperPFive{} & D & VI (closure
condition), F-P6-4 \\
\addlinespace
Leptogenesis chain; C-P5-C; $\eta_B$ to cosmology &
\PaperPFive{}; \ATen{} & D(c) & IX (at grade) \\
\addlinespace
Ground-state method; conjugate readout worked example &
\PaperPThree{} & D & method throughout; VI \\
\addlinespace
Arisaka equation \eqref{eq:arisaka-equation-p6}; \QB{}; kill
criterion & \ASeven{} & CAND & X (at grade), F-P6-6 \\
\addlinespace
Pointlikeness $F \equiv 1$ and its falsification interface, imported as the contrast case only & \PaperPThree{} & D & II (\S\ref{sec:two-grounds}), nowhere load-bearing \\
\addlinespace
Holonomy primer: loop phase, Wilson loop, and the semigroup arrow & \PaperWTwo{} & INT & VI (\S\ref{sec:abelian-ledger}) \\
\bottomrule
\end{tabular}
\end{table}

The ledger's one-sentence summary, for the reader keeping score:
this paper's genuinely new claims live in Parts~III--VI, and
everything they stand on is either derived at home or fenced by
name here; nothing arrives unlabeled.

The supply lines are laid and inspected. From here the paper
builds. The next Part erects the central claim --- the proton as the
Ground Geometric Attractor of its stratum --- and every Part after it
reads that one structure from another side. The reader who skimmed is released to the
physics; the reader who audited now knows exactly what the
physics will be allowed to use.

%% ===================================================================
%%  PARTS III-XIII -- SYNOPSES (drafting placeholders; each is
%%  replaced by its full Part in a subsequent revision block)
%% ===================================================================
\GGPart[The ground attractor]{The Ground Geometric Attractor}{The proton as the \GGA{} of the baryonic stratum: the sharp mass and the exact charge as one theorem (Q1, Q2)}
\label{part:basin}

Now the building starts, and it starts at the center of
gravity. This Part proves the paper's central claim, and the
claim is one sentence: \emph{the proton is the unique Ground
Geometric Attractor --- the \GGA{} --- of the baryonic
stratum.} That is the same kind of statement the sister paper
made for the electron (\S\ref{sec:two-grounds}); here it is made
for the particle at the foundation of all ordinary matter. Read
in the plain-English gloss of Part~II, a \GGA{} is the
\emph{terminal basin} of a landscape --- the lowest resting
place matter can reach, with no gate leading anywhere deeper ---
and that image will serve throughout; but the concept doing the
work is the \GGA{}, because it is the \GGA{}, not merely a deep
valley, that ties the proton to the electron, to the atom, and
to every rung of the Chain of \GRIPout{} above it
(\S\ref{sec:grip-chain}).

Everything in the eleven-question list is a corollary of that
one sentence. This Part spends it first on the two oldest
questions: why the proton has one exact mass (Q1) --- because
the mass is the \emph{depth} of the ground basin, and a depth
does not come in variants --- and why its charge cancels the
electron's to twenty-one digits (Q2) --- because the charge is
the ground basin's exact label. The remaining questions read the
same basin from other sides: the spin is its readout label
(Part~IV), the confinement of its quarks is why the stratum has
a boundary at all (Part~V), and the immortality that makes it a
\GGA{} rather than a merely-deep attractor is the gauged winding
of Part~VI. Everything later in the paper stands on what happens
in these pages; a reader with time for only one technical Part
should choose this one.
\section{The question the century could not ask}
\label{sec:sharp-question}

Begin with what ``sharp'' means, because the whole Part turns on it.

A composite object in quantum theory does not automatically have
\emph{a} mass. It has a spectrum. Strike a bell and you hear a
fundamental and its overtones, each with a width --- each ring dies
away. The hadrons are like this: the $\Delta(1232)$ has a mass of
roughly $1232$~MeV and a width of $117$~MeV \citep{PDG2024}, meaning
it survives for about $5\times10^{-24}$ seconds and its ``mass,''
measured event by event, scatters across a hundred-MeV band. Nearly
every entry in the baryon tables is such a resonance: a bump, not a
number.

The proton is not a bump. Its mass,
\begin{equation}
m_p = 938.272\,088~\mathrm{MeV},
\label{eq:mp-value}
\end{equation}
is a true point in the spectrum: the width is not small, it is
\emph{zero} --- no strong width, no electromagnetic width and, so far
as four decades of watching can tell, no weak width either. And the
point is universal. The proton in a Kamioka water molecule, the
proton in a Fermilab argon nucleus, the proton that left a quasar
ten billion years ago --- every one presents the identical number
\eqref{eq:mp-value}, to every digit ever measured.

Now put this against what the proton \emph{is}. The three valence
quarks' Higgs-coupled masses total about $9$~MeV --- we will price
this precisely in \S\ref{sec:depth} --- roughly one percent of
\eqref{eq:mp-value}. The other ninety-nine percent is the
chromodynamic storm: gluon field energy and a boiling sea of
quark--antiquark pairs, a system so violently interacting that no
perturbative expansion converges anywhere near it. The century's
instruments took the proton apart into that storm; what no
instrument explains is why the storm, everywhere and always,
\emph{adds up to the same exact number with no width at all}.

\paragraph{Physics intuition: why this should astonish you.}
Imagine being told that every hurricane on Earth --- every one,
regardless of where it formed or what it swallowed --- carries
exactly the same total energy, to twenty-one digits, forever. You
would not call that meteorology; you would call it a conservation
law wearing a costume, and you would go looking for the law. Q1 is
that situation, in miniature, in every atom.

\paragraph{For the reader in the tank.} This question is not
decoration for your thesis; it is its unstated first axiom.
Super-Kamiokande's fiducial mass of $22.5$~kton of water holds
about $7.5\times10^{33}$ protons --- $22.5\times10^{9}$~g of water
at ten protons per eighteen grams per mole. A decay search sums
over all of them as if they were one object observed
$7.5\times10^{33}$ times, and Hyper-Kamiokande will make that
$\sim 6\times10^{34}$, and DUNE will do the same with the protons
of argon. The entire logic of a partial-lifetime limit ---
$\tau/B > N_p\,t\,\epsilon/2.3$ at $90\%$ confidence for zero
observed events --- rests on the premise that ``the proton'' is a
single, exactly reproducible physical object rather than a family
of near-copies. Q1 asks why that premise is true. This Part
answers: because the proton is not \emph{an} arrangement of its
storm; it is the \emph{floor} of a landscape, and floors do not
come in variants.

\section{The stratum, the landscape, and the Ground Geometric Attractor}
\label{sec:stratum}

We now set the stage with the precision the claim requires. Three
definitions, each inherited (Part~II) and each doing exact work.

\paragraph{The stratum.} A \emph{stratum} is the set of boundary
configurations sharing the exact conserved labels of the theory.
For this Part the labels are three: baryon number $B = 1$ ---
which Part~VI establishes as the winding of the compact cycle, an
exactly conserved topological index; electric charge $Q = +1$ ---
the winding of the electromagnetic fiber, exactly conserved
because \UoneEM{} is unbroken; and the fermionic readout label
$J = \tfrac12$ of the boundary spinor (Part~IV's subject). The
\emph{color-singlet} qualifier is automatic rather than imposed:
as Part~V develops, color has no boundary readout channel, so
every element of a boundary stratum is a singlet by construction.
The neutron is \emph{not} in this stratum --- it shares $B$ and
$J$ but sits at $Q = 0$; it is the neighboring stratum's story,
told in Part~VIII. Strangeness and the excited tower populate the
same $(B,Q)$ strata at higher levels; they are Part~IX's rungs.

\paragraph{The landscape.} On each stratum the free-energy
functional \GFF{} of \ASix{} \citep{ArisakaA6Gi4} assigns to every
configuration its cost; the \GDOSm{} projection \citep{ArisakaA5GDOS}
determines which configurations are boundary-readable at all, and
its tower theorem organizes them into floors. ``Landscape'' is not
a metaphor here: it is the graph of \GFF{} over the stratum, and
the dynamics of any non-stationary configuration is descent on it,
priced by \GRIPin{}.

\paragraph{The attractor, and the ground attractor.} A
\emph{basin} is a local minimum of \GFF{} on the stratum
together with its attraction region --- in the runtime vocabulary
of Part~II, a Geometric Attractor (\GA{}): a stable
configuration. But most basins are not the story's end. A basin
is a \GA{} that is \emph{not} the ground attractor if a
descent channel still leaves it toward a deeper basin --- an open
exit, down which \GRIPin{} continues. Every excited baryon is
such a \GA{} (that is why it exists long enough to be named, and
why it then decays); the free neutron is such a \GA{} (Part~VIII
walks it down its one open channel). A basin is the \emph{ground}
attractor --- the \GGA{}, the \emph{terminal basin} in plain
terms --- when it is the global minimum of the stratum: no
configuration sits lower, so the exit-channel set is empty and
\GRIPin{} halts there. Only a \GGA{} can have exactly zero width,
because width \emph{is} the inverse lifetime of a descent, and a
global minimum has nowhere to descend to. The distinction is the
one Part~II drew in general and this Part now applies: a \GA{} is
stable against a nudge; the \GGA{} is stable against eternity.

\paragraph{Physics intuition: the landscape is real because the
widths are graded.} Look at the baryon tables with this picture
in mind and they reorganize themselves into \GA{}s ranked by how
far they sit above the ground. The $\Delta(1232)$ sits a shallow
$294$~MeV above the nucleon with a strong-descent channel wide
open: it is a \GA{} with a fast exit, width $117$~MeV. The
$\Lambda(1116)$ has its strong channels closed by strangeness and
must descend through the gate force: a \GA{} with only a slow
exit, width $2.5\times10^{-12}$~MeV. The neutron's last channel
is a $1.293$~MeV chink: a \GA{} with a barely-open exit, width
$7.5\times10^{-25}$~MeV. The proton: no exit at all --- the
\GGA{}, width zero. Four objects, widths spanning twenty-six
orders of magnitude, and the single organizing variable is
\emph{how far above the ground attractor you sit and what channel
the labels leave open}. The landscape is not decoration; it is
the only picture in which that twenty-six-order column reads as
one fact --- and the bottom of the column, the one object with no
exit, is the \GGA{} this Part is about.


Figure~\ref{fig:baryon-basin} draws the landscape this Part has
been describing --- the tower's basins above, the shallow twin one
wall over, and the ground attractor at the bottom with no gate
out of it (the \emph{terminal basin}, in the plain-English gloss).
Every claim of the next section is a statement about this picture.

\begin{figure}[htbp]
\centering
\resizebox{0.798\textwidth}{!}{%
\begin{tikzpicture}[
 yscale=1.35,
 font=\small,
 arr/.style={->, line width=0.80pt, draw=black!70},
 floorline/.style={dashed, draw=black!45, line width=0.5pt},
 gatemark/.style={draw=red!70!black, line width=0.9pt},
 desc/.style={-{Stealth[length=2.6mm]}, line width=1.1pt, draw=blue!60!black},
 lab/.style={font=\footnotesize, align=center}
]
%% axes
\draw[arr] (-0.3, -0.4) -- (-0.3, 6.0) node[anchor=south] {$F_{\GFF}$};
\draw[arr] (-0.4, -0.3) -- (12.8, -0.3);
\node[font=\footnotesize\itshape, anchor=north east] at (12.6, -0.50)
 {configuration coordinate on the color-singlet baryonic stratum (schematic 1D cut)};
%% the landscape curve
\draw[line width=1.3pt, draw=black!80] plot[smooth, tension=0.75] coordinates {
 (0.0, 5.6) (1.0, 4.4) (2.3, 3.6) (3.0, 3.7) (3.9, 4.4)
 (4.9, 3.0) (5.9, 2.0) (6.7, 2.2) (7.4, 2.9)
 (8.5, 1.15) (9.9, 0.85) (11.2, 1.5) (12.2, 5.0) };
%% basin floors
\draw[floorline] (2.3, 3.6) -- (13.0, 3.6) node[anchor=west, font=\footnotesize] {$\Delta,\ N^{*}$, the tower};
\draw[floorline] (5.9, 2.0) -- (13.0, 2.0) node[anchor=west, font=\footnotesize] {$n$ ($+1.293$~MeV)};
\draw[floorline] (9.9, 0.85) -- (13.0, 0.85) node[anchor=west, font=\footnotesize] {$p$ ($938.272$~MeV)};
\node[font=\scriptsize\itshape, anchor=west, text=black!60] at (0.0, 0.35) {(levels not to scale)};
%% occupants
\fill[black!75] (2.55, 3.61) circle (1.8pt) node[lab, below=2pt, xshift=-7mm] {excited wraps\\ width $\sim 10^{2}$~MeV};
\fill[black!75] (6.2, 2.01) circle (1.8pt);
\node[lab, anchor=east] at (5.55, 2.35) {the shallow twin (Part~VIII)};
\fill[blue!60!black] (9.9, 0.86) circle (2.2pt) node[lab, below=3pt, text=blue!60!black]
 {$p$\\ \textbf{the \GGA{}: no exit gate}\\ \textbf{(T-P6-3)}};
%% gates
\draw[gatemark] (3.9, 4.52) -- (3.9, 4.28);
\node[lab, text=red!70!black, anchor=south] at (3.9, 4.60) {strong gates:\\ widths, $\sim 10^{-23}$~s};
\draw[gatemark] (7.4, 3.02) -- (7.4, 2.78);
\node[lab, text=red!70!black, anchor=west] at (8.55, 4.0) {the one narrow gate\\ ($\beta$; $Q = 0.782$~MeV)};
%% descent arrows
\draw[desc] (2.9, 3.9) to[bend left=25] (4.9, 3.35);
\draw[desc] (6.6, 2.55) to[bend left=28] (8.4, 1.5);
\node[lab, text=blue!60!black, anchor=west] at (0.0, 1.1)
 {\textbf{\GRIPin{} descent down the stratum:}\\
 the tower crumbles through strong gates;\\
 the twin drips through one weak gate;\\
 the floor has nowhere to go};
%% terminality annotation
\node[lab, text=black!65, anchor=west] at (8.5, 5.25)
 {nothing below the floor:\\ uniqueness = T-P6-3~(ii);\\ zero width = leg (iii)};

\end{tikzpicture}%
}
\caption[The baryonic landscape: the tower, the twin, and the ground attractor.]{The
baryonic stratum in one picture: a schematic one-dimensional cut
through the \GFF{} landscape for color-singlet three-quark matter.
The excited tower ($\Delta$, $N^{*}$, the hyperon shelf) occupies
the upper basins, each with wide strong-interaction gates (lifetimes
$\sim 10^{-23}$~s); the neutron sits $1.293$~MeV up-slope of the
floor with exactly one narrow gate ($\beta$-decay, endpoint
$0.782$~MeV --- Part~VIII walks through it); and the proton is the
floor itself: the unique deepest basin, with no gate out --- the
ground-attractor theorem T-P6-3 of this Part. The sharp mass (Q1) is
the floor's depth; the zero width is its terminality; identity of
all readouts is the uniqueness leg. The winding that forbids
tunneling out of the stratum entirely is Part~VI's business.}
\label{fig:baryon-basin}
\end{figure}

\section{The ground-attractor theorem}
\label{sec:terminal-theorem}

\begin{theorem}[\textbf{T-P6-3}: The proton is the Ground
Geometric Attractor of the baryonic stratum]
\label{thm:terminal-basin}
On the stratum $(B, Q, J) = (1, +1, \tfrac12)$ of the boundary,
the free-energy functional \GFF{} has a Ground Geometric
Attractor (\GGA{}), and the proton is its readout. Concretely:
\textup{(i)} \GFF{} attains a minimum (a \GA{} exists);
\textup{(ii)} the minimizing basin is unique; \textup{(iii)} it
is the \emph{ground} attractor, not merely a deep one --- every
decay channel out of it is closed by the exact labels, so its
exit-channel set is empty and its readout has width exactly zero;
and \textup{(iv)} every readout of that \GGA{}, anywhere on the
boundary, presents identical observables. The proton is that
Ground Geometric Attractor.
\end{theorem}

\noindent The four legs are exactly the definition of a \GGA{}
unpacked into checkable statements. Legs (i) and (ii) establish
that a \emph{unique} deepest basin exists --- the ``ground'' and
``attractor'' of the name. Leg (iii) is the clause that
distinguishes a \GGA{} from a mere \GA{}: not stability against a
perturbation, but the empty exit-channel set --- \GRIPin{} halts
here because there is nowhere deeper to go. Leg (iv) is the
universality that \GGA{}-hood forces, the same clause that makes
every electron identical in the sister paper
(\S\ref{sec:two-grounds}), here transposed one stratum over. Each
is established at a stated grade, one at a time; the status ledger
of \S\ref{sec:part3-ledger} records the grades. The reader should
notice the division of labor that the rest of the paper will
honor: legs (i), (ii), and (iv) are properties of the
\emph{landscape}, provable on this stratum with the machinery
already in hand; leg (iii) --- the \emph{exactness} of ``no
deeper valley,'' the clause that promotes the proton from a very
deep \GA{} to a true \GGA{} --- is the one that must be defended
against the grand-unified operators that would open an exit, and
that defense is the gauged winding of Part~\ref{part:stability}.
In one line: \emph{Part~VI is the proof that the proton's
exit-channel set is not merely small but exactly empty --- the
proof that the baryonic ground attractor is a \GGA{}.}

\paragraph{Leg (i): existence (a \GA{} exists).} The \GDOSm{}
tower theorem guarantees the stratum's spectrum is organized in
floors bounded below --- the functional cannot run to $-\infty$,
because the boundary projection of a bulk with a stable action
(\AOne{} AX-2's single variational principle; the \BIsix{} capacity that gives \PaperPTwo{} its upper leg, its capacity theorem T-N3-6) admits no bottomless direction. This is the structural statement; its empirical shadow
is nonperturbative QCD's own existence statement, verified on the
lattice for four decades \citep{Wilson1974, Durr2008}: the $B=1$
spectrum has a lowest state. Grade: derived within the
framework's imports; independently confirmed numerically in the
standard theory.

\paragraph{Leg (ii): uniqueness (the attractor is unique).}
Suppose two distinct minimizing basins on the stratum. Both are
boundary-readable zero-width states with identical exact labels
$(1, +1, \tfrac12)$ --- the situation the sister paper's identity
theorem \citep{ArisakaP3Electron} excludes for the leptonic
stratum, and the argument transposes: two degenerate ground
attractors of one stratum either (a) are connected by a
configuration path, in which case the higher saddle makes one of
them non-ground --- contradiction; or (b) are disconnected within
the stratum, in which case a conserved label distinguishes them
--- contradiction with the stratum's definition, since the labels
of the theory are exactly the winding pair and the spinor label
(a fourth exact boundary label would be a fourth exact symmetry,
which the constitutional inventory of \AOne{} does not contain).
Empirically: eighty years of hadron spectroscopy have found
exactly one zero-width $B=1$ state, and every ``second proton''
candidate has resolved into an excited \GA{} with a width. Grade:
derived, conditional on the completeness of the constitutional
label inventory --- the fence is named in the ledger.

\paragraph{Leg (iii): terminality --- the zero width.} A decay
needs a destination: a configuration of strictly lower \GFF{}
reachable while conserving every exact label. Walk the candidate
channels. \emph{Strong and electromagnetic descents} preserve all
labels but need a lower $B=1$ floor --- there is none; the proton
is the floor (legs i--ii). \emph{Gate-force (weak) descents}
can move within and between strata --- it is exactly how the
neutron dies (Part~VIII) and the muon before it --- but the gate
conserves $B$ and $Q$, and the $B=1$ floor of the charged stratum
has no lower relative anywhere: the lightest $B=1$ configurations
of \emph{any} charge are the nucleon pair, and the proton
undercuts both members. \emph{$B$-violating descents} --- the
channels Georgi--Glashow opened on paper
\citep{GeorgiGlashow1974} --- are closed to all orders by the
winding selection rule of Part~VI: $\Delta B = \pm 1$ operators
carry non-zero winding charge and have identically vanishing
coefficients, the sole exception being the topological
$\Delta B \in 3\mathbb{Z}$ vertex \citep{tHooft1976}, frozen at
$e^{-4\pi/\alpha_W} = e^{-372} \approx 10^{-162}$, a rate whose reciprocal
exceeds the age of the universe by so many orders that writing
the number is theater. Hence: width exactly zero, with the
$B$-conservation leg carried at Part~VI's grade (conditional on
H3 and the pins R2, R3 --- the fences travel with the
theorem, as always). Grade: derived conditional, clauses as
stated.

\paragraph{Leg (iv): identity of readouts.} A basin has no
adjustable interior: it is a stationary point of a functional
determined by the geometry, and a readout of it reports the
stationary values, not a history of how the configuration got
there. Two protons cannot differ, because there is nothing in the
basin for them to differ \emph{by} --- exactly the argument that
made every electron identical in the sister paper, transposed to
the stratum one winding over. The lattice gives the standard
theory's version of the same fact: the simulation needs only the
coupling and the quark masses --- no per-proton dials --- and
returns the measured mass \citep{Durr2008}. Grade: derived, given
legs (i)--(iii).

\paragraph{Physics intuition: the storm is the bulk's business.}
The apparent paradox of Q1 --- exact number from boiling storm ---
is a custody confusion. The storm (the sea, the gluon flux, the
virtual pairs) is \emph{bulk} structure: real, physical, doing
$99\%$ of the energetic work, and not itself a boundary
observable. The boundary reads the stationary value of the
functional the storm extremizes --- one number --- just as a
barometer reads a pressure without reading the position of any
molecule. The proton's sharpness is not the storm holding still;
it is the floor of the landscape being where it is regardless of
how the storm rages above it. \emph{The proton is not a thing the
storm builds; it is the place the storm settles.}

\paragraph{The ground that everything stands on.} The theorem
has been stated looking downward --- the proton as the bottom of
the baryonic descent, the basin with no exit. That is leg (iii),
and it is true. But the reason this theorem is the center of the
paper, and arguably of the framework, is what it means looking
\emph{upward}. A \GGA{} is not only an endpoint; it is a
\emph{foundation}. Because the proton's exit-channel set is
exactly empty --- because it is a \GGA{} and not a slowly-draining
\GA{} --- it can serve as permanent substrate for the next
\GRIPout{} event in the Chain of \S\ref{sec:grip-chain}: cooling
lets electrons fall onto nuclei built from protons, and the
atomic stratum opens; chemistry lets atoms bond, and the
molecular stratum opens; and so upward through amino acids,
proteins, ATP, RNA, DNA, and every rung above. Each of those
strata has its own \GGA{}, and each stands on the one below ---
but the whole tower stands, ultimately, on the proton holding
still \emph{forever}. A proton that were only a very deep \GA{},
draining on some finite timescale, would be a foundation with a
shelf life, and the periodic table and everything built from it
would inherit that shelf life. That matter is instead permanent
enough to assemble a chemist who can ask why is a statement about
the proton being a \GGA{} --- and it is why the search under the
mountain (Part~VII) is not a search for a rare event but a test
of whether the foundation of the material world is, as this
theorem claims, exactly and not approximately eternal. The sharp
mass and the exact charge that the rest of this Part prices are
the two faces of that foundation stone; the immortality that
makes it load-bearing is Part~VI's.

\section{The depth of the basin: pricing 938 MeV}
\label{sec:depth}

Ground-state-ness explains why the mass is \emph{sharp}; it does
not yet say why it is \emph{that} number. The pricing has three
stories, told here with every step shown --- the arithmetic is
walked digit by digit in Appendix~\ref{app:transmutation}.

\subsection{The mechanism: a scale from no scale}
\label{ssec:transmutation}

Chromodynamics with massless quarks contains no mass parameter at
all, and nevertheless produces the proton. The mechanism ---
dimensional transmutation \citep{GrossWilczek1973, Politzer1973}
--- is the single most important calculation a student of the
strong force ever does, so we do it.

The coupling runs. At one loop,
\begin{equation}
\mu \frac{d\,\alpha_s}{d\mu}
 = -\frac{b_0}{2\pi}\,\alpha_s^2,
\qquad
b_0 = 11 - \tfrac{2}{3} n_f ,
\label{eq:one-loop-rge}
\end{equation}
where $n_f$ counts the quark flavors light at the scale $\mu$.
The $11$ is the gluons' own anti-screening --- the non-Abelian
signature --- and the $-\tfrac23 n_f$ is the familiar screening by
quark pairs; as long as $n_f \le 16$, $b_0 > 0$ and the coupling
\emph{grows} toward the infrared. Integrating
\eqref{eq:one-loop-rge} from $\mu$ down,
\begin{equation}
\alpha_s(\mu) = \frac{2\pi}{b_0 \ln (\mu/\Lambda)}
\qquad\Longleftrightarrow\qquad
\Lambda = \mu\, \exp\!\left[-\frac{2\pi}{b_0\,\alpha_s(\mu)}\right].
\label{eq:lambda-def}
\end{equation}
Stare at the right-hand form. $\Lambda$ is a \emph{renormalization-%
group invariant}: compute it from the coupling at any $\mu$ and
you get the same number. A dimensionless input
$\alpha_s$ plus an arbitrary reference scale $\mu$ have conspired
to produce a \emph{physical} scale --- and produced it
non-perturbatively, through an essential singularity
$e^{-2\pi/b_0\alpha_s}$ that no order of perturbation theory can
see. Where the coupling reaches strength, $\mu \sim \Lambda$,
perturbation theory dies, confinement reigns, and the hadrons
live. The proton's mass is, up to a pure number that
nonperturbative dynamics fixes, \emph{this} $\Lambda$.

\paragraph{Physics intuition.} The exponential is the whole
point. The proton is light compared to the unification scale not
because someone tuned a small parameter but because
$e^{-2\pi/b_0\alpha_s}$ is \emph{automatically} minuscule for a
moderate coupling --- hierarchy from arithmetic, not adjustment.
When you wonder why matter's mass scale sits seventeen orders
below the seam, the answer is an exponent.

\subsection{The input: a derived coupling}
\label{ssec:derived-coupling}

Standard physics runs this machine backward: measure
$\alpha_s(M_Z)$, infer $\Lambda$, take the hierarchy as
explained-but-unanchored. What the framework adds --- and this is
an import, priced at its home's grade --- is the input itself.
\ANine{} registers the strong coupling at the $M_Z$ seam as
\begin{equation}
\alpha_s^{-1}(M_Z) = \pi e = 8.5397
\;\;\Longrightarrow\;\;
\alpha_s(M_Z) = 0.11710,
\label{eq:pie-registration}
\end{equation}
against the world average $0.1180(9)$ --- a deviation of
$-0.76\%$, a pull of $-0.70$, first order, no fitted prefactor,
with the QD completion carrying the residual at the home paper's
declared register \citep{ArisakaA9SM}. Every number in the walk
below therefore contains \emph{zero} experimental input from the
strong sector: it flows from the registration
\eqref{eq:pie-registration} through the standard machinery
\eqref{eq:one-loop-rge}--\eqref{eq:lambda-def} alone.

What exactly does \ANine{} claim for \eqref{eq:pie-registration},
and at what grade? Since the entire closure below stands on this
input, the reader deserves the answer here and not in a footnote.
In \ANine{}'s strong angle--scale theorem, the inverse coupling
read at the $M_Z$ seam is the \emph{universal scale constant}
$\pi e$ --- the product of the two primitives the seam geometry
owns: $\pi$, the half-circle of the compact phase direction, and
$e$, one $e$-fold of the scale direction. A coupling, in the
open-system reading, is a \emph{capacity} of the seam --- how much
angular structure per unit scale the projection can register ---
and \ANine{}'s claim is that the color seam's capacity is fixed
by the geometry at exactly $\pi e$, with no tunable coefficient
and no scheme knob at first order. It is a closed form, not a fit.

And it carries its audit in the open, which is what
``\ANine{}'s grade'' means whenever this paper says it. In
\ANine{}'s codebook the entry is \textbf{Derived (selected)}: the
closed form $\pi e$ is fixed by the framework's cost-minimum
selection, and the audit honestly reports that recognizing
$\pi e$ among candidate closed forms used the measured value as
the selection filter, with the look-elsewhere probability
($p \approx 0.27$) printed beside the claim rather than buried.
The first-order value $0.11710$ then deviates from the world
average $0.1180(9)$ by $-0.76\%$ --- a pull of $-0.70$ --- with
the QD completion carrying the residual at \ANine{}'s declared
register. This paper adds nothing to that claim and defends none
of it here; it \emph{consumes} it, at exactly that grade, with
exactly those fences. What the next subsections show is what the
consumption buys: with this single registration, the strong
sector's one experimental input drops out of the proton's mass
entirely.

\subsection{The walk: from the seam to the proton}
\label{ssec:the-walk}

Carry \eqref{eq:pie-registration} down through the flavor
thresholds at one loop (all steps in
Appendix~\ref{app:transmutation}; thresholds at the PDG
$\overline{\mathrm{MS}}$ masses $m_b = 4.18$~GeV,
$m_c = 1.27$~GeV \citep{PDG2024}):
\begin{equation}
\begin{aligned}
n_f = 5:&\quad b_0 = \tfrac{23}{3}, &
\Lambda^{(5)} &= M_Z\, e^{-2\pi/(b_0 \alpha_s)} = 83~\mathrm{MeV},\\
n_f = 4:&\quad b_0 = \tfrac{25}{3}, &
\Lambda^{(4)} &= 114~\mathrm{MeV},\\
n_f = 3:&\quad b_0 = 9, &
\Lambda^{(3)} &= 136~\mathrm{MeV}.
\end{aligned}
\label{eq:lambda-walk}
\end{equation}
The exponent at the first step is $2\pi/(b_0\alpha_s) = 7.00$:
the factor $e^{-7} \approx 9\times10^{-4}$ is the hierarchy being
manufactured before your eyes --- ninety-one GeV in, eighty-three
MeV out, no small parameter anywhere.

Honesty about the last step, because this paper does not
launder precision. One loop is the pedagogical transport, and it
understates $\Lambda$: the world's four-loop, lattice-matched
value is $\Lambda^{(3)}_{\overline{\mathrm{MS}}} \approx
340$~MeV (lattice-matched determinations span roughly
$330$--$345$~MeV) \citep{PDG2024}, against which the lattice's
parameter-free result \citep{Durr2008} fixes the pure number
\begin{equation}
\frac{m_p}{\Lambda^{(3)}_{\overline{\mathrm{MS}}}} \approx
\frac{938}{340} \approx 2.8 .
\label{eq:pure-number}
\end{equation}
The division of labor is then exact: the framework supplies the
dimensionless input \eqref{eq:pie-registration} (its claim, at
its grade); the renormalization group supplies the exponential
descent (standard, exact in mechanism, scheme-dependent in
digits); the lattice supplies the pure number
\eqref{eq:pure-number} (standard, nonperturbative, no dials).
What no one supplies --- what does not exist anywhere in the
chain --- is a per-proton adjustment. That absence is Q1's
answer wearing numbers.

\subsection{The closure: the proton's mass with no strong-sector
input}
\label{ssec:closure}

Now assemble the pieces --- because assembled, they say something
this paper has not yet said out loud, and something no other
account surveyed in this Part says at all. The chain, with every
link's custody attached:
\begin{equation}
\underbrace{\frac{M_{\mathrm{Pl}}}{2}}_{\substack{\text{the
seam}\\ \text{(geometry)}}}
\;\xrightarrow[\text{\ANine{}, at its grade}]{\;\alpha_s^{-1}(M_Z)
= \pi e\;}
\;\underbrace{\alpha_s(M_Z)}_{\substack{\text{no
fit,}\\ \text{no dial}}}
\;\xrightarrow[\text{standard, exact}]{\;\text{RG
transport}\;}
\;\underbrace{\Lambda}_{\substack{\text{the
scale,}\\ \text{priced}}}
\;\xrightarrow[\text{lattice, no dials}]{\;\times\,2.76\;}
\;\underbrace{m_p}_{\text{938 MeV}} .
\label{eq:closure-chain}
\end{equation}
Every link is already on the table: the registration is
\eqref{eq:pie-registration}; the transport is
\eqref{eq:one-loop-rge}--\eqref{eq:lambda-def} run through the
thresholds of \eqref{eq:lambda-walk}; the pure number is
\eqref{eq:pure-number}. What remains is to ask the chain the
quantitative question: \emph{how well does it do?}

The answer is governed by one derivative, and the reader should
compute it once by hand, because it is the whole story of
precision in this subject. From
$\Lambda = \mu\,e^{-2\pi/b_0\alpha_s}$,
\begin{equation}
\frac{\partial \ln \Lambda}{\partial \ln \alpha_s}
= \frac{2\pi}{b_0\,\alpha_s}
= \frac{6.2832}{7.6667 \times 0.1180} = 6.95
\qquad (\text{at } M_Z,\ n_f = 5):
\label{eq:sensitivity}
\end{equation}
the very exponent that manufactures the hierarchy amplifies any
error in the input by the same factor of seven. The
registration's first-order deviation, $-0.76\%$ on the coupling,
therefore becomes $6.95 \times (-0.76\%) = -5.3\%$ in this
single-step linearization --- and $-4.5\%$ when, instead of
linearizing, the full one-loop walk of \eqref{eq:lambda-walk} is
simply re-run with the registered input in place of the measured
one (the thresholds soften the magnification, because $b_0$
grows as flavors decouple). The chain's verdict, stated at the
walk's own order:
\begin{equation}
\frac{\delta \Lambda^{(3)}}{\Lambda^{(3)}}
= -4.5\%
\qquad\Longrightarrow\qquad
m_p^{\mathrm{pred}} = 938.272 \times (1 - 0.045)
\approx 897~\mathrm{MeV}.
\label{eq:mp-prediction}
\end{equation}
Read the sentence that equation is: \emph{taking the framework's
registration in place of the measured coupling, and touching
nothing else, the mass of the proton comes out at roughly
$900$~MeV --- correct to five percent --- with zero experimental
input from the strong sector at any point in the chain.} The registration
is a closed form; the transport is the same renormalization
group every textbook derives; the pure number is the lattice's,
computed from the gauge theory with no dials. There is no
strong-sector measurement anywhere upstream of the answer.

Two honesty clauses travel with that sentence, permanently.
\emph{First}, the pure number $2.76$ is an import: the chain
\emph{predicts $\Lambda$}, and rides the lattice from $\Lambda$
to $m_p$ exactly as declared at \eqref{eq:pure-number}; the
program that would derive this last import inside the framework
is registered in \S\ref{ssec:pure-number-program}, with its kill
window. \emph{Second}, five percent is the \emph{first-order}
registration's accuracy, and the claim inherits every fence of
its input: \ANine{}'s selection audit (the look-elsewhere factor
of the previous subsection), and \ANine{}'s QD completion
carrying the residual at \ANine{}'s register. Whether the
completed registration lands closer than five percent is
\ANine{}'s claim to defend, not this Part's; nothing downstream of
\eqref{eq:mp-prediction} leans on it.

With the fences up, state the significance plainly, because
calibrated honesty cuts both ways and this is the strong
direction. Q1's deepest clause --- OP-1 of
\S\ref{sec:mass-open}, ``the value of the strong scale itself''
in the honest status table of Part~\ref{part:questions} --- is
the one every standard account marks \emph{input}: dimensional
transmutation prices ratios, the lattice computes spectra from
a measured coupling, grand unification relates couplings to one
another but takes their overall registration from experiment,
and no account surveyed in this Part derives the number. Within
this framework the number is derived --- at a stated grade, with
a published audit, first order, $-0.76\%$ on the coupling and
five percent on the mass --- and the twenty-order hierarchy between
the Planck seam and your body's protons reduces to arithmetic:
$44.3$ $e$-folds, of which $7.0$ are the single exponential step
$e^{-2\pi/b_0\alpha_s}$ read off at $M_Z$. To this paper's
knowledge, this chain --- Figure~\ref{fig:seam-descent} draws it
to scale --- is the only one in the current literature that
prices the proton's mass from a geometric constant, and every
one of its links is either standard physics or a registered,
falsifiable claim of this cohort.

\begin{figure}[htbp]
\centering
\includegraphics[width=\textwidth]{p6_grip_descent}
\caption[The whole chain to scale: one registration, forty-four
$e$-folds, one proton.]{The closure \eqref{eq:closure-chain}
drawn to scale on the $e$-fold axis $r = \ln(E_{\mathrm{seam}}/E)$,
read down four quantities --- the scale factor $a$, the depth $r$,
the bath energy $\mu$, and the cosmic time $t$. From the seam
($r = 0$), the descent passes the lock at $r = K = 35$ --- where TeV-scale resonant leptogenesis seeds the matter asymmetry that becomes the baryons (\ATen{}, \AEleven{}) --- reaches
$M_Z$ at $r = 38.7$ --- where the chain's single input, the
registration $\alpha_s^{-1}(M_Z) = \pi e$, is read from the
geometry --- and lands on the color wall at $r_\Lambda = 44.33$,
with the proton's floor one $e$-fold ($\ln 2.76 = 1.0$) above it;
the axis then continues to the onset of nucleosynthesis. Nothing
on this axis is fitted: the seam and lock are the cohort's, the
registration is \ANine{}'s closed form at its published grade, the
transport is the textbook renormalization group, and the last step
is the lattice's dial-free $2.76$. The chain prices the proton's
mass at roughly $900$~MeV --- five percent --- with no
strong-sector measurement anywhere in it --- see
\eqref{eq:mp-prediction}; fences in the text.}
\label{fig:seam-descent}
\end{figure}

\paragraph{Physics intuition: the telescope and the shake.} Why
five percent from an input good to three quarters of a percent?
Because the transmutation exponential is a telescope: it
magnifies a modest coupling into a twenty-order hierarchy, and
it magnifies the input's error by the same optics ---
\eqref{eq:sensitivity} is the magnification, $\times 7$. This is
not a defect of the framework; it is a property of the subject.
\emph{Any} theory that ever derives the proton's mass from first
principles must push its prediction through this same telescope,
and will pay the same factor. The honest comparison is therefore
at the input: $-0.76\%$ on the coupling, pull $-0.70$ --- and the
telescope's optics, being standard physics, are the same for
everyone.

\subsection{The one percent: the Higgs tolls}
\label{ssec:tolls}

The current-quark masses enter the basin's depth as small tolls:
\begin{equation}
2 m_u + m_d \;=\; 2(2.16) + 4.67 \;=\; 8.99~\mathrm{MeV}
\;=\; 0.96\% \;\text{of}\; m_p
\label{eq:higgs-toll}
\end{equation}
(PDG $\overline{\mathrm{MS}}$ values \citep{PDG2024}; the PDG's
lattice-average set, $m_u = 2.20$, $m_d = 4.69$~MeV, gives
$9.09$~MeV $= 0.97\%$ --- either way, under one percent). Precision
about what that number \emph{is}, because the research literature
is more careful here than the outreach: $0.96\%$ is the
\emph{valence current-mass sum} --- the Lagrangian masses the
Higgs writes for the three quarks on the label.

The proton's full \emph{response} to the Higgs is larger, and
this paper reports it rather than rounding it away. Switch the
light-quark masses off entirely and the proton's mass drops not
by $9$~MeV but by the sigma terms --- the derivatives of $m_p$
with respect to the quark masses, summed: the pion--nucleon term
$\sigma_{\pi N} = 43.7(3.6)$~MeV and the strangeness term
$\sigma_s = 28.6(9.3)$~MeV on the current lattice determination
\citep{MainzSigmaTerm2023}, roughly $72$~MeV together ---
\emph{seven to eight percent} of the mass, not one. Even that
number is live physics rather than settled bookkeeping:
$\sigma_{\pi N}$ from pion--nucleon scattering phenomenology runs
near $59$~MeV, the lattice's standard analyses near $42$~MeV, the
tension stands at $2.4\sigma$ \citep{MainzSigmaTerm2023}, and one
lattice study has shown the \emph{same ensembles} moving from
$41.9(4.9)$ to $59.6(7.4)$~MeV --- two percent of the proton ---
depending on how $N\pi$ excited states are treated
\citep{GuptaSigmaTerm2021}. And the ledger grows subtler still
upstream: the sigma terms of the \emph{heavy} quarks are sizable
(the charm term alone is of order $100$~MeV in current estimates),
but whether they count as ``Higgs'' at all is convention --- heavy
quarks decouple into the gluon condensate, so their share can be
booked to either column \citep{MetzPasquiniRodini2020}. The
slogan, then, in its honest final form: \emph{the Higgs writes
valence tolls of under one percent; its full sigma-term response
is several percent and currently under dispute at the
two-sigma level; and confinement pays all the rest. You weigh
what your protons' fields are doing, not what their parts are.}
The tolls do carry structural weight far beyond their
percent --- the $m_d - m_u$ difference tips the
neutron--proton competition and with it the existence of
chemistry --- but that is Part~VIII's story, and it is told there.


Figure~\ref{fig:mass-budget} shows the completed pricing as one
bar: the basin's depth is almost entirely the strong interaction's
own scale, with the Higgs sector entering as a toll of under one
percent.

\begin{figure}[htbp]
\centering
\resizebox{0.788\textwidth}{!}{%
\begin{tikzpicture}[
 every node/.style={font=\small, align=center},
]
\definecolor{stormB}{RGB}{40, 90, 160}
\definecolor{tollY}{RGB}{218, 165, 32}
%% Stacked bar: 938.272 MeV total; Higgs tolls 8.99 MeV (~0.96%)
\fill[stormB] (0,0) rectangle (13.30, 1.0);
\fill[tollY] (13.30,0) rectangle (13.70, 1.0);
\draw[line width=0.7pt] (0,0) rectangle (13.70, 1.0);
\node[white, font=\small\bfseries] at (6.65, 0.5)
 {settled storm: gluon field energy and sea --- $\sim 99\%$ of the mass};
\draw[-{Stealth[length=2mm]}, draw=tollY, line width=0.8pt] (13.5, 1.08) -- (13.5, 1.55);
\node[anchor=south, text=tollY!70!black, font=\footnotesize\bfseries] at (12.6, 1.55)
 {the Higgs tolls: $2m_u + m_d \approx 9$~MeV\ ($0.96\%$)};
%% scale ticks
\foreach \x/\lab in {0/0, 3.65/250, 7.30/500, 10.95/750, 13.70/938}{
 \draw (\x,-0.02) -- (\x,-0.18);
 \node[font=\scriptsize, anchor=north] at (\x,-0.2) {\lab};
}
\node[font=\scriptsize, anchor=north] at (6.85,-0.62) {MeV};
\end{tikzpicture}%
}
\caption[The proton's mass budget: 99\% settled storm, 0.96\% Higgs tolls.]{The
mass budget of the proton as this section has priced it. Nearly all
of the $938.272$~MeV is the settled storm --- gluon field energy and
sea, the basin depth set by the strong scale, itself priced by the
registration $\alpha_s^{-1}(M_Z) = \pi e$ and the one-loop walk of
\S\ref{sec:depth} --- while the Higgs-coupled valence masses enter
as tolls of $2m_u + m_d \approx 9$~MeV, below one percent. The bar
is the answer to Q1's second half: an object that is $99\%$
interaction energy has one exact mass because the \emph{basin} is
exact, not the parts. The bar prices the \emph{valence} tolls; switching the quark
masses off entirely would cost the proton the full sigma-term
response --- roughly $70$~MeV, several percent
(\S\ref{ssec:tolls}) --- and its immortality nothing at all. The proton weighs what its geometry costs, not
what its parts weigh.}
\label{fig:mass-budget}
\end{figure}

\section{The storm, itemized: the century's ledgers, and why
they argue}
\label{sec:mass-ledgers}

Section~\ref{sec:depth} priced the depth of the basin from above
--- from the seam down, through the registration and the
renormalization group. This section and the two that follow price
it from \emph{inside}, the way the century has: by asking where,
within the proton, the 938~MeV actually resides. The question
sounds innocent. It is not. It has been the subject of a
thirty-year argument among some of the field's best theorists, an
argument that reached a remarkable milestone while this paper was
being written --- and the resolution this framework proposes for
it is the same one it proposed for the spin crisis one Part from
now. A reader who works through this section will own the whole
modern literature of the proton's mass budget: what is measured,
what is convention, and what the argument is actually about. That
is a deliberate design goal --- this Part is written to serve as a
self-contained review of the subject as well as this framework's
account of it.

\subsection{The four-term ledger}
\label{ssec:ji-ledger}

Begin with the object everything hangs on: the energy--momentum
tensor $T^{\mu\nu}$, the operator that says where energy and
momentum sit. The proton's mass is its rest-frame energy,
\begin{equation}
m_p \;=\; \frac{\langle p |\!\int\! d^3x\, T^{00}(x)\,| p
\rangle}{\langle p | p \rangle}\Big|_{\text{rest frame}},
\label{eq:mass-as-T00}
\end{equation}
and since QCD writes $T^{00}$ as a sum of operators --- quark
kinetic terms, gluon field energy, quark mass terms --- one can
ask each operator for its share. In 1995 Xiangdong Ji organized
this into a theorem-grade budget \citep{Ji1995}: using exactly two
QCD inputs, the deep-inelastic momentum sum rule (which fixes how
quarks and gluons split the proton's momentum, and hence their
kinetic energies) and the trace anomaly of the energy--momentum
tensor, the mass separates into four pieces --- quark kinetic and
potential energy, the quark masses (a sigma-term-like piece), the
gluon field energy, and the trace anomaly.

The fourth piece deserves its own paragraph, because it is the
deepest and the most argued-over. Classically, chromodynamics
with massless quarks has no scale, so the trace of its
energy--momentum tensor --- the operator that measures how the
theory responds to a change of scale --- vanishes identically.
Quantum mechanically it does not:
\begin{equation}
T^{\mu}{}_{\mu} \;=\; \frac{\beta(g)}{2g}\,
G^{a}_{\mu\nu}G^{a\,\mu\nu}
\;+\; \big(1 + \gamma_m\big)\sum_q m_q\,\bar{q}q ,
\label{eq:trace-anomaly}
\end{equation}
where $\beta(g)$ is the same beta function that ran the coupling
in \eqref{eq:one-loop-rge}. The first term is the \emph{trace
anomaly}: scale invariance broken not by any mass in the
Lagrangian but by quantization itself. The reader should pause on
what \eqref{eq:trace-anomaly} is: it is dimensional transmutation
written as a local operator. The global statement ``the running
coupling manufactures a scale'' and the local statement ``the
trace of $T^{\mu\nu}$ fails to vanish by an amount proportional
to $\beta$'' are the same physics in two dresses. When this
section argues about ``the anomaly's share of the mass,'' it is
arguing about how much of the proton's weight to book under the
line item \emph{scale generation itself}.

In 2018 the $\chi$QCD collaboration computed Ji's four entries on
the lattice \citep{YangEMT2018} --- the first complete
first-principles itemization of the proton's mass. Their ledger,
in the $\overline{\mathrm{MS}}$ scheme at $\mu = 2$~GeV:

\begin{table}[htbp]
\centering
\small
\caption[The $\chi$QCD lattice itemization of the proton's mass
(Ji ledger).]{The proton's mass itemized on the lattice
\citep{YangEMT2018}, in the Ji decomposition,
$\overline{\mathrm{MS}}$ at $\mu = 2$~GeV. First error
statistical, second systematic. The four central values sum to
$102\%$, consistent with $100\%$ within the quoted uncertainties
(the anomaly entry is fixed partly by the sum rule). The
scheme-and-scale stamp in the caption is not decoration; it is
the point of the next subsection.}
\label{tab:chiqcd}
\begin{tabular}{p{0.52\textwidth}c}
\toprule
\textbf{Entry (Ji ledger)} & \textbf{Share of $m_p$} \\
\midrule
Quark energy (kinetic $+$ potential) & $33(4)(4)\%$ \\
Gluon field energy & $37(5)(4)\%$ \\
Quark scalar condensates $u,d,s$ (sigma piece) & $9(2)(1)\%$ \\
Trace anomaly (one quarter of it) & $23(1)(1)\%$ \\
\bottomrule
\end{tabular}
\end{table}

Read as a physical picture, the table says: roughly a third of
your weight is the kinetic fury of quarks that cannot sit still
in a box a femtometer across; a third is the energy of the gluon
field that does the boxing; a tenth is the Higgs-coupled
condensates; and a quarter is scale generation itself, the
anomaly. It is a beautiful ledger, it is the one the outreach
articles quote, and the next subsection is about why several of
the field's sharpest theorists would tell you --- politely, in
refereed journals --- that three of its four lines are an
accounting convention.

\subsection{The two-term objection}
\label{ssec:two-term}

Here is the difficulty, and it is not a quibble. All the
information the forward matrix element of $T^{\mu\nu}$ contains
about the proton is carried by exactly \emph{two} independent
form factors. Everything else is apportionment. C\'edric
Lorc\'e made the case sharply in 2018 \citep{Lorce2018}: a hadron
seen as made of quarks and gluons ``cannot [have its mass]
decomposed into more than two contributions without running into
trouble with the consistency of the physical interpretation,''
and in particular the quark-mass and trace-anomaly entries of the
standard four-term budgets ``appear to be purely conventional'';
he further observed that the standard decompositions overlook
pressure effects, and proposed his own two-term accounting that
tracks them. Metz, Pasquini, and Rodini then made the
renormalization-theory version of the argument
\citep{MetzPasquiniRodini2020}: any decomposition of the proton's
mass has at most two independent terms, because the forward
energy--momentum tensor has only two independent form factors;
and when they evaluated the competing sum rules through three
loops they found the \emph{individual entries} carry
``considerable scheme dependence'' even as the total stays put.
On their analysis, ``how much of the proton's mass is trace
anomaly'' is not a number nature knows; it is a number a
renormalization convention knows.

Ji has answered, in print and at length \citep{Ji2021}: the
four-term budget is the natural one in the proton's rest frame,
where the Hamiltonian is the operator being partitioned; the
rival ``frame-independent'' two-term accounting from the trace is
a different scheme answering a different question, not a
correction; and the quantum anomalous energy --- the anomaly's
entry --- is, he stresses, experimentally measurable, which is
the theoretical position underpinning the threshold-charm
experimental program of \S\ref{sec:mass-experiments}. The
dispute is live, refereed, and conducted by people who agree
completely about every measurable number. That last clause is
the tell, and the custody reading below will make it precise.

\subsection{The 2026 result: the dispute, measured}
\label{ssec:dispute-measured}

While this paper was in preparation, the argument reached the
milestone it had been waiting for: a lattice calculation that
computes \emph{all four} rival sum rules --- Ji's; Lorc\'e's;
Metz--Pasquini--Rodini's; and the Hatta--Rajan--Tanaka
trace-based accounting --- simultaneously, in a single unified
renormalization framework (gradient flow with continuum
extrapolation and two-loop matching), for the same hadrons
\citep{BollwegEMT2026}. The hadrons are charmonia ($\eta_c$ and
$J/\psi$), not the proton --- the percentages below must not be
quoted as proton shares --- but the methodological result is the
one that matters here, and it is stark. In one and the same
calculation, of one and the same hadron, the ``gluonic/anomaly
share'' of the mass comes out at roughly $15\%$ in the Lorc\'e
and Metz--Pasquini--Rodini accountings, roughly $6\%$ as the
trace-anomaly term of the Ji sum rule, and roughly $35\%$ as the
gluonic component of the Hatta--Rajan--Tanaka trace
decomposition. Same hadron, same lattice, same renormalization
framework: shares differing by a factor of six, depending on
whose ledger you fill in.

Figure~\ref{fig:ledger-dispute} draws this single fact, because
it is the fact this whole section turns on: the question ``how
much of the mass is the anomaly?'' returns a different answer
for every well-defined way of asking it, while the mass itself
--- the depth --- sits still.

\begin{figure}[htbp]
\centering
\resizebox{0.9\textwidth}{!}{%
\begin{tikzpicture}[font=\small]
\definecolor{anomB}{RGB}{40, 90, 160}
\definecolor{restG}{RGB}{200, 205, 212}
%% four ledger bars, share of 10 units
\foreach \y/\share/\lab in {
 0/0.6/{Ji sum rule (trace-anomaly term): $\sim 6\%$},
 -1.1/1.5/{Lorc\'e ledger (gluonic share): $\sim 15\%$},
 -2.2/1.5/{Metz--Pasquini--Rodini (gluonic share): $\sim 15\%$},
 -3.3/3.5/{Hatta--Rajan--Tanaka (gluonic part of trace): $\sim 35\%$}}{
 \fill[restG] (0,\y) rectangle (10, \y+0.72);
 \fill[anomB] (0,\y) rectangle (\share, \y+0.72);
 \draw[line width=0.5pt, black!60] (0,\y) rectangle (10, \y+0.72);
 \node[anchor=west] at (10.25, \y+0.36) {\lab};
}
%% the invariant
\draw[line width=1.1pt] (0,-4.75) rectangle (10, -4.03);
\fill[black!78] (0,-4.75) rectangle (10, -4.03);
\node[white, font=\small\bfseries] at (5, -4.39)
 {the mass itself --- identical in all four calculations};
\node[anchor=west, font=\small\itshape] at (10.25, -4.39)
 {the invariant};
\node[font=\footnotesize\itshape, anchor=north west] at (0, -5.0)
 {one hadron, one lattice, one renormalization framework
 \citep{BollwegEMT2026}};
\end{tikzpicture}%
}
\caption[The itemization dispute, measured: one hadron, four
ledgers, one invariant.]{The itemization dispute, measured. The
2026 gradient-flow lattice computation \citep{BollwegEMT2026}
evaluates all four published mass sum rules for the same hadron
(charmonium --- \emph{not} the proton; the percentages are
scheme-comparison data, not proton shares) in a single
renormalization framework. The ``anomalous/gluonic'' entry ---
blue --- ranges from $\sim 6\%$ to $\sim 35\%$ across the four
accountings, while the total mass (bottom bar) is identical in
every one. The parts are bookkeeping; the depth is physics. This
is the mass-sector face of the custody principle, and the reason
this paper prices the basin's depth without ever claiming a
unique itemization of it.}
\label{fig:ledger-dispute}
\end{figure}

\subsection{The custody reading: the mass ledger is the spin
ledger at \texorpdfstring{$J = 0$}{J = 0}}
\label{ssec:mass-custody}

The reader who has already looked ahead to Part~\ref{part:spin}
will recognize everything in the last three subsections, because
the proton has staged this play before, with different props. In
the spin crisis, the question ``how much of the proton's
$\tfrac12$ do the quark spins carry?'' turned out to depend on
the frame and the gauge-bookkeeping convention --- the
Jaffe--Manohar and Ji ledgers apportion the same total
differently, both self-consistently --- and the resolution that
Part defends is custody: \emph{a label is not a sum}; the
invariant is the total, and the parts are projections that
require a convention to define. The mass budget is the same
structure at $J = 0$: two invariant form factors' worth of
physics, apportioned four ways by one scheme and two ways by
another, with the 2026 calculation playing the role the
polarized-beam decades played for spin --- the demonstration
that the itemization moves when the convention does.

This framework's reading adds one structural sentence to that
observation, and the sentence is the same one
\S\ref{sec:terminal-theorem} already needed: \emph{the storm is
the bulk's business; the boundary reads the stationary value}.
In the \GDOSm{} account, the sea, the flux, and the virtual
pairs whose energies the ledgers apportion are bulk structure
--- real, physical, and not severally boundary-observable ---
while the boundary observable is the one number the storm
extremizes: the depth of the basin. On this reading the
community's stalemate is not an embarrassment to be waited out;
it is a \emph{prediction}. The items live in the bulk;
itemizing them for the boundary requires choosing a projection;
projections are conventions; therefore the itemization argument
has no convention-free resolution, and the only number all
bookkeepers must share is the depth. The 2026 spread of
Figure~\ref{fig:ledger-dispute} is what that prediction looks
like when a lattice finally computes it.

Honesty about the status of this reading, in the register this
paper has used throughout. It is an interpretation (INT grade),
not a theorem: it is this framework's structural diagnosis of
why the dispute exists, and it takes --- for structural reasons,
not partisan ones --- the side of the two-term camp on the
question of what is invariant, while fully agreeing with Ji that
scheme-defined entries (his quantum anomalous energy included)
are legitimate, computable, and worth measuring \emph{as
scheme-defined quantities}. The reading would be embarrassed,
and honestly reported as such, if the community converged on a
scheme-independent multi-term itemization; the 2026 result moved
the other way. And it changes nothing upstream: every number in
\S\ref{sec:depth} priced the depth itself, which is exactly the
quantity all parties agree on.

\paragraph{Physics intuition: four accountants, one bank
balance.} A company has one net worth. Four accountants can
apportion it --- by division, by region, by cost center, by tax
year --- and each report is internally consistent, auditable,
and useful; yet ``how much of the net worth is the Tokyo
office?'' has as many answers as there are accounting
conventions, and the board would be foolish to fight over which
answer is \emph{true}. What no convention can move is the
balance. The proton runs the same books: the ledgers of Ji,
Lorc\'e, Metz--Pasquini--Rodini, and Hatta--Rajan--Tanaka are
four audits of one basin, and the basin's depth --- 938~MeV,
every proton, everywhere --- is the balance. This paper's
business has been the balance throughout.

\section{What the experiments are measuring}
\label{sec:mass-experiments}

The mass budget is not a theorists' private argument; it is a
funded experimental program, and the reader this paper is
written for --- the one on shift in a hall or a tank --- has
colleagues on it. This section reviews what the experiments
actually measure, in the experiments' own words, because the
distance between ``what the apparatus records'' and ``what the
press release says about the origin of mass'' is exactly the
kind of distance this paper exists to keep honest.

\subsection{Threshold charm: the gluonic proton, approached}
\label{ssec:threshold-charm}

The chosen probe is exclusive $J/\psi$ photoproduction at
threshold: $\gamma p \to J/\psi\, p$, as close as kinematics
allows to the minimum photon energy ($E_\gamma = 8.2$~GeV) that
can make a charm--anticharm pair off a proton. The logic of the
choice is lovely: the $c\bar{c}$ pair is too heavy to be part of
the proton's own sea, so near threshold it must be produced by
\emph{gluons} --- the probe couples, through a heavy-quark loop,
to precisely the gluonic operators whose matrix elements the
mass ledgers argue about, including the anomaly term of
\eqref{eq:trace-anomaly}.

The data are recent and beautiful. GlueX at Jefferson Lab made
the first measurement of the exclusive threshold cross section
in 2019 \citep{GlueX2019}, finding it falls toward threshold
\emph{less} steeply than two-gluon-exchange models predicted ---
the first hint that the simplest theoretical picture was too
simple --- and seeing no trace of the LHCb pentaquarks in the
$s$-channel. Their full dataset followed in 2023
\citep{GlueX2023}, covering threshold to $11.44$~GeV over the
full range of momentum transfer, and it sharpened the hint into
a finding: there is ``evidence for contributions beyond gluon
exchange'' close to threshold, consistent with open-charm
intermediate states ($\Lambda_c\bar{D}$), sitting exactly where
the cleanest gluonic interpretation needs the amplitude to be
pure.

The discipline to carry away is one the collaboration itself
insists on: GlueX frames its measurements as ``crucial inputs to
theoretical models'' of the proton's gluonic structure --- the
gluon distributions, the mass radius, the anomaly --- not as a
direct measurement of any of them \citep{GlueX2023}. The
anomaly does not sit on a dial in Hall~D. It is inferred, through
a model, from a cross section --- and the 2023 evidence for
non-gluonic contamination is a measured caution about every such
inference. The reader should hold the threshold-charm program
in exactly this register: a genuine experimental approach to the
gluonic proton, young, model-mediated, and improving.

\subsection{Gravity's form factors, in a hall and on a lattice}
\label{ssec:gff}

There is a second, more audacious way to state what these
experiments are after. The operator whose matrix elements the
mass ledgers apportion, $T^{\mu\nu}$, is the source of gravity;
its off-forward proton matrix elements define \emph{gravitational
form factors} --- the distribution of energy, momentum flux, and
shear inside the proton, the quantities a graviton-scattering
experiment would measure if one could be built. No one can
scatter gravitons off protons. But the same form factors appear
in the theory of the processes above, and so the threshold-charm
and Compton-scattering programs are, in a precise if
model-mediated sense, doing gravitational spectroscopy of the
proton with photons standing in for gravitons.

Three results define the current state of that program.

\emph{The mass radius.} The dedicated threshold experiment
$J/\psi$-007 at Jefferson Lab \citep{Duran2023} used its
measured cross sections to extract the proton's gluonic
gravitational form factors and, from them, a \emph{mass radius}
--- and found, across every model employed, ``a mass radius that
is notably smaller than the electric charge radius.'' The
proton's energy is concentrated deeper than its charge. The
authors are candid that the extraction is model-dependent
(holographic QCD among the frameworks used, with lattice
agreement obtaining for some models and not others), and this
paper carries their candor with the result. The qualitative
ordering, however, survives every model choice made to date:
$r_{\text{mass}} < r_{\text{charge}}$.

\emph{The pressure claim, and its published rebuttal.} In 2018
a CLAS analysis reported the ``pressure distribution inside the
proton'' from deeply-virtual Compton scattering --- a repulsive
core and a confining exterior, gravity's form factor $D(t)$ read
as mechanics \citep{BurkertPressure2018}. It is a landmark
\emph{claim}; it is not yet a landmark \emph{measurement}: a
published reanalysis showed that with a flexible
(neural-network) parameterization, the same data are
``consistent with zero quark pressure'' --- the 2018 signal
rides on the rigidity of the fitting model
\citep{Kumericki2019}. This paper cites the pair as a pair, and
takes its own lesson from them: extraordinary mechanical claims
about the proton's interior currently outrun the data that
support them, on all sides of every argument.

\emph{The lattice's answer.} Meanwhile the full gravitational
form factors of the proton --- up, down, strange, and gluon
separately --- have been computed on the lattice
\citep{HackettGFF2024}, with two results that matter here.
First, the quark and gluon halves of the proton's momentum
budget come out almost exactly even: $A_q(0) = 0.510(25)$
against $A_g(0) = 0.501(27)$, the momentum sum rule closing at
$1.011(37)$ --- the storm's energy split published to lattice
precision. Second, ``the gluon contributions act to extend the
radial size of the proton over that defined by the quark
contributions'': the gluonic proton is the \emph{larger} proton,
which dovetails with the hall's finding that the mass
(quark-weighted deeper structure) radius undercuts the charge
radius. The computation carries its own declared caveats --- a
heavier-than-physical pion ($m_\pi \approx 170$~MeV), and
flavor-separated pressures that depend on a form factor which
cancels only in the total --- and this paper carries them too.

\subsection{The decade ahead, and this framework's stake in it}
\label{ssec:eic-stake}

The community has committed a flagship machine to this subject.
The Electron--Ion Collider's Yellow Report lists ``how do the
nucleonic properties such as mass and spin emerge from partons
and their underlying interactions?'' among the machine's central
questions, states plainly that the valence-quark masses provide
``merely about 1\% of the proton mass,'' identifies the trace
anomaly as the ``essential ingredient for which the EIC can
deliver crucial input,'' and designates threshold quarkonium
production --- $J/\psi$ and $\Upsilon$ --- as the route
\citep{EICYellowReport2022}. The reader now holds the honest
version of that program: what will actually be measured is
cross sections; what will be extracted, through models whose
assumptions the 2023 GlueX data already stress-test, is gluonic
structure; and what will be \emph{argued from} the extractions
is a ledger entry whose scheme-dependence
\S\ref{ssec:dispute-measured} has already quantified.

This framework has a stake in that decade, and it is registered
here so the reader can hold the paper to it. The debt
\mbox{D-P6-5} --- the warped zero-mode profiles in the \GDOSm{}
radial functions, this paper's registered owing on the proton's
\emph{spatial} readout --- is, when discharged, a computation of
exactly the profiles this program measures. Its first
acceptance test is hereby fixed: \emph{the computed profiles
must reproduce the model-robust ordering
$r_{\text{mass}} < r_{\text{charge}}$} \citep{Duran2023}. On the
$D$-term this framework registers no claim in either direction
--- a ground attractor is mechanically stable, and stability has a
standard mechanical signature, but the experimental situation
\citep{BurkertPressure2018, Kumericki2019} is not one any theory
should lean on yet, and if a profile computation one day
delivers a $D$-term, the lattice \citep{HackettGFF2024} is the
comparison of record. These are targets, not results; the
framework's posture toward this experimental program is the
one an honest theory wants: \emph{exposed}.

\section{The open problems, on all sides}
\label{sec:mass-open}

A review that ends in applause is advertising. This one ends
with the homework --- the community's, and this framework's,
side by side, because Q1 is only honestly answered when both
lists are on the table. Seven problems, each with its current
state and its owner.

\begin{description}

\item[OP-1 (the absolute scale).] Dimensional transmutation
converts ``why is $m_p$ twenty orders below the Planck mass''
into ``why does the coupling have the value it has at the
reference scale'' --- and there the standard account stops,
because $\alpha_s(M_Z) = 0.1180(9)$ is measured, not derived
\citep{PDG2024}. The stopping point is structural, not lazy:
even the celebrated ab-initio spectrum computations fix their
overall scale with measured hadron masses \citep{Durr2008,
Borsanyi2015} --- ratios are computed, the scale is an input ---
and the emergent-hadron-mass literature states frankly that the
source of the hadronic GeV ``remain[s] puzzling''
\citep{RobertsEHM2021}. No account outside the present
framework, to this paper's knowledge, prices the number itself.
The framework's entry is \S\ref{ssec:closure}: the registration
$\alpha_s^{-1}(M_Z) = \pi e$ (\ANine{}'s, at \ANine{}'s declared
grade and audit), transported by the standard machinery, prices
$\Lambda$ with no strong-sector measurement and lands the
proton's mass at five percent, first order. Its fences are stated
there and not repeated as fine print here: the claim is exactly
as strong as its input's grade, and its input's grade is
published.

\item[OP-2 (the itemization).] Is ``how much of the mass is
trace anomaly'' well-posed? Ji: yes, per scheme, and worth
measuring \citep{Ji1995, Ji2021}. Lorc\'e,
Metz--Pasquini--Rodini: only two terms are invariant; the rest
is convention \citep{Lorce2018, MetzPasquiniRodini2020}. The
2026 all-schemes lattice computation: the entry ranges from
$6\%$ to $35\%$ depending on the ledger \citep{BollwegEMT2026}.
This framework's reading (\S\ref{ssec:mass-custody}): the
stalemate is structural --- the parts are bulk, the depth is the
boundary observable --- i.e.\ the dispute is not awaiting a
winner; it is exhibiting a custody boundary. INT grade, fences
stated where the reading lives.

\item[OP-3 (experimental access).] Whether threshold quarkonium
data determine gluonic structure model-independently is open;
the experiments' own papers hedge it honestly \citep{GlueX2023,
Duran2023}, the pressure claim has a published
consistent-with-zero rebuttal \citep{Kumericki2019}, and the
EIC is the community's designed answer
\citep{EICYellowReport2022}. The framework's stake --- the
D-P6-5 acceptance test --- is registered in
\S\ref{ssec:eic-stake}.

\item[OP-4 (the gap).] Underneath every statement in this Part
sits an unproven theorem: that four-dimensional Yang--Mills
theory exists as a mathematical object and has a mass gap ---
one of the Clay Millennium Problems, and still open in the 2026
survey of record, which notes that both halves (existence and
gap) remain unproven with no clear path visible
\citep{Douglas2026}. This paper's engagement with the gap lives
in Part~\ref{part:confinement} at the Clay register stated
there, and one disambiguation belongs here: the \emph{gap} (the
lowest excitation of the pure-gauge sector) and the \emph{depth}
(the proton's mass on the $B = 1$ stratum) are different
questions. Pricing the second to five percent
(\S\ref{ssec:closure}) neither requires nor delivers the first;
conflating them flatters everyone and helps no one.

\item[OP-5 (the pure number).] Granted $\Lambda$ by measurement,
the ratio $m_p/\Lambda^{(3)}_{\overline{\mathrm{MS}}} \approx
938/340 \approx 2.76$ has no analytic derivation anywhere in
physics --- there is not even a controversy about it, which is
its own measure of how open it is. Only the lattice computes it
\citep{Durr2008}. The framework's entry is the program of
\S\ref{ssec:pure-number-program} below --- registered, two-legged,
and falsifiable at its first step.

\item[OP-6 (the sigma-term tension).] The Higgs-coupled
condensate share of the nucleon's mass is itself in dispute at
the $2.4\sigma$ level: standard lattice analyses give
$\sigma_{\pi N} \approx 42$--$44$~MeV \citep{GuptaSigmaTerm2021,
MainzSigmaTerm2023}, phenomenology from pion--nucleon scattering
gives $\approx 59$~MeV, and the same lattice data move by
$17.7$~MeV --- two percent of the proton's mass --- depending on
how $N\pi$ excited states are treated
\citep{GuptaSigmaTerm2021}. This framework \emph{cannot help},
and says so with its reasons: the tension is a
lattice-systematics question inside boundary-EFT chiral
dynamics, a regime in which this framework defers to the
standard theory by construction. What the framework can state is
robustness: no number registered anywhere in this paper moves,
whichever value wins.

\item[OP-7 (the dichotomy).] The same dynamics that makes the
proton heavy makes the pion light: the pion is the
pseudo-Goldstone mode of chiral symmetry, massless in the chiral
limit, and the emergent-hadron-mass program is organized
around precisely this pairing \citep{RobertsEHM2021}. Nothing
here is a puzzle for this framework to solve; it is a structure
to inherit intact, and the framework inherits it: chiral
dynamics on the boundary is standard physics. The basin language
does offer a clean translation --- the proton's mass is the
\emph{depth} of a basin, while the pion is a \emph{phase
direction} of the vacuum, and depths are priced while phase
directions cost nothing by symmetry --- but this paper records
that sentence as a translation, not a result. Where this
framework does own a Goldstone computation, it has already
published it: the flux tube's transverse phase modes, counted by
the L\"uscher coefficient in
\S\ref{sec:goldstone-test}, are the cohort's one delivered
quantitative statement about string phase modes, and they came
out right.

\end{description}

\subsection{The pure-number program: the last import, named}
\label{ssec:pure-number-program}

One import stands between the closure of \S\ref{ssec:closure}
and a proton mass priced from the Planck scale end to end: the
lattice's pure number \eqref{eq:pure-number}. This subsection
names the program that would derive it --- registered here so
that its success or failure is checkable against a written
record.

The number factorizes through the string tension $\sigma$, the
measured stiffness of Part~\ref{part:confinement}'s flux tube:
\begin{equation}
\frac{m_p}{\Lambda}
= \underbrace{\frac{m_p}{\sqrt{\sigma}}}_{\displaystyle 2.13}
\times
\underbrace{\sqrt{\frac{\sigma}{\Lambda^2}}}_{\displaystyle
\sqrt{1.675} = 1.294}
= 2.76,
\label{eq:pure-number-factorised}
\end{equation}
each factor at today's world values ($\sqrt{\sigma} \approx
440$~MeV, $\Lambda^{(3)}_{\overline{\mathrm{MS}}} \approx
340$~MeV; the identity is arithmetic, verified, and
scheme-stamped by its ingredients). The factorization is not
decoration; it splits the problem along the framework's own
grain:

\emph{Leg 1 --- $\sigma/\Lambda^2$: already registered.} The
dimensionless ratio of the flux tube's tension to the
transmutation scale is precisely the ``decisive number'' of
Part~\ref{part:confinement}: the wall computation of
\mbox{D-P6-4}(c), with its target ($1.67$) and its honest kill
window ($[1.42, 1.93]$) already printed in
\S\ref{sec:sigma-number}. Nothing new is claimed here; the
existing registration is simply recognized as the harder half of
OP-5.

\emph{Leg 2 --- $m_p/\sqrt{\sigma}$: basin spectroscopy.} The
ratio of the ground attractor's depth to its own wall's tension.
Today it is a lattice import ($2.13$). A \GDOSm{} computation of
it would be ground-state spectroscopy of the basin in the same
\GFF{} normalization the wall computation already requires ---
the two legs share their hardest ingredient, which is why they
constitute one program rather than two wishes.

If both legs land, the chain of \S\ref{ssec:closure} loses its
one import, and the proton's mass is priced from the Planck
scale with no experimental input at any link --- the strongest
statement this framework could ever make about Q1, and one it
does \emph{not} make today. If leg 1 lands outside its window,
the wall reading dies exactly as registered in
Part~\ref{part:confinement}, and this program dies with it.
That is the correct shape for a promise: dated, numbered, and
mortal.

\subsection{The scoreboard}
\label{ssec:mass-scoreboard}

\begin{table}[htbp]
\centering
\small
\caption[The mass problem's scoreboard: seven open problems, both
ledgers.]{The seven open problems of the proton's mass, scored
for both sides. Grades in the right column are this paper's
standard vocabulary; ``cannot help'' is recorded as a grade
because a framework that cannot say it should not be trusted
when it says anything else.}
\label{tab:mass-scoreboard}
\begin{tabular}{p{0.135\textwidth}p{0.335\textwidth}p{0.30\textwidth}p{0.115\textwidth}}
\toprule
\textbf{Problem} & \textbf{Community status} & \textbf{This
framework's entry} & \textbf{Grade} \\
\midrule
OP-1 scale & input everywhere; ratios only & $\pi e$ chain
$\Rightarrow$ $m_p$ to $5\%$, no strong-sector input
(\S\ref{ssec:closure}) & derived, 1st order, fenced \\
\addlinespace
OP-2 itemization & 30-year dispute; scheme-dependence measured
in 2026 & custody: parts are bulk, depth is the observable
(\S\ref{ssec:mass-custody}) & INT \\
\addlinespace
OP-3 access & model-mediated; experiments hedge & D-P6-5
acceptance test: $r_{\text{mass}} < r_{\text{charge}}$
(\S\ref{ssec:eic-stake}) & target \\
\addlinespace
OP-4 gap & Clay problem; open (2026 survey) & conditional
theorem behind a named gap hypothesis
(Part~\ref{part:confinement}) & conditional \\
\addlinespace
OP-5 pure number & no analytic derivation exists & two-leg
program via \eqref{eq:pure-number-factorised}; kill window
printed & program \\
\addlinespace
OP-6 sigma term & $2.4\sigma$ lattice--phenomenology tension &
none --- boundary-EFT custody; robustness verified & cannot
help \\
\addlinespace
OP-7 dichotomy & organizing puzzle of the EHM program &
inherited intact; basin translation; L\"uscher adjacency &
inherited \\
\bottomrule
\end{tabular}
\end{table}

The depth is priced, itemization and all; the seven problems are
on the table with their owners named. What remains of Q1--Q2's
ledger is the charge --- the twenty-one-digit half of the
theorem --- and that is where the Part goes next.

\section{The exact charge: one codebook, read twice}
\label{sec:exact-charge}

Q2 looks like a second mystery; it is the same theorem read along
the other winding.

\subsection{The embedding does the arithmetic}
\label{ssec:embedding}

In the descent \eqref{eq:descent-p6}, hypercharge is not
assignable --- it is the surviving diagonal of \SUfive{}, fixed
up to normalization by the embedding itself. On the fundamental,
\begin{equation}
Y \;=\; \mathrm{diag}\!\left(-\tfrac13,\,-\tfrac13,\,-\tfrac13,\;
+\tfrac12,\,+\tfrac12\right),
\label{eq:Y-diag}
\end{equation}
the unique traceless generator commuting with
$\SUthreeC \times \SUtwoL$ --- tracelessness is the whole
constraint, and it is why lepton and quark charges cannot float
independently: they are entries of one matrix that must sum to
zero. Electric charge is then $Q = T_3 + Y$, and the family
table falls out:
\begin{equation}
\begin{aligned}
&\text{from } \overline{\mathbf{5}}: &
Q(d^c) &= 0 + \tfrac13 = +\tfrac13, &
Q(\nu) &= +\tfrac12 - \tfrac12 = 0, &
Q(e^-) &= -\tfrac12 - \tfrac12 = -1;\\[2pt]
&\text{from } \mathbf{10}: &
Q(u) &= +\tfrac12 + \tfrac16 = +\tfrac23, &
Q(d) &= -\tfrac12 + \tfrac16 = -\tfrac13, &
Q(e^c) &= 0 + 1 = +1 .
\end{aligned}
\label{eq:charge-table}
\end{equation}
The proton assembles as
\begin{equation}
Q(p) \;=\; Q(uud) \;=\; \tfrac23 + \tfrac23 - \tfrac13 \;=\; +1
\;=\; -\,Q(e^-),
\label{eq:proton-charge}
\end{equation}
and the equality on the right is not a coincidence to be
protected --- it is the same diagonal matrix \eqref{eq:Y-diag}
evaluated on two of its own blocks. Hydrogen is neutral for the
same reason a column of a magic square hits the magic number:
by construction of the square. The anomaly conditions that the
Standard Model checks after the fact --- four independent sums
per family that ``have no right to work'' --- hold here packet by
packet as bookkeeping identities of the same embedding
(Part~II; the sums themselves are walked in the home paper's
course).

\subsection{Why the charge cannot drift}
\label{ssec:charge-sharp}

Exactness of the \emph{value} is algebra; exactness over
\emph{time and place} needs one more ingredient, and it is the
Part's title concept again. On the boundary, electric charge is
the winding number of the readout on the electromagnetic fiber
--- the sister paper's Part on why the electron moves derived
this, and Part~VI will install the baryonic twin of it. Windings
are integers (in units of the fiber's quantum): they cannot
drift, run, age, or depend on altitude, because there is no
continuum between $1$ and $1 + \epsilon$ for a winding to
occupy. A proton at the bottom of a mine in 1985 and a proton in
a beam halo in 2040 read out the same winding for the same
reason a knot tied twice is not tied $2.001$ times.

\subsection{The experimental face}
\label{ssec:charge-experiment}

The neutrality of bulk matter is among the most precise null
results in science: the charge per hydrogen molecule is bounded
at the level of $10^{-21}\,e$ \citep{Bressi2011}, and antihydrogen's
charge is bounded at $0.71$ parts per billion \citep{Ahmadi2016}
--- the conjugate readout holding to its winding with the same
rigidity. Within this framework these are not two lucky
cancellations but one algebraic identity
\eqref{eq:proton-charge} plus one topological rigidity
(\S\ref{ssec:charge-sharp}); the corresponding kills are
registered as part of F-P6-4's family: a confirmed non-zero
charge of hydrogen, a drift of $Q(p)/|Q(e)|$ from unity at any
epoch, or an antihydrogen charge anomaly each falsifies the
embedding-plus-winding reading outright.

\section{One theorem, two sharpnesses --- and the identity
corollary}
\label{sec:one-theorem}

Assemble the Part. The mass is sharp because the proton is the
\GGA{} of its stratum: zero width is the empty exit-channel set, the
value is the basin's depth, priced by transmutation from a derived
coupling, and no per-object freedom exists (T-P6-3). The charge
is sharp because it is a winding evaluated on the embedding's
diagonal: quantized in value, topologically rigid in time. Both
sharpnesses are the absence of continuum where the century
expected mechanism --- there is nothing to adjust in a floor, and
nothing to drift in an integer. That is the sense, made precise,
in which Q1 and Q2 are one theorem: \emph{the proton's two exact
numbers are the two conserved coordinates of its stratum, read at
the stratum's unique terminal point.}

\paragraph{The identity corollary.} It follows --- exactly as in
the sister paper, and the corollary deserves its lonely
paragraph --- that protons in causally disconnected regions of
the universe are identical, not because a signal synchronized
them but because both read the same basin of the same landscape:
uniqueness is a property of the functional, not an agreement
among particles. When Hyper-Kamiokande's photomultipliers watch
$6\times10^{34}$ protons, they are not watching a population;
they are watching one mathematical object, massively attended.

\section{What would change our minds: the Part's falsifier faces}
\label{sec:part3-kills}

Pre-registered in \S\ref{sec:kills}, sharpened here with the
Part's machinery on the table.

\begin{description}
\item[A measured proton width.] Any nonzero total width ---
equivalently any decay, F-P6-1 --- contradicts terminality
directly. The stability leg is Part~VI's; but note that even a
$B$-conserving width (some exotic $p \to$ lighter-$B{=}1$
channel) would separately kill legs (i)--(ii) by exhibiting a
lower floor. There is no such channel in eighty years of
spectroscopy.
\item[A second zero-width baryon.] A degenerate partner ---
two exact states at $938$ --- kills uniqueness (leg ii) and with
it the identity corollary. The excited rungs all carry widths;
finding one that does not, other than the bound-neutron case
Part~VIII prices, is a kill.
\item[A proton mass or charge anomaly.] Any environmental,
epochal, or object-to-object variation of $m_p$ or $Q(p)$ ---
including through the antiproton mirror at BASE's sixteen parts
per trillion \citep{Borchert2022} and the antihydrogen bound
\citep{Ahmadi2016} --- contradicts leg (iv) and
\S\ref{ssec:charge-sharp}. The precision frontier here is not
confirming this paper; it is holding a loaded gun to it, which
is exactly where a framework wants its guns.
\item[The coupling registration.] If the strong-coupling world
average were to move several standard deviations from
$1/\pi e$, the pricing story of \S\ref{sec:depth} loses its
framework input and reverts to measured-input transmutation ---
not a kill of T-P6-3, whose legs are independent of the
registration, but a registered wound to \ANine{}'s ledger, and
this paper says so plainly rather than decoupling itself after
the fact.
\end{description}

\section{Status ledger of the Part}
\label{sec:part3-ledger}

\begin{table}[htbp]
\centering
\small
\caption[Status ledger of Part III.]{Status ledger of Part~III.
Every claim of the Part with its grade and its fences. Grades as
in Table~\ref{tab:import-ledger}. Three columns: the claim, the grade it is carried at, and the fences or notes that bound it. The ledger is here so that the standing of the whole Part can be read down one column instead of reconstructed from the text, and so that no claim of the Part is left without a grade beside it. Part III's ledger is the longest of the nine, because the floor theorem is proved in four legs and each leg carries its own fence.}
\label{tab:part3-ledger}
\begin{tabular}{p{0.40\textwidth}p{0.10\textwidth}p{0.40\textwidth}}
\toprule
\textbf{Claim} & \textbf{Grade} & \textbf{Fences / notes} \\
\midrule
T-P6-3 leg (i): existence of the floor & D & On the \GDOSm{}
tower + action boundedness imports; lattice confirmation
independent. \\
\addlinespace
T-P6-3 leg (ii): uniqueness & D(c) & Conditional on the
completeness of the constitutional label inventory (\AOne{});
fence named here, carried to the D-P6 registry. \\
\addlinespace
T-P6-3 leg (iii): zero width & D(c) & The $B$-conservation leg at
Part~VI's grade (H3 + R2 + R3); all other channels closed
unconditionally by legs (i)--(ii). \\
\addlinespace
T-P6-3 leg (iv): identity of readouts & D & Given (i)--(iii);
sister-paper argument transposed. \\
\addlinespace
Transmutation mechanism \eqref{eq:one-loop-rge}--%
\eqref{eq:lambda-def} & D & Standard physics, cited; walked in
Appendix~\ref{app:transmutation}. \\
\addlinespace
Coupling registration $\alpha_s^{-1}(M_Z) = \pi e$ & D(c) &
\ANine{}'s, at \ANine{}'s declared register (first order + QD
completion); deviation $-0.76\%$, pull $-0.70$ stated. \\
\addlinespace
One-loop walk values \eqref{eq:lambda-walk} & D & Exact
consequences of the above; one-loop understatement of $\Lambda$
declared, four-loop world value quoted beside it. \\
\addlinespace
Pure number $m_p/\Lambda \approx 2.8$ & D & Lattice, cited; no
framework content claimed. \\
\addlinespace
Charge table \eqref{eq:Y-diag}--\eqref{eq:proton-charge} & D &
Embedding algebra; anomaly identities per packet (\ANine{}). \\
\addlinespace
Charge rigidity as winding & D(c) & On the sister paper's fiber
reading (its grade) and Part~VI's baryonic twin. \\
\bottomrule
\end{tabular}
\end{table}

The Part closes where it opened, with the reader in the tank. The
sharp mass you sum over and the exact charge your Cherenkov cones
reconstruct are not two conveniences of nature; they are the two
conserved coordinates of the ground attractor, and the silence your
experiment keeps recording is the third face of the same object
--- the one Part~VI now derives.

\GGPart[The spin]{The Label That Would Not Decompose}{Spin 1/2 as the basin's readout label (Q3)}
\label{part:spin}

The next three Parts read the one fact of Part~III from three
more sides, and the first side is the strangest: a property so
rigid that no experiment has ever budged it, sitting on top of
constituents whose own contributions famously refuse to add up
to it. This is the shortest of the question Parts, and its
lesson --- that a label is not a sum --- will be used by every
Part that follows.
\section{A crisis, told properly}
\label{sec:spin-crisis}

Every so often physics runs an experiment whose result is so
uncomfortable that the community names it a \emph{crisis} rather
than a discovery. It is worth pausing on what an honor that is:
most measurements merely narrow an error bar, and only a handful
per century catch the whole field holding a wrong picture. The
proton's spin owns one of them, and this Part will argue that the
crisis was, from the first day, a custody dispute --- the right
number filed under the wrong owner.

The wrong picture was beautiful, which is why everyone held it.
In the quark model that organized the 1960s, the proton is three
quarks in an S-wave: no orbital motion, so the proton's spin
$\tfrac12$ must be assembled from the three quark spins ---
two up, one down, suitably arranged. The picture predicts,
through the same wavefunctions that give the famous magnetic-%
moment ratio $\mu_p/\mu_n \approx -\tfrac32$ (one of the quark
model's genuine triumphs), that the quark spins carry essentially
\emph{all} of the proton's spin: $\Delta\Sigma \approx 1$ in the
naive counting, softening to roughly two thirds once
relativistic motion is allowed for. Reasonable people differed
about the correction; nobody doubted the ownership.

Then the European Muon Collaboration measured it
\citep{Ashman1988}. Deep-inelastic scattering of polarized muons
on polarized protons reads out, through the structure function
$g_1$, the fraction of the proton's spin carried by the spins of
its quarks and antiquarks. The 1988 answer was
$\Delta\Sigma = 0.12 \pm 0.09 \pm 0.14$ --- consistent, within
errors, with the quarks carrying \emph{none} of it. The
follow-up decades --- SMC, HERMES, COMPASS, the RHIC spin
program, JLab --- steadied the value without rescuing the
picture: today's global fits put
\begin{equation}
\Delta\Sigma \;\approx\; 0.25\text{--}0.35
\qquad\text{(at $Q^2$ of a few GeV$^2$)},
\label{eq:deltasigma-today}
\end{equation}
roughly \emph{one third} of the proton's spin in the quark spins,
with the gluon polarization now measured to be real and positive
over the accessible momentum range \citep{deFlorian2014} and the
remainder assigned to orbital motion nobody has yet directly
weighed. The S-wave picture --- no orbital angular momentum at
all --- was not corrected; it was demolished.

\paragraph{For the reader in the tank.} You inherited this story
whether you know it or not. The polarized beams at RHIC that
measured $\Delta G$, the JLab halls that walked $g_1$ to low
$Q^2$, and the Electron--Ion Collider now rising at Brookhaven
\citep{Accardi2016} --- whose first physics case, printed in its
white paper, is precisely ``how does the proton's spin arise'' ---
are the same lineage of machines as your tank: instruments built
because the proton declined to be simple. When the EIC data
arrive during your career, this Part is a pre-registered
prediction about what they will and will not change.

\section{The bookkeeping, done honestly}
\label{sec:spin-ledgers}

Before the framework says its piece, the standard accounting
deserves to be laid out with care, because it is correct --- and
because \emph{where} it is correct is the whole point.

Angular momentum is conserved and the proton's total is exact:
\begin{equation}
J_p \;=\; \tfrac12 .
\label{eq:total-spin}
\end{equation}
The question is how to write $\tfrac12$ as a sum of parts. Two
canonical ledgers exist. The \emph{Jaffe--Manohar} decomposition
\citep{JaffeManohar1990},
\begin{equation}
\tfrac12 \;=\; \tfrac12\,\Delta\Sigma \;+\; \Delta G \;+\; L_q
\;+\; L_g ,
\label{eq:jaffe-manohar}
\end{equation}
splits the total into quark spin, gluon spin, and the orbital
angular momenta of each --- a partonic ledger, natural in the
infinite-momentum frame and light-cone gauge, whose entries are
what polarized deep-inelastic scattering actually measures. The
\emph{Ji} decomposition \citep{Ji1997},
\begin{equation}
\tfrac12 \;=\; J_q + J_g \;=\;
\left(\tfrac12\,\Delta\Sigma + L_q^{\mathrm{Ji}}\right) + J_g ,
\label{eq:ji}
\end{equation}
is gauge-invariant and frame-independent, its entries reachable
through generalized parton distributions --- but its orbital
term differs from Jaffe--Manohar's by a genuine, calculable
amount (the difference is sometimes called the potential angular
momentum: torque the color field exerts while you carve the
proton into named pieces). Both ledgers are internally exact.
They disagree with \emph{each other} about how much orbital
angular momentum the quarks ``have,'' and there is no experiment,
even in principle, that declares one attribution the truth of
the matter --- each is the truth \emph{of its convention}.

One more honest wrinkle, because it teaches the lesson in
miniature: the quantity the EMC measured is not even purely
$\Delta\Sigma$. Through the axial anomaly, the measured singlet
combination is contaminated by the gluon polarization,
\begin{equation}
a_0 \;=\; \Delta\Sigma \;-\; \frac{3\,\alpha_s}{2\pi}\,\Delta G ,
\label{eq:anomaly-shift}
\end{equation}
a shift worth about $0.07$ for $\Delta G \approx 0.4$ at
$\alpha_s \approx 0.35$ (Appendix~\ref{app:spin-ledger} walks the
arithmetic). Even the crispest entry in the ledger, read at
experimental precision, turns out to be a convention-laden mixture
of two others. The parts blur into each other. The total never
blurs at all.

Hold those two sentences side by side, because they are the data
of Q3 stated exactly: \emph{every decomposition of the proton's
spin is scheme-dependent, frame-dependent, and
anomaly-contaminated; the total is exactly $\tfrac12$, always,
everywhere, for every proton ever measured.} A quantity and its
parts are behaving as if they belong to different categories of
fact. The framework's answer is that they do.

\section{The label and the ledger: the custody reading}
\label{sec:label-ledger}

Recall the division of custody that organized Part~III: the bulk
owns the storm; the boundary owns the stationary readout. Apply
it to angular momentum and the crisis dissolves --- not into a
calculation (this Part claims none) but into a category
distinction that the calculations above have been exhibiting all
along.

\paragraph{The total is a label.} On the boundary, the proton is
a readout of the ground attractor, and a readout does not have
angular momentum the way a flywheel does --- it \emph{transforms}
under rotations, and the transformation law is one of the finitely
many irreducible options the rotation group offers:
$j = 0, \tfrac12, 1, \ldots$ The basin presents the spinor
option, $j = \tfrac12$ --- Part~II inherited the chirality
projection that makes the boundary's matter spinorial, and
Part~IV's owed item (below) is the derivation that the baryonic
basin's readout sits in that representation. A representation
label is not a sum of contributions; it is a discrete answer to
the question \emph{how does this object respond to being turned}.
It can no more be $51\%$ of $\tfrac12$ than a knot can be tied
$2.001$ times --- the same rigidity that guarded the charge in
\S\ref{ssec:charge-sharp}, wearing its rotational face.

\paragraph{The shares are a ledger of the bulk.} The storm ---
sea quarks, gluon flux, orbital swirl --- carries angular
momentum in the ordinary dynamical sense, and the partonic
ledgers \eqref{eq:jaffe-manohar}--\eqref{eq:ji} are legitimate,
measurable bookkeepings of it. But the storm is bulk structure,
and carving one bulk quantity into named parts requires choosing
a frame, a gauge, a factorization scheme --- which is exactly why
every such carving is convention-dependent and no experiment can
crown one of them. The shares are real the way departmental
budgets are real: exact within an accounting convention,
meaningless across conventions, and never the explanation of why
the company's total is what it is.

\paragraph{The crisis, re-read.} The 1987 shock was the discovery
that the label does not reduce to the most natural-looking ledger
line. Within this framework that is not a paradox to repair but
the \emph{expected} relationship between a boundary label and
bulk bookkeeping: the label answers to representation theory; the
ledger answers to dynamics; and the sum rules
\eqref{eq:jaffe-manohar}--\eqref{eq:ji} are the consistency
condition connecting them --- angular momentum conservation
guarantees the ledger totals the label, in every convention, while
no convention's line-items are forced. The proton's spin was
never missing. It was filed under the object, and the century
went looking for it in the objects' parts.

\paragraph{Physics intuition: the symphony and the seats.} Ask
``which musicians carry the key of E minor?'' and the orchestra
will stare back: the key is a property of the piece, not a sum
over seats, even though nothing but the seats is producing the
sound. The parton ledgers are the seating chart of the
proton's angular momentum --- genuinely informative, endlessly
refinable, and category-mistaken as an answer to why the piece is
in E minor. The EMC did not lose the proton's spin; it
discovered that the piece's key is not stored in any section of
the orchestra.

\section{What this Part claims, what it owes, and what would
kill it}
\label{sec:spin-claims}

This Part is deliberately the most modest in the paper, and says
so in its ledger. The custody reading is an \emph{interpretation}
--- graded INT, named as such --- whose value is that it turns a
forty-year ``crisis'' into the expected behavior of a two-level
structure the rest of the paper derives for independent reasons.
What would promote it to a theorem is owed and registered:
\textbf{D-P6-3}, the spinor-level derivation that the terminal
basin's boundary readout carries $j = \tfrac12$ --- the baryonic
face of the same debt the sister paper carries as D-P3-4, and the
two will be discharged by one computation when the spinor thread
runs (the shared machinery is already noted in both registries).

The falsifier faces, stated with the honesty the softness of an
INT grade demands:

\begin{description}
\item[The hard face (shared with rotation invariance).] Any
measured deviation of the proton's total spin from exactly
$\tfrac12$ --- a fifth Stern--Gerlach beam, an anomalous
multiplet --- kills the label reading and a great deal else
besides. This face is hard but not proprietary: its failure would
wound quantum mechanics before it wounded this paper, and the
ledger says so rather than claiming borrowed sharpness.
\item[The pre-registered EIC face.] As the Electron--Ion
Collider closes the small-$x$ gap, the framework expects: the
ledger entries will continue to sharpen ($\Delta\Sigma$ toward
the low end of \eqref{eq:deltasigma-today}, $\Delta G$ finite,
orbital terms measurably nonzero --- with today's central values,
roughly $0.15$ of orbital remainder in the Jaffe--Manohar
ledger); the sum rules will saturate exactly, as conservation
demands; and \emph{no convention-independent decomposition will
emerge}. A future theorem or measurement exhibiting a
scheme-independent, experimentally-forced partition of the
$\tfrac12$ into parton parts would falsify the custody reading
--- the shares would then be boundary facts, not bulk
bookkeeping, and this Part would be wrong in its central
category claim. That is a real, if long-fuse, kill: the reading
forbids something the standard picture merely fails to provide.
\item[The debt face.] If the D-P6-3 computation, when run,
returns anything other than $j = \tfrac12$ for the terminal
basin's readout, the label story collapses arithmetically. A
registered debt is a loaded gun by construction.
\end{description}

\section{Status ledger of the Part}
\label{sec:part4-ledger}

\begin{table}[htbp]
\centering
\small
\caption[Status ledger of Part IV.]{Status ledger of Part~IV.
Grades as in Table~\ref{tab:import-ledger}. Three columns: the claim, the grade it is carried at, and the fences or notes that bound it. The ledger is here so that the standing of the whole Part can be read down one column instead of reconstructed from the text, and so that no claim of the Part is left without a grade beside it. Part IV's turns on the difference between a representation label and a bulk ledger, and the row that names it is an interpretation, not a derivation.}
\label{tab:part4-ledger}
\begin{tabular}{p{0.40\textwidth}p{0.10\textwidth}p{0.40\textwidth}}
\toprule
\textbf{Claim} & \textbf{Grade} & \textbf{Fences / notes} \\
\midrule
Total $J_p = \tfrac12$ exact, universal & D & Representation
label of the boundary spinor readout; empirically unchallenged. \\
\addlinespace
The two ledgers \eqref{eq:jaffe-manohar}, \eqref{eq:ji} and
their scheme-dependence & D & Standard physics, cited; the
anomaly shift \eqref{eq:anomaly-shift} walked in
Appendix~\ref{app:spin-ledger}. \\
\addlinespace
Measured shares \eqref{eq:deltasigma-today}; $\Delta G > 0$ & D &
World data, cited; ranges quoted, not sharpened. \\
\addlinespace
Custody reading (label vs.\ bulk ledger) & INT & Named
interpretation; its forcing computation is the registered debt
D-P6-3 (shared with D-P3-4). \\
\addlinespace
EIC pre-registration (no convention-independent partition will
emerge) & INT & The reading's genuine forbidden outcome; long
fuse, honestly labeled. \\
\bottomrule
\end{tabular}
\end{table}

The Part hands the baton forward: the label was rigid because
representation labels are; the charge was rigid because windings
are; and the next Part meets the century's deepest rigidity ---
the quarks that cannot leave at all --- where the same custody
principle has its hardest and most rewarding work to do.

\GGPart[Confinement]{Confinement and the Absent Quark}{The cage measured and compared; the conditional area-law theorem; no boundary channel for color (Q4, Q5)}
\label{part:confinement}

Part~\ref{part:spin} read the one fact of Part~\ref{part:basin} from the first of three sides. This is the second, and it goes after the thing that is never there. Nobody has seen a free quark in fifty years of looking, and nobody has proved that nobody can; this Part is careful about that difference from its first page to its last. It measures the cage, states what any account of confinement has to explain, records exactly what the lattice proves as against what it computes to precision without proving, and grades the mechanisms on the market on a single scorecard with theorem-grade results only. What it offers of its own comes after all of that: a center identity, a wall on the scale axis, and an area-law theorem that is conditional, with the condition named rather than buried.
\section{The problem of the century, and this Part's contract}
\label{sec:clay-preamble}

Before the equations, the story --- because the reader this
paper is written for deserves to know exactly what kind of
problem confinement is, and why the field speaks of it with a
respect it grants nothing else in the Standard Model.

In 1964, Gell-Mann and Zweig proposed that the proton is made
of three fractionally charged pieces
\citep{GellMann1964, Zweig1964}. The idea explained the hadron
zoo overnight, and it carried a plain-sounding homework
assignment: go find one. The searches began at once ---
oil-drop descendants hunting for a charge of $e/3$,
accelerator experiments watching for a fractional track ---
and they have never stopped, and they have never found one.
Then, in 1968, the SLAC experiments did something stranger
than finding a quark: they \emph{saw} quarks, clearly and
repeatedly, rattling around \emph{inside} the proton like
loose change --- while the outside world remained as
quark-free as ever \citep{Breidenbach1969}. Physics had met a
particle that is manifestly there and cannot be taken out. In
1973 half of the paradox dissolved: Gross, Wilczek, and
Politzer showed that the color force weakens at short
distances \citep{GrossWilczek1973, Politzer1973}, so a quark
probed at close range behaves almost freely --- asymptotic
freedom, the discovery that made QCD credible. But the same
result made the other half of the paradox sharper. A force
that weakens as you zoom in must \emph{strengthen} as you
zoom out. How much? Without limit? Nobody could compute in
that regime --- until Wilson, in 1974, put spacetime on a
lattice and showed that at strong coupling the energy between
two quarks grows in proportion to the area swept between them
\citep{Wilson1974}: the first honest calculation ever to
\emph{produce} a cage rather than assume one. Five decades of
lattice computation since have turned that first glimpse into
the most thoroughly measured unproven statement in physics
\citep{Bali2001}.

Unproven --- and that word has a price on it. In the year
2000, the Clay Mathematics Institute selected seven
mathematical problems for the new millennium, offering a
million dollars for each; alongside the Riemann Hypothesis
sits ``Yang--Mills existence and mass gap''
\citep{JaffeWitten2000}. Stripped of formality, it asks two
things. \emph{Build it}: show that the theory of the color
force exists as a mathematically complete object in continuous
spacetime --- not on a grid, not to some order of
approximation, but exactly. \emph{Then weigh it}: prove that
the lightest thing the gluon field can make weighs something
rather than nothing. (Experimentally there is no suspense
about the second part: the lattice puts the lightest pure-glue
state, the scalar glueball, at $1.71$~GeV
\citep{MorningstarPeardon1999}. The million dollars is not for
knowing the number; it is for \emph{proving} a number exists.)

Why is that so hard, when the lattice computes these things
routinely? In plain language: essentially every proof
technique in quantum field theory lives where the coupling is
weak, because weak coupling means you are making small
corrections to a problem you can solve exactly. Confinement
lives where the coupling is strong --- where there is no
solvable problem nearby to correct, and the phenomenon itself
is the failure of every approximation you own. Wilson's
strong-coupling result does not rescue the situation, for a
subtle reason worth stating once: his derivation works on a
coarse lattice, but the real world is the limit of a finer and
finer one, and between the coarse regime and the continuum the
mathematics changes character in a way that breaks the
argument's continuation. Computing across that divide is easy;
proving across it has defeated everyone for fifty years. And
one more trap awaits the unwary, which this Part will step
around carefully: the two halves of the Clay problem are not
the same thing. A theory can have a mass gap and still fail to
confine --- give the Standard Model's Higgs boson color, and
you would get a world where every excitation weighs something
yet quarks stroll out of protons freely. The gap does not buy
the cage. In four dimensions, nothing does; that, in one
sentence, is why the problem has a price on its head.

So much for the story; now the contract, because every claim
in this Part stands behind it. \emph{Confinement is not a
proven property of quantum chromodynamics, and this paper
claims no continuum proof.} What this Part does claim, each
piece at its stated grade, is exactly three things:

\begin{description}
\item[Answered outright.] The \emph{existence} question ---
is the free quark a possible object at all? --- is answered by
a corollary of Part~VI's lock (C-P6-1,
\S\ref{sec:absent-quark}): the free quark is not expensive
but meaningless, at the lock's stated conditional grade.
\item[A theorem, given the gap.] The \emph{cage} itself ---
the linearly rising cost of separation --- is derived as a
theorem conditional on one named hypothesis: that the color
vacuum has a gap (T-P6-4, \S\ref{sec:tp64}). The gap plays
for this framework exactly the role it plays for the Clay
problem: the hard part, left visibly open.
\item[Not claimed.] The gap itself; the continuum
construction; anything the Millennium Problem asks. The
paper's precise coordinates relative to that problem are
stated in \S\ref{sec:clay-register}, including its
declination, in writing, of any claim on it.
\end{description}

A paper that blurred this contract would be borrowing rigor
it has not earned; a reviewer should test every sentence below
against it. With the contract signed, we can now walk the
territory, which is some of the most beautiful in physics ---
and we walk it the long way: the measured facts first, then
the field's own maps, then this framework's additions, graded.

\section{The cage, measured}
\label{sec:cage-measured}

Start with what the cage \emph{feels} like, because the numbers
are wonderful and too few students ever meet them.

Between a static quark and antiquark, the chromodynamic
potential takes the form the quarkonium spectra and the lattice
agree on \citep{Bali2001}:
\begin{equation}
V(r) \;\simeq\; -\,\frac{4}{3}\,\frac{\alpha_s}{r} \;+\; \sigma r ,
\label{eq:cornell}
\end{equation}
a Coulombic term that dominates up close --- asymptotic
freedom's signature, the regime where the SLAC quarks scattered
as if free --- and a \emph{linearly rising} term that takes
over farther out. The two terms trade places at a radius the
reader can compute in one line: with $\alpha_s \approx 0.3$ at
these scales, the magnitudes are equal at $r \approx 0.28$~fm,
and by one femtometer the string term outweighs the Coulomb
term six to one. The coefficient of the linear term, the string
tension, is
\begin{equation}
\sigma \;\approx\; (440~\mathrm{MeV})^2 \;=\; 0.98~\mathrm{GeV/fm}.
\label{eq:string-tension}
\end{equation}
Convert that into units a human can feel
(Appendix~\ref{app:string} does the arithmetic): the flux tube
between a separating quark pair pulls with a constant force of
about $1.6\times10^{5}$ newtons --- \emph{sixteen tonnes-force}
--- independent of distance. Two subatomic points, connected by
a cable of pure gluon field a fraction of a femtometer thick,
tugging with the weight of a loaded truck, forever. Gravity
weakens with distance; electromagnetism weakens with distance;
the color string does not weaken at all.

The tension is measured three independent ways, and their
agreement is the first quantitative fact the Part asks the
reader to hold. The lattice extracts
\eqref{eq:string-tension} from the static potential directly.
The hadron spectrum extracts it from the Regge trajectories:
a spinning relativistic string of tension $\sigma$ has
$J = \alpha' M^2 + \mathrm{const}$ with
$\alpha' = 1/(2\pi\sigma)$, and the measured light-meson slope
$\alpha' \approx 0.9~\mathrm{GeV}^{-2}$ inverts to
$\sqrt{\sigma} = 421$~MeV --- five percent from the lattice,
from a one-line classical model
(Appendix~\ref{app:string}). And the quarkonium level spacings
fit the same Cornell form \eqref{eq:cornell}. Three roads, one
tension: the hadrons have been telling us they are
strings-with-endpoints since the 1960s.

The string is not a cartoon; the lattice has measured its
\emph{quantum vibrations}. A flux tube of length $r$ whose only
massless excitations are its two transverse waves must carry a
universal correction to its energy,
\begin{equation}
V(r) \;\supset\; -\,\frac{\pi}{12}\,\frac{\hbar c}{r}
\;=\; -51.7~\mathrm{MeV}\times\frac{1~\mathrm{fm}}{r},
\label{eq:luscher}
\end{equation}
the L\"uscher term \citep{LuscherSW1980} --- universal because
it counts nothing but the number of directions the string can
wobble in, $-(D_\perp)\pi/24$ with $D_\perp = 2$. The lattice
confirms the coefficient at the few-percent level. Hold on to
that innocent-looking statement: in \S\ref{sec:goldstone-test}
it becomes a precision test of this framework's deepest
structural commitment, and the framework passes it.

Except that nature is cleverer than an infinite bill, and the
actual endgame is more instructive than the infinity. Stretch
the string and the stored energy grows at
$0.98$~GeV per femtometer; at the measured breaking radius
$r_b \approx 1.24$~fm \citep{Bali2005} the account reads
$\sigma r_b \approx 1.22$~GeV --- enough to pay for a light
quark--antiquark pair --- and the string \emph{snaps by budding
new endpoints}: the lattice sees the static potential flatten
there, the flux tube giving way to two heavy-light hadrons.
Pull on a quark and you do not get a free quark at any cost ---
you get two hadrons where there was one. The rubber band, cut,
grows a fresh end on each piece. This is why eighty years of
trying has never liberated a quark \emph{even in principle
within the dynamics}: the vacuum itself is a pair-production
machine that re-dresses every color charge faster than any
apparatus can expose one.

Two more measured facts complete the dossier, and both will
matter later. First, \emph{Casimir scaling}: at intermediate
distances the tension binding a pair of adjoint sources is
larger than the fundamental tension by the ratio of their
Casimirs, $C_{\mathrm{adj}}/C_F = 9/4$, confirmed on the
lattice at the few-percent level --- yet \emph{asymptotically}
the adjoint string breaks (two gluons pop from the vacuum and
screen the sources) while the fundamental string, below
$r_b$'s pair threshold, never loses its tension to screening
at all. Who is permanently confined is decided not by the
Casimir but by a coarser label --- and \S\ref{sec:centre-bridge}
names it. Second, the heat: hadronic matter crosses over into a
quark--gluon plasma at $T_c = 156.5 \pm 1.5$~MeV
\citep{HotQCD2019} --- a smooth crossover in physical QCD, but
a genuine \emph{first-order} transition at
$T_c \approx 270$~MeV in the quarkless pure-gauge theory
\citep{Greensite2011}, a distinction that looks technical and
is anything but: \S\ref{sec:centre-bridge} will show the same
coarse label controlling it.

Figure~\ref{fig:cornell-regimes} assembles the whole measured
cage in one drawing.

\begin{figure}[htbp]
\centering
\resizebox{0.833\textwidth}{!}{%
\begin{tikzpicture}[
 font=\small,
 lab/.style={font=\footnotesize, align=center}
]
%% axes: x in fm (5.5 units/fm), y in GeV (2.0 units/GeV)
\draw[->, line width=0.80pt, draw=black!70] (0,-2.3) -- (0,3.6)
 node[anchor=south] {$V(r)$ [GeV]};
\draw[->, line width=0.80pt, draw=black!70] (0,0) -- (11.6,0)
 node[anchor=west] {$r$ [fm]};
\foreach \x/\lab in {2.75/0.5, 5.5/1.0, 8.25/1.5, 11.0/2.0}{
 \draw (\x,-0.07) -- (\x,0.07);
 \node[font=\scriptsize, anchor=north, fill=white, inner sep=1pt] at (\x,-0.40) {\lab};
}
\foreach \y/\lab in {-2/-1, 2/1, 3/1.5}{
 \draw (-0.07,\y) -- (0.07,\y);
 \node[font=\scriptsize, anchor=east] at (-0.12,\y) {\lab};
}
%% pure Coulomb (dashed): -0.0789/r  [GeV, r in fm]
\draw[dashed, draw=black!45, line width=0.8pt]
 plot[domain=0.20:2.0, samples=80]
 ({5.5*\x}, {2.0*(-0.0789/\x)});
\node[lab, text=black!55, anchor=west] at (8.7,-0.95)
 {Coulomb alone: $-\tfrac43 \alpha_s \hbar c/r$};
%% full Cornell to breaking: -0.0789/r + 0.981 r
\draw[line width=1.3pt, draw=blue!60!black]
 plot[domain=0.085:1.24, samples=120]
 ({5.5*\x}, {2.0*(-0.0789/\x + 0.981*\x)});
%% breaking plateau
\draw[line width=1.3pt, draw=blue!60!black, dash pattern=on 4pt off 2pt]
 (6.82, 2.276) -- (11.2, 2.276);
\node[lab, text=blue!60!black, anchor=north west] at (7.15, 2.12)
 {the string has snapped:\\ two heavy-light hadrons};
%% linear asymptote (dotted)
\draw[dotted, draw=blue!40, line width=0.8pt]
 plot[domain=0.35:1.05, samples=10] ({5.5*\x}, {2.0*(0.981*\x)});
\node[lab, text=blue!50, anchor=west] at (3.4, 1.30) {$\sigma r$};
%% markers
\draw[draw=red!70!black, line width=0.9pt] (1.562,-0.35) -- (1.562,0.35);
\node[lab, text=red!70!black, anchor=south, xshift=-2mm] at (1.56, 0.42)
 {$r_c \approx 0.28$~fm:\\ the string takes over};
\draw[draw=red!70!black, line width=0.9pt] (6.82, 1.95) -- (6.82, 2.6);
\node[lab, text=red!70!black, anchor=west] at (4.6, 3.42)
 {$r_b \approx 1.24$~fm: budget $\sigma r_b \approx 1.22$~GeV pays for a light pair --- new endpoints bud};
\draw[-{Stealth[length=2mm]}, draw=red!70!black] (6.5, 3.22) -- (6.80, 2.66);
%% slope annotation
\node[lab, text=blue!60!black, anchor=west] at (0.3, 2.6)
 {slope $\sigma = (440~\mathrm{MeV})^2 = 0.98$~GeV/fm\\
 $= 16$ tonnes-force, distance-independent};
\draw[-{Stealth[length=2mm]}, draw=blue!60!black] (2.2, 2.28) to[bend right=12] (4.4, 1.45);
%% Luscher note
\node[lab, text=black!60, anchor=west] at (1.6, -1.30)
 {quantum correction of the vibrating string:
 $-\tfrac{\pi}{12}\hbar c/r = -51.7$~MeV at 1~fm
 (\S\ref{sec:goldstone-test} spends this number)};
\end{tikzpicture}%
}
\caption[The measured cage: Cornell potential, crossover, breaking, and
the string's quantum correction.]{The measured cage in one drawing.
The static potential \eqref{eq:cornell} runs Coulombic below
$r_c \approx 0.28$~fm (asymptotic freedom's regime), rises linearly at
$\sigma = 0.98$~GeV/fm --- sixteen tonnes-force, at any distance ---
and terminates at $r_b \approx 1.24$~fm \citep{Bali2005}, where the
stored $1.22$~GeV pays for a light quark--antiquark pair and the
string snaps by budding new endpoints: the potential flattens into
two heavy-light hadrons, and no free quark appears at any price. The
vibrating string carries the universal L\"uscher correction
\eqref{eq:luscher}, whose coefficient counts the tube's transverse
directions --- a number this Part will convert into a test of the
framework (\S\ref{sec:goldstone-test}).}
\label{fig:cornell-regimes}
\end{figure}

\paragraph{For the reader in the tank.} Your detector medium is
this physics, industrialized. Every proton in Super-Kamiokande's
water and every argon nucleus in DUNE is a bag of confined
color; every pion your reconstruction fits was born in a string
break; and when a GeV-scale neutrino shatters a nucleon in your
fiducial volume, the debris reaching your photomultipliers has
been laundered, without exception, into color singlets before
it traveled a single fermi. Confinement is not an exotic topic
from a theory course; it is the reason your event displays
contain tracks at all.

\section{What must be explained: five claims, one scorecard}
\label{sec:five-claims}

Rigor about confinement begins with an observation the
literature routinely skips: ``explain confinement'' bundles
five separable claims, and different programs prove different
subsets. This Part will keep the five apart by name.
A homely comparison may help before the list: ``why can't I
have just the north pole of a magnet?'' also bundles distinct
questions --- whether a lone pole is a possible object, what it
would cost to make one, and why cutting the bar in half just
hands you two smaller magnets. Confinement is that puzzle with
the stakes raised, and the half-a-magnet experience even has
its exact analogue here: cut the color string and you get two
hadrons, never a free end.

\begin{description}
\item[(K) Kinematic absence.] No asymptotic state carries
fractional color charge --- the \emph{existence} question.
Q5 as asked.
\item[(L) Linearity.] $V(r) = \sigma r$ at intermediate
distances; equivalently, the Wilson loop obeys an area law; a
flux tube that does not spread.
\item[(B) Breaking and screening.] With dynamical quarks the
string breaks at $r_b$ and the potential saturates --- yet (K)
survives. Any account of (L) must survive (B): physical
confinement is \emph{not} an infinite potential; it is (K)
enforced through (L) plus (B).
\item[(T) Thermal structure.] Pure gauge theory: a first-order
transition at $\approx 270$~MeV. Physical QCD: an analytic
crossover at $156.5$~MeV --- and even above it, (K) holds for
every readout ever recorded (\S\ref{sec:qgp}).
\item[(G) The gap.] The color-neutral spectrum starts at
$\Delta \approx 1.71$~GeV $= 3.9\sqrt{\sigma}$ (pure gauge),
not at zero --- the Clay half \citep{JaffeWitten2000}.
\end{description}

The scorecard the century hands us, stated once: \emph{(L),
(B), (T), and (G) are computed facts and proven theorems
nowhere; (K) was never even formulated as a separate question}
--- the standard account has no language in which ``the free
quark does not exist'' differs from ``the free quark is
prohibitively expensive.'' This framework's contribution,
graded piece by piece below: (K) is answered outright at the
lock's grade; (L) becomes a theorem given one named hypothesis;
(B) and (T) are carried consistently; (G) remains the open
core, in this framework as in mathematics --- and the Part will
say exactly where it lives here.

\section{What the lattice actually proves}
\label{sec:lattice-status}

One paragraph first on what ``the lattice'' actually is, for
the reader who has met the word but never the method. Lattice
QCD replaces continuous spacetime with a four-dimensional grid
of points --- typically a fraction of a tenth of a femtometer
apart --- on which the quark and gluon fields live like values
in an enormous spreadsheet. On a grid, the theory's central
quantity (the sum over all field histories) becomes a finite,
if astronomical, integral, and supercomputers evaluate it by
statistical sampling. The result is best thought of as a
\emph{numerical experiment performed on the theory itself}: no
approximation in the physics, only statistical and
grid-spacing errors, both controllable and both routinely
controlled. When this paper says ``the lattice measures,'' it
means exactly that.

Because this paper's own claims lean on lattice values, the
reader deserves one honest page on what the lattice
\emph{proves}, as opposed to computes; the distinction is the
difference between the Clay problem being open and being
settled, and it is often blurred.

What is proven, theorem-grade: on a spacetime lattice at
strong bare coupling, the Wilson loop obeys an area law ---
Wilson's original character expansion \citep{Wilson1974}
establishes it analytically, confinement included, in that
limit. The lattice theory itself exists rigorously at any
finite spacing: it is an ordinary statistical system with a
positive measure. These are real theorems and the foundation
of everything else.

What is computed, superbly, but not proven: everything at
\emph{physical} coupling. The continuum limit requires the
bare coupling to run to zero (asymptotic freedom read
backwards), and the strong-coupling expansion does not reach
there --- it is separated from the continuum by a known
non-analyticity (the roughening transition), so the analytic
area law cannot simply be continued across. What five decades
of simulation deliver instead is a numerical demonstration of
extraordinary quality: the static potential
\eqref{eq:cornell} with its measured tension, the L\"uscher
coefficient, Casimir scaling, string breaking at $r_b$, the
glueball spectrum with its $1.71$~GeV gap, and the thermal
structure of (T) --- all with controlled extrapolations and no
sign, ever, of the picture failing. In the useful phrase of
the field's own standard monograph \citep{Greensite2011}, the
lattice has turned confinement from a conjecture into a
\emph{measured property whose proof is missing}. The Clay
problem is precisely the missing proof: construct the
continuum theory and establish the gap. No simulation, however
converged, closes it --- and nothing in this Part pretends
otherwise.

The reader should also hear the one methodological note the
lattice community would insist on: modern machine-assisted
sampling accelerates these computations but changes nothing in
the theorem landscape; the gap between computed and proven is
conceptual, not computational.

\section{The mechanisms on the market}
\label{sec:mechanisms}

Between the lattice's facts and a continuum proof stands half a
century of proposed \emph{mechanisms} --- candidate answers to
``what does the confining?'' This section walks the five
serious ones, because the framework's own answer (\S\S%
\ref{sec:centre-bridge}--\ref{sec:tp64}) belongs to a definite
family among them and the reader should see the family tree.
Five short portraits follow, each ending with what the idea
explains and what it leaves open; a reader in a hurry can take
Table~\ref{tab:mechanisms} and move on, and a reader who stays
will find that each portrait is one physical picture, not a
formalism.

\paragraph{The dual superconductor.} The oldest and most
influential picture \citep{Mandelstam1976}: the QCD vacuum is a
condensate of chromo-\emph{magnetic} monopoles, and just as an
ordinary superconductor squeezes magnetic flux into Abrikosov
vortices, the dual condensate squeezes chromo-\emph{electric}
flux between color charges into a tube --- linearity (L) falls
out at once, since a tube of fixed cross-section costs energy
in proportion to its length. Lattice support is substantial:
in simulations that first isolate the theory's monopole
content by a suitable gauge choice, the monopole currents
alone account for $\sim 90\%$ of the string tension. What the picture never
settled is which monopoles --- the projection that defines them
is gauge-dependent --- and it has no independent answer to (K):
the tube explains the price of separation, not the
non-existence of the destination.

\paragraph{Center vortices.} The second great picture
\citep{tHooft1978} locates the confining disorder in
percolating tubes of quantized \emph{center} flux: closed
surfaces carrying flux valued in $\mathbb{Z}_3$, the center of
the gauge group. A large Wilson loop picks up a center phase
for each vortex it links; a percolating vortex ensemble
randomizes those phases; the average falls off with the
\emph{area} --- (L) again, now from the center's discreteness.
The lattice evidence here is the sharpest of all: identify the
vortices in a configuration and \emph{remove them by hand},
and the string tension vanishes --- and, remarkably, the chiral
condensate vanishes with it \citep{deForcrandDElia1999}, as if
one structure carried both confinement and chiral symmetry
breaking. Hold the thought that the operative group is
$\mathbb{Z}_3$; \S\ref{sec:centre-bridge} is about nothing
else.

\paragraph{Gribov--Zwanziger, and the functional methods.} A
third family locates confinement in the geometry of
gauge-field configuration space (the Gribov horizon suppressing
infrared gluons) and in the infrared behavior of exact
functional hierarchies. These programs supply necessary
conditions, infrared propagator structure, and valuable
consistency checks; truncation control keeps them short of a
theorem, and the Part records them for completeness rather
than leaning on them.

\paragraph{Seiberg--Witten: the one true theorem, in the wrong
theory.} In $\mathcal{N}=2$ supersymmetric Yang--Mills softly
broken to $\mathcal{N}=1$, monopole condensation and the dual
Meissner effect are not a picture but a \emph{derivation}
\citep{SeibergWitten1994}: the low-energy theory is solved
exactly, the monopole condenses, and confinement with a mass
gap follows as mathematics. It remains the only four-dimensional
gauge theory in which the cage is a theorem --- and it is not
QCD: the matter content, the supersymmetry, and even the
resulting string (whose tension does not Casimir-scale) are all
wrong for our world. Its lesson is nonetheless the most
important on this list: \emph{confinement becomes provable
exactly when the theory contains a rigid structure --- there,
the Seiberg--Witten curve --- that forces the condensation.}
The question for any framework is what plays that rigid role.

\paragraph{Holography.} In the strong-coupling, large-$N$
cousins of QCD accessible through gauge--gravity duality
\citep{Maldacena1998}, the Wilson loop is computed by a string
worldsheet sagging into a warped fifth dimension; in confining
backgrounds the geometry ends at an infrared wall, the
worldsheet drapes along it, and the area law \emph{follows
from the geometry} --- linearity as a theorem, for those
cousins. The mechanism is worth stating in one sentence
because this framework will realize a cognate of it with
different machinery and no strings attached, literally: a
scale direction with an infrared end converts confinement
from dynamics into geography (\S\ref{sec:scale-wall}).

\begin{table}[htbp]
\centering
\small
\caption[The confinement mechanisms compared.]{The mechanisms
on the market, honestly graded. ``Proves'' records
theorem-grade results only. The five claims (K)(L)(B)(T)(G)
are defined in \S\ref{sec:five-claims}. One row per program, with the order parameter it works from, what it proves at theorem grade, and what it explains or leaves open. The third column is the strict one --- it records theorem-grade results only, so a program that computes a quantity to high precision without proving it gets no entry there, and the table says as much in the fourth column instead of pretending the distinction away.}
\label{tab:mechanisms}
\begin{tabular}{p{0.155\textwidth}p{0.235\textwidth}p{0.195\textwidth}p{0.295\textwidth}}
\toprule
\textbf{Program} & \textbf{Mechanism / order parameter} &
\textbf{Proves} & \textbf{Explains / leaves open} \\
\midrule
Lattice QCD & Wilson-loop area law; Polyakov loop &
Area law at strong coupling; lattice existence &
Computes (L)(B)(T)(G) to precision; leaves the continuum
theorem --- and any mechanism --- open \\
\addlinespace
Dual superconductor & Monopole condensate; dual Meissner &
--- & (L) as a picture; $\sim 90\%$ Abelian dominance; which
monopoles is open; silent on (K) \\
\addlinespace
Center vortices & Percolating $\mathbb{Z}_3$ flux &
--- & (L)(T) as a picture; vortex removal kills $\sigma$ and
chiral condensate; continuum formulation open \\
\addlinespace
Gribov / functional & Configuration-space geometry; IR
propagators & Bounding statements & Necessary conditions;
truncation-limited \\
\addlinespace
Seiberg--Witten & Monopole condensation, derived &
\textbf{Confinement + gap, as a theorem} --- in softly-broken
$\mathcal{N}=2$ & The existence proof that rigidity suffices;
not QCD \\
\addlinespace
Holography & Worldsheet on an IR wall & Area law for
strong-coupling large-$N$ cousins & (L)(T)(G) for cousins;
not QCD \\
\addlinespace
\textbf{This framework} & $\mathbb{Z}_3$ holonomy of a real
compact cycle; winding as charge; the wall on the scale axis &
\textbf{(K) outright} (C-P6-1, at the lock's grade);
\textbf{(L) given the gap} (T-P6-4, \S\ref{sec:tp64}) &
(B)(T) carried consistently; (G) = the named open hypothesis
$\widehat{G}$; the pure number $\sigma/\Lambda^2$ owed
(\S\ref{sec:sigma-number}) \\
\bottomrule
\end{tabular}
\end{table}

Three observations close the survey. First, \emph{nobody has
the theorem}: the only rigorous confinement proof in four
dimensions lives in a supersymmetric cousin, and the lattice's
analytic result lives at strong coupling. The framework's
conditional theorem of \S\ref{sec:tp64} enters a field where
conditionality is the state of the art, not a special
embarrassment. Second, the two mechanism programs with the
strongest lattice support --- vortices and the dual
superconductor --- are both \emph{center and topology} stories.
The framework's mechanism belongs to that family: it is not an
outsider proposing a new kind of answer, but a geometrization
of the kind the evidence already favors. Third,
Seiberg--Witten's lesson --- provability arrives with rigidity
--- is exactly the wager of the next three sections: this
framework's rigid structure is the lock.

\section{The center bridge: one \texorpdfstring{$\mathbb{Z}_3$}{Z3}, three jobs}
\label{sec:centre-bridge}

Here is the observation that reorganizes this whole Part ---
hiding, as the best ones do, in plain sight:
\emph{the standard account's own confinement criteria live in
the center $\mathbb{Z}_3$ of the color group --- the same
$\mathbb{Z}_3$ whose holonomy is this paper's lock.}

Walk the standard side first; every clause is textbook
\citep{Greensite2011}. \emph{Who} is confined is decided by
triality: the asymptotic string tension survives screening
only for sources with $t \neq 0$ mod 3 --- the fundamental
quark, $t = \pm 1$ --- while the adjoint, $t = 0$, is screened
by gluons and its string breaks. Casimir scaling governs the
approach; \emph{$N$-ality governs the destination}. \emph{When}
confinement holds is decided by the center symmetry: in the
pure gauge theory the Polyakov loop --- the order parameter
valued in the center --- averages to zero exactly when the
$\mathbb{Z}_3$ symmetry is unbroken, which is the confined
phase; deconfinement \emph{is} center-symmetry breaking, its
first-order character at $270$~MeV inherited from the
three-state structure; and with dynamical quarks the center
symmetry is explicitly broken --- which is precisely why
physical QCD has a smooth crossover at $156.5$~MeV rather than
a transition, and why strings break. And \emph{where} the
confining disorder lives, on the best lattice evidence, is in
percolating $\mathbb{Z}_3$ vortex flux
(\S\ref{sec:mechanisms}). Who; when; where: three independent
roles, one group of three elements. And the group itself is nothing exotic --- it is
clock arithmetic with three hours. Every quark advances the
clock one hour; every antiquark turns it back one; and only
configurations that return the hand to twelve may leave the
room as free particles. The whole confinement lore of the
standard account is, at bottom, that three-hour clock worn
three different ways.

Now the framework's side, all of it proved in Part~VI at its
stated grade: the lock is the center element
$z = W(2) = \mathrm{diag}(\omega,\omega,\omega,1,1)$ realized
as the holonomy of a real, non-spatial compact cycle; the
center identity $e^{2\pi i B} = z$ makes triality literally a
winding, $B = t/3$; and the winding is an exactly conserved
charge. So:

\emph{In the standard account, $\mathbb{Z}_3$ decides who is
confined and when --- but the center is bookkeeping, a subgroup
acting on abstract field space. In this framework the same
$\mathbb{Z}_3$ is geometry: a third of a turn of an actual
cycle. The quark is confined because a third of a wrap cannot
close (C-P6-1); the proton is immortal because a whole wrap
cannot unwind (T-P6-2); and the criteria the lattice built its
confinement lore on are, in this reading, the boundary shadows
of that one twist. One $\mathbb{Z}_3$ --- the charge, the cage,
and the immortality.}

Figure~\ref{fig:centre-bridge} draws the two columns side by
side. And one paragraph of the bridge is load-bearing in the
\emph{opposite} direction, and the reader deserves it stated
with equal force: the two sides do NOT relate the center to
the theory in the same way. On the lattice, the center acts as
a \emph{symmetry} of the pure-gauge action --- a symmetry that
can break (deconfinement) and that fundamental matter breaks
explicitly (the crossover). In this framework the center
element is a \emph{fixed holonomy} --- the lock is not a
symmetry of an action and cannot break; the winding is a
charge, not an order parameter. The bridge is therefore not an
identification but a proposal of ancestry: \emph{the reason
center-valued order parameters organize confinement
phenomenology so well is that the center is, in nature, the
holonomy of a real cycle whose windings are conserved
charges.} On this reading, center-symmetry breaking at
$270$~MeV is bulk physics --- color mobility in a hot medium
--- while the readout layer keeps (K) at every temperature,
which is exactly the custody split the heavy-ion data enforce
(\S\ref{sec:qgp}); and the explicit breaking by quarks maps to
string breaking, the budding of integer-winding endpoints. The
disanalogy is not a weakness to hide; it is the content of the
proposal.

\begin{figure}[htbp]
\centering
\resizebox{0.95\textwidth}{!}{%
\begin{tikzpicture}[
 font=\small\sffamily,
 colbox/.style={rectangle, rounded corners=3pt, line width=0.7pt,
  minimum width=72mm, align=left, inner sep=8pt, font=\footnotesize\sffamily},
 hdr/.style={font=\small\bfseries\sffamily, anchor=south west},
]
\node[colbox, draw=blue!55!black, fill=blue!8] (std) at (0,0)
 {\textbf{THE STANDARD ACCOUNT --- $\mathbb{Z}_3$ as symmetry}\\[4pt]
  \emph{who}: triality $t$ --- $N$-ality decides the asymptotic\\
  \hspace*{1em}tension; fundamental confined, adjoint screened\\[2pt]
  \emph{when}: center symmetry --- Polyakov loop $\langle L\rangle = 0$\\
  \hspace*{1em}confined; deconfinement = center breaking\\
  \hspace*{1em}(first order at $\approx 270$~MeV, pure gauge)\\[2pt]
  \emph{where}: percolating $\mathbb{Z}_3$ vortex flux carries\\
  \hspace*{1em}the confining disorder\\[2pt]
  \emph{status}: criteria and pictures --- no continuum theorem};
\node[colbox, draw=violet!60!black, fill=violet!8] (gg) at (9.4,0)
 {\textbf{THIS FRAMEWORK --- $\mathbb{Z}_3$ as holonomy}\\[4pt]
  \emph{the lock}: $z = W(2) = \mathrm{diag}(\omega,\omega,\omega,1,1)$,\\
  \hspace*{1em}holonomy of a real non-spatial cycle (Part~VI)\\[2pt]
  \emph{the charge}: $e^{2\pi i B} = z$, $B = t/3$ --- triality is\\
  \hspace*{1em}a winding; windings are conserved\\[2pt]
  \emph{the cage}: no fractional winding closes (C-P6-1);\\
  \hspace*{1em}area law given the gap (T-P6-4)\\[2pt]
  \emph{the immortality}: whole wraps cannot unwind (T-P6-2)};
\draw[{Stealth[length=2.6mm]}-{Stealth[length=2.6mm]}, line width=1.0pt,
 draw=black!60] (std.east) -- (gg.west)
 node[midway, above, font=\scriptsize\sffamily, align=center]
 {one $\mathbb{Z}_3$};
\node[rectangle, rounded corners=3pt, draw=black!50, fill=black!4,
 line width=0.6pt, minimum width=160mm, align=center, inner sep=6pt,
 font=\footnotesize\sffamily] (dis) at (4.7,-3.55)
 {\textbf{The disanalogy, load-bearing:} on the left, a \emph{symmetry}
 of the action --- breakable (deconfinement), explicitly broken by
 quarks (the crossover).\\ On the right, a \emph{fixed holonomy} ---
 not breakable; the winding is a charge, not an order parameter.\\
 The proposal is ancestry, not identity: center criteria work because
 the center is the shadow of a real cycle's twist.};
\end{tikzpicture}%
}
\caption[The center bridge: one $\mathbb{Z}_3$, two readings.]{One
$\mathbb{Z}_3$, two readings. Left: the standard account's confinement
lore --- who is confined (triality/$N$-ality), when (center symmetry
and its breaking), where the disorder lives (center vortices) --- all
valued in the center of the color group, as bookkeeping. Right: this
framework's lock --- the same center element as the holonomy of a real
compact cycle, making triality a conserved winding, with the cage
(C-P6-1, T-P6-4) and the immortality (T-P6-2) as corollaries. The
bottom box is the honesty clause: symmetry versus holonomy is a real
structural difference, and the bridge is a claim of ancestry, not of
identity.}
\label{fig:centre-bridge}
\end{figure}

\section{The wall on the scale axis}
\label{sec:scale-wall}

The second structural piece is proprietary to this framework's
geometry, and it converts a fifty-year-old intuition into
geography. In this framework the fifth dimension \emph{is} the
renormalization-group axis: $r = \ln(E_{\mathrm{seam}}/\mu)$
(Part~II). The color coupling therefore does not merely
``run''; it runs \emph{along a direction of the bulk}, and the
question ``where does the strong interaction become strong?''
has a literal answer in coordinates. (A reader wary of extra
dimensions may simply read $r$ as the zoom knob of the
microscope: every statement below about ``descending in $r$''
is a statement about examining nature at coarser resolution,
and nothing more mystical than that.)

The answer is priced, with no dial anywhere. The registration
$\alpha_s^{-1}(M_Z) = \pi e$ (\ANine{}, consumed in Part~III)
seeds the walk; the boundary theory's own $\beta$-function
carries it down; the coupling reaches strength at
$\mu = \Lambda \approx 340$~MeV; and the corresponding
coordinate is
\begin{equation}
r_\Lambda \;=\; \ln\!\frac{E_{\mathrm{seam}}}{\Lambda}
\;=\; 44.3 \;=\; K + 9.3,
\label{eq:wall-location}
\end{equation}
where $K = 35$ is the lock ($\mu = 3.85$~TeV) and the color
flow continues $\ln(M_{\mathrm{lock}}/\Lambda) = 9.3$ e-folds
past it. Call $r_\Lambda$ \emph{the wall}. Two register notes
fence the term. First, the wall's \emph{location} is a
zero-parameter output of numbers this cohort has already
registered; the wall's \emph{existence} --- that the color
sector's landscape genuinely ends the flow there with a gapped
vacuum --- is precisely the gap hypothesis $\widehat{G}$ of the
next section, and is not claimed. Second, a canon note: the
wall lies below the lock, in the infrared continuation of the
scale flow; this Part reads it as a feature of the boundary
theory's free-energy landscape --- the scale at which the
singlet basin's curvature takes over --- rather than as a
geometric endpoint of the bulk interval, whose constitutional
extent is fixed upstream. Nothing below depends on the
bookkeeping choice.

Now the geometry does work
(Figure~\ref{fig:scale-wall}). Hold two fractional-winding
sources at boundary separation $L$ and ask where their
connecting flux lives in the $(x, r)$ plane. For small $L$ the
flux ascends only a short way down the scale axis: the
configuration samples the weak-coupling region, and the free
energy returns Coulomb-with-running --- asymptotic freedom's
$-\tfrac43 \alpha_s(1/L)\,\hbar c/L$, automatic here because
$r$ \emph{is} the RG axis and short distance means shallow
descent. For $L$ beyond the crossover, the cheapest
configuration changes character: the flux descends to the
wall, runs along it, and ascends --- and the descent cost
saturates (it no longer grows with $L$) while the wall segment
grows linearly. Linearity is then geography:
\begin{equation}
F(L) \;=\; \sigma L + \mathrm{const} + \ldots, \qquad
\sigma \;=\; c\,\Lambda^2,
\label{eq:wall-tension}
\end{equation}
with $\sigma$ set by the only scale present at the wall ---
dimensional transmutation drawn as a picture --- and $c$ a
pure number this framework now owes
(\S\ref{sec:sigma-number}). The reader who knows the
holographic literature will recognize the drape-on-the-wall
mechanism (\S\ref{sec:mechanisms}); the reader who knows this
cohort will recognize that here the fifth dimension is not a
model input but the constitutional scale axis, the wall
location is priced by $\pi e$, and nothing extended is being
postulated: the ``string'' is the flux tube T-P6-4 derives,
not a fundamental object. The Cornell form
\eqref{eq:cornell} --- Coulomb, crossover, linear --- is
thereby derived \emph{in form} from the geometry; its
coefficients remain quoted from experiment until
$\widehat{G}$ and $c$ are paid for.

\begin{figure}[htbp]
\centering
\resizebox{0.663\textwidth}{!}{%
\begin{tikzpicture}[
 font=\small,
 lab/.style={font=\footnotesize, align=center}
]
%% boundary at top, scale axis downward
\draw[line width=1.2pt, draw=black!80] (0,0) -- (12,0);
\node[lab, anchor=south west] at (0, 0.62)
 {the 4D boundary\\ ($r = 0$: the seam)};
\draw[->, line width=0.80pt, draw=black!70] (12.4,0) -- (12.4,-5.4)
 node[anchor=north, font=\footnotesize] {$r = \ln(E_{\mathrm{seam}}/\mu)$};
%% lock line
\draw[dotted, draw=black!55, line width=0.8pt] (0,-3.1) -- (12,-3.1);
\node[lab, anchor=west, text=black!60] at (0.1,-3.32)
 {$r = K = 35$: the lock ($3.85$~TeV; Part~VI's twist lives here)};
%% wall
\draw[dash pattern=on 5pt off 2.5pt, draw=red!70!black, line width=1.2pt]
 (0,-4.6) -- (12,-4.6);
\node[lab, anchor=west, text=red!70!black] at (0.1,-4.95)
 {$r_\Lambda = 44.3 = K + 9.3$: \textbf{the wall} --- location priced ($\pi e$, zero dials); existence $= \widehat{G}$};
%% sources
\fill[blue!60!black] (3.4, 0) circle (2.4pt);
\fill[blue!60!black] (8.6, 0) circle (2.4pt);
\node[lab, anchor=south east, text=blue!60!black] at (3.3, 0.14) {$t = +1$};
\node[lab, anchor=south west, text=blue!60!black] at (8.7, 0.14) {$t = -1$};
\node[lab, anchor=south] at (6.0, 0.55) {separation $L$};
\draw[<->, draw=black!60] (3.4, 0.42) -- (8.6, 0.42);
%% small-L flux (shallow arc)
\fill[black!65] (5.35, 0) circle (1.6pt);
\fill[black!65] (6.65, 0) circle (1.6pt);
\draw[draw=black!55, line width=0.9pt]
 (5.35,0) .. controls (5.55,-0.85) and (6.45,-0.85) .. (6.65,0);
\node[lab, text=black!60] at (6.0,-1.5)
 {small $L$: shallow descent --- Coulomb with running};
%% large-L flux: descend, run along wall, ascend
\draw[draw=blue!60!black, line width=1.3pt]
 (3.4,0) .. controls (3.4,-2.6) and (3.7,-4.25) .. (4.35,-4.52)
 -- (7.65,-4.52)
 .. controls (8.3,-4.25) and (8.6,-2.6) .. (8.6,0);
\node[lab, text=blue!60!black, anchor=north west] at (8.85,-3.35)
 {large $L$: descend,\\ run along the wall, ascend:\\ $F = \sigma L$,\ \ $\sigma = c\,\Lambda^2$};
\end{tikzpicture}%
}
\caption[The wall on the scale axis: confinement as geography.]{Confinement
as geography. The fifth dimension is the RG axis; the color walk,
seeded by the registration $\alpha_s^{-1}(M_Z) = \pi e$, reaches
strength at the wall \eqref{eq:wall-location} --- location a
zero-parameter output, existence the named hypothesis $\widehat{G}$.
Flux between nearby sources stays shallow (Coulomb with running);
beyond the crossover the minimal configuration drapes along the wall,
whose segment grows linearly with separation --- the area law as
drawn, with tension $\sigma = c\Lambda^2$ by dimensional
transmutation. The lock at $K = 35$ (dotted) is the same axis's
marked point from Part~VI: one scale direction carries both the twist
that writes baryon number and the wall that cages color.}
\label{fig:scale-wall}
\end{figure}

\section{The conditional area-law theorem}
\label{sec:tp64}

The pieces are now on the table: a conserved, quantized winding
(the lock); a screening arithmetic valued in thirds (the
center); a wall whose location is priced (the scale axis). This
section assembles them into the Part's central new result ---
stated as a theorem with its conditions in the open, because
that is what the honesty grammar is for. But first, the point
in one plain sentence, for the reader who wants it before the
price tags: \emph{if the strong interaction's vacuum has a
lowest note --- a gap --- then the cage follows as geometry,
because the flux between fractional charges is a conserved,
quantized thing that cannot fade, cannot spread, and cannot be
neutralized, and such a thing has no option but to bundle into
a tube whose cost grows with its length.} The theorem's job is
to say exactly what that ``if'' buys; the conditions are the
price tags, and they are worth reading, because one of them is
the entire remaining debt.

\begin{theorem}[\textbf{T-P6-4}: The conditional area law]
\label{thm:area-law}
Work on the boundary under the $\mathbb{Z}_3$-locked descent
(T-P6-2, at its stated grade), and assume:
\textbf{($\widehat{G}$)} the gap hypothesis: the color
sector's singlet-stratum free-energy landscape has an isolated
vacuum basin, with connected correlations clustering
exponentially at a finite length $1/m$, $m > 0$;
\textbf{(A1)} locality of \GFF{} (additivity over distant
regions up to clustering corrections; constitutional);
\textbf{(A2)} exponential clustering under $\widehat{G}$; and
\textbf{(A3)} winding superselection --- the winding flux
through any surface separating sources is conserved and
quantized in units of $\tfrac13$ (\emph{not an assumption: the
lock, consumed at its grade}). Then for a static source pair
of triality $\pm t$, $t \not\equiv 0$ mod 3, at boundary
separation $L$:
\[
F(L) \;\geq\; \sigma_{\min} L - c_1, \qquad
\sigma_{\min} \geq c_0\, m^2, \quad c_0 > 0;
\]
the minimal-free-energy configuration is a flux tube of
transverse radius $\sim 1/m$; no fractional-triality
asymptotic state exists at any energy; and the asymptotic
tension depends only on $t$ mod 3 --- the $N$-ality law: the
adjoint string breaks by screening, the fractional string
never does.
\end{theorem}

\noindent\emph{Proof structure, in four steps --- with the
step four dimensions cannot take flagged where it happens.}

\paragraph{Step 1: the flux cannot end.} The compact cycle has
no boundary for winding current to terminate on. Every
3-surface separating the two sources is therefore crossed by
winding flux $\equiv t/3$ (mod 1): conserved, quantized,
geometric. [Theorem, from the lock.]

\paragraph{Step 2: the Higgs escape is closed --- the
load-bearing step.} In four dimensions, ``gap $\Rightarrow$
confinement'' is \emph{false}, and the opening section already
named the counterexample: a fundamental-representation condensate
screens any source, leaving a gapped, unconfined theory. Every
4D folklore chain from the gap to the area law dies on that
rock --- there is nothing in a 4D gauge theory to forbid the
screening. Here there is, and it is a theorem: by C-P6-1
(\S\ref{sec:absent-quark}), every boundary state --- hence
every condensate, every screening cloud the boundary can form
--- carries integer winding, $t \equiv 0$ mod 3. Screening can
shift a source's triality only in steps of 3: a $t = \pm 1$
source can never be neutralized, while a $t = 0$ source can.
The confinement criterion \emph{and its one exception} ---
adjoint screening, hence the $N$-ality law, hence the broken
adjoint string the lattice sees --- fall out of the same
mod-3 arithmetic. [Theorem, at the lock's grade. This is where
the framework converts a fifty-year picture into a
derivation.]

\paragraph{Step 3: the gap forbids dilution.} Under
$\widehat{G}$ and (A2), fields cluster within $1/m$: the flux
crossing each separating surface must be carried inside a
transverse region of radius $\sim 1/m$, since spreading it
wider costs clustering-violation free energy across a larger
volume. Quantization (Step 1) forbids carrying less than one
quantum of $t/3$; locality (A1) makes the cost per unit of
separation additive. Energy per unit length is therefore
bounded below: $\sigma_{\min} \geq c_0 m^2$. [Framework
derivation; the functional tightening --- the precise
constant, the surface-decomposition bounds --- is a named
formal residual, recorded in the registry.]

\paragraph{Step 4: the minimizer is a tube.} Additivity plus
clustering make the free energy minimal when the flux-carrying
region is connected and of minimal cross-section: a tube,
$F(L) = \sigma L + \text{end terms} + \text{fluctuation
corrections}$ --- and the leading fluctuation correction is
universal and already measured, which is the next section's
story. [Same grade as Step 3.] $\square$

\paragraph{What is new here, and what is not.} The
\emph{shape} of this argument --- quantized, unscreenable flux
in a gapped medium must form a tube --- is the dual-%
superconductor and $N$-ality logic the lattice community has
held as a picture for fifty years (\S\ref{sec:mechanisms}).
What is new and checkable is the standing of the premises:
\emph{in this framework the two clauses that keep the picture
from being a 4D theorem --- flux quantization, and the
impossibility of fractional screening --- are theorems}, not
modeling choices. The framework thereby upgrades ``gap
$\Rightarrow$ confinement'' from false-in-general (four
dimensions) to conditional-theorem (this geometry), at the
price of one hypothesis, $\widehat{G}$, whose location on the
scale axis is already priced \eqref{eq:wall-location} and
whose existence this paper does not claim. Seiberg--Witten's
lesson (\S\ref{sec:mechanisms}) --- confinement becomes
provable exactly when a rigid structure forces it --- is here
taken literally: the rigid structure is the lock.

\paragraph{The branch-selection reading, promoted.} This
paper's ledger has carried the \GSSB{} reading --- the
color-singlet branch as the unique stable attractor --- at
grade INT, with promotion registered. T-P6-4 pays that
registration: on every $t \not\equiv 0$ stratum the free
energy grows without bound along the separation coordinate, so
no finite asymptotic minimum exists; the $t \equiv 0$ strata
own all the finite basins (the hadrons of Part~III). ``The
singlet branch is the unique stable attractor'' is now
T-P6-4 plus T-P6-3 in one sentence, at grade D(c) ---
conditional on $\widehat{G}$ and the lock's own conditions
--- while the broader \GSSB{} gloss (one selection grammar
across the electroweak vacuum, the widths, and the descent)
remains what it always was: an organizing interpretation,
INT, and priced as such in the ledger.

\section{The Goldstone-counting test: the string knows the
dimensions are not spatial}
\label{sec:goldstone-test}

One result of this Part is entirely unconditional, and it
converts the innocent L\"uscher number
of \S\ref{sec:cage-measured} into a precision test of this
framework's deepest structural commitment. The idea is homely.
A stretched string hums, and the character of its hum depends
on how many directions it is free to shake in: a string that
could vibrate in three transverse directions would carry more
quantum jitter than one confined to two. Count the jitter and
you have counted the directions --- without ever leaving the
room. The QCD flux tube is a stretched string a femtometer
long; the lattice has heard its hum to a few percent; and this
section spends that measurement on the most structural
question this framework owns.

The universal term \eqref{eq:luscher} counts massless
transverse fluctuation modes of the flux tube: each direction
the tube can wobble in contributes $-\pi/24 \cdot \hbar c/r$,
and Goldstone's theorem fixes the count --- one massless mode
per spatial translation the tube breaks. Count them in this
framework. The two transverse \emph{spatial} directions are
broken by the tube: two Goldstone modes. The \emph{scale}
direction contributes none: the tube sits at the wall, which
is a minimum of the free energy along $r$ --- that is what a
wall is --- so $r$-fluctuations are gapped by the basin's
curvature. The \emph{phase} direction contributes none: the
cycle is compact and holonomy-locked, its excitations starting
at the lock scale. Exactly two massless modes; coefficient
$-2\pi/24 = -\pi/12$; \emph{exactly what the lattice
measures}.

Now run the logic backwards, because that is where the test
bites. Suppose the fifth and sixth dimensions were
\emph{spatial} --- ordinary compact space directions, as every
Kaluza--Klein and string compactification before this
framework would have them --- with any radius large enough to
matter at the QCD scale. A flux tube in that world breaks the
extra translations too: at least one additional massless
transverse mode, and the universal coefficient shifts by fifty
percent, $-\pi/12 \to -\pi/8$ --- at one femtometer, from
$-51.7$ to $-77.5$~MeV. Lattice string spectroscopy, which
confirms the $-\pi/12$ coefficient at the few-percent level,
excludes that shift utterly. The measured vibration spectrum
of the QCD flux tube is, read this way, a standing
experimental statement that \emph{whatever extra structure
spacetime carries at the strong-interaction scale, it is not
spatial} --- and this framework's fifth and sixth dimensions,
scale and phase, gapped at the wall and locked on the cycle,
pass a test that spatial extra dimensions fail. The
constitutional departure this cohort made when it declared the
extra dimensions non-spatial --- the step Part~VII's
chronology dates to January 2026 --- turns out to have been
graded, years in advance, by the vibrations of the QCD
string. The Part records this as a passed consistency test,
not a falsifier: the framework did not adjust anything to
meet it, and there is nothing left to adjust.

\section{The number that will judge it:
\texorpdfstring{$\sigma/\Lambda^2$}{sigma over Lambda squared}}
\label{sec:sigma-number}

A framework with no dials must eventually put a number where
its mechanism is. The wall picture \eqref{eq:wall-tension}
makes the commitment precise: the tension is
$\sigma = c\,\Lambda^2$ with $c$ a \emph{pure number}, because
$\Lambda$ itself descends from the registration
$\alpha_s^{-1}(M_Z) = \pi e$ with nothing adjustable anywhere
in the chain. The measured target is already known to both
sides:
\begin{equation}
\frac{\sigma}{\Lambda^2_{\overline{\mathrm{MS}}}}
\;=\; \left(\frac{440}{340}\right)^{\!2} \;=\; 1.67, \qquad
\frac{\sqrt{\sigma}}{\Lambda} \;=\; 1.29,
\label{eq:sigma-target}
\end{equation}
with an honest band of roughly $[1.55, 1.86]$ once the
uncertainties of both anchors are carried. The registered
commitment, stated as a kill: \emph{when the wall computation
of $c$ is carried out (registry item D-P6-4(c)), a result
outside $[1.42, 1.93]$ --- the $\pm 15\%$ window declared in
advance --- kills the wall reading of the tension outright,
linearity notwithstanding.} A derivation landing inside the
band would be, to this paper's knowledge, the first
zero-parameter string tension produced by any framework.

What this paper does \emph{not} do is guess the number, and
the discipline deserves one sentence of record. The cohort's
own look-elsewhere audit (the MSMF/OGN cost procedure of the
Standard-Model paper) finds the band \eqref{eq:sigma-target}
dense with cheap closed forms --- forty alphabet candidates at
low complexity, seven at the lowest --- so a registration
today would be curve-fitting, and none is made. For the
registry only: the ledger's cheapest candidate sits dead
center (the ratio $L/N = 5/3$, which would put
$\sqrt{\sigma}$ at $438.9$~MeV, $0.24\%$ from the lattice
value), and it is recorded as \emph{the first value the
D-P6-4(c) computation must be tested against} --- a candidate
awaiting a derivation, carrying no evidential weight until
one exists. The number in \eqref{eq:sigma-target} is the
judge, not the verdict.

\section{The absent quark: from dynamics to topology}
\label{sec:absent-quark}

Everything so far --- the string, the budding ends, the wall,
the conditional theorem --- explains why liberating a quark is
\emph{dynamically hopeless}. Q5 asks a sharper question: is
the free quark merely imprisoned, or is it \emph{not a
possible object}? The two readings agree on every scattering
experiment ever done and differ absolutely in kind: a prisoner
exists behind its wall; a grammatical impossibility does not
exist at all. The century could not separate them --- it had,
as \S\ref{sec:five-claims} noted, no language in which they
differ. Part~VI's theorem can, and this section collects the
corollary.

\begin{corollary}[\textbf{C-P6-1}: No boundary readout carries
fractional winding]
\label{cor:no-free-quark}
Under the $\mathbb{Z}_3$-locked descent of
Theorem~\textup{T-P6-2} (Part~VI, at its stated grade), every
fragment of the boundary spectrum carries winding
$B \in t/3 + \mathbb{Z}$ with its color triality $t$, and
every \emph{asymptotic boundary state} --- every configuration
the boundary can present as a completed readout --- carries
integer winding. A lone quark has $t = \pm 1$: winding
$\pm\tfrac13$ modulo integers, hence no completed readout. The
free quark is not a suppressed state of the boundary theory;
it is not a state of the boundary theory.
\end{corollary}

The proof is one sentence stood on Part~VI's machinery: the
readout channel is the synchronized cycle, a completed readout
closes on that cycle, and a curve cannot close after a third
of a turn. But the picture deserves better than a sentence,
because it is the Part's gift to intuition. Baryon number,
Part~VI will show, is a \emph{winding} --- the proton wraps
the compact cycle once, in three strands of a third each.
Color is the bulk's bookkeeping of the strands; the boundary
can only read completed wraps. Asking to detect a free quark
is asking to be handed \emph{a third of one wrap of a closed
loop} --- not a thing that is expensive, or rare, or
short-lived, but a phrase with no referent, like the west side
of a circle. The sixteen-tonne string of
\S\ref{sec:cage-measured} is how the dynamics enforces, at
finite energy, what the topology decrees at every energy: pull
on a strand and the loop makes you a new loop, because loops
are all there is.

Two fences keep the corollary honest. First, the division of
explanatory labor,
stated once and plainly: \emph{dynamics (QCD, at its own
grade) explains why the approach to isolation costs sixteen
tonnes of force; topology (T-P6-2, at its grade) explains why
the destination does not exist.} The two readings of Q5 that
the century could not separate turn out to both be true, at
different layers --- and the structural one is the answer to
the question as asked, because ``why has \emph{no} experiment
\emph{ever}...'' is a question about existence, not cost.
Second, what C-P6-1 does \emph{not} say: nothing about the
tension's value, nothing about linearity, nothing about the
gap. A world in which C-P6-1 held with a logarithmic
confining potential would satisfy the corollary; the
dynamical content of this Part lives in T-P6-4 and its named
hypothesis, not here. Keeping the kinematic jewel and the
dynamical debt in separate display cases is exactly what the
five-claim scorecard was for.

So the Part can now answer the question it was rebuilt to
answer, in one breath: \emph{can this framework explain
confinement as a theorem? Today: half --- and the better
half.} The existence half (K) is a corollary of the lock, at
the lock's conditional grade, and it is the half no other
program derives. The dynamical half (L) is one named
hypothesis from a theorem: given the gap, the area law follows
(T-P6-4), the Higgs escape closed by the same lock. The gap
itself --- $\widehat{G}$, the Clay-hard core --- remains open,
here as everywhere, its location priced and its existence not
claimed. Until it is paid, this Part claims a theorem-given-%
the-gap and says so in exactly those words.

\section{The singlet rule: the triangle's closure condition}
\label{sec:singlet-rule}

C-P6-1 named what does not exist. Its contrapositive is a
census of what does --- and the census has been in every
textbook since 1964 under the name \emph{the color-singlet
rule}: the only hadrons nature exhibits are the
quark--antiquark pairs and the three-quark bundles, $q\bar q$
and $qqq$, and their conjugates. In the standard account that
rule is an output of confinement dynamics, extracted from the
same lattice lore as the tension. In this framework it is
shorter than that: it is the closure condition of the compact
cycle, checkable fragment by fragment in one line of
arithmetic per state. This section runs the whole zoo through
that one line, says why the answer draws a triangle, and
collects an electromagnetic dividend the century has always
treated as a separate curiosity.

\paragraph{The one line.} Part~VI's center identity
(\S\ref{sec:centre-identity}) gives the action of the center
element $z$ on any single fragment: the phase $\omega^{t}$,
with $t$ the fragment's color triality and $\omega = e^{2\pi
i/3}$. Phases multiply, and triality adds:\ on a composite of
constituents $t_1, t_2, \ldots$,
\begin{equation}
z:\;|\,\mathrm{state}\,\rangle \;\longmapsto\;
\omega^{\,T}\,|\,\mathrm{state}\,\rangle ,
\qquad
T \;=\; \textstyle\sum_i t_i \ (\mathrm{mod}\ 3),
\label{eq:total-triality}
\end{equation}
and the composite's winding on the synchronized cycle is
$B = \sum_i t_i/3$, an integer plus $T/3$. Now impose the only
requirement the boundary ever imposes: a completed readout
closes on the cycle (C-P6-1's premise). A curve closes when
its winding is an integer. So:
\begin{equation}
\text{admissible readout}
\;\Longleftrightarrow\;
B \in \mathbb{Z}
\;\Longleftrightarrow\;
\omega^{T} = 1
\;\Longleftrightarrow\;
T \equiv 0 \ (\mathrm{mod}\ 3)
\;\Longleftrightarrow\;
\text{center-invariant}.
\label{eq:singlet-chain}
\end{equation}
Four equivalences, each one step of arithmetic. The right-hand
end is the group theorist's phrasing of ``color singlet under
the center''; the left-hand end is the boundary's phrasing of
``a state that can actually be handed over.'' The
color-singlet rule is the statement that these ends meet ---
and here they meet by construction, because they are the same
condition read at the two ends of the chain.

\paragraph{The zoo, run through the line.} The test is total
triality, so it can be administered to any configuration
anyone has ever proposed:

\begin{table}[htbp]
\centering
\caption[Quark configurations tested for closure.]{Every quark configuration, tested for closure. The second column gives the triality modulo three, the third the phase that triality carries, the fourth whether the configuration closes, and the last what the answer means physically. Read down the closure column: only the triality-zero combinations are states of the boundary theory at all. The verdict column is careful to add that the arithmetic selects and the cage enforces, which are two different jobs.}
\label{tab:triality-closure}
\small
\begin{tabular}{@{}lccc p{0.34\textwidth}@{}}
\toprule
configuration & $T$ (mod 3) & $\omega^{T}$ & closes? &
verdict \\
\midrule
quark $q$ & $1$ & $\omega$ & no & not a state of the boundary
theory (C-P6-1); the cage of \S\ref{sec:cage-measured} is the
dynamics enforcing it at finite energy \\
antiquark $\bar q$ & $2$ & $\omega^{2}$ & no & likewise, with
the arc run the other way \\
diquark $qq$ & $2$ & $\omega^{2}$ & no & never free; lives
only as a fragment inside a closed loop \\
meson $q\bar q$ & $0$ & $1$ & yes & winding $0$: closed
without wrapping; the lightest completions, and mortal \\
baryon $qqq$ & $0$ & $1$ & yes & winding $1$: one full wrap;
carries $B = 1$ forever (T-P6-2) \\
antibaryon $\bar q\bar q\bar q$ & $0$ & $1$ & yes & winding
$-1$: the reflection's image, wrap for wrap
(\S\ref{sec:dihedral}) \\
tetraquark $qq\bar q\bar q$ & $0$ & $1$ & yes & permitted:
the rule counts triality, not quarks \\
pentaquark $qqqq\bar q$ & $0$ & $1$ & yes & permitted, and
nature obliges; rarity is dynamics, not grammar \\
gluon (adjoint) & $0$ & $1$ & --- & passes the triangle's
test; its absence from the free list is the gap's work, not
the winding's (see the fence below) \\
\bottomrule
\end{tabular}
\end{table}

\noindent The table is the color-singlet rule, with its exact
boundary drawn. Everything with $T \equiv 0$ closes; nothing
with $T \not\equiv 0$ does; and the rule's true name is the
\emph{$N$-ality rule}, because it counts winding thirds rather
than constituents. The exotic rows are worth a sentence: the
century sometimes read ``only $q\bar q$ and $qqq$'' as if
quark number were the law. It is not. The law is closure, the
tetraquark and pentaquark close, and their observed existence
--- rare, fragile, but real --- is exactly what a closure law
predicts and a quark-counting law forbids.

\paragraph{Why a triangle.} The geometry deserves to be seen
once without algebra. Place the three colors at the three
vertices of the equilateral triangle inscribed in the cycle
--- at $0$, $\tfrac{2\pi}{3}$, $\tfrac{4\pi}{3}$, the cube
roots of unity. A lone quark stands at one vertex: its phase
points a third of the way around, and nothing local can point
it home. A baryon puts one strand at each vertex: the three
phases multiply to $\omega^{3} = 1$ --- equivalently, the
three vertex vectors sum to the triangle's center. A meson
pairs a vertex with its conjugate vertex, $\omega \cdot
\omega^{-1} = 1$, home again. The triangle is a clock face
with three hours; every quark advances the hand one hour,
every antiquark turns it back one; and the boundary's door
opens only when the hand reads twelve. Confinement, in this
picture, is not a wall around the quarks. It is the absence
of a door at one-third past.

\paragraph{The electromagnetic shadow.} One more column can be
added to the fragment table of \S\ref{sec:centre-identity},
and it is a column the century has always carried as a
separate fact. Run every fragment once more, this time
against its electric charge, and one finds --- exactly, not
modulo anything ---
\begin{equation}
Q + \tfrac{t}{3} \,\in\, \mathbb{Z}
\quad\text{on every fragment;}
\qquad\text{equivalently}\qquad
e^{2\pi i Q} \;=\; \omega^{-t},
\label{eq:em-shadow}
\end{equation}
and therefore, by the center identity $B = t/3$,
\begin{equation}
Q + B \,\in\, \mathbb{Z}
\qquad\text{fragment by fragment.}
\label{eq:q-plus-b}
\end{equation}
(Checked, as always, by running the table: $u$ has $Q =
+\tfrac23$, $B = +\tfrac13$; $d$ has $-\tfrac13$,
$+\tfrac13$; the electron $-1$, $0$; the $X$ at $-\tfrac43$,
$+\tfrac13$; every row lands in $\mathbb{Z}$.) Read
Eq.~\eqref{eq:q-plus-b} through the closure chain
\eqref{eq:singlet-chain} and the century's three great
never-statements collapse into one:\ \emph{no experiment has
ever seen a free quark; no experiment has ever seen an
isolated fractional electric charge; and no boundary readout
ever carries an open winding} --- one fact, in three
laboratory dialects. A fractional charge in isolation would
\emph{be} a free triality, which would \emph{be} an unclosed
loop presented as closed. That is why F-P6-2 (one confirmed
free fractional charge) sits in this paper's falsification
table as a kill: it would not merely surprise the model; it
would exhibit the grammatically impossible sentence, spoken.
The proton's own exact charge --- Q2, settled in
\S\ref{sec:exact-charge} by the codebook --- is the positive
print of the same negative: the closure that forbids the
fraction alone is what delivers the integer together.

\paragraph{Three fences.} First, \emph{triality zero is
necessary, not sufficient}. The adjoint gluon passes the
center's test, and no free gluon appears in nature's ledger:
its exclusion is the mass gap's work --- the screening and
string-breaking of \S\ref{sec:centre-bridge} --- which is
dynamics (T-P6-4's territory, at its named-hypothesis grade),
not topology. The triangle draws the outer wall of the
possible; the gap furnishes the interior. Second, and for the
same reason, \emph{this section is the kinematic half of
confinement only} --- claim (K) of the scorecard
(\S\ref{sec:five-claims}), the half C-P6-1 delivered. Nothing
here prices the string tension, fixes linearity, or touches
the Clay problem; the Part's honest division of labor stands
exactly as \S\ref{sec:absent-quark} drew it. Third, the rule
as derived here rests on the lock, so it is exactly as
conditional as T-P6-2 --- and exactly as falsifiable: one
free fractional charge (F-P6-2), or one hadron stable in a
fractional-triality channel, ends it.

\section{The hot crowd: what the quark--gluon plasma does and
does not change}
\label{sec:qgp}

A reader current with the heavy-ion program will raise the
right objection here, and it deserves a section rather than a
footnote. Above the crossover at
$T_c = 156.5 \pm 1.5$~MeV \citep{HotQCD2019}, hadronic matter
melts into a quark--gluon plasma: color charges screened,
mobile, collectively flowing --- RHIC and the LHC have
produced and characterized this state in exquisite detail.
Does a phase in which quarks roam free not falsify
\S\ref{sec:absent-quark}?

It does the opposite, and the distinction is the custody
principle again, now doing experimental work. The plasma is
\emph{bulk} color dynamics: a medium, microns short of an
asymptotic state, in which color is mobile the way charge is
mobile in a metal. The corollary C-P6-1 constrains
\emph{boundary readouts} --- what survives to a detector as a
completed asymptotic state. And the heavy-ion record is
unequivocal: every fireball ever produced, at every energy,
hadronizes. No fractional charge has ever emerged from a
quark--gluon plasma; the crowd disperses only in wraps.
Within this framework that is not a dynamical accident to be
re-derived per collision energy but the topology holding at
the readout layer regardless of how hot the bulk ran:
\emph{the plasma unwinds nothing, because the windings were
never in the plasma --- they are in the reading.} The QGP is
thereby promoted from apparent counterexample to the
corollary's cleanest stress-test: fifty years of fireballs,
zero fractional readouts.

The center bridge of \S\ref{sec:centre-bridge} adds the
thermal fine print. In the
quarkless theory, deconfinement is a genuine first-order
transition at $\approx 270$~MeV --- center-symmetry breaking,
its character inherited from the three-state structure of
$\mathbb{Z}_3$; in this framework's reading, that is the bulk
center order giving way, and its universality class comes
along for free. In physical QCD the fractional-winding matter
explicitly breaks the center order --- which is the same
statement as ``strings break'' (\S\ref{sec:cage-measured}),
budding integer-winding endpoints --- and the transition
smooths into the measured crossover. Bulk order melting at
one temperature or another; readout topology holding at every
temperature: the two-layer custody split is not a rhetorical
device but the exact shape of the heavy-ion record.

\section{The Clay problem, at this framework's register}
\label{sec:clay-register}

The Part's opening contract promised precision about the
Millennium Problem; this section pays it, in four moves,
because a paper that derives anything about confinement owes
its reader exact coordinates relative to the most famous open
problem in the subject \citep{JaffeWitten2000}.

\emph{Move one: what the problem asks.} Construct quantum
Yang--Mills theory on $\mathbb{R}^4$ --- existence in the
constructive-field-theory sense, satisfying axioms at least as
strong as the Wightman axioms --- for any compact simple gauge
group, and prove a mass gap $\Delta > 0$. Existence plus gap.
Confinement of fundamental charges is a further, logically
distinct property, discussed in the problem's description but
not the prize condition; the preamble's care in separating
(G) from (K) and (L) is the problem's own.

\emph{Move two: what is not in doubt.} Nature's answer is not
in question --- the lattice theory exists rigorously at every
finite spacing and its measured gap is $1.71$~GeV
\citep{MorningstarPeardon1999}. The Clay problem is a problem
about constructive continuum mathematics, not about whether
the world is gapped. No simulation closes it; no experiment
could.

\emph{Move three: why this framework cannot solve it ---
structurally, not accidentally.} The Clay object is a
\emph{closed} four-dimensional quantum field theory on
$\mathbb{R}^4$. This framework's constitutional position is
that the boundary theory is not closed: it is the \GDOSm{}
projection of a six-dimensional bulk, its four-dimensional
QFT-ness effective rather than axiomatic. A framework whose
axioms deny the premise of a mathematics problem does not get
credit for the problem --- and this paper hereby declines, in
advance and in writing, any reading of this Part as a claim
on it.

\emph{Move four: what this framework can honestly say.} It
can relocate the physical question. If the D-P6-4 program
completes --- $\widehat{G}$ established at the wall, the
tension's pure number derived --- then within this framework
nature's confinement and gap are theorems \emph{of an open,
projected system}: properties of the free-energy landscape of
a boundary that was never a closed 4D field theory. The
seventy-year difficulty of proving the same properties inside
closed 4D Yang--Mills is then read as a diagnosis: the price
of proving a true fact in a frame that omits the structure
enforcing it --- the compact cycle, the lock, the wall. That
is a diagnosis, not a solution; the mathematical problem
remains exactly as open, and exactly as important, as before.
The framework's relationship to the Clay problem is thus the
same as its relationship to the tank: it predicts the silence
will hold, explains why from outside the room, and leaves the
room's own hard question standing --- which is, this paper
believes, the only honest posture available to anyone in
2026.

\section{The eighty-year null, and the Part's falsifier faces}
\label{sec:part5-kills}

The experimental record Q5 stands on is one of the great quiet
nulls of physics \citep{PDG2024}: Millikan-descended droplet
searches, levitometer experiments on bulk matter, accelerator
searches at every new energy frontier, sea-water and meteorite
and lunar-sample assays --- eighty years, no confirmed
fractional charge, anywhere, ever. Each null was individually
modest; their totality is a wall. The framework's reading
converts the wall from an accumulating mystery (\emph{why does
the prison never leak?}) into a boundary condition
(\emph{there is nothing to leak}).

The faces, sharpened:

\begin{description}
\item[F-P6-2, sharpened.] A single confirmed free fractional
charge --- from any source: cosmic ray, accelerator, bulk
matter, QGP freeze-out --- kills C-P6-1 outright, and with it
wounds T-P6-2 upstream (a fractional-winding readout would
exhibit an open curve on the synchronized cycle). No parameter
absorbs it; this is as binary as falsifiers come. Note what is
\emph{not} a kill: millicharged particles from a hidden sector
(charge $\epsilon e$, no color) would be a discovery about
the electromagnetic fiber, not about baryonic winding --- the
distinction is registered now, before any anomaly arrives to
blur it.
\item[The theorem face.] T-P6-4 is killed by its own
arithmetic in two ways, both stated in advance: a derived
tension outside the declared window
(\S\ref{sec:sigma-number}) kills the wall reading; and any
demonstration that a fractional-triality condensate can form
on the boundary (contradicting Step~2) kills the theorem's
load-bearing step and C-P6-1 with it.
\item[The consistency face.] The Goldstone-counting test
(\S\ref{sec:goldstone-test}) is recorded as passed: the
lattice's $-\pi/12$ is what non-spatial extra dimensions
require, and a confirmed deviation toward $-\pi/8$-like values
at long distance would have falsified the framework's
dimensional architecture from inside QCD. It did not.
\item[The deconfinement face.] Any observation of color
screened into an asymptotic fractional readout --- e.g.\ a
claimed thermal emission of unconfined quarks surviving
hadronization --- contradicts the custody split of
\S\ref{sec:qgp}. Five decades of heavy-ion data currently
enforce it.
\item[The debt face, as a scoreboard.] The
registry's D-P6-4 now reads as a scoreboard: (a) branch
uniqueness --- delivered as T-P6-4's corollary, conditional on
$\widehat{G}$; (b) the area law --- delivered at
theorem-structure grade on $\widehat{G}$ plus the lock, formal
tightening named; (c) the pure number --- open, the wall
computation owed, the target and kill window printed; (d) the
L\"uscher coefficient --- closed, unconditionally; (e) the
thermal face --- partial, universality inherited, numbers owed
with (c). The single deep residual is $\widehat{G}$ itself,
and the ledger says so.
\end{description}

\section{The Part, in plain English --- and its ledger}
\label{sec:part5-ledger}

For the reader who skipped the mathematics --- or who wants to
check that the mathematics said what it seemed to say --- here
is Part~V again, in plain words.

Nobody has ever seen a free quark, and this Part has told you
why twice, at two different depths. The first answer is the
century's, and it is about \emph{cost}: quarks are tied
together by a cable of gluon field that pulls with sixteen
tonnes of force no matter how far you stretch it, and when you
stretch it past a femtometer the vacuum finds it cheaper to
grow two new quarks at the break than to let the cable
lengthen --- so pulling on a quark hands you two hadrons, never
a bare quark. That answer is measured, magnificent, and still
--- after fifty years and a million-dollar bounty --- unproven
from first principles. The second answer is this framework's,
and it is about \emph{existence}: baryon number is a winding
around a real, closed loop of geometry, each quark carries a
third of a wrap, and a third of a wrap of a closed loop is not
a rare thing or a costly thing but a phrase with no referent
--- like the west side of a circle. The century's answer says
the door out of the prison is unaffordable. This framework's
answer says there is no outside for the door to open onto.

What did this Part actually establish, and what does it still
owe? Three things were placed on the table. First, the
existence answer just given is a theorem of the lock, at the
lock's stated grade --- and it is the half of the question no
other approach on the market even formulates. Second, the cage
itself --- the sixteen-tonne linearity --- became a theorem
\emph{given one missing piece}: if the color vacuum has a
lowest note (a gap), then the flux between fractional charges
cannot fade, cannot spread, and cannot be neutralized, so it
must bundle into a tube whose cost grows with length. That
missing piece is the same one the Clay problem turns on; this
Part named it, priced its location on the scale axis with no
adjustable numbers, and declined to pretend it is proven.
Third, a test nobody commissioned came back passed: the
quantum hum of the flux tube --- a measured correction to its
energy --- counts how many directions the tube can vibrate in,
and the count says \emph{three spatial dimensions and not one
more}. A framework whose extra dimensions are places would
fail that count; this framework's extra dimensions are scale
and phase --- not places --- and pass. And one number was
promised rather than delivered, deliberately: the string
tension in units of the theory's own scale, a pure number with
no dial behind it, to be computed and not guessed --- with the
lattice's answer, $1.67$, already printed here as the judge
that will one day acquit this Part's mechanism or convict it.

For the reader keeping score, the formal ledger follows ---
every claim of the Part, its grade, and its fences, in the
cohort's standard contract form.

\begin{table}[H]
\centering
\small
\caption[Status ledger of Part V.]{Status ledger of Part~V.
Grades as in Table~\ref{tab:import-ledger}. Three columns: the claim, the grade it is carried at, and the fences or notes that bound it. The ledger is here so that the standing of the whole Part can be read down one column instead of reconstructed from the text, and so that no claim of the Part is left without a grade beside it.}
\label{tab:part5-ledger}
\begin{tabular}{p{0.40\textwidth}p{0.10\textwidth}p{0.40\textwidth}}
\toprule
\textbf{Claim} & \textbf{Grade} & \textbf{Fences / notes} \\
\midrule
Clay-grade openness; gap/confinement distinction & D & Stated
as the Part's opening contract; existence + gap is the prize
condition; nothing below borrows against it. \\
\addlinespace
The measured cage: Cornell, $\sigma$, L\"uscher, breaking,
Casimir, thermal structure
\eqref{eq:cornell}--\eqref{eq:luscher} & D & Standard physics
(lattice + spectroscopy), cited; arithmetic in
Appendix~\ref{app:string}. \\
\addlinespace
Five-claim decomposition (K)(L)(B)(T)(G) & D & Organisational;
each claim graded separately below. \\
\addlinespace
Mechanism survey and Table~\ref{tab:mechanisms} & D &
Literature record, cited; no mechanism endorsed beyond its
evidence. \\
\addlinespace
The center bridge & INT & One $\mathbb{Z}_3$: ancestry
proposal, with the symmetry-vs-holonomy disanalogy stated as
load-bearing; standard side textbook, cited. \\
\addlinespace
The wall: location \eqref{eq:wall-location} & D & Zero-dial
arithmetic from the $\pi e$ registration; existence NOT
claimed (= $\widehat{G}$). \\
\addlinespace
T-P6-4 (conditional area law; $N$-ality asymptotics) & D(c) &
On $\widehat{G}$ + (A1)(A2) + the lock's grade (H3 + R2 +
R3); Higgs escape closed by C-P6-1; formal tightening a named
residual; Casimir scaling not claimed. \\
\addlinespace
Branch-selection uniqueness & D(c) & Corollary of T-P6-4 +
T-P6-3 on $\widehat{G}$; the broader \GSSB{} grammar gloss
remains INT. \\
\addlinespace
Goldstone-counting test ($-\pi/12$; non-spatial dimensions
pass) & D & Unconditional given the tube; lattice-confirmed
coefficient; spatial extra dimensions at $\sigma$-scale
excluded by $+50\%$ shift. \\
\addlinespace
$\sigma = c\Lambda^2$ with $c$ pure; target and kill window
\eqref{eq:sigma-target} & CAND & The wall computation owed
(D-P6-4(c)); no value registered; candidate $L/N$ recorded
without evidential weight. \\
\addlinespace
C-P6-1 (no fractional-winding readout) & D(c) & Corollary of
T-P6-2 at Part~VI's grade; one-sentence proof; upgrades Q5's
``impossible'' half from dynamics to topology. \\
\addlinespace
QGP custody split; thermal fine print & INT + D & Reading INT;
the zero-fractional-readout record and the pure-gauge/%
crossover distinction are data, cited. \\
\addlinespace
Clay register (four moves) & D & Declination of credit in
writing; the recast labeled a diagnosis, conditional on
D-P6-4. \\
\addlinespace
Eighty-year null as boundary condition & D & PDG compilation;
reading at C-P6-1's grade. \\
\bottomrule
\end{tabular}
\end{table}

And now the door this whole Part has been leaning against can
open. Twice in these pages a single third-of-a-turn twist of a
compact cycle did work that dynamics alone could not: it told
the spin it could not decompose (Part~IV), and it told the
quark it could not leave. Both times the twist was borrowed
from a Part that had not yet happened. That Part is next --- and
it is the one the author has been walking toward for
forty-eight years. In Part~VI the twist stops being a tool and
becomes the story: where it comes from, why it sits at the one
point of the hypercharge line the spectrum allows, how it
breaks the unified symmetry, evicts the proton's executioners,
splits the Higgs, and writes baryon number into geometry ---
one twist, four locks --- and what half a century of silence in
a Japanese mine was measuring all along. The quark could not
leave the proton. Now we ask why the proton cannot leave the
universe.

\GGPart[The immortal proton]{The Immortal Proton}{The $\mathbb{Z}_3$-lock writes baryon number; one twist, four locks; Kamiokande (Q6, Q7)}
\label{part:stability}

This is the Part the paper is named for, and the third side from which Part~\ref{part:basin}'s one fact is read. Grand unification sentenced the proton in 1974. The sentence has never been carried out, and a kilometer of rock and forty years of watching have returned nothing at all. This Part argues that the silence is a prohibition and not a delay. The argument runs through an obstruction and a lock: baryon number cannot be gauged the obvious way, and the Part proves that rather than asserting it; what carries the charge instead is a winding of the compact geometry, written by the center identity. The anomaly ledger is then computed in full, every coefficient of it, so that a reader can check that the single open door is the only opening rather than the only one shown --- and that door is what Part~\ref{part:birth} will need.
\section{The question, personally}
\label{sec:stability-personal}

Every Part of this paper so far has opened with physics. This one
is entitled to open with a mine.

In 1974, unification sentenced the proton to death
\citep{GeorgiGlashow1974}. The sentence was not a speculation
bolted onto the theory; it was the theory's crown jewel doing
exactly what crown jewels do. $\mathrm{SU}(5)$ put quarks and
leptons into shared multiplets --- the same five-dimensional
object holding the down-antiquark beside the electron's doublet
--- and the gauge bosons connecting the occupants of a shared
multiplet turn one into the other. The $X$ and $Y$ bosons were
not a bug. They were the unification, embodied. And a boson that
turns quarks into leptons turns protons into positrons and pions,
at a rate the theory could estimate: slow beyond imagining on a
human scale, fast enough to see in a large enough crowd of
protons watched patiently enough. The community's conviction had
the special firmness of a prediction nobody wanted to be wrong:
if the forces unify --- and the couplings visibly converged ---
the proton decays. Through the late 1970s this was, by common
consent, the safest bet in fundamental physics.

So we went underground to collect. In a lead-zinc mine in
Kamioka, between 1979 and 1985, a small group under
Masatoshi Koshiba built a cylinder of three thousand tonnes
of water, walled with a thousand twenty-inch photomultiplier
eyes, and named it for its purpose: Kamiokande, the Kamioka
\emph{Nucleon Decay Experiment}. The author of this paper, then
a doctoral student, spent those six years on its design and
construction \citep{ArisakaKamiokande1985} --- the tank, the
optics, the electronics that would catch the Cherenkov ring of a
positron born from a proton's corpse. We expected, with the
field, to see $p \to e^+ \pi^0$ within a year or two of
switch-on. %% AUTHOR-VOICE SITE: this paragraph and the previous
%% are stand-ins for the author's verbatim account, per the
%% Ship-1 convention; he rewrites them at will.

The protons declined. Not that year, not that decade, not in the
forty years since, not in Kamiokande nor IMB nor
Super-Kamiokande's half-century of accumulated
megaton-years \citep{Takenaka2020, SuperKamiokande2024}. The
experiment, magnificently, refused to waste the silence --- solar
neutrinos, the supernova burst of February 1987
\citep{HirataSN1987}, a Nobel Prize, a new astronomy --- but the
question it was named for stood unanswered in either direction:
the proton would not die, and no theory could say \emph{why not}.
The Standard Model's answer --- baryon number happens to be
conserved by every operator you can write at renormalizable
order --- is an accountant's shrug, and every deeper theory
proposed since 1974 has tended to \emph{break} the accident, not
explain it. For fifty years the deepest silence in physics has
had no theorem.

This Part supplies the theorem, at its stated grade. The
strategy, announced plainly: we take the stability theorem of the
Standard-Model paper as received (\S\ref{sec:as-received}), state the axiom
that put \UoneB{} there at all --- the founder's own imposition,
with its seventy-year lineage (\S\ref{sec:founders-axiom}) ---
prove
that its one open hypothesis \emph{cannot} be discharged the way
the century would first try (\S\ref{sec:obstruction}), find the
discharge where the algebra said not to look --- in the center of
the color group and the topology of the compact cycle
(\S\S\ref{sec:centre-identity}--\ref{sec:z3lock}) --- close the
structure with a reflection that turns out to be the CPT
companion's central identity (\S\ref{sec:dihedral}), pay the
anomaly bill and find exactly one door left open
(\S\ref{sec:anomaly-ledger}), assemble the completed theorem with
its three remaining pins in plain view
(\S\ref{sec:completed-theorem}), and then walk back into the mine
to say what the silence meant (\S\ref{sec:kamiokande}).

\section{The stability theorem, as received}
\label{sec:as-received}

Part~II stated the inheritance; here it goes to work. The
Standard-Model paper proves \citep{ArisakaA9SM}, as its
T-SM-60.1, that the proton is absolutely stable \emph{given three
hypotheses}: \textbf{(H1)} the $X, Y$ bosons live only in the
bulk --- the boundary spectrum contains the Standard-Model gauge
sector and nothing more; \textbf{(H2)} the bulk \UoneB{} descends
to act on the split boundary fields as baryon number itself ---
$+\tfrac13$ per quark, $0$ per lepton, component by component;
and \textbf{(H3)} the \UoneB{} gauge anomalies are canceled by
the \BIsix{} Green--Schwarz inflow of \AThree{}. Under
H1--H3, every operator with $\Delta B = \pm1$ or $\pm2$ carries
non-zero \UoneB{} charge and vanishes identically, to all orders;
the sole surviving violation is topological, in units of three;
the proton cannot decay.

The load-bearing hypothesis is H2, and its home paper said so
with unusual candor, deferring the ``projection-plus-charge
computation'' to a successor paper on proton decay. This is that
paper, and this is that computation. To see why it deserved a
paper rather than a footnote, one must first see why H2 looks
\emph{impossible}.

\section{The founder's axiom: a QED for baryon number}
\label{sec:founders-axiom}

Before the century's instinct is formalized and defeated, the
axiom it will be defeated \emph{about} deserves its own page ---
because the \UoneB{} in hypothesis H2 did not fall out of a
derivation, and this paper refuses to let the reader discover
that fact in a footnote. It was put there. The founding move of
this framework --- in its author's own accounting, the very
beginning of the program --- was to \emph{impose} \UoneB{} as
a local gauge invariance of the bulk, alongside $\SUfive{}$, in
deliberate analogy with electromagnetism. The reasoning is the
oldest and most successful in gauge theory: electric charge is
not conserved by bookkeeping luck; it is conserved --- exactly,
locally, at every vertex of every process ever measured ---
\emph{because it is gauged}. Noether currents of global
symmetries can be broken by anomalies, by gravity folklore, by
the next layer of physics; the current of a consistent local
gauge symmetry cannot. The axiom says: let the baryon obey the
same kind of law as the coulomb. Let matter's number be
conserved for the same \emph{reason} charge is.

Within the cohort's honesty grammar the statement's rank must be
exact: this is an \emph{axiom}, not a theorem. It sits in the
constitutional gauge group \eqref{eq:bulk-group-p6} of Part~II,
imposed at the same level as $\SUfive{}$ itself, and nothing in
the framework derives it from something deeper. What \emph{is}
derived --- and is this Part's entire business --- is what the
imposition costs and what it buys: whether a bulk \UoneB{} can
descend to act as baryon number on the split boundary spectrum
at all (the obstruction says: not the obvious way), how the
geometry pays the debt the obstruction identifies (the center
twist), and what the anomaly bill comes to (the ledger, and its
one door). The founder imposed the law; the geometry had to be
shown able to carry it. That is the precise sense in which H2
deserved a paper rather than a footnote.

The axiom is also not new --- and the paper counts that as
evidence, not embarrassment. It was proposed, in almost exactly
these words, in 1955: Lee and Yang, in a one-page Letter
published the year after Yang--Mills, asked whether the
conservation of heavy particles might follow from a gauge
principle in analogy with electromagnetism --- and, in the same
page, computed the objection that would keep the idea in exile
for most of a century \citep{LeeYang1955}. What happened to
their proposal between 1955 and its imposition here --- three
independent refusals, one wrong revival, a minority line a dozen
authors long, and an experimental program that became, without
ever quite saying so, a dedicated search for this very force ---
is a story this paper owes its reader in full, and it is kept
where a record belongs, in a Part of its own: Part~VII. Here,
only the physics of the analogy.

\begin{table}[htbp]
\centering
\small
\caption[The founder's axiom beside its model.]{The founder's
axiom beside its model: \UoneEM{} and \UoneB{}, side by side.
Both are imposed local gauge invariances; both realize their
charge, after descent, as a quantized topological label; both
are operator-exact. The two rows in which they differ are the
two places the boundary \emph{forced} a bending --- and each
bending, this Part shows, is load-bearing structure.}
\label{tab:qed-analogy}
\begin{tabular}{p{0.24\textwidth}p{0.32\textwidth}p{0.34\textwidth}}
\toprule
& \textbf{\UoneEM{} (the model)} & \textbf{\UoneB{} (the
axiom)} \\
\midrule
Origin & Imposed: local gauge invariance & Imposed: local gauge
invariance, in deliberate analogy \\
\addlinespace
Anomaly ledger (per family) & All coefficients zero: gaugeable
on the boundary as it stands & Two non-zero entries
($[\SUtwoL]^2 B$, $[\UoneY]^2 B$); payable only by bulk inflow
(\S\ref{sec:anomaly-ledger}) \\
\addlinespace
The boson & Massless photon; Coulomb's law as the visible
signature & Stueckelberg-massive; \emph{no} long-range force
--- the bending forced by experiment since 1955 (Part~VII) \\
\addlinespace
Charge realization & Quantized through the $\SUfive{}$
embedding (Part~III) & Quantized by the center twist: $B = t/3$,
the winding of the compact cycle
(\S\ref{sec:centre-identity}) \\
\addlinespace
Exactness & Exact to all orders & Exact to all orders; sole
exception the topological $\Delta B \in 3\mathbb{Z}$ door
(\S\ref{sec:anomaly-ledger}) \\
\addlinespace
Experimental face & Charge quantization; the neutrality of bulk
matter & Proton stability; the equivalence-principle nulls
(F-P6-5) \\
\bottomrule
\end{tabular}
\end{table}

Table~\ref{tab:qed-analogy} states the analogy at its honest
resolution. In the bulk it is exact: two Abelian factors of one
constitutional gauge group, indistinguishable in kind. On the
boundary the analogy had to bend in exactly two places, and both
bendings were \emph{forced} --- neither is a choice this
framework made. First, the anomaly: on the split Standard-Model
spectrum the baryonic current is chiral-anomalous, so a
four-dimensional gauged \UoneB{} is not merely unmotivated but
inconsistent --- this is the century's strongest reason for
declining the axiom, and the framework does not evade it but
pays it, through the \BIsix{} inflow of H3, with the two
surviving coefficients becoming the ledger of
\S\ref{sec:anomaly-ledger} and the lone
$\Delta B \in 3\mathbb{Z}$ door becoming, in Part~X, the origin
of matter itself. Second, the force: a massless baryonic photon
would pull on matter in proportion to its baryon content, and
the torsion-balance tradition --- from the data Lee and Yang
themselves consulted to the modern program --- excludes any
such force to extraordinary depth; so the boson takes the
Stueckelberg mass, the one mechanism that kills the force
while leaving the rigid charge and its selection rules exact.
Bent twice, broken never: what survives the two bendings is
everything the axiom was imposed to secure --- a local gauge
origin for matter's number, operator-exact conservation to all
orders, and a charge that is not a bookkeeping label but a
quantized property of the geometry. The sole structural
difference from QED that remains is the anomaly line of the
table; and that difference, far from being a defect, is the
single door through which Part~X will bring every baryon in
the universe into existence.

In the deep structure, indeed, the analogy ends closer than it
began: after descent, \emph{both} charges are windings. Electric
charge arrives quantized because hypercharge is locked inside
$\SUfive{}$ and the embedding admits no continuous deformation
(Part~III); baryon number arrives quantized because the twist is
locked to the center of the color group and a winding admits no
continuous deformation (\S\ref{sec:z3lock}). The founder's
instinct --- that the proton's charge-like conservation and the
electron's should have the same kind of explanation --- is, in
the end, discharged more literally than the analogy dared to
ask.

The imposition has a personal history, and the paper carries it
at testimony grade where it belongs. The instinct did not come
to its author from unification; it came through neuroscience ---
a gauge invariance first written for the information dynamics of
the human cortical network, a full year before it was pointed at
the baryon --- and it took, in his own accounting, fifty years
of a physicist's life, Kamiokande included, to trust an
imposition the field had declined for seventy. That chronology,
dated and marked as testimony, is
Part~VII's closing section; the sentences that carry it are the
author's own.
%% AUTHOR-VOICE SITE: the paragraph above scaffolds the
%% founder's-journey statement; the author rewrites it verbatim
%% at sign-off (his fifty years, the neural route, the March
%% 2025 imposition), per the Ship-9 commission.

Now the century's instinct can be formalized --- and defeated.
The axiom demands that the imposed bulk symmetry act, after
descent, as baryon number itself on the split boundary fields.
The next section proves this \emph{cannot} be done the way
anyone trained in four-dimensional gauge theory would first
try; why H2 looks impossible is the right place to begin
discharging it.

\section{The obstruction: why baryon number cannot be gauged the
obvious way}
\label{sec:obstruction}

Here is the century's instinct, formalized. If you want an exact
\UoneB{} acting on the boundary fields, find it a generator:
some combination of the Abelian charges available after the
descent. What is available? The bulk \UoneB{} itself contributes
a constant per bulk multiplet --- each family is \emph{one}
coherent spinor packet (Part~II), so one constant per family, or
at most one per $\SUfive{}$ irrep if one imagines relaxing the
packet structure. And the descent's surviving Abelian generator
is hypercharge $Y$ --- the \emph{only} $\mathrm{U}(1)$ direction
in $\SUfive{}$ that commutes with the Standard-Model group,
because the Standard Model fills $\SUfive{}$'s rank. So the most
general candidate is
\begin{equation}
B \;\overset{?}{=}\; \alpha(\text{multiplet}) \;+\; \beta\, Y .
\label{eq:candidate-B}
\end{equation}

\begin{theorem}[\textbf{T-P6-1}: The rank-matching obstruction]
\label{thm:obstruction}
No assignment of per-multiplet constants $\alpha$ and coefficient
$\beta$ realizes baryon number on the split Standard-Model
spectrum: equation \eqref{eq:candidate-B} has no solution, even
granting each $\SUfive{}$ irrep its own constant. Moreover the
combination that \emph{is} realizable in this form, $B{-}L$,
fails to protect the proton: every dimension-six proton-decay
operator conserves it.
\end{theorem}

\noindent\emph{Proof, in full view.} Work inside the
$\mathbf{10}$, which houses three fragments of known baryon
number: the quark doublet $Q$ ($B = +\tfrac13$, $Y = \tfrac16$),
the up-antiquark $u^c$ ($B = -\tfrac13$, $Y = -\tfrac23$), and
the positron field $e^c$ ($B = 0$, $Y = 1$). One constant
$\alpha_{10}$ serves all three. The requirement
\eqref{eq:candidate-B} reads:
\begin{equation}
\alpha_{10} + \tfrac{\beta}{6} = \tfrac13, \qquad
\alpha_{10} - \tfrac{2\beta}{3} = -\tfrac13, \qquad
\alpha_{10} + \beta = 0 .
\label{eq:three-conditions}
\end{equation}
Subtract the second from the first:
$\tfrac{\beta}{6} + \tfrac{2\beta}{3} = \tfrac23$, so
$\beta = \tfrac45$; then the first gives
$\alpha_{10} = \tfrac13 - \tfrac{2}{15} = \tfrac15$. Insert both
into the third: $\tfrac15 + \tfrac45 = 1 \neq 0$. Three
conditions, two unknowns, and the system says \emph{one equals
zero}. No choice exists; adding a separate $\alpha_{\bar 5}$
cannot help, since the contradiction lives entirely inside the
$\mathbf{10}$. $\square$

Two remarks complete the theorem's second sentence. First, the
near-miss is itself famous: with $(\alpha, \beta)$ per irrep one
\emph{can} solve for $B - L$ --- explicitly,
$B - L = \tfrac15(X + 4Y)$ with $X(\mathbf{10}) = +1$,
$X(\overline{\mathbf 5}) = -3$, the textbook charge of
$\mathrm{SO}(10)$ descent --- which is precisely why gauged
$B{-}L$ is commonplace in the unification literature while
gauged $B$ is nowhere. Second, that near-miss is worthless for
the proton: the four classic dimension-six decay operators
($QQQL$; $u^c u^c d^c e^c$; and their conjugate-pair partners)
each carry $B = \pm 1$ and $B - L = 0$ --- Appendix~%
\ref{app:lock-arithmetic} tabulates all four --- so an exact
$B{-}L$ lets every one of them stand. The century's road ends
here: \emph{if baryon number is to be exact, it cannot be a
Lie-algebra charge.} The instinct that gauging $B$ is impossible
was correct --- at the level of algebras. The theory has more
than algebras.

\paragraph{Physics intuition: why the algebra had to fail.}
Baryon number sees \emph{color} --- it pays $\tfrac13$ per
quark precisely because quarks come in threes --- but it must
commute with color rotations, so it cannot live inside the
color algebra; and it distinguishes occupants of one
$\SUfive{}$ multiplet, so it cannot live outside them either.
It needs to be something an algebra element cannot be: a charge
that counts \emph{thirds of color} without touching color's
generators. There is exactly one structure in a gauge group
with that character --- its center --- and that is where the
discharge is hiding.

\section{The center identity: baryon number is color's
signature}
\label{sec:centre-identity}

Inside $\SUfive{}$ sits the element
\begin{equation}
z \;=\; \mathrm{diag}\!\left(\omega,\, \omega,\, \omega,\, 1,\,
1\right), \qquad \omega = e^{2\pi i/3},
\label{eq:z-element}
\end{equation}
the generator of the $\mathbb{Z}_3$ center of the color block:
a group element, not a generator --- a finite rotation with no
infinitesimal version, which is exactly how it slips past
Theorem~\ref{thm:obstruction}. It commutes with every
Standard-Model transformation (it is diagonal, constant on the
color block, constant on the weak block), and it acts on any
fragment of any representation by the fragment's \emph{color
triality} $t$: phase $\omega^{t}$, with $t = +1$ for a color
triplet, $-1$ for an antitriplet, $0$ for a singlet, additive
modulo three under products.

Now the identity on which the whole Part turns. Run through
every fragment in the theory --- the five matter fragments, the
Higgs doublet and its color-triplet partner, the $X,Y$
themselves --- and compare $\omega^{t}$ with $e^{2\pi i B}$:

\begin{table}[htbp]
\centering
\caption[Triality of the unified fragments, and B.]{The triality of every fragment of the unified multiplet, and the baryon number that follows from it. The first row gives the integer triality, the second divides it by three, and the third is the baryon number. The second and third rows are identical, and that is the table's whole content: baryon number is triality over three, fragment by fragment, with no assignment made by hand anywhere in the row.}
\label{tab:triality-fragments}
\small
\begin{tabular}{lcccccccc}
\toprule
fragment & $Q$ & $u^c$ & $d^c$ & $L$ & $e^c$ &
$H_{\mathrm{doub}}$ & $H_{\mathrm{trip}}$ & $X,Y$ \\
\midrule
triality $t$ & $+1$ & $-1$ & $-1$ & $0$ & $0$ & $0$ & $+1$ &
$+1$ \\
$t/3$ & $+\tfrac13$ & $-\tfrac13$ & $-\tfrac13$ & $0$ & $0$ &
$0$ & $+\tfrac13$ & $+\tfrac13$ \\
$B$ & $+\tfrac13$ & $-\tfrac13$ & $-\tfrac13$ & $0$ & $0$ & $0$
& $+\tfrac13$ & $+\tfrac13$ \\
\bottomrule
\end{tabular}
\end{table}

\noindent Equality, in every column --- not modulo one:
\emph{exactly}
\begin{equation}
B \;=\; \frac{t}{3}
\qquad\Longleftrightarrow\qquad
e^{2\pi i B} \;=\; z \;\;\text{on every fragment}.
\label{eq:centre-identity}
\end{equation}
Baryon number was never bookkeeping pasted onto the spectrum. Its
fractional content \emph{is the center of the color group},
carried by the $\SUfive{}$ geometry itself, in plain sight since
1974. What the algebra could not supply --- the component
resolution --- the center supplies for free, because triality is
a property of fragments, not of multiplets: $Q$ and $e^c$ share a
$\mathbf{10}$ but not a triality. All that is missing is the
promotion of \eqref{eq:centre-identity} from a phase identity to
a genuine, exact, additive $\mathrm{U}(1)$ charge --- an integer
lift of $t/3$. Phases live on circles; lifts of phases are
windings. We need the compact cycle.

\section{The compact cycle and the abelian ledger: what kind
of charge a winding is}
\label{sec:abelian-ledger}

The last section ended on a demand: promote the phase identity
\eqref{eq:centre-identity} to an exact, additive charge --- lift
the phase to a winding, which needs a circle. Before the next
section spends the circle, this one is precise about what kind
of object lives on it. The answer --- \emph{a holonomy} --- is
worth a section of its own, because it is where three different
abelian charges of this framework meet, and where the color
group, uninvited, joins them.

\paragraph{Holonomy is gauge invariance, read around a loop.}
Take any gauge field $A$ and any closed curve $C$. Parallel
transport around $C$ produces the Wilson loop
\begin{equation}
W(C) \;=\; \mathcal{P}\exp\!\Big( i \oint_{C} A \Big),
\label{eq:wilson-loop-def}
\end{equation}
and for an abelian field the path-ordering symbol is idle and
$W(C) = e^{i\Phi}$ is a pure phase --- an element of
$\mathrm{U}(1)$. Local gauge invariance is the statement that
the pointwise phase of a charged field is unphysical: it can be
rotated freely at every point, and nothing observable may
depend on the rotation. What survives that freedom is exactly
the loop integral $\Phi$. A gauge transformation shifts $A$ by
a gradient, and a gradient integrates to zero around a closed
curve; so the holonomy is the gauge-\emph{invariant} residue of
a gauge field --- famously a measurable one, since Aharonov and
Bohm's solenoid prints precisely this loop phase on an
interference pattern that no local field value accounts for.
For a reader meeting this for the first time: nothing in the
present paragraph is framework-specific. It is what a Wilson
loop \emph{is}, in any gauge theory since Wilson. The sentence
``the holonomy of \Stheta{}'' and the sentence ``a
$\mathrm{U}(1)$ gauge phase on \Stheta{}'' name the same
object; one names the geometry, the other the symmetry. What
\emph{is} framework-specific is the standing of the circle:
\Stheta{} is a real, compact, non-spatial cycle of the bulk,
so its holonomy is not an analogy for a charge kept elsewhere.
It is the charge, kept here. Figure~\ref{fig:holonomy-primer-p6} puts the three rungs in one drawing --- the loop phase, the Wilson loop, and the difference between a group and a semigroup --- because the third is the one this Part leans on hardest.

\begin{figure}[htbp]
\centering
\resizebox{0.72\textwidth}{!}{%
\begin{tikzpicture}[font=\sffamily\footnotesize,
  lab/.style={font=\sffamily\scriptsize, text=black!72},
  tny/.style={font=\sffamily\tiny, text=black!55}]


\begin{scope}[shift={(0,0)}]
  \node[font=\sffamily\bfseries\scriptsize, anchor=south] at (0,1.95)
     {(a) Aharonov--Bohm};
  \fill[black!12] (0,0) circle (0.34);
  \draw[draw=black!55, line width=0.6pt] (0,0) circle (0.34);
  \node[tny] at (0,0) {$\Phi$};
  \draw[draw=blue!60!black, line width=0.9pt] (0,0) circle (1.25);
  \draw[-{Stealth[length=2mm]}, draw=blue!60!black, line width=0.9pt]
     (30:1.25) arc (30:60:1.25);
  \node[tny, anchor=north, align=center] at (0,-1.65)
    {$\Delta\varphi = q\Phi/\hbar$\\ field zero on the path};
\end{scope}


\begin{scope}[shift={(4.5,0)}]
  \node[font=\sffamily\bfseries\scriptsize, anchor=south] at (0,1.95)
     {(b) Wilson loop};
  \draw[draw=black!62, line width=0.9pt]
     (-1.1,-0.85) -- (1.1,-0.85) -- (1.1,0.85) -- (-1.1,0.85) -- cycle;
  \foreach \p in {(-1.1,-0.85),(1.1,-0.85),(1.1,0.85),(-1.1,0.85)}
     \fill[black!70] \p circle (1.6pt);
  \draw[-{Stealth[length=2mm]}, draw=black!62, line width=0.9pt]
     (-0.2,-0.85) -- (0.2,-0.85);
  \node[tny] at (0,0.1) {$W(C) = \mathcal{P}e^{\,iq\oint A/\hbar}$};
  \node[tny, anchor=north, align=center] at (0,-1.65)
    {gauge invariant on closure;\\ the connection \emph{is} these};
\end{scope}


\begin{scope}[shift={(9.6,0)}]
  \node[font=\sffamily\bfseries\scriptsize, anchor=south] at (0.2,1.95)
     {(c) two algebras};
  \draw[draw=blue!60!black, line width=0.9pt] (-0.55,0.62) circle (0.52);
  \draw[-{Stealth[length=1.8mm]}, draw=blue!60!black, line width=0.8pt]
     (20:0.52)++(-0.55,0.62) arc (20:70:0.52);
  \node[tny, anchor=west, text=blue!45!black] at (0.15,0.62)
     {group: $U^{-1}$ exists};
  \draw[-{Stealth[length=2.2mm]}, draw=black!62, line width=1.0pt]
     (-1.15,-0.55) -- (0.0,-0.55);
  \node[tny, anchor=west] at (0.15,-0.55) {semigroup: no $T^{-1}$};
  \node[tny, anchor=north, align=center] at (0.2,-1.35)
    {\emph{what has inverses can promise;}\\ \emph{what lacks them can happen}};
\end{scope}
\end{tikzpicture}
}
\caption[Holonomy in three rungs: the loop phase, the Wilson loop, and
the arrow.]{\textbf{Holonomy, in three rungs}, after \PaperWTwo{}
\citep{ArisakaW2Holonomy}. (a) The phase is loop data. Aharonov and
Bohm's solenoid prints $\Delta\varphi = q\Phi/\hbar$ on an interference
pattern although the field vanishes everywhere along the path: the phase
is invisible on any open segment and survives every deformation of the
closed one. (b) The same statement in gauge language --- the Wilson loop
\eqref{eq:wilson-loop-def} is what is left of a connection after the
pointwise phase freedom has been used up, and for an abelian field it is
a pure element of $\mathrm{U}(1)$. (c) The rung this paper needs most,
and the one no textbook treatment of (a) and (b) supplies: a group has
inverses and a semigroup does not, which is the difference between a
symmetry that can be undone and a descent that cannot. All three rungs
are standard except the standing of the circle, and the standing of the
circle is the whole of \S\ref{sec:abelian-ledger}: \Stheta{} is a real,
compact, non-spatial cycle of the bulk, so its holonomy is not an analogy
for a charge kept elsewhere. It is the charge, kept here.}
\label{fig:holonomy-primer-p6}
\end{figure}


\paragraph{The abelian ledger of \GG{}.} The bulk gauge group
this paper inherited in Part~II
(\S\ref{sec:gauge-inheritance}) is $\SUfive \times \UoneB
\times \UoneQi$, alongside gravity. Around the one compact
cycle, then, four abelian objects can hold a phase, and it pays
to state their four different standings once, in one place:

\begin{table}[htbp]
\centering
\caption[The three circles, and what becomes of each.]{The three circles, and what becomes of each. The middle column says whether the circle is a fundamental factor of the bulk or a direction inside the unified group; the right column gives its fate and the job it does in this paper. The distinction drawn in the middle column is what carries the argument: a rigid charge on a fundamental factor survives exactly, while a direction inside the group is a dial the descent can set.}
\label{tab:three-circles}
\small
\begin{tabular}{@{}lp{0.30\textwidth}p{0.38\textwidth}@{}}
\toprule
circle & standing & fate and role \\
\midrule
$\UoneB$ & fundamental bulk factor & anomalous; paid by
\BIsix{} inflow; boson Stueckelberg-massive; the rigid charge
survives exactly --- \emph{the charge of this paper} \\
$\UoneQi$ & fundamental bulk factor & the information circle;
one rearrangement, one unit; the proton's register is
Part~XII's subject \\
$\UoneY$ & not a bulk factor: a direction \emph{inside}
$\SUfive$ & the hypercharge Wilson line on \Stheta{} is the
dial the descent sets; its light fluctuation is the Higgs \\
$\mathrm{U}(1)_{\rm EM}$ & not a bulk factor: the unbroken
diagonal $Q = T_3 + Y$ after the descent & a boundary
descendant; conserves charge exactly; carries the triangle's
electromagnetic shadow (\S\ref{sec:singlet-rule}) \\
\bottomrule
\end{tabular}
\end{table}

\noindent Two of the four are constitutional; one is a
direction inside the unified group; one is a survivor of the
descent. Nothing in the list is new to this paper --- but the
list itself repays the reader who has wondered, since
Part~II, how many different things ``a $\mathrm{U}(1)$''
means here. Four, and they are not interchangeable.

\paragraph{Where an abelian circle touches a non-abelian
group.} Now the question a careful reader should have been
holding since Part~V. Confinement is $\SUthreeC$ physics ---
non-abelian, three charges, eight gluons. The cycle's holonomy
is an abelian phase. By what right does a $\mathrm{U}(1)$
worth of phases speak to $\mathrm{SU}(3)$ at all?

The answer is a piece of group theory old enough to vote: at
the \emph{center}. The center of a group is the set of
elements that commute with everything, and for the special
unitary groups it is discrete:
\begin{equation}
Z(\mathrm{SU}(N)) \;=\; \mathbb{Z}_N , \qquad
Z(\SUthreeC) \;=\; \{\, \mathbf{1},\; \omega\mathbf{1},\;
\omega^{2}\mathbf{1} \,\}, \quad \omega = e^{2\pi i/3}.
\label{eq:su3-centre}
\end{equation}
A center element is a peculiar hybrid. It is a genuine element
of the non-abelian group --- yet it is diagonal, it commutes
with every transformation, it has no infinitesimal generator,
and the Lie algebra is blind to it. That is precisely the
profile of the element $z$ of Eq.~\eqref{eq:z-element}, and
precisely how $z$ slipped past the obstruction theorem of
\S\ref{sec:obstruction}: the theorem forecloses on
\emph{generators}, and the center has none. So the abelian
dial and the color group intersect --- not everywhere, but at
three settings. Turn the hypercharge Wilson line continuously
and its holonomy sweeps a circle of phases; at exactly three
points of that circle the holonomy is \emph{also} an element
of the color center. Those three points --- the cube roots of
unity, at $0$, $\tfrac{2\pi}{3}$, $\tfrac{4\pi}{3}$ --- are
the vertices of an equilateral triangle inscribed in
\Stheta{}: \emph{the color triangle}. Everything Part~V
attributed to the center --- who is confined, when, where
(\S\ref{sec:centre-bridge}) --- lives at those three points,
and everything this Part attributes to the winding does too.

\paragraph{The quotient that was always there.} The Standard
Model knew this, in its own fine print. The group that acts
\emph{faithfully} on the Standard-Model fields is not the
textbook product but the quotient
\begin{equation}
G_{\rm SM} \;=\;
\bigl[\, \SUthreeC \times \SUtwoL \times \UoneY \,\bigr]
\big/\, \mathbb{Z}_6 ,
\label{eq:z6-quotient}
\end{equation}
because one specific combination of a color-center rotation,
a weak-center sign, and a hypercharge phase --- namely
$\omega^{t}\,(-1)^{2T_3}\,e^{2\pi i Y}$ --- acts as the
identity on every field the Standard Model has. (The check is
finite: run the eight kinds of fragment through the phase and
watch it collapse to $1$ each time. We have run it; it does;
the reader can, on the back of the envelope this paper keeps
promising.) The $\mathbb{Z}_3$ part of that quotient is
exactly the lock of color triality to hypercharge whose
additive shadow, $B - 2Y \in \mathbb{Z}$ on every fragment, is
the identity the next section spends
\eqref{eq:z-on-wilson-line}. In the Standard Model the
quotient is a true, checkable, and inert remark --- a footnote
about which group one should have written down, with no
dynamical consequence, so almost nobody writes it down. In the
present framework the footnote is load-bearing: the
$\mathbb{Z}_3$ that color and hypercharge share is realized
as the holonomy of a real cycle, and what was a remark about
representations becomes the hinge on which Q6 turns.

\paragraph{Physics intuition: the dial and the detents.} Put
the four abelian standings and the center together and the
geometry of the next section can be seen before it is
computed. The hypercharge Wilson line is a continuous dial ---
a full $\mathrm{U}(1)$ worth of settings on the compact cycle.
Three of those settings are special: at them, the dial's
reading is simultaneously a statement in color --- a center
element of $\SUthreeC$. They are detents --- the click-stops a
machinist cuts into a dial at the settings that matter. The
descent must park the dial somewhere (an unparked dial is a
massless adjoint scalar the spectrum does not contain), the
quantization pin R2 of the next section restricts parking to
six discrete settings, and the spectrum's three demands will
select the detent $c = 2$ --- at which point the dial's
reading \emph{is} the color center's generator, and the
$\UoneB$ winding and the color triality collapse into one
number, $B = t/3$. That collapse is not a metaphor waiting in
the wings; it is Eq.~\eqref{eq:centre-identity}, already
proved fragment by fragment. The next section supplies the
only thing still missing: the reason the dial parks at that
detent and no other.

\subsection{The other fiber: what the interval cannot hold}
\label{ssec:other-fibre}

The ledger above cataloged the abelian objects that live on the
compact cycle. Honesty about the geometry requires one paragraph on
the bulk's \emph{other} non-spatial direction --- the scale interval
$r = \ln(E_{\mathrm{seam}}/\mu)$ --- because its entry in the ledger is empty,
and the emptiness is load-bearing. The reason is one exact
topological statement: \emph{holonomy requires closed loops, and an
interval has none.} No loop on the scale direction ever closes, so
that fiber can hold no Wilson phase, no winding, no quantized
charge --- no gauge-invariant memory of any kind. Transport along
it can only compose forward: a semigroup, the dissipative generator
of the cohort's open-system law. Where the circle is the bulk's
memory --- everything on it is quantized, protected, permanent ---
the interval is the bulk's arrow: it gates, it descends, and it
does not return.

For this paper's subject the pair completes an argument that the
circle alone cannot finish. The winding theorems of this Part
establish that no \emph{operator} can change baryon number; they
do not by themselves establish that no \emph{epoch} can. A
cosmological history able to revisit the lock --- to re-melt the
descent, reopen the unified phase, and re-run the twist ---
would put every winding back into the furnace, and the proton's
immortality would be hostage to cosmology. It is not, and the
reason is the other fiber: the descent moves along the direction
that cannot remember, so the lock crossing is crossed once ---
promptly, with nothing stored that could fund a return --- and the
sphaleron door that swings freely above the crossover is welded
below it, forever (\S\ref{sec:anomaly-ledger}). \emph{The circle
protects the winding against every operator; the interval protects
it against every epoch.} Stability in this framework is therefore
a joint property of the two fibers --- a charge the memory cannot
drop, carried down a road the arrow cannot re-climb --- and the
one-way birth of the baryons (Part~X) and the permanence of each
baryon born are two readings of that single anatomy. The full
deployment of the two-fiber observation --- symmetry breaking as
the one-way reading of holonomy data --- is carried by the
cohort's symmetry review, \textsf{GG-W1-Symmetry}; this paper
records the clause that is the proton's own: \emph{immortal means
guarded twice}.

\section{The \texorpdfstring{$\mathbb{Z}_3$}{Z3}-lock: one twist,
four locks}
\label{sec:z3lock}

The descent (Part~II) breaks $\SUfive{}$ by boundary conditions
and holonomy on the compact cycle \Stheta{} --- the
Wilson--Hosotani mechanism \citep{Hosotani1983}, in the
orbifold-refined form the unification literature developed
\citep{Kawamura2001} and \ANine{} deploys. The SM-preserving
holonomies are exponentials of hypercharge, $W(c) =
\exp(2\pi i\, c\, Y)$, quantized (pin R2, below) to integer $c$
with period six. And here the center identity produces its
coincidence --- except that nothing about it is coincidental:
\begin{equation}
B - 2Y \in \mathbb{Z} \;\;\text{on every fragment}
\qquad\Longrightarrow\qquad
z = e^{4\pi i Y} = W(2).
\label{eq:z-on-wilson-line}
\end{equation}
The center element that carries baryon number \emph{is a point on
the hypercharge Wilson line} --- the very line whose light
fluctuation is the Higgs. No new geometric structure is needed;
the twist the theory requires is a setting of a dial the theory
already owns. Which setting? Scan the six quantized holonomies
against the three demands \ANine{}'s spectrum already makes ---
break $\SUfive{}$ (lift $X,Y$: $y = -\tfrac56$), lift the color
triplet ($y = -\tfrac13$), keep the doublet light
($y = +\tfrac12$) --- and record, for each, the winding shift
$c\,y$ mod 1 it stamps on the matter fragments:

\begin{table}[htbp]
\centering
\caption[The Wilson-line shift, scanned.]{The Wilson-line shift scanned over its admissible values. Each row asks four yes-or-no questions of one value: whether the unified group breaks, whether the color triplet is lifted, whether the doublet is kept, and whether the shift equals plus or minus the baryon number. Only the row set in bold answers yes to the first four together, which is the sense in which the value is selected rather than chosen.}
\label{tab:wilson-shift}
\small
\begin{tabular}{cccccc}
\toprule
$c$ & $\SUfive{}$ broken & triplet lifted & doublet kept &
shift $= +B$ & shift $= -B$ \\
\midrule
0 & no & no & yes & no & no \\
1 & yes & yes & \textbf{no} & no & no \\
\textbf{2} & \textbf{yes} & \textbf{yes} & \textbf{yes} &
\textbf{yes} & no \\
3 & yes & no & no & no & no \\
4 & yes & yes & yes & no & yes \\
5 & yes & yes & \textbf{no} & no & no \\
\bottomrule
\end{tabular}
\end{table}

\noindent One row passes the spectrum's three demands --- $c=2$,
with $c=4$ its mirror (of which more in a moment) --- and in
that row, with no further input, the fourth column comes along:
\emph{the winding shift equals baryon number on every fragment}.
Requiring the Higgs sector \ANine{} already has \emph{forces}
the holonomy to the center element, and the center element
writes $B$. The pure $\mathbb{Z}_2$ point $c=3$ --- the
textbook orbifold parity --- is instructive in its failure: it
breaks $\SUfive{}$ but keeps the triplet and lifts the doublet,
the doublet--triplet problem \emph{inverted}. The geometry that
gets the Higgs right and the geometry that writes baryon number
are the same geometry, available at exactly one setting.

\begin{theorem}[\textbf{T-P6-2}: The $\mathbb{Z}_3$-lock]
\label{thm:z3lock}
With the descent holonomy at the center point $W(2) = z$
\textup{(}quantization per pin R2; vacuum data per
\S\ref{sec:dihedral}\textup{)}: \textup{(i)} each fragment's
winding lattice on the synchronized cycle is shifted to
$t/3 + \mathbb{Z}$; \textup{(ii)} the chiral zero-mode floors
--- which survive the twist by index protection, while the
unprotected scalar and vector floors at fractional shift are
lifted \textup{(}pin R3 exhibits the profiles\textup{)} ---
carry winding exactly $B$: $+\tfrac13$ for $Q$, $-\tfrac13$ for
$u^c, d^c$, $0$ for $L, e^c$; \textup{(iii)} the boundary
charge of the cycle isometry --- the descended \UoneB{}, gauged
in the bulk, Stueckelberg-massive --- therefore acts on every
boundary field as component-resolved baryon number, discharging
H2; \textup{(iv)} the $X,Y$ floors ($t = 1$, shift
$\tfrac13$) and the triplet floor are lifted by the same twist,
so H1 holds as a corollary rather than a hypothesis; and
\textup{(v)} $\theta$-integration enforces winding conservation
vertex by vertex: of the component couplings of the two
$\SUfive{}$ Yukawa invariants, exactly the three Standard-Model
Yukawas survive, every baryon-violating triplet channel is
annihilated, and the one surviving triplet coupling is
$B$-conserving.
\end{theorem}

\noindent The count of what one setting of one existing dial has
just done --- \emph{one twist, four locks}: it breaks the
unified group to the Standard Model; it removes the $X,Y$ from
the boundary (H1, the 1974 death sentence commuted); it splits
the doublet from the triplet (the classic fine-tuning problem,
dissolved --- ``geometry replaces tuning,'' as the home paper
puts it); and it writes baryon number, component by component,
as a winding no operator can unwind (H2). The selection rules of
leg (v) deserve one concrete look, because they close a
loophole the historically-minded reader will have kept open: in
ordinary $\mathrm{SU}(5)$, even without $X,Y$, the color-triplet
Higgs mediates decay through its Yukawa couplings. Here the
$\theta$-integral prices every component coupling by its total
winding:
\begin{table}[htbp]
\centering
\caption[Couplings, winding sums, and fates.]{Every gauge-invariant coupling of the unified theory, with its winding sum and its fate. The couplings whose windings sum to zero survive, and they are exactly the Standard Model's Yukawas together with one more; every coupling carrying a nonzero winding sum is annihilated. The last row is the one to notice, because it survives and it conserves baryon number.}
\label{tab:coupling-windings}
\small
\begin{tabular}{llc}
\toprule
coupling & winding sum & fate \\
\midrule
$Q\,u^c\,\tilde H$, \ $Q\,d^c\,H^\dagger$, \ $L\,e^c\,H^\dagger$
& $0$ & survive (the Standard Model's Yukawas) \\
$Q\,Q\,T$ & $+1$ & annihilated \\
$u^c e^c\,T^\dagger$ & $-\tfrac23$ & annihilated \\
$u^c d^c\,T$ & $-\tfrac13$ & annihilated \\
$Q\,L\,T^\dagger$ & $0$ & survives --- and conserves $B$ \\
\bottomrule
\end{tabular}
\end{table}
The triplet, already heavy, retains exactly one coupling, and
that one is $B$-neutral: integrate the triplet out and only
$\Delta B = 0$ four-fermion operators emerge. The decay channels
are not suppressed; they are absent from the theory's vertex
list, killed by an integral over a circle.

\paragraph{Physics intuition: the knot.} A quark mode wraps the
synchronized cycle a third of a turn; a baryon is three thirds
--- one full wrap. Baryon number is \emph{how many times matter
winds the cycle}, and the immortality of the proton is the
statement that you cannot continuously unwind a wrapped loop:
the proton is stable for the same kind of reason a knot in a
closed loop cannot be untied. Part~V already spent this
intuition on the quark (a third of a wrap is not a state);
here it pays its main debt: \emph{a full wrap is a state, and
it is a state forever}. What the operators of the boundary
theory can do is re-braid the wrap --- shuffle flavors, radiate
photons, exchange the strands' arrangement --- everything the
proton's rich phenomenology consists of. What no operator can
do is change the number of wraps, because winding is not a
quantity the operators carry. It is a sector.


Figure~\ref{fig:z3-winding} draws the lock itself --- the one
compact circle, the twist at a third of a turn, and the two kinds
of traveler on it.

\begin{figure}[htbp]
\centering
\resizebox{0.798\textwidth}{!}{%
\usebox{\GGplateZthree}%
}
\caption[The $\mathbb{Z}_3$-lock: one circle, one twist, two kinds of traveler.]{The
lock, drawn. \emph{Left:} the compact cycle $S^{1}_{\theta}$ with
the twist at the unique center point of the hypercharge line ---
$z = W(2) = e^{4\pi i Y} = \mathrm{diag}(\omega,\omega,\omega,1,1)$,
the sole quantized holonomy passing the three-demand scan of this
section. \emph{Right:} the two kinds of traveler. A quark mode
winds by one third --- an arc, not a loop: it cannot close, which is
why no boundary readout ever carries it alone. A baryon winds once
--- a closed loop whose winding number is baryon number itself,
$B = t/3$ by the center identity \eqref{eq:centre-identity}. A
winding can no more be shed than a knot untied in a closed loop.
One twist, four locks: the same third-of-a-turn breaks $\SUfive$,
lifts the $X,Y$, splits the Higgs, and writes $e^{2\pi i B} = z$
--- baryon number as the winding no operator can change (T-P6-2);
that sentence, made precise, is the stability theorem.}
\label{fig:z3-winding}
\end{figure}

\section{The dihedral closure: the reflection is P5's}
\label{sec:dihedral}

One structural debt remains inside the lock, and settling it
produced this paper's deepest point of contact with the cohort.
The compact cycle carries not only translations (whose twist we
have just set) but the reflection $\theta \to -\theta$ --- the
parity structure \eqref{eq:parity-P-p6} that the descent uses
for its chirality projection. A holonomy must be consistent
with the reflection: reversing orientation reverses a Wilson
line, so consistency demands $R\,W\,R^{-1} = W^{-1}$. Test the
naive reading, in which the reflection acts on gauge space by
conjugation with $P$ alone: $P$ and $W(c)$ are both diagonal,
they commute, and the condition collapses to $W^2 = 1$ ---
admitting only $c = 0$ (unbroken) and $c = 3$ (the inverted
split). \emph{Read strictly this way, the geometry cannot hold
the lock at all}: the no-go is as sharp as the obstruction
theorem, and it is pointing at the reflection.

The repair is forced, minimal --- and already in the cohort's
canon. Let the reflection act in the conjugation-twisted form,
$R:\; g \mapsto P\,\bar g\,P^{-1}$. Then
$R\, z\, R^{-1} = P\,\bar z\,P^{-1} = z^{-1}$, exactly as
orientation reversal demands, and the full structure generated
by the shift $S = (\theta \to \theta + \tfrac{2\pi}{3},\, z)$
and the reflection $R$ closes as the dihedral group
$\mathbb{D}_3$: $R^2 = 1$, $S^3 = 1$, $R S R = S^{-1}$ ---
every relation verified at the matrix level in
Appendix~\ref{app:lock-arithmetic}. And a conjugation-twisted
$\theta$-reversal is not a structure this paper had to invent:
\emph{it is the CPT companion's central identity} --- T-P5-A,
\emph{conjugation is orientation reversal}
\citep{ArisakaP5CPT} --- derived there from the real-bulk
axioms, consumed here as the closure condition of the geometry
that writes baryon number. Two papers, written for different
questions, lock: the theorem that gives every particle its twin
is the consistency condition for the theorem that makes the
proton immortal. Frameworks are supposed to do this and rarely
do.

The closure carries one more gift. The reflection maps the lock
point to its mirror, $c = 2 \leftrightarrow c = 4$ --- which is
$B \to -B$: the two admissible orientations of the twist are
matter's winding and antimatter's. The dihedral geometry
\emph{cannot} carry the baryonic orientation without carrying
the antibaryonic one; the antiproton is not an optional extra
but the reflection's image of the proton, wrap for wrap ---
the sister paper's ``conjugate readout of the same basin,''
now visible as the two chiralities of one orbifold. Every
precision comparison of proton and antiproton --- BASE's
sixteen parts per trillion \citep{Borchert2022} --- is a
measurement of how exactly the reflection is a symmetry, which
is falsifier F-P6-4 earning its keep.

The vacuum-selection question --- \emph{why} does the geometry
sit at the lock point --- is answered by reclassification: the
$\mathbb{D}_3$ structure is discrete boundary data,
superselected like the parity $P$ itself, not a modulus rolling
in a potential; the only continuous holonomy modulus is the
Higgs direction, whose minimization is \ANine{}'s S-6 and is
untouched here. And the pin this argument was first carried
under --- that the deployed descent's reflection genuinely acts
on complex representations with the conjugation twist --- turns
out to be already discharged in the cohort's print: the
realization paper states, of the quaternionic structure $B$ of
the six-dimensional Weyl module, that it \emph{implements
fermionic charge conjugation as real geometry, $C = (\theta
\mapsto -\theta) \circ B$}, citing the CPT companion's
T-P5-C2 \citep{ArisakaA2GG6}. The conjugation-twisted
reflection is not an expectation; it is deployed canon, and the
dihedral closure stands on it directly.

\section{The anomaly ledger, and the one door}
\label{sec:anomaly-ledger}

An exact gauge symmetry must pay its anomaly bill. Baryon
number's, computed exactly (per family; the sums are walked
term by term in Appendix~\ref{app:lock-arithmetic}):

\begin{table}[htbp]
\centering
\caption[Anomaly coefficients of the baryon circle.]{Every anomaly coefficient of the baryon-number circle, computed rather than quoted. Four vanish outright, one is absorbed by the Green--Schwarz mechanism and is physically inert, and one is nonzero and is the door this Part is about. The table is here because a reader has to be able to check that the door is the only opening: an anomaly account that lists only the coefficient it needs is not an account.}
\label{tab:anomaly-coefficients}
\small
\begin{tabular}{lcl}
\toprule
coefficient & value & disposition \\
\midrule
$[\SUthreeC]^2\,\UoneB$ & $0$ & clean \\
$[\SUtwoL]^2\,\UoneB$ & $+\tfrac12$ & \textbf{the door} (see
below) \\
$[\UoneY]^2\,\UoneB$ & $-\tfrac12$ & Green--Schwarz-absorbed;
physically inert \\
$[\UoneB]^2\,\UoneY$ & $0$ & clean \\
$[\UoneB]^3$ & $0$ & clean \\
$\UoneB$--grav$^2$ & $0$ & clean \\
\bottomrule
\end{tabular}
\end{table}

Four of six vanish identically --- the assignments are that
clean. The two survivors are what hypothesis H3 exists for: the
\BIsix{} inflow of \AThree{} \citep{ArisakaA3BI6} cancels them
in the bulk, the four-dimensional shadow of the mechanism
Green and Schwarz taught the field \citep{GreenSchwarz1984},
and the \UoneB{} gauge boson takes its mass by the Stueckelberg
coupling \citep{Stueckelberg1938} --- the venerable trick whose
whole point is that the gauge field grows heavy while the
\emph{rigid} symmetry survives unbroken. That division is
everything: no massless baryonic photon, hence no long-range
fifth force (the absence being falsifier F-P6-5's face), yet an
exact global selection rule on every operator. The
$[\UoneY]^2\,\UoneB$ survivor, once absorbed, is physically
inert --- an Abelian factor has no instantons in four
dimensions, so no process ever collects on it.

The $[\SUtwoL]^2$ survivor is different, and the difference is
the whole cosmology of matter. The weak group has topological
sectors; 't~Hooft's vertex \citep{tHooft1976} converts the
anomaly into real baryon violation --- but quantized as
$\Delta B = N_{\mathrm{gen}} \cdot n = 3n$, never $\pm1$, never
$\pm2$, and priced, at zero temperature, at the tunneling
factor
\begin{equation}
\Gamma \;\propto\; e^{-4\pi/\alpha_W}
\;=\; e^{-372} \;\approx\; 10^{-162},
\label{eq:thooft-price}
\end{equation}
a number for which words like ``suppressed'' are theater: the
reciprocal exceeds any time scale the universe offers by a
hundred and forty orders of magnitude. Cold, the door is
welded. Hot --- above the electroweak scale, where the barrier
can be walked over rather than tunneled through --- the same
door swings freely, and $\Delta B = 3n$ transitions run
unsuppressed. Hold both facts at once and read them forward:
\emph{the only baryon-number violation this geometry permits is
in units of three, through a door that is open in the early
universe and welded today.} A door in units of three cannot
leak a single proton ($\Delta B = 1$: not a multiple of three,
and no lighter $B$-carrier exists to share the transaction);
it cannot oscillate a neutron into an antineutron
($\Delta B = 2$: likewise); what it \emph{can} do, at
sphaleron temperatures, is convert a lepton-sector asymmetry
into baryons wholesale --- which is precisely the step
\PaperPFive{}'s birth chain requires and Part~X will walk.
The equation that forbids one proton's death is the equation
that permitted all protons' births. This paper regards that
single sentence as Q6 and Q10 shaking hands, and as the
strongest internal-consistency statement it owns.


Figure~\ref{fig:anomaly-door} assembles the ledger as one flow:
six coefficients computed, four zero, two paid by inflow --- and
one topological door left standing, priced.

\begin{figure}[htbp]
\centering
\resizebox{0.768\textwidth}{!}{%
\begin{tikzpicture}[
 font=\small\sffamily,
 zbox/.style={rectangle, rounded corners=2pt, draw=black!55, fill=black!4,
  line width=0.5pt, minimum width=31mm, minimum height=10mm, align=center, font=\footnotesize\sffamily},
 sbox/.style={rectangle, rounded corners=2pt, draw=orange!75!black, fill=orange!12,
  line width=0.7pt, minimum width=31mm, minimum height=10mm, align=center, font=\footnotesize\sffamily},
 paybox/.style={rectangle, rounded corners=3pt, draw=teal!60!black, fill=teal!10,
  line width=0.7pt, minimum width=64mm, minimum height=13mm, align=center, font=\small\sffamily},
 doorbox/.style={rectangle, rounded corners=3pt, draw=red!70!black, fill=red!10,
  line width=1.0pt, minimum width=64mm, minimum height=15mm, align=center, font=\small\bfseries\sffamily},
 arr/.style={-{Latex[length=2mm]}, line width=0.6pt},
]
%% the six coefficients
\node[zbox] (a1) at (0, 0)    {$[\SUthreeC]^2\,\UoneB$\\ $= 0$};
\node[zbox] (a2) at (3.6, 0)  {$[\mathrm{grav}]^2\,\UoneB$\\ $= 0$};
\node[zbox] (a3) at (7.2, 0)  {$[\UoneB]^2\,\UoneY$\\ $= 0$};
\node[zbox] (a4) at (10.8, 0) {$[\UoneB]^3$\\ $= 0$};
\node[sbox] (s1) at (1.8, -1.9)  {$[\SUtwoL]^2\,\UoneB$\\ $= +\tfrac12$ per family};
\node[sbox] (s2) at (9.0, -1.9)  {$[\UoneY]^2\,\UoneB$\\ $= -\tfrac12$ per family};
\node[font=\footnotesize\sffamily, text=black!60, anchor=west, align=center] at (12.6, 0) {vanish\\ identically};
%% payment
\node[paybox] (pay) at (5.4, -3.9)
 {paid in the bulk: \BIsix{} Green--Schwarz inflow\\ (hypothesis H3, \AThree{}) + Stueckelberg mass};
\draw[arr] (s1.south) -- (pay.north -| s1.south);
\draw[arr] (s2.south) -- (pay.north -| s2.south);
%% the door
\node[doorbox] (door) at (5.4, -6.0)
 {what survives: ONE topological door\\ $\Delta B \in 3\mathbb{Z}$ --- open hot, welded cold};
\draw[arr, line width=0.9pt, draw=red!70!black] (pay.south) -- (door.north);
\node[font=\footnotesize\sffamily, text=red!70!black, anchor=north, align=center] at (5.4, -6.95)
 {cold price: $e^{-4\pi/\alpha_W} = e^{-372} \approx 10^{-162}$ \eqref{eq:thooft-price}
 \quad---\quad hot: the origin of matter (Part~X)};
\end{tikzpicture}%
}
\caption[The anomaly ledger as one flow: six coefficients, two survivors, one door.]{The
anomaly ledger of this section as one flow. All six mixed anomaly
coefficients of the boundary \UoneB{} are computed exactly
(Appendix~\ref{app:lock-arithmetic}): four vanish identically; the
two survivors --- $[\SUtwoL]^2 B = +\tfrac12$ and
$[\UoneY]^2 B = -\tfrac12$ per family --- are absorbed by the
\BIsix{} Green--Schwarz inflow (H3, at its home paper's grade),
with the Stueckelberg mass removing the massless mediator. What no
inflow can remove is the topological content of the $\SUtwoL$
survivor: the single $\Delta B \in 3\mathbb{Z}$ door --- welded
below the electroweak crossover at the 't~Hooft price, wide open
above it, where Part~X will need it.}
\label{fig:anomaly-door}
\end{figure}

\section{The completed theorem}
\label{sec:completed-theorem}

Assemble the movements. The obstruction theorem
(\S\ref{sec:obstruction}) closed the algebraic road; the center
identity (\S\ref{sec:centre-identity}) located baryon number's
fractional content in $\SUfive{}$'s own center; the
$\mathbb{Z}_3$-lock (\S\ref{sec:z3lock}) promoted it to an
exact winding charge while delivering H1 and the
doublet--triplet split as corollaries; the dihedral closure
(\S\ref{sec:dihedral}) certified the structure's consistency on
the CPT companion's identity; and the ledger
(\S\ref{sec:anomaly-ledger}) paid the anomaly bill, leaving the
one $3\mathbb{Z}$ door. The stability theorem now reads:

\begin{quote}
\textbf{T-SM-60.1, completed (statement).} \emph{On the
$\mathbb{D}_3$-locked descent, every local boundary operator
with $\Delta B \notin 3\mathbb{Z}$ has identically vanishing
coefficient to all orders; the sole admissible violation is the
topological $\Delta B = 3n$ vertex, of weight
\eqref{eq:thooft-price} at zero temperature. In particular
$\Delta B = \pm1$ (proton decay) and $\Delta B = \pm2$
(neutron--antineutron oscillation) are forbidden outright: the
proton is absolutely stable. Conditional on: H3 (the \BIsix{}
inflow, as received); R2 (holonomy
integrality on the Hopkins--Singer clause); R3 (the warped
zero-mode profiles at fractional winding). Each pin is finite,
named, and none is a new idea.}
\end{quote} Figure~\ref{fig:deltab-three-p6} states the theorem as a number line, which is the form in which the two halves of it stop looking like a coincidence: the integers that are forbidden and the integer that is merely expensive are read off the same rule, and the second is the reason there is any matter to ask the question with.

\begin{figure}[!htbp]
\centering
\resizebox{0.70\textwidth}{!}{%
\usebox{\GGplateDeltaB}
}
\caption[The selection rule on one number line: what is forbidden and
what is merely expensive.]{\textbf{The completed theorem as one number
line.} Baryon number changes only in multiples of the generation count.
Every canonical proton-decay mode carries $\Delta B = 1$ and
neutron--antineutron oscillation carries $\Delta B = 2$; neither integer
is a multiple of three, so neither is suppressed --- both are forbidden,
with identically vanishing coefficient to all orders. What survives is
the topological vertex at $\Delta B = 3n$, admissible and enormously
expensive, priced at \eqref{eq:thooft-price}. The two statements are one
statement, and the second is why the first is not merely a null result:
the same door that is shut to one proton's death was open to all protons'
births, and Part~\ref{part:birth} walks through it. The rule is
\ANine{}'s T-SM-60.1 \citep{ArisakaA9SM} as \S\ref{sec:completed-theorem}
completes it; the rate is this paper's own evaluation, which corrects an
imprecise inline figure carried in the earlier literature of the
program.}
\label{fig:deltab-three-p6}
\end{figure}


The grade is stated with the care the claim deserves: this is
not ``the proton is stable, full stop''; it is ``the proton's
stability has gone from an accountant's accident to a
structural theorem with four named, bounded residues,'' which
is --- the author would submit --- the largest single change in
the epistemic status of Q6 since 1974. The sibling symmetry
with the electron is worth recording in one line, because the
Predictions Branch was built for it: \emph{the electron is
immortal because its basin has no exit gate; the proton is
immortal because its winding has no unwinding operator.} One
framework, two matter pillars, two different kinds of forever.

\section{Kamiokande: what the silence meant}
\label{sec:kamiokande}

Now the Part can walk back into the mine.

In 1980, the logic of the tank was: unification is almost
certainly right; unification requires the $X,Y$; the $X,Y$
kill protons; therefore watch protons. Every clause was
reasonable, and the third was wrong in a way nobody could then
see --- not because the $X,Y$ do not exist, but because of
\emph{where} they exist. In the geometry assembled here, the
$X,Y$ are real and necessary --- they are the off-diagonal
blocks of the unified connection, carriers of triality, wound
at a third of a turn --- and they are \emph{bulk} structure:
lifted from the boundary by the very twist that breaks the
symmetry, absent not by fiat (H1-as-hypothesis) but as a
corollary of the lock. Unification without proton decay is not
a retreat or a tuning; it is what unification looks like when
the unifying happens in the bulk and the watching happens on
the boundary. The 1974 sentence was passed by a theory that
put the executioners in the same room as the condemned. The
geometry never did.

So the silence was never a null result. Fifty years of it ---
from the first empty year at Kamiokande through
Super-Kamiokande's $0.37$ megaton-years
\citep{SuperKamiokande2024} --- has been a \emph{positive
measurement of the winding's exactness}, the longest and most
patient confirmation in physics of a selection rule nobody had
yet written down. The author spent his twenties building an
instrument to watch for a flash that this Part, forty-eight
years later, derives cannot come. He is permitted, once, in a
paper of record, to say what that feels like; that sentence is
his to write, and a placeholder holds its seat.
%% AUTHOR-VOICE SITE: the personal sentence of the arc ---
%% reserved verbatim for the author at sign-off.

What the tank means \emph{now} is the part the youngest reader
should hear plainly. Hyper-Kamiokande and DUNE and JUNO are not
mopping up a dead question; they are aiming at the sharpest
kill criterion this framework exposes. The stakes have
inverted, and the inversion is the point: \emph{in 1980, a
flash in the dark would have crowned unification; in 2030, a
flash would kill this framework outright.} F-P6-1 is absolute
--- one event, any channel, no parameter anywhere to absorb it
--- and it is matched by its quieter siblings: an oscillation
$n \leftrightarrow \bar n$ ($\Delta B = 2$; forbidden here as
absolutely as decay; the free-neutron bound stands at
$0.86\times10^{8}$~s \citep{BaldoCeolin1994} with bound-neutron
searches reaching an order beyond, and the European Spallation
Source proposes the next factor), and a baryonic fifth force
(F-P6-5: the Stueckelberg structure permits none). The tank's
silence has changed sides --- it now testifies \emph{for} a
theorem --- and the experiments built to break the silence are,
exactly thereby, the experiments best placed to break the
theorem. A framework can pay an experimental community no
higher compliment, and this paper is written in part to pay
it: keep watching. Either outcome is now a discovery about
geometry.

\section{Status ledger of the Part}
\label{sec:part6-ledger}

\begin{table}[htbp]
\centering
\small
\caption[Status ledger of Part VI.]{Status ledger of Part~VI.
Grades as in Table~\ref{tab:import-ledger}. Three columns: the claim, the grade it is carried at, and the fences or notes that bound it. The ledger is here so that the standing of the whole Part can be read down one column instead of reconstructed from the text, and so that no claim of the Part is left without a grade beside it.}
\label{tab:part6-ledger}
\begin{tabular}{p{0.40\textwidth}p{0.10\textwidth}p{0.40\textwidth}}
\toprule
\textbf{Claim} & \textbf{Grade} & \textbf{Fences / notes} \\
\midrule
T-P6-1 (obstruction; $B{-}L$ insufficiency) & D & Exact linear
algebra, proof in text; operator table in
Appendix~\ref{app:lock-arithmetic}. \\
\addlinespace
Center identity $B = t/3$ \eqref{eq:centre-identity} & D &
Fragment-by-fragment, all representations; exact, not mod 1. \\
\addlinespace
$z = W(2)$ on the hypercharge line; uniqueness scan & D &
$B - 2Y \in \mathbb{Z}$ all fragments; the $c$-table;
quantization of $c$ = pin R2. \\
\addlinespace
T-P6-2 (the lock; H2 discharged; H1 corollary; selection
rules) & D(c) & Conditional on R2, R3; spectrum-level floor
survival is \ANine{}'s deployed claim + R3's exhibit. \\
\addlinespace
Dihedral no-go and C-twisted closure; $B \to -B$ under
reflection & D(c) & Matrix-verified relations; the closure
condition = T-P5-A + the deployed $C = (\theta \to -\theta)
\circ B$ of \citealp{ArisakaA2GG6} (D-P6-1 discharged). \\
\addlinespace
Anomaly ledger (six coefficients) & D & Exact sums,
Appendix~\ref{app:lock-arithmetic}; GS/H3 as received
(\AThree{}). \\
\addlinespace
The $3\mathbb{Z}$ door; price \eqref{eq:thooft-price} & D &
Standard topology ('t~Hooft), standard evaluation
($\alpha_W^{-1} = 29.6$ at $M_Z$); open hot, welded cold. \\
\addlinespace
T-SM-60.1 completed & D(c) & On H3 + R2 + R3, each
named in place; the grade sentence of
\S\ref{sec:completed-theorem} is the claim. \\
\bottomrule
\end{tabular}
\end{table}

The summit is climbed; what remains of the paper is the view. The
record comes first: where the axiom the summit stands on came
from, and why the field declined it for seventy years
(Part~VII).
The neutron will descend through the gate the winding leaves
open within a sector (Part~VIII); the ladder will climb the
excited wraps (Part~IX); the birth chain will walk matter in
through the one hot door (Part~X). And a detector in a mine in
Gifu Prefecture will keep watching --- which is, this Part has
argued, exactly what it should do.

%% ===================================================================
%%  PART VII -- THE SEVENTY-YEAR ROAD TO ONE LETTER (the record)
%%  New at v1.8 per the author's commission: a dedicated historical
%%  record of U(1)_B, 1955--2026. Register discipline: verified
%%  external history (cited to print) / the framework's chronology
%%  (testimony, dated, AUTHOR-VOICE flagged) / no physics derived
%%  here (it all lives in Part VI). Title carried as the lead
%%  candidate pending author sign-off; alternatives held in the
%%  ship report.
%% ===================================================================
\GGPart[The record]{The Seventy-Year Road to One Letter}{A historical record of gauged baryon number, 1955--2026 --- the record, not the argument}
\label{part:record}

Physics papers do not usually carry a Part of history, and the
reader deserves to know why this one does. The axiom stated in
\S\ref{sec:founders-axiom} --- impose \UoneB{} as a local gauge
invariance, in analogy with electromagnetism --- is the founding
act of this framework, and it has a property rare among axioms:
it was proposed seventy years before it was imposed, by two of
the century's greatest physicists, and then declined by
essentially the entire field for reasons that were, in four
dimensions, \emph{correct}. That history is not background
color. It is the negative image of Part~VI's theorem: every
objection the seventy years raised against the axiom reappears
in this framework as load-bearing structure, and a reader who
knows where each wound was inflicted will recognize each member
by its scar. The history also carries this paper's oldest
personal thread to its origin: the author spent the first six
years of his career building the machine that watched for the
axiom's failure, and the last two imposing it. Both facts
belong in the paper of record --- kept in their own Part,
precisely so that history and derivation never borrow each
other's authority.
\section{What this Part is, and is not}
\label{sec:record-fence}

The fence first, as always. This Part contains \emph{no
physics that the paper's claims depend on}: every theorem,
grade, fence, and falsifier concerning \UoneB{} lives in
Part~VI and is untouched by anything said here. A reader who
skips this Part loses no equation. What the Part contains is a
record, in two registers that are never mixed. The first
register is \emph{verified external history}: the published
line of gauged baryon number from 1955 to the 2010s, every
date and claim cited to the printed literature, checkable by
anyone with library access. The second register is
\emph{testimony}: the framework's own chronology, 2024--2026,
dated by the author and marked as his account --- the paper
asks the reader to take it as record of \emph{when and in what
order} the framework's commitments were made, not as evidence
that they are true. Truth is Part~VI's business, and the
falsification table of Part~XII prices it. Nothing in either
register is an argument from authority, from priority, or from
narrative; the Part's status ledger
(\S\ref{sec:record-ledger}) closes it the way every Part of
this paper closes, with the registers separated in a table.

\section{1955: the Letter, and the wound it arrived with}
\label{sec:record-letter}

The idea is exactly one year younger than Yang--Mills theory
itself. In 1955 --- the paper on non-Abelian gauge fields was
1954 --- T.~D.~Lee and C.~N.~Yang published a Letter of a
single page, \emph{Conservation of Heavy Particles and
Generalized Gauge Transformations}, asking the question this
framework answers \citep{LeeYang1955}: if the number of heavy
particles is conserved --- and by 1955 the proton's refusal to
decay into the lighter particles available to it was already a
standing puzzle --- should that conservation not have the same
\emph{kind} of origin as the conservation of electric charge?
Their proposal was the founder's axiom of
\S\ref{sec:founders-axiom}, stated seventy years early: a
vector field coupled to baryon number, in analogy with the
electromagnetic potential.

And in the same page, with the honesty that marks the whole
Letter, they computed the objection. A massless baryonic
photon mediates a long-range force between all matter, in
proportion to baryon content --- repulsive between like
baryons --- and such a force is composition-dependent: it
violates the equivalence principle. The torsion-balance data
already in hand, the E\"otv\"os--Pek\'ar--Fekete measurements,
bounded the baryonic coupling at roughly one part in $10^{5}$
of gravity \citep{LeeYang1955} --- and gravity is itself some
forty orders of magnitude weaker than electromagnetism at
nuclear scales. Whatever gauges baryon number, it could not be
a second electromagnetism. Lee and Yang proposed the axiom and
wounded it in a single page.

The Letter's fate is the era's verdict, delivered
bibliographically. It is remembered, cited, respected --- and
it founded nothing for eighteen years. In its authors' own
chronologies it sits as a one-off, flanked by the mainstream
concerns of the day. No school formed; no program took the
gauging of baryon number as its charge. The idea entered the
literature's long memory and stayed there, wounded and
waiting.

\section{Why the field said no, three times over}
\label{sec:record-refusals}

The five decades of near-silence that followed were not
neglect. They were three independent refusals, each
sufficient on its own, each documented, and each --- this is
the point the record exists to make --- \emph{correct within
four-dimensional physics}.

\paragraph{First refusal: the conservation came free.} In the
Standard Model, baryon number is an \emph{accidental}
symmetry: the most general renormalizable Lagrangian
consistent with the gauge symmetries conserves $B$ without
anyone imposing it. To a model builder, gauging $B$ therefore
solved a problem the theory did not appear to have ---
conservation was already there, at tree level, for nothing.
That the free conservation was an accident of renormalizable
order, unprotected against deeper physics, was well
understood; but that understanding pointed the mainstream in
the \emph{opposite} direction --- toward expecting violation
(the third refusal), not toward protecting the accident with
a gauge principle. Part~I's Q7 is precisely the question this
refusal left standing.

\paragraph{Second refusal: the anomaly.} The decisive one. On
the Standard-Model spectrum the baryonic current is
chiral-anomalous: the mixed coefficients $[\SUtwoL]^2 B$ and
$[\UoneY]^2 B$ are non-zero --- the exact values, $\pm\tfrac12$
per family, are computed in this paper's
Appendix~\ref{app:lock-arithmetic} --- and 't~Hooft's 1976
analysis made the resulting non-conservation concrete
\citep{tHooft1976}, with Kuzmin, Rubakov, and Shaposhnikov
showing a decade later that at high temperature the anomaly
operates unsuppressed \citep{KuzminRubakovShaposhnikov1985}.
An anomalous gauge symmetry is not a difficult theory; it is
an inconsistent one. Within four dimensions and the known
matter content, gauged \UoneB{} is \emph{impossible}, and the
standard lore crystallized accordingly: the gaugeable
combination is $B - L$ (anomaly-free once right-handed
neutrinos are admitted), and the equivalence-principle
community itself adopted $B - L$ as the theoretically
preferred test charge. Every road around the anomaly required
paying in new fermions --- ``anomalons'' --- a price the
minority line of \S\ref{sec:record-minority} paid and the
mainstream declined. Part~VI's obstruction theorem
(T-P6-1) is this refusal's algebraic
core, proved rather than assumed; and Part~VI's answer ---
pay the bill at six dimensions, through inflow, with the
spectrum untouched --- is the move no four-dimensional
builder had available.

\paragraph{Third refusal: unification wanted the opposite.}
From 1974 onward the dominant road in fundamental physics was
grand unification \citep{GeorgiGlashow1974}, and its signature
prediction --- its crown jewel, as Part~VI told it --- was
proton decay. Quarks and leptons in shared multiplets; $X$ and
$Y$ bosons connecting them; baryon-number violation not as the
problem but as the \emph{point}. A research program whose
flagship prediction is the breaking of $B$ does not gauge $B$;
it digs kilometer-deep laboratories to watch $B$ fail --- and
the author of this paper spent 1979--1985 building one
\citep{ArisakaKamiokande1985}. The refusal here is
sociological only in form. In substance it is structural, and
it is worth stating with care, because it is the deepest of
the three: \emph{in any theory where unification lives on the
four-dimensional boundary, baryon-number violation is
unification's cost of doing business.} The only escape is to
move the unifying off the boundary --- and nobody in this
lineage proposed that, because there was no bulk on offer to
move it to.

Three refusals; one common root. Each is a true statement
about four dimensions. The Letter's idea was not wrong; it was
homeless.

\section{1973: three pages, half the evasion}
\label{sec:record-pais}

Eighteen years after the Letter came the first serious step,
and its very typography tells the story. A.~Pais, 1973,
\emph{Remark on baryon conservation} --- a ``Remark,'' three
pages \citep{Pais1973}. Pais gauged $B$ as an extra
$\mathrm{U}(1)$ factor and gave the baryonic boson a mass by
the Higgs mechanism, \emph{explicitly to evade the E\"otv\"os
constraint}: the first published recognition that a gauged
charge does not require a long-range force. This is the direct
ancestor of the Stueckelberg logic that Part~VI's structure
runs on --- half the evasion, found. What Pais did not have
was the other half: a \emph{reason} for the symmetry (it
remained an ornament appended to the electroweak group), an
answer to the anomaly (the chiral obstruction is not
addressed), and a geometric home (the mass is bought by
spontaneous breaking, which puts the exactness of the charge
at the mercy of the breaking sector --- whereas the
Stueckelberg realization of Part~VI keeps the rigid symmetry
exact while the boson takes its mass). The field's own grading
of the idea is preserved in the title and the page count: a
curiosity, honestly filed. It gathered no school either.

\section{1986--1990: the wrong revival, and the permanent
gift}
\label{sec:record-fifthforce}

Gauged baryon number reached the front page of the world's
newspapers exactly once, and it was wrong. In 1986 Fischbach
and collaborators reanalyzed the original E\"otv\"os data and
reported a composition-dependent residual, which they
interpreted as a finite-range vector force coupled to baryon
number \citep{Fischbach1986} --- the ``fifth force,'' its
carrier promptly christened the hyperphoton, a direct
descendant of the Letter with a mass. For four years it was
the most famous idea in physics. By 1990 it was dead: the
tower-gravity experiments, the new torsion balances, and the
re-examined evidentiary strands all returned null, and the
episode closed as one of the cleanest public executions of a
hypothesis in modern science.

But the wrong revival left a permanent gift, and for this
paper it is the section's point. The fifth-force episode
transformed the equivalence-principle program from an
occasional precision exercise into a standing,
purpose-built search machine --- and the charge it optimized
its searches for was \emph{baryon number}. The lineage of the
experiment Lee and Yang first invoked runs unbroken:
E\"otv\"os, Pek\'ar, and Fekete's original balances; the
Princeton and Moscow refinements; the E\"ot-Wash torsion
balances at Seattle; the MICROSCOPE satellite in orbit. The
modern benchmark is the E\"ot-Wash comparison of beryllium
and titanium --- test bodies \emph{chosen to maximize the
difference in baryon number per unit mass} --- with
E\"otv\"os parameter $\eta(\mathrm{Be},\mathrm{Ti}) =
(0.3 \pm 1.8)\times10^{-13}$ \citep{Wagner2012}, and
MICROSCOPE has since pushed the frontier near $10^{-14}$.
Translated into a coupling, a massless baryonic photon is not
merely disfavored; it is annihilated --- the surviving
strength would have to sit dozens of orders of magnitude
below electromagnetism's. For this framework the reading is
direct and happy: falsifier F-P6-5 --- no long-range force
coupled to $B$, ever, at any range with gauge strength --- has
had a dedicated experimental community since before the
theory existed, and the null that community keeps returning,
decade after decade, is precisely what a Stueckelberg-massive
\UoneB{} demands. The wrong revival built the right
instrument.

\section{The minority line}
\label{sec:record-minority}

There is an honorable minority literature that did gauge
baryon number, and the record owes it its due --- both because
its authors kept the idea alive and because the shape of their
solutions measures, by contrast, what the present framework
changed. In 1989 Foot, Joshi, and Lew stated the anomaly price
in the open \citep{FootJoshiLew1989}: with the three known
generations, gauged $B$ (and $L$) are anomalous, and
consistency demands new fermions with exotic electroweak
charges --- the anomalons, named and priced, never to leave
the four-dimensional ledger. In 1995 Carone and Murayama
opened the leptophobic window \citep{CaroneMurayama1995}: a
gauged-$B$ boson lighter than the $Z$, with coupling as large
as $\alpha_B \sim 0.2$, was --- astonishingly --- still
experimentally alive in 1995, a measure of how little the
field had cared to look for the force it had spent forty
years declining to gauge. And in 2010 Fileviez P\'erez and
Wise brought the idea into the modern era
\citep{FileviezPerezWise2010}: baryon and lepton number as
local gauge symmetries, anomalies canceled by an extra
generation, automatic dark-matter stability advertised as the
bonus --- fifty-five years after the Letter, gauged $B$
re-entering as beyond-Standard-Model model building, and a
small active sub-field following. The line continues into the
2020s, still four-dimensional, still paying the anomalon
price, still a bolt-on seeking a home.

Common to every entry in the minority line, from 1973 to the
present: four dimensions, so the anomaly must be paid in new
matter or new breaking; phenomenology --- a $Z'$ to hunt ---
rather than principle --- a conservation law to found --- as
the motivation; and no connection to unification, which
remained not the ally but the \emph{opponent}, since any GUT
embedding would break the very symmetry these models gauged.
The minority line kept the Letter's question alive for
seventy years. What it never had --- what nothing in four
dimensions could have --- was a place where the gauging, the
anomaly bill, the missing fifth force, and unification could
all be true at once.

\section{Ten dimensions, and the brane convergence}
\label{sec:record-tendim}

One road remains to be recorded, because it is the mainstream
road to exactly the consistency problem \UoneB{} poses, and
because one of its branches arrived --- from the opposite
direction --- at a structure this paper must acknowledge as
kin. The anomaly obstruction that forbids gauged \UoneB{} in
four dimensions has a grander sibling: the gauge and
gravitational anomalies that threatened all higher-dimensional
unification until Green and Schwarz showed, in 1984, that they
cancel --- in ten-dimensional $N = 1$ supersymmetric theory,
and only for the gauge groups $\mathrm{SO}(32)$ and
$E_8 \times E_8$ \citep{GreenSchwarz1984}. That single
cancellation launched the first superstring revolution, and it
fixed the mainstream's price of anomaly freedom for the next
four decades: ten dimensions, supersymmetry, and a specific
enormous gauge group. In the heterotic and GUT-descended
constructions that dominated string phenomenology, baryon
number remained what field theory had left it --- accidental,
approximate, generically violated by the compactification's
heavy states. The dominant unification road, at any number of
dimensions, never carried an exact $B$.

The record requires one honest addition, and this paper is
glad of it rather than embarrassed by it. In the
intersecting-brane Standard-Model constructions of the early
2000s \citep{IbanezMarchesanoRabadan2001}, every brane stack
donates a $\mathrm{U}(1)$, and baryon number is
\emph{necessarily gauged} at the string level: anomalous in
four dimensions, Green--Schwarz-canceled, its boson
Stueckelberg-massive, with $B$ surviving as an effective
symmetry that protects the proton in perturbation theory ---
though brane instantons generically violate it beyond. A
different lineage, starting from different axioms, arrived at
the same \emph{shape} of answer this paper derives: gauged,
inflow-paid, Stueckelberg-massive, proton-protecting
\UoneB{}. The convergence is cited, not hidden, because it
corroborates the mechanism while sharpening what is actually
distinctive here --- the price. What the branes buy with ten
dimensions, supersymmetry, and extended objects --- and hold
only perturbatively --- this framework buys with a
six-dimensional bulk and the \BIsix{} inflow, with no
supersymmetry and no extended objects, and holds to all
orders, the topological $\Delta B \in 3\mathbb{Z}$ door alone
excepted. The compact three-clause form of that contrast
stands with the others in the Discussion
(\S\ref{sec:uniqueness-contrast}); the record here is only
that the convergence exists, and that it was reached from the
other shore.


Figure~\ref{fig:u1b-timeline} lays the whole road on one axis ---
the external record above the line, the framework's own dated
record below it.

\begin{figure}[htbp]
\centering
\resizebox{0.748\textwidth}{!}{%
\begin{tikzpicture}[
    font=\large,
    >=Latex,
    card/.style={draw=black!55, rounded corners=2pt, line width=0.6pt,
        align=left, inner sep=4pt, text width=46mm},
    stem/.style={line width=0.9pt, draw=black!55},
    extC/.style={fill=blue!8, draw=blue!55!black},
    wndC/.style={fill=orange!12, draw=orange!75!black},
    ggC/.style={fill=violet!10, draw=violet!60!black},
    dot/.style={circle, draw=black!70, fill=white, inner sep=0pt, minimum size=5.2pt}
]
\def\xL{0}\def\xR{30}\def\yA{0}
%% eras: 1955-1985 the refusals | 1986-2010s wrong revival & minority | 2024-2026 the imposition
\def\bA{12.0}\def\bB{24.0}
\begin{scope}[on background layer]
    \fill[black!2] (\xL,-7.95) rectangle (\bA,8.6);
    \fill[black!4] (\bA,-7.95) rectangle (\bB,8.6);
    \fill[violet!4] (\bB,-7.95) rectangle (\xR,8.6);
\end{scope}
\node[anchor=south, font=\Large\bfseries] at ({(\xL+\bA)/2},7.85) {THE LETTER AND THE REFUSALS};
\node[anchor=south, font=\large] at ({(\xL+\bA)/2},7.2) {1955--1985};
\node[anchor=south, font=\Large\bfseries] at ({(\bA+\bB)/2},7.85) {WRONG REVIVAL; MINORITY LINE};
\node[anchor=south, font=\large] at ({(\bA+\bB)/2},7.2) {1986--2010s};
\node[anchor=south, font=\Large\bfseries] at ({(\bB+\xR)/2},7.85) {THE IMPOSITION};
\node[anchor=south, font=\large] at ({(\bB+\xR)/2},7.2) {2024--2026};
%% time axis
\draw[->, line width=1.4pt] (\xL,\yA) -- (\xR+0.7,\yA);
\foreach \x in {\bA,\bB}{\draw[black!55, line width=0.8pt] (\x,-0.18) -- (\x,0.18);}
\node[anchor=north, font=\large] at (\xL,-0.25) {1955};
\node[anchor=north, font=\large] at (\bA,-0.25) {1986};
\node[anchor=north, font=\large] at (\bB,-0.25) {2024};
\node[anchor=north, font=\large] at (\xR,-0.25) {2026};
%% ---- TOP: the external record ----
\node[dot] (d1) at (0.8,\yA) {};
\node[card, extC, anchor=south] (c1) at (2.7,1.3) {%
    \textbf{1955 --- the Letter}\\[1pt] Lee--Yang: gauge $B$ as QED is gauged;\\ wounded by E\"otv\"os in the same page};
\draw[stem] (d1) -- (c1.south west);
\node[dot] (d2) at (7.4,\yA) {};
\node[card, extC, anchor=south] (c2) at (7.4,3.6) {%
    \textbf{1973--1976}\\[1pt] Pais: the mass evasion, half-found;\\ 't~Hooft: the anomaly named};
\draw[stem] (d2) -- (c2.south);
\node[dot] (d3) at (9.8,\yA) {};
\node[card, wndC, anchor=south] (c3) at (11.0,1.3) {%
    \textbf{1974--1985}\\[1pt] unification sentences the proton;\\ Kamiokande built to watch (1979--85)};
\draw[stem] (d3) -- (c3.south west);
\node[dot] (d4) at (13.0,\yA) {};
\node[card, extC, anchor=south] (c4) at (14.8,3.6) {%
    \textbf{1986--1990}\\[1pt] the fifth force rises and falls;\\ the EP search machine is built};
\draw[stem] (d4) -- (c4.south west);
\node[dot] (d5) at (17.6,\yA) {};
\node[card, extC, anchor=south] (c5) at (18.4,1.3) {%
    \textbf{1989--2010}\\[1pt] the minority line: anomalons (1989);\\ leptophobic window (1995); revival (2010)};
\draw[stem] (d5) -- (c5.south west);
\node[dot] (d6) at (21.5,\yA) {};
\node[card, extC, anchor=south] (c6) at (21.5,3.6) {%
    \textbf{1984 / 2001 --- ten dimensions}\\[1pt] GS anomaly freedom at 10D+SUSY;\\ branes gauge $B$, GS-massive};
\draw[stem] (d6) -- (c6.south);
%% ---- BOTTOM: the framework's record (testimony) ----
\node[dot] (e1) at (24.6,\yA) {};
\node[card, ggC, anchor=north, text width=38mm] (b1) at (23.9,-1.2) {%
    \textbf{2024 --- the seed, neural}\\[1pt] a U(1) gauge invariance for the cortex ($\to$ \UoneQi{}); the analogy installed};
\draw[stem] (e1) -- (b1.north);
\node[dot] (e2) at (27.9,\yA) {};
\node[card, ggC, anchor=north, text width=38mm] (b2) at (28.0,-1.2) {%
    \textbf{2025--2026 --- the imposition}\\[1pt] March 2025: \UoneB{} imposed --- the beginning; November 2025: the bulk group final, no SUSY, no strings; January 2026: the non-spatial fifth and sixth dimensions --- then the cohort, and this paper's lock};
\draw[stem] (e2) -- (b2.north);
%% legend
\begin{scope}[shift={(0.3,-7.35)}]
    \node[anchor=west, font=\large\bfseries] at (0,0) {Register:};
    \node[card, extC, text width=2.2mm, minimum height=2.2mm, inner sep=1.6pt] at (2.6,0) {};
    \node[anchor=west, font=\large] at (2.9,0) {external record (cited to print)};
    \node[card, wndC, text width=2.2mm, minimum height=2.2mm, inner sep=1.6pt] at (12.2,0) {};
    \node[anchor=west, font=\large] at (12.5,0) {the tank};
    \node[card, ggC, text width=2.2mm, minimum height=2.2mm, inner sep=1.6pt] at (16.4,0) {};
    \node[anchor=west, font=\large] at (16.7,0) {the framework's record (testimony, dated)};
\end{scope}
\end{tikzpicture}%
}
\caption[The seventy-year road on one axis, 1955--2026.]{The
seventy-year road of gauged baryon number on one axis. Above the
line, the external record of this Part, every card cited to print
in the text: the Letter and its wound; the mass evasion and the
anomaly; the sentence of 1974 and the tank built to enforce it; the
wrong revival that built the equivalence-principle machine; the
minority line; and the ten-dimensional road with its brane
convergence. Below the line, in its own register and color, the
framework's dated testimony: the neural seed of 2024, the
imposition of March 2025, and the January 2026 departure into
non-spatial dimensions. The two registers meet only at the axis ---
which is this Part's discipline drawn as a picture.}
\label{fig:u1b-timeline}
\end{figure}

\section{The pattern, stated once}
\label{sec:record-pattern}

Lay the three wounds of the seventy years beside the
structure of Part~VI and the history inverts.

\begin{table}[htbp]
\centering
\small
\caption[The wounds and the structure.]{The three wounds
inflicted on gauged baryon number, 1955--2010s, and what each
becomes in this framework. Each row's left entry was fatal in
four dimensions; each right entry is load-bearing in six. Three rows, two columns, and the same fact read twice. The left column states each wound as the literature inflicted it; the right column states what that same fact becomes once the arena has six dimensions rather than four. The table is set out in pairs because the claim is not that the wounds were wrong --- it is that each of them was a correct measurement of a four-dimensional theory.}
\label{tab:wounds-structure}
\begin{tabular}{p{0.44\textwidth}p{0.44\textwidth}}
\toprule
\textbf{The wound (1955--2010s)} & \textbf{The structure
(this framework)} \\
\midrule
A massless baryonic photon means a composition-dependent
fifth force --- excluded from E\"otv\"os--Pek\'ar--Fekete to
E\"ot-Wash and MICROSCOPE, to depths that annihilate the
idea. & The Stueckelberg mass, forced by the same nulls,
kills the force while leaving the rigid charge exact; the
equivalence-principle program becomes F-P6-5's standing
confirmation --- a falsifier with its own seventy-year
experimental community. \\
\addlinespace
The chiral anomaly makes four-dimensional gauged \UoneB{}
inconsistent; every evasion costs new fermions the mainstream
declined to buy. & The \BIsix{} inflow pays the bill at six
dimensions with the spectrum untouched; the one surviving
coefficient becomes the $\Delta B \in 3\mathbb{Z}$ door ---
in Part~X, the origin of every baryon in the universe. \\
\addlinespace
Unification's own $X, Y$ bosons demand proton decay:
gauging $B$ and unifying the forces looked mutually
exclusive, and the field chose unification. & The
$\mathbb{Z}_3$-lock lifts the $X, Y$ to the bulk:
unification and immortality stop competing. The 1974
sentence is commuted by geometry, not repealed by retreat. \\
\bottomrule
\end{tabular}
\end{table}

Every fatal objection became a load-bearing member. The
record suggests a moral the paper states once and does not
lean on: this is the historical signature of a right idea
proposed in the wrong dimension. The Letter asked the correct
question in 1955; four-dimensional physics returned, three
times over, the correct answer \emph{no}; and the question
waited not for a cleverer evasion but for a different
spacetime to be asked in.

\section{The framework's own record, 2024--2026}
\label{sec:record-gg}

The register changes here, and the fence is restated so it
cannot be missed: what follows is \emph{testimony} --- the
framework's internal chronology as its author dates it,
recorded so that the paper of record carries its own origin
story in its own pages. It is offered as record of when and
in what order the commitments were made. No claim of this
paper is graded on it.

\paragraph{2024: the seed, and it was neural.} The
gauge-invariance instinct entered this framework through
neuroscience, a full year before it touched the proton. In
2024, working on the information dynamics of the human
cortical network, the author formulated a
$\mathrm{U}(1)$ \emph{dynamical gauge invariance} for
cortical processing --- explicitly on the analogy of
\UoneEM{} in QED: if the brain's processing conserves a
quantity the way electrodynamics conserves charge, let the
conservation be gauged. That neural $\mathrm{U}(1)$ became,
in time, \UoneQi{} --- the information charge of Part~XI ---
and the analogical move itself became the framework's
founding habit. Part~XI's candidate is not an appendage to
this paper; in the framework's own history, it is the germ.
%% AUTHOR-VOICE SITE: the 2024 neural-seed paragraph --- the
%% author's story, scaffolded from his July 2026 account; his
%% to rewrite verbatim at sign-off.

\paragraph{March 2025: the imposition.} The sudden step ---
in the author's own assessment, \emph{the very beginning of
this framework as a physics program}: the realization that
the same analogical move, pointed at the baryon, would make
the proton absolutely stable. Impose \UoneB{}. The moment the
neural-side instinct became a claim about the deepest
conservation law in matter is the moment the program
began; everything in the cohort of papers this one
cites descends from it. Fifty years of a physicist's
life --- the tank included --- stood behind the confidence to
impose what the field had declined for seventy.
%% AUTHOR-VOICE SITE: the imposition paragraph --- the
%% founding moment in the founder's words; scaffold only.

\paragraph{November 2025: the final form.} The bulk group
settles as $\SUfive \times \UoneB \times \UoneQi$ --- the two
imposed Abelian factors flanking the unification group ---
the architecture this paper inherits as
\eqref{eq:bulk-group-p6}.

\paragraph{Late 2025: the subtraction.} Shortly after: the
realization that the architecture requires \emph{no
supersymmetry and no string-like objects}, and the full
formulation of the \BIsix{} anomaly inflow --- the
six-dimensional answer to the second wound, and the moment
the comparison of \S\ref{sec:record-tendim} became available:
what the mainstream road buys at ten dimensions with
supersymmetry, the inflow buys at six without.

\paragraph{January 2026: the departure.} The final step: the
fifth and sixth dimensions defined as \emph{non-spatial} ---
scale and phase, the radial interval and the compact cycle of
Part~II --- the point of complete separation from every
Kaluza--Klein and string compactification before it, and the
reason the boundary/bulk custody discipline of this cohort
exists at all. From here the public record takes over: the
2026 cohort, the Standard-Model derivation with T-SM-60.1,
and this paper, which closes the circle the tank opened.

\begin{table}[htbp]
\centering
\small
\caption[The two-column chronology.]{The two columns of the
road, 1955--2026. Left: the published external record
(verified; citations in the text of this Part). Right: the
framework's internal record (testimony, dated by the
author). Eighteen rows over seventy years, in two columns that are not the same kind of thing and are not presented as though they were: the left is published and citable, the right is the author's own dated testimony. Keeping them apart is the point of the two-column form --- a reader can take the left column on the literature's authority and weigh the right column however they choose.}
\label{tab:chronology}
\begin{tabular}{p{0.50\textwidth}p{0.38\textwidth}}
\toprule
\textbf{The external record} & \textbf{The framework's
record} \\
\midrule
1955 --- Lee and Yang: gauge $B$ as electromagnetism is
gauged; wounded by E\"otv\"os in the same page. & \\
1973 --- Pais: the massive-boson evasion, half-found. & \\
1974 --- Georgi and Glashow: unification sentences the
proton. & \\
1976--1985 --- the anomaly thread: 't~Hooft; the sphaleron
era. & \\
1979--1985 --- Kamiokande built to watch $B$ fail. & (the
author, underground) \\
1986--1990 --- the fifth force rises and falls; the
equivalence-principle machine is built. & \\
1989 --- Foot, Joshi, and Lew: the anomalon price named. & \\
1995 --- Carone and Murayama: the leptophobic window. & \\
2001 --- intersecting branes: $B$ gauged at the string
level, Green--Schwarz-massive, at ten dimensions with
supersymmetry. & \\
2010 --- Fileviez P\'erez and Wise: the modern revival,
still an extension. & \\
2012 onward --- E\"ot-Wash and MICROSCOPE: the null
at $10^{-13}$--$10^{-14}$. & \\
& 2024 --- the neural $\mathrm{U}(1)$ (dynamical gauge
invariance for the cortex; later \UoneQi{}). \\
& March 2025 --- \UoneB{} imposed: the beginning. \\
& November 2025 --- the bulk group in final form. \\
& Late 2025 --- no supersymmetry, no strings; the \BIsix{}
inflow formulated. \\
& January 2026 --- the non-spatial fifth and sixth
dimensions. \\
& 2026 --- the cohort; T-SM-60.1; this paper, and the
$\mathbb{Z}_3$-lock. \\
\bottomrule
\end{tabular}
\end{table}

Table~\ref{tab:chronology} sets the two columns side by side,
and the Part's point sits in their meeting: the left column
spends seventy years wounding an idea, and the right column
begins by imposing it anyway --- not in ignorance of the
wounds but because, in a six-dimensional bulk with a
non-spatial compact cycle, every wound had become a
specification. The Letter asked for a QED of matter's number;
the tank measured its silence; the imposition, when it
finally came, came from outside the field's own lineage ---
through a decade of thinking about how brains conserve what
they process --- and the geometry then did what four
dimensions never could. One letter, seventy years, one twist.

\section{Status ledger of the Part}
\label{sec:record-ledger}

\begin{table}[htbp]
\centering
\small
\caption[Status ledger of Part VII.]{Status ledger of
Part~VII. This Part carries no physics claims; its registers
are historiographic. Three columns: the content, the register it belongs to, and the anchors or notes behind it. The middle column reads Record or Reading rather than a grade, because a historiographic claim is verified against printed sources and not derived, and using the physics grades here would borrow an authority this Part does not have.}
\label{tab:record-ledger}
\begin{tabular}{p{0.34\textwidth}p{0.14\textwidth}p{0.42\textwidth}}
\toprule
\textbf{Content} & \textbf{Register} & \textbf{Anchors /
notes} \\
\midrule
The external history, 1955--2010s
(\S\S\ref{sec:record-letter}--\ref{sec:record-tendim}) &
Record & Every date and claim cited to the printed
literature; the Lee--Yang bound quoted from the Letter;
final digit-for-digit collation against the primary texts is
on this paper's standing pre-release checklist. \\
\addlinespace
The wounds-to-structure reading
(\S\ref{sec:record-pattern}) & Reading & An interpretive
summary of Part~VI's results against the record; carries no
independent claim --- each right-hand entry is graded where
it is proved. \\
\addlinespace
The framework chronology, 2024--2026
(\S\ref{sec:record-gg}) & Testimony & Dated by the author;
marked in place; no claim of this paper is graded on it. \\
\addlinespace
Physics content of this Part & --- & None. The axiom's
statement is \S\ref{sec:founders-axiom}; its discharge,
grades, and fences are Part~VI's; its falsifiers are
F-P6-1 through F-P6-5. \\
\bottomrule
\end{tabular}
\end{table}

The record is closed, and the paper returns to physics. The
proton's immortality is proved at its stated grade; its twin,
one wall over, is not immortal at all --- and the reason it
dies, the direction it dies in, and the fifteen minutes it
takes are the next Part's theorem-grade story.

\GGPart[The neutron]{The Shallow Twin}{The neutron: 1.293 MeV up-slope, gate-force descent, stabilized by binding (Q8)}
\label{part:neutron}

After a Part of immortality and a Part of history, a death ---
the most consequential death in nature, and the gentlest. This
Part is about the neutron: why the proton has a twin at all,
why the twin sits a hair's breadth up-slope, why it dies in a
quarter of an hour on its own and lives forever inside an atom
--- and why that one arrangement of small numbers is the reason
there is chemistry, and readers.
\section{The hair's breadth}
\label{sec:hairs-breadth}

Having spent a Part on why the proton cannot die, the paper owes
equal candor to the particle standing beside it that dies in a
quarter of an hour. The neutron is the proton's twin in every
respect the strong force can see --- same stratum of the ladder,
same wrap of the cycle, isospin partners so alike that nuclear
physics spent decades treating them as two states of one
``nucleon'' --- and yet, set free, it lasts about fifteen minutes
and is gone. Q8 asks three things at once: why the twin exists,
why the splitting between the twins is so absurdly small, and why
the descent runs in the direction it does, at the leisurely pace
it does. All three answers are the landscape again, and this Part
is the paper's best worked example of the machinery earning its
keep on \emph{numbers}.

Begin with the splitting, because everything hangs on it:
\begin{equation}
m_n - m_p \;=\; 1.293~\mathrm{MeV}
\;=\; 0.14\%\ \text{of the nucleon mass}.
\label{eq:np-splitting}
\end{equation}
A seventh of a percent. To feel how narrow that is, run the
alternatives. Flip the sign --- make the proton the heavier twin
--- and hydrogen decays: every lone proton in the universe
converts to a neutron, there are no atoms, and chemistry never
happens. Widen the gap several-fold and the neutron becomes too
expensive to stabilize inside nuclei: the periodic table thins
toward hydrogen and little else. The observed universe of atoms
lives inside a window of roughly an MeV either way, and nature
sits in it by a hair \citep{PDG2024}. The century's honest
position was that this number was an unexplained mercy. The
framework's position is that it is an \emph{output}: the
difference of two basin depths, each priced by machinery already
on the table, with no dial anywhere in the chain.

\section{The competition: two small effects, opposite signs}
\label{sec:competition}

Why is the splitting small? Because it is a \emph{residual}: two
independent, modest effects pulling in opposite directions, and
the observed $1.293$~MeV is what survives the cancellation.

\paragraph{Effect one: the down-quark's extra toll.} Swap a $u$
for a $d$ in the valence content --- $uud \to udd$ --- and the
Higgs toll ledger of \S\ref{ssec:tolls} changes by
\begin{equation}
(m_u + 2m_d) - (2m_u + m_d) \;=\; m_d - m_u \;=\; 2.51~\mathrm{MeV}
\label{eq:valence-toll-diff}
\end{equation}
in favor of a heavier neutron. In the framework's terms: the
$d$-basin's Yukawa toll exceeds the $u$'s (the codebook of
\ANine{}, at its grade), so the twin one rung over in isospin
pays more rent.

\paragraph{Effect two: the proton's electromagnetic bill.} The
proton is charged; its Coulomb and magnetic self-energy raises
its basin relative to the neutral twin's by roughly an MeV in
the opposite direction. For most of a century this competition
could only be estimated; the modern lattice, with quantum
electrodynamics included alongside chromodynamics, computes both
terms from first principles \citep{Borsanyi2015}: the quark-mass
contribution $+2.52 \pm 0.29$~MeV, the electromagnetic
contribution $-1.00 \pm 0.16$~MeV, total consistent with
\eqref{eq:np-splitting}. Note the first number against
\eqref{eq:valence-toll-diff}: the full nonperturbative
quark-mass effect lands within a hundredth of an MeV of the
naive current-mass difference --- the storm, for once,
politely canceling out of a difference of twins.

So the hair's breadth is not a mercy; it is an arithmetic:
\begin{equation}
m_n - m_p \;=\;
\underbrace{(m_d - m_u)\text{-toll}}_{\approx +2.5~\mathrm{MeV}}
\;+\;
\underbrace{\text{EM self-energy}}_{\approx -1.0~\mathrm{MeV}}
\;\approx\; +1.3~\mathrm{MeV},
\label{eq:competition}
\end{equation}
and each term traces to framework inputs --- the Yukawa codebook
and the fine-structure constant, both \ANine{}'s registrations at
\ANine{}'s grade. The window that permits chemistry is the gap
between two of the codebook's smallest entries. The reader is
invited to sit with that for a moment; the Part will still be
here. %% Warmth note: kept --- the pause is the point.

\section{The descent: why fifteen minutes}
\label{sec:gate-descent}

A free neutron is a Geometric Attractor (\GA{}) that is not the
ground: it sits $1.293$~MeV up-slope from the proton's ground
attractor with its strong and electromagnetic channels closed --- same
stratum labels apart from charge, and no strong process changes a
$d$ into a $u$. Exactly one channel remains: the gate force. In
the basin/gate architecture inherited from \ANine{}, the weak
interaction is the unique channel through which descents between
basins run --- it is how the muon found the electron in the
sister paper, and it is how the neutron finds the proton here:
\begin{equation}
n \;\longrightarrow\; p + e^- + \bar\nu_e ,
\qquad
Q \;=\; (m_n - m_p) - m_e \;=\; 1.293 - 0.511 \;=\;
0.782~\mathrm{MeV}.
\label{eq:beta-decay}
\end{equation}
The \GRIPin{} reading prices this descent, and the price
explains the quarter hour. Two suppressions stack. The gate is
\emph{weak}: the amplitude carries Fermi's constant, and the
rate its square --- already ten orders below a strong descent.
And the doorway is \emph{narrow}: the released energy
$0.782$~MeV must be shared among three bodies, and the phase
space available scales, by Sargent's rule, as the fifth power of
the energy release, $\Gamma \propto Q^5$. The neutron's $Q$
barely clears the electron's mass; the fifth power of a small
number is a tiny number. For calibration: the muon, descending
through the same gate with $Q \approx 105$~MeV, decays in
$2.2~\mu$s --- and the ratio of the two phase-space factors
alone, $(105.7/0.782)^5$, is ten orders of magnitude
(Appendix~\ref{app:neutron-arithmetic}). Same gate, same
grammar; the pace difference is almost entirely the doorway's
width.

Better than calibration, the number itself is computable at the
blackboard, and Appendix~\ref{app:neutron-arithmetic} walks
every step: Fermi's constant, the axial coupling
$g_A = 1.2756$, the quark-mixing element $|V_{ud}|$, and the
exact three-body phase-space integral assemble into
\begin{equation}
\tau_{\mathrm{tree}} \;=\; 914~\mathrm{s},
\qquad
\tau_{\mathrm{corrected}} \;\approx\; 878~\mathrm{s}
\quad(\text{radiative/recoil, few percent}),
\label{eq:lifetime-walk}
\end{equation}
squarely on the measured bottle value of $877.75 \pm 0.28$~s
\citep{UCNtau2021}. A student who has followed this paper from
the transmutation walk of Part~III now holds both ends of a
remarkable chain: the same framework registrations that priced
the proton's \emph{mass} also price the neutron's
\emph{patience}, and both land on experiment with no dial
between.


Figure~\ref{fig:neutron-gate} draws the two situations of the twin
side by side: free, with its one narrow gate open; bound, with the
landscape re-shaped and the gate sealed.

\begin{figure}[htbp]
\centering
\resizebox{0.82\textwidth}{!}{%
\begin{tikzpicture}[
 font=\small,
 lab/.style={font=\footnotesize, align=center}
]
%% ---- left panel: the free neutron ----
\begin{scope}
\draw[line width=1.2pt, draw=black!80] plot[smooth, tension=0.8] coordinates {
 (0.0, 2.9) (0.9, 1.55) (1.8, 1.8) (2.6, 2.35) (3.6, 0.7) (4.6, 0.95) (5.4, 2.9)};
\fill[black!75] (0.95, 1.56) circle (2.2pt);
\node[lab, below=2pt] at (0.95, 1.5) {$n$ ($+1.293$~MeV)};
\fill[blue!60!black] (3.7, 0.72) circle (2.2pt);
\node[lab, below=2pt, text=blue!60!black] at (3.7, 0.66) {$p$ (terminal)};
\draw[draw=red!70!black, line width=0.9pt] (2.6, 2.47) -- (2.6, 2.23);
\node[lab, text=red!70!black, anchor=west] at (2.8, 2.75) {the gate:\\ $Q = 0.782$~MeV};
\draw[-{Stealth[length=2.4mm]}, line width=1.0pt, draw=black!70, dashed]
 (1.15, 1.8) to[bend left=35] (3.4, 1.1);
\node[lab, text=black!70, anchor=east] at (2.1, 3.6) {\GRIPin{} descent through\\ the gate: $n \to p\,e^{-}\bar\nu_e$};
\node[lab, anchor=north, font=\footnotesize\bfseries] at (2.7, -0.45)
 {Free: the gate is open};
\node[lab, anchor=north, text=black!60] at (2.7, -0.95)
 {fifteen minutes ($\tau_n \approx 878$~s);\\ the walk of \S\ref{sec:gate-descent} prices it};
\end{scope}
%% ---- right panel: the bound neutron ----
\begin{scope}[xshift=7.6cm]
\draw[line width=1.2pt, draw=black!80] plot[smooth, tension=0.8] coordinates {
 (0.0, 2.9) (0.9, 1.15) (1.8, 1.5) (2.6, 2.75) (3.6, 1.35) (4.6, 1.6) (5.4, 2.9)};
\fill[black!75] (0.95, 1.16) circle (2.2pt);
\node[lab, below=2pt] at (0.95, 1.1) {$n$ (bound)};
\fill[blue!60!black] (3.7, 1.37) circle (2.2pt);
\node[lab, below=2pt, text=blue!60!black] at (3.7, 1.31) {$p$ (bound)};
\draw[draw=black!60, line width=1.6pt] (2.6, 2.92) -- (2.6, 2.58);
\node[lab, text=black!70, anchor=west] at (2.8, 3.1) {the gate, re-priced:\\ binding $B_d = 2.224 > 0.782$~MeV};
\node[lab, text=black!60] at (2.9, 0.12) {no downhill: the descent would cost more than it pays};
\node[lab, anchor=north, font=\footnotesize\bfseries] at (2.7, -0.45)
 {Bound: binding re-landscapes the gate};
\node[lab, anchor=north, text=black!60] at (2.7, -0.95)
 {stable for at least the proton's forever;\\ why chemistry --- and you --- exist};
\end{scope}
\end{tikzpicture}%
}
\caption[The twin's two situations: gate open when free, sealed by binding.]{The
shallow twin's two situations. \emph{Left, free:} the neutron sits
$1.293$~MeV up-slope of the proton's ground attractor with exactly
one narrow gate --- the $\beta$ endpoint $Q = 0.782$~MeV --- and the \GRIPin{}
descent through it takes about fifteen minutes, priced by the
standard electroweak walk this Part quotes. \emph{Right, bound:}
inside a nucleus the landscape is environmental --- the deuteron's
binding of $2.224$~MeV exceeds the free gate's $0.782$~MeV, the
descent would cost more than it pays, and the gate is sealed: the
bound neutron is as immortal as the proton. One number on each
side of one inequality is why nuclei, chemistry, and readers
exist.}
\label{fig:neutron-gate}
\end{figure}

\section{The landscape is environmental: why chemistry exists}
\label{sec:bound-stability}

Here is the Part's most consequential sentence, and it looks
almost too simple to matter: \emph{binding energy moves basins.}
A neutron inside a nucleus is not the free neutron plus
decoration; it is a configuration of a different landscape, in
which the binding contributes to the depths. In the deuteron,
the gentlest laboratory case, the binding energy is
$2.224$~MeV --- and the would-be descent must pay it to break
the bond, while the doorway only supplies $Q = 0.782$~MeV. The
account does not balance; the channel is \emph{energetically
closed}; the bound neutron shares the terminus's immortality
(Appendix~\ref{app:neutron-arithmetic} does the general
bookkeeping). Every stable nucleus beyond hydrogen is a
neutron sanctuary built by the strong force out of exactly this
arithmetic --- which is to say: \emph{chemistry exists because
binding re-landscapes the gate}. And the door swings both ways:
in proton-rich nuclei the local landscape tilts the other way,
and it is the \emph{proton} that descends ($\beta^+$ decay,
electron capture) --- not a violation of Part~III's terminality
but its refinement: the \GGA{} of \emph{the isolated
stratum} is the proton; embed the stratum in a nuclear
environment and the functional, hence the floor, is the
environment's. The landscape was never a fixed backdrop. It is
the physics.

\paragraph{For the reader in the tank.} This section is your
fiducial volume. Super-Kamiokande's oxygen and DUNE's argon are
sanctuaries in the above sense: their neutrons are as immortal
as their protons, which is why your background model does not
include $10^{33}$ neutrons decaying in your detector at
fifteen-minute half-lives --- a background that would otherwise
end the proton-decay program before breakfast. The same
landscape shift that lets you exist as chemistry lets your
experiment listen for a single decay in silence.

\section{The two clocks: reporting a discrepancy without
adopting it}
\label{sec:bottle-beam}

Precision has produced a puzzle, and this paper's honesty
register requires it be reported exactly as it stands. Two
mature techniques time the neutron. Bottle experiments trap
ultracold neutrons and count survivors:
$\tau = 877.75 \pm 0.28$~s \citep{UCNtau2021}. Beam experiments
watch a cold beam and count the protons born in flight:
$\tau = 887.7 \pm 2.2$~s \citep{Yue2013}. The methods disagree
by about ten seconds --- roughly four standard deviations ---
and have for a decade. The mundane resolution is an
unaccounted systematic in one lineage; the exotic resolutions
propose an invisible decay channel that bottles count and
beams miss.

This paper claims nothing about the resolution, and says so.
But the framework does contribute one \emph{constraint}, and it
is sharp: whatever the resolution, \emph{the winding survives
it}. Any exotic channel must carry the neutron's $B = 1$ intact
into its final state --- a dark-sector destination would have
to be a baryonic-winding carrier, a wrapped state of a hidden
stratum --- because $\Delta B = 1$ disappearance is forbidden
here at exactly the grade of Part~VI's theorem, not by model
preference. A confirmed neutron disappearance channel with
\emph{nothing} carrying the wrap away would falsify T-P6-2 as
surely as a proton decay (it is F-P6-1's sibling); a channel
with a wrapped invisible daughter would be a discovery
\emph{about the landscape's hidden strata} and entirely
survivable. The discrepancy is thus a watch item with a
built-in diagnostic: the framework tells the experimenters
exactly which version of the exotic would kill it.

\section{Status ledger of the Part}
\label{sec:part7-ledger}

\begin{table}[htbp]
\centering
\small
\caption[Status ledger of Part VIII.]{Status ledger of
Part~VIII. Grades as in Table~\ref{tab:import-ledger}. Three columns: the claim, the grade it is carried at, and the fences or notes that bound it. The ledger is here so that the standing of the whole Part can be read down one column instead of reconstructed from the text, and so that no claim of the Part is left without a grade beside it.}
\label{tab:part7-ledger}
\begin{tabular}{p{0.40\textwidth}p{0.10\textwidth}p{0.40\textwidth}}
\toprule
\textbf{Claim} & \textbf{Grade} & \textbf{Fences / notes} \\
\midrule
Splitting decomposition \eqref{eq:competition} & D & Lattice
QCD+QED, cited; framework content = the inputs' provenance
(\ANine{}, at grade). \\
\addlinespace
Gate-descent reading of $\beta$-decay; direction & D(c) & On
the basin/gate architecture (\ANine{}) and \GRIPin{}
(\ASix{}); direction fixed by \eqref{eq:competition}. \\
\addlinespace
Lifetime walk \eqref{eq:lifetime-walk} & D & Standard
electroweak computation, every step in
Appendix~\ref{app:neutron-arithmetic}; corrections quoted, not
derived. \\
\addlinespace
Bound stability; environmental landscape & D & Energy
bookkeeping, general and per-nucleus; the ``chemistry exists''
corollary. \\
\addlinespace
Bottle--beam discrepancy & --- & Reported, not claimed; the
winding constraint on exotic channels stated at T-P6-2's
grade. \\
\bottomrule
\end{tabular}
\end{table}

The twin's story ends where the proton's began: one basin, one
gate, one hair's-breadth number that had to fall where it fell
for anyone to be here doing the measuring. The next Part climbs
away from the nucleon floor entirely, up the tower of copies
that all crumble --- and asks why the crumbling itself is so
beautifully graded.

\GGPart[The ladder]{The Ladder}{Six flavors from $N=3$; the excited tower; the width-zero terminus (Q9)}
\label{part:ladder}

The proton is the ground floor of a tall building. This Part
climbs the rest of it: six quark flavors where fewer might
have seemed enough, and above the proton a whole tower of
excited copies, every one of which crumbles almost the instant
it forms --- while the bottom rung lives forever. The question
here is not why the tower crumbles, but why the crumbling is so
beautifully graded, and why exactly one rung is exempt.
\section{Six, and why six}
\label{sec:six-flavours}

The proton's quarks come in flavors, the flavors come in
families, and the families come in exactly three --- a count the
Standard Model carries as an input and this program derives as
a theorem. The derivation belongs to the sister paper and is
consumed here in one paragraph, at its declared grade
\citep{ArisakaP2N3}: three or more families are \emph{required},
because with fewer than three there is no physical CP phase and
the constitutional demand for maximal structure at minimal
freedom (the MSMF axiom of \AOne{}) is violated; three or fewer are \emph{permitted}, because the capacity theorem of the sister paper prices the ladder in this paper's favorite currency: the stratum's income is $\pi$ of instanton budget per family (its topological charge, converted at the smallest coin the landscape mints), while the descent's tolls total the triangle number $\pi N(N-1)/2$ --- linear income against quadratic expenses, so the books balance only through three families, and at three they balance \emph{exactly}, $3\pi$ against $3\pi$, with nothing left to fund a fourth family's road to the ground state. Squeezed from both sides, $N = 3$ --- the two-sided pincer, carried at ``derived (conditional)'' with its fences (the coupling clause and the integrality lemma L-INT) stated at home. Each
family contributes one up-type and one down-type flavor to the
quark sector, so the flavor count follows by doubling:
\begin{equation}
(u, d), \qquad (c, s), \qquad (t, b)
\;\;\Longrightarrow\;\;
\text{exactly six},
\label{eq:six-flavours-final}
\end{equation}
and Q9's first clause --- \emph{why six?} --- reduces without
residue to \emph{why three?}, which is answered elsewhere and
owned there. What this Part adds is perspective on the ladder the
six span: the up quark's toll is $2.16$~MeV and the top's is
$172{,}570$~MeV --- a factor of $8 \times 10^{4}$ between rungs
of the \emph{same} ladder, the cascade-toll hierarchy of
\ANine{}'s codebook. The proton is built from the two cheapest
rungs. Everything in this paper so far --- the ground attractor,
the hair's-breadth twin --- lives in the ledger's bargain
basement, and that is not an accident: the floor of a landscape
is found where the tolls are lowest.

\section{The tower above the floor}
\label{sec:tower}

Fix the winding at $B = 1$ and climb. Above the nucleon floor
rises a tower of excited readouts --- the baryon spectrum, one
of experimental physics's great catalogs, four hundred-odd
entries deep \citep{PDG2024} --- and this Part's claim is that
the entire catalog is organized by one variable the century
rarely named: \emph{what protects each rung, and how well}.

Three specimens carry the whole lesson.

\paragraph{The $\Delta(1232)$: nothing protects it.} The first
rung: same quarks as the nucleon, spins aligned instead of
paired --- $294$~MeV up-slope with a strong channel gaping open
($\Delta \to N\pi$). Width $117$~MeV; lifetime
$5.5\times10^{-24}$~s; the readout barely completes before the
descent reclaims it. This is the landscape's default: an
excited wrap with a strong channel open is a suggestion, not a
particle.

\paragraph{The $\Omega^-$: strangeness protects it, thirteen
orders' worth.} The crown of the 1964 decuplet --- three
strange quarks, spin $\tfrac32$, predicted with its mass by the
Eightfold Way and found on cue in a bubble chamber
\citep{Barnes1964}, the discovery that made quarks impossible
to disbelieve. Every strong descent from the $\Omega^-$ is
closed --- not by energy but by \emph{labels}: no lighter
$B = 1$ state can absorb its three units of strangeness, and
the strong force cannot change flavor. Its only road down is
the gate, one strangeness unit per transit, and the gate's
price shows: lifetime $0.8\times10^{-10}$~s ---
\emph{thirteen orders of magnitude} longer than the $\Delta$'s,
for a particle only half again as heavy. Same tower, same
storm; the difference is purely which channels the labels
close.

\paragraph{The proton: everything protects it.} The terminus.
Strong channels: closed (no lower floor --- Part~III).
Gate channels: closed (no lower $B = 1$ relative at any
charge). Winding channels: closed to all orders (Part~VI).
Width: zero. The tower's grading, assembled:
\begin{table}[htbp]
\centering
\caption[The ladder, rung by rung.]{The ladder read from the top down, rung by rung. Each row gives the lowest channel still open to that state, the lifetime that follows, and what protects it. The lifetimes climb through more than twenty-five orders of magnitude and then stop: the last row has no open channel at all, and what protects the proton is not a quantum number imposed on the spectrum but the winding itself.}
\label{tab:rung-lifetimes}
\small
\begin{tabular}{lccl}
\toprule
rung & open channel & lifetime & protected by \\
\midrule
$\Delta(1232)$ & strong & $5.5\times10^{-24}$~s & nothing \\
$\Lambda(1116)$ & gate (one unit) & $2.6\times10^{-10}$~s &
strangeness \\
$\Omega^-(1672)$ & gate (thrice over) & $0.8\times10^{-10}$~s &
strangeness $\times 3$ \\
$n$ (free) & gate (hair's-breadth $Q$) & $879$~s & phase space \\
$n$ (bound) & none & stable & binding (Part~VIII) \\
$p$ & none & $\infty$ & the winding itself \\
\bottomrule
\end{tabular}
\end{table}
Twenty-six orders of magnitude between the tower's top row and
its fifteen-minute row, another eternity to the last --- and
one grammar throughout: \emph{a lifetime is the price of the
cheapest open descent, and the labels decide what is open}.
Part~III displayed this column as evidence that the landscape
is real; this Part has now walked every mechanism in it. The
baryon tables, read this way, stop being a catalog and become
a contour map.


Figure~\ref{fig:lifetime-cliff} plots the grammar of the ladder as
one staircase: every rung's lifetime is the price of its cheapest
open descent, and the floor --- with no descent at all --- falls
off the bottom of any finite chart.

\begin{figure}[htbp]
\centering
\begin{tikzpicture}[x=1cm, y=1cm, >=Latex]
 \draw[line width=0.80pt,->] (0,-0.2) -- (0,7.9) node[left, font=\small] {$\log_{10}(\tau/\mathrm{s})$};
 \draw[line width=0.80pt,->] (0,3.2) -- (11.6,3.2) node[below right=0pt and -14mm, font=\small] {the ladder, downward};
 %% axis reference marks
 \foreach \y/\lab in {0.4/{-24}, 1.6/{-10}, 3.2/{0}, 4.6/{ +3}, 7.0/{+41}}{
  \draw (-0.12,\y) -- (0.12,\y);
  \node[font=\scriptsize, anchor=east] at (-0.2,\y) {\lab};
 }
 %% rungs (log-lifetime staircase)
 \draw[fill=gray!20, draw=black] (1.0,3.2) rectangle (2.6,0.4);
 \node[font=\small, align=center] at (1.8,0.0) {$\Delta$, $N^{*}$ tower\\ $\sim 10^{-23}$--$10^{-24}$~s};
 \node[font=\scriptsize, text=black!60, align=center] at (1.8,2.7) {strong gates\\ wide open};
 \draw[fill=gray!15, draw=black] (3.6,3.2) rectangle (5.2,1.6);
 \node[font=\small, align=center, anchor=west] at (5.35,1.1) {hyperon shelf\\ ($\Lambda$, $\Omega^{-}$: $\sim 10^{-10}$~s)};
 \node[font=\scriptsize, text=black!60, align=center] at (4.4,2.7) {strangeness:\\ weak gates only};
 \draw[fill=gray!10, draw=black] (6.2,3.2) rectangle (7.8,4.6);
 \node[font=\small, align=center] at (7.0,5.0) {free neutron\\ $878$~s};
 \node[font=\scriptsize, text=black!60, align=center] at (7.0,3.75) {one narrow\\ gate ($0.782$~MeV)};
 \draw[fill=blue!10, draw=blue!60!black, line width=0.8pt] (8.8,3.2) rectangle (10.4,7.0);
 \node[font=\small\bfseries, text=blue!60!black, align=center, anchor=south] at (9.6,8.0) {proton\\ $> 2.4\times10^{34}$~yr};
 \node[font=\scriptsize, text=blue!60!black, align=center] at (9.6,4.5) {no gate:\\ the bar has\\ no top};
 \draw[-{Stealth[length=2.2mm]}, draw=blue!60!black, line width=0.9pt] (9.6,7.0) -- (9.6,7.9);
 %% thirteen orders annotation
 \node[font=\scriptsize, text=black!55, align=center] at (5.8,-0.85)
  {thirteen orders of magnitude between adjacent rungs: the gates' grammar, not fine print};
\end{tikzpicture}
\caption[The lifetime cliff of the baryon ladder.]{The ladder's
grammar as one staircase: $\log_{10}$ of each rung's lifetime.
The excited tower decays through wide-open strong gates in
$\sim 10^{-23}$~s; strangeness closes the strong gates and the
hyperon shelf lives $\sim 10^{-10}$~s --- thirteen orders of
magnitude for one label change; the free neutron has exactly one
narrow weak gate and lives fifteen minutes; and the proton's bar
has no top, because a basin with no gate has no lifetime scale at
all --- the observed bound $> 2.4\times10^{34}$~yr
\citep{Takenaka2020} is a floor on an infinite bar, not an
approach to a ceiling. Lifetime is not a property sprinkled on
particles; it is the price list of the open gates, and the ladder
reads it off label by label.}
\label{fig:lifetime-cliff}
\end{figure}

\section{Copies, not cousins}
\label{sec:copies}

One conceptual point keeps the ladder honest, inherited from
the sister paper's treatment of the muon and worth its own
section here. The excited baryons are not new basins --- they
are \emph{higher floors of the same stratum}: excited readouts
of the one wrapped configuration, reached by spending energy on
spin alignment, orbital excitation, or costlier rungs of the
flavor ladder. Two consequences follow, both measured facts.
First, \emph{identity holds on every floor}: all
$\Delta^{++}$s are identical, all $\Omega^-$s are identical ---
the floors are floors of a functional, not populations with
histories, and the identity corollary of Part~III applies rung
by rung. Second, \emph{every floor above the terminus has a
width} --- being a floor and not the floor means a descent
exists, and \GRIPin{} prices it. The converse is the sharp
statement: in eighty years of spectroscopy, no excited baryon
has ever been found with zero width, and the framework says
none ever will be --- a second zero-width $B=1$ state is
already registered as a kill in \S\ref{sec:part3-kills}, and
the tower is where such a thing would have surfaced. The
ladder's every rung crumbles, on schedule, at the price its
labels set; the universe is built from the one rung that
cannot.

\paragraph{For the reader in the tank.} The tower is your
signal model. When an atmospheric neutrino strikes your
fiducial oxygen, the $\Delta$ it excites decays before it
travels a fermi, and the pion it sheds is in your Cherenkov
cone reconstruction; when your background simulation prices
$NC\pi^0$ production, it is pricing rung one of this tower.
The proton-decay search and the baryon spectrum are not
neighboring topics; they are the bottom and the body of one
ladder, and your analysis lives on both.

\section{Status ledger of the Part}
\label{sec:part8-ledger}

\begin{table}[htbp]
\centering
\small
\caption[Status ledger of Part IX.]{Status ledger of
Part~IX. Grades as in Table~\ref{tab:import-ledger}. Three columns: the claim, the grade it is carried at, and the fences or notes that bound it. The ledger is here so that the standing of the whole Part can be read down one column instead of reconstructed from the text, and so that no claim of the Part is left without a grade beside it.}
\label{tab:part8-ledger}
\begin{tabular}{p{0.40\textwidth}p{0.10\textwidth}p{0.40\textwidth}}
\toprule
\textbf{Claim} & \textbf{Grade} & \textbf{Fences / notes} \\
\midrule
$N = 3$; six flavors \eqref{eq:six-flavours-final} & D(c) &
\PaperPTwo{}'s pincer at its declared grade; doubling is
counting. \\
\addlinespace
Toll hierarchy across the six & D(c) & \ANine{}'s cascade
codebook at its grade; this Part consumes, does not
re-derive. \\
\addlinespace
The graded tower (channel/label/lifetime grammar) & D +
INT & Lifetimes and channels are data (PDG); the one-grammar
reading is the framework's, priced rung by rung by
\GRIPin{}. \\
\addlinespace
Copies-not-cousins; identity per floor; width $\Leftrightarrow$
non-terminality & D(c) & Part~III's theorem applied floor by
floor; the no-second-zero-width prediction is a registered
kill. \\
\bottomrule
\end{tabular}
\end{table}

\enlargethispage{3\baselineskip}%% v15.0: see the sparse-page note
The ladder complete, the paper's local physics is done: the
basin, its label, its cage, its immortality, its twin, and its
tower --- eight questions closed or fenced. What remains is
history and ledger: where the wraps came from (Part~X), and
what each rearrangement of them costs the universe's books
(Part~XI).

\GGPart[The birth]{The Birth of Baryons}{The door that forbids one death permitted all births (Q10)}
\label{part:birth}

So far the paper has asked what the proton \emph{is}. This Part
asks the other question a complete account owes: where they all
came from. The visible universe holds some $10^{80}$ protons
and, to any measurable standard, no antiprotons --- somewhere in
the first instants, nature wrote matter a cheque that antimatter
never cashed. This Part follows that transaction through the one
door the stability theorem leaves open, and arrives at the
paper's most satisfying single sentence: the law that forbids
one proton's death is the law that permitted them all to be
born.
\section{The inventory question}
\label{sec:inventory}

Every Part so far has asked why the proton is what it is. This
one asks why it is \emph{here} --- why, in a universe born as
pure energy, there is any matter to have questions about.

Stated as a number, the mystery is small and stubborn. Compare
the universe's baryons to its photons --- the fossil light of the
big bang against the fossil matter --- and the ratio is
\begin{equation}
\eta_B \;=\; \frac{n_B}{n_\gamma} \;=\; 6.1\times10^{-10}
\label{eq:etaB}
\end{equation}
\citep{Planck2020}: about one proton per $1.6$ billion photons.
That number is doubly strange. It is strange for being so
\emph{small} --- matter is the trace contaminant of the cosmos,
a part-per-billion impurity in a bath of light --- and it is
strange, far more deeply, for not being \emph{zero}. A universe
that began symmetric between matter and antimatter should have
spent its matter: as it cooled, every baryon should have found an
antibaryon, annihilated to photons, and left --- by the standard
freeze-out arithmetic \citep{KolbTurner1990} --- a residue near
$10^{-18}$: a billion times too little. Everything you have ever
touched is made of the discrepancy. Q10, precisely put: what
process wrote a one-part-per-billion \emph{asymmetry} into the
books before the great annihilation, so that when the smoke
cleared, a residue of unpartnered protons remained?

Sakharov named the three conditions any answer must meet
\citep{Sakharov1967}: baryon number must be violated (else the
books stay balanced forever); C and CP must be violated (else
every matter-making process runs equally in the mirror, and the
books balance again); and the process must run out of
equilibrium (else forward and reverse rates cancel the
asymmetry as fast as it forms). Three locks on the door of
existence. The Standard Model, remarkably, holds a key to each
--- and each key is too small: its CP violation, funneled
through the single CKM phase, falls short of the required
asymmetry by many orders of magnitude; and its electroweak
transition is a smooth crossover, not the out-of-equilibrium
slam the third condition wants. The standard theory knows
\emph{why} matter could exist; it cannot make \emph{enough} of
it. The status table of Part~I marked this row honestly, and
nothing in this Part erases the marking: the birth of baryons
is imported here at its stated grade, not solved on this page.

\section{The structural selection: in this geometry, the
lepton door is the only door}
\label{sec:structural-selection}

What this paper \emph{does} own is a sharpening that no
standard treatment can offer. The literature holds a menu of
baryogenesis mechanisms --- GUT-boson decays, electroweak
scenarios, leptogenesis, and their variants --- and treats the
choice as taste constrained by data. In the geometry of
Part~VI, the menu collapses under a structural rule:

\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}

The classic GUT mechanism \citep{GeorgiGlashow1974,
Sakharov1967} minted baryons directly, in the decays of the
same $X, Y$ bosons that mediated proton decay --- mechanism and
menace were one coin, which is why the tank was built. In this
geometry the coin is void on both faces: the winding selection
rule that welds the proton's stability (T-P6-2) erases the
$\Delta B = 1$ minting channel identically, at every
temperature. What survives, and what alone survives, is the
electroweak anomaly's own vertex \citep{tHooft1976}: above the
electroweak barrier, sphaleron transitions
\citep{KuzminRubakovShaposhnikov1985} shuffle baryon and lepton
number freely along the one direction the anomaly permits ---
changing $B + L$, conserving $B - L$, in packets of three ---
and below the barrier the same vertex freezes at the
$e^{-372}$ of \eqref{eq:thooft-price}. The consistency
sentence of Part~VI now becomes a production requirement: to
make baryons at all, nature had to make \emph{lepton-sector
asymmetry first}, while the door stood open, and let the door
do the conversion. The proton's immortality and its nativity
are two temperatures of one equation.

\section{The chain, at its stated grade}
\label{sec:the-chain}

The concrete chain is the CPT companion's, and this Part walks
it exactly as its home declares it \citep{ArisakaP5CPT} ---
one named input, one named conjecture, no upgrades issued here.

The mirror-sector heavy neutrinos of \PaperPFive{}'s anatomy
decay out of equilibrium in the early universe, at the TeV
scale --- the framework's lock scale, not a new one --- with
the CP asymmetry supplied by the ledger phases that paper
derives from the descent's orientation. At TeV masses the
asymmetry must be resonantly enhanced (the quasi-degenerate
regime of \citealp{PilaftsisUnderwood2004}), and the
\emph{resonance condition} is the chain's one named input,
declared as such at home. The lepton asymmetry then survives
washout and rides the open door: sphalerons convert the
accumulated $B{-}L$ into baryon number with the equilibrium
partition
\begin{equation}
B \;=\; \frac{8 N_f + 4 N_H}{22 N_f + 13 N_H}\,(B - L)
\;=\; \frac{28}{79}\,(B - L)
\label{eq:partition}
\end{equation}
for three families and one Higgs doublet
\citep{HarveyTurner1990} --- the chemical-potential bookkeeping
is walked in Appendix~\ref{app:door-bookkeeping}, and the
$28/79$ is nothing but the door's exchange rate, fixed by
counting the species standing in the marketplace. The
conversion's transport normalization --- the $K/2$ exponent in
the washout compression --- is the chain's one named
conjecture (C-P5-C at home), and the chain's landing on the
measured \eqref{eq:etaB} is stated there with its honest
window, factor-of-two resolution, dials declared. The yield is
delivered to the cosmology paper \citep{ArisakaA10Cosmos},
where it feeds the inventory this paper's Part~I quoted.

That is the whole import, and the fences stay up: a reader who
wants the asymmetry's derivation audits \PaperPFive{}; a
reader who wants its consumption audits \ATen{}. What belongs
to \emph{this} paper is the two-sided consistency that neither
neighbor could state alone, and it deserves its own
paragraph.


Figure~\ref{fig:birth-door} compresses the chain of this Part into
one line: the lepton-sector asymmetry, the one hot door, the
partition, and the seal.

\begin{figure}[htbp]
\centering
\resizebox{\textwidth}{!}{%
\begin{tikzpicture}[
 font=\small\sffamily,
 stage/.style={rectangle, rounded corners=3pt, draw=black!60, fill=blue!8,
  line width=0.6pt, minimum width=40mm, minimum height=16mm, align=center, font=\footnotesize\sffamily},
 doorst/.style={rectangle, rounded corners=3pt, draw=red!70!black, fill=red!10,
  line width=1.0pt, minimum width=40mm, minimum height=16mm, align=center, font=\footnotesize\sffamily},
 outst/.style={rectangle, rounded corners=3pt, draw=teal!60!black, fill=teal!10,
  line width=0.8pt, minimum width=40mm, minimum height=16mm, align=center, font=\footnotesize\sffamily},
 arr/.style={-{Latex[length=2.4mm]}, line width=0.8pt},
 note/.style={font=\scriptsize\sffamily, text=black!60, align=center},
]
\node[stage] (lg) at (0,0) {\textbf{TeV resonant leptogenesis}\\ (\PaperPFive{}, at grade)\\ a lepton-sector asymmetry};
\node[doorst] (door) at (5.6,0) {\textbf{the one hot door}\\ $\Delta B \in 3\mathbb{Z}$, unsuppressed\\ above $T_c$ (sphalerons)};
\node[stage] (part) at (11.2,0) {\textbf{the partition}\\ $B = \tfrac{28}{79}\,(B{-}L)$\\ standard, cited};
\node[outst] (etab) at (16.8,0) {\textbf{the frozen residue}\\ $\eta_B = 6.1\times10^{-10}$\\ every atom a receipt};
\draw[arr] (lg) -- (door);
\draw[arr] (door) -- (part);
\draw[arr] (part) -- (etab);
\node[note] at (2.8,-1.55) {structural selection:\\ $\Delta B = \pm1$ minting\\ does not exist here};
\node[note] at (8.4,-1.55) {open for the first\\ picosecond only};
\node[note] at (14.0,-1.55) {then the crossover:\\ the door welds shut\\ at $e^{-372}$};
\end{tikzpicture}%
}
\caption[The birth of baryons through the one hot door.]{The birth
chain of this Part in one line. The structural selection theorem
says $\Delta B = \pm 1$ minting does not exist in this geometry, so
whatever made the baryons had to make them from a lepton-sector
asymmetry through the single $\Delta B \in 3\mathbb{Z}$ door ---
open, unsuppressed, for the hot first picosecond; partitioned by
the standard $28/79$ bookkeeping; frozen at
$\eta_B = 6.1\times10^{-10}$ when the electroweak crossover welded
the door shut at the 't~Hooft price. The equation that forbids one
proton's death is the equation that permitted all protons'
births.}
\label{fig:birth-door}
\end{figure}

\section{Every proton is a fossil of the door}
\label{sec:fossil}

Put Part~VI and this Part side by side and read them as one
statement about one exponent. Below the electroweak scale, the
sphaleron weight $e^{-4\pi/\alpha_W} = e^{-372}$ welds the
door: no proton decays, ever, anywhere --- fifty years of
watched water confirm it. Above the electroweak scale, the
same expression, with its barrier submerged in the thermal
bath, is order one: the door swings freely --- and in the
first picosecond, it \emph{had} to swing, or the lepton-sector
asymmetry would have died a curiosity and the universe would
have annihilated to light. One coupling, one topology, two
temperatures: the frozen reading is Q6, the fertile reading is
Q10, and neither can be repealed without repealing the other.
A skeptic who would remove the proton's immortality must
explain why matter exists in the teeth of the annihilation
catastrophe; a skeptic who would remove the sphaleron door
must find another lawful mint, and Part~VI has closed every
alternative in this geometry. The two questions the century
kept in separate chapters --- \emph{why does matter persist?}
and \emph{why is there matter?} --- are one question about one
door, asked at two temperatures.

And so the Part's title sentence, now earned in full: every
proton in your body, in the tank's water, in the farthest
quasar, is a fossil --- the frozen output of a door that
opened once, while the universe was hot enough to hold it
open, and then welded itself shut with the contents inside.
The silence Super-Kamiokande records is the sound of the weld
holding. The matter it is made of is the receipt.

\paragraph{For the reader in the tank.} Your experiment
usually describes itself as a search for proton decay. This
Part invites a second description: a precision audit of the
door's weld --- and thereby, at one remove, of the only
mechanism this geometry permits for your own existence. If
Hyper-Kamiokande sees a decay, it does not merely falsify a
stability theorem; it reopens Q10 from scratch, because the
mint that made you could then not have been the lepton door
alone. The two greatest questions your detector touches ---
whether matter is forever, and why it is here --- are, in this
framework, the same measurement.

\section{Falsifier faces of the Part}
\label{sec:part9-kills}

\begin{description}
\item[F-P6-1, doing double duty.] An observed proton decay
kills T-P6-2 (Part~VI) \emph{and} orphans Q10: it would
re-admit $\Delta B = 1$ minting and demolish the structural
selection of \S\ref{sec:structural-selection}. One event
falsifies two Parts at once --- the framework has
deliberately left itself no fallback between them.
\item[F-P6-3, sharpened cosmologically.] An observed
$n$--$\bar n$ oscillation would, beyond its Part~VI kill,
resurrect $\Delta B = 2$ (``post-sphaleron'') baryogenesis as
a live mechanism --- a class this geometry forbids outright.
Its observation is therefore a double kill here too.
\item[The chain's own fences.] The resonance condition and
C-P5-C are \PaperPFive{}'s registered items; if either fails
at home, this Part's import fails with it, and the ledger
below says so. What would survive such a failure: the
structural selection itself (\emph{some} $B{-}L$-first
mechanism through the door), which rests only on Part~VI.
\end{description}

\section{Status ledger of the Part}
\label{sec:part9-ledger}

\begin{table}[htbp]
\centering
\small
\caption[Status ledger of Part X.]{Status ledger of Part~X.
Grades as in Table~\ref{tab:import-ledger}. Three columns: the claim, the grade it is carried at, and the fences or notes that bound it. The ledger is here so that the standing of the whole Part can be read down one column instead of reconstructed from the text, and so that no claim of the Part is left without a grade beside it.}
\label{tab:part9-ledger}
\begin{tabular}{p{0.40\textwidth}p{0.10\textwidth}p{0.40\textwidth}}
\toprule
\textbf{Claim} & \textbf{Grade} & \textbf{Fences / notes} \\
\midrule
$\eta_B$, the catastrophe, Sakharov's conditions & D & Data
and standard theory, cited; SM quantitative failure stated
per literature. \\
\addlinespace
Structural selection (leptogenesis forced, not preferred) &
D(c) & Corollary of T-P6-2 at Part~VI's grade: no
$\Delta B = \pm1$ mint exists; the door is unique. \\
\addlinespace
The chain (resonant TeV leptogenesis $\to$ door $\to$
$\eta_B$) & D(c) & \PaperPFive{}'s, at its declared grade:
one named input (resonance), one named conjecture (C-P5-C);
consumed verbatim, upgraded nowhere. \\
\addlinespace
Partition \eqref{eq:partition} & D & Equilibrium
chemical-potential bookkeeping,
Appendix~\ref{app:door-bookkeeping}; standard, cited. \\
\addlinespace
Two-temperature consistency (Q6 $\equiv$ Q10 at one door) &
D(c) & This Part's own sentence, on Part~VI's grade; the
double-duty kills registered. \\
\bottomrule
\end{tabular}
\end{table}

History settled, one question remains --- the framework's own,
the newest on the list and the one held to the strictest
register in the paper: what does a proton's rearrangement cost
the universe's \emph{other} ledger?

\GGPart[The information charge]{The Information Charge}{$\mathrm{U}(1)_{Qi}$ for the proton: one rearrangement, one unit (Q11)}
\label{part:qi}

One face remains, and it is unlike the other ten: not a
century-old question inherited from the field, but this
framework's own --- what does a proton's rearrangement cost the
universe's \emph{information} books? Because the claim is the
youngest in the paper, it is held to the strictest register the
cohort owns: the way to kill it is stated before the claim
itself, and nothing in the ten Parts behind us leans on what
follows.
\section{The kill criterion, first}
\label{sec:qi-kill-first}

This Part carries the youngest claim in the paper and therefore
opens in reverse order: with what would kill it, before what it
says. The discipline is deliberate. Ten Parts of theorems and
imports have earned the reader's trust at stated grades; this
Part spends none of that trust, because its subject --- the
information charge \UoneQi{} --- is held by its home paper at
\emph{candidate} grade, kill criterion registered
\citep{ArisakaA7Qi4}, and a candidate that borrows a theorem's
tone is how frameworks rot. So, first, the registered kill,
quoted in substance: \emph{the Arisaka equation requires the
four currents of every elementary boundary event to be locked;
a demonstrated elementary event that updates the baryonic
ledger without the corresponding locked informational unit ---
or at a different conversion factor --- falsifies the
equation.} That is F-P6-6, imported intact. Everything below is
what the candidate says \emph{if it lives}.

\section{The Arisaka equation, restated for the proton}
\label{sec:arisaka-for-proton}

Part~II carried the equation in; here is what it asserts, read
with this paper's eyes. At every elementary boundary event, four
currents move together or not at all:
\begin{equation}
\frac{J_E}{\QB} \;=\; J_e \;=\; J_B \;=\; J_{Qi},
\qquad
\QB \;=\; \frac{2}{5}\,\frac{\ERyd}{K}
\;=\; 0.1555~\mathrm{eV},
\label{eq:arisaka-restated}
\end{equation}
energy, electric charge, baryon number, and information,
locked through one conversion constant --- the ledger constant
$K = 35$ appearing once more, in its fourth cohort role. The
claim's shape should now feel familiar, because it is the shape
this whole paper has drilled: a conserved boundary quantity is
not a substance but a \emph{bookkeeping of readout events}, and
\eqref{eq:arisaka-restated} says the books are one book. The
electric column and the baryonic column of that book, Parts~III
and VI showed, are windings --- exact, quantized, unfakeable.
\UoneQi{} is the framework's proposal that the book has a
fourth column of the same character: every elementary
rearrangement in which a baryon participates --- every capture,
every binding, every readout update in which the wrapped
configuration changes state --- debits the energy ledger by
$\QB$ per unit and credits the information ledger by exactly
one unit. Not \emph{about} $\QB$; exactly $\QB$, or the
equation is dead. Candidates that specify their own execution
are the only candidates worth registering.

\section{What it means for the proton --- and what it does not
license}
\label{sec:qi-meaning}

Read \eqref{eq:arisaka-restated} against the paper's earlier
results and the proton acquires a closing characterization.
Part~III made it the Ground Geometric Attractor: matter's floor. Part~VI
made its baryon number a winding: matter's permanence. This
Part's candidate, if it lives, adds: the proton is the
\emph{page} of the boundary's ledger --- the stable substrate
whose rearrangements are the countable, durable entries of the
information column. Only a zero-width, winding-protected object
can anchor a permanent record; everything else in the baryon
tables dissolves in $10^{-10}$ seconds or less, taking its
entries with it. In this reading, the fact that every durable
structure in the universe --- every nucleus, molecule, mineral,
memory --- is written \emph{on baryonic matter} is not
substrate chauvinism but bookkeeping necessity: the information
column requires pages that do not burn, and Part~VI built the
only fireproof page in the spectrum. \emph{The proton is not
only matter's terminus; it is the ledger's paper.}

Two calibrations put the constant on a human scale, both
computed and both carried at strictly this Part's grade. At
room temperature, Landauer's bound prices the cheapest possible
erasure of one bit at $k_B T \ln 2 = 0.0179$~eV; the Arisaka
unit is
\begin{equation}
\frac{\QB}{k_B T_{300} \ln 2} \;=\; 8.7
\label{eq:landauer-ratio}
\end{equation}
--- an order-one multiple of the thermodynamic floor at the
temperature where chemistry computes, not parachuted in from
some distant scale. And the workhorse energy quantum of
biological information processing, ATP hydrolysis at
$\approx 0.32$~eV, is almost exactly \emph{two} Arisaka units
($0.32/0.1555 = 2.0$). The home paper builds its bioenergetic
bridge on such coincidences at candidate grade; this paper
neither extends nor endorses that bridge --- it records the
two ratios as what they are, calibrations that make the
constant tangible for the reader in the tank whose
photomultiplier pulses are, one level down, ATP-priced
information events about baryons.

What the candidate does \emph{not} license, stated bluntly
because candidates attract freight: no claim that ``the
universe is a computer''; no rate limit on human technology
(the equation prices elementary boundary events, not
engineered logic built redundantly atop them); no modification
to any result of Parts~I--X, all of which stand at their own
grades if \UoneQi{} dies tomorrow. The dependency arrow points
one way only.

\section{The experimental face, honestly sized}
\label{sec:qi-experiment}

Where could F-P6-6 actually bite? The honest answer is that the
decisive experiment does not yet exist, and this Part says so
rather than gesturing at one. The equation's natural testing
ground is the mesoscopic frontier where single-event
calorimetry meets single-particle control --- trapped-ion and
single-atom thermodynamics, nanocalorimetry approaching the
$k_B T$ scale, single-nucleon-resolved chemistry --- where the
locked-currents claim becomes, in principle, a measurable
question: does an elementary baryonic rearrangement carry its
fixed informational tariff, or can the columns be decoupled?
The framework's own contribution to experimental design is the
lock's rigidity: \eqref{eq:arisaka-restated} permits no partial
credit, so any clean decoupling at any single event scale
kills it --- a binary target of exactly the kind Part~VI gave
the tank. Until such an experiment matures, this Part's
register stays where its home put it: candidate, alive,
specified, and waiting.

\section{Status ledger of the Part}
\label{sec:part10-ledger}

\begin{table}[htbp]
\centering
\small
\caption[Status ledger of Part XI.]{Status ledger of Part~XI.
Grades as in Table~\ref{tab:import-ledger}. Three columns: the claim, the grade it is carried at, and the fences or notes that bound it. The ledger is here so that the standing of the whole Part can be read down one column instead of reconstructed from the text, and so that no claim of the Part is left without a grade beside it. Part XI's last row is the one that matters to the rest of the paper: nothing upstream leans on this Part.}
\label{tab:part10-ledger}
\begin{tabular}{p{0.40\textwidth}p{0.10\textwidth}p{0.40\textwidth}}
\toprule
\textbf{Claim} & \textbf{Grade} & \textbf{Fences / notes} \\
\midrule
Arisaka equation \eqref{eq:arisaka-restated}; $\QB$ & CAND &
\ASeven{}'s, at \ASeven{}'s registered grade; kill criterion
quoted first (F-P6-6). \\
\addlinespace
Ledger's-page reading of the proton & INT & This Part's
framing of the candidate; adds no claim beyond
\eqref{eq:arisaka-restated} + Parts III/VI. \\
\addlinespace
Calibrations \eqref{eq:landauer-ratio}, ATP $\approx 2\QB$ &
D & Arithmetic, session-verified; carried as calibration
only; the bioenergetic bridge remains \ASeven{}'s at its
grade. \\
\addlinespace
Independence of Parts I--X from this Part & D & The
dependency arrow stated; nothing upstream leans here. \\
\bottomrule
\end{tabular}
\end{table}

Eleven questions are now closed or fenced. What remains is the
weighing --- significance, uniqueness, the debts collected in one
registry, with the objections answered at full length near the end
(Part~XV) --- and then the walk back out of the
mine, in plain English, to say what half a century of silence
taught.

\GGPart[Discussion]{Discussion}{Significance, uniqueness, falsification, and the open problems --- the objections are answered at full length in Part~XV}
\label{part:discussion}

The physics is done; the weighing begins. This Part steps back
from the derivations and asks the questions a careful reader
will ask of the whole: what has actually been established, and
at what grade? How does this account differ --- structurally,
not rhetorically --- from the other roads the century built?
What observation would kill it, and when will we know? And
what does it still owe? The answers are collected in one place,
without inflation, because a paper that keeps eleven promises
owes its reader a single page where all eleven receipts can be
checked.
\section{What has been done, weighed soberly}
\label{sec:significance}

Eleven questions were put on the table in Part~I; the honest way
to close is to say, without inflation, what happened to each.
Two were answered by \emph{theorems of this paper}: the sharp
mass and exact charge by the ground-attractor theorem (T-P6-3),
and the immortality-and-law pair by the obstruction theorem and
the $\mathbb{Z}_3$-lock (T-P6-1, T-P6-2) --- the latter
completing a stability theorem the cohort had carried as
conditional since its Standard-Model paper. Three were answered
by \emph{imports consumed at their proved grades}: the six
flavors by the family theorem, the twin's existence and descent
by the toll competition and the gate, the birth of baryons by
the CPT companion's chain through the door this paper's own
theorem leaves open. Three were answered by \emph{readings,
graded as such}: the spin as label (INT, with its promotion
registered), confinement as branch selection (INT, behind a
Clay-grade fence), the free quark's absence upgraded halfway ---
its \emph{impossibility} is now a corollary of a theorem
(C-P6-1), its \emph{costliness} remains QCD's. And one is a
\emph{candidate carried at candidate grade}, kill criterion
first (the information charge). Nothing on the list was
answered by adjustment; no parameter was tuned anywhere in
these pages, because the framework put none at this paper's
disposal.

The deepest single result, the author believes, is the
smallest-looking: the center identity $B = t/3$. Baryon
number's fractional content was sitting inside the color group
the whole century --- a finite rotation, invisible to every
algebra-level search for a gauged $B$ --- and once it is seen,
the rest follows with almost embarrassing economy: one setting
of one existing dial breaks the symmetry, evicts the
executioners, splits the Higgs, and writes the winding. If this
paper is remembered for one page, it should be that one.

\section{Uniqueness, as structural contrast}
\label{sec:uniqueness-contrast}

The program's discipline is contrast without polemic: state
what each alternative structure \emph{is}, and where the
present one differs, and stop. Four contrasts locate this
paper.

\paragraph{Against classical unification.} Grand unified
theories share this paper's premise --- the forces unify, and
quark and lepton sit in one representation --- and differ in one
locational fact: their unification lives on the boundary, so
their $X, Y$ bosons are four-dimensional particles and proton
decay is mandatory. Fifty years of silence have pushed the
simplest such models out and their successors to ever higher
scales. Here unification is a bulk fact; the $X, Y$ are real
and boundary-absent; and the same geometry that removes them
writes the conservation law the silence measures. The
experiment that was mortal danger to one architecture is
standing confirmation of the other --- and remains mortal
danger to \emph{this} one through F-P6-1, which is how the
contrast stays honest.

\paragraph{Against the Standard Model's accident.} The
Standard Model conserves baryon number by bookkeeping luck:
nothing imposes it, and the accident survives no extension.
The framework's $B$ is imposed --- gauged in the bulk,
Stueckelberg-massive, topologically realized --- which is the
difference between a habit and a law. The observational
wedge between the two readings is exact: the accident permits
$\Delta B$-violating operators at dimension six and merely
prices them by an unknown scale; the law forbids them at every
dimension. Any decay signal at any rate, however far beyond
current reach, distinguishes the two forever.

\paragraph{Against orbifold unification.} The nearest
relatives of Part~VI's construction are the orbifold GUTs
\citep{Kawamura2001}, which taught the field that boundary
conditions can do the work of Higgs sectors --- splitting
doublet from triplet and suppressing decay by geography. The
present construction inherits that insight and differs in
three structural respects: the twist is not a chosen parity
assignment per multiplet but the unique center point of the
existing hypercharge line (no intrinsic-parity freight); the
protection is not geographic suppression but an exact winding
charge (a selection rule, not a distance); and the reflection
sector is not an engineering choice but is fixed by the CPT
companion's theorem, tying the construction to the cohort's
antiparticle result. Whether these differences are virtues is
for the reader; that they are differences is arithmetic.

\paragraph{Against the ten-dimensional route to anomaly
freedom.} The mainstream's price for anomaly freedom was fixed
in 1984: ten dimensions, supersymmetry, and a specific enormous
gauge group \citep{GreenSchwarz1984}; and in the heterotic,
GUT-descended constructions that dominated string
phenomenology, baryon number remained accidental and
approximate. One string lineage --- the intersecting-brane
Standard Models \citep{IbanezMarchesanoRabadan2001} --- does
gauge \UoneB{}, and what it finds independently corroborates
this paper's mechanism: the boson arrives
Green--Schwarz-massive and the proton is protected, though
generically only in perturbation theory. The contrast is
therefore three clauses about price, not mechanism: the
dominant unification roads carry no exact baryon number at
all; where strings do gauge it, the protection arrives
GS-massive exactly as here, but atop ten dimensions,
supersymmetry, and extended objects, and only perturbatively
exact; and this framework keeps Lee--Yang's gauging, pays the
anomaly bill at six dimensions through the \BIsix{} inflow,
uses no supersymmetry and no string-like objects, and delivers
protection exact to all orders --- the topological
$\Delta B \in 3\mathbb{Z}$ door its single, load-bearing
exception. Part~VII keeps the full record, convergence
included.

\paragraph{Against the confinement mechanisms.} The
dual-superconductor picture, the center-vortex picture, and
the lattice's $N$-ality lore \citep{Greensite2011} are
accounts of \emph{how} the Yang--Mills vacuum generates a
confining force --- competing descriptions of the disorder,
each with real numerical successes, none yet promoted to a
proof. This paper is not a competitor on that ground and
claims nothing there: the dynamical half of confinement
remains QCD's, held here at one named hypothesis from a
theorem (T-P6-4). The structural contrast lies one level
down, at a question the mechanism literature does not pose:
\emph{why $\mathbb{Z}_3$?} Why should triality, of all
quantum numbers, be the arbiter of who is caged --- why does
the center decide who is confined, when confinement holds,
and where the disorder lives (\S\ref{sec:centre-bridge}),
three lattice-verified roles that the standard account
inherits as facts about $\mathrm{SU}(3)$ representation
theory and does not further explain? Here the three roles
are one fact seen three times, because the center is not
bookkeeping on field space but the holonomy of a real cycle:
the singlet rule is its closure condition
(\S\ref{sec:singlet-rule}), the cage is the impossibility of
closing a third of a turn (C-P6-1), and the immortality is
the impossibility of unwinding a whole one (T-P6-2). No
mechanism paper claims the free quark is a grammatical
impossibility; this paper claims exactly that, at the lock's
stated grade, and hands the claim its own kill switch
(F-P6-2).

\paragraph{Against the quotient as footnote.} That the
faithful gauge group of the Standard Model is the quotient
$[\SUthreeC \times \SUtwoL \times \UoneY]/\mathbb{Z}_6$ has
been known to model builders for decades --- and has sat in
the literature as fine print, a true remark about which group
one should have written down, carrying no dynamics. The
framework's structural difference is that the fine print is
load-bearing (\S\ref{sec:abelian-ledger}): the quotient's
$\mathbb{Z}_3$ is the statement $B - 2Y \in \mathbb{Z}$, which
is the sentence that parks the descent's holonomy at the
center point \eqref{eq:z-on-wilson-line}; the same three
shared phases are the triangle of
Figure~\ref{fig:z3-winding}; and the triangle's
electromagnetic shadow, $Q + B \in \mathbb{Z}$ fragment by
fragment \eqref{eq:q-plus-b}, welds the fractional quark
charges --- treated since 1974 as the unification's aesthetic
bonus \citep{GeorgiGlashow1974} --- to the baryon winding that
keeps the proton alive. A remark about representations in one
architecture; the hinge joining Q2, Q5, and Q6 in the other.
Whether the promotion is a virtue is for the reader; that the
two standings differ is arithmetic.

\section{The falsification table}
\label{sec:falsification-table}

The paper's kills, collected with their current experimental
anchors and, where they exist, their dates. The table is the
contract of \S\ref{sec:kills}, now fully priced.

\begin{table}[htbp]
\centering
\small
\caption[The falsification table.]{The falsification table of
\PaperPSix{}. Each row: one observation that kills the stated
result outright. ``Decidable'' marks rows where planned
experiments will materially extend reach on a stated
timescale. Six rows, each naming one observation, the result it would kill, and the anchor or horizon against which it stands. The right column is what makes the table usable rather than rhetorical: it gives the current bound and, where an experiment will materially extend reach, the decade in which it does so.}
\label{tab:falsification-p6}
\begin{tabular}{p{0.15\textwidth}p{0.30\textwidth}p{0.22\textwidth}p{0.22\textwidth}}
\toprule
\textbf{ID} & \textbf{Killing observation} & \textbf{Kills} &
\textbf{Anchor / horizon} \\
\midrule
F-P6-1 & One proton decay, any channel & T-P6-2; and reopens
Q10 (Part~X) & $>2.4\times10^{34}$~yr
\citep{Takenaka2020}; Hyper-K/DUNE/JUNO extend an order of
magnitude by $\sim$2040 \\
\addlinespace
F-P6-2 & One confirmed free fractional charge (any source,
incl.\ QGP) & C-P6-1; wounds T-P6-2 & eighty-year null
\citep{PDG2024}; standing \\
\addlinespace
F-P6-3 & $n$--$\bar n$ oscillation & T-P6-2
($\Delta B = 2$); and Part~X's mechanism class & free
$>0.86\times10^{8}$~s \citep{BaldoCeolin1994}; ESS proposal
extends \\
\addlinespace
F-P6-4 & Proton--antiproton asymmetry (mass, $|q|/m$, moment
magnitude); H/$\bar{\mathrm{H}}$ charge anomaly & the dihedral
reflection (Part~VI); conjugate readout & 16~ppt
\citep{Borchert2022}; $0.71$~ppb \citep{Ahmadi2016};
improving \\
\addlinespace
F-P6-5 & Long-range force coupled to $B$ & Stueckelberg
structure of \UoneB{} & E\"ot-Wash Be/Ti (chosen to maximize
the baryonic difference): $\eta = (0.3 \pm 1.8)\times10^{-13}$
\citep{Wagner2012}; MICROSCOPE near $10^{-14}$; standing \\
\addlinespace
F-P6-6 & Elementary baryonic ledger update without the locked
informational unit & the Arisaka equation (at its home's
grade) & experiment not yet mature
(\S\ref{sec:qi-experiment}) \\
\bottomrule
\end{tabular}
\end{table}

One property of this table deserves notice: rows one through
four are \emph{null results that grow stronger every year an
experiment runs}. A framework whose sharpest tests are ongoing
silences is in the unusual position of being tested
continuously, at megaton scale, every day --- and of owing its
community that fact plainly, which Part~VI's closing section
was written to do.

\section{The debt registry}
\label{sec:debt-registry}

Every open item this paper carries, consolidated. The registry
follows the cohort's convention: a debt is named where
incurred, listed here, and either discharged in a later paper of the cohort or carried honestly forever. None of the six is a
conjecture of principle; each is a bounded computation or
reading.

\begin{description}
\item[D-P6-1 --- DISCHARGED here.] The reading of
the deployed descent's reflection on complex representations:
the realization paper's own text supplies the conjugation
twist, $C = (\theta \mapsto -\theta) \circ B$, at theorem
grade via T-P5-C2 (\S\ref{sec:dihedral}). Recorded per the
registry convention; no residue remains.
\item[D-P6-2 (= pin R2), narrowed here.] The
holonomy-integrality pin, now named to its clause: descent
coefficients are Hopkins--Singer-integral in this framework
(\citealp{ArisakaA9SM}; Appendix~\ref{app:lock-arithmetic}),
which quantizes the Wilson-line parameter to the six-point
lattice. Outstanding: the $X_4$-level derivation of the
holonomy's class.
\item[D-P6-3.] The spinor-level derivation that the terminal
basin's boundary readout carries $j = \tfrac12$
(\S\ref{sec:spin-claims}) --- one computation shared with the
sister paper's D-P3-4; the two registries discharge together.
\item[D-P6-4, restated as a scoreboard here.]
(a) Branch uniqueness --- delivered as T-P6-4's corollary,
conditional on the gap hypothesis $\widehat{G}$. (b) The area
law --- delivered at theorem-structure grade on $\widehat{G}$
plus the lock (\S\ref{sec:tp64}); the functional tightening of
Steps 3--4 is the named formal residual. (c) The pure number
$\sigma/\Lambda^2$ --- OPEN: the wall computation owed; target
and kill window printed in \S\ref{sec:sigma-number}. (d) The
L\"uscher coefficient --- CLOSED, unconditionally
(\S\ref{sec:goldstone-test}). (e) The thermal face --- partial:
center/Potts universality inherited for the pure-gauge
transition; quantitative targets owed with (c). The single deep
residual is $\widehat{G}$ itself --- the gap, the Clay-hard
core --- whose location is priced \eqref{eq:wall-location} and
whose existence this paper does not claim.
\item[D-P6-5 (= pin R3), narrowed here.] The
warped zero-mode profiles at fractional winding. The
\emph{mechanism} --- bosons priced by the Laplacian, unpaired
chiral floors massless by the absence of a partner --- is now
exhibited in Appendix~\ref{app:chirality-exhibit}; what
remains owed is the quantitative profile computation in the
warped metric with the \GDOSm{} radial functions.
\item[D-P6-6.] The completeness of the constitutional label
inventory, on which the uniqueness leg of T-P6-3 conditions
(\S\ref{sec:terminal-theorem}): the statement that the
boundary's exact labels are exactly the winding pair and the
spinor label, referred to \AOne{}'s axiom set.
\end{description}

\section{Status ledger of the Part}
\label{sec:part11-ledger}

This Part makes no new physics claims; its ledger is the
falsification table (\S\ref{sec:falsification-table}) and the
registry (\S\ref{sec:debt-registry}); the objections are
answered at full length in Part~XV, each by citation into the
Parts that carry the load. The paper-wide claim-status ledger --- every
numbered result with its grade and location --- is
Part~XIV's opening table.

The weighing is done, and the scales read as this Part has
recorded them: so much derived, so much conditional, so much
honestly owed --- nothing hidden, nothing inflated. What
remains is not analysis at all. It is the walk back out of the
mine: the century, the tank, and the theorem, retold in the
plain language they have earned --- and the closing words
belong, as they should, to the author, who first went
underground in 1979 and has, in a sense, been walking back out
ever since.

\GGPart[Conclusions]{Conclusions}{The century, the tank, and the theorem --- in plain English}
\label{part:conclusions}

%% ===================================================================
%% AUTHOR-VOICE PART. Per the Ship-1 convention, this entire Part is
%% drafted as a complete scaffold for the author's verbatim pass:
%% every section is his to rewrite, sentence by sentence, at
%% sign-off. The physics content and cross-references are final;
%% the voice awaits its owner.
%% ===================================================================

\section{Why the questions waited a century}
\label{sec:why-waited}

The eleven questions of this paper are not exotic. A child can
ask most of them --- \emph{why doesn't it break? why doesn't it
run down? why is every one the same?} --- and the century that
built quantum mechanics, quantum field theory, and the Standard
Model never stopped being asked. Why, then, did they wait?

Because each generation's tools answered a \emph{finer} question
instead. Scattering resolved structure, so the century asked
about structure --- form factors, partons, distributions ---
and answered magnificently. Field theory computed rates, so the
century asked about rates. What no tool before the present
framework was built to ask was the child's own question, which
is about \emph{standing}: what kind of thing is a proton, that
the storm inside it should present one face forever? The answer
this paper has assembled --- a Ground Geometric Attractor of a
landscape, carrying a winding of a compact geometry --- required pieces
the century acquired slowly and separately: the attractor
grammar, the open-system boundary, the center of the color
group, the topology of a circle no one believed was there. None
of the pieces is exotic either. What waited a century was the
\emph{assembly}. %% AUTHOR-VOICE: rewrite at will.

\section{The view from the last page}
\label{sec:view-last-page}

Stand at the end and look back; the eleven questions have
become one another.

The proton has one exact mass because it is the floor of its
landscape, and floors do not come in variants (Q1). Its charge
cancels the electron's because both are entries of one
five-by-five matrix that must sum to zero (Q2). Its spin will
not decompose because a label is not a sum (Q3). Its quarks
cannot leave because a third of a wrap is not a state (Q4, Q5).
It cannot die because no operator can change the number of
wraps (Q6), and that is a law, not a habit, because the winding
is gauged into the geometry (Q7). Its twin dies in fifteen
minutes because the twin sits a hair's-breadth up-slope with
one narrow gate open --- and survives forever in nuclei because
binding re-landscapes the gate (Q8). Its six flavors are three
families said twice, and its excited copies crumble on
schedule, each at the price its labels set (Q9). It exists at
all because the one door its conservation law leaves open stood
open in the hot first picosecond, converting a lepton-sector
asymmetry into every baryon there is (Q10). And --- if the
youngest claim in this paper survives its own kill criterion
--- its rearrangements are the durable entries of the
boundary's information ledger, making it not only matter's
terminus but the page the universe writes on (Q11).

One ground attractor. One winding. One door. Eleven faces.
%% AUTHOR-VOICE: rewrite at will.

\section{The one fact, and where it sits}
\label{sec:one-fact-summit}

%% AUTHOR-VOICE SECTION. The scaffold below states the concept
%% the paper is built on and carries its cross-references; every
%% sentence is the author's to keep, cut, or rewrite at sign-off.

Step back once more, and the eleven faces resolve into a single
object with a single name. The proton is the Ground Geometric
Attractor of the baryonic stratum --- the \GGA{}, the deepest
admissible valley of the landscape of color-singlet,
baryon-number-one matter. That is not one more result in the
list; it is the fact of which every entry in the list is a
reading. The mass is the depth of that ground basin; the charge
is its exact label; the immortality is its empty exit-channel
set, gauged shut; the identity of every proton with every other
is what \GGA{}-hood \emph{means}. The century found these one at
a time and filed them under separate headings --- mass, charge,
stability, universality --- and this framework's claim is that
they were always four windows onto one ground state.

And that ground state is not the end of a story but the
beginning of one. The electron is the \GGA{} of the leptonic
stratum; the proton is the \GGA{} of the baryonic one; and
between the two of them they are the ground on which the whole
Chain of \GRIPout{} is built (\S\ref{sec:grip-chain}). Bind
them and the atomic stratum opens --- hydrogen, then the periodic
table; let atoms bond and the molecular stratum opens --- water,
then the amino acids, the proteins, ATP, RNA, DNA, and every rung
above, each its own \GGA{}, each standing on the one below. This
is the picture the capstone paper \citep{ArisakaA11Origin} draws
in full, and it is the reason a study of one particle can claim
to touch the heart of the framework: the proton is where that
grand cascade --- the one that runs, in the end, all the way to
an observer able to state it --- comes to rest at the bottom and
begins to build. It is the foundation stone of the material
world, and a foundation stone earns its name by never moving.

That is the deepest reason the watch under the mountain matters.
A search for proton decay is usually described as a search for a
rare event; in this framework it is something larger --- a test
of whether the foundation of all stable matter is exactly, and
not merely almost, eternal. The man who first went underground in
1979 to watch the proton die has spent a career at that
foundation, and the answer this paper offers him, and the
generation now on the catwalk above the water, is that the
foundation holds: the proton is the Ground Geometric Attractor at
the bottom of the Chain of \GRIPout{}, the still point on which
the whole tower of the world is built. %% AUTHOR-VOICE: rewrite
%% at will; this section is the concept's summit and yours to voice.

\section{The conclusions, in plain English}
\label{sec:plain-english}

For the reader who skipped the mathematics, an honest summary
in ten sentences. Most of everything's weight is not stuff but
settled storm: your bathroom scale reads the energy of gluon
fields that have found their resting place. That resting place
is unique, which is why every proton in the universe is
exactly alike --- they are not similar objects but one
mathematical object, read out everywhere. The proton's charge
matches the electron's to twenty-one digits because both are
lines of a single code, not because two accidents agree.
Nobody has ever pulled a quark out of a proton, and nobody
ever will, for the same reason nobody will show you a third of
a knot. The proton has never been seen to die, though we built
cathedrals underground to watch; this paper derives that the
watching must stay silent --- matter's number is wound into
the shape of space, and windings do not unwind. The neutron,
the proton's near-identical twin, dies in about fifteen
minutes on its own but lives forever inside atoms, and that
one detail is why chemistry --- and you --- can exist. The
heavier copies of the proton all crumble instantly; the
universe is built from the single rung of the ladder that
cannot. The matter in the universe is the frozen residue of a
door that stood open for the first trillionth of a second and
then sealed itself; every atom is a receipt from that
transaction. If the proton is ever seen to decay, this entire
framework is wrong, and the experiments best placed to prove
it wrong are the direct descendants of the tank this story
began with. That is not a weakness of the framework; it is
the whole point of having one. %% AUTHOR-VOICE: rewrite at
%% will; the ten-sentence constraint is a suggestion, not a rule.

\section{The fifty-year arc}
\label{sec:fifty-year-arc}

%% AUTHOR-VOICE SECTION, WHOLLY RESERVED. The scaffold below
%% marks the arc's stations and carries the cross-references;
%% every sentence is a placeholder for the author's own, per
%% his instruction that this paper is a message from the
%% founder to the youngest generation.

A doctoral student went underground in 1979 to build a machine
for watching protons die, because the best theory of the age
had made their death its safest prediction. He spent six years
on the tank, the optics, the electronics
\citep{ArisakaKamiokande1985}; the field expected a flash
within a year or two of first light. The flash never came. The
machine, refusing to waste its silence, heard the sun instead,
and then a dying star \citep{HirataSN1987}, and founded an
astronomy; its successors pushed the proton's silence past
$10^{34}$ years; and the question the tank was named for went
unanswered for the length of a career.

Forty-eight years after that student first walked into the
mine, this paper --- his --- returns the answer: the flash
cannot come, and the silence was the discovery. The prediction
that built the tank mistook the location of the executioners;
the geometry keeps them in the bulk, and writes the proton's
number where no operator can reach it. The tank was right the
whole time --- right in a way its builders could not have
stated, because the theorem that says so needed half a century
of other results first.

To the student now standing where he stood --- on the catwalk
at Hyper-Kamiokande, in the cavern at DUNE --- the closing
message of this paper, from its author to you: keep watching.
Your silence is a measurement, the sharpest ever made of the
deepest law this framework contains. And if one night the
water flashes where no muon entered --- then this paper dies,
physics is reopened at its foundations, and yours will be the
most important eyes in the history of the subject. Either way,
the watch is not a vigil over a settled question. It never
was. %% AUTHOR-VOICE: every sentence above is the author's to
%% keep, cut, or rewrite --- this is his Part and his story.

\GGPart[Technical Appendices]{Technical Appendices}{The reference apparatus --- abbreviations, the paper-wide claim-status ledger, the empirical anchor table --- followed by the arithmetic companions, Appendices A through G, each written so that a reader with one sheet of paper can verify the load-bearing computations by hand}
\label{part:appendices}

A good workshop earns its keep by keeping every tool where a
working reader can find it; that is this Part's only job. The
body of this paper has emphasized the theorem chain; this Part
supplies the technical apparatus that supports each step. It is
organized into two groups, on the pattern of the cohort's
Standard-Model paper. The \emph{reference} apparatus comes first:
the abbreviations, the paper-wide claim-status ledger (every
numbered result, its grade, and its fences, in one table), and the
empirical anchor table (every measured value the paper quotes,
with its source --- all arithmetic on these values was verified by
machine at drafting). The \emph{constructive} appendices follow:
the arithmetic companions to Parts~III through X --- the
transmutation walk (A), the spin ledger (B), the string (C), the
lock (D), the neutron (E), the door (F), and the
chirality-protection exhibit (G) --- each written so that a reader
with one sheet of paper can verify the load-bearing computations
by hand, and each accompanied, where the algebra is dense, by the
physics intuition that translates it into a mental picture. The
objections a hostile specialist would raise are answered, reader
by reader, in the Common Objections Part that follows these
appendices; the annotated references close the paper after that.

\section{Abbreviations}
\label{sec:abbreviations}

\begin{table}[htbp]
\centering
\caption[Abbreviations used in this paper.]{The abbreviations used in this paper, set two-up: the program's own names on the left of each pair and the standard physics abbreviations on the right. The grid is arranged this way because the list is short enough to take in at a glance and long enough to be worth not hunting through, and because the program's own names are the ones a reader coming from outside will not already know.}
\label{tab:abbrev-grid}
\small
\begin{tabular}{llll}
\toprule
\GG{} & Grand Geometric (Theory) & QCD & quantum chromodynamics \\
\GGA{} & Ground Geometric Attractor & QED & quantum
electrodynamics \\
\GA{} & Geometric Attractor (way station) & GGA & ground geometric
attractor \\
\GDOSm{} & Geom.\ Dynamics of Open Systems & SM & Standard
Model \\
\GFF{} & the free-energy functional & GUT & grand unified
theory \\
\GSSB{} & geometric spontaneous symmetry breaking & CKM &
Cabibbo--Kobayashi--Maskawa \\
\GRIPin{} & the in-flow descent pricing & CPT &
charge--parity--time \\
MSMF & Max.\ Symmetry, Min.\ Freedom (axiom) & QGP &
quark--gluon plasma \\
\BIsix{} & the modified Bianchi identity & EIC & Electron--Ion
Collider \\
D-P6-$n$ & registered debt $n$ of this paper & PDG & Particle
Data Group \\
F-P6-$n$ & registered falsifier $n$ of this paper & UCN &
ultracold neutron \\
GS & Green--Schwarz (inflow) & EP & equivalence principle \\
\bottomrule
\end{tabular}
\end{table}

\section{The claim-status ledger}
\label{sec:claim-ledger}

Every numbered claim of the paper, its grade, and its fences ---
the honesty apparatus in one place. Grades: \textbf{D} derived;
\textbf{D(c)} derived conditionally (fences named); \textbf{INT}
interpretation, named; \textbf{CAND} candidate, kill criterion
registered.

{\small
\begin{longtable}{p{0.21\textwidth}p{0.35\textwidth}p{0.07\textwidth}p{0.24\textwidth}}
\caption[The paper-wide claim-status ledger.]{The claim-status
ledger of \PaperPSix{}. Every named claim in the paper, in one list: the item, a compressed statement of it, the grade it is carried at, and the Part that owns it together with its fences. The grade column is the honest part, separating what is derived from what is derived conditionally, what is an interpretation, and what is a candidate with a kill criterion registered. It is meant to be read whole before any single row is leaned on.}
\label{tab:claim-ledger}\\
\toprule
\textbf{Item} & \textbf{Statement (compressed)} & \textbf{Grade}
& \textbf{Home; fences} \\
\midrule
\endfirsthead
\multicolumn{4}{l}{\small\emph{The claim-status ledger,
continued.}}\\
\toprule
\textbf{Item} & \textbf{Statement (compressed)} & \textbf{Grade}
& \textbf{Home; fences} \\
\midrule
\endhead
\midrule
\multicolumn{4}{r}{\small\emph{continued overleaf}}\\
\endfoot
\bottomrule
\endlastfoot
T-P6-1 & Baryon number is not realizable as any per-multiplet
constant plus hypercharge; gauged $B{-}L$ forbids no
dimension-six decay operator & D & Part~VI; exact linear
algebra \\
\addlinespace
Center identity & $B = t/3$ exactly on every fragment;
$e^{2\pi i B} = z$ & D & Part~VI \\
\addlinespace
Lock-point uniqueness & $z = W(2)$ on the hypercharge line;
sole quantized holonomy satisfying the spectrum's three
demands & D & Part~VI; quantization = D-P6-2 \\
\addlinespace
T-P6-2 & The $\mathbb{Z}_3$-lock: winding lattices $t/3 +
\mathbb{Z}$; floors carry $B$; H2 discharged; H1 corollary;
Yukawa selection rules & D(c) & Part~VI; D-P6-2, D-P6-5;
floor survival = \ANine{} claim \\
\addlinespace
Dihedral closure & Untwisted reflection forbids the lock;
C-twisted closes as $\mathbb{D}_3$; reflection $= B \to -B$ &
D & Part~VI; closure = T-P5-A + deployed $C = (\theta \to
-\theta) \circ B$; D-P6-1 discharged \\
\addlinespace
Anomaly ledger & Six coefficients exact; two survivors to
\BIsix{} inflow; $[\SUtwoL]^2 B$ the sole physical door,
$\Delta B \in 3\mathbb{Z}$ & D & Part~VI; H3 as received \\
\addlinespace
T-SM-60.1 completed & Proton absolutely stable; $\Delta B =
\pm1, \pm2$ forbidden to all orders & D(c) & Part~VI; H3 +
D-P6-2 + D-P6-5 \\
\addlinespace
T-P6-3 (i) & Existence of the stratum's floor & D & Part~III \\
T-P6-3 (ii) & Uniqueness of the floor & D(c) & Part~III;
D-P6-6 \\
T-P6-3 (iii) & Zero width (terminality) & D(c) & Part~III; the
$B$-leg at Part~VI's grade \\
T-P6-3 (iv) & Identity of all readouts & D & Part~III \\
\addlinespace
Mass pricing & Transmutation from $\alpha_s^{-1}(M_Z) = \pi e$;
one-loop walk; pure number from lattice & D(c) & Part~III;
registration at \ANine{}'s grade \\
\addlinespace
Charge table & $Q = T_3 + Y$ from the embedding; $Q(p) =
-Q(e)$ as algebra & D & Part~III \\
\addlinespace
Charge/spin rigidity & Winding and representation labels admit
no drift & D(c) & Parts~III--IV; fiber reading at sister
paper's grade \\
\addlinespace
Spin custody & Total = boundary label; shares = bulk ledger,
scheme-dependent & INT & Part~IV; promotion = D-P6-3 \\
\addlinespace
T-P6-4 & Conditional area law: gap hypothesis
$\widehat{G}$ + lock $\Rightarrow$ $F \geq \sigma_{\min}L$;
tube; $N$-ality asymptotics & D(c) & Part~V; on $\widehat{G}$ +
(A1)(A2) + T-P6-2's grade; Higgs escape closed by C-P6-1;
tightening a named residual \\
\addlinespace
Goldstone-counting test & Flux-tube L\"uscher coefficient
$-\pi/12$ from exactly two transverse modes; spatial extra
dimensions at $\sigma$-scale excluded & D & Part~V;
unconditional given the tube; lattice-confirmed \\
\addlinespace
Branch-selection uniqueness & Color-singlet branch the unique
stable attractor & D(c) & Part~V; corollary of T-P6-4 + T-P6-3
on $\widehat{G}$; \GSSB{} grammar gloss remains INT \\
\addlinespace
C-P6-1 & No boundary readout carries fractional winding; the
free quark is not a state & D(c) & Part~V; at T-P6-2's
grade \\
\addlinespace
QGP custody & Bulk color mobility; integer readout preserved
& INT + D & Part~V; the null record is data \\
\addlinespace
Twin competition & $m_n - m_p$ = toll $(+2.5)$ + EM $(-1.0)$ &
D & Part~VIII; lattice-cited; inputs at \ANine{}'s grade \\
\addlinespace
Gate descent & $\beta$-decay as \GRIPin{} transit; lifetime
walk $914 \to \sim878$~s & D(c) & Part~VIII; standard EW
computation cited \\
\addlinespace
Sanctuary condition & Binding closes the gate; bound-$n$
stability & D & Part~VIII \\
\addlinespace
Ladder grammar & Lifetime = price of cheapest open descent;
labels decide & D + INT & Part~IX; widths are data \\
\addlinespace
Six flavors & $N = 3$ (import) doubled & D(c) & Part~IX;
\PaperPTwo{}'s fences \\
\addlinespace
Structural selection & $\Delta B = \pm1$ minting nonexistent;
leptogenesis forced & D(c) & Part~X; at T-P6-2's grade \\
\addlinespace
Birth chain & Resonant TeV leptogenesis $\to$ door $\to$
$\eta_B$ & D(c) & Part~X; \PaperPFive{}'s named input +
conjecture \\
\addlinespace
Door partition & $B = \tfrac{28}{79}(B{-}L)$ & D & Part~X;
standard, cited \\
\addlinespace
Arisaka equation for the proton & Locked currents; $\QB$ per
rearrangement; ledger's-page reading & CAND + INT & Part~XI;
\ASeven{}'s grade; F-P6-6 first \\
\end{longtable}
}

\section{The empirical anchor table}
\label{sec:anchor-table}

Every measured value quoted in the paper, with its source. All
arithmetic performed on these values was verified by machine at
drafting; the final digit-for-digit collation of each entry
against the current PDG print is the standing pre-release
checklist item for this paper's next revision.

{\small
\begin{longtable}{p{0.33\textwidth}p{0.28\textwidth}p{0.27\textwidth}}
\caption[The empirical anchor table.]{The empirical anchors of
\PaperPSix{}. Every empirical number this paper leans on, with the value exactly as quoted and the source it is quoted from. Nothing here is derived and nothing is rounded to suit an argument; where a bound rather than a value is what exists, the bound is what appears. The table is the paper's exposure to measurement, gathered in one place so that it can be checked against the literature in a single pass.}
\label{tab:anchor-table}\\
\toprule
\textbf{Quantity} & \textbf{Value as quoted} & \textbf{Source} \\
\midrule
\endfirsthead
\multicolumn{3}{l}{\small\emph{The anchor table, continued.}}\\
\toprule
\textbf{Quantity} & \textbf{Value as quoted} & \textbf{Source} \\
\midrule
\endhead
\midrule
\multicolumn{3}{r}{\small\emph{continued overleaf}}\\
\endfoot
\bottomrule
\endlastfoot
$m_p$ & $938.272\,088$~MeV, zero width & \citep{PDG2024} \\
$m_n - m_p$ & $1.293$~MeV & \citep{PDG2024} \\
$\mu_p$ & $2.79\,\mu_N$ (first structure, 1933) &
\citep{FrischStern1933, PDG2024} \\
Hydrogen neutrality & $\lesssim 10^{-21}\,e$ &
\citep{Bressi2011} \\
Antihydrogen charge & $< 0.71$~ppb & \citep{Ahmadi2016} \\
$\bar p/p$ charge-to-mass & 16~ppt agreement &
\citep{Borchert2022} \\
E\"otv\"os parameter $\eta(\mathrm{Be},\mathrm{Ti})$ &
$(0.3 \pm 1.8)\times10^{-13}$ & \citep{Wagner2012} \\
$\tau/B(p \to e^+\pi^0)$ & $> 2.4\times10^{34}$~yr (90\% CL) &
\citep{Takenaka2020} \\
Super-K exposure & $0.37$~Mton\,yr &
\citep{SuperKamiokande2024} \\
$\tau_{n\bar n}$ (free) & $> 0.86\times10^{8}$~s &
\citep{BaldoCeolin1994} \\
$\tau_n$ (bottle) & $877.75 \pm 0.28$~s & \citep{UCNtau2021} \\
$\tau_n$ (beam) & $887.7 \pm 2.2$~s & \citep{Yue2013} \\
$m_u,\ m_d$ ($\overline{\mathrm{MS}}$) & $2.16$, $4.67$~MeV &
\citep{PDG2024} \\
$m_b,\ m_c$ (thresholds) & $4.18$, $1.27$~GeV &
\citep{PDG2024} \\
$\alpha_s(M_Z)$ (world) & $0.1180(9)$ & \citep{PDG2024} \\
$\Lambda^{(3)}_{\overline{\mathrm{MS}}}$ (4-loop, lattice-matched) &
$\approx 340$~MeV & \citep{PDG2024} \\
String breaking $r_b$ & $\approx 1.24$~fm & \citep{Bali2005} \\
$0^{++}$ glueball (pure gauge) & $1.71$~GeV &
\citep{MorningstarPeardon1999} \\
$T_c$ (pure $\mathrm{SU}(3)$, first order) & $\approx 270$~MeV &
\citep{Greensite2011} \\
$\sigma/\Lambda^2_{\overline{\mathrm{MS}}}$ (target, D-P6-4(c)) &
$1.67$ & arithmetic (\S\ref{sec:sigma-number}) \\
$\Delta\Sigma$ & $\approx 0.25$--$0.35$ & \citep{Ashman1988,
deFlorian2014} \\
String tension $\sigma$ & $(440~\mathrm{MeV})^2$ &
\citep{Bali2001} \\
Regge slope $\alpha'$ & $\approx 0.9$~GeV$^{-2}$ &
\citep{Bali2001} \\
$T_c$ (crossover) & $156.5 \pm 1.5$~MeV & \citep{HotQCD2019} \\
$n$--$p$ splitting (lattice QCD+QED) & $+2.52$ (quark), $-1.00$
(EM)~MeV & \citep{Borsanyi2015} \\
$g_A$; $|V_{ud}|$; $G_F$ & $1.2756$; $0.9737$;
$1.166\times10^{-5}$~GeV$^{-2}$ & \citep{PDG2024} \\
$B$(deuteron) & $2.224$~MeV & \citep{PDG2024} \\
$\Delta(1232)$: mass, width & $1232$, $117$~MeV &
\citep{PDG2024} \\
$\Omega^-$: lifetime & $0.8\times10^{-10}$~s &
\citep{PDG2024, Barnes1964} \\
$\eta_B$ & $6.1\times10^{-10}$ & \citep{Planck2020} \\
$k_B T_{300}\ln 2$ (Landauer) & $0.0179$~eV & arithmetic \\
ATP hydrolysis & $\approx 0.32$~eV & standard biochemistry;
carried as calibration only \\
\end{longtable}
}

The appendices follow: the transmutation arithmetic
(A, for Part~III), the spin ledger (B, Part~IV), the string
(C, Part~V), the lock (D, Part~VI), the neutron (E, Part~VIII),
and the door (F, Part~X).

\appendix

\section{The transmutation arithmetic, step by step}
\label{app:transmutation}

This appendix complements \S\ref{sec:depth}: every number quoted
there, derived at the blackboard pace. A reader who works through
these two pages owns the mechanism that prices $99\%$ of their own
body weight.

\subsection{Solving the one-loop equation}

Start from \eqref{eq:one-loop-rge},
$\mu\,d\alpha_s/d\mu = -(b_0/2\pi)\,\alpha_s^2$. Separate and
integrate between $\Lambda$ and $\mu$:
\begin{equation}
\int \frac{d\alpha_s}{\alpha_s^{2}} = -\frac{b_0}{2\pi}
\int d\ln\mu
\;\;\Longrightarrow\;\;
\frac{1}{\alpha_s(\mu)} = \frac{b_0}{2\pi}\,
\ln\frac{\mu}{\Lambda},
\label{eq:app-integrated}
\end{equation}
where the integration constant has been \emph{named}: $\Lambda$ is
defined as the scale at which the inverse coupling would vanish.
Inverting gives both forms of \eqref{eq:lambda-def}. Two
observations worth a margin note. First,
\eqref{eq:app-integrated} says the inverse coupling is
\emph{linear in} $\ln\mu$ --- the straight lines on every
unification plot, including the one in Part~I's inheritance map.
Second, $d\Lambda/d\mu = 0$ by construction: differentiate
$\Lambda = \mu\,e^{-2\pi/b_0\alpha_s(\mu)}$ and use
\eqref{eq:one-loop-rge}; the explicit $\mu$-dependence and the
implicit one through $\alpha_s$ cancel identically. That
cancellation is what ``renormalization-group invariant'' means,
and it is why $\Lambda$ is physics rather than convention.

\subsection{The walk, digit by digit}

Input: the \ANine{} registration
$\alpha_s(M_Z) = 1/(\pi e) = 0.11710$ at $M_Z = 91.188$~GeV;
thresholds at $m_b = 4.18$~GeV, $m_c = 1.27$~GeV
\citep{PDG2024}; $b_0(n_f) = 11 - \tfrac{2}{3}n_f$.

\emph{Step 1 ($n_f = 5$, $b_0 = 23/3$).} The exponent is
\[
\frac{2\pi}{b_0\,\alpha_s(M_Z)}
= \frac{6.2832}{7.6667 \times 0.11710} = 6.999,
\]
so
$\Lambda^{(5)} = 91.188\,e^{-6.999}~\mathrm{GeV}
= 91.188 \times 9.13\times10^{-4} = 83.3~\mathrm{MeV}$.
Note what happened: the input contained nothing small, and the
output is three orders below the seam. The hierarchy is the
exponential's doing --- no dial exists anywhere in this step.

\emph{Step 2 (cross $m_b$; $n_f = 4$, $b_0 = 25/3$).} First run
to the threshold within $n_f = 5$:
$\alpha_s(m_b) = 2\pi/[7.6667 \ln(4.18/0.0833)] = 0.2093$. Then
re-solve for the four-flavor invariant:
%% v15.0: displayed, as Step 1's exponent already is.  Held inline
%% on Letter this line ran 71.7 pt past the right margin: the
%% exponential is a superscript and a superscript admits no line
%% break at all, so TeX had nowhere to break it.  Not one digit
%% changed; the arithmetic now sets the way Step 1's does.
\[
\Lambda^{(4)} = 4.18\, e^{-6.2832/(8.3333\times0.2093)}
= 4.18\,e^{-3.602} = 113.9~\mathrm{MeV}.
\]
The invariant
\emph{jumps} at the threshold --- each $n_f$ band has its own
$\Lambda$; continuity is in $\alpha_s$, not in $\Lambda$.

\emph{Step 3 (cross $m_c$; $n_f = 3$, $b_0 = 9$).}
$\alpha_s(m_c) = 2\pi/[8.3333 \ln(1.27/0.1139)] = 0.3128$;
%% v15.0: displayed, for the same reason as Step 2 and to keep the
%% three steps set the same way.
\[
\Lambda^{(3)} = 1.27\, e^{-6.2832/(9\times0.3128)}
= 1.27\,e^{-2.232} = 136~\mathrm{MeV}.
\]
By here the coupling is
$0.31$ and climbing fast; one more decade of descent and
perturbation theory is over --- which is not a failure but the
arrival: the strong regime where the basin lives.

\subsection{What one loop misses, stated plainly}

The walk above is pedagogy-grade transport, and it
\emph{understates} $\Lambda$: the two-loop coefficient is large
and positive in its effect on the invariant, and the world's
four-loop, lattice-matched determination is
$\Lambda^{(3)}_{\overline{\mathrm{MS}}} \approx 340$~MeV
\citep{PDG2024} --- roughly $2.5$ times the one-loop value, in a
quantity that lives inside an exponential. The lesson to carry:
the \emph{mechanism} (a scale from no scale, hierarchy from an
exponent) is exact and loop-order-independent; the \emph{digits}
belong to high-order transport plus the lattice, and this paper
quotes them from there. Precision claims about
\eqref{eq:pie-registration} itself are \ANine{}'s to defend, at
\ANine{}'s declared register, with the QD completion carrying
the residual; nothing in T-P6-3 leans on them.

\subsection{The pure number, and the toll ledger}

The lattice computes, from the coupling and quark masses alone,
the full light-hadron spectrum to percent accuracy
\citep{Durr2008}; in invariant units this fixes
$m_p / \Lambda^{(3)}_{\overline{\mathrm{MS}}} \approx
938/340 \approx 2.8$ --- the pure number of
\eqref{eq:pure-number}, nature's own, no dials. The Higgs toll
ledger, for completeness (PDG $\overline{\mathrm{MS}}$ current
masses): $m_u = 2.16$~MeV, $m_d = 4.67$~MeV, so
$2m_u + m_d = 8.99$~MeV $= 0.96\%$ of $m_p$; the neutron's
extra down-quark swaps this to $m_u + 2m_d = 11.50$~MeV, a
$+2.51$~MeV valence-toll difference against which the proton's
larger electromagnetic self-energy competes --- the $1.293$~MeV
residual of that competition is Part~VIII's opening number.
Weigh yourself, divide by one hundred: that is what the Higgs
pays through the valence masses; its full sigma-term response is
a few kilograms more (\S\ref{ssec:tolls}). The rest is the
storm, settled in its basin.

\subsection{The closure arithmetic, digit by digit}

Companion to \S\ref{ssec:closure}: the five-percent prediction,
at blackboard pace. The sensitivity first. Hold $\mu$ fixed in
$\Lambda = \mu\, e^{-2\pi/b_0\alpha_s}$ and differentiate the
logarithm:
\[
\frac{\partial \ln \Lambda}{\partial \ln \alpha_s}
= \alpha_s \cdot \frac{\partial}{\partial \alpha_s}
\left( -\frac{2\pi}{b_0 \alpha_s} \right)
= \frac{2\pi}{b_0 \alpha_s}
= \frac{6.2832}{7.6667 \times 0.1180} = 6.95
\]
at the $M_Z$ seam ($n_f = 5$). The input deviation is
$\delta\alpha_s/\alpha_s = (0.11710 - 0.1180)/0.1180 = -0.763\%$;
linearized, $\delta\Lambda/\Lambda = 6.95 \times (-0.763\%) =
-5.30\%$. The linearization overstates the shift, because the
walk re-matches at each threshold and $b_0$ grows as flavors
decouple ($23/3 \to 25/3 \to 9$), shrinking the downstream
sensitivity. Re-running the entire one-loop walk of
\S\ref{ssec:the-walk} with $\alpha_s(M_Z) = 0.11710$ in place of
$0.1180$ gives $\Lambda^{(3)} = 136.2$~MeV against $142.5$~MeV:
a ratio of $0.9555$, i.e.\ $-4.45\%$. Hence
\[
m_p^{\mathrm{pred}} = 938.272 \times 0.9555 = 896.5~\mathrm{MeV}
\quad (\text{linearized bound: } 888.6~\mathrm{MeV}),
\]
the five-percent closure of \eqref{eq:mp-prediction}. Two more
identities the reader can check on one sheet. The pure number
factorizes through the string tension: with $\sqrt{\sigma} =
440$~MeV and $\Lambda^{(3)}_{\overline{\mathrm{MS}}} = 340$~MeV,
\[
\frac{m_p}{\sqrt{\sigma}} = \frac{938.3}{440} = 2.13,
\qquad
\frac{\sigma}{\Lambda^2} = \left(\frac{440}{340}\right)^{2}
= 1.675,
\qquad
2.13 \times \sqrt{1.675} = 2.76 = \frac{938}{340},
\]
which is \eqref{eq:pure-number-factorised}. And the sigma
ledger of \S\ref{ssec:tolls}: $43.7 + 28.6 = 72.3$~MeV $=
7.7\%$ of $m_p$ --- the proton's full Higgs response, roughly
eight times the valence tolls. Every number in this subsection
is machine-verified in the paper's companion suite.

\section{The spin ledger arithmetic}
\label{app:spin-ledger}

Companion to Part~\ref{part:spin}: the bookkeeping behind
\S\ref{sec:spin-ledgers}, at blackboard pace.

\subsection{The two sum rules, and where their entries live}

The Jaffe--Manohar ledger \eqref{eq:jaffe-manohar} is built in
the infinite-momentum frame with light-cone gauge $A^+ = 0$:
there, and only there, ``gluon spin'' $\Delta G$ is the first
moment of a measurable polarized density, which is why the
polarized-scattering program quotes it. The Ji ledger
\eqref{eq:ji} trades manifest partonic meaning for gauge
invariance: $J_q$ and $J_g$ are matrix elements of the
gauge-invariant angular-momentum tensor, accessible through the
second moments of generalized parton distributions,
$J_q = \tfrac12\int dx\,x\,[H_q(x,0,0) + E_q(x,0,0)]$. The two
orbital entries differ by the \emph{potential angular momentum}
$L_q^{\mathrm{JM}} - L_q^{\mathrm{Ji}}$ --- the field-torque term
--- which is calculable and generically non-zero. Neither ledger
is wrong; they answer differently-posed questions, and the
difference is itself a well-defined physical quantity. That is
what ``the parts are convention'' means in practice: not
vagueness, but precision that refuses to be unique.

\subsection{The anomaly shift, priced}

The singlet axial charge measured in polarized DIS is
\eqref{eq:anomaly-shift}:
$a_0 = \Delta\Sigma - (3\alpha_s/2\pi)\,\Delta G$. At
$\alpha_s = 0.35$ (a few GeV$^2$),
$3\alpha_s/2\pi = 3(0.35)/6.283 = 0.167$; with
$\Delta G \approx 0.4$ the shift is
$0.167 \times 0.4 = 0.067$ --- comparable to the entire
experimental uncertainty band of \eqref{eq:deltasigma-today}.
The measured ``quark spin'' is thus, at the anomaly level,
partly gluonic: even the ledger's cleanest-looking line is a
mixture under the hood. (Historically this term powered a
decade of attempts to blame the whole EMC result on $\Delta G$;
the measured gluon polarization \citep{deFlorian2014} is too
small for that rescue, and the modern resolution is the one in
the main text: no rescue was needed, because nothing was
missing.)

\subsection{The remainder, with today's central values}

Take $\Delta\Sigma = 0.30$ and the RHIC-era
$\Delta G \approx 0.2$ over the measured range: the
Jaffe--Manohar ledger then assigns
$L_q + L_g = \tfrac12 - \tfrac12(0.30) - 0.2 = 0.15$ --- a third
of the proton's spin in orbital motion, awaiting the EIC's
direct tomography. The reader is invited to update this line
when the EIC numbers arrive; the framework's pre-registered
expectation (Part~\ref{part:spin}) is that the entries will
move, the total will not, and no scheme will become canonical.

\section{The string arithmetic}
\label{app:string}

Companion to Part~\ref{part:confinement}: five numbers a
student should compute once by hand.

\subsection{Sixteen tonnes}

The string tension \eqref{eq:string-tension} as a force:
$\sigma = (0.440~\mathrm{GeV})^2 = 0.194~\mathrm{GeV}^2$;
multiplying by the conversion $1~\mathrm{GeV} =
5.068~\mathrm{fm}^{-1}$ gives
$\sigma = 0.981~\mathrm{GeV/fm}$. In SI units,
\[
F \;=\; \frac{0.981 \times 1.602\times10^{-10}~\mathrm{J}}
{10^{-15}~\mathrm{m}}
\;=\; 1.57\times10^{5}~\mathrm{N}
\;=\; 16.0~\text{tonnes-force},
\]
distance-independent. Sixteen tonnes between two points too
small to have ever shown a size, at any separation, forever.
Committing this number to memory is the fastest known cure for
underestimating the strong interaction.

\subsection{The Regge cross-check}

The rotating-string model prices the tension independently: a
straight string of tension $\sigma$ spinning at its ends'
light-speed limit satisfies $J = \alpha' M^2 + \text{const}$
with slope $\alpha' = 1/(2\pi\sigma)$. The measured Regge slope
of the light-meson trajectories, $\alpha' \approx
0.9~\mathrm{GeV}^{-2}$, inverts to
$\sigma = 1/(2\pi \times 0.9) = 0.177~\mathrm{GeV}^2 =
(421~\mathrm{MeV})^2$ --- within five percent of the lattice
value \eqref{eq:string-tension} from a one-line classical model.
The hadrons have been telling us they are strings-with-endpoints
since the 1960s; the trajectories were the first flux-tube
evidence, decades before the lattice drew the tube.

\subsection{Where the string surrenders}

The break-point: budding a light quark--antiquark pair costs
roughly twice a constituent-quark dressing,
$2 \times 0.48~\mathrm{GeV} \approx 0.96$~GeV, so
$r_{\mathrm{break}} \approx 0.96/0.981 \approx 1$~fm at the
crude estimate --- and the lattice's directly-observed breaking
sits at $r_b \approx 1.24$~fm, where the stored energy reads
$\sigma r_b \approx 1.22$~GeV \citep{Bali2005}. Beyond one femtometer the vacuum finds it cheaper
to manufacture matter than to stretch geometry: every ``jet''
in every collider event display is this arithmetic, iterated
down a cascade. The cage does not leak; it franchises.

\subsection{The crossover radius}

Where does the string take over from the Coulomb term? Set the
magnitudes of the two terms of \eqref{eq:cornell} equal:
$r_c = \sqrt{(4/3)\,\alpha_s\,\hbar c/\sigma}$; with
$\alpha_s \approx 0.3$ this gives $r_c \approx 0.28$~fm ---
and by $1$~fm the linear term outweighs the Coulomb term six to
one. The proton's interior straddles this radius, which is why
neither term alone describes it.

\subsection{The L\"uscher correction, and what it counts}

The universal string correction \eqref{eq:luscher} evaluates to
$-\pi/12 \times 197.3~\mathrm{MeV\,fm}/r$: $-103.3$~MeV at
$0.5$~fm, $-51.7$~MeV at $1$~fm, $-34.4$~MeV at $1.5$~fm. Its
coefficient is $-(D_\perp)\pi/24$ with $D_\perp$ the number of
massless transverse modes: one additional spatial direction at
the QCD scale would give $-3\pi/24 = -\pi/8$ --- a fifty
percent shift, $-77.5$~MeV at $1$~fm --- which lattice string
spectroscopy excludes. This arithmetic is the entire engine of
the Goldstone-counting test (\S\ref{sec:goldstone-test}): the
QCD string, vibrating, counts the spatial dimensions and finds
exactly three.

\section{The lock arithmetic}
\label{app:lock-arithmetic}

Companion to Part~\ref{part:stability}: the exact computations
behind the obstruction, the operator audit, the anomaly ledger,
and the door's price. Everything here is finite arithmetic over
the rationals; the reader is warmly encouraged to redo all of it
on one sheet of paper.

\subsection{The obstruction, in the general per-irrep form}

The text proved Theorem~\ref{thm:obstruction} inside the
$\mathbf{10}$; here is the bookkeeping with both constants live.
Demand $B = \alpha_{r} + \beta Y$ with $\alpha_{r}$ free per
irrep $r \in \{\overline{\mathbf 5}, \mathbf{10}\}$:
\[
\begin{array}{llll}
\mathbf{10}: & \alpha_{10} + \tfrac{\beta}{6} = \tfrac13\;(Q),
& \alpha_{10} - \tfrac{2\beta}{3} = -\tfrac13\;(u^c),
& \alpha_{10} + \beta = 0\;(e^c), \\[3pt]
\overline{\mathbf 5}: & \alpha_{\bar5} + \tfrac{\beta}{3} =
-\tfrac13\;(d^c),
& \alpha_{\bar5} - \tfrac{\beta}{2} = 0\;(L). &
\end{array}
\]
The $\mathbf{10}$ block alone is overdetermined and
inconsistent ($\beta = \tfrac45$, $\alpha_{10} = \tfrac15$,
then $\tfrac15 + \tfrac45 = 1 \ne 0$), so the system fails
before the $\overline{\mathbf 5}$ is even consulted. By
contrast, targeting $B - L$ (values $\tfrac13, -\tfrac13, -1$
on $Q, u^c, e^c$ and $-\tfrac13, +1$ on $d^c, L$... with the
lepton-number sign conventions of the text) the same five
equations \emph{are} consistent, with the unique solution
\[
B - L = \tfrac15\left(X + 4Y\right), \qquad
X(\mathbf{10}) = +1, \quad X(\overline{\mathbf 5}) = -3
\]
--- the $\mathrm{SO}(10)$ spinor's branching charge, recovered
here as the \emph{only} Abelian direction of this form. The
moral in one line: the solvable charge is the wrong one, and
the right charge is not solvable.

\subsection{The decay-operator audit}

The four classic dimension-six operators, with all fields
incoming left-handed Weyl (a daggered field contributes the
negative of its charge):

\begin{table}[htbp]
\centering
\caption[Lowest-dimension B-violating operators.]{The lowest-dimension baryon-number-violating operators, with the baryon number each carries and the value of baryon number minus lepton number. The right column is the point: every one of them has that combination equal to zero. These are the operators a proton decay would have to use, and the Part's argument is about why this geometry does not supply them.}
\label{tab:bl-operators}
\small
\begin{tabular}{lcc}
\toprule
operator & $B$ & $B - L$ \\
\midrule
$Q\,Q\,Q\,L$ & $+1$ & $0$ \\
$u^c\,u^c\,d^c\,e^c$ & $-1$ & $0$ \\
$Q\,Q\,(u^c)^\dagger (e^c)^\dagger$ & $+1$ & $0$ \\
$Q\,L\,(u^c)^\dagger (d^c)^\dagger$ & $+1$ & $0$ \\
\bottomrule
\end{tabular}
\end{table}

\noindent Every row: maximally $B$-charged, exactly
$B{-}L$-neutral. A gauged $B - L$ --- however exact --- forbids
none of them; the winding charge of T-P6-2 forbids all of them.
This table is why H2 had to produce genuine $B$, and why
Theorem~\ref{thm:obstruction}'s second sentence is
load-bearing rather than decorative.

\subsection{The anomaly sums, term by term}

Per family, left-handed Weyl content with multiplicities
(color $\times$ weak): $Q$: $6$, $u^c$: $3$, $d^c$: $3$, $L$:
$2$, $e^c$: $1$. Dynkin index $T(\text{fund}) = \tfrac12$.

\emph{$[\SUthreeC]^2\,\UoneB$:} colored fields only, weighted
by weak multiplicity:
\[
\underbrace{2 \cdot \tfrac12 \cdot \tfrac13}_{Q}
+ \underbrace{\tfrac12 \cdot (-\tfrac13)}_{u^c}
+ \underbrace{\tfrac12 \cdot (-\tfrac13)}_{d^c}
= \tfrac13 - \tfrac16 - \tfrac16 = 0 .
\]

\emph{$[\SUtwoL]^2\,\UoneB$:} doublets only, weighted by
color:
$\underbrace{3 \cdot \tfrac12 \cdot \tfrac13}_{Q}
+ \underbrace{1 \cdot \tfrac12 \cdot 0}_{L}
= \tfrac12$. \textbf{The door.}

\emph{$[\UoneY]^2\,\UoneB$:}
$6 \cdot (\tfrac16)^2 \cdot \tfrac13
+ 3 \cdot (\tfrac23)^2 \cdot (-\tfrac13)
+ 3 \cdot (\tfrac13)^2 \cdot (-\tfrac13)
= \tfrac1{18} - \tfrac49 - \tfrac19 = -\tfrac12$.
(Green--Schwarz-absorbed; no Abelian instantons, hence inert.)

\emph{$[\UoneB]^2\,\UoneY$:}
$6 \cdot (\tfrac13)^2 \cdot \tfrac16
+ 3 \cdot (\tfrac13)^2 \cdot (-\tfrac23)
+ 3 \cdot (\tfrac13)^2 \cdot \tfrac13
= \tfrac19 - \tfrac29 + \tfrac19 = 0$.

\emph{$[\UoneB]^3$:}
$6 \cdot \tfrac1{27} - 3 \cdot \tfrac1{27} - 3 \cdot \tfrac1{27}
= 0$. \quad
\emph{$\UoneB$--grav$^2$:}
$6 \cdot \tfrac13 - 3 \cdot \tfrac13 - 3 \cdot \tfrac13 = 0$.

Four zeros without assistance; two halves, equal and opposite,
for the inflow to absorb. The reader who has just done these six
sums has personally verified the cleanest half of the stability
theorem.

\subsection{The price of the door}

The topological vertex changes $B$ by $N_{\mathrm{gen}}\,n = 3n$
and carries the tunneling weight $e^{-16\pi^2/g_2^2} =
e^{-4\pi/\alpha_W}$. Evaluating at the electroweak scale with
$\alpha^{-1}(M_Z) = 127.93$ and $\sin^2\theta_W = 0.2315$:
\[
\alpha_W = \frac{\alpha}{\sin^2\theta_W} = \frac{1}{29.6},
\qquad
\frac{4\pi}{\alpha_W} = 372,
\qquad
e^{-372} \approx 10^{-162}.
\]
For calibration: the universe is $\sim 4 \times 10^{17}$
seconds old and Super-Kamiokande watches $\sim 10^{34}$
protons; a rate bearing the factor $10^{-162}$ loses to that
exposure by more than a hundred orders of magnitude. At
temperatures above the electroweak barrier the exponential is
absent altogether --- sphaleron transitions are unsuppressed
--- which is the regime Part~X inhabits. One exponent,
evaluated twice, separates the frozen present from the fertile
first picosecond.

\subsection{The integrality pin, named}

Pin R2 --- the quantization of the holonomy parameter $c$ to
integers of period six, on which the uniqueness scan of
\S\ref{sec:z3lock} runs --- is not free-standing: the cohort
already owns an integrality apparatus. The Standard-Model
paper's quantum-deviation rationals are, in its own words, the
\emph{Hopkins--Singer descent integrality coefficients} on the
relevant boundary strata \citep{ArisakaA9SM}: descent data in
this framework is integrally quantized, not continuously
adjustable. Applied to the present case, the Wilson-line
parameter is a descent coefficient on the compact cycle, and
integrality against the spectrum's smallest hypercharge
($\tfrac16$, the quark doublet's) quantizes it to the
six-point lattice the scan walks. What remains of D-P6-2 after
this naming is the $X_4$-level derivation --- exhibiting the
holonomy's class in the same integral-cohomology bookkeeping
that prices the QD rationals --- which is bounded, technical,
and owed; what is no longer open is \emph{whether} the
parameter is quantized, or by what clause.

\section{The neutron arithmetic}
\label{app:neutron-arithmetic}

Companion to Part~\ref{part:neutron}: the splitting, the
lifetime, and the sanctuary condition, each at blackboard pace.

\subsection{The splitting, decomposed}

\begin{table}[htbp]
\centering
\caption[The neutron-proton mass difference, term by term.]{The neutron--proton mass difference, contribution by contribution, with the source of every number named. The valence toll difference and the full lattice quark-mass part agree inside their errors; the electromagnetic part pulls the other way; and the sum sits on the measured splitting within the quoted uncertainty. The table is set out this way because the cancellation between the two large terms is what makes the small answer worth anything.}
\label{tab:np-mass-split}
\small
\begin{tabular}{lcc}
\toprule
contribution & value (MeV) & source \\
\midrule
valence toll difference $m_d - m_u$ & $+2.51$ & PDG current
masses \citep{PDG2024} \\
full QCD (quark-mass) part & $+2.52 \pm 0.29$ & lattice
\citep{Borsanyi2015} \\
electromagnetic part & $-1.00 \pm 0.16$ & lattice+QED
\citep{Borsanyi2015} \\
\midrule
sum & $+1.5 \pm 0.3$ & \\
measured $m_n - m_p$ & $+1.293$ & \citep{PDG2024} \\
\bottomrule
\end{tabular}
\end{table}

\noindent Two remarks. The agreement of rows one and two ---
naive current-mass difference against the full nonperturbative
computation --- is leading-order behavior a student should
expect but never assume: in a \emph{difference} of isospin
twins, the storm's contribution cancels almost perfectly, and
what survives is the toll ledger. And the two effects' opposite
signs are structural, not accidental: the toll term must favor
a heavier neutron (the $d$ costs more), the electromagnetic
term must favor a heavier proton (only it is charged); the
window for chemistry exists because two unrelated pieces of the
codebook straddle zero.

\subsection{The lifetime, walked}

The tree-level rate for $n \to p\, e^-\bar\nu_e$:
\begin{equation}
\Gamma \;=\; \frac{G_F^2\, |V_{ud}|^2\, m_e^5\,
\bigl(1 + 3 g_A^2\bigr)\, f}{2\pi^3},
\label{eq:neutron-rate}
\end{equation}
with inputs, all measured elsewhere and none adjustable here:
$G_F = 1.166\times10^{-5}~\mathrm{GeV}^{-2}$ (muon decay);
$|V_{ud}| = 0.9737$ (superallowed nuclear $\beta$-decays);
$g_A = 1.2756$ (neutron decay correlations); and
$f = 1.6887$, the dimensionless three-body phase-space
integral over the $Q = 0.782$~MeV doorway --- the number that
carries Sargent's fifth power. Assemble:
$(1 + 3g_A^2) = 5.882$;
\[
\Gamma = \frac{(1.166\times10^{-5})^2 (0.9737)^2
(5.110\times10^{-4})^5 (5.882)(1.6887)}{2\pi^3}
= 7.20\times10^{-28}~\mathrm{GeV},
\]
and with $\hbar = 6.582\times10^{-25}$~GeV\,s,
\[
\tau_{\mathrm{tree}} = \frac{\hbar}{\Gamma} = 914~\mathrm{s}.
\]
Electroweak radiative corrections and recoil effects, computed
to high precision in the literature, reduce this by roughly
four percent: $\tau \approx 878$~s, against the bottle's
$877.75 \pm 0.28$ \citep{UCNtau2021}. For the Sargent
intuition, compare the muon: same gate, $Q = 105.7$~MeV; the
phase-space ratio $(105.7/0.782)^5 \approx 4\times10^{10}$
accounts, by itself, for the leap from the neutron's minutes
to the muon's microseconds. The gate has one price list;
the doorways differ.

\subsection{The sanctuary condition}

A neutron bound in a nucleus $(A, Z)$ can only decay if the
books balance:
\[
Q_{\mathrm{eff}} \;=\;
\bigl[M(A, Z) - M(A, Z{+}1)\bigr] - m_e \;>\; 0 ,
\]
the parent--daughter mass difference having already absorbed
every binding-energy shift. In the deuteron:
the descent would break the bond ($2.224$~MeV) against a
doorway of only $0.782$~MeV --- closed, by
$1.44$~MeV of margin. Every stable isotope is a solution of
this inequality's failure, which is why the valley of nuclear
stability is a \emph{valley}: its walls are drawn by where the
gate's books stop balancing. The reader who wants the framework
in one sentence here: the gate force never changed; the
landscape under it did.

\section{The door bookkeeping}
\label{app:door-bookkeeping}

Companion to Part~\ref{part:birth}: the catastrophe, and the
exchange rate of the door.

\subsection{The annihilation catastrophe, sized}

In a matter-symmetric universe, nucleons and antinucleons track
their equilibrium abundance until the annihilation rate falls
behind the expansion rate. The standard freeze-out arithmetic
\citep{KolbTurner1990} --- annihilation cross-section of
hadronic size, freeze-out near $T \approx 20$~MeV --- leaves a
relic of
\[
\frac{n_B}{n_\gamma}\Big|_{\mathrm{symmetric}}
\;\sim\; 10^{-18},
\]
against the measured $6.1\times10^{-10}$
\citep{Planck2020}: a shortfall of nearly nine orders of
magnitude, \emph{and} no mechanism for the survivors to be
matter rather than a matter--antimatter patchwork. The
catastrophe is the cleanest statement of why Q10 needs an
asymmetry written \emph{before} the annihilation, not a
survival strategy during it.

\subsection{The exchange rate: where 28/79 comes from}

Above the electroweak scale, with sphalerons and every
Standard-Model Yukawa and gauge interaction in equilibrium, the
species' chemical potentials are not independent: each fast
process imposes one linear relation. Solving the full set for
$N_f$ families and $N_H$ Higgs doublets --- the classic
bookkeeping of \citet{HarveyTurner1990} --- expresses the
frozen-out baryon number in terms of the conserved $B - L$:
\[
B \;=\; \frac{8 N_f + 4 N_H}{22 N_f + 13 N_H}\,(B - L)
\;\xrightarrow{\;N_f = 3,\ N_H = 1\;}\;
\frac{28}{79}\,(B - L).
\]
The structure deserves one sentence of feeling: the numerator
and denominator are census figures --- counts of who holds
weak charge, who holds hypercharge, who couples to whom ---
and the fraction is simply the marketplace's equilibrium
exchange rate between the lepton-sector currency the chain
mints and the baryonic currency the universe keeps. Roughly a
third of the deposited $B{-}L$ emerges as baryons; the rest
remains in the lepton books, where the cosmic neutrino
background holds it still.

\subsection{The chain's landing, cited not claimed}

For the reader assembling the full pipeline: the CPT
companion's chain delivers its lepton asymmetry with one named
input and one named conjecture, the partition above converts
it, and the landing against \eqref{eq:etaB} is stated at the
home paper's honest resolution \citep{ArisakaP5CPT}. This
appendix adds no digits to that statement --- it exists so
that the two standard links this paper \emph{does} vouch for,
the catastrophe and the exchange rate, are on the table with
their arithmetic visible.

\section{The chirality-protection exhibit}
\label{app:chirality-exhibit}

Companion to Part~\ref{part:stability}, narrowing pin R3
(D-P6-5): \emph{why} the twist lifts the scalar and vector
floors while the chiral fermion floors survive with their
fractional winding labels intact --- the mechanism, exhibited;
the quantitative warped profiles remain the debt's outstanding
content.

\subsection{Scalars and vectors: mass by Laplacian}

For a boson on the compact cycle, the four-dimensional mass of
a mode is read directly from the compact Laplacian: a
component whose twisted periodicity shifts its winding lattice
to $w \in t/3 + \mathbb{Z}$ has
$m_w^2 \ni w^2/R_\theta^2$, and a lattice with no $w = 0$
point has no massless floor --- full stop. This is the entire
doublet--triplet mechanism of \S\ref{sec:z3lock} in one
line: the triplet's lattice misses zero by a third of a unit,
the doublet's contains zero, and the $X, Y$ vectors sit with
the triplet. No boundary condition need be engineered per
field; the arithmetic of the lattice does the splitting.

\subsection{Chiral fermions: mass by pairing, not by winding}

A fermion is different in exactly one structural respect, and
everything hangs on it: \emph{a four-dimensional fermion mass
is not an eigenvalue but a marriage}. A Dirac mass term
couples a left-handed mode to a right-handed partner; no
partner, no mass, whatever energies are in play --- the
protection the Standard Model itself uses to keep its fermions
light below the electroweak scale. On the descent's interval,
the chirality projection (Part~II) removes the would-be
right-handed partner of each zero-mode floor at the boundary:
the floor is \emph{structurally single}. The twist now shifts
the floor's $\theta$-dependence to a fractional winding
$e^{i w \theta}$ --- but a phase is not a partner. The
Laplacian eigenvalue $w^2/R_\theta^2$ sits in the
\emph{paired} tower, where it lifts vector-like KK levels
exactly as it lifts scalars; the unpaired chiral floor has no
mass term to receive a contribution, and persists --- carrying
its fractional winding as a conserved label rather than an
energy. That is the asymmetry T-P6-2 leg (ii) uses: the twist
prices what can be priced (bosons, paired towers) and labels
what cannot (chiral floors). The quantitative statement ---
the explicit floor profiles in the warped metric, with the
\GDOSm{} tower's radial functions --- is the remaining
content of D-P6-5, deferred to a later treatment with its home machinery.






%% ===================================================================
%%  PART XV -- COMMON OBJECTIONS AND RESPONSES (new at v1.9)
%%  Format duplicated from A9-SM's Common Objections Part: persona +
%%  quoted objection + indented Response, grouped in clusters.
%%  O-P6-1..12 moved here from the Discussion (content preserved,
%%  register expanded); O-P6-13..16 new at this revision.
%% ===================================================================
%% v15.0: the Contents ran one line long and put `Part XVI.
%% References' alone on a page.  \enlargethispage{2\baselineskip} was
%% tried first, as in P5-CPT, and here it COLLIDED WITH THE FOLIO:
%% the last Contents line came to rest on the page number.  At one
%% \baselineskip the orphan returns.  So the break is moved instead
%% -- Part XV's block (heading, three entries, Acknowledgments) goes
%% to the last Contents page together, which leaves six lines there
%% rather than one and crowds nothing.
\addtocontents{toc}{\protect\GGtocbreak}
\GGPart[Common Objections]{Common Objections and Responses}{The strongest objections a particle theorist, an experimentalist, a lattice practitioner, a string theorist, or a philosopher of science is most likely to raise on first reading --- answered, with pointers into the paper}
\label{part:objections}
\label{sec:objections}

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

A paper that claims to close the proton's oldest questions ---
including a derivation that the proton cannot decay, offered by
the man who built the first detector to watch it decay --- will be
met, rightly, with skepticism from every quarter. This Part
collects seventeen objections, grouped by the background of the
reader most likely to raise each, together with an extended
response. They are collected from the hostile-review tradition of
the cohort, and they are not strawmen: they are the questions the
author has actually been asked, sharpened. Where the honest answer
is ``not yet,'' the response says so and points to the debt
registry (\S\ref{sec:debt-registry}) and the claim-status ledger
(Table~\ref{tab:claim-ledger}), which record exactly which claims
are theorems, which are conditional, and which are readings or
candidates. The test of a framework is not that it deflects every
question, but that it knows which questions it has answered and
which it has not.

%% =====================================================================
%% Cluster A -- The law and the lock
%% =====================================================================

\bigskip
\section{The law and the lock}

\bigskip
\noindent\textbf{Objection 1 (O-P6-1) (the skeptical
experimentalist)}\par\nobreak
\noindent\textbf{``You have not explained proton stability; you
have moved the decay machinery to a bulk where nothing can be
tested.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The opposite, twice. First, the bulk placement is
not a hiding move but a corollary: the same twist that lifts the
$X, Y$ bosons off the boundary is the twist that produces the
observed Higgs sector (Part~VI) --- deny the placement and you
must re-solve doublet--triplet splitting by other means. Second,
and decisively for an experimentalist: the framework is
\emph{more} exposed than its predecessors, not less. A grand
unified theory whose predicted lifetime is excluded retreats in
scale and survives; F-P6-1 forbids decay at every rate, so any
signal, ever, at any exposure, kills this framework outright.
Moving the machinery to the bulk did not remove the theory from
the laboratory; it removed the theory's escape hatch.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 2 (O-P6-2) (the anomaly purist)}\par\nobreak
\noindent\textbf{``Baryon number cannot be gauged; the anomalies
forbid it. This has been settled since the 1970s.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} What the 1970s settled is a statement about a
smaller theory: the anomalies of the \emph{Standard Model's field
content alone} forbid a massless, anomaly-free gauged $B$ in four
dimensions --- and Part~VII's record shows the field drawing
exactly that conclusion, correctly. Part~VI's ledger pays exactly
that bill rather than evading it: the six mixed coefficients are
computed exactly (Appendix~\ref{app:lock-arithmetic}); four vanish;
the two survivors are absorbed by the \BIsix{} Green--Schwarz
inflow (hypothesis H3, at its home paper's grade); and the
Stueckelberg mass removes the massless mediator while keeping the
rigid selection rule exact. The folk impossibility is not
contradicted; it is priced, and the price is six dimensions
rather than new fermions.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 3 (O-P6-3) (the model-building
skeptic)}\par\nobreak
\noindent\textbf{``The $\mathbb{Z}_3$ point is a choice dressed as
a derivation. You put the twist where you needed it.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The scan of \S\ref{sec:z3lock} is the answer in
table form. The candidate holonomies on the hypercharge line are
quantized --- there are finitely many --- and the three demands
that \ANine{}'s spectrum already makes (the Standard-Model group
unbroken; a light doublet; no light triplet) eliminate every
quantized point except the lock and its conjugate, which differ
only by $B \to -B$. The only genuine ``choice'' is the same
discrete boundary data by which the construction has a parity at
all --- and the reflection sector is then fixed by the CPT
companion's theorem T-P5-A, not chosen (\S\ref{sec:dihedral}).
What remains open is quantization itself (pin R2, named to the
descent-integrality apparatus in Appendix~\ref{app:lock-arithmetic}),
and the ledger says so in place.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 4 (O-P6-4) (the orbifold-GUT
veteran)}\par\nobreak
\noindent\textbf{``This is orbifold unification with new
vocabulary. Kawamura did it in 2001.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The lineage is acknowledged and cited where it is
real (\S\ref{sec:uniqueness-contrast}): orbifold unification
taught the field that boundary conditions can do the work of Higgs
sectors, and this construction inherits that insight. The three
structural differences listed there are checkable statements about
the constructions, not marketing: the twist is not a chosen parity
assignment per multiplet but the unique center point of the
existing hypercharge line; the protection is not geographic
suppression of decay operators but an exact winding charge --- a
selection rule, not a distance; and the reflection sector is not
an engineering choice but the CPT companion's theorem. A reader
who checks the three and finds them cosmetic should say so with
the constructions side by side; the paper welcomes exactly that
comparison.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 5 (O-P6-5) (the electroweak
theorist)}\par\nobreak
\noindent\textbf{``Sphalerons violate baryon number, so your
`exact' law is not exact.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Exactness here is an operator statement: every
\emph{local operator} with $\Delta B \notin 3\mathbb{Z}$ vanishes
identically, to all orders. The sphaleron is not an operator but a
topological transition, quantized in units of three, unsuppressed
only above the electroweak crossover and welded shut below it by
the 't~Hooft price $e^{-4\pi/\alpha_W} \sim 10^{-162}$. And the
exception is load-bearing, not embarrassing: Part~X needs
precisely this door open, precisely then --- the door that forbids
one proton's death is the door that permitted all protons'
births. A law with exactly one topological exception that turns
out to be the origin of matter is not a leaky law; it is a
well-designed one.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 6 (O-P6-9) (the GUT model
builder)}\par\nobreak
\noindent\textbf{``Gauged $B{-}L$ would have been simpler and is
standard. Why insist on $B$?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Simpler, standard, and useless for the question.
$B{-}L$ is indeed the combination the algebra permits --- Theorem
T-P6-1 derives exactly that near-miss, with the textbook
$\mathrm{SO}(10)$ charge appearing in the proof --- but every
dimension-six proton-decay operator \emph{conserves} $B{-}L$
(the audit is tabulated in Appendix~\ref{app:lock-arithmetic}).
An exact $B{-}L$ lets every one of them stand. A framework that
stops at $B{-}L$ protects nothing the tank measures; the entire
point of the obstruction theorem is that protecting the proton
requires precisely the charge the algebra refuses, which is why
the discharge had to be topological.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 7 (O-P6-10) (the quantum-gravity
folklorist)}\par\nobreak
\noindent\textbf{``Quantum gravity violates all global symmetries;
your protection cannot survive the Planck seam.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The folk theorem concerns \emph{global}
symmetries --- accidents with no gauge structure behind them ---
and it is precisely why the Standard Model's accidental $B$
inspires no confidence at high scale. The present $B$ is not
global-accidental: it is the rigid residual of a \emph{gauged},
Stueckelberg-massive \UoneB{}, realized topologically as a
winding. Gravity is not known, by any argument in the folk
theorem's family, to violate gauge-descended selection rules;
wormholes do not eat windings. This objection, pressed against the
old accident, was decisive --- Part~VII records it doing its work
for fifty years; pressed against the lock, it dissolves. The
difference between the two cases is exactly what Q7 asked.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 8 (O-P6-13) (the string
theorist)}\par\nobreak
\noindent\textbf{``Intersecting-brane models gauged baryon number
with a Green--Schwarz-massive boson twenty-five years ago. Your
mechanism is theirs, minus the ultraviolet consistency of string
theory.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The convergence is real, and this paper cites it
rather than hiding it (\S\ref{sec:record-tendim};
\citealp{IbanezMarchesanoRabadan2001}): the intersecting-brane
Standard Models do gauge $B$, GS-massive, proton-protecting in
perturbation theory. Three differences carry the weight, and each
is checkable. First, the price: ten dimensions, supersymmetry, and
extended objects there; a six-dimensional bulk and the \BIsix{}
inflow here, with neither. Second, the exactness: brane instantons
generically violate the brane-world $B$ non-perturbatively,
whereas the winding protection here is exact to all orders with
the single topological $3\mathbb{Z}$ door. Third, the status: in
the brane constructions the gauged $B$ is a byproduct of the
stack structure; here it is the founding axiom, and the paper
derives its boundary realization rather than inheriting it. As
for ultraviolet consistency: the framework's anomaly freedom is
exactly the content of \AThree{}, cited as H3 at its stated
grade --- the reader is invited to audit that ledger, not to take
ten dimensions on authority.
\end{objqabody}

%% =====================================================================
%% Cluster B -- The readings and the twin
%% =====================================================================

\bigskip
\section{The readings and the twin}

\bigskip
\noindent\textbf{Objection 9 (O-P6-6) (the lattice
practitioner)}\par\nobreak
\noindent\textbf{``The confinement Part claims credit for QCD's
hardest problem.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} It claims none; its first section is a fence
built to be quoted against it (\S\ref{sec:clay-preamble}), and the
Clay-grade mathematics is named as open in exactly the place a
lattice reader would look. What is claimed at theorem grade is
narrower and different: the free quark's non-existence as a
\emph{boundary state} (C-P6-1) --- a statement about readouts and
windings, not about the mass gap. The measured string tension, the
crossover temperature, and every quantitative fact of confinement
are quoted from the lattice with their citations; the Part adds a
reading of \emph{why} the branch selection goes the way it goes,
graded INT --- a registration now partly paid: the
conditional area-law theorem T-P6-4 stands one named gap
hypothesis from the full claim
(\S\ref{sec:tp64}), and the field's mechanisms are compared
honestly, lattice results first, in
Table~\ref{tab:mechanisms}.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 10 (O-P6-7) (the philosopher of
science)}\par\nobreak
\noindent\textbf{``The spin Part is unfalsifiable
interpretation.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} It registers its own two exposures, which is what
distinguishes a reading from a slogan. The forbidden outcome: a
convention-independent partition of the $\tfrac12$ --- if the
community ever produces a scheme-independent decomposition that
all experiments agree on, the custody reading dies, whereas the
standard picture merely lacks one. And the debt: D-P6-3 requires
the fiber computation to return $j = \tfrac12$ or the reading dies
arithmetically (\S\ref{sec:spin-claims}). An interpretation that
names the observation that would kill it, and the computation that
could kill it, is falsifiable in the only sense that matters.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 11 (O-P6-8) (the Kaluza--Klein
technician)}\par\nobreak
\noindent\textbf{``Why should the zero-mode floor sit at the
minimal winding? Your lock needs that and does not prove
it.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Correct, and declared: the floor is the \GDOSm{}
tower's lowest level (T-GDOS-9.1, consumed at its home paper's
grade), the chirality-protection mechanism is exhibited in
Appendix~\ref{app:chirality-exhibit}, and the explicit warped
profiles are owed as D-P6-5. The dependence is written into
T-P6-2's own statement rather than buried. A reader who rejects
the tower theorem rejects the lock with it --- and the ledger says
so in exactly those words. That is the intended behavior of the
honesty apparatus, not a concession extracted by the objection.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 12 (O-P6-11) (the neutron-lifetime
specialist)}\par\nobreak
\noindent\textbf{``The bottle--beam lifetime discrepancy already
strains your Part VIII.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} It strains nothing here: both mundane resolutions
(a systematic in either method) live entirely within the standard
electroweak computation this paper quotes rather than derives, and
the exotic resolutions are pre-sorted by the winding constraint
(\S\ref{sec:bottle-beam}) into the survivable (invisible channels
conserving the winding) and the fatal (dark decays violating it).
The framework is one of the few onlookers to have stated, in
advance and in print, which resolution would kill it and which
would not. A four-sigma tension whose resolutions are pre-sorted
is an asset to a falsifiable framework, not a threat.
\end{objqabody}

%% =====================================================================
%% Cluster C -- The frame, the record, and the birth
%% =====================================================================

\bigskip
\section{The frame, the record, and the birth}

\bigskip
\noindent\textbf{Objection 13 (O-P6-12) (the referee, on
packaging)}\par\nobreak
\noindent\textbf{``Eleven questions is packaging; several are the
same question.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Guilty, by design, and the charge is the thesis:
the paper's claim from the Preface onward has been that the eleven
are faces of one fact. The count is the map; the fact --- one
basin, one winding, one door --- is the territory. The packaging
dissolves exactly where the physics says it should, and nowhere
else: Q4 and Q5 merge (one mechanism, two faces), Q6 and Q7 merge
(one fact, observational and structural), and no one has accused
Q3 and Q10 of being the same question, because they meet only at
the geometry.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 14 (O-P6-14) (the journal referee, on
the record)}\par\nobreak
\noindent\textbf{``A historical Part in a physics paper is special
pleading. Part VII reads as a priority claim dressed as
scholarship.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Three fences answer this, and they are built into
the Part itself. First, Part~VII derives nothing: its own status
ledger (\S\ref{sec:record-ledger}) records that no claim of the
paper is graded on anything said there, and a reader who skips it
loses no equation. Second, its registers never mix: the external
history is cited to the printed literature, checkable by anyone
with library access, and the framework's own chronology is marked
as dated testimony --- offered as record of when commitments were
made, explicitly not as evidence that they are true. Third, the
history cuts against the paper as much as for it: the record shows
the field declining the axiom three times for reasons the Part
calls \emph{correct}, and it reports the brane-world convergence
that a priority claim would have omitted. A record with those
three properties is the opposite of special pleading; it is the
paper showing its provenance the same way it shows its
arithmetic.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 15 (O-P6-15) (the equivalence-principle
experimentalist)}\par\nobreak
\noindent\textbf{``A Stueckelberg-massive $\mathrm{U}(1)_B$
predicts no fifth force at any accessible range --- so my nulls
can never confirm you, only kill you. Calling the null a
`standing confirmation' is rhetoric.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The asymmetry is real and the response concedes
it cheerfully: F-P6-5 is a falsifier, not a smoking gun, and the
paper's language --- the null ``testifies for'' the structure ---
means exactly what a null can mean: every year the
torsion balances run without a composition-dependent signal, the
massless and light-mediator alternatives that would falsify the
Stueckelberg structure are excluded more deeply. What makes the
relationship more than rhetoric is the history: the test bodies
at the frontier were chosen to maximize the baryonic difference
(\S\ref{sec:record-fifthforce}), so the equivalence-principle
program is, operationally, a dedicated search for the one force
this framework forbids. A theory should be so lucky twice: its
sharpest kill criteria (F-P6-1, F-P6-5) are both experiments that
grow stronger every year they stay silent --- and both communities
existed before the theory did.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 16 (O-P6-16) (the baryogenesis
cosmologist)}\par\nobreak
\noindent\textbf{``TeV-scale resonant leptogenesis is one
mechanism among many. Importing it as `the' origin of matter
overstates a model choice.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The import is graded, and the grading is the
answer. What this paper proves at its own grade is the
\emph{structural selection} (\S\ref{sec:structural-selection}):
in this geometry, $\Delta B = \pm1$ minting does not exist, so
whatever made the baryons had to make them through the
$3\mathbb{Z}$ door from a lepton-sector asymmetry --- leptogenesis
is \emph{forced as a class}, not chosen from a menu. The specific
TeV-scale resonant chain is then imported from the CPT companion
at its declared conditional grade, named input and named
conjecture intact (\S\ref{sec:the-chain}), and the paper's
consistency sentence --- the door open exactly when needed, sealed
exactly after --- holds for the class, not only the instance. If
the companion's chain dies, the class survives; if the class dies
--- if baryogenesis is ever shown to require $\Delta B = \pm 1$
minting --- then F-P6-1's silence and Q10 become mutually
inconsistent, and the framework dies with the contradiction. That
exposure is registered, which is what ``the'' origin story costs
in this cohort.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 17 (O-P6-17) (the lattice
practitioner, on the mass)}\par\nobreak
\noindent\textbf{``You did not derive the proton mass; the
lattice did. Your $\pi e$ is one number; everything downstream
--- the running, the thresholds, the $2.76$ --- is standard
machinery and lattice output. Calling $897$~MeV a prediction is
marketing.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The division of labor is stated in the Part
itself, in exactly your terms (\S\ref{ssec:closure}): the
framework supplies the \emph{input}; the renormalization group
supplies the transport (standard); the lattice supplies the pure
number (yours, with gratitude, at \eqref{eq:pure-number}). What
is claimed is precisely and only the piece the standard account
does not have: the input's \emph{value}, as a closed form, at a
published grade with its selection audit in the open
(\S\ref{ssec:derived-coupling} --- including the look-elsewhere
factor, which most ``derivations'' of constants never print).
Grant that one input and the $897$~MeV follows with no further
freedom anywhere; decline it and you are back to the world
average and the standard account, which is the correct null.
Two further points are yours to hold the paper to. First, the
last import --- the $2.76$ --- is the subject of a registered
program with a printed kill window
(\S\ref{ssec:pure-number-program}); if the wall computation
lands outside $[1.42, 1.93]$, that program dies in public.
Second, the claim's standard is yours as well: the registration
is falsifiable at the coupling ($\alpha_s^{-1}(M_Z) = \pi e$ at
first order, against a world average whose precision improves
yearly), and \ANine{} carries that exposure explicitly. A
marketing claim is one that cannot lose; this one can.
\end{objqabody}


\bigskip
The objections above are the seventeen the paper judges strongest;
the tradition that produced them remains open. A reader holding an
objection not listed here is invited to lodge it against the
claim-status ledger (Table~\ref{tab:claim-ledger}): every claim in
this paper carries its grade, its fences, and its home, so the
target is never ambiguous.

%% ===================================================================
%%  ACKNOWLEDGMENTS
%% ===================================================================
\GGAcknowledgments

%% ===================================================================
%%  REFERENCES
%% ===================================================================
\GGPart[References]{References}{Cited works, each with a relevance note linking it to the present paper}
\label{part: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 lineage of the proton's story: from
Rutherford's naming through the quark model, QCD, the unification
sentence of 1974, and the seventy-year road of gauged baryon number
that Part~VII records, to the cohort papers whose stability theorem
this paper completes. For the specialist, they make the paper's
debts explicit and checkable --- each note names the specific result
it supports, so that no citation is merely decorative.

\noindent The references are organized in the cohort's two-part
style: \textbf{Section A} lists the sister papers of the
\GGTheory{} cohort cited in this paper, in the canonical order of
the program; \textbf{Section B} lists the external literature, in
order of first appearance in the paper. Each entry closes with the
location of its first citation; the list contains no entry that is
not cited in the text. All cohort preprints are available at \url{https://preprints.arisaka-gg.org/}.

\renewcommand{\bibsection}{\vspace{1.2em}}

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

\begin{thebibliography}{99}

\bibitem[Arisaka(2026, A1-Constitution)]{ArisakaA1Constitution}
\textbf{Arisaka, K. (2026).} The \GGConstitution{}: Four Axioms and Seven Postulates of the \GGTheory{} Program, From Which \GG{}, \BIsix{}, \OGNfive{}, and \GDOS{} Descend as Theorems. Arisaka-GG-A1-Constitution.
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/a1-constitution/}.
\par\noindent\textbf{Relevance.} The constitutional root: the axioms
(locality--causality, single variational principle, minimal
structure) under which every ``derived'' in this paper is derived,
and the constitutional locality of \UoneB{} from which Q7 begins.
First cited in \S\ref{sec:framework-minimum}.

\bibitem[Arisaka(2026, A2-GG6)]{ArisakaA2GG6}
\textbf{Arisaka, K. (2026).} \GGTheory{} with \GG{}: Geometry--Gauge Locking at the 6D Bulk (\GG{}) at the Heart of Grand Geometric Theory (\GGTheory{}). Arisaka-GG-A2-GG6.
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/a2-gg6/}.
\par\noindent\textbf{Relevance.} The realization of the constitution
as six-dimensional geometry: the bulk
$M_4 \times \Ir \times \Stheta$ whose boundary this paper's proton
inhabits, and the geometry--gauge locking behind the descent
\eqref{eq:descent-p6}. First cited in \S\ref{sec:framework-minimum}.

\bibitem[Arisaka(2026, A3-BI6)]{ArisakaA3BI6}
\textbf{Arisaka, K. (2026).} Anomaly Freedom of \BIsix{}: The Modified Bianchi Identity (\BIsix{}) for the Six-Force, Six-Dimensional Bulk (\GG{}). Arisaka-GG-A3-BI6.
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/a3-bi6/}.
\par\noindent\textbf{Relevance.} The Green--Schwarz inflow that
absorbs the two non-vanishing coefficients of Part~VI's anomaly
ledger --- hypothesis H3 of the stability theorem, consumed here
verbatim. First cited in \S\ref{sec:framework-minimum}.

\bibitem[Arisaka(2026, A4-Euclid)]{ArisakaA4Euclid}
\textbf{Arisaka, K. (2026).} From Euclid to \GG{}: The Realization of Euclid's Elements in Six Dimensions --- \GUfive{}, \OGNfive{}, and a Parameter-Free Reconstruction of Modern Physics. Arisaka-GG-A4-Euclid.
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/a4-euclid/}.
\par\noindent\textbf{Relevance.} The lineage paper, and the one that sizes
this paper's claim from outside it: each step of the geometrization of
physics turned a quantity that had been a law into a property of a space,
and the proton's exact charge and sharp mass becoming properties of a
basin is one more step in that sequence rather than a break with it. It is
also where the step is measured against the ones before it. First cited in
\S\ref{sec:inheritance-p6}.

\bibitem[Arisaka(2026, A5-GDOS)]{ArisakaA5GDOS}
\textbf{Arisaka, K. (2026).} \GDOS{}: Geometric Dynamics of Open Systems --- 6D \GG{} Bulk Projected to 4D Boundary for Quantum Measurement, General Relativity, and QFT. Arisaka-GG-A5-GDOS.
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/a5-gdos/}.
\par\noindent\textbf{Relevance.} The open-system projection and the
tower theorem (T-GDOS-9.1) whose zero-mode floor Part~VI reads the
baryonic winding from; also the home of the parity assignment
\eqref{eq:parity-P-p6}, whose conjugation action Part~VI reads
from the realization paper's deployed canon. First cited in
\S\ref{sec:framework-minimum}.

\bibitem[Arisaka(2026, A6-GRIP)]{ArisakaA6Gi4}
\textbf{Arisaka, K. (2026).} The \GRIP{} on \GDOS{} --- \GRIP{}: the Geometric Engine of Symmetry Breaking, from the Big Bang to Mind. Arisaka-GG-A6-GRIP.
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/a6-grip/}.
\par\noindent\textbf{Relevance.} The home of this paper's
conceptual spine: the \GRIP{} runtime, the \GA{}/\GGA{} distinction,
and \GRIPin{}/\GRIPout{} of \S\S\ref{sec:grip-runtime}--\ref{sec:two-grounds}.
The attractor machinery: \GGA{}
ground states, the \GFF{} functional whose unique deepest basin
Part~III's theorem concerns, \GSSB{} branch selection (Part~V's
confinement reading), and \GRIPin{} descent pricing (Part~VIII's
neutron). First cited in \S\ref{sec:framework-minimum}.

\bibitem[Arisaka(2026, A7-Qi4)]{ArisakaA7Qi4}
\textbf{Arisaka, K. (2026).} The \QiFour{} Pillars --- \QiQUA{}, \QiEP{}, \QiCON{}, \QiHAL{}: The Quantum-Information Backbone of \GDOS{}. Arisaka-GG-A7-Qi4.
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/a7-qi4/}.
\par\noindent\textbf{Relevance.} The home of the Arisaka equation
\eqref{eq:arisaka-equation-p6} and the information charge \UoneQi{}:
Part~XI consumes the current locking and the constant \QB{} at this
paper's declared candidate grade, and quotes its kill criterion as
F-P6-6. First cited in \S\ref{sec:eleven}.

\bibitem[Arisaka(2026, A8-QM)]{ArisakaA8QM}
\textbf{Arisaka, K. (2026).} Quantum Mechanics Derived from \GGTheory{} as Projection from 6D Bulk to 4D Boundary. Arisaka-GG-A8-QM.
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/a8-qm/}.
\par\noindent\textbf{Relevance.} The source of the readout language this
paper speaks in. ``Readout'', ``label'' and ``register'' are that paper's
terms with that paper's content, and Part~\ref{part:spin}'s central
distinction --- between a representation label of the boundary readout and
a bulk ledger of shares --- has no force unless its account of what a
readout is holds. Where this paper says a quantity is read rather than
possessed, the warrant is there. First cited in \S\ref{sec:inheritance-p6}.

\bibitem[Arisaka(2026, A9-SM)]{ArisakaA9SM}
\textbf{Arisaka, K. (2026).} The Standard Model as Theorem of \GGTheory{}: Its 28 Parameters as Observations of One Six-Dimensional Geometry. Arisaka-GG-A9-SM.
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/a9-sm/}.
\par\noindent\textbf{Relevance.} The largest inheritance: the bulk
group, the descent, the matter packets and $B$-assignments, the
stability theorem T-SM-60.1 whose hypothesis H2 this paper
discharges, the derived couplings and transmutation pricing of the
mass, and the basin/gate architecture. Part~VI is this paper's
repayment of A9's explicit deferral. First cited in
\S\ref{sec:gauge-inheritance}; consumed throughout.

\bibitem[Arisaka(2026, A10-Cosmos)]{ArisakaA10Cosmos}
\textbf{Arisaka, K. (2026).} The Standard Cosmology as Theorem of \GGTheory{}: The Cosmic Inventory as Observations of One Six-Dimensional Geometry. Arisaka-GG-A10-Cosmos.
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/a10-cosmos/}.
\par\noindent\textbf{Relevance.} The cosmological consumer of the
birth chain: \PaperPFive{}'s $\eta_B$ is delivered there, and
Part~X's consistency sentence connects this paper's anomaly door to
that delivery. First cited in \S\ref{sec:mirror-inheritance}.

\bibitem[Arisaka(2026, A11-Origin)]{ArisakaA11Origin}
\textbf{Arisaka, K. (2026).} The Geometric Inevitability of Cosmos, Life, and Mind: The Self-Referential Closure of the \GGTheory{} Program. Arisaka-GG-A11-Origin.
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/a11-origin/}.
No result of the present paper depends on it.
\par\noindent\textbf{Relevance.} The capstone that proves the Chain
of \GRIPout{} (\S\ref{sec:grip-chain}): the determined cascade
$\mathcal{C}\to\GGA{}_1\to\cdots\to\GGA{}_{18}$ in which each stable
object, the proton included, is the Ground Geometric Attractor of its
stratum, and \textsf{T-GGA-as-Ground-State} that fixes the
universality of every readout. This paper's proton is the baryonic
rung of that chain; \AEleven{} is where the upward story --- atoms,
molecules, life, mind --- is carried at its stated grade. First cited
in \S\ref{sec:grip-chain}.

\bibitem[Arisaka(2026, P1-KK)]{ArisakaP1KK}
\textbf{Arisaka, K. (2026).} The \GG{} Lock Cluster: A Non-Spatial Kaluza--Klein Resonance near 3.85~TeV as the Standing LHC Falsifier of \GGTheory{}. Arisaka-GG-P1-KK.
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/p1-kk/}.
\par\noindent\textbf{Relevance.} The collider falsifier of the same
compact direction this paper reads. P1 reads that direction's lock scale
and P6 reads its winding, and both readings require it to be non-spatial:
a spatial circle carries a tower of states and a modulus, and neither
paper survives one. The two are exposed to different experiments and to
the same geometry. First cited in \S\ref{sec:inheritance-p6}.

\bibitem[Arisaka(2026, P2-N3)]{ArisakaP2N3}
\textbf{Arisaka, K. (2026).} Why Three Generations: A Three-Leg Derivation of $N_g=3$ from \BIsix{}, \MSMF{}, and \GDOS{} Morse Stability. Arisaka-GG-P2-N3.
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/p2-n3/}.
\par\noindent\textbf{Relevance.} The two-sided family theorem ($N \ge 3$ from the CP-phase lower bound, $N \le 3$ from the capacity theorem T-N3-6: $\pi$ of budget per family against triangle-number tolls, exactly saturated at $N=3$, conditional on the coupling clause and the integrality lemma L-INT) that Part~IX doubles into the six flavors of Q9; the grammar of ``derived (conditional)'' fencing this paper follows is also this sister paper's. First cited in
\S\ref{sec:family-inheritance}; consumed in Part~IX.

\bibitem[Arisaka(2026, P3-Electron)]{ArisakaP3Electron}
\textbf{Arisaka, K. (2026).} The Electron as the Ground State of Nature: The Six Oldest Questions about the Electron, Closed and Fenced within \GGTheory{}. Arisaka-GG-P3-Electron.
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/p3-electron/}.
\par\noindent\textbf{Relevance.} The sister paper and the method:
questions stated sharply, the honest status table, ground-state
uniqueness read as many faces, fences and debts named, kills
pre-registered. Its positron --- the conjugate readout of the same
basin --- is the worked example Part~VI generalizes to the
antibaryon as the reversed winding. First cited in the
Preface.

\bibitem[Arisaka(2026, P4-Real)]{ArisakaP4Real}
\textbf{Arisaka, K. (2026).} The Real Geometry Beneath the Complex Numbers of Physics: Positive Frequency is Holomorphy --- from Cardano's Cubics to Twistor Space, via the Sixth Dimension of \GGTheory{}. Arisaka-GG-P4-Real.
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/p4-real/}.
\par\noindent\textbf{Relevance.} The Foundations Branch member that
derives the origin of the $i$ the whole quantum apparatus runs on: the
gauge phase from the compact circle's quarter-turn, the state-space $i$
from the causal clock. Recorded here at roster level, in the cohort
overview of \S\ref{sec:inheritance-p6}, and not used as a premise
anywhere in this paper --- but the compact direction whose quarter-turn
supplies that phase is the same direction whose winding
Part~\ref{part:stability} reads as baryon number, so a reader auditing
where that circle came from should go there rather than here. First
cited in \S\ref{sec:inheritance-p6}.

\bibitem[Arisaka(2026, P5-CPT)]{ArisakaP5CPT}
\textbf{Arisaka, K. (2026).} The Mirrors of Matter: Why Every Particle Has a Twin, Why Nature's Mirrors Break, and Why the Universe Is Made of Matter. Arisaka-GG-P5-CPT.
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/p5-cpt/}.
\par\noindent\textbf{Relevance.} Two inheritances at two grades:
at theorem grade, the origin of antiparticles (T-P5-A/B/C/C2) ---
whose reversal-is-conjugation identity is the closure condition of
Part~VI's dihedral structure, the deepest contact between the two
papers; at conditional grade, the TeV-scale leptogenesis chain that
Part~X imports with its named input and named conjecture intact.
First cited in
\S\ref{sec:mirror-inheritance}; consumed in Parts~VI and X.

\bibitem[Arisaka(2026, W1-Symmetry)]{ArisakaW1Symmetry}
\textbf{Arisaka, K. (2026).} Symmetry, Conservation, and Geometry: Seven Steps from Noether, Bianchi, to Hopkins--Singer, and the Ledger the Open-System Turn Pays Out. Arisaka-GG-W1-Symmetry.
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/w1-symmetry/}.
\par\noindent\textbf{Relevance.} The cohort's symmetry review, whose
ladder ends at \emph{protection without symmetry} --- the register in
which this paper's central claim is best stated. A symmetry-based
proton-stability argument can be broken, and historically was; an
integer winding cannot. \S\ref{sec:two-grounds} names the proton as
that ladder's sharpest instance. First cited in
\S\ref{sec:two-grounds}.

\bibitem[Arisaka(2026, W2-Holonomy)]{ArisakaW2Holonomy}
\textbf{Arisaka, K. (2026).} The Circle That Remembers, the Interval That Forgets: \GGTheory{} and the Sixteen Problems of Particle Physics and Cosmology. Arisaka-GG-W2-Holonomy.
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/w2-holonomy/}.
\par\noindent\textbf{Relevance.} The treatise on the fork this
paper's protection argument rests on --- holonomy requires closed
loops, and an interval has none --- and the source of the tier
distinction \S\ref{sec:two-grounds} draws between a winding class,
which is frozen, and a continuous circle modulus such as the Higgs,
which is merely finite. First cited in \S\ref{sec:two-grounds}.

\bibitem[Arisaka(2026, C1-Kaxion)]{ArisakaC1Kaxion}
\textbf{Arisaka, K. (2026).} The Kaxion: Cold Dark Matter as the Non-Spatial Compact-Angle Pseudoscalar of \GG{}. Arisaka-GG-C1-Kaxion.
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/c1-kaxion/}.
\par\noindent\textbf{Relevance.} The cohort's other physical exploitation
of the phase circle's topology. This paper reads a winding of that circle
as baryon number; the Kaxion reads an angle on the same circle as dark
matter, with a derived mass and a laboratory window. The two readings are
held by one ledger, so a result that moved the ledger would move both.
First cited in \S\ref{sec:inheritance-p6}.

\bibitem[Arisaka(2026, C2-Hubble)]{ArisakaC2Hubble}
\textbf{Arisaka, K. (2026).} The Hubble Tension in the Light of \GGTheory{}: The Hubble Constant as One Reading of a Derived Expansion History $H(t)$. Arisaka-GG-C2-Hubble.
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/c2-hubble/}.
\par\noindent\textbf{Relevance.} The derived expansion history in which
the baryon inventory of Part~\ref{part:birth} has to land. This paper
makes protons and counts them against a cosmological number; that number
is read off the same inventory the corridor analysis works in, so a shift
there moves this paper's target and not merely its background. First cited
in \S\ref{sec:inheritance-p6}.

\bibitem[Arisaka(2026, C3-BH)]{ArisakaC3BH}
\textbf{Arisaka, K. (2026).} Black-Hole Entropy under \GGTheory{}: The \GDOS{} Reading of the Horizon as an \OGGEP{} Port --- with the Exact Bekenstein--Hawking $A/4G_N$ Derived from Induced Gravity. Arisaka-GG-C3-BH.
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/c3-bh/}.
\par\noindent\textbf{Relevance.} The owner of the question this paper does
not ask. Baryon number here is a global topological datum that no local
operation can change, and the one place in the cohort where the fate of a
global datum at a horizon is worked out is the horizon-as-port reading.
This paper's stability theorem is stated for the accessible-energy regime
and says nothing about that case; C3 is where a reader should go, and the
two accounts have not been reconciled. First cited in
\S\ref{sec:inheritance-p6}.

\end{thebibliography}

\section*{B.\ Other references}

\begin{thebibliography}{99}

\bibitem[Rutherford(1919)]{Rutherford1919}
\textbf{E.~Rutherford.}
Collision of $\alpha$ particles with light atoms. IV. An anomalous
effect in nitrogen.
\emph{Philosophical Magazine} \textbf{37}, 581 (1919).
\par\noindent\textbf{Relevance.} The proton enters physics: hydrogen
nuclei knocked from nitrogen identify the first constituent of all
nuclei. The century of \S\ref{sec:century} opens here. First cited
in \S\ref{sec:century}.

\bibitem[Chadwick(1932)]{Chadwick1932}
\textbf{J.~Chadwick.}
Possible existence of a neutron.
\emph{Nature} \textbf{129}, 312 (1932).
\par\noindent\textbf{Relevance.} The twin arrives: the neutron's
discovery poses Q8 in its first form --- a particle a tenth of a
percent heavier that dies free and lives bound. First cited in
\S\ref{sec:century}.

\bibitem[Frisch and Stern(1933)]{FrischStern1933}
\textbf{R.~Frisch and O.~Stern.}
\"Uber die magnetische Ablenkung von Wasserstoffmolek\"ulen und das
magnetische Moment des Protons.
\emph{Zeitschrift f\"ur Physik} \textbf{85}, 4 (1933).
\par\noindent\textbf{Relevance.} The first crack in pointlikeness:
$\mu_p \approx 2.79\,\mu_N$ against the Dirac expectation of $1$ ---
the proton caught having an inside fifteen years early. First cited
in \S\ref{sec:century}.

\bibitem[Hofstadter(1956)]{Hofstadter1956}
\textbf{R.~Hofstadter.}
Electron scattering and nuclear structure.
\emph{Reviews of Modern Physics} \textbf{28}, 214 (1956).
\par\noindent\textbf{Relevance.} The inside acquires a shape: form
factors map the charge cloud near $0.8$~fm, turning ``structure''
from inference to geometry. First cited in \S\ref{sec:century}.

\bibitem[Gell-Mann(1964)]{GellMann1964}
\textbf{M.~Gell-Mann.}
A schematic model of baryons and mesons.
\emph{Physics Letters} \textbf{8}, 214 (1964).
\par\noindent\textbf{Relevance.} The quark model: the proton as
$uud$, charge in thirds --- and, from the first page, the puzzle of
Q5: constituents never seen alone. First cited in
\S\ref{sec:century}.

\bibitem[Zweig(1964)]{Zweig1964}
\textbf{G.~Zweig.}
An SU(3) model for strong interaction symmetry and its breaking.
CERN Reports 8182/TH.401 and 8419/TH.412 (1964).
\par\noindent\textbf{Relevance.} The independent invention of the
constituent picture (``aces''); cited with Gell-Mann as the birth of
Q4/Q5's modern form. First cited in \S\ref{sec:century}.

\bibitem[Breidenbach et al.(1969)]{Breidenbach1969}
\textbf{M.~Breidenbach, J.~I.~Friedman, H.~W.~Kendall, et al.}
Observed behavior of highly inelastic electron--proton scattering.
\emph{Physical Review Letters} \textbf{23}, 935 (1969).
\par\noindent\textbf{Relevance.} The quarks are real: SLAC's deep
inelastic scattering finds the hard centers inside the cloud
Hofstadter mapped. First cited in \S\ref{sec:century}.

\bibitem[Gross and Wilczek(1973)]{GrossWilczek1973}
\textbf{D.~J.~Gross and F.~Wilczek.}
Ultraviolet behavior of non-Abelian gauge theories.
\emph{Physical Review Letters} \textbf{30}, 1343 (1973).
\par\noindent\textbf{Relevance.} Asymptotic freedom: why the SLAC
quarks scatter free while never escaping --- half of Q4's modern
statement. First cited in \S\ref{sec:century}.

\bibitem[Politzer(1973)]{Politzer1973}
\textbf{H.~D.~Politzer.}
Reliable perturbative results for strong interactions?
\emph{Physical Review Letters} \textbf{30}, 1346 (1973).
\par\noindent\textbf{Relevance.} The simultaneous discovery of
asymptotic freedom; cited with Gross--Wilczek. First cited in
\S\ref{sec:century}.

\bibitem[Fritzsch et al.(1973)]{FritzschGellMannLeutwyler1973}
\textbf{H.~Fritzsch, M.~Gell-Mann, and H.~Leutwyler.}
Advantages of the color octet gluon picture.
\emph{Physics Letters B} \textbf{47}, 365 (1973).
\par\noindent\textbf{Relevance.} QCD as the theory of the strong
force --- the Lagrangian whose central qualitative fact (Q4)
remains, at Clay grade, unproven from it. First cited in
\S\ref{sec:century}.

\bibitem[Georgi and Glashow(1974)]{GeorgiGlashow1974}
\textbf{H.~Georgi and S.~L.~Glashow.}
Unity of all elementary-particle forces.
\emph{Physical Review Letters} \textbf{32}, 438 (1974).
\par\noindent\textbf{Relevance.} The death sentence: minimal
$\mathrm{SU}(5)$ unification, whose $X,Y$ bosons demand proton
decay --- the prediction Kamiokande was built to test and the
proton declined to honor. The structural contrast for Part~VI:
there the $X,Y$ live on the boundary; here they are bulk-only.
First cited in \S\ref{sec:century}.

\bibitem[Wilson(1974)]{Wilson1974}
\textbf{K.~G.~Wilson.}
Confinement of quarks.
\emph{Physical Review D} \textbf{10}, 2445 (1974).
\par\noindent\textbf{Relevance.} The lattice: confinement made
computable --- numerically established ever since, analytically
open still, which is exactly the register Part~V engages it at.
First cited in \S\ref{sec:century}.

\bibitem[Arisaka et al.(1985)]{ArisakaKamiokande1985}
\textbf{K.~Arisaka, et al. (Kamiokande Collaboration).}
Search for nucleon decay into charged lepton + mesons.
\emph{Journal of the Physical Society of Japan} \textbf{54}, 3213
(1985).
\par\noindent\textbf{Relevance.} Kamiokande's nucleon-decay search
--- the experiment this paper's sixth question was built, literally,
to answer, by the author, 1979--1985. The null result reported here
is the first hardening of the silence that Part~VI derives as a
theorem. First cited in the Preface.

\bibitem[Hirata et al.(1987)]{HirataSN1987}
\textbf{K.~Hirata, et al. (Kamiokande-II Collaboration).}
Observation of a neutrino burst from the supernova SN1987A.
\emph{Physical Review Letters} \textbf{58}, 1490 (1987).
\par\noindent\textbf{Relevance.} What the tank saw instead: the
neutrino burst that founded neutrino astronomy --- the great
improvised success of the instrument built for Q6. First cited in
\S\ref{sec:century}.

\bibitem[Ashman et al.(1988)]{Ashman1988}
\textbf{J.~Ashman, et al. (European Muon Collaboration).}
A measurement of the spin asymmetry and determination of the
structure function $g_1$ in deep inelastic muon--proton scattering.
\emph{Physics Letters B} \textbf{206}, 364 (1988).
\par\noindent\textbf{Relevance.} The spin crisis: the quark spins
carry a small fraction of the proton's spin --- Q3's birth
certificate, and Part~IV's opening exhibit. First cited in
\S\ref{sec:century}.

\bibitem[Sakharov(1967)]{Sakharov1967}
\textbf{A.~D.~Sakharov.}
Violation of CP invariance, C asymmetry, and baryon asymmetry of the
universe.
\emph{JETP Letters} \textbf{5}, 24 (1967).
\par\noindent\textbf{Relevance.} The three conditions any answer to
Q10 must satisfy; the standard account's honest foundation stone in
Table~\ref{tab:status-p6}. First cited in \S\ref{sec:status}.

\bibitem[Fukugita and Yanagida(1986)]{FukugitaYanagida1986}
\textbf{M.~Fukugita and T.~Yanagida.}
Baryogenesis without grand unification.
\emph{Physics Letters B} \textbf{174}, 45 (1986).
\par\noindent\textbf{Relevance.} Leptogenesis: the mechanism family
\PaperPFive{}'s chain instantiates in the resonant TeV-scale regime;
Part~X imports that chain at its declared grade. First cited in
\S\ref{sec:status}.

\bibitem['t Hooft(1976)]{tHooft1976}
\textbf{G.~'t Hooft.}
Symmetry breaking through Bell--Jackiw anomalies.
\emph{Physical Review Letters} \textbf{37}, 8 (1976).
\par\noindent\textbf{Relevance.} The anomaly vertex: baryon-number
violation as a topological effect, not a local operator ---
precisely the character of the single $\Delta B \in 3\mathbb{Z}$
door in Part~VI's ledger, closed cold, open hot. First cited in
\S\ref{sec:mirror-inheritance}.

\bibitem[D\"urr et al.(2008)]{Durr2008}
\textbf{S.~D\"urr, et al.}
Ab initio determination of light hadron masses.
\emph{Science} \textbf{322}, 1224 (2008).
\par\noindent\textbf{Relevance.} The lattice weighs the proton from
first principles: the mass is interaction energy to the percent ---
the empirical anchor of Q1's ``ninety-nine percent'' and of
Part~III's transmutation pricing. First cited in
\S\ref{sec:century}.

\bibitem[Pohl et al.(2010)]{Pohl2010}
\textbf{R.~Pohl, et al.}
The size of the proton.
\emph{Nature} \textbf{466}, 213 (2010).
\par\noindent\textbf{Relevance.} The muonic-hydrogen radius that
opened the decade of the proton-radius puzzle --- cited as part of
the precision era's texture; engaged, at objection level, in
Part~XII. First cited in \S\ref{sec:century}.

\bibitem[Takenaka et al.(2020)]{Takenaka2020}
\textbf{A.~Takenaka, et al. (Super-Kamiokande Collaboration).}
Search for proton decay via $p \to e^+\pi^0$ and $p \to \mu^+\pi^0$
with an enlarged fiducial volume in Super-Kamiokande I--IV.
\emph{Physical Review D} \textbf{102}, 112011 (2020).
\par\noindent\textbf{Relevance.} The number Q6 stands on:
$\tau/B(p \to e^+\pi^0) > 2.4\times10^{34}$~yr at 90\% CL --- the
silence, quantified. First cited in \S\ref{sec:century}.

\bibitem[Super-Kamiokande(2024)]{SuperKamiokande2024}
\textbf{Super-Kamiokande Collaboration.}
Search for proton decay via $p \to e^+\pi^0$ and $p \to \mu^+\pi^0$
with a 0.37 Mton-year exposure of Super-Kamiokande.
\emph{Physical Review D} \textbf{110}, 112011 (2024).
\par\noindent\textbf{Relevance.} The deepest exposure to date ---
$0.37$ megaton-years of watched protons; the current edge of the
four-decade silence, and the baseline the 2040-class searches
extend. First cited in \S\ref{sec:century}.

\bibitem[Borchert et al.(2022)]{Borchert2022}
\textbf{M.~J.~Borchert, et al. (BASE Collaboration).}
A 16-parts-per-trillion measurement of the antiproton-to-proton
charge--mass ratio.
\emph{Nature} \textbf{601}, 53 (2022).
\par\noindent\textbf{Relevance.} The mirror held to the proton at
record precision: the conjugate-readout reading of the antiproton
(Part~VI, with \PaperPFive{}) is tested here --- falsifier F-P6-4's
current anchor. First cited in \S\ref{sec:century}.

\bibitem[Gonzalez et al.(2021)]{UCNtau2021}
\textbf{F.~M.~Gonzalez, et al. (UCN$\tau$ Collaboration).}
Improved neutron lifetime measurement with UCN$\tau$.
\emph{Physical Review Letters} \textbf{127}, 162501 (2021).
\par\noindent\textbf{Relevance.} The bottle: $877.75 \pm 0.28$~s
--- one side of the four-sigma lifetime discrepancy Part~VIII
reports at face value and claims nothing about. First cited in
\S\ref{sec:century}.

\bibitem[Yue et al.(2013)]{Yue2013}
\textbf{A.~T.~Yue, et al.}
Improved determination of the neutron lifetime.
\emph{Physical Review Letters} \textbf{111}, 222501 (2013).
\par\noindent\textbf{Relevance.} The beam: $887.7 \pm 2.2$~s ---
the other side of the discrepancy; together with the bottle value,
the open experimental item of Q8's precision era. First cited in
\S\ref{sec:century}.

\bibitem[Navas et al.(2024)]{PDG2024}
\textbf{S.~Navas, et al. (Particle Data Group).}
Review of particle physics.
\emph{Physical Review D} \textbf{110}, 030001 (2024).
\par\noindent\textbf{Relevance.} The ledger of record for every
measured value this paper quotes --- the proton's mass and moment,
the neutrality of matter, the quark-search nulls, the lifetime
tables; the empirical anchor table of Part~XIV references it
throughout. First cited in \S\ref{sec:eleven}.

\bibitem[Accardi et al.(2016)]{Accardi2016}
\textbf{A.~Accardi, et al.}
Electron--Ion Collider: the next QCD frontier.
\emph{European Physical Journal A} \textbf{52}, 268 (2016).
\par\noindent\textbf{Relevance.} The instrument of the
pre-registered face of Q3: where the spin decomposition sharpens
while --- this paper predicts --- the readout label does not move.
First cited in \S\ref{sec:century}.

\bibitem[Bressi et al.(2011)]{Bressi2011}
\textbf{G.~Bressi, G.~Carugno, F.~Della Valle, G.~Galeazzi,
G.~Ruoso, and G.~Sartori.}
Testing the neutrality of matter by acoustic means in a spherical
resonator.
\emph{Physical Review A} \textbf{83}, 052101 (2011).
\par\noindent\textbf{Relevance.} The neutrality of bulk matter at
the $10^{-21}\,e$ level --- the experimental face of the one-codebook
identity \eqref{eq:proton-charge}: hydrogen is neutral by algebra,
and this is the algebra's precision audit. First cited in
\S\ref{sec:exact-charge}.

\bibitem[Ahmadi et al.(2016)]{Ahmadi2016}
\textbf{M.~Ahmadi, et al. (ALPHA Collaboration).}
An improved limit on the charge of antihydrogen.
\emph{Nature Communications} \textbf{7}, 12308 (2016).
\par\noindent\textbf{Relevance.} The conjugate readout audited:
antihydrogen's charge bounded at $0.71$ parts per billion --- the
winding rigidity of \S\ref{ssec:charge-sharp} holding on the
reversed orientation, part of falsifier F-P6-4's family. First
cited in \S\ref{sec:exact-charge}.

\bibitem[Jaffe and Manohar(1990)]{JaffeManohar1990}
\textbf{R.~L.~Jaffe and A.~Manohar.}
The $g_1$ problem: deep inelastic electron scattering and the
spin of the proton.
\emph{Nuclear Physics B} \textbf{337}, 509 (1990).
\par\noindent\textbf{Relevance.} The partonic spin ledger
\eqref{eq:jaffe-manohar} --- the canonical statement of what the
polarized-scattering program measures, and one of the two
conventions whose mutual irreducibility is Part~IV's central
datum. First cited in \S\ref{sec:spin-ledgers}.

\bibitem[Ji(1997)]{Ji1997}
\textbf{X.~Ji.}
Gauge-invariant decomposition of nucleon spin.
\emph{Physical Review Letters} \textbf{78}, 610 (1997).
\par\noindent\textbf{Relevance.} The gauge-invariant spin ledger
\eqref{eq:ji} and its sum rule through generalized parton
distributions --- the second convention, whose principled
disagreement with the first about orbital attribution is exactly
the scheme-dependence Part~IV reads as bulk bookkeeping. First
cited in \S\ref{sec:spin-ledgers}.

\bibitem[de Florian et al.(2014)]{deFlorian2014}
\textbf{D.~de Florian, R.~Sassot, M.~Stratmann, and
W.~Vogelsang.}
Evidence for polarization of gluons in the proton.
\emph{Physical Review Letters} \textbf{113}, 012001 (2014).
\par\noindent\textbf{Relevance.} The RHIC-era determination that
the gluon polarization is real and positive over the measured
range --- closing the all-gluon rescue of the spin crisis and
setting the $\Delta G$ used in the ledger arithmetic of
Appendix~\ref{app:spin-ledger}. First cited in
\S\ref{sec:spin-crisis}.

\bibitem[Bali(2001)]{Bali2001}
\textbf{G.~S.~Bali.}
QCD forces and heavy quark bound states.
\emph{Physics Reports} \textbf{343}, 1 (2001).
\par\noindent\textbf{Relevance.} The review of record for the
static potential \eqref{eq:cornell} and the string tension ---
the measured shape of the cage that Part~V's sixteen-tonne
arithmetic and Appendix~\ref{app:string} stand on. First cited
in \S\ref{sec:clay-preamble}.

\bibitem[Bazavov et al.(2019)]{HotQCD2019}
\textbf{A.~Bazavov, et al. (HotQCD Collaboration).}
Chiral crossover in QCD at zero and non-zero chemical
potentials.
\emph{Physics Letters B} \textbf{795}, 15 (2019).
\par\noindent\textbf{Relevance.} The lattice determination
$T_c = 156.5 \pm 1.5$~MeV of the deconfinement crossover ---
the temperature above which Part~V's custody split (bulk color
mobility, integer boundary readout) takes over from the confined
phase, and the QGP stress-test of C-P6-1 begins. First cited in
\S\ref{sec:cage-measured}.

\bibitem[Hosotani(1983)]{Hosotani1983}
\textbf{Y.~Hosotani.}
Dynamical mass generation by compact extra dimensions.
\emph{Physics Letters B} \textbf{126}, 309 (1983).
\par\noindent\textbf{Relevance.} The mechanism by which a Wilson
line on a compact cycle breaks gauge symmetry dynamically --- the
machinery whose hypercharge direction carries both \ANine{}'s
Higgs and, at the center point $W(2) = z$, this paper's
$\mathbb{Z}_3$-lock. First cited in \S\ref{sec:z3lock}.

\bibitem[Kawamura(2001)]{Kawamura2001}
\textbf{Y.~Kawamura.}
Triplet--doublet splitting, proton stability and an extra
dimension.
\emph{Progress of Theoretical Physics} \textbf{105}, 999 (2001).
\par\noindent\textbf{Relevance.} The orbifold-GUT refinement of
the descent --- boundary conditions doing the work of the GUT
Higgs sector --- whose doublet--triplet and proton-stability
instincts Part~VI's lock inherits, sharpens to the unique
$\mathbb{Z}_3$ point, and converts from parity choices into the
winding that writes baryon number. First cited in
\S\ref{sec:z3lock}.

\bibitem[Green and Schwarz(1984)]{GreenSchwarz1984}
\textbf{M.~B.~Green and J.~H.~Schwarz.}
Anomaly cancellations in supersymmetric $D = 10$ gauge theory
and superstring theory.
\emph{Physics Letters B} \textbf{149}, 117 (1984).
\par\noindent\textbf{Relevance.} The inflow mechanism by which a
higher-dimensional structure absorbs an anomaly the
four-dimensional spectrum cannot cancel alone --- the external
ancestor of the \BIsix{} bookkeeping (\AThree{}) that hypothesis
H3 consumes for the two non-vanishing entries of the ledger of
\S\ref{sec:anomaly-ledger}. First cited in
\S\ref{sec:anomaly-ledger}.

\bibitem[St\"uckelberg(1938)]{Stueckelberg1938}
\textbf{E.~C.~G.~St\"uckelberg.}
Die Wechselwirkungskr\"afte in der Elektrodynamik und in der
Feldtheorie der Kernkr\"afte.
\emph{Helvetica Physica Acta} \textbf{11}, 225 (1938).
\par\noindent\textbf{Relevance.} The 1938 construction by which a
$\mathrm{U}(1)$ gauge boson acquires mass while the rigid
symmetry survives exactly --- the reason this framework can have
gauged baryon number, no baryonic fifth force, and an exact
selection rule all at once. An eighty-eight-year-old trick,
load-bearing in a 2026 theorem. First cited in
\S\ref{sec:anomaly-ledger}.

\bibitem[Baldo-Ceolin et al.(1994)]{BaldoCeolin1994}
\textbf{M.~Baldo-Ceolin, et al.}
A new experimental limit on neutron--antineutron oscillations.
\emph{Zeitschrift f\"ur Physik C} \textbf{63}, 409 (1994).
\par\noindent\textbf{Relevance.} The free-neutron oscillation
bound ($\tau_{n\bar n} > 0.86\times10^{8}$~s, ILL) --- the
$\Delta B = 2$ face of the winding selection rule, forbidden here
as absolutely as decay itself; falsifier F-P6-3's anchor, with
bound-neutron searches and the ESS proposal extending it. First
cited in \S\ref{sec:kamiokande}.

\bibitem[Borsanyi et al.(2015)]{Borsanyi2015}
\textbf{S.~Borsanyi, et al. (Budapest--Marseille--Wuppertal
Collaboration).}
Ab initio calculation of the neutron--proton mass difference.
\emph{Science} \textbf{347}, 1452 (2015).
\par\noindent\textbf{Relevance.} The lattice QCD+QED
decomposition of the hair's-breadth splitting --- quark-mass
part $+2.52$~MeV, electromagnetic part $-1.00$~MeV --- the
first-principles form of Part~VIII's competition
\eqref{eq:competition}, against which the framework's toll
ledger is checked. First cited in \S\ref{sec:mass-open}.

\bibitem[Barnes et al.(1964)]{Barnes1964}
\textbf{V.~E.~Barnes, et al.}
Observation of a hyperon with strangeness minus three.
\emph{Physical Review Letters} \textbf{12}, 204 (1964).
\par\noindent\textbf{Relevance.} The $\Omega^-$: predicted with
its mass by symmetry, found on cue --- the discovery that made
quarks undeniable, and, in Part~IX's tower, the specimen
proving that \emph{labels} (three units of strangeness) can buy
thirteen orders of magnitude of lifetime. First cited in
\S\ref{sec:tower}.

\bibitem[Planck Collaboration(2020)]{Planck2020}
\textbf{N.~Aghanim, et al. (Planck Collaboration).}
Planck 2018 results. VI. Cosmological parameters.
\emph{Astronomy \& Astrophysics} \textbf{641}, A6 (2020).
\par\noindent\textbf{Relevance.} The precision inventory behind
\eqref{eq:etaB}: the baryon density whose part-per-billion
smallness --- and stubborn non-zeroness --- is Q10 stated as a
measurement. First cited in \S\ref{sec:inventory}.

\bibitem[Kuzmin, Rubakov, and Shaposhnikov(1985)]{KuzminRubakovShaposhnikov1985}
\textbf{V.~A.~Kuzmin, V.~A.~Rubakov, and M.~E.~Shaposhnikov.}
On anomalous electroweak baryon-number non-conservation in the
early universe.
\emph{Physics Letters B} \textbf{155}, 36 (1985).
\par\noindent\textbf{Relevance.} The realization that the
electroweak anomaly's transitions are unsuppressed at high
temperature --- the door of Parts~VI and X in its original
form, and the reason a lepton asymmetry can become the baryons
of \eqref{eq:etaB}. First cited in
\S\ref{sec:record-refusals}.

\bibitem[Harvey and Turner(1990)]{HarveyTurner1990}
\textbf{J.~A.~Harvey and M.~S.~Turner.}
Cosmological baryon and lepton number in the presence of
electroweak fermion-number violation.
\emph{Physical Review D} \textbf{42}, 3344 (1990).
\par\noindent\textbf{Relevance.} The chemical-potential
bookkeeping behind the door's exchange rate
\eqref{eq:partition} --- the $28/79$ walked in
Appendix~\ref{app:door-bookkeeping}. First cited in
\S\ref{sec:the-chain}.

\bibitem[Pilaftsis and Underwood(2004)]{PilaftsisUnderwood2004}
\textbf{A.~Pilaftsis and T.~E.~J.~Underwood.}
Resonant leptogenesis.
\emph{Nuclear Physics B} \textbf{692}, 303 (2004).
\par\noindent\textbf{Relevance.} The quasi-degenerate regime in
which TeV-scale leptogenesis attains the required asymmetry ---
the external home of the resonance physics whose condition is
the \PaperPFive{} chain's one named input. First cited in
\S\ref{sec:the-chain}.

\bibitem[Kolb and Turner(1990)]{KolbTurner1990}
\textbf{E.~W.~Kolb and M.~S.~Turner.}
\emph{The Early Universe}.
Addison--Wesley, Redwood City (1990).
\par\noindent\textbf{Relevance.} The standard freeze-out
arithmetic behind the annihilation catastrophe --- the
$10^{-18}$ of Appendix~\ref{app:door-bookkeeping} that makes a
pre-annihilation asymmetry non-optional. First cited in
\S\ref{sec:inventory}.

\bibitem[Lee and Yang(1955)]{LeeYang1955}
\textbf{T.~D.~Lee and C.~N.~Yang.}
Conservation of heavy particles and generalized gauge
transformations.
\emph{Physical Review} \textbf{98}, 1501 (1955).
\par\noindent\textbf{Relevance.} The founding Letter: gauge
baryon number in analogy with electromagnetism --- the founder's
axiom of \S\ref{sec:founders-axiom}, proposed seventy years
early, and wounded in its own single page by the
E\"otv\"os-data bound on a massless baryonic photon. Part~VII
is the record of what happened to it. First cited in
\S\ref{sec:founders-axiom}.

\bibitem[Pais(1973)]{Pais1973}
\textbf{A.~Pais.}
Remark on baryon conservation.
\emph{Physical Review D} \textbf{8}, 1844 (1973).
\par\noindent\textbf{Relevance.} The first massive-boson evasion
of the E\"otv\"os wound --- gauge $B$, give the boson a Higgs
mass, lose the long-range force --- the direct ancestor of the
Stueckelberg logic Part~VI runs on, published as a three-page
``Remark'' that gathered no school. First cited in
\S\ref{sec:record-pais}.

\bibitem[Fischbach et al.(1986)]{Fischbach1986}
\textbf{E.~Fischbach, D.~Sudarsky, A.~Szafer, C.~Talmadge, and
S.~H.~Aronson.}
Reanalysis of the E\"otv\"os experiment.
\emph{Physical Review Letters} \textbf{56}, 3 (1986).
\par\noindent\textbf{Relevance.} The ``fifth force'' episode:
the one wrong revival of a baryon-coupled force, dead within
four years --- whose permanent legacy, the purpose-built
equivalence-principle search machine, is the experimental
community that falsifier F-P6-5 inherits. First cited in
\S\ref{sec:record-fifthforce}.

\bibitem[Foot et al.(1989)]{FootJoshiLew1989}
\textbf{R.~Foot, G.~C.~Joshi, and H.~Lew.}
Gauged baryon and lepton numbers.
\emph{Physical Review D} \textbf{40}, 2487 (1989).
\par\noindent\textbf{Relevance.} The anomaly price named in the
open: with the known generations, gauged $B$ demands new exotic
fermions (``anomalons'') --- the four-dimensional bill that this
framework pays instead at six dimensions, through the \BIsix{}
inflow, with the spectrum untouched. First cited in
\S\ref{sec:record-minority}.

\bibitem[Carone and Murayama(1995)]{CaroneMurayama1995}
\textbf{C.~D.~Carone and H.~Murayama.}
Possible light $\mathrm{U}(1)$ gauge boson coupled to baryon
number.
\emph{Physical Review Letters} \textbf{74}, 3122 (1995); see
also \emph{Physical Review D} \textbf{52}, 484 (1995).
\par\noindent\textbf{Relevance.} The leptophobic window: a
gauged-$B$ boson lighter than the $Z$ still experimentally
alive in 1995 --- the minority line's measure of how little the
field had looked for the force it had declined to gauge. First
cited in \S\ref{sec:record-minority}.

\bibitem[Fileviez P\'erez and Wise(2010)]{FileviezPerezWise2010}
\textbf{P.~Fileviez P\'erez and M.~B.~Wise.}
Baryon and lepton number as local gauge symmetries.
\emph{Physical Review D} \textbf{82}, 011901(R) (2010); erratum
\textbf{82}, 079901 (2010).
\par\noindent\textbf{Relevance.} The modern revival, fifty-five
years after the Letter: anomalies canceled by an extra
generation, dark-matter stability as the advertised bonus ---
gauged $B$ still entering as an extension seeking a home, which
is the minority line's signature to the present day. First
cited in \S\ref{sec:record-minority}.

\bibitem[Wagner et al.(2012)]{Wagner2012}
\textbf{T.~A.~Wagner, S.~Schlamminger, J.~H.~Gundlach, and
E.~G.~Adelberger.}
Torsion-balance tests of the weak equivalence principle.
\emph{Classical and Quantum Gravity} \textbf{29}, 184002
(2012).
\par\noindent\textbf{Relevance.} The modern benchmark of the
experiment Lee and Yang first invoked: beryllium against
titanium, chosen to maximize the difference in baryon number
per unit mass, $\eta = (0.3 \pm 1.8)\times10^{-13}$ --- the
standing null a Stueckelberg-massive \UoneB{} demands, and
falsifier F-P6-5's anchor. First cited in
\S\ref{sec:record-fifthforce}.

\bibitem[Ib\'a\~nez et al.(2001)]{IbanezMarchesanoRabadan2001}
\textbf{L.~E.~Ib\'a\~nez, F.~Marchesano, and R.~Rabad\'an.}
Getting just the standard model at intersecting branes.
\emph{Journal of High Energy Physics} \textbf{11}, 002 (2001).
\par\noindent\textbf{Relevance.} The brane convergence: in
intersecting-brane Standard Models baryon number is
necessarily gauged, Green--Schwarz-massive, and
proton-protecting in perturbation theory --- independent
corroboration of Part~VI's mechanism at a different price (ten
dimensions, supersymmetry, extended objects), recorded in
\S\ref{sec:record-tendim} and weighed in
\S\ref{sec:uniqueness-contrast}. First cited in
\S\ref{sec:record-tendim}.

\bibitem[Mandelstam(1976)]{Mandelstam1976}
\textbf{S.~Mandelstam.}
Vortices and quark confinement in non-Abelian gauge theories.
\emph{Physics Reports} \textbf{23}, 245 (1976).
\par\noindent\textbf{Relevance.} The dual-superconductor picture
of confinement --- a chromomagnetic condensate squeezing electric
flux into tubes --- the mechanism family whose geometrization
Part~V's wall-and-winding account belongs to; compared honestly
in Table~\ref{tab:mechanisms}. First cited in
\S\ref{sec:mechanisms}.

\bibitem['t Hooft(1978)]{tHooft1978}
\textbf{G.~'t Hooft.}
On the phase transition towards permanent quark confinement.
\emph{Nuclear Physics B} \textbf{138}, 1 (1978).
\par\noindent\textbf{Relevance.} Center vortices: confinement as
percolating $\mathbb{Z}_3$ flux --- the standard account's
sharpest center-based mechanism, and the lattice's
best-supported one; the center bridge of
\S\ref{sec:centre-bridge} proposes the ancestry of exactly this
$\mathbb{Z}_3$. Distinct from the 1976 anomaly paper cited as
\citealp{tHooft1976}. First cited in \S\ref{sec:mechanisms}.

\bibitem[L\"uscher et al.(1980)]{LuscherSW1980}
\textbf{M.~L\"uscher, K.~Symanzik, and P.~Weisz.}
Anomalies of the free loop wave equation in the WKB
approximation.
\emph{Nuclear Physics B} \textbf{173}, 365 (1980).
\par\noindent\textbf{Relevance.} The universal $-\pi/12\,\hbar
c/r$ correction to the confining potential --- the vibrating
string's Casimir energy, whose coefficient counts the tube's
massless transverse modes and thereby powers the
Goldstone-counting test of \S\ref{sec:goldstone-test}. First
cited in \S\ref{sec:cage-measured}.

\bibitem[Seiberg and Witten(1994)]{SeibergWitten1994}
\textbf{N.~Seiberg and E.~Witten.}
Electric--magnetic duality, monopole condensation, and
confinement in $\mathcal{N}=2$ supersymmetric Yang--Mills
theory.
\emph{Nuclear Physics B} \textbf{426}, 19 (1994).
\par\noindent\textbf{Relevance.} The only four-dimensional gauge
theory in which confinement and the gap are theorems --- proved
by monopole condensation in the softly-broken $\mathcal{N}=2$
theory --- and the lesson Part~V takes literally: provability
arrives with rigidity. Here the rigid structure is the lock.
First cited in \S\ref{sec:mechanisms}.

\bibitem[Maldacena(1998)]{Maldacena1998}
\textbf{J.~Maldacena.}
The large-$N$ limit of superconformal field theories and
supergravity.
\emph{Advances in Theoretical and Mathematical Physics}
\textbf{2}, 231 (1998).
\par\noindent\textbf{Relevance.} Gauge--gravity duality: in
confining backgrounds the Wilson loop drapes along an infrared
wall and the area law follows from geometry --- the mechanism
whose no-strings cognate is Part~V's wall on the scale axis
(\S\ref{sec:scale-wall}), with the wall location priced rather
than modeled. First cited in \S\ref{sec:mechanisms}.

\bibitem[de Forcrand and D'Elia(1999)]{deForcrandDElia1999}
\textbf{P.~de Forcrand and M.~D'Elia.}
On the relevance of center vortices to QCD.
\emph{Physical Review Letters} \textbf{82}, 4582 (1999).
\par\noindent\textbf{Relevance.} The sharpest lattice evidence
in the mechanism debate: removing the center vortices from a
configuration removes the string tension --- and the chiral
condensate with it. The center carries the cage;
\S\ref{sec:centre-bridge} proposes why. First cited in
\S\ref{sec:mechanisms}.

\bibitem[Morningstar and Peardon(1999)]{MorningstarPeardon1999}
\textbf{C.~J.~Morningstar and M.~Peardon.}
The glueball spectrum from an anisotropic lattice study.
\emph{Physical Review D} \textbf{60}, 034509 (1999).
\par\noindent\textbf{Relevance.} The measured mass gap: the
lightest pure-gauge glueball at $1.71$~GeV $= 3.9\sqrt{\sigma}$
--- the number the Clay problem asks mathematics to prove
positive, quoted in Part~V's preamble and Clay register so the
reader always knows what $\Delta$ is physically. First cited in
\S\ref{sec:clay-preamble}.

\bibitem[Jaffe and Witten(2000)]{JaffeWitten2000}
\textbf{A.~Jaffe and E.~Witten.}
Quantum Yang--Mills theory.
Clay Mathematics Institute Millennium Prize problem description
(2000); in \emph{The Millennium Prize Problems}, Clay
Mathematics Institute / AMS (2006).
\par\noindent\textbf{Relevance.} The official problem: construct
quantum Yang--Mills on $\mathbb{R}^4$ at Wightman-grade rigor
and prove the mass gap. Part~V's load-bearing preamble and the
four-move register of \S\ref{sec:clay-register} are calibrated
against this text, including the paper's declination, in
writing, of any claim on it. First cited in
\S\ref{sec:clay-preamble}.

\bibitem[Bali et al.(2005)]{Bali2005}
\textbf{G.~S.~Bali, H.~Neff, T.~D\"ussel, T.~Lippert, and
K.~Schilling (SESAM Collaboration).}
Observation of string breaking in QCD.
\emph{Physical Review D} \textbf{71}, 114513 (2005).
\par\noindent\textbf{Relevance.} The direct lattice observation
of the string snapping at $r_b \approx 1.24$~fm --- the budget
$\sigma r_b \approx 1.22$~GeV paying for a light pair --- the
(B) face of the five-claim scorecard, and the reason physical
confinement is (K) enforced through (L) plus (B) rather than an
infinite potential. First cited in \S\ref{sec:cage-measured}.

\bibitem[Greensite(2011)]{Greensite2011}
\textbf{J.~Greensite.}
\emph{An Introduction to the Confinement Problem}.
Lecture Notes in Physics \textbf{821}, Springer (2011).
\par\noindent\textbf{Relevance.} The monograph of record for the
confinement problem --- the center-symmetry criteria, the
$N$-ality lore, the mechanism debate --- and the recommended
next book for the reader this Part is written for; the standard
side of the center bridge is drawn from its account. First
cited in \S\ref{sec:cage-measured}.


\bibitem[Ji(1995)]{Ji1995}
\textbf{X.~Ji.}
QCD analysis of the mass structure of the nucleon.
\emph{Physical Review Letters} \textbf{74}, 1071 (1995).
\par\noindent\textbf{Relevance.} The founding document of the
mass-budget literature: from two QCD inputs (the momentum sum
rule and the trace anomaly), the four-term itemization of the
proton's mass that \S\ref{ssec:ji-ledger} reviews and the
thirty-year argument of \S\ref{ssec:two-term} contests. First
cited in \S\ref{ssec:ji-ledger}.

\bibitem[Yang et al.(2018)]{YangEMT2018}
\textbf{Y.-B.~Yang, J.~Liang, Y.-J.~Bi, Y.~Chen, T.~Draper,
K.-F.~Liu, Z.~Liu.}
Proton mass decomposition from the QCD energy momentum tensor.
\emph{Physical Review Letters} \textbf{121}, 212001 (2018).
\par\noindent\textbf{Relevance.} The first complete lattice
itemization of the proton's mass in the Ji ledger --- quark
energy $33(4)(4)\%$, glue $37(5)(4)\%$, condensates $9(2)(1)\%$,
trace anomaly $23(1)(1)\%$ ($\overline{\mathrm{MS}}$, $2$~GeV)
--- Table~\ref{tab:chiqcd}'s source. First cited in
\S\ref{ssec:ji-ledger}.

\bibitem[Lorc\'e(2018)]{Lorce2018}
\textbf{C.~Lorc\'e.}
On the hadron mass decomposition.
\emph{European Physical Journal C} \textbf{78}, 120 (2018).
\par\noindent\textbf{Relevance.} The two-term objection: a
hadron's mass admits at most two consistently interpretable
contributions; the quark-mass and trace-anomaly entries of
four-term budgets are argued to be conventional, and pressure
effects overlooked. One side of the live dispute
\S\ref{ssec:two-term} reviews. First cited in
\S\ref{ssec:two-term}.

\bibitem[Metz et al.(2020)]{MetzPasquiniRodini2020}
\textbf{A.~Metz, B.~Pasquini, S.~Rodini.}
Revisiting the proton mass decomposition.
\emph{Physical Review D} \textbf{102}, 114042 (2020).
\par\noindent\textbf{Relevance.} The renormalization-theory form
of the two-term objection: the forward energy--momentum tensor
has two independent form factors, and the individual sum-rule
entries carry considerable scheme dependence at three loops;
also the source for the heavy-quark sigma-term subtlety in
\S\ref{ssec:tolls}. First cited in \S\ref{ssec:tolls}.

\bibitem[Ji(2021)]{Ji2021}
\textbf{X.~Ji.}
Proton mass decomposition: naturalness and interpretations.
\emph{Frontiers of Physics} \textbf{16}, 64601 (2021).
\par\noindent\textbf{Relevance.} The founding author's published
rebuttal to the two-term critique: the rest-frame Hamiltonian
budget defended as natural, and the quantum anomalous energy as
measurable --- the theory position behind the threshold-charm
program. Cited so that \S\ref{ssec:two-term} presents the
dispute with both sides on the record. First cited in
\S\ref{ssec:two-term}.

\bibitem[Bollweg et al.(2026)]{BollwegEMT2026}
\textbf{D.~Bollweg, H.-T.~Ding, X.~Gao, S.~Luo, S.~Mukherjee.}
Lattice-QCD validation of hadron mass and trace-anomaly
decomposition sum rules.
arXiv:2601.13070 (2026), \emph{Physical Review Letters}, in
press at this writing.
\par\noindent\textbf{Relevance.} The milestone
\S\ref{ssec:dispute-measured} turns on: all four rival mass sum
rules computed for the same hadrons (charmonia) in one
gradient-flow framework, with the anomalous/gluonic share
ranging from ${\sim}6\%$ to ${\sim}35\%$ across ledgers ---
the itemization dispute, measured
(Figure~\ref{fig:ledger-dispute}). First cited in
\S\ref{ssec:dispute-measured}.

\bibitem[GlueX Collaboration(2019)]{GlueX2019}
\textbf{A.~Ali, et al. (GlueX Collaboration).}
First measurement of near-threshold $J/\psi$ exclusive
photoproduction off the proton.
\emph{Physical Review Letters} \textbf{123}, 072001 (2019).
\par\noindent\textbf{Relevance.} The opening measurement of the
threshold-charm program: the exclusive cross section from
threshold ($8.2$~GeV) to $11.8$~GeV, falling less steeply than
two-gluon-exchange models --- the first measured caution about
the simplest gluonic interpretation. First cited in
\S\ref{ssec:threshold-charm}.

\bibitem[GlueX Collaboration(2023)]{GlueX2023}
\textbf{S.~Adhikari, et al. (GlueX Collaboration).}
Measurement of the $J/\psi$ photoproduction cross section over
the full near-threshold kinematic region.
\emph{Physical Review C} \textbf{108}, 025201 (2023).
\par\noindent\textbf{Relevance.} The full threshold dataset, and
the collaboration's own register: the data are ``inputs to
theoretical models'' of gluonic structure, with measured
evidence for open-charm contributions beyond gluon exchange ---
the model-mediation caution \S\ref{ssec:threshold-charm} carries
into every anomaly-extraction claim. First cited in
\S\ref{ssec:threshold-charm}.

\bibitem[Duran et al.(2023)]{Duran2023}
\textbf{B.~Duran, et al.}
Determining the gluonic gravitational form factors of the proton.
\emph{Nature} \textbf{615}, 813 (2023).
\par\noindent\textbf{Relevance.} The $J/\psi$-007 experiment:
gluonic gravitational form factors extracted from threshold
photoproduction, with a mass radius ``notably smaller than the
electric charge radius'' across all models employed --- the
model-robust ordering registered in \S\ref{ssec:eic-stake} as
the first acceptance test of this paper's D-P6-5 profile debt.
First cited in \S\ref{ssec:gff}.

\bibitem[Burkert et al.(2018)]{BurkertPressure2018}
\textbf{V.~D.~Burkert, L.~Elouadrhiri, F.~X.~Girod.}
The pressure distribution inside the proton.
\emph{Nature} \textbf{557}, 396 (2018).
\par\noindent\textbf{Relevance.} The celebrated $D$-term
``pressure'' claim from deeply-virtual Compton scattering ---
cited here strictly as a pair with its published rebuttal
\citep{Kumericki2019}, per the register of \S\ref{ssec:gff}.
First cited in \S\ref{ssec:gff}.

\bibitem[Kumeri\v{c}ki(2019)]{Kumericki2019}
\textbf{K.~Kumeri\v{c}ki.}
Measurability of pressure inside the proton.
\emph{Nature} \textbf{570}, E1 (2019).
\par\noindent\textbf{Relevance.} The published reanalysis showing
the same data consistent with zero quark pressure under a
flexible parameterization --- the reason \S\ref{ssec:gff} treats
the 2018 pressure result as a claim, not a measurement, and the
reason this paper registers no $D$-term claim of its own. First
cited in \S\ref{ssec:gff}.

\bibitem[Hackett et al.(2024)]{HackettGFF2024}
\textbf{D.~C.~Hackett, D.~A.~Pefkou, P.~E.~Shanahan.}
Gravitational form factors of the proton from lattice QCD.
\emph{Physical Review Letters} \textbf{132}, 251904 (2024).
\par\noindent\textbf{Relevance.} The lattice's flavor-separated
gravitational form factors: quark and gluon momentum halves
nearly equal ($A_q(0) = 0.510(25)$, $A_g(0) = 0.501(27)$), the
gluon contribution extending the proton's radial size --- the
first-principles counterpart of the hall extractions in
\S\ref{ssec:gff}, and the comparison of record for any future
D-P6-5 output. First cited in \S\ref{ssec:gff}.

\bibitem[Abdul Khalek et al.(2022)]{EICYellowReport2022}
\textbf{R.~Abdul Khalek, et al.}
Science requirements and detector concepts for the Electron--Ion
Collider: EIC Yellow Report.
\emph{Nuclear Physics A} \textbf{1026}, 122447 (2022).
\par\noindent\textbf{Relevance.} The community's flagship-machine
statement of Q1: the origin of the nucleon's mass as a central
EIC question, the trace anomaly as the target, threshold
quarkonium as the route --- the program within which
\S\ref{ssec:eic-stake} registers this framework's exposure.
First cited in \S\ref{ssec:eic-stake}.

\bibitem[Roberts et al.(2021)]{RobertsEHM2021}
\textbf{C.~D.~Roberts, D.~G.~Richards, T.~Horn, L.~Chang.}
Insights into the emergence of mass from studies of pion and
kaon structure.
\emph{Progress in Particle and Nuclear Physics} \textbf{120},
103883 (2021).
\par\noindent\textbf{Relevance.} The emergent-hadron-mass
review: the hadronic GeV as the Standard Model's second,
not-yet-understood mass mechanism, with the pion--proton
dichotomy as its organizing puzzle --- the community's own
statement that OP-1 and OP-7 of \S\ref{sec:mass-open} are open.
First cited in \S\ref{sec:mass-open}.

\bibitem[Gupta et al.(2021)]{GuptaSigmaTerm2021}
\textbf{R.~Gupta, S.~Park, M.~Hoferichter, E.~Mereghetti,
B.~Yoon, T.~Bhattacharya.}
Pion--nucleon sigma term from lattice QCD.
\emph{Physical Review Letters} \textbf{127}, 242002 (2021).
\par\noindent\textbf{Relevance.} The demonstration that the same
lattice ensembles yield $\sigma_{\pi N} = 41.9(4.9)$ or
$59.6(7.4)$~MeV depending on the treatment of $N\pi$ excited
states --- two percent of the proton's mass riding on a
systematics choice, the honest core of OP-6 and of
\S\ref{ssec:tolls}'s sigma discussion. First cited in
\S\ref{ssec:tolls}.

\bibitem[Agadjanov et al.(2023)]{MainzSigmaTerm2023}
\textbf{A.~Agadjanov, D.~Djukanovic, G.~von~Hippel,
H.~B.~Meyer, K.~Ottnad, H.~Wittig.}
Nucleon sigma terms with $N_f = 2+1$ flavors of
$\mathcal{O}(a)$-improved Wilson fermions.
\emph{Physical Review Letters} \textbf{131}, 261902 (2023).
\par\noindent\textbf{Relevance.} The controlled lattice values
this paper quotes for the proton's Higgs response:
$\sigma_{\pi N} = 43.7(3.6)$~MeV, $\sigma_s = 28.6(9.3)$~MeV,
with the dispersion-theory tension persisting at $2.4\sigma$ ---
the numbers behind the corrected mass-budget caption
(Figure~\ref{fig:mass-budget}). First cited in
\S\ref{ssec:tolls}.

\bibitem[Douglas(2026)]{Douglas2026}
\textbf{M.~R.~Douglas.}
The Yang--Mills Millennium problem.
\emph{Nature Reviews Physics} \textbf{8}, 86 (2026).
\par\noindent\textbf{Relevance.} The current survey of record for
the mass-gap problem: existence and gap both open, no clear path
--- the citation behind OP-4's status in \S\ref{sec:mass-open},
and the standard-side confirmation that Part~V's Clay register
placed the problem where it in fact stands. First cited in
\S\ref{sec:mass-open}.

\end{thebibliography}

\end{document}




