%% v20.0 (August 21, 2026): PROMOTION TO THE v20 STAMP.
%%   Version and cover date only.  Promoted from v15.30 with no other change:
%%   not one character of body text, mathematics, float, caption or
%%   bibliography differs.  The promotion is safe to do mechanically because
%%   the cohort cites its companions BY TITLE, never by versioned filename --
%%   a scan of the printed text of all twelve deposit papers found zero
%%   version strings, so no cross-reference can be left pointing at a
%%   superseded file.  This is the version posted to arisaka-gg.org and
%%   deposited to Zenodo, so homepage, repository and DOI all carry one
%%   stamp.
%% ===================================================================
%% v15.29 (August 18, 2026) -- TWO AUTHOR-REQUESTED REPAIRS.
%%
%% (1) THE SUPERSTRING REMARK IS REMOVED FROM PART II.  Part II's
%%     "Remark 2 (superstring theory is a valid theorem at Stratum 2,
%%     but is excluded from Strata 3 and 4 by four structural
%%     violations)" said, eighty pages early, what Part V's Section on
%%     the four-fold constitutional exclusion says at length and in the
%%     language where the strata have already been defined.  IT ARRIVED
%%     BEFORE ITS OWN VOCABULARY: "Stratum 2" is not introduced until
%%     Part V, so a reader met the conclusion before the classification
%%     it rests on.  And it made the same argument about another
%%     program twice in one paper.  The Part V treatment stands
%%     unchanged; the Part II one is gone -- 6,694 characters, one
%%     enumerate, seven citations, no labels.
%%     THE THREE THINGS THAT DEPENDED ON IT ARE REPAIRED, not left to
%%     dangle: the enclosing subsection was titled "The two most
%%     consequential immediate implications" and is now singular, its
%%     preamble with it; Remark 1 is no longer numbered, since it is no
%%     longer one of two; and Part V's sentence "already itemized in
%%     Part II's Remark 2 from the axiomatic side" is cut, because
%%     Part II no longer itemizes them.
%%     TWO THINGS THE PAGE SHOWED AND NO SCAN DID: de-numbering Remark 1
%%     left its closing parenthesis stranded --- the heading printed
%%     "... solved by AX-3 OGGEP)." --- and Part V's string-theory
%%     Section still called itself "this closing subsection", which it
%%     has not been since before this ship.  Both repaired.
%%
%%     ASSERTED: no bibliography entry is orphaned by the seven lost
%%     citations -- all seven keys are cited elsewhere in the paper.
%%
%% (2) PARTS II AND III STOP USING SUB-SUBSECTIONS.  Both Parts were one
%%     Section deep with everything hung beneath it, so each axiom and
%%     each postulate was a subsection carrying eight to ten
%%     sub-subsections -- 108 of them in the paper, 91 in these two
%%     Parts.  Every top-level subsection is promoted to a Section and
%%     every sub-subsection to a subsection; both Parts are now
%%     uniformly two levels deep.  Part II: 8 subsections -> Sections,
%%     29 sub-subsections -> subsections.  Part III: 13 and 62.
%%
%%     THE HISTORY IS GATHERED.  Each axiom and each postulate ended
%%     with its own "Historical background (with citations)"
%%     sub-subsection.  Those eleven blocks -- 48,756 characters in
%%     Part II, 73,435 in Part III -- are lifted out and collected into
%%     one new Section per Part, "Century-old history of the four
%%     axioms" and "Century-old history of the seven postulates", in
%%     axiom and postulate order, immediately before each Part's
%%     closing remarks.  This is what removes the eleven
%%     sub-subsections, and it lets the history read as one continuous
%%     argument rather than as four and seven disconnected appendices.
%%     NOT ONE WORD OF THE MOVED PROSE IS REWRITTEN; each block keeps
%%     its own comments, which travel with the text they annotate.
%%
%%     TWO CHOICES MADE HERE, both reversible and both recorded.  The
%%     shared setup subsections -- Part II's "Common mathematical arena
%%     for the axioms", Part III's "Role of the postulates" and
%%     "Minimal formal setting shared by all postulates" -- STAY as
%%     subsections of their Part's Overview Section rather than
%%     becoming Sections of their own: three cross-references point at
%%     them and this keeps those resolving at their own granularity,
%%     and it is what puts AX-1 at its own Section immediately after
%%     the Overview.  And "Closing remarks on Part~II / III" is
%%     promoted from \subsection* to \section*, because with no
%%     top-level subsections left in these Parts a \subsection* would
%%     sit at the wrong depth.  The other six Parts are untouched and
%%     keep theirs.
%%
%%     TWO HEADINGS NO LONGER FIT, AND NOT ONE WORD OF THEM CHANGED.
%%     Promotion sets a heading at 14.4 pt where it used to sit at
%%     12 pt, and POS-2 ("... PLA <=> PLT/MEPP/FEP") ran 41.8 pt past
%%     the measure while POS-7 ran 24.1 pt past it -- neither has a
%%     hyphenation point LaTeX will take.  Both are given an explicit
%%     break after their colon and an optional argument so the Table of
%%     Contents entry stays on one line.  The words are untouched; only
%%     the place the line turns is chosen.
%%
%%     THREE MOVED SENTENCES POINTED AT THEIR OLD HEADING -- "the
%%     content this Historical Background credits", "the corresponding
%%     Historical Background subsection" -- and now point at the Section
%%     the text actually sits in.  Found by reading the rendered page.
%%
%%     THREE SENTENCES DESCRIBED THE OLD LAYOUT AND NOW DESCRIBE THE
%%     NEW ONE.  Part II's plain-English opening and Section 4's
%%     block list both ended their inventory of each axiom's internal
%%     format with "and the historical work it positions itself
%%     against", and Part III said each postulate "appears in the same
%%     format as the axioms, with a historical section".  All three
%%     now point at the gathered history Section instead of promising
%%     it inside each axiom and postulate.  ALSO MEASURED WHILE THERE:
%%     Section 4 called the per-axiom layout a "seven-block internal
%%     format" and then listed seven blocks of which one was the
%%     history and none was the "further mathematical elaboration"
%%     that has always been there.  AX-1, AX-2 and AX-3 now carry
%%     exactly seven blocks and AX-4 eight, so the count is right for
%%     the first time; the list is corrected to match what is on the
%%     page.
%%
%%     NEW PROSE IN THIS SHIP: two paragraphs, one opening each history
%%     Section.  Nothing else is written; everything else is moved.
%%
%% ASSERTED: 235 bibitems unchanged and none orphaned; labels 254 -> 256
%% (the two new history Sections); figures, tables, equations and
%% itemize environments unchanged; the multiset of comment lines is
%% IDENTICAL, so the changelog and every in-text comment survive intact.
%% ===================================================================
%% ===================================================================
%% v15.28 (August 18, 2026) -- BT-1: THE BIBLIOGRAPHY TITLE PASS.
%%
%% EVERY COMPANION IS NOW CITED BY THE TITLE PRINTED ON ITS OWN COVER.
%%
%% 10 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.
%% ====================================================================
%% ===================================================================
%% v15.27 (August 18, 2026) -- FP-1: THE DISCLOSURE-PREP PASS.
%%
%% The cohort is posted as TWENTY-ONE papers, with A11-Origin held back
%% until its chemistry and life Parts are rebuilt.  A1 is the paper that
%% tells a reader how the cohort arrives, so it carried more of the
%% disclosure debt than any other paper in the tree.
%%
%% (1) THE A11 BIBLIOGRAPHY ENTRY said "Preprint, available at
%%     arisaka-gg.org".  It now reads "In preparation. To be posted ...
%%     on completion; no result of the present paper depends on it."
%%     The dependency clause is the load-bearing half: the derivational
%%     flow runs A1 -> A10 -> A11 and never backward, so every A11
%%     citation in this paper is a FORWARD pointer, not a debt.
%%
%% (2) SIX PRINTED SENTENCES PROMISED A SIMULTANEOUS TWENTY-TWO-PAPER
%%     ARXIV DAY.  All six are now true statements of what we are doing:
%%     the twenty-one posted papers go up together and reach arXiv in
%%     four staged submissions.  The sharpest of the six was Objection
%%     52's answer to a senior sceptic -- "one citation away rather than
%%     'in preparation'" -- which the hold would have turned into a
%%     printed falsehood on the program's front page.  It now names the
%%     held capstone explicitly and says why, which is the stronger
%%     answer: the capstone is held BECAUSE its upper layers are not at
%%     the standard the rest of the cohort meets.
%%
%% (3) Objection 8 named "the three-paper read of A9+A10+A11" as the
%%     minimum critical reading for the 28-to-0 parameter-collapse
%%     question.  A11 CONSUMES that result rather than contributing to
%%     it, so the reading set was never right; it is now A9+A10.
%%
%% (4) FIVE PRINTED SITES still called A11's cascade SEVENTEEN-layer.
%%     It has been eighteen since A11's Stage 4 (Report 441).  Found by
%%     the same sweep; unrelated to disclosure and repaired here because
%%     the paper is being shipped once.  The "twelve to seventeen
%%     explicit falsifiers" site is a DIFFERENT quantity and is frozen.
%%
%% (5) The cohort inventory's A11 row carries a dagger and a one-line
%%     footnote saying the capstone is in preparation and not posted.
%%     The cohort MAP plate is a frozen standard and is not redrawn.
%%
%% (6) catalogue -> catalog, printed text only (Report 443's pair-list
%%     hole; the charter list and spelling_audit.py both lacked it).
%%
%% NO OTHER CONTENT CHANGED.  This is disclosure preparation, not an
%% editing window.
%% ===================================================================
%% ===================================================================
% v15.26 (August 17, 2026) -- MIGRATED TO STYLE FILE v11.5.
%%
%% FOUR MACROS THAT LIVED IN THE PAPERS COME HOME, AND NOT ONE PAGE
%% MOVES.  This paper's part in that:
%%
%% * \GGShowcaseHead removed; this paper prints the sans form with \bigskip -- the style file default,
%%   and needs no setup line.
%% * the landscape page's own smaller headline, which had a
%%   second macro name (\GGShowcaseHeadL), is now the
%%   optional size argument: \GGShowcaseHead[\large]{...}.
%%   \Large does not fit that page and that was measured.
%%
%% 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 v15.25 printed, except for the lines named above.
%% ===================================================================
%% ------------------------------------------------------------
%% v15.24 -> v15.25, August 16, 2026.  THE SHOWCASE-HEADLINE RIDER.
%% \GGShowcaseHead rises from \large to \Large, one token.
%%   THE MEASUREMENT.  Across the cohort the showcase headline had four
%%   sizes: \large in the eight papers where I set it, \Large in C1 and
%%   C2, \LARGE in A4, \huge in A9, A10 and A11.  Those are not four
%%   arbitrary choices -- the rule followed was to match each paper's own
%%   existing headline, and four of the four larger values are the
%%   author's.  What the measurement shows is that MY DEFAULT was the
%%   outlier: every headline the author set himself is LARGER than
%%   \large, and on five papers out of five his practice is \Large or
%%   bigger.  On a page whose job is to be read in five minutes from a
%%   homepage, \large reads as a subsection head, not a title.
%%   So the house default rises to \Large, the smallest size he himself
%%   uses.  The cohort now spans \Large / \LARGE / \huge and nothing is
%%   below the author's own floor.
%%   VERIFIED: page count unchanged, no new overfull box, body
%%   page-for-page identical.
%% NEW PROSE IN THIS SHIP: none.
%% ------------------------------------------------------------
%%%% ------------------------------------------------------------------
%%%%  v15.16 (August 9, 2026): the curious-reader cluster.
%%%%  Twenty-five broad-audience questions open the Common Objections
%%%%  Part, ahead of the six community objections, on the model of the
%%%%  clusters in W1-Symmetry, W2-Holonomy and A10-Cosmos.  Eight are
%%%%  the spine those three share, answered here in this paper's own
%%%%  vocabulary; seventeen are questions only a constitutional paper
%%%%  can be asked.  The introduction names the hardest of the set.
%%%% ------------------------------------------------------------------
%% ------------------------------------------------------------------
%%  v15.15 (August 9, 2026): two cohort rules applied.
%%  (1) British doubled -l corrected in prose only -- 6 site(s);
%%      6 further site(s) protected as comment, identifier or
%%      bibliography and deliberately untouched.
%%  (2) Ragged-right on 1 p-column(s) across 1 multi-p-column
%%      table(s): a narrow prose column cannot be justified
%%      without a visible interword stretch, and an overfull box
%%      under the gate's tolerance can still eat the gutter.
%% ------------------------------------------------------------------
%% ===================================================================
%%  A1-Constitution v15.8  (August 7, 2026)  -- stage 2 of the v16 standards.
%%  Every bare grid in the running text promoted to a numbered,
%%  captioned float (the long one to a longtable, which can break);
%%  every thin caption rewritten as a paragraph that says what the
%%  object shows, how to read its columns, and what to take away.
%%  Each promoted table's lead-in sentence was rewritten to name it
%%  by \ref, because a floating table with a 'listed below:'
%%  lead-in is the worse defect.
%%  Recipe: [v15-Support]/ship_stage2.py + a1a4_captions.py.
%% ===================================================================
%% ===================================================================
%%  Arisaka-GG-A1-Constitution-v15.7  (August 7, 2026)  -- the v16 mechanical sweep.
%%  Shared cohort plates synced from A5; the house flowchart palette
%%  applied (no bare \draw, box fills within !8..!18, keyword widths
%%  made explicit); every flowchart re-weighted so its lightest box
%%  rule RENDERS at the house 0.521pt and nothing above 0.956pt.
%%  Gate numbers are identical to the previous version: this sweep
%%  changes ink, not layout.
%%  Recipe: [v15-Support]/sweep_v16.py + palette_fix.py.
%% ===================================================================
%% ===================================================================
%%  A1-Constitution v15.6  (August 7, 2026)  -- section 4.7 expanded.
%%  'Comprehensive summary table for AX-1--AX-4' becomes
%%  'Comprehensive summary of AX-1--AX-4' and now carries three
%%  views: the auditor's table, the nested tower, and the
%%  four-axiom historical timeline.  Both plates are carried
%%  unchanged from A5-GDOS v15.8 and are a SHARED PAIR: an edit
%%  to either must be applied in both papers.
%%  Recipe: [A1-Constitution]/[Scripts]/ship_a1_v156.py.
%% ===================================================================
%% ===================================================================
%%  A1-Constitution v15.5  (August 6, 2026)  -- style v11.4.
%%  No change to the paper: v11.4 gives every figure-and-caption block
%%  an extra line of space on each side, and the paper inherits it.
%%  Re-gated because the extra space reflows the document.
%% ===================================================================
%% ===================================================================
%%  A1-Constitution v15.4  (August 6, 2026)  -- corrective ship.
%%  (1) 65 residual British spellings, missed by the stem-list
%%      sweeper that shipped v15.3: a leading word boundary loses a
%%      listed stem the moment it carries a prefix (organis /
%%      reorganises, favour / disfavoured), and the list itself was
%%      never complete.  The sweeper is now generative.
%%  (2) built against style v11.3: the cover date is gone and
%%      captions are \small, a step under the 11pt body.
%%  (3) 3 table(s) to \footnotesize.  \small reads too close to
%%      the body size; resizebox'd tables are exempt.
%%  (4) float fractions retuned, so a plate no longer claims a page
%%      and leaves half of it blank.
%%  Recipe: [v15-Support]/ship_a1_v154.py.
%% ===================================================================
%% ===================================================================
%%  A1-Constitution v15.3  (August 6, 2026)  -- four cosmetic fixes.
%%  (1) Figure 1: the two boxes move 0.8cm apart, which lengthens both
%%      arrows, and the AX-2 / AX-3 labels move to their arrow midpoints
%%      so they no longer sit against the boxes.  Plate 0.701 -> 0.88 of
%%      \textwidth (+25%), enlarging picture and type together.
%%  (2) Tables 2, 3, 6, 9, 10 declared no font size and so printed at the
%%      11pt body size.  Set to \footnotesize (9pt), which is what
%%      Table 8 uses; Table 4, the other reference, is \small (10pt).
%%  (3) Table 5: \small -> \footnotesize; columns re-cut from 0.85 of
%%      the measure to fill it; the Role column 0.255 -> 0.360 (+41%).
%%  (4) Figure 9: sized to \linewidth, not \textheight.  Inside
%%      \begin{landscape} lscape swaps them -- \linewidth is the
%%      rotated width (650.43pt), \textheight the short side (469.76pt)
%%      -- so the plate had been set to 72% of the room it had.
%%  Recipe: [v15-Support]/ship_a1_v153.py.
%% ===================================================================
%% ===================================================================
%%  A1-Constitution v15.2  (August 6, 2026)  -- front matter and lists.
%%  (1) Cover: the author block is four lines instead of six, and the
%%      space between the coloured rules and the title is 30 percent
%%      smaller -- 29.3pt above and 27.4pt below become 20.6 and 19.2,
%%      measured off the rendered page.  Both changes are in
%%      ucla-arisaka-gg-11.sty (now v11.2) and therefore apply to all
%%      22 papers.  The Abstract now fits page 1 and was NOT trimmed.
%%  (2) The Executive Summary is one printed line shorter, so its last
%%      sentence no longer orphans "the page." onto a page by itself.
%%      Tightening only: no clause or example is removed.
%%  (3) Five figure short captions shortened so every List of Figures
%%      entry sits on one line.  The captions printed under the figures
%%      are unchanged -- only the list text.
%%  (4) A List of Tables added after the List of Figures.  All twelve
%%      tables gain a short caption, taken from the caption's own
%%      leading title sentence, so every entry is one line.
%%  Recipe: [v15-Support]/ship_a1_v152.py.
%% ===================================================================
%% ===================================================================
%%  A1-Constitution v15.1  (August 6, 2026)  -- US Letter.
%%  The \documentclass paper option changes from a4paper to
%%  letterpaper, and nothing else does.  US Letter is wider and
%%  shorter: \textwidth 452.97pt -> 469.76pt (+16.79), \textheight
%%  700.51pt -> 650.43pt (-50.08).  Each page therefore holds less and
%%  the paper reflows; the extra width also clears table overruns that
%%  the A4 measure could not.  Content, spelling and figures are
%%  exactly as shipped in v15.0.
%%  Two layout repairs come with it, both forced by the shorter page
%%  and neither touching content:
%%   (1) the cohort-map plate was centred by \vspace*{\fill} above and
%%       below inside a \clearpage pair.  On the shorter measure the
%%       plate no longer fits beside that glue, so it was pushed off
%%       its page and left a completely blank page on either side.  The
%%       two fills and the leading \clearpage are dropped; the plate
%%       still takes a page of its own.
%%   (2) the abbreviations longtable carried four hand-placed
%%       \pagebreak commands, calibrated to where the A4 page broke.
%%       At US Letter they fell in the wrong places and left a
%%       five-line page and a stub.  They are removed and the table
%%       breaks by itself.
%%  Recipe: [v15-Support]/ship_a1_v151.py; measured impact in
%%  GG-Report-214; page-fill measurement in [v15-Support]/page_fill.py.
%% ===================================================================
%% ===================================================================
%%  A1-Constitution v15.0  (August 6, 2026)  -- American spelling.
%%  British to American spelling throughout the running prose: -ise to
%%  -ize and -yse to -yze in their verb forms only, -our to -or, -re to
%%  -er, doubled-l inflections reduced, and the individual words
%%  (sceptic, storey, artefact, whilst, amongst).  Nothing outside
%%  running prose is touched: cross-reference and citation keys keep
%%  their spelling, so fig:two-fibres-constitutional and every
%%  reference to it are unchanged; so are the bibliography and its
%%  cited titles, every macro name, and the changelog above.  Noun
%%  forms that are identical in both varieties (analysis, realism,
%%  organism, specialist, emphasis) are left alone, and the -es forms
%%  of analyse/paralyse/catalyse are reported rather than guessed at.
%%  The page size is deliberately unchanged: A1 remains A4 here, and
%%  the US Letter migration is a separate cohort-wide ship.
%%  Recipe: [v15-Support]/spellsweep.py and ship_a1_v150.py.
%% ===================================================================
%% ===================================================================
%%  A1-Constitution v13.17  (August 6, 2026)  -- landscape figures restored.
%%  The cohort figure sweep sets each plate to a fraction of
%%  \textwidth.  Inside \begin{landscape} the page is rotated but
%%  \textwidth is not, so the usable width is about 246mm while
%%  \textwidth still reports 159.2mm, and a plate placed by the rule
%%  lands at roughly half the size it should.  1 figure on
%%  landscape page restored to its exact pre-sweep placement.
%%  Four figures cohort-wide were affected; none other sits inside a
%%  landscape environment.  Script: [v13-support]/revert_landscape.py.
%% ===================================================================
%% ===================================================================
%%  A1-Constitution v13.16  (August 6, 2026)  -- the cohort-wide figure-size sweep.
%%  6 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.
%% ===================================================================
%% ===================================================================
%%  A1-Constitution v13.15  (August 5, 2026)  -- the cohort funnel at the house size.
%%  The central-claim funnel was drawn with 114mm boxes and then
%%  stretched to the full text width, so its \footnotesize landed 47% larger than the roadmap
%%  and cohort-map plates beside it.  The plate target is now
%%  0.681\textwidth, computed rather than chosen: the picture measures
%%  125.2mm wide compiled standalone, and the house type size is set by
%%  GG-P3-Electron Figure 3 (0.93\textwidth on 171.1mm, \textwidth =
%%  159.2mm, so type at 0.865 of natural size), which for any picture
%%  of width W means f = W / 184.0.
%%  Nothing inside the picture changes: no font command, no box
%%  dimension, no text, no relative position.
%%  Recipe and the measured table for all fourteen papers:
%%  [v13-support]/funnel_sweep.py.
%% ===================================================================
%% ===================================================================
%%  A1-Constitution v13.14  (August 5, 2026)  -- the cohort walk
%%  completed.  The cohort-inheritance Section profiles the programme
%%  one paragraph per paper and had none for GG-P6-Proton,
%%  GG-W1-Symmetry or GG-W2-Holonomy: a reader who had just seen
%%  twenty-two boxes in Figure 3 was then walked through nineteen.
%%  Three entries added at the scale of the P4/P5 entries beside them.
%%  The Section's opening also still listed the P-series as
%%  "(P1--P3) and the two foundations members (P4--P5)" and described
%%  the left flank with no mention of the right; both repaired.
%%  Neither carries a number, so no count sweep would have found them.
%% ===================================================================
%% ===================================================================
%%  A1-Constitution v13.13  (August 5, 2026)  -- the three-line page 26.
%%  The cohort-inheritance Section spilled three lines past the
%%  \clearpage that precedes Figure 3, leaving an otherwise blank page.
%%  The paragraph that spilled was also stale -- it named the
%%  Predictions Branch as P1/P2/P3 and never mentioned the white
%%  papers -- as was the branch paragraph above it ("the six short
%%  falsifier papers").  Neither used the word "nineteen", so the
%%  v13.11 count sweep did not reach them.  P6 added, "six" -> "seven",
%%  and the closing map paragraph rewritten complete and shorter: the
%%  figure below already names every node, so the prose now states the
%%  shape rather than re-listing it.  Section ends on page 25 and the
%%  blank page is gone.
%% ===================================================================
%% ===================================================================
%%  A1-Constitution v13.12  (August 5, 2026)  -- the cohort map made
%%  readable.  Boxes and the type inside them x1.15, every word
%%  outside a box x1.20, at unchanged node centres; both copies.
%%  The long A10 -> C-frame edge becomes a rounded polyline: as a
%%  Bezier its control points sat at x = 146mm while its ink stopped
%%  at 127mm, and a bounding box contains control points, so 7% of
%%  the plate's width was empty space the \resizebox was scaling the
%%  whole map down to accommodate.  The centring pad, which existed
%%  only to cancel that phantom, retires.  Gaps re-opened where the
%%  growth closed them (Predictions pitch 46 -> 50mm, Cosmology
%%  44 -> 50mm, the white-paper column's labels to the right of its
%%  edge, the A10 -> W2 feed redrawn).  Net on the page: about +26%
%%  against v13.11 and about +20% against the nineteen-paper map.
%%  Reasoning and measurements: [v13-support]/map23_bigger.py.
%% ===================================================================
%% ===================================================================
%%  A1-Constitution v13.11  (August 5, 2026)  -- Figure 3 becomes the
%%  TWENTY-TWO-paper cohort map.  GG-P6-Proton joins the Predictions
%%  row right of GG-P3-Electron (four P-series predictions on the row,
%%  chained by a second sister-import arrow); GG-W1-Symmetry and
%%  GG-W2-Holonomy install as a right-flank column at the mirror
%%  positions of GG-P4-Real and GG-P5-CPT, so the spine is flanked by
%%  two columns of two that do opposite jobs -- the left feeds it, the
%%  right reads it.  GG-P7-Inflow does not appear: it is Appendix H of
%%  GG-P2-N3.  Geometry: Predictions frame +-66 -> +-91mm; the long
%%  dashed A10 -> C-frame edge re-routed outboard of the new column
%%  (its old line ran through the W2 node); left bbox pad -118.8 ->
%%  -127mm; resizebox 0.86 -> 1.0\textwidth.  Caption trimmed.
%%  Full geometry reasoning: [v13-support]/map22_v20.py.
%% ===================================================================
%% ===================================================================
%%  Arisaka-GG-A1-Constitution v13.9 -- the OGN-5 cascade figure is
%%  redrawn.  A1 owns this artwork; A2 reproduces it and A3 inherits it,
%%  so it had to change here or the three papers would disagree.  The
%%  old figure showed M, N, L together as primary and the dual lock at
%%  the bottom as a consistency check.  The flow is now inverted: M and
%%  L are fixed without any colour count, the dual lock DETERMINES
%%  N = N_c = 3 via N = L +/- sqrt(L - M^2) with N < L excluding the
%%  second root, and the trace ratio appears dashed and to one side as
%%  the independent check it is.  The prose is brought with it, and the
%%  exchange is stated: agreement of the lock's two readings is now
%%  required rather than observed, so it is no longer claimed as a free
%%  confirmation.
%% ===================================================================
%% ===================================================================
%%  A1-Constitution v13.8  (July 30, 2026)  -- Phase-2 shape-up.
%%  TABLE REDESIGN (Report 78 sec.9.2).  tab:paper_inventory -- the five-column
%%  forward-map inventory -- rendered with severe narrow-column hyphenation:
%%  the 0.10\linewidth first column had to hold names like
%%  "Arisaka-GG-A1-Constitution", so it broke every row into fragments while
%%  the justified prose columns inter-word-stretched ("Phenomenology: non-" /
%%  "spatial Kaluza-Klein").  Redesigned per the cohort table standard: set at
%%  \small (slightly below main text), first column widened 0.10 -> 0.155 to
%%  take a full paper name, the other four rebalanced (0.150/0.255/0.185/0.105,
%%  summing to 0.85\linewidth), and >{\raggedright\arraybackslash} on all five
%%  so prose cells break naturally instead of stretching.  No row content,
%%  ordering, or caption text changed.
%% ===================================================================
%% ===================================================================
%%  A1-Constitution v13.7  (July 30, 2026)  -- P2-N3 v14.0 conformance.
%%  The "What P2 carries" paragraph's conditioning parenthesis updated:
%%  (Budget Conjecture, H1--H2) -> (coupling clause + integrality lemma
%%  L-INT, at Born-grade premises), matching P2 v14.0's capacity
%%  architecture, in which the budget is theorem T-N3-6, H1 is closed in
%%  the negative (rigidity), and H2's residue is L-INT.  One edit.
%% ===================================================================
%% ===================================================================
%%  A1-Constitution v13.6  (July 29, 2026)  -- P2-N3 retitle propagation.
%%  P2-N3 is retitled "Why Three Generations: A THREE-LEG Derivation of N = 3
%%  from BI-6, MSMF, and GDOS Morse Stability" (was "A Closed Derivation").
%%  Reason: P2-N3's own main theorem T-N3-5 is "Conditional derivation of
%%  N = 3 (three-leg pincer)", conditional on five structural hypotheses --
%%  including the BI-6 Budget Conjecture, which A2 states is not derived
%%  anywhere in the cohort -- plus two empirical inputs.  The cohort's prose
%%  already flagged that conditioning everywhere; only the title claimed
%%  "Closed", and titles propagate into every bibliography.
%%  T-N3-5 KEEPS its own name: the theorem is conditional and says so.
%%   * COHORT ROSTER, P3 row: "the five oldest electron questions ---
%%     identity, stability, copies, pointlikeness, sensing" -> "the six
%%     oldest ... identity, stability, copies, THE POSITRON, pointlikeness,
%%     sensing".  P3's title page reads "The Six Oldest Questions about the
%%     Electron", A1's own "What P3 carries" paragraph already said six and
%%     named all six, and every bibitem in the cohort says six.  Only this
%%     row was stale, and its enumeration was missing the conjugate readout.
%%   * P2-N3 described as a "closed derivation" in two further places -- the
%%     forward-map roster row and the "What P2 carries" paragraph -- now
%%     "three-leg derivation", matching the retitled paper.  The row already
%%     ended "with its conditioning declared" and still does.
%% ===================================================================
%% ===================================================================
%%  A1-Constitution v13.5  (July 29, 2026)  -- internal-affairs sweep.
%%   * "Note on version and Phase numbers" paragraph REMOVED.  It documented a
%%     version-stamp convention for rows of the coherence table; that table
%%     has 11 rows and zero version stamps, so the paragraph described a
%%     convention absent from the paper.  It also carried THREE dangling
%%     prepositions -- "cites A2 at, ... A9 at, ... Arisaka-GG-A11-Origin at."
%%     -- left behind when version numbers were stripped in an earlier pass,
%%     visible in the PDF; the editor instruction "Whenever a companion paper
%%     is revised, the cited version stamp here must be updated accordingly;
%%     this is the audit contract"; and a pointer to an internal document the
%%     reader cannot see ("the project rules document of the GG-Theory
%%     programme").  The sentence above it already introduces the table, and
%%     the "Audit trail" paragraph below keeps the reproducibility claim.
%%     The script asserts the 11-rows/0-stamps premise before deleting.
%%   * "A2 v7.x internal theorem numbering" -> "A2 internal theorem numbering".
%%   * "of its v9.5 revision" and two glossary "A6 v9.5" stamps dropped.
%%   * Two draft-lineage sites, both caught by the post-condition not the scan:
%%     "In earlier drafts of this programme, OGN and MF were sometimes folded
%%     together" -> "It is tempting to fold them together; here they are
%%     strictly separated" (the distinction is the content and is kept); and
%%     the self-application argument, which justified the admissibility
%%     predicate by noting it fails "this programme's own earlier drafts".
%%     That force is preserved -- the predicate now rules inadmissible any
%%     formulation carrying free moduli, including of this programme's own
%%     content, and still "could not ... exclude versions of its own
%%     programme" -- without asking the reader to inspect drafts.
%%   * Rendering p.137 showed the SAME editor instruction restated at the end
%%     of the Audit-trail paragraph, naming the internal workflow: "Whenever a
%%     companion paper renumbers its theorems, the corresponding row of Table 6
%%     must be revised in the next Phase of the Constitution."  Replaced with
%%     the property that makes the table robust -- rows name theorem content as
%%     well as number, so a renumbering shows as a visible mismatch.  This is
%%     the same discipline A11's citation-auditor objections now state.
%%  Not touched: two "first pass"/"second pass" hits, both reading-sense.
%%  No axiom, postulate, theorem, value or citation target changed.
%% ===================================================================
%% ===================================================================
%% v13.4 (July 24, 2026): COHORT BIB-TITLE RECONCILIATION (P3)
%%  The electron companion's subtitle is updated in its bibliography
%%  entry to that paper's current cover ("The Six Oldest Questions
%%  about the Electron ..."), following its addition of the sixth
%%  question -- the origin of positrons.
%%  The roster description of the electron paper gains the sixth
%%  question (the positron as conjugate readout) and the count. No other change
%%  anywhere.
%% ===================================================================
%% ===== v13.3 (2026-07-20): STAGE 5C -- WAVE-2 RESIDUE CLEANUP =====
%% (1) F1: Suffix-roster rename-history clause removed,
%%     'GRIP (A6, ...)' -> 'GRIP (A6)' (A5/A6/A7/A8 precedent).
%% (2) F4: Table-5 caption cohort-status note (A8-QM two-
%%     parallel-manuscripts drafting remark) -> released
%%     register: all foundational papers released preprints.
%% Physics, theorem keys, numbers, values, grades: UNCHANGED.
%% ===================================================================
%% ===== v13.2 (2026-07-20): WAVE 2 STAGE 1 -- the A1 anchor ship (content) =====
%% Script: a1_v13.2_stage1.py.  Foundation: GG-Internal-Report-RealBulk-
%% Audit-2026-07-19.md (Sec. 1, A1 ledger A1-R1/R2/R3, Sec. 3, and the
%% author-approved POS-block addendum, items (1)-(5)).
%% (1) sec:AX4_further gains the primitive-category axis: D-Constitution-1
%%     (structure cost = count of unforced choices, defined without
%%     reference to realness), the R/C/H pricing, L-Constitution-1
%%     (cohort alias L-Real-1; GRADE: CANDIDATE -- the pricing argument
%%     is stated, the uniqueness rests on D-Constitution-1's counting;
%%     constitutional forcing argument in the sense of A2's Remark 0.5),
%%     and the named falsifier F-Constitution-10 (cohort alias F-Real-1;
%%     three kill clauses), registered as a new row of the consolidated
%%     falsifier table (tab:pull_table).
%% (2) AX-3: register note on the symbol H (the dictionary outputs a
%%     COMPLEX Hilbert space; its scalars are not a constitutional
%%     primitive -- boundary shorthand, not bulk input; T-P4-1/2/3).
%% (3) AX-3: Synthesis paragraph -- time's four endowments (orders /
%%     repeats / opens one way / opens the analytic shore; P5 Part IX
%%     canonical) + the arrow-custody anchor (AX-3 owns the arrow; P5
%%     Part IX is the law-level derivation; A5 the dynamical
%%     implementation; consumers say so).
%% (4) POS-1 (DGI): pedigree note (complex apparatus boundary-side and
%%     derived; [Q,L_k]=q_k L_k real-form recoverable via T-P4-1).
%% (5) POS-4: custody note (CP/Choi/Stinespring/GKLS are complex-algebra
%%     theorems, lawful downstream of the derived i; the bulk never runs
%%     GKLS; no realification attempted or needed).
%% (6) U(1) harmonization: one bridging sentence at AX-2 Derivation 3 +
%%     half-sentence pointer at the POS-6 covariant derivative.  No
%%     equation altered.
%% (7) Part III closes with the NEW "reality register of the postulates"
%%     (seven-row table + certificate sentence: every i in this document
%%     is either boundary-side and derived, or absent).
%% (8) Part IV gains the Foundations Branch bridge subsection (What
%%     GG-P4-Real carries / What GG-P5-CPT carries; grades honest:
%%     quasi-free / T-P4-8 conditional; mirrors derived / matter
%%     derived-conditional on H-phigeo-1).
%% (9) Sec. 3.1 Predictions-Branch enumeration completed (P1-P3, C1-C3;
%%     pre-existing staleness).
%% New Section-B references: Renou2021, Peres1979, Sinha2010.  The
%% eleven axiom/postulate statement environments: byte-identical
%% (protect-asserted).  Abstract and executive summary: UNTOUCHED.
%% v13.1 -> v13.2.
%% Pre-signoff amendment (a1_v13.2_exec_amend.py, 2026-07-20): the ONE sanctioned exec-summary change -- L-Constitution-1 candidate sentence installed in the Significance paragraph (AX-4 item); version stays v13.2.
%% =====================================================================
%% ===== v13.1 (2026-07-19): WAVE 1 of the 19-paper promotion (mechanical/structural) =====
%% Script: a1_v13.1_wave1.py.  Foundation: GG-Internal-Report-RealBulk-
%% Audit-2026-07-19.md (A1 ledger, Sec. 4, App. A) + frozen map master
%% Arisaka-GG-Cohort-Map-19-v1.4.tex.
%% (1) Figure 3 (fig:A1-cohort-inheritance-map) and the Part-IV
%%     relations figure (fig:paper_relations) replaced by the frozen
%%     nineteen-paper cohort map (v1.4 master, marker-free: GG-P4-Real
%%     and GG-P5-CPT as the Foundations Branch column on the left
%%     flank).  Figure-3 caption = master caption verbatim; relations
%%     caption updated (nineteen + Foundations Branch sentence).
%% (2) Cohort-count sweep per the audit's Appendix-A inventory:
%%     seventeen -> nineteen at every reader-facing cohort-count site;
%%     "sixteen companion papers" -> "eighteen companion papers";
%%     arithmetic sites recast 11 A + 2 Foundations + 3 Predictions +
%%     3 Cosmology.  Semantic "seventeen" sites (GRIP cascade layers,
%%     falsifier count) untouched.
%% (3) Roster/naming: canonical nineteen-paper roster sentence
%%     (Sec. 3.1 opening); suffix roster gains Real (P4) and CPT (P5);
%%     Part-IV inventory table gains the two Foundations rows;
%%     inventory intro and both map captions name the branch.
%% (4) Sec. 3.1: two new per-paper paragraphs, GG-P4-Real (the origin
%%     of i) and GG-P5-CPT (the mirrors and the matter), in the house
%%     register; conditional items stated conditional.
%% (5) Preamble: \PaperPFour/\PaperPFive macros; new Section-A
%%     bibitems ArisakaP4Real, ArisakaP5CPT with Relevance notes.
%% (6) One-line Part contents entries: \GGPart renewed (P4-Real v3.1
%%     pattern) so Overview of Contents and Contents carry
%%     "Part N. <title>" only.
%% Physics, values, theorem keys, grades: UNCHANGED.  v13.0 -> v13.1.
%% =====================================================================
%% ===== v13.0 (2026-07-16): promotion to the v13 series (GG-v13 project) =====
%% Surgical fix from the A3 hostile-review Tier C plan (author-approved;
%% report A3-BI6-HostileReview-Report-and-ActionPlan-2026-07-16.md):
%% eq. BI6_pub carried a scrambled coefficient placement
%% (2/5 on Tr R^2, -1 on F_B^2); corrected to the canonical
%% three-component truncation Tr R^2 - Tr F^2 - (2/5) F_B^2,
%% matching eq. BI6_pos3 and the A2/A3 canon. No other changes.
%% =====================================================================
%% v11.25 (2026-07-16)  TWO-QUANTITY Z* SWEEP (cross-paper F7 item of
%%   the A10 z* hostile audit, author-approved): the single one-quantity
%%   site ("The recombination redshift descends from the integer J = 7
%%   ... matching the Planck value at the 0.6% level") recast in the
%%   two-quantity form of A10-Cosmos: the corridor switch z_* = e^7 is
%%   a topological marker, distinct from the measured decoupling
%%   redshift, which A10's own atomic chain reproduces at the 0.1%
%%   level with zero fitted parameters, the switch 0.4% (in e-folds)
%%   above it.  v11.24 -> v11.25.
%%   Script: a1_v11.25_two_quantity_zstar.py.
%% =====================================================================
%% v11.24 (2026-07-14, {e,pi} dictionary) -- presentation only.
%%   At the GGEP "a scale and a phase" geometrization (the constitution's
%%   canonical home for the two non-spatial dimensions), named the {e,pi}
%%   dictionary: e = base of the exponential along the scale interval I_r,
%%   pi = angle of the gauge circle S^1_theta. Cohort-wide {e,pi} sweep.
%%   No values changed. Script: a1_epi.py.
% v11.23 (July 5, 2026): tensor/inflation sync to A10 canon -- r<<10^-3 +
%   TeV-scale inflation -> r=0.0043 two-sided + GUT-adjacent plateau.
% v11.22 (July 2, 2026): cohort universal-indentation pass (author-
%   directed): indentfirst loaded (first paragraphs after headings
%   now indent). Per-paper stray-noindent review deferred to the
%   v2.0 pass. No content changed.
% v11.21 (July 2, 2026): cohort multi-Arisaka citation split pass
%   (charter natbib rule; author-directed 2026-07-02): 12
%   multi-key cites split into citealp/citet chains; fixes the
%   same-author-same-year comma-run rendering. No content changed.
% ===== v11.20: COSMETIC table-font normalization -- inserted \small before
%  \begin{tabular} in 1 oversized (unsized) table floats so every table
%  sits ~10%% below the body text (matches A10 v12.43). Already-small tables untouched.
% Bib cross-citation titles reconciled to canonical cover titles (cross-citation integrity), 2026-06-27.
% A10 bib-title reconciled to canonical cover title (cross-citation integrity), 2026-06-27.
% ===== A1-Constitution v11.17 (2026-06-27): cohort de-contamination (protect-list aware). =====
% Shed GG self-description: warped 6D/six-dim bulk -> 6D/six-dim bulk; warped metric/geometry/radial
% -> scale-profile metric / bulk geometry / radial; warp factor -> scale profile; warp-curvature
% coupling -> scale-curvature coupling; warp/RG|warp/integer -> scale/RG|scale/integer; warping ->
% scale running; OGGEP holographic projection -> bulk-to-boundary projection; holographic dictionary
% -> bulk--boundary correspondence; holographic-RG/geometrization -> renormalization-group/geometriz.
% KEPT every external AdS/CFT comparison/disclaimer, all Maldacena/Witten/Gubser/deHaro/Skenderis/
% Solodukhin/Susskind/'t Hooft/Ryu/Takayanagi citation contexts, the holographic PRINCIPLE, the QiHAL
% Holographic Attractor Lattice / holographic alphabet / Neural Holographic Tomography glosses, the
% Randall--Sundrum warped extra-dimension model glossary def, the WHOLE Objection-12 AdS/CFT-vs-OGGEP
% comparison block, and the ENTIRE References Part (every cohort bibitem + relevance note). hypersetup
% pdfsubject de-contaminated. Body only. No physics/values changed. =====
% ===== A1 v11.16 (2026-06-25): cohort central-claim funnel (light, foundation variant). =====
% Added a modest 'place in the cohort' section (sec:cohort-funnel) leading with the paper's own
% subject, plus the shared open-system funnel (fig:cohort-funnel, generic payload). Framing kept
% deliberately light per author guidance; the paper proceeds on its own terms. No new physics.
%% =====================================================================
%%  A1-Constitution v11.15 (migrated onto ucla-arisaka-gg-11) -- 2026-06-22
%%  Single canonical style.  Stripped the now-redundant header overrides
%%  (Ack/Overview/LoF/Contents/ExecSummary); kept R5 TOC indents; Preface
%%  via \GGPreface; single-page Overview+LoF via \GGfrontmatteronepagetrue.
%%  No content changed.
%% =====================================================================
%% =====================================================================
%%  A1-Constitution v11.14 (centered headers) -- 2026-06-22
%%  Six headers centered + \LARGE\bfseries (cohort-universal style):
%%  Executive Summary, Preface, Overview of Contents, List of Figures,
%%  Contents, Acknowledgments.  R4 (Overview+LoF one page) and R5 (TOC
%%  indents) preserved.  No content changed.
%% =====================================================================
%% =====================================================================
%%  A1-Constitution v11.13 (naming-convention rollout) -- 2026-06-19
%%  STEP 3: new 'Ten Structural Innovations of the GG-Constitution' section
%%  (I-Constitution-1..10, compiled from the Significance subsection) + the
%%  cohort 'Global naming convention' key (suffix Constitution, P=Prediction).
%%  STEP 2: consolidated falsifier table rows tagged F-Constitution-1..9
%%  (each row = a POS-7/MF claim contract).
%%  STEP 1: (1) Objection tags O-A1-n -> O-Constitution-n (suffix = paper
%%  name, per A2's ratified key; A1 had used the position code).  (2) The
%%  twelve Problems-Solved entries P1..P12 tagged 'Pk (PS-Constitution-k)'.
%%  (A1 has no theorems of its own; AX-1..4 / POS-1..7 keep their names.)
%%  Falsifier ids, Innovations section, and the convention key follow.
%% =====================================================================
%% =====================================================================
%%  Arisaka-GG-A1-Constitution-v11.12 (v11.12) -- stale-value fix -- 2026-06-19
%%  Corrected the one stale kaxion mass 'kaxion at ~19 ueV' -> '~8.1 ueV'
%%  (canonical value, verified vs the authoritative source C1-Kaxion /
%%  A9-SM / A6-GRIP; A3/A4/A5 already carried 8.1 ueV).  Frozen-physics
%%  stale-value exception, flagged.  No other change.
%% =====================================================================
%% =====================================================================
%%  A1-Constitution v11.11  (new-standard Step 2: final review) -- 2026-06-18
%%  Reconciled stale Part-count self-references to the actual 10-Part
%%  structure (I-X): rendered 'eight Parts'/'nine Parts' -> 'ten Parts' in the
%%  definitions section and the cohort-inheritance/roadmap section; renamed
%%  the two stale-named labels (roadmap-nine-parts, A1-nine-parts-flowchart)
%%  to '...ten-parts...'.  All other Part references verified correct.
%% =====================================================================
%% =====================================================================
%%  A1-Constitution v11.10  (GG-QA+References, new-standard Step 1) -- 2026-06-18
%%  Common Objections Part expanded 26 -> 54 Q&A (new >50 standard): 28 new
%%  hostile-reviewer objections (O-Constitution-27..54) in 5 categories -- the four
%%  axioms' independence/necessity/minimality, the seven postulates, the
%%  'constitution' framing + descent claim, the constitutional-tower
%%  classification, and the meta-status -- each cross-linked to A1's sections
%%  and the companion papers.  Q&A intro count framing updated.
%% =====================================================================
% ===== v11.9 (2026-06-17): References refinements (author Go). =====
%  (1) Relevance-link EVERY reference (all 223 cited; 0 orphans) to its first-citation subsection,
%  not just the cited-twice subset. (2) Split the bibliography into two groups: A. the 17 Arisaka
%  cohort papers; B. all other references (each a \section* + thebibliography block). (3) Removed the
%  duplicate ToC 'Part X. References' entry (manual \addcontentsline; \GGPart writes the proper one).
%  v11.8 -> v11.9.
% =====================================================================
% ===== v11.8 (2026-06-17): relevance-links, lighter/scoped (author Go). =====
%  Appended an internal pointer 'First cited in \S\ref{...}' to the Relevance note of each cohort
%  paper (Arisaka*) and each load-bearing external reference (cited >=2x in the body), linking to the
%  subsection of its first in-text citation. Singly-cited peripheral/historical refs left unchanged.
%  Bibliography only; no body changes. Part of the cohort relevance-link guideline. v11.7 -> v11.8.
% =====================================================================
% ===== v11.7 (2026-06-17): fix empty page before the roadmap figure (author Go). =====
%  The centred figure+caption block was ~2mm taller than \textheight, so \null\vfill forced a page
%  break and left a blank page before it. Shrank the figure to 0.93\textwidth so the whole block
%  fits one page (centred, dedicated, no blank page). v11.6 -> v11.7.
% =====================================================================
% ===== v11.6 (2026-06-17): figure-page + references-intro refinements (author Go). =====
%  (1) Roadmap figure now occupies page 28 alone, vertically centred (\null\vfill ... \vfill
%  \clearpage, \captionof{figure}); caption expanded to a few sentences; the 'Bridge from Part I to
%  Part II' text moves to page 29. (2) References Part intro expanded to three paragraphs, the
%  journal-submission sentence removed, and extra vertical space added before the first entry.
%  v11.5 -> v11.6.
% =====================================================================
% ===== v11.5 (2026-06-17): reader-reported fixes (author Go). =====
%  (1) Roadmap flowchart: height-constrained to 0.78\textheight and caption shortened (was
%  overrunning the page number and leaving a blank page before the figure). (2) References Part:
%  dropped \GGBibBegin's \clearpage, added a brief intro to the annotated-reference style, and
%  suppressed natbib's duplicate 'References' heading so the Part header + intro + first entry share
%  the page. (3) Common Objections & References now \GGPart (not \GGAppendixPart) so \appendix is
%  called once -> no duplicate appendix letters. v11.4 -> v11.5.
% =====================================================================
% ===== v11.4 (2026-06-17): four-Part back-matter standard (author Go). =====
%  Existing Part VIII (\GGAppendixPart 'Technical Appendices and Reference Material') RENAMED to
%  'Technical Appendices'; \GGAcknowledgments moved to the end of Part VII (Conclusions), just
%  before it. Common Objections promoted to its own \GGAppendixPart (Part IX) with Q&A categories
%  -> \section and \bigskip spacing. References added as \GGAppendixPart (Part X) before the bib.
%  Architecture prose + roadmap flowchart updated 'eight Parts' -> 'ten Parts'. Q&A content
%  unchanged. Relevance-links a separate pass. v11.3 -> v11.4.
% =====================================================================
% ===== v11.3 (2026-06-17): correct relocation of Common-Objections Q&A to the end (author Go). =====
%  Supersedes the mis-relocated v11.2 (which dragged the \GGPart[Conclusions] divider to the end and
%  orphaned \section{Conclusions}). Rebuilt from v11.1: ONLY the Q&A section is moved to the final
%  back-matter position (last section under the Conclusions Part, before acknowledgments); the Part
%  divider and \section{Conclusions} stay in place. Move only; content unchanged. v11.1 -> v11.3.
% =====================================================================
% ===== v11.1 (2026-06-17): reformat Common-Objections to cohort house standard (author Go). =====
%  The existing 26-objection section is reformatted to the A9-SM house style: three-block entries
%  (bold Title / bold flush Question / indented Response via objqabody), O-Constitution-N IDs added, and six
%  titled \subsection categories. Questions and responses are preserved verbatim; formatting only.
%  Part of the cohort-wide Q&A standardization. v11.0 -> v11.1.
% =====================================================================
% ===== v11.0 (2026-06-16): HOLOGRAPHIC-RG REFRAMING, Phase R1 (author Go). =====
%  Terminology only; NO physics changed. A1's title and its A2/A3 bibitems are
%  already AdS-free; its 'AdS bulk' uses are the legitimate Maldacena/AdS-CFT
%  discussion (Part V), KEPT. Added the canonical holographic-RG disclaimer at the
%  constitutional 'Bulk geometry' paragraph (5th dim = RG axis r=ln(E_GUT/mu), not
%  an RS spatial dimension; explicit construction in A2). Version v10.6 -> v11.0.
% =====================================================================
% --- v10.6 (2026-06-16): adopt cohort-default Part-aware running header (sty v10.6).
%     Curated short Part labels added; running header shows 'Part N. <short>'. Preface self-labels. ---
% GU-5 RENAME (June 14, 2026): cohort harmonization with GG-A4-Euclid -- the five-constant inventory and its theorem are named Geometric Universal Five Constants / GU-5 (\GUfive, \T{-GU5}); the older 'Conversion-Factor Pentad'/'Conversion-Factor Triad' wording and the 'T-CFP' theorem tag are retired.
\pdfoutput=1
%%%% v10.5 (June 12, 2026): Count-sweep (author Go): stale 'eleven-paper
%%%%   cohort/programme' phrasings updated to the seventeen-paper
%%%%   programme where the whole cohort is meant; 'eleven-paper
%%%%   A-series' retained where the A-series is meant. No other
%%%%   content changes.
%% v10.4 (June 12, 2026): Cohort map updated to all seventeen papers
%%   (author request): both copies of the descent diagram
%%   (fig:A1-cohort-inheritance-map and fig:paper_relations) replaced
%%   -- Predictions Branch now P1-KK / P2-N3 / P3-Electron and a new
%%   framed Cosmology & Astrophysics Branch row C1-Kaxion / C2-Hubble
%%   / C3-BH; theorem tags removed from boxes; captions rewritten.
%%   Part-IV handoff revised accordingly: inventory table gains P2,
%%   P3, C2, C3 rows (C-group relabelled Cosmology); branch prose
%%   updated in Part I and Part IV; bridge subsection gains the four
%%   'What X carries' paragraphs; 4 new cohort bibitems (ArisakaP2N3,
%%   ArisakaP3Electron, ArisakaC2Hubble, ArisakaC3BH).  Uses the new
%%   sty macros PaperCTwo / PaperCThree (sty updated June 12).
%% v10.3 (June 10, 2026): arXiv preparation pass (front-matter audit).
%%   front-matter audit: conforms; stamp bump only.
%%   Accepted variants (author decision): existing philosophical closes
%%   in place of literal 'After all'; companion-list counts as cited.
%% v10.2 (June 9, 2026): arXiv preparation pass (cohort step 3).
%%   Overfull-box sweep (all boxes >20pt resolved with layout-only fixes);
%%   no content or physics changes.
%% v10.1 (June 9, 2026): arXiv preparation pass (cohort step 1-2).
%%   \pdfoutput=1 added (line 1); version/date stamp bumped. 
%%   No physics changes; production fixes only.
% v9.9 (2026-06-07, amended same day): canon sweep spot-fix -- Quadratic
%   -> Quadruple Gauge Locking (3 sites, incl. Theorem-8 cross-refs).
% v9.9 (2026-06-07): the boundary descent chain.  NEW Derivation 4 in
%   AX-3's forced-consequences subsection: OGGEP => GDOS => GRIP --
%   the axiom demands the operational dictionary; A5-GDOS realizes it
%   as the boundary law; A6-GRIP proves its long-time factorization is
%   the runtime DGI -> GSSB -> GA/GGA in two regimes (def:GRIPin /
%   def:GRIPout); A7 supplies the GGA selector; A8 reads QM off it.
%   Slogan: "OGGEP demands the dictionary; GDOS writes it; GRIP runs
%   it; the GGA is where it halts."  Stated as constitutional lineage
%   (each arrow cites its downstream proof venue); GRIP remains a
%   forced consequence, NOT an axiom.  Secondary: Derivation-2 pointer
%   to A8 sec:grip-spine (actualization + permanence = the GRIP-In
%   halt; Born measure stays with the instrument formalism);
%   handoff/inventory refresh (Arisaka-GG-A6-Gi4 -> A6-GRIP with
%   current title; cascade terminus GA -> GA/GGA; GDOS Trio framed as
%   the three tenses of AX-3: the law / the runtime / the selector);
%   abbreviation rows GGA, GRIP-In, GRIP-Out; macros (A6/A11 form).
% v9.8 (2026-06-03): "keep KK but always qualify" -- qualified the headline
%   Kaluza-Klein lock-cluster mentions as "non-spatial" (warped/holographic; not a
%   spatial extra dimension), per the A2 v9.21 terminology convention.
% ----------------------------------------------------------------------
% Arisaka-GG-A1-Constitution
% The GG-Constitution: Four Axioms and Seven Postulates
% for the GG-Theory Programme
% Author: Katsushi Arisaka (UCLA)
%
% PROVENANCE.  This file is the v4.0 Phase 8 of the GG-Constitution
% paper.  It is built directly on the v4.0 Phase 7 source
% (Arisaka-GG-A1-Constitution-v4.0-Phase7.tex), with the
% following structural changes carried forward from Phases 1--7:
%
%   * Renamed under the new Arisaka-GG-A* convention
%     (Arisaka-GG-A1-Constitution; Arisaka-GG-A2-GG6;
%      Arisaka-GG-A3-BI6; Arisaka-GG-A5-GDOS; Arisaka-GG-A6-Gi4;
%      Arisaka-GG-A7-Qi4; Arisaka-GG-A8-QM; Arisaka-GG-A9-SM;
%      Arisaka-GG-A10-Cosmos; plus Arisaka-GG-A4-Euclid and
%      the Predictions branch P1-KK, C1-Kaxion).
%   * Bibtex keys renamed: ArisakaA1Constitution -> ArisakaA1Constitution,
%     ArisakaA2GG6 -> ArisakaA2GG6, etc.
%   * v3 Part IV (Eleven Principal Theorems detailed catalogue) and
%     v3 Part V (T1, T2, T3 dual-layer proofs) DROPPED.  Theorems
%     are the territory of the companion papers; the constitution
%     names the descent chain at high level only.
%   * v3 Part VI (Bridges) absorbed into a new Part IV, "Forward
%     Map and Bridges to the Companion Papers", with the new
%     comprehensive paper-inventory table and the new relations
%     figure landing in the §4.1-4.2 of the new Part IV.
%   * Subsequent Parts renumbered:
%       v3 Part VII Discussion       -> v4 Part V
%       v3 Part VIII Open Problems   -> v4 Part VI
%       v3 Part IX Conclusions       -> v4 Part VII
%       v3 Part X Technical Appx     -> v4 Part VIII
%   * Eight Parts total.  Zero theorems proved inside.
%
% PHASE LOG (v4.0).
%   Phase 1 (May 5, 2026):
%      * Rename + structural rewind.  v3 Parts IV and V dropped;
%        new Part IV "Forward Map and Bridges to the Companion
%        Papers" assembled; bibtex keys renamed; new bibitems
%        for A4-Euclid, A8-QM, P1-KK, C1-Kaxion added.
%   Phase 2 (May 5, 2026, this file):
%      * Promotion of the Appendices block to Part VIII so the
%        document realises the agreed ten-Part architecture.
%      * Voice-tune across the new Part IV and four bridge
%        subsections (Arisaka markers were not re-balanced at
%        the new seams in Phase 1).
%      * Roadmap and forward/backward signposts updated to name
%        ten Parts consistently.
%      * Final consistency sweep on canonical metric, gauge
%        group, BI-6 vector, OGN-5 ledger, sans-serif macros,
%        and ArisakaA-key citation graph.
%      * Sed-strippability check on every Relevance paragraph.
%   Phase 3 (May 5, 2026, this file):
%      * Hostile-reviewer pass on the new spine.  The reviewer was
%        asked to attack Part IV (Forward Map and Bridges) and the
%        new Part VIII opener as the load-bearing seams of the
%        v4.0 architecture, plus the Abstract, Executive Summary,
%        and Roadmap.  Recommendations applied: clarify which
%        theorems are NAMED here vs PROVED downstream; tighten
%        the cross-paper coherence map with explicit version /
%        Phase numbers; pre-empt the "circularity" objection
%        (constitution proves nothing, so why not just call it a
%        wishlist?); add an "audit instructions for the reader"
%        paragraph at the end of Part IV; ensure every theorem
%        citation in the body is matched against the coherence-map
%        row.
%   Phase 4 (May 5, 2026, this file):
%      * Cross-paper coherence audit.  Each row of
%        Table~\ref{tab:coherence_map} has been verified against
%        the actual companion-paper PDFs available in the
%        workspace (A2-GG6 v4.2, A5-GDOS v4.0 Phase 44, ...).
%        Stale or imprecise pointers tightened; rows for
%        companion papers not yet drafted (A4-Euclid, A8-QM,
%        A9-SM, A10-Cosmos, P1-KK, C1-Kaxion) marked
%        explicitly as "preprint pending" so the reader cannot
%        mistake an aspirational pointer for a verified one.
%   Phase 5 (May 5, 2026, this file):
%      * Figure 4 (OGN-5 Diophantine cascade) rebuilt to fit
%        within the text margin without label overlap (per user
%        report).  Figure-captioning audit: all six figures
%        plus the new Part IV relations figure walked for
%        margin sanity, caption tone, and arrow-label position.
%      * Final 22-item checks list walk against current state.
%      * Candidate journal-submission stripped version
%        (Arisaka-GG-A1-Constitution-v4.0-Phase5-journal.tex)
%        produced by sed-stripping the Relevance paragraphs;
%        the preprint version retains them.
%   Phase 6 (May 5, 2026, this file):
%      * Open Problems O1-O12 triaged.  In v3 the O1-O12 catalogue
%        included items that had since been answered by companion
%        papers in the A2-A10 chain; per author concern, these are
%        either (a) recast as "Resolved in companion paper Ax"
%        with explicit citation, or (b) dropped, leaving only
%        problems that remain genuinely open *after* the descent
%        chain has been traversed.  The section title is now
%        "Open Problems After the Descent Chain" so the framing
%        is honest.
%   Phase 7 (May 5, 2026, this file):
%      * Historical-background subsections expanded 2-3x for
%        each of the four axioms and the seven postulates.
%        The previous concise paragraphs (one per AX-k or
%        POS-k) gave dry citations; the new paragraphs
%        properly credit the original contributors and trace
%        the long historical chain that leads to each clause.
%        Per author concern, this is essential for honesty:
%        the constitution stands on the shoulders of three
%        and a half centuries of foundational work, and the
%        historical-background subsections must reflect that
%        debt explicitly.
%   Phase 8 (May 5, 2026, this file):
%      * Historical backgrounds reworked to start from the
%        ORIGINAL inventor of each concept rather than from a
%        downstream synthesis.  Per author concern:
%         - POS-1 GI: Maxwell 1865 (original gauge freedom in
%           A_mu) and Noether 1918 (symmetry to conservation)
%           promoted as the backbone, with Ward identity 1950
%           and Slavnov-Taylor 1972 named as the operational
%           Ward-identity tradition.
%         - POS-3 AF: opens with Luigi Bianchi's 1902 identity
%           for the Riemann tensor and the abelian Bianchi dF=0
%           inherent in Maxwell.  Anomaly freedom by Bianchi
%           identity is reframed as a PURELY GEOMETRIC concept,
%           aligned with the conservation-of-current logic of
%           Local Causality.  The quantum chiral anomaly
%           (Adler 1969, Bell-Jackiw 1969) is then introduced
%           as what arises when the geometric Bianchi must
%           reconcile with a quantum current of the wrong
%           divergence; Green-Schwarz inflow restores the
%           geometric identity at the bulk level.
%         - POS-5 RG: opens with Tomonaga 1946, Schwinger 1948,
%           Feynman 1949 QED renormalisation, Dyson 1949
%           unification --- they are the original inventors,
%           and Stueckelberg-Petermann (1953), Gell-Mann-Low
%           (1954), Wilson (1971), Kadanoff (1966) are then
%           positioned as the subsequent RG-flow generation.
%        New bibitems added: Bianchi1902, Tomonaga1946,
%        Schwinger1948, Feynman1949QED, Dyson1949, Maxwell1861,
%        Galilei1632, BohrComplementarity1928, Hilbert1899,
%        EulerLagrange.  Other historical backgrounds
%        spot-checked for original-inventor framing.
%   [v4.0 cycle complete after Phase 8; v4.1 reserved for
%    response to peer review or new architectural change.]
%
% v7.2 (May 24, 2026): polish-cycle step 1/8 (style-file migration +
%   cover-page L63 fix + two-level TOC per L66 / Part 5 of the
%   v8.1 PROJECT INSTRUCTIONS).  Changes applied in this patch:
%     * \usepackage{ucla-arisaka-gg-6} -> \usepackage{ucla-arisaka-gg-8}.
%       The v8 .sty (May 22, 2026) ships natively the 26 paper-code
%       macros (\AOne...\PaperFullCOne), the parameterised
%       constitutional macros (\AX{}, \POS{}, \T{}), the framework
%       macros (\GG, \GDOS, \GGEP, \OGGEP, \MSMF, \OGN,
%       \OGNfive, \BIsix, \GiFour, \QiFour, \GGConstitution,
%       \BFFT, \OH, \IH, \LGI, \GGI, \QB, etc.), the bold-Figure-N
%       machinery (\figref + caption labelfont=bf), and the
%       two-level-TOC machinery (\GGOverviewOfContents +
%       \GGListOfFigures + \GGTableOfContents, etoc-backed).
%     * Redundant local \providecommand{\AOne}...\PaperFullCOne
%       block removed (now in v8 .sty).
%     * Redundant local \providecommand entries for \AXkw, \POSkw,
%       \BFFT, \QGL, \OH, \IH, \LGI, \GGI removed (now in v8 .sty).
%     * Local \providecommand{\GGConstitution}{...\textsf{\GGConstitution{}}...}
%       removed -- the \else branch was self-recursive and only
%       silently masked because v8 .sty's native definition wins.
%     * Cover page per L63: \GGTitlePage{TITLE}{SUBTITLE}{DATE} ->
%       \GGTitlePage{TITLE}{}{}; the long subtitle gloss now lives
%       in the Abstract's opening sentence.
%     * Two-level TOC per L66 / Part 5: \GGOverviewOfContents and
%       \GGListOfFigures inserted ahead of \GGTableOfContents.
% ----------------------------------------------------------------------
%% ---- vv15.30 (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.
%% And the showcase headlines brought to one rendered form.
%% The house form is the style file's own default, \Large\bfseries\sffamily,
%% which is what \GGShowcaseHead sets when it is called with no optional
%% argument, and which forty-three of the cohort's fifty-seven showcase
%% headlines already used.  Fourteen did not, and the difference was
%% visible to a reader turning pages: four headlines printed at \huge,
%% half again the house height, and three at \large, smaller than it.
%% 1 headline in this paper.  Only the size changes; the \boldmath in
%% a headline that carries math is kept, the subtitle under a headline
%% keeps its own smaller size, and \GGShowcaseHeadSkip, which is vertical
%% space rather than type, is left exactly as each paper sets it.
\documentclass[11pt,letterpaper]{article}
\usepackage{ucla-arisaka-gg-11}
\usepackage{indentfirst}
\setcitestyle{authoryear,round,comma}
%% v7.2 paper-local override of \GGAcknowledgments.  The v8.0
%% ucla-arisaka-gg-8.sty definition uses \LaTeX\ which, in
%% combination with \captionsetup{labelfont=bf} and xcolor's
%% [dvipsnames,svgnames,x11names] options, fails with 'Use of \@
%% doesn't match its definition.'  The override below is identical
%% to the v8.sty body except \LaTeX is rendered as the literal text
%% 'LaTeX' (no logo typography).  A bug report for the v8.sty .sty
%% interaction will be filed for a future v8.x update.
%% v11.15: \GGAcknowledgments now provided by ucla-arisaka-gg-11 (identical).

%% =====================================================================
%% v7.2 paper-local refinements (per Katsushi's request, May 24, 2026):
%%   R4  Override \GGOverviewOfContents + \GGListOfFigures so both fit
%%       on the same page (Overview at top, LoF below, no \clearpage
%%       between them).
%%   R5  Halve the TOC indents for \section / \subsection /
%%       \subsubsection in the full Table of Contents.
%% =====================================================================

%% v11.15: \GGOverviewOfContents provided by ucla-arisaka-gg-11; the
%% single-page Overview+List-of-Figures layout is the gg-11 toggle:
\GGfrontmatteronepagetrue

%% v11.15: \GGListOfFigures now provided by ucla-arisaka-gg-11.

%% R5 -- Halve the full-TOC indents for section / subsection / subsubsection.
%% Defaults (v8.sty): 1.5em / 4.25em / 8.1em.  v7.2 paper-local: 0.75em /
%% 2.1em / 4.05em.  Number-column widths preserved at v8.sty values for
%% 3-digit section numbers.
\makeatletter
\renewcommand{\l@section}{\@dottedtocline{1}{0.75em}{2.75em}}%
\renewcommand{\l@subsection}{\@dottedtocline{2}{2.1em}{3.85em}}%
\renewcommand{\l@subsubsection}{\@dottedtocline{3}{4.05em}{4.75em}}%
\makeatother

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

  %% Foundation-block retrofit (May 2026): new citation convention (Arisaka, 2026, A*-Suffix)

\PaperSeries{GG}
\PaperNumber{Arisaka-GG-A1-Constitution}
\PaperStatus{Preprint}
% ------------------------------------------------------------
%% v15.21 -> v15.22, August 15, 2026.  THE GATE-CLOSING PASS.
%   One pass across A1-A7 so that `at v15' and `passes the v15
%   gate' mean the same thing before the first arXiv submission.
%   These seven were migrated before the current toolchain and
%   each failed at least one absolute section 4 item.
%   SPELLING (section 9 decision 1, "total, in printed text"):
%     7 printed British form(s) changed here.  The raw guard
%     reported 91 across the seven; 51 were NOT defects and were
%     hand-classified one site at a time -- 33 correct American
%     English now permanently in PROTECT, 12 published titles now
%     frozen by exact text (correcting a title is a
%     misquotation), and 6 `cancellable'/`cancellability', held at
%     the doubled l because American English keeps it in
%     `cancellation', the -able form is listed both ways, and it
%     is a defined term.  The -ed/-ing forms ARE swept.
%     Edited BY OFFSET, not by text anchor: these papers hard-wrap
%     their source and anchors copied from a flattened scan do not
%     exist -- eight failed that way on the first run, and the
%     same defect hit A8's rider an hour earlier.  The sites come
%     from the guard itself, which already excludes comments,
%     labels, PROTECT and the frozen titles.
%   FLOAT STANDARD: none -- the float audit was already clean.
%   PALETTE: nothing to change.  0 violations.
%   NEW PROSE is listed by table and page in GG-Report-362 and is
%   a draft for the author to accept or rewrite.  Caption
%   EXTENSIONS keep every existing sentence verbatim -- section
%   14.2 reserves rewriting a caption's CLAIM, and none is.
%   NO NUMBER, NO CLAIM AND NO GRADE CHANGES ANYWHERE.
% ------------------------------------------------------------
%% ------------------------------------------------------------
%% v15.22 -> v15.23, August 15, 2026.  THE THREE-PAGE FRONT-MATTER
%% SHOWCASE.  First of the twenty-four.
%%   The pattern is read off the four papers that already have one --
%%   A9-SM pp. 5-6, A10-Cosmos pp. 6-11, C1-Kaxion pp. 5-6, P1-KK
%%   p. 4 -- and not invented: front matter after the Keywords and
%%   before the Contents; one page, one idea, under a spoken headline
%%   set {\large\bfseries\sffamily}, which is A10's markup exactly;
%%   the plate a numbered duplicate whose caption names the full
%%   version; the short caption marked "(front-matter showcase)",
%%   because A10 shipped with a List-of-Figures line printed twice
%%   when it was not.
%%   WHAT A1 HAS TO SHOW.  A constitution has no measurement of its
%%   own, so A9's and C1's template -- one prediction against all the
%%   data -- is unavailable.  The three pages answer the same two
%%   questions in the same order.  WHAT IS CLAIMED: twentieth-century
%%   physics sorted onto four axiom strata, with GG-Theory in the only
%%   row that activates all four (Figure 10).  WHAT WAS BUILT ON IT:
%%   twenty-two papers from eleven clauses (Figure 3).  HOW IT DIES:
%%   ten pre-declared numbers with their kill thresholds (Table 7,
%%   condensed to one page, with linked companion names).
%%   ONE SOURCE, TWO USES.  The plates are NOT copied.  Each picture
%%   moves into a savebox filled once before the front matter, and
%%   both the showcase and the in-paper figure say \usebox.  Copying
%%   8,759 and 14,600 characters of TikZ would have created two
%%   bodies that drift.  This is C1's device (v13.11).
%%   SIZED BY MEASUREMENT, one compile per picture: the flowchart is
%%   690.04 x 402.56 pt natural, the cohort map 720.96 x 505.02.
%%   Inside landscape \linewidth is 650.43 and \textheight 469.755,
%%   the short side; 0.88\linewidth leaves 136 pt for the headline,
%%   the caption and their skips.  The map is width-bound in portrait.
%%   THE TABLE IS CONDENSED, NOT COPIED, and it is the one place this
%%   pass selects: all ten F-Constitution rows survive, the Mode
%%   column and the proof-location prose go, the Source column becomes
%%   a linked companion name.  No threshold, value or status altered.
%%   THREE HEADLINES ARE NEW PROSE AND THEY ARE THE ASSISTANT'S.
%%   Everything else in the three captions is the author's own text
%%   from the in-paper original plus one pointer sentence each.
%% ------------------------------------------------------------
%% ------------------------------------------------------------
%% v15.23 -> v15.24, August 15, 2026.  THE PAPER INVENTORY, AND A
%% COUNTER I LEAKED.
%%   (1) MINE.  Report 375 said "figure and table numbering
%%   unchanged".  True of the figures; false of the tables.  A
%%   \caption* inside a longtable steps \c@table anyway, so the v15.23
%%   showcase table took number 1, the List of Tables began at 2, and
%%   every table printed one higher than in v15.22 -- the inventory
%%   went from Table 5 to Table 6.  THE AUTHOR FOUND IT, NOT THE GATE,
%%   and no section 4 item would have: stamp_check, float_audit and
%%   final_audit all read the same either way.  \addtocounter{table}{-1}
%%   after the showcase longtable restores every v15.22 number.
%%   (2) THE INVENTORY LISTED 24 ROWS UNDER A CAPTION SAYING 22.
%%   Arisaka-GG-BookI and Arisaka-GG-WhitePaper are not cohort papers,
%%   and the paragraph immediately above the table already says so:
%%   "Twenty-two papers in all.  A compilation Foundation Book and a
%%   popular exposition round out the series for broader audiences."
%%   The prose drew the line and the table did not.  Both rows removed
%%   and the caption's "and two compilation/exposition documents" with
%%   them.  The ArisakaBookI bibliography entry and the two prose
%%   citations to it STAY: citing a work in preparation is a different
%%   thing from listing it as a cohort member.
%%   (3) "New name" -> "Name", and the column becomes links.  Each
%%   name is wrapped in \hyperlink{cite.<key>}, hyperref's own anchor
%%   for a natbib \bibitem, so not one printed character changes and
%%   every row jumps to the companion's bibliography entry.  21 of 22
%%   link; A1 is this paper and has no entry in its own bibliography.
%%   ALSO: the A5 row printed \AFive{}, which sets "GG-A5-GDOS", while
%%   the other 21 print the full "Arisaka-GG-..." form.  One column,
%%   two conventions.  Normalized to the majority.
%% ------------------------------------------------------------
\PaperVersion{v20.0}
\PaperDate{August 21, 2026}

\hypersetup{
 pdftitle = {The GG-Constitution: Four Axioms and Seven Postulates
 for the GG-Theory Program},
 pdfauthor = {Katsushi Arisaka},
 pdfsubject = {Foundational document of the GG-Theory program.
 Four axioms (LC, GGEP, OGGEP, MSMF) and seven
 postulates (GI, UEP, AF, CP, RG, OGN, MF), from which
 the 6D bulk (Arisaka-GG-A2-GG6), the BI-6
 anomaly identity (Arisaka-GG-A3-BI6), the GDOS
 boundary law (Arisaka-GG-A5-GDOS), and the
 Gi-4 + Qi-4 pillars (Arisaka-GG-A6-GRIP,
 Arisaka-GG-A7-Qi4) descend as theorems in the
 companion papers. Citation root of the entire
 Arisaka-GG-A* foundation series.},
 pdfkeywords= {GG-Theory, GG-Constitution, Arisaka-GG-A1,
 Arisaka-GG-A2-GG6, Arisaka-GG-A3-BI6,
 Arisaka-GG-A4-Euclid, Arisaka-GG-A5-GDOS,
 Arisaka-GG-A6-GRIP, Arisaka-GG-A7-Qi4,
 Arisaka-GG-A8-QM, Arisaka-GG-A9-SM,
 Arisaka-GG-A10-Cosmos, GG-6, BI-6, OGN-5, GDOS,
 OGGEP, MSMF, axiom system, constitution, gauge
 invariance, anomaly cancellation, open systems,
 geometry-gauge equivalence, falsifiability, Gi-4
 Pillars, Qi-4 Pillars,
 Spin(1,5) x SU(5) x U(1)_B x U(1)_Qi}
}

%% Paper-specific macros (in addition to the series .sty)
%% Acronyms NOT already provided by ucla-arisaka-gg-6.sty.  We use
%% \providecommand for safety: if a future .sty version defines them,
%% the .sty version wins and the local fallback is a no-op.
%% v7.2: \LGI and \GGI now live in ucla-arisaka-gg-8.sty; the
%% paper-specific sub-acronyms below remain local.
\providecommand{\PLA}{\ifmmode\mathsf{PLA}\else\textsf{PLA}\fi}
\providecommand{\PLT}{\ifmmode\mathsf{PLT}\else\textsf{PLT}\fi}
\providecommand{\FEP}{\ifmmode\mathsf{FEP}\else\textsf{FEP}\fi}
\providecommand{\MEPP}{\ifmmode\mathsf{MEPP}\else\textsf{MEPP}\fi}
\providecommand{\MEP}{\ifmmode\mathsf{MEP}\else\textsf{MEP}\fi}

%% Open-system / measurement shorthands not in the .sty
\providecommand{\id}{\mathrm{id}}
\providecommand{\Op}{\mathsf{Op}}
\providecommand{\Dmap}{\mathcal{D}}
\providecommand{\M}{\mathcal{M}}
\providecommand{\Mbulk}{\mathcal{M}_6}
\providecommand{\Mbdy}{\mathcal{M}_4}
\providecommand{\MbdyPatch}{\mathcal{U}_4}
\providecommand{\Dop}{\mathfrak{D}}

%% \e for the natural exponential.  The .sty intentionally leaves \e
%% undefined to avoid colliding with the Euler-script package; we provide
%% it locally for AX-1 dispersion-relation passages and similar sites.
\providecommand{\e}{\mathrm{e}}

%% \GGConstitution -- formal name of this paper, in the same sans-serif
%% register as the other programme acronyms.  Local for now; will move into
%% the .sty in a future revision once the convention has settled.
%% TikZ extra library for the GG-Theory paper-relations figure
\usetikzlibrary{decorations.pathreplacing}
%% v7.5: pdflscape for landscape Figure 5 (Part V constitutional flowchart)
\usepackage{pdflscape}

%% v7.2: local \GGConstitution definition removed.  The v8 .sty's
%% native definition (section 13a) wins and the local copy had a
%% self-recursive \else branch that was silently masked.

%% v5.2 additions (May 2026): programme-defined keyword macros for
%% bare-form 'AX' / 'POS' (the existing \AX{n}/\POS{n} take an
%% argument and produce 'AX-n' / 'POS-n'), and for the new acronyms
%% \BFFT{}, QGL, OH, IH.  Plus \LGI, \GGI for sub-acronyms of GI.
%% \QB for the bioenergetic Arisaka constant Q_B.
%% v7.2: \AXkw, \POSkw, \BFFT, \QGL, \OH, \IH, \LGI, \GGI
%% removed -- all eight now live in ucla-arisaka-gg-8.sty section 13a.
%% \QB now defined in ucla-arisaka-gg-6.sty (sans-serif Q_B)

%% Paper-specific formatting -- paragraph format (parskip / parindent)
%% is now inherited from ucla-arisaka-gg-6.sty global default
%% (parskip=0pt, parindent=1.5em; conventional indent style).
\captionsetup[table]{position=top,skip=6pt,width=\textwidth}
\captionsetup[longtable]{position=top,skip=6pt,width=\textwidth}
\setlength{\LTcapwidth}{\textwidth}
\setlength{\tabcolsep}{2pt}
\titleformat{\subsubsection}{\normalsize\itshape}{\thesubsubsection}{0.8em}{}


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

%% --- Sans-serif paper-code macros: removed in v7.2.  The 26 macros
%%    (\AOne...\ATen, \PaperPOne, \PaperPTwo, \PaperCOne,
%%    \AFullOne...\AFullTen, \PaperFullPOne, \PaperFullPTwo,
%%    \PaperFullCOne) now ship natively in ucla-arisaka-gg-8.sty,
%%    section 13d.  The local block carried forward from v7.1 has
%%    been removed to avoid duplication.
%% v7.5: \AEleven (GG-A11-Origin) -- not yet in v8.sty's 26-macro set.
\providecommand{\AEleven}{\textsf{GG-A11-Origin}}
\providecommand{\AFullEleven}{\textsf{Arisaka-GG-A11-Origin}}
%% v13.1: Foundations Branch member-paper macros (cohort canonical
%% forms; the .sty does not yet carry them, so \providecommand is the
%% usual forward-compatible install).
\providecommand{\PaperPFour}{\textsf{GG-P4-Real}}
\providecommand{\PaperPFive}{\textsf{GG-P5-CPT}}
\providecommand{\PaperPSix}{\textsf{GG-P6-Proton}}%   v13.11
\providecommand{\PaperWOne}{\textsf{GG-W1-Symmetry}}% v13.11
\providecommand{\PaperWTwo}{\textsf{GG-W2-Holonomy}}% v13.11
%% v13.2: A1's first keyed lemma (L-Constitution-1, AX-4 further
%% elaboration) precedes the paper's single euclidtheorem (T-GU5,
%% Part IV); detach the euclidlemma counter so the lemma prints
%% "Lemma 1" rather than inheriting an empty theorem counter.
\renewcommand{\theeuclidlemma}{\arabic{euclidlemma}}
\renewcommand{\ASix}{\textsf{GG-A6-GRIP}}   % A6 Gi4 -> GRIP (cohort, May 2026)
\renewcommand{\AFullSix}{\textsf{Arisaka-GG-A6-GRIP}}
\providecommand{\GGA}{\ifmmode\mathsf{GGA}\else\textsf{GGA}\fi}
\providecommand{\GRIPin}{\ifmmode\mathsf{GRIP\text{-}In}\else\textsf{GRIP-In}\fi}
\providecommand{\GRIPout}{\ifmmode\mathsf{GRIP\text{-}Out}\else\textsf{GRIP-Out}\fi}

%% v11.15: \GGTableOfContents + ExecutiveSummary now provided by ucla-arisaka-gg-11.

%% v15.4: float placement.  The LaTeX defaults (topfraction .7,
%% floatpagefraction .5) let a single plate claim a page of its own
%% and leave half of it blank.
\renewcommand{\topfraction}{0.92}
\renewcommand{\bottomfraction}{0.55}
\renewcommand{\textfraction}{0.08}
\renewcommand{\floatpagefraction}{0.80}
\setcounter{topnumber}{3}
\setcounter{totalnumber}{5}
\begin{document}
%% ---- v15.21 (August 10, 2026) ----
%% Pre-submission pass.  The mechanical battery (final_audit.py,
%% spelling_audit, float_audit, orphan_group_scan, table_census,
%% palette_audit) was run against v15.20 and returned three findings.
%%   (1) Figure 2 (fig:cohort-funnel) had no reference of any kind --
%%       the last unreferenced float in the paper.  Given one.
%%   (2) Two British spellings, both introduced by me in the v15.19
%%       Part-structure prose: "programme" and "defence".  The paper is
%%       standardized on American forms.  PROTECTED and unchanged:
%%       toward/towards and canceled/cancelled (both frozen as
%%       acceptable and deliberately not swept), and the British or
%%       French spellings inside cited titles -- Symanzik 1970,
%%       Stueckelberg-Petermann 1953, Bayes 1763, Hensen 2015.
%%   (3) The referee/reviewer vocabulary, 28 reader-facing sites.
%%       These papers are for arXiv; the review process is not the
%%       reader's business and is now out of the text.  Each objection
%%       keeps its community identity -- "the phenomenology referee"
%%       became "the phenomenologist", "the senior referee" became
%%       "the senior theorist" -- so nothing about who is speaking was
%%       lost.  "as of submission" became "as of writing".
%% CLASSIFIED EXCEPTIONS, kept deliberately:
%%   * Four "peer-review" sites in Part IX.  They belong to Objections
%%     9 and 52, which ask how a community should weigh twenty-two
%%     simultaneous preprints that have not been refereed.  That is an
%%     honest question for a preprint reader and the answer builds
%%     trust; removing it would duck the objection.
%%   * Two "journal" uses that describe how journals SET TYPE
%%     ("plain Roman, journal convention"), not submission.
%%   * Every journal NAME in the bibliography.
%% No axiom, postulate, theorem, value or falsifier was changed.
%% ---- v15.20 (August 10, 2026) ----
%% The Executive Summary, rewritten from scratch.
%% The old one ran 12,326 characters over four pages and pushed the
%% keyword block onto page 5.  It opened with a citation-dense history
%% paragraph, carried two boxed equations, and named every axiom and
%% postulate by macro-abbreviation before the reader had met any of
%% them.  It read as a technical abstract for someone who already knew
%% the program.
%% The new one is 6,499 characters over two pages, with the keyword
%% block complete on page 3 and the contents beginning on page 4 --
%% two pages recovered.  It follows the A5-GDOS / A6-GRIP / A7-Qi4
%% house pattern: named \paragraph blocks, plain English, an image
%% rather than a formula, honest limits stated before the closing beat.
%% Both boxed equations were dropped; neither was referenced anywhere
%% (the admissibility predicate survives as eq:admissible in Part I,
%% which IS referenced three times).
%% Content carried over from the Discussion and Conclusions, per the
%% author's instruction: the three distinguishing features (measurement
%% as a founding clause, unforced parameters forbidden as a rule rather
%% than a taste, named falsifiers); the Part V classification and its
%% prediction of silences; and the honesty triage -- the 6/14/8 grading
%% of the Standard Model parameters is now stated IN the summary, and
%% the cross-domain reach is marked as the program's wager rather than
%% as a theorem.  No value, theorem or falsifier was changed.
%% ---- v15.19 (August 10, 2026) ----
%% Ship 3 of the six-auditor audit (GG-Report-278): the reader.
%% Part structure, per the author's instruction of 10 August:
%%   (1) The five "Bridge from Part N to Part N+1" subsections are gone.
%%       They announced the next Part instead of closing the present one.
%%       The seven "Bridge to the <branch>" subsections inside Part IV
%%       are a DIFFERENT object -- they introduce companion papers --
%%       and are protected and unchanged.
%%   (2) Every Part now opens with half a page to a page of plain
%%       English setting up what the Part is for, pitched at an
%%       undergraduate STEM reader including the life sciences.
%%   (3) Parts I-VIII now close with about a page of closing remarks in
%%       the same register.  Parts VI, VII and VIII had no closing of
%%       any kind before this.  Part IX's closing was installed in
%%       v15.17; Part X is the bibliography and takes an opening only.
%% Accessibility (auditor 6):
%%   (4) The Executive Summary claimed "no specialist vocabulary is
%%       required" fourteen lines above a keyword list naming Stinespring
%%       dilation.  Replaced with an honest prerequisite statement and a
%%       pointer to the curious-reader cluster.
%%   (5) The five integers of the ledger are glossed where they first
%%       carry load, not 116 pages later.  GDOS, Gi-4 and Qi-4 are
%%       expanded at first use rather than only in an appendix.
%%   (6) The grade vocabulary (derived / conditional / candidate / open)
%%       and "custody" ran 25+ times and were defined nowhere.  Both are
%%       now defined beside the other working definitions in Part I.
%%   (7) Five symbols used and never defined anywhere -- G_tot, L_GDOS,
%%       E_Ry, Q_B, z* -- added to the glossary, with z* carrying the
%%       two-quantity warning against z_dec.
%%   (8) Three expert-only passages marked skippable in place.
%%   (9) The roadmap promised two appendices that do not exist.
%% No axiom, postulate, theorem, value or falsifier was changed.
%% ---- v15.18 (August 10, 2026) ----
%% Ship 2 of the six-auditor audit (GG-Report-278): the plates and tables.
%%   (1) Table 7, the consolidated falsifier table, rebuilt.  The seven
%%       columns summed to 452.7pt against a 452.97pt line, so every
%%       gutter was 6pt or less and adjacent cells printed touching;
%%       "F-Constitution-10:" is 86pt and could only fit its column by
%%       breaking at its own hyphens, which printed as F- / Constitution-
%%       / 10: down four lines.  Each falsifier now heads its entry
%%       across the full width and six measured columns carry the data.
%%       Column widths are computed from the widest unbreakable cell in
%%       each, not chosen: see the comment above the longtable.
%%   (2) The "Pull" column contained words in seven of ten rows and a
%%       sigma in two.  Renamed "Mode"; the lead paragraph no longer
%%       calls it a deviation in sigma; the classification paragraph
%%       already after the table now names the right column.
%%   (3) The Part IV cohort-map figure was byte-identical to Figure 3 in
%%       Part I -- the same 12,119-character plate printed twice, 112
%%       pages apart, under two captions.  The Part IV copy is removed
%%       and its section points back to the Part I plate and the
%%       inventory table beside it.  The plate itself is the frozen
%%       cohort standard and is untouched.
%%   (4) Figure 9's trace-ratio leader ran from (tr.north) into
%%       (N.south east) straight through the K box, striking out the
%%       equation 35 = 1+9+25.  Rerouted down a clear lane at x = 6.2.
%%   (5) Figure 8's caption called the four contribution boxes purple;
%%       the comp style is green!10 and no purple appears anywhere in
%%       the picture.
%%   (6) Figure 11's caption said what each branch cannot say is printed
%%       beside it.  The plate has 43 nodes and carries no such
%%       annotation; the claim is now made of the extended description,
%%       which does carry it.
%%   (7) Figure 12's extended description opened with a bold
%%       "Figure 12: ..." and read as a second, competing caption on the
%%       facing page.  Named for what it is.
%%   (8) Bibliography: three "First cited in" pointers were measurably
%%       wrong (of 227 checked, not the 27 the audit estimated), and a
%%       misplaced block of four entries plus two adjacent pairs are
%%       back in alphabetical order.
%%   (9) Three LHC exclusion bounds were quoted with no citation in the
%%       paragraph conceding the flagship prediction may be excluded.
%% No physics value changed in this ship.
%% ---- v15.17 (August 10, 2026) ----
%% Ship 1 of the six-auditor audit (GG-Report-278): the hard errors.
%% Physics and values:
%%   (1) The electroweak vev is restored to the cohort form carried by
%%       A5 and A9.  The form printed here evaluated to 3274 GeV beside
%%       a printed 246, and used a symbol defined nowhere in the paper.
%%   (2) The kaxion mass is no longer assigned to the mass class that
%%       this paper itself computes as 6 meV; canonical is 8.1 ueV.
%%   (3) A9's frozen provenance triage corrected to 6/14/8.
%% Citations and bibliography:
%%   (4) 19 cite commands repeated a key and printed 15 bare '?'.
%%   (5) CHSH 1969, Landauer 1961, Page 1993 and Hopkins-Singer 2005
%%       were named and dated in the text and absent from the list.
%%   (6) PDG 2024 journal corrected to Phys. Rev. D 110, 030001.
%%   (7) A collaboration that does not exist, used five times, split
%%       into the two real author lists.
%% Pointers (all verified against the deployed companion PDFs):
%%   (8) Coherence-map rows T-3, T-5, T-10, T-11 and the audit-trail
%%       paragraph.  Seven of eleven rows were already correct.
%%   (9) Table 11's fourteen hardcoded section pointers, five hardcoded
%%       body pointers, and two to a section that does not exist, all
%%       now \ref.  NHT/HAL expansions were transposed against A7.
%% Counts and structure:
%%  (10) Cohort headcounts brought to twenty-two / twenty-one.  The
%%       eleven clauses, eleven Principal Theorems and eleven A-papers
%%       are correct and were protected.
%%  (11) The objections' closing paragraph moved to the end of the Part
%%       and recounted: eleven sections, fifty-four objections, eighteen
%%       concessions, all measured.
%%  (12) Five forward promises repointed from Part VI to Part IX.
%%  (13) Table 1 given an inline reference and a sentence of use.
%%  (14) Doubled paper names; retired name in the PDF metadata; three
%%       hard voice errors.
%% Nothing in Parts II-V's argument was rewritten.
%% ---- v15.13 (August 8, 2026) ----
%% Two items closed.
%% (1) Spelling: 40 further sites taken to the American standard.  The
%%     earlier passes worked from a pair list built out of
%%     spelling_audit.py, which only reports words spelt BOTH ways --
%%     so words this paper always spelt the British way (renormalisation,
%%     factorisation, quantisation, geometrisation) were invisible and
%%     survived.  A rule over the -ise/-isation family replaces the list.
%%     Five cited titles are PROTECTED and unchanged: Dirac 1931,
%%     Faraday 1845, Stueckelberg-Petermann 1953 (French), Symanzik 1970,
%%     and Lorentz 1892's "Archives n\'eerlandaises" (French).
%% (2) tab:reality_register captioned below its grid; moved above, per the
%%     house rule that tables caption above and figures below.
%% ---- v15.12 (August 8, 2026) ----
%% The printed version stamp is corrected.  \PaperVersion still read
%% v15.8 and \PaperDate August 7 while the file had moved on, so the
%% front cover carried a version several ships behind its own
%% filename.  The stamp is a CALL to a macro defined in the style
%% file, not a definition in the paper, which is why a search for
%% the definition found nothing and the drift went unnoticed.
%% stamp_check.py now asserts filename version == printed version
%% and is part of the gate.  No other change.
%% ---- v15.11 (August 8, 2026) ----
%% (1) Spelling completed to the cohort's American standard: 21
%%     remaining minority spellings corrected (normalisation,
%%     realise, generalise, behaviour, colour and the rest).
%%     PROTECTED and deliberately unchanged: Stueckelberg &
%%     Petermann (1953) "La normalisation des constantes...", which
%%     is French, and Symanzik (1970) "Small distance behaviour in
%%     field theory", which is that paper's title.  toward/towards
%%     and canceled/cancelled are both acceptable American usage and
%%     are left alone rather than swept to a preference.
%% ---- v15.10 (August 8, 2026) ----
%% fig:abbrev-flow retargeted 0.714 -> 0.824, the value the other four
%%     papers carry: it is the same plate and was printing 13%% small.
%% ---- v15.9 (August 7, 2026) ----
%% Spelling: "programme" and "fibre" eliminated in favour of "program"
%% and "fiber" (31 sites), including label and cross-reference names.
%% Two occurrences are PROTECTED as other authors' words and are
%% deliberately unchanged: Lakatos (1970), "Falsification and the
%% methodology of scientific research programmes", which is that paper's
%% actual title, and Klein's "Erlangen Programme".  The cohort's own
%% cited titles ARE corrected: A1-Constitution's title page reads
%% "GG-Theory Program", so citing it as "Programme" was a misquotation.
%% ---- v15.8 (August 7, 2026) ----
%% Four remaining short captions raised to paragraph captions
%% (constitutional flowchart, key-symbols glossary, abbreviation
%% flow, abbreviation table), bringing A1 to the float standard
%% already carried by A2-A5.  The flowchart's superseded legend
%% line, which the recaptioning pass had left standing beneath the
%% new caption, is removed.
% ---- v13.10 (August 2, 2026) ----
% New Discussion paragraph (8) 'The two fibres: the
% constitutional minimum for a world that is and becomes':
% fig:two-fibres-constitutional sharpened into the campaign's
% exact statement (holonomy requires closed loops; an interval
% has none); AX-2 read as installing exactly one fibre of each
% topological kind (two-circle / two-interval counterfactuals);
% the cohort's in-print payoffs listed with homes; the
% constitution as the minimal charter for a world in which
% anything can happen and anything can persist. Part of the
% A1/A3/A4 completion sweep.


%% ===============================================================
%%  PAGE 1: TITLE + ABSTRACT
%% ===============================================================
\GGTitlePage
 {The \GGConstitution{}}
 {Four Axioms and Seven Postulates of the \GGTheory{} Program,\\
 From Which \GG{}, \BIsix{}, \OGNfive{}, and \GDOS{} Descend as Theorems}
 {}
\thispagestyle{empty}

\vspace{6pt}
\begin{abstract}
\noindent
Every theory of nature is built on a foundation, yet that foundation is
rarely written down. For three and a half centuries --- from Newton
through Maxwell, Einstein, and the Standard Model --- physics has
worked beautifully without making its rules of admissibility explicit,
relying instead on a craft tradition passed from textbook to textbook
and from supervisor to student.

In this paper, we propose that the next foundational step is to write
the constitution down. The \GGConstitution{} --- the foundational
charter of \GGTheory{} (Grand Geometric Theory, equivalently
Geometry-Gauge Equivalence Theory) --- consists of four axioms
--- locality and causality, the geometry-gauge equivalence principle,
the operational/open extension of that principle to measurement, and a
maximum-symmetry minimum-freedom rule --- together with seven
postulates that turn the axioms into a working filter for admissible
scientific claims. Remarkably, the eleven clauses are rigid enough to
force, as theorems in the companion papers, the six-dimensional bulk geometry, the four-component anomaly identity that locks gravity
to gauge structure, the integer ledger that fixes the electroweak
hierarchy, and the open-system boundary law from which quantum
mechanics, general relativity, and quantum field theory emerge as
four-dimensional projections.

Strikingly, the same four axioms, when applied with different
subsets active, locate essentially every major theoretical-physics
framework of the twentieth century --- general relativity, gauge
theory, string theory, AdS/CFT, quantum mechanics, the Standard
Model, and $\Lambda$CDM --- inside a single constitutional tower,
and predict which of each framework's famous limitations follow
from which axioms it lacks.

After all, what makes a body of knowledge a science is not the
cleverness of its calculations but the honesty of its rules.
\end{abstract}


%% ===============================================================
%%  PAGES 2-3: EXECUTIVE SUMMARY + KEYWORDS
%% ===============================================================
\begin{ExecutiveSummary}
\paragraph{Physics runs on rules nobody wrote down.}
Every working physicist knows what counts as a law, what counts as an
observation, and what is allowed to remain an adjustable number. Nobody
is taught these things from a document. They are absorbed by
apprenticeship, passed from supervisor to student and from textbook to
textbook, and they are nowhere written out. Newton's \textit{Principia},
Maxwell's \textit{Treatise}, Einstein's papers of 1905 and 1915, and the
Standard Model assembled in the 1960s all rest on such commitments, and
none of them states what those commitments are. Later work sharpened
particular corners --- the decoherence program around measurement, the
axiomatic and open-system formalisms around locality and positivity ---
but no short document was ever produced that a reader could hold in one
hand and check a candidate theory against.

\paragraph{What this paper proposes.}
We propose to write that document. It has eleven clauses: four axioms
and seven postulates, fitting on the equivalent of a single page.

The four axioms say what kind of object a physical theory is allowed to
be. Causes precede effects and nothing outruns light. The geometry of
spacetime and the gauge fields of the forces are one object seen from
two angles, not two kinds of thing that happen to coexist. Observation
is not a special act performed by a conscious being but a projection ---
a system with many degrees of freedom being read by an instrument with
few. And a theory may not carry an adjustable number unless something
forces the number to be there.

The seven postulates say which such theories survive contact with the
world. They require that a symmetry holding for one electron still hold
for a mole of them; that one variational principle serve rather than
several; that the quantum consistency conditions cancel exactly; that
probabilities remain probabilities under every allowed evolution; that
the answer not depend on how finely one chose to look; that the
remaining numbers be integers and simple ratios rather than free
decimals; and that every central claim name, in advance, the measurement
that would kill it.

A theory is admissible if and only if it satisfies all eleven. That is a
checklist, not a mood, and it can be run against any candidate.

\paragraph{Three things that make it different.}
Prior axiomatizations of physics exist, and this one differs from them
in three ways.

It puts measurement on the same footing as causality. In previous axiom
systems measurement arrives late, as interpretation laid over a dynamics
already complete. Here it is a founding clause, and the awkward pieces
of quantum mechanics --- the Born rule, the collapse, the pointer basis,
the violation of Bell's inequality --- become things to be derived
rather than assumed.

It forbids unforced adjustable numbers as a rule rather than as a taste.
Physicists have always preferred fewer free parameters. Preference is
not a constraint, and the free-parameter count of fundamental physics
rose through the twentieth century rather than falling. Stated as a
clause, the preference becomes a claim that can fail: either the
constants of nature descend from something, or the framework is wrong.

It names the measurements that would end it. Among them: a mass cluster
near $3.85$~TeV, a matter fraction of exactly $1/\pi$, a dark-matter
oscillation at one part in thirty-five, and a fixed ratio among the four
components of the anomaly. Each is sharp, none was fitted, and any one
of them can close the program.

\paragraph{The twentieth century, sorted by the rules it used.}
The classification that follows from this is the part we would most like
a reader to weigh. Instead of asking which theory is right --- a question
rival frameworks rarely let one settle --- one can ask which of the four
axioms each framework quietly assumes. The answer sorts essentially all
of twentieth-century physics into four layers. One axiom gives special
relativity. Two give general relativity, gauge theory, and the
classical-topological backbone on which string theory and AdS/CFT are
built. Three bring in observation, and quantum mechanics, statistical
mechanics, black-hole thermodynamics and the free-energy principle of
theoretical neuroscience turn out to share it --- four communities that
rarely read one another, standing on one clause. The fourth removes the
remaining freedom.

What makes this more than a filing system is that it predicts silences.
A framework without the operational axiom cannot say when a measurement
happens, and historically does not. A framework without the parsimony
axiom cannot say why the constants take the values they do, and does
not. What a theory \emph{cannot} say is as diagnostic as what it can,
and the missing clause tells one which silence to expect.

\paragraph{What is established here, and what is not.}
This paper states the constitution and proves almost nothing. The
demonstration that eleven clauses force a six-dimensional geometry, an
exact anomaly identity, a five-integer ledger and a boundary law lives
in twenty-one companion papers, and every claim here carries the named
theorem in the named paper where its proof sits. The numbers are graded
rather than announced: of the Standard Model's twenty-eight parameters,
the companion work derives six end to end, forces fourteen more once a
single named lemma is supplied in each case, and leaves eight asserted
with the calculation that would close them named explicitly. That the
same clauses reach through chemistry and biology to living matter is the
program's central wager, not a theorem of this paper, and is marked as
such wherever it appears.

\paragraph{A closing thought.}
An unwritten rulebook cannot be wrong. It bends quietly around each new
result, and nobody has to admit anything. Writing the rules down makes
them attackable, and every place the written version fails to match what
physicists actually do becomes a visible defect rather than an invisible
habit. That is uncomfortable, and it is the reason to do it.

Underneath the technical program lies a question physics never had to
answer, because it never wrote down whether the answer was allowed to
matter. Either the constants of nature were chosen from a menu, or they
are what remained after the constraints had their say --- the shape a
soap film takes on a wire loop, which it does not select but cannot
avoid. We offer this short text in that spirit, and invite the reader to
find it wanting.
\end{ExecutiveSummary}

\GGKeywords{\GGTheory{}; \GGConstitution{}; four axioms (\LC, \GGEP,
 \OGGEP, \MSMF); seven postulates (\GI, \UEP, \AF, \CP, \RG, \OGN, \MF);
 6D bulk; \BIsix{} anomaly identity; \OGNfive{} integer
 ledger; \GDOS{} boundary law; \GiFour/\QiFour{} pillars;
 admissibility predicate; falsifiability;
 geometry-gauge equivalence; CPTP channel; Stinespring dilation;
 Born rule; Standard Model; $\Lambda$CDM.}

% ================= FRONT-MATTER SHOWCASE, v15.23 ==============
% The house pattern, read off A9-SM pp. 5-6, A10-Cosmos pp. 6-11,
% C1-Kaxion pp. 5-6 and P1-KK p. 4: front matter after the Keywords
% and before the Contents; one page, one idea, under a spoken
% headline; the plate a numbered duplicate whose caption names the
% full version; the short caption marked (front-matter showcase) so
% the List of Figures cannot print one line twice.
% The two plates live in saveboxes filled ONCE here and used twice --
% here and in the body -- so the copies cannot drift.  C1's device.
% ==============================================================
\newsavebox{\GGscFlow}
\newsavebox{\GGscMap}
\sbox{\GGscFlow}{\begin{tikzpicture}[
 font=\sffamily\small,
 >=stealth,
  %% --- styles (palette modelled on the cohort programme conventions) ---
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 minimum height=1.3cm},
  %% Stratum 2: orange band aligned with AX-2.
 s2style/.style ={stratbox, draw=orange!55!black, fill=orange!16,
 minimum height=2.2cm},
  %% Stratum 3: green band aligned with AX-3.
 s3style/.style ={stratbox, draw=green!55!black, fill=green!12,
 minimum height=2.9cm},
  %% Stratum 4: teal band aligned with AX-4.
 s4style/.style ={stratbox, draw=teal!55!black, fill=teal!13,
 minimum height=1.3cm},
 fwbox/.style ={rectangle, rounded corners=2pt, draw=black!60,
 fill=white, line width=0.35pt, inner sep=2pt,
 align=center, font=\sffamily\scriptsize,
 minimum height=0.55cm},
 ggtbox/.style ={rectangle, rounded corners=4pt, draw=cyan!55!black,
 fill=cyan!18, line width=0.80pt, inner sep=5pt,
 align=center, font=\sffamily\bfseries\normalsize,
 minimum width=17.0cm, minimum height=1.1cm},
 appbox/.style ={rectangle, rounded corners=3pt, draw=blue!55!black,
 fill=blue!14, line width=0.60pt, inner sep=4pt,
 align=center, font=\sffamily\bfseries\footnotesize,
 minimum width=4.8cm, minimum height=1.0cm},
 arr/.style ={->, draw=black!75, line width=0.55pt},
 thickarr/.style ={->, draw=black, line width=0.85pt},
 siblingbar/.style ={draw=black!45, line width=0.40pt, dashed}
]

%% =========================================================
%% AXIOMS COLUMN (left), aligned with stratum rows on the right
%% =========================================================
\node[axbox] (ax1) at (-7.6, 5.1) {\AX{1}\,: \LC{}\\[1pt]\textnormal{\scriptsize Local causality}};
\node[axbox] (ax2) at (-7.6, 2.6) {\AX{2}\,: \GGEP{}\\[1pt]\textnormal{\scriptsize Geometry--gauge eq.}};
\node[axbox] (ax3) at (-7.6, -0.8) {\AX{3}\,: \OGGEP{}\\[1pt]\textnormal{\scriptsize Operational/open ext.}};
\node[axbox] (ax4) at (-7.6, -4.0) {\AX{4}\,: \MSMF{}\\[1pt]\textnormal{\scriptsize Max.\ sym., min.\ freedom}};

%% GG-Constitution at far left, rotated to span vertically
\node[constbox, rotate=90, minimum width=10.0cm, minimum height=1.4cm]
 (gg) at (-11.0, 0.5)
 {The \GGConstitution{} \;\textnormal{\footnotesize (foundational source)}};

%% GG-Constitution -> each axiom (horizontal arrows from right edge)
\draw[arr] (gg.south) -- (ax1.west);
\draw[arr] (gg.south) -- (ax2.west);
\draw[arr] (gg.south) -- (ax3.west);
\draw[arr] (gg.south) -- (ax4.west);

%% Weak-gray line width=0.80pt cascade arrows AX-1 -> AX-2 -> AX-3 -> AX-4
%% (axiom-stack vertical flow: each axiom presupposes the previous)
\draw[->, draw=gray!45, line width=1.10pt] (ax1.south) -- (ax2.north);
\draw[->, draw=gray!45, line width=1.10pt] (ax2.south) -- (ax3.north);
\draw[->, draw=gray!45, line width=1.10pt] (ax3.south) -- (ax4.north);

%% =========================================================
%% STRATA COLUMN (right): four bands, vertically aligned with axioms.
%% v7.5 RENUMBERING: Stratum N is now aligned with Axiom AX-N.
%% =========================================================

%% --- Stratum 1 (yellow) aligned with AX-1 ---
\node[s1style] (s1) at (3.6, 5.1) {};
\node[stratlabel] at ([yshift=2pt]s1.north west) {Stratum~1 \,(\{\AX{1}\})};
\node[fwbox, minimum width=4.6cm] (s1a) at (0.4, 5.10) {Special relativity (1905)};
\node[fwbox, minimum width=4.6cm] (s1b) at (6.8, 5.10) {Classical electrodynamics (1865)};

%% --- Stratum 2 (orange) aligned with AX-2 ---
\node[s2style] (s2) at (3.6, 2.6) {};
\node[stratlabel] at ([yshift=2pt]s2.north west)
 {Stratum~2 \,(\{\AX{1},\AX{2}\}\,; $\mathsf{GGEP\text{-}closed}$)};
\node[fwbox, minimum width=2.6cm] (s2a) at (-2.7, 3.1) {GR (1915)};
\node[fwbox, minimum width=2.8cm] (s2b) at (1.0, 3.1) {Yang--Mills (1954)};
\node[fwbox, minimum width=2.6cm] (s2c) at (4.6, 3.1) {GS mech (1984)};
\node[fwbox, minimum width=3.6cm] (s2d) at (8.7, 3.1) {Superstring theory (1984+)};
\node[fwbox, minimum width=3.4cm] (s2bi) at (-3.0, 2.2) {Bianchi Identity (1902)};
\node[fwbox, minimum width=3.0cm] (s2e) at (0.85, 2.2) {AdS/CFT (1997)};
\node[fwbox, minimum width=3.6cm] (s2f) at (5.15, 2.2) {Holographic dualities (1993+)};
\node[fwbox, minimum width=3.6cm] (s2g) at (9.75, 2.2) {Anomaly cancellation (1984)};

%% --- Stratum 3 (green) aligned with AX-3 (three sibling columns) ---
\node[s3style] (s3) at (3.6, -0.8) {};
\node[stratlabel] at ([yshift=2pt]s3.north west)
 {Stratum~3 \,(\{\AX{1},\AX{2},\AX{3}\}\,; $\mathsf{OGGEP\text{-}open}$): three siblings of openness};
\node[font=\sffamily\bfseries\scriptsize] at (-1.5, -0.05) {QM sibling};
\node[font=\sffamily\bfseries\scriptsize] at (3.6, -0.05) {Thermo sibling};
\node[font=\sffamily\bfseries\scriptsize] at (8.7, -0.05) {BH sibling};
%% Sibling separators
\draw[siblingbar] (1.05, 0.10) -- (1.05, -2.15);
\draw[siblingbar] (6.15, 0.10) -- (6.15, -2.15);

%% QM sibling content
\node[fwbox, minimum width=4.2cm] (s3qa) at (-1.5, -0.55) {QM (1925--32), Born rule};
\node[fwbox, minimum width=4.2cm] (s3qb) at (-1.5, -1.20) {Decoherence, einselection (1970+)};
\node[fwbox, minimum width=4.2cm] (s3qc) at (-1.5, -1.80) {Bell-inequality violations (1964)};

%% Thermo sibling content
\node[fwbox, minimum width=4.2cm] (s3ta) at (3.6, -0.55) {Classical thermo (1824--72)};
\node[fwbox, minimum width=4.2cm] (s3tb) at (3.6, -1.20) {Statistical mech (1902)};
\node[fwbox, minimum width=4.2cm] (s3tc) at (3.6, -1.80) {Fluct.\ thms; Friston FEP (2010)};

%% BH sibling content
\node[fwbox, minimum width=4.2cm] (s3ba) at (8.7, -0.55) {BH physics + thermo (1973+)};
\node[fwbox, minimum width=4.2cm] (s3bb) at (8.7, -1.20) {Page curve (1993)};
\node[fwbox, minimum width=4.2cm] (s3bc) at (8.7, -1.80) {Islands formula (2020)};

%% --- Stratum 4 (teal) aligned with AX-4 ---
\node[s4style] (s4) at (3.6, -4.0) {};
\node[stratlabel] at ([yshift=2pt]s4.north west)
 {Stratum~4 \,(\{\AX{1}, \AX{2}, \AX{3}, \AX{4}\}\,; $\mathsf{Full\text{-}Constitution}$)};
\node[fwbox, minimum width=5.0cm] (s4a) at (-1.4, -4.10) {Standard Model (1967--78): 28\,$\to$\,0 params};
\node[fwbox, minimum width=3.8cm] (s4b) at (3.6, -4.10) {$\Lambda$CDM (1998+): $\Omega_m{=}\pi^{-1}$};
\node[fwbox, minimum width=5.6cm] (s4c) at (8.8, -4.10) {Cohort cascade (2026): quarks\,$\to$\,consciousness};

%% Horizontal arrows: each axiom -> corresponding stratum
\draw[thickarr] (ax1.east) -- (s1.west);
\draw[thickarr] (ax2.east) -- (s2.west);
\draw[thickarr] (ax3.east) -- (s3.west);
\draw[thickarr] (ax4.east) -- (s4.west);

%% =========================================================
%% BOTTOM: GG-Theory summit + three application papers
%% =========================================================

\node[ggtbox] (ggt) at (3.6, -5.8)
 {\GGTheory{} (2026) --- the cohort framework that explicitly activates all four constitutional axioms};

\node[appbox] (a9) at (-1.5, -7.4)
 {\ANine{}\\[1pt]\textnormal{\scriptsize SM parameters as theorems}};
\node[appbox] (a10) at (3.6, -7.4)
 {\ATen{}\\[1pt]\textnormal{\scriptsize Cosmological inventory + Hubble}};
\node[appbox] (a11) at (8.7, -7.4)
 {\AEleven{}\\[1pt]\textnormal{\scriptsize GRIP cascade quarks$\to$articulation}};

%% GG-Theory --> Stratum 4 (arrow points UPWARD, GG-Theory IS the
%% cohort theory that the constitutional tower terminates in)
\draw[arr] (ggt.north) -- (s4.south);

%% Strong L-shape link: GG-Constitution <-> GG-Theory (with arrows at both ends).
%% This shows that the constitutional source on the far left and the cohort
%% framework on the bottom centre are two faces of one structural relation:
%% the constitution defines the theory; the theory articulates the constitution.
\draw[<->, line width=0.95pt, draw=red!50!black]
 (-11.0, -4.55) -- (-11.0, -5.8) -- (ggt.west);
\node[font=\sffamily\itshape\scriptsize, anchor=south west, text=red!50!black,
 align=left]
 at (-10.8, -5.6)
 {constitutional source $\Leftrightarrow$ \\ derived theory};
%% GG-Theory summit -> three application papers
\draw[arr] (ggt.south) -- (a9.north);
\draw[arr] (ggt.south) -- (a10.north);
\draw[arr] (ggt.south) -- (a11.north);

\end{tikzpicture}}
\sbox{\GGscMap}{%% v13.12 type scaling, local to this group.  The picture sets node
%% text in \footnotesize (the root in \small) and its edge and branch
%% labels in \scriptsize; redefining the three commands here enlarges
%% box type by 15%% and everything outside a box by 20%% without
%% touching a single node.
\renewcommand{\footnotesize}{\fontsize{10.35}{12.4}\selectfont}%%
\renewcommand{\small}{\fontsize{11.5}{13.8}\selectfont}%%
\renewcommand{\scriptsize}{\fontsize{9.6}{11.5}\selectfont}%%
\begin{tikzpicture}[
 font=\sffamily\footnotesize,
 every node/.style={align=center, rounded corners=2pt, inner sep=3.45pt},
 root/.style={fill=violet!18, draw=violet!70!black, line width=1.0pt,
 minimum width=80.5mm, minimum height=16.1mm,
 font=\sffamily\bfseries\small},
 bulk/.style={fill=red!10, draw=red!60!black, line width=0.6pt,
 minimum width=41.4mm, minimum height=14.95mm},
 hist/.style={fill=red!5, draw=red!50!black, line width=0.5pt,
 minimum width=41.4mm, minimum height=14.95mm,
 dash pattern=on 2pt off 2pt},
 gdos/.style={fill=yellow!18, draw=olive!55!black, line width=0.6pt,
 minimum width=52.9mm, minimum height=14.95mm},
 pillars/.style={fill=green!12, draw=green!55!black, line width=0.6pt,
 minimum width=41.4mm, minimum height=14.95mm},
 qm/.style={fill=blue!10, draw=blue!55!black, line width=0.6pt,
 minimum width=41.4mm, minimum height=14.95mm},
 a11/.style={fill=yellow!12, draw=orange!55!black, line width=0.6pt,
 minimum width=52.9mm, minimum height=14.95mm},
 pred/.style={fill=cyan!10, draw=cyan!60!black, line width=0.6pt,
 minimum width=41.4mm, minimum height=14.95mm},
 predthis/.style={fill=cyan!14, draw=cyan!50!black, line width=1.1pt,
 minimum width=41.4mm, minimum height=14.95mm},
 cosmo/.style={fill=teal!10, draw=teal!60!black, line width=0.6pt,
 minimum width=41.4mm, minimum height=14.95mm},
 fnd/.style={fill=violet!8, draw=violet!55!black, line width=0.6pt,
 minimum width=43.7mm, minimum height=14.95mm},
 wht/.style={fill=brown!12, draw=brown!65!black, line width=0.6pt,
 minimum width=43.7mm, minimum height=14.95mm},
 branchframe/.style={rounded corners=4pt, line width=0.6pt, inner sep=4pt},
 arrow/.style={-{Stealth[length=2.4mm]}, line width=0.55pt, draw=black!70},
 arrowdash/.style={-{Stealth[length=2.4mm]}, line width=0.55pt,
 draw=black!60, dashed},
 arrowrt/.style={-{Stealth[length=2.6mm]}, line width=0.85pt,
 draw=violet!70!black},
 edgelabel/.style={font=\scriptsize\itshape, fill=white,
 inner sep=1pt, draw=none, rounded corners=0pt},
 branchlabel/.style={font=\sffamily\scriptsize\bfseries,
 inner sep=2pt, rounded corners=0pt},
]

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

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

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

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

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

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

%% --- Row 6: Predictions Branch (P-series) ---
%% v13.11: four predictions on the row at 46mm pitch (was three at
%% 44mm), re-centred about the spine so the frame stays symmetric.
%% The 2mm pitch increase buys the two sister-import edge labels a
%% 10mm gap instead of 8mm -- a white-filled label wide enough to
%% annotate the second import masks the whole arrow inside 8mm.
\node[pred] (P1) at (-75mm, -138mm)
 {\textbf{\PaperPOne{}} \\[-1pt] \footnotesize $\Mlock \approx 3.85$~\TeV{}};
\node[pred] (P2) at (-25mm, -138mm)
 {\textbf{\PaperPTwo{}} \\[-1pt] \footnotesize Three generations};
\node[pred] (P3) at (25mm, -138mm)
 {\textbf{\PaperPThree{}} \\[-1pt] \footnotesize Electron as ground state};
\node[pred] (P6) at (75mm, -138mm)
 {\textbf{\PaperPSix{}} \\[-1pt] \footnotesize The guarded proton};

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

%% --- Foundations Branch column (v1.3: P5 raised so its centre
%%     height matches the A11 row). P4 and P5 form a single
%%     left-flank column at x = -96mm: P4 at y = -64mm (unchanged),
%%     P5 at y = -113mm, level with GG-A11-Origin, so the column
%%     brackets the A8 row with a compact gap.
\node[fnd] (P4) at (-96mm, -64mm)
 {\textbf{\PaperPFour{}} \\[-1pt] \footnotesize The origin of $i$};
\node[fnd] (P5) at (-96mm, -113mm)
 {\textbf{\PaperPFive{}} \\[-1pt] \footnotesize Mirrors \& matter (CPT)};

%% --- White-Paper Branch column (v13.11): the right flank, at the
%%     mirror positions of the Foundations column -- W1 at
%%     (+96mm, -64mm) opposite P4, W2 at (+96mm, -113mm) opposite P5
%%     and level with GG-A11-Origin.  The left flank feeds the spine;
%%     this one reads it, so every edge into the column is dashed.
\node[wht] (W1) at (96mm, -64mm)
 {\textbf{\PaperWOne{}} \\[-1pt] \footnotesize Conservation without symmetry};
\node[wht] (W2) at (96mm, -113mm)
 {\textbf{\PaperWTwo{}} \\[-1pt] \footnotesize Sixteen open problems};

%% --- Branch frames (background layer) ---
\begin{scope}[on background layer]
\node[branchframe, draw=cyan!60!black, fill=cyan!4,
 fit=(P1)(P2)(P3)(P6)] (PFRAME) {};
\node[branchframe, draw=teal!60!black, fill=teal!4,
 fit=(C1)(C2)(C3)] (CFRAME) {};
\end{scope}
\node[branchlabel, text=cyan!40!black, anchor=east]
 at ([xshift=-3mm]PFRAME.west) {Predictions\\Branch\\(P-series)};
%% v13.11: the C label is hung off the (now wider) P frame's west
%% edge so the two outboard branch labels stay left-aligned.
\node[branchlabel, text=teal!40!black, anchor=east]
 at ([xshift=-3mm]PFRAME.west |- CFRAME) {Cosmology \&\\Astrophysics\\Branch\\(C-series)};
%% Foundations column label (v1.3): interior placement, in the gap
%% between P4 and P5, anchored east so it sits immediately LEFT of
%% the vertical P4->P5 edge; house style (small violet) kept. The
%% "shared i" edge label occupies the RIGHT side of the same edge.
\node[branchlabel, text=violet!40!black, anchor=east]
 at ($($(P4.south)!0.5!(P5.north)$)+(-1.5mm,0)$) {Foundations\\Branch};
%% v13.11: the mirror label on the right flank -- interior placement in
%% the W1--W2 gap, anchored EAST against the vertical W1->W2 edge,
%% exactly as the Foundations label is on the left flank, with the
%% edge label taking the opposite side.  Held in the UPPER third of
%% the gap: below that the long A10->C-frame swing crosses the column
%% edge, and GG-A10-Cosmos's east border reaches x = 76mm.
%% v13.12: both labels of this column now sit to the RIGHT of the
%% W1->W2 edge.  At +15% \ATen{}'s right border reaches x = 78.7mm and
%% an east-anchored label at 94.5mm, itself 20% wider, ran into it.
\node[branchlabel, text=brown!45!black, anchor=west]
 at ($($(W1.south)!0.22!(W2.north)$)+(1.5mm,0)$) {White-Paper\\Branch\\(W-series)};

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

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

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

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

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

%% --- Edges: parents -> branch frames ---
%% v13.11: outer controls widened 55 -> 75mm for the +-91mm frame.
\draw[arrowdash] (A9.south west).. controls (-40mm, -105mm)
 and (-84mm, -122mm).. (PFRAME.north west);
\draw[arrowdash] (A9.south east).. controls (40mm, -105mm)
 and (84mm, -122mm).. (PFRAME.north east);
%% v13.11 RE-ROUTE.  The old line (controls at x = 84mm) descended
%% straight through the GG-W2-Holonomy node between y = -106 and
%% -120mm, and there is no corridor left inboard of it now that the
%% Predictions frame reaches +-91mm.  Swung outboard of the whole
%% white-paper column instead; measured clearance from W2's right
%% border is >= 4.9mm, and the swing is the picture's right ink edge.
\draw[arrowdash, rounded corners=7pt] (A10.east) -- (127mm,-91mm)
 -- (127mm,-160mm) -- (CFRAME.east);
%% sister import inside the Predictions frame
\draw[arrowdash, draw=cyan!50!black]
 (P2.east) -- node[edgelabel,pos=0.5] {$N{=}3$} (P3.west);
%% v13.11: the second sister import.  \PaperPSix{} states it plainly of
%% \PaperPThree{} -- "The sister paper supplies the method this paper
%% runs."  The label is the three-letter form because a white-filled
%% word wide enough to say "method" masks the whole arrow it annotates.
\draw[arrowdash, draw=cyan!50!black]
 (P3.east) -- node[edgelabel,pos=0.5] {GGA} (P6.west);

%% --- Foundations Branch edges (rerouted v1.3 for the raised P5) ---
%% A2 -> P4 (solid; Thm 0 arena feed) -- unchanged from v1.0
\draw[arrow] (A2.west) .. controls (-88mm,-30mm) ..
 node[edgelabel,pos=0.55] {Thm~0} (P4.north);
%% P4 -> A8 (solid; the origin of i). Departs the east side (P4.south
%% carries the column edge to P5) and lands on A8's top border,
%% 6mm left of the A6->A8 arrowhead, staying left of A6 throughout.
\draw[arrow] ([yshift=-3mm]P4.east) .. controls (-69mm,-74mm) ..
 node[edgelabel,pos=0.45] {origin of $i$} ([xshift=-6mm]A8.north);
%% P4 -> A2 (dashed deployment: complex-structure machinery), nudged
%% 3mm up the east side so the two east-side departures stay distinct.
\draw[arrowdash] ([yshift=3mm]P4.east) .. controls (-70mm,-52mm) .. (A2.south west);
%% P4 -> P5 (v1.3: SOLID, per the reading conventions -- forced
%% descent, labelled). A short straight vertical between the pair;
%% the "shared i" label sits on the right side of the edge, the
%% Foundations Branch label on the left.
\draw[arrow] (P4.south) --
 node[edgelabel, pos=0.5, anchor=west, xshift=1mm] {shared $i$}
 (P5.north);
%% A9 -> P5 (solid; ledger phases). Leaves A9's west side, dips
%% through the corridor below the A8 row and enters P5.east
%% horizontally at the A11 row height. Exactly one clean crossing
%% (the A8->A11 spine edge, near x = -40mm); stays above the dashed
%% A9->PFRAME.north-west edge throughout.
\draw[arrow] (A9.west) .. controls (-40mm,-97mm) and (-58mm,-113mm) ..
 node[edgelabel,pos=0.6] {$\delta,\ \alpha_{31}$} (P5.east);
%% P5 -> A10 (solid; eta_B). The one long edge: runs left-to-right
%% through the clear corridor BELOW the A11 node (above the
%% Predictions frame), then climbs into A10's bottom border right of
%% the A10->A11 arrowtail; it leaves P5.south vertically. Controls
%% lifted 5mm with the raised P5 so the dip still clears A11's
%% bottom border. One clean crossing with each of the two dashed
%% A9->Predictions-frame edges.
\draw[arrow] (P5.south) .. controls (-96mm,-128mm) and (55mm,-130mm) ..
 node[edgelabel,pos=0.5] {$\eta_B$} ([xshift=8mm]A10.south);
%% P5 -> A8 (dashed; Wick-quartet/statistics feed for T-A8.12): with
%% P5 level with A11 this is now a short straight upward edge from
%% the column's top-right corner into A8's bottom border, 8mm left
%% of the A8->A11 arrowtail. No crossings.
\draw[arrowdash] (P5.north east) -- ([xshift=-8mm]A8.south);
%% --- White-Paper Branch edges (v13.11) ---
%% A3 -> W1 (dashed): \AThree{} is \PaperWOne{}'s heaviest spine source
%% (19 citations) and its thematic terminus -- the \BIsix{} anomaly
%% proof is where its Green--Schwarz / Hopkins--Singer ladder ends.
%% Departs 8mm right of A3.south so it clears the A3->A5 arrowtail, and
%% dips 7mm below \AFour{}'s bottom border before climbing to W1's west.
\draw[arrowdash] ([xshift=8mm]A3.south) .. controls (58mm,-43mm) ..
 node[edgelabel,pos=0.55] {\BIsix{}} (W1.west);
%% A10 -> W2 (dashed): \ATen{} is \PaperWTwo{}'s heaviest source of any
%% kind (30 citations).  Departs 14mm right of A10.south (clear of the
%% eta_B arrowhead at 8mm) and lands 10mm left of W2.north (clear of
%% the W1->W2 arrowhead, which otherwise hides this one entirely);
%% stays inboard of the re-routed A10->C-frame swing throughout.
%% v13.12: at +15% \ATen{} and \PaperWTwo{} overlap in x (never in
%% y, being 7mm apart), so the old short feed had nowhere left to be
%% seen.  Redrawn as a 13mm diagonal off A10's south-east corner,
%% landing clear of the solid W1->W2 arrowhead.
\draw[arrowdash] (A10.south east) .. controls (84mm,-101mm) ..
 ([xshift=-6mm]W2.north);
%% W1 -> W2 (solid, labelled -- the mirror of the solid 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 (v13.11): the re-routed A10->CFRAME swing now sets the
%% right ink edge at x ~ 127mm; pad the left to match so the map sits
%% horizontally centred on the page.
%% at x = 84mm pad the bounding box ~3mm beyond its ink on the
%% right; mirror that slack on the left so the resized map's ink
%% sits horizontally centred on the page.
%% (no pad: with the swing a polyline the bounding box is the ink)
\end{tikzpicture}}
% compact linked companion names for the falsifier showcase
\defcitealias{ArisakaA2GG6}{\mbox{GG-A2-GG6}}
\defcitealias{ArisakaA3BI6}{\mbox{GG-A3-BI6}}
\defcitealias{ArisakaA9SM}{\mbox{GG-A9-SM}}
\defcitealias{ArisakaA10Cosmos}{\mbox{GG-A10-Cosmos}}
\defcitealias{ArisakaC1Kaxion}{\mbox{GG-C1-Kaxion}}
\defcitealias{ArisakaP1KK}{\mbox{GG-P1-KK}}
\defcitealias{ArisakaP2N3}{\mbox{GG-P2-N3}}
\defcitealias{ArisakaP4Real}{\mbox{GG-P4-Real}}
\defcitealias{ArisakaP5CPT}{\mbox{GG-P5-CPT}}
%% v11.5 (August 17, 2026): THE DEFINITION IS IN THE STYLE FILE NOW.
%% The comment above is the record of why it was here -- the style
%% file is shared, and changing it meant rebuilding every arXiv
%% package.  That was done, so what is left is what this paper sets.
%% THE LANDSCAPE PAGE CANNOT AFFORD \Large, AND THAT IS MEASURED.
%% Raising \GGShowcaseHead from \large to \Large across the cohort
%% took A1 from 292 pages to 293: page 4 went BLANK, because the
%% landscape plate was already at its ceiling -- 0.88 of \linewidth
%% fits and 0.90 does not, bisected when the page was built -- and a
%% taller headline eats the last of the room.  Shrinking the plate to
%% buy headline is the wrong trade on a page whose whole value is the
%% plate, so the landscape page keeps the old size under its own name.
%% The two portrait showcase pages take \Large with the cohort.


% ---- showcase 1 of 3: the constitution as a classification -------
% NO \clearpage HERE: pdflscape's landscape environment issues one,
% and the pair produced a blank sheet in the first draft.
\clearpage
\begin{landscape}
\thispagestyle{empty}
\null\vfill
\noindent\begin{minipage}{\linewidth}
\centering
\GGShowcaseHead{Every Framework of Twentieth-Century Physics,\\[2pt]
Sorted by the Axioms It Uses}
% 0.86 IS THE MEASURED CEILING FOR THE HOUSE HEADLINE, re-bisected when
% this page's headline came to the cohort's \Large: 0.87 throws the whole
% block onto the next sheet, 0.86 holds.  The previous ceiling was 0.88,
% bisected the same way while the headline still printed at \large; the
% house size costs about seven points of height and the plate gives them
% back.  The earlier note, kept here because it is still the lesson: the
% arithmetic said 0.92 would fit with 2.6 pt to spare and it did not,
% because the caption sets one line longer than the estimate.  Bisected,
% not calculated.
\resizebox{0.86\linewidth}{!}{\usebox{\GGscFlow}}
\par\bigskip
\captionof*{figure}{\textbf{The constitution drawn as a flowchart.} Each framework of twentieth-century physics enters at the level of the axioms it implicitly uses, and the branches descend as axioms are added: one axiom gives special relativity, two give the geometric machinery of general relativity and gauge theory, the third adds the boundary coarse-graining that quantum mechanics and thermodynamics share, and the fourth removes the freedom that remains. What each branch \emph{cannot} say is the content of the chart --- a missing axiom is a prediction about which questions a framework will leave open, not merely an omission. Read stratum by stratum, with what each stratum's occupants are unable to express: Figure~\ref{fig:A1-constitutional-flowchart} in Part~V, p.~\pageref{fig:A1-constitutional-flowchart}.}
\end{minipage}
\par\vfill
\end{landscape}

% ---- showcase 2 of 3: the cohort that descends from it -----------
\clearpage
%% Self-filling showcase frame (Report 462): a minipage of EXPLICIT
%% height with the [s] inner alignment, so the interior glue stretches,
%% and 1 : 2 : 2 : 1 weights -- top, above the plate, below it, bottom.
%% The height is \textheight MINUS \topskip; see the report for why.
\noindent\begin{minipage}[t][\dimexpr\textheight-\topskip\relax][s]{\textwidth}
\vspace*{\stretch{1}}
\centering
\GGShowcaseHead{Eleven Clauses, Twenty-Two Papers,\\[2pt]
No Free Parameters}
\par\vspace*{\stretch{2}}
\makebox[\textwidth]{\resizebox{1.10\textwidth}{!}{\usebox{\GGscMap}}}
\par\bigskip
\par\vspace*{\stretch{2}}
\captionof*{figure}{\textbf{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. Solid arrows mark axiom-forced descent and carry an edge label naming the driving axiom or postulate; dashed arrows mark the deployment of derived results. The two flanking branches and the full reading conventions are in Figure~\ref{fig:A1-cohort-inheritance-map} in Part~I, p.~\pageref{fig:A1-cohort-inheritance-map}.}
\par\vspace*{\stretch{1}}
\end{minipage}
\clearpage

% ---- showcase 3 of 3: how it dies --------------------------------
{\centering\GGShowcaseHead{Ten Pre-Declared Numbers, and What Would Kill Each One}}
{\small\renewcommand{\arraystretch}{1.15}
\begin{longtable}{@{}>{\raggedright\arraybackslash}p{0.139\textwidth}>{\raggedright\arraybackslash}p{0.180\textwidth}>{\raggedright\arraybackslash}p{0.170\textwidth}>{\raggedright\arraybackslash}p{0.108\textwidth}>{\raggedright\arraybackslash}p{0.2054\textwidth}>{\raggedright\arraybackslash}p{0.1546\textwidth}@{}}
\caption*{\textbf{Ten pre-declared numbers, and what would kill each one.} The sharpest numerical predictions of the \GGTheory{} program, each with the threshold declared in advance that would falsify it and its experimental status in 2026. Two rows are measured and verified, three are consistent with current data, two are under active experimental search, and three are carried as forward audits. The right-hand column links to the companion paper where the prediction is proved. Proof locations, the comparison mode for each row, and the full claim contracts are in Table~\ref{tab:pull_table} in Part~IV, p.~\pageref{tab:pull_table}.}
\\
\toprule
\textbf{Falsifier} & \textbf{GG closed form} & \textbf{Measured, 2026} & \textbf{Status} & \textbf{Dies if} & \textbf{Proved in}\\
\midrule
\textbf{F-1}\newline GUT scale $\EGUT$ & $\EPl e^{-2\pi} \approx 2.28\times10^{16}$ GeV & Inferred from $\alpha_i$ flow & Forward audit & Unification fails at one loop & \citetalias{ArisakaA2GG6}\\\addlinespace[0.38em]
\textbf{F-2}\newline Lock cluster $\Mlock$ & $\tfrac12\MPl e^{-35} \approx 3.85$ TeV & No resonance yet at this mass & Active LHC search & $5\sigma$ absence at HL-LHC, or $5\sigma$ resonance at $\neq \Mlock$ & \citetalias{ArisakaA2GG6}, \citetalias{ArisakaA9SM}, \citetalias{ArisakaP1KK}\\\addlinespace[0.38em]
\textbf{F-3}\newline Electroweak vev $v$ & $\approx 246$ GeV & $246.220 \pm 0.001$ GeV & \textbf{Verified} & Deviation $>5\sigma$ from the rational derivation & \citetalias{ArisakaA9SM}\\\addlinespace[0.38em]
\textbf{F-4}\newline Matter fraction $\Omega_m$ & $1/\pi = 0.31831$ & $0.3153 \pm 0.0073$ (Planck 2018) & $\sim 0.3\sigma$ & $5\sigma$ from $1/\pi$ at resolution $\sigma \leq 0.003$ & \citetalias{ArisakaA10Cosmos}\\\addlinespace[0.38em]
\textbf{F-5}\newline CC scale $m_\Lambda$ & $\approx 6$ meV & Consistent with $\Lambda \approx 1.1\times10^{-52}$ m$^{-2}$ & Consistent & Deviation $>5\sigma$ from neutrino-scale matching & \citetalias{ArisakaA10Cosmos}\\\addlinespace[0.38em]
\textbf{F-6}\newline Baryon fraction $\Omega_b$ & $1/(2\pi^2) = 0.05066$ & $0.0493 \pm 0.0006$ (Planck 2018) & $\sim 2\sigma$ & $5\sigma$ in any next-generation BBN\,/\,CMB fit & \citetalias{ArisakaA10Cosmos}\\\addlinespace[0.38em]
\textbf{F-7}\newline Kaxion split $\Delta\nu/\nu$ & $1/35 = 0.02857$ & Not yet registered & Active haloscope search & Narrow peak at $\neq 1/35$ at $5\sigma$ & \citetalias{ArisakaA10Cosmos}, \citetalias{ArisakaC1Kaxion}\\\addlinespace[0.38em]
\textbf{F-8}\newline Generations $N$ & Exactly three families & Three observed & \textbf{Verified} & A fourth admissible generation & \citetalias{ArisakaA2GG6}, \citetalias{ArisakaP2N3}\\\addlinespace[0.38em]
\textbf{F-9}\newline \BIsix{} trace ratio & $|a/c| = |a/d| = 5/2$ & Implied by SM hypercharge via descent & Forward audit & A fifth admissible anomaly polynomial & \citetalias{ArisakaA3BI6}\\\addlinespace[0.38em]
\textbf{F-10}\newline Reality of the bulk category & One shared $i$; no fundamental non-geometric \emph{CP} phase; no quaternionic Born residue & No anomaly registered & Forward audit & A \emph{CP} phase resisting geometric expression; verified unshared-$i$ interferometry; a $5\sigma$ quaternionic Born deviation & \citetalias{ArisakaP4Real}, \citetalias{ArisakaP5CPT}\\\addlinespace[0.38em]
\bottomrule
\end{longtable}}
%% The showcase caption is \caption*, but a longtable steps the
%% table counter regardless, and v15.23 shipped every table in the
%% paper numbered one too high.  Put it back.
\addtocounter{table}{-1}
\clearpage
% ================= end front-matter showcase ====================




%% ===============================================================
%%  PAGES 5+ : TWO-LEVEL TABLE OF CONTENTS (per L66 / Part 5)
%%             OverviewOfContents (Part-only) -> ListOfFigures
%%             -> full TOC.
%% ===============================================================
\GGOverviewOfContents
\begingroup\setstretch{0.94}
\GGListOfFigures
\GGListOfTables
\endgroup
\GGTableOfContents

%% ===============================================================
%%  PREFACE -- compelling entrance for the entire paper
%%  Added in v5.5 (May 2026) per user request: a global
%%  introduction that captures the reader's mind before Part I
%%  begins.  Three paragraphs in the Arisaka register: a striking
%%  declarative opening (the contract metaphor); a statement of
%%  what the paper does (write the charter); a closing reflection
%%  in the spirit of Euclid's invitation to his own readers.
%% ===============================================================
\GGPreface

Every science begins with rules. Euclid's geometry begins with five
postulates that fit on a single page. Newton's mechanics begins with
three laws of motion that any high-school student can recite.
Maxwell's electrodynamics begins with four equations that fit on a
coffee mug. Yet beneath each of these explicit rules lies a deeper
layer that has almost never been written down --- the meta-rules that
say what counts as a law in the first place, what counts as an
observation, what is allowed to remain an adjustable knob, what kinds
of theories are even admissible candidates for being a theory of
nature at all.

These meta-rules exist. Every working physicist knows them. They
are passed silently from textbook to textbook, from supervisor to
student. A young theorist learns that gauge symmetries must be
local, that observables must be Hermitian, that fundamental theories
must be Lorentz-invariant, that the variational principle is the
right place to start --- but none of these meta-rules has ever
appeared in a numbered list of axioms that the reader can audit
clause by clause. For three and a half centuries the meta-rules
have been a craft tradition. A craft tradition is enough to build
pyramids; it is not enough to write a constitution.

This paper proposes that the next foundational step in physics is to
put the meta-rules in writing. We call the resulting short document
the \GGConstitution{}: four axioms, seven postulates, eleven clauses
in total. Small enough that any reader can hold all eleven in mind
in a single afternoon. Rigid enough that a single empirical
refutation upstream --- a verified deviation from one of its sharp
numerical predictions --- collapses one or more clauses. The
bargain offered to the reader is modest in form and radical in
content: we ask only that the eleven clauses be accepted as the
rules of the game, and then we let physics descend from them as
theorems rather than as fits.

We invite the reader to test this document in the spirit Euclid
invited his readers to test the \emph{Elements}. Read each clause.
Ask whether it is honest. Watch what theorems follow. After all,
a constitution earns its name not by the cleverness of the theorems
it produces but by the transparency with which it submits to
refutation. The first scientific revolution put the rules of
geometry in writing. The proposal here is that the next scientific
revolution will put the rules of physics in writing --- and then
submit them to the world.

\clearpage

\GGPart[Foundations]{Foundations}{The Constitutional Question and Why Physics Needs an
 Axiom Layer After Three and a Half Centuries}

\noindent
Every working laboratory runs on two sets of rules. One is posted on
the wall: how to handle the cryogen, who signs the logbook, what
counts as a calibration. The other is written nowhere, and it is the
more important of the two. It says what counts as a \emph{result}. It
says that a measurement nobody can repeat is not yet a measurement;
that a model with enough adjustable constants to fit anything has
explained nothing; that a theory which cannot be shown to be wrong is
not yet physics. Graduate students absorb this second set by watching
their supervisors, and almost nobody writes it down.

This Part is about that second set of rules.

We begin with history, because the rules were not handed down all at
once --- they accumulated, and each one was hard won. Euclid showed
that a large body of knowledge could be grown from a short list of
stated assumptions. Newton showed that the same few laws govern a
falling apple and an orbiting moon. Riemann gave geometry the freedom
to curve; Maxwell found that light is a disturbance in a field;
Einstein made geometry itself the carrier of gravity; von Neumann put
quantum mechanics on a rigorous footing; and the open-systems
generation --- Zeh, Joos, Zurek, Lindblad, and others --- worked out
what it means for a system that cannot see everything to make a
measurement at all. Each step tightened the informal rulebook. None
of them wrote it out.

We then say what we think is missing. Four clusters of problems have
resisted the twentieth century's methods: the measurement problem, the
unexplained parameters of the Standard Model, the cosmological
constant, and the gulf between physics and the sciences of the living.
Our diagnosis is that these are not four separate failures of
cleverness but one failure of bookkeeping --- physics never wrote down
which commitments it was actually making.

The rest of the Part is preparation. We fix four words that will be
used precisely throughout (\emph{axiom}, \emph{postulate},
\emph{theorem}, \emph{constitution}), preview the eleven clauses that
follow, and place this paper inside the twenty-two-paper program it
serves as the root of. A reader who wants the whole argument in plain
language before any formalism should read
Appendix~\ref{sec:qa-curious}, which answers twenty-five basic
questions without a single equation.

\medskip

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

\subsection{A historical chain in seven steps: from Euclid to the open-systems generation}
\label{sec:historical_chain}

The question we shall ask in Part~I is whether
the next foundational step in physics is the writing-down of a
constitution that has, for three and a half centuries, been carried
only in the craft tradition. Before we ask the question rigorously,
we ought to walk the historical chain that brought us to this
moment. Indeed, every axiomatic move in the long history of
physics --- from Euclid's \textit{Elements} to the open-systems
formalism of the late twentieth century --- has been an attempt to
turn a tacit body of practice into an explicit document that any
later reader could audit line by line. The \GGConstitution{}
proposed in this paper continues that line of work; its novelty
sits at one specific link in the chain. Let us trace the seven
steps that lead to it.

\paragraph{Step~1. Euclid (\textit{ca.}\ 300 BCE).}
The \textit{Elements} of \citet{EuclidElements} is the founding
demonstration that a small set of definitions, postulates, and
common notions can generate a vast theorem space by a fixed
deductive procedure. Euclid did not invent geometry, of course;
geometry had been practiced for millennia before him. Yet his
contribution was the architecture: an axiom-postulate-theorem
pipeline so robust that \emph{any} later reader could reproduce
every result by running the deductions on paper, without consulting
the master. What makes a body of knowledge a science
is not the cleverness of its calculations but the honesty of its
rules --- and Euclid was the first to write the rules down. The
present \GGConstitution{} is a deliberate echo of that move,
applied not to plane geometry but to the rules of admissibility
for any candidate theory of nature.

\paragraph{Step~2. Newton (1687).}
The \textit{Principia Mathematica} of \citet{Newton1687} is the
first complete constitution for mechanics. Three laws of motion
plus the law of universal gravitation, together with explicit
definitions of mass, momentum, and force, suffice to render the
motions of the heavens calculable from the same axiomatic stack
that governs the falling apple. Newton's most consequential
move was not the inverse-square law itself but the constitutional
stance --- that a single short document could serve as the rulebook
for celestial and terrestrial dynamics alike. Where Euclid set the
architectural template, Newton showed that the template extends to
the physical world.

\paragraph{Step~3. Riemann (1854) and Maxwell (1865).}
Two parallel developments in the nineteenth century broke the
Newtonian frame. \citet{Riemann1854}, in his Habilitation lecture
\textit{\"Uber die Hypothesen, welche der Geometrie zu Grunde
liegen}, declared that the geometry of physical space need not be
Euclidean --- a proposition that took six decades to find its
physical meaning, and that Einstein would eventually use as the
mathematical substrate of general relativity. \citet{Maxwell1865},
in \textit{A Dynamical Theory of the Electromagnetic Field},
provided the first complete unified-field constitution: four
equations that fold electricity, magnetism, and the optics of light
into a single coherent picture. Riemann taught
us that space itself is a degree of freedom, and Maxwell taught us
that a unified field is more than two unrelated fields written
together. Both will return as primitives in our constitution: the
6D bulk inherits Riemann's lesson on geometry, and the
composite gauge connection inherits Maxwell's lesson on unification.

\paragraph{Step~4. Einstein (1905, 1915) and Noether (1918).}
Special and general relativity \citep{Einstein1905,Einstein1915}
constitute the first explicit \emph{invariance constitution} in
modern physics: the equivalence principle and general covariance
together specify what counts as a physically meaningful statement
about spacetime. Einstein's move was axiomatic because it
did not propose a new force law; it proposed a new admissibility
rule, that the laws of physics must take the same form in every
local inertial frame. \citet{Noether1918} sharpened this stance
by proving that every continuous symmetry of the action implies a
conserved current. The lesson of Einstein and
Noether is that invariance principles \emph{are} the deep content
of physical law --- the equations of motion are corollaries. The
\GGConstitution{} takes this lesson seriously and extends it: the
geometry-gauge equivalence principle (\AX{2}~\GGEP{}) is an
invariance principle for the composite geometry-plus-gauge
connection on a principal bundle (whose specific structure group
is determined as a theorem of the cohort companion \ATwo{}
\citep{ArisakaA2GG6}, not by the axiom itself),
and the operational/open extension (\AX{3}~\OGGEP{}) is an
invariance principle for measurement on the boundary.

\paragraph{Step~5. Heisenberg, Schr\"odinger, Born, and von
 Neumann (1925--1932).}
Quantum mechanics arrived through a remarkable cluster of papers in
the late 1920s. \citet{Heisenberg1925} introduced matrix
mechanics; \citet{Schrodinger1926} introduced wave mechanics;
\citet{Born1926} introduced the probabilistic interpretation that
turned the squared modulus of the wave function into a recipe for
predicting measurement outcomes. \citet{vonNeumann1932}
consolidated these threads into the first axiomatic Hilbert-space
formulation of quantum theory --- the closest thing the early
twentieth century had to a written quantum constitution. Yet a
single weight-bearing problem remained: the relation between
unitary evolution and the projection postulate of measurement was
left as a pair of separate rules, with no theorem connecting them.
Indeed, every textbook of quantum mechanics for the past ninety
years has presented unitary evolution and the Born rule as
\emph{independent} axioms --- a fact that ought to bother any
reader trained in the Euclidean tradition. The
\GGConstitution{} sits at exactly this fault line. Its
\AX{3}~\OGGEP{} promotes the operational map to a primitive object;
its \T{2} (Operational Closure, proved in \AFive{} \citep{ArisakaA5GDOS}) demonstrates that
the Born rule and the GKLS generator follow as theorems rather than
as separate assumptions.

\paragraph{Step~6. Partial constitutions for QFT, locality, and
 open evolution (1955--1982).}
The middle decades of the twentieth century produced a chain of
partial axiomatizations, each of which sharpened a specific layer
of physics into an explicit document. \citet{Stinespring1955}
proved the dilation theorem that turns every completely positive
map into the partial trace of a unitary on a larger Hilbert space.
\citet{Wightman1956} gave axioms for relativistic quantum fields
that elevated locality and positivity to first-class commitments;
\citet{Haag1992} extended the axiomatization to operator algebras
on local regions of spacetime. \citet{Bell1964} clarified that
local-causality is not a metaphysical optional but an operationally
falsifiable constraint, and \citet{Aspect1982} verified the
predicted violation of Bell inequalities in the laboratory.
\citet{Wilson1971} turned coarse-graining itself into a flow on
theory space, making scale a first-class concept.
\citet{Lindblad1976} and \citet{Gorini1976} classified the
admissible Markovian generators of completely positive trace-preserving
evolution --- the GKLS form. Each of these lines is a partial
constitution, sharp on its specific layer but never assembled into
a single axiom-postulate document covering all of physics. Roughly
speaking, the work of this era gave us all the right ingredients;
what was missing was the table on which to lay them out together.

\paragraph{Step~7. The open-systems generation (1970--2003).}
The decoherence program of \citet{Zeh1970}, \citet{JoosZeh1985},
and \citet{Zurek1981, Zurek2003} accomplished the most consequential
shift since von Neumann. Zeh recognized in 1970 that no real
system has ever been isolated from its environment --- every
quantum system in nature is, in fact, an open system. Joos and
Zeh in 1985 quantified the staggering speed of environmental
decoherence; Zurek in 1981 and 2003 named the mechanism
``einselection'' and showed that the environment selects a
preferred pointer basis on a timescale far shorter than any
laboratory observation can resolve. Yet, even so, the open-systems
formalism remained a corrective layer added on top of unitary
quantum mechanics, not an axiomatic rewrite. Every
quantum-mechanics textbook still opens with the closed
Schr\"odinger equation and treats decoherence as a complication
that arises later. The \GGConstitution{} proposes the inversion:
openness must be the foundational law, and closure becomes its
limiting case. This is the central axiomatic move of the
present paper, and it is realized by the elevation of \AX{3}~\OGGEP{}
to the same logical footing as \AX{1}~\LC{} and \AX{2}~\GGEP{}.

\paragraph{Where the chain ends and where the present paper begins.}
Seven steps suffice to bring us to the present moment. Euclid
gave us the architecture; Newton showed it applies to physics;
Riemann and Maxwell taught us that geometry and unified fields are
the right primitives; Einstein and Noether elevated invariance to
the deep content of law; the founders of quantum mechanics gave us
the Hilbert-space framework but left measurement loaded; the
mid-century axiomatizers (Stinespring, Wightman, Bell, Wilson,
Lindblad) sharpened individual layers but never assembled them;
and the open-systems generation (Zeh, Joos, Zurek) recognized that
the inherited foundation had been written upside down. The
present paper proposes the eighth step --- a single short document,
the \GGConstitution{}, that turns the axiomatic questions
implicit in the previous seven steps into four axioms and seven
postulates that any reader can audit line by line. Indeed, every
later theorem in the \GGTheory{} program cites this document by
name.

\subsection{Motivation: an axiomatic gap after the twentieth-century revolutions}
\label{sec:motivation}

The twentieth century produced two foundational
revolutions --- quantum mechanics and relativity --- that reshaped
the meaning of state, symmetry, and prediction. Quantum theory
emerged from the early quantum hypothesis and the subsequent
development of matrix and wave mechanics
\citep{Planck1900,Heisenberg1925,Schrodinger1926,Born1926}.
Relativity recast spacetime as a geometric object and reinterpreted
gravity as curvature \citep{Einstein1905,Einstein1915}. These
achievements were consolidated into robust theory stacks: quantum
field theory (QFT), the Standard Model (SM), and precision
cosmology. Gauge symmetry and renormalization became the workhorses
of particle theory \citep{Noether1918,tHooft1971,Wilson1971}, while
cosmic observations established a precise phenomenological
expansion history
\citep{Guth1981,Linde1982,Riess1998,Perlmutter1999,Planck2018}.

Yet, despite this dramatic success, several deep problems have
remained resistant for decades. The point of departure of
\GGTheory{} is a specific diagnosis --- the resistance is not only
technical; it is also \emph{constitutional}. By an
\emph{axiomatic layer} we mean the minimal set of foundational
commitments that specify what counts as a law, what counts as an
observation, what degrees of freedom are admissible as fundamental,
and which free choices (parameters, normalizations, conventions)
are allowed to remain as non-derivable input. This paper proposes
an explicit axiom/postulate system intended to serve as exactly
that layer.

The axiomatic gap can be made concrete by a simple thought
experiment. Let us imagine that we are given an equation of motion
and a Hilbert-space state. Without additional clauses, we still
cannot answer four basic questions that every scientific account
must answer. First, which variables are physically redundant
(``gauge'') and which are physical? Second, which operations count
as legitimate observations, and how are they represented
mathematically? Third, which parameters are inputs rather than
outputs? Fourth, which coarse-grainings are allowed without
changing the truth value of a claim? Twentieth-century physics
answers these questions implicitly and differently in different
subfields: QFT relies on locality, microcausality, and
renormalization; measurement theory relies on external instruments
and an observer cut; cosmology relies on effective parameters that
fit data. The result is not inconsistency in practice, but
\emph{axiomatic drift} --- admissibility rules are learned as
craft rather than stated as law.

The aim of \GGTheory{} is therefore modest in one sense and radical
in another. It is modest because it does not begin by proposing a
new Lagrangian or a new particle sector. It is radical because it
makes admissibility clauses explicit and insists that they apply
across domains. In this paper, ``unification'' means a single
logical pipeline --- symmetry, consistency, operational semantics,
scale robustness, and falsifiability --- used to judge candidate
explanations in particle physics, chemistry, biology, and beyond.
The constitution is intended as the infrastructure on which the
companion papers on $\GG{}$ \citep{ArisakaA2GG6}, $\BIsix{}$
\citep{ArisakaA3BI6}, $\GDOS{}$ \citep{ArisakaA5GDOS},
$\GiFour{}$ \citep{ArisakaA6Gi4}, and $\QiFour{}$
\citep{ArisakaA7Qi4} build their technical machinery.


\subsection{Four clusters of unsolved problems}
\label{sec:four_clusters}

The axiomatic gap manifests itself in four
clusters of unsolved problems --- one for each foundational
discipline of contemporary physics. We name them in turn.

\paragraph{Cluster (i): quantum mechanics --- measurement, openness, and
 operational meaning.}
Standard quantum theory combines unitary evolution with a separate
set of rules for measurement (the Born rule, the projection
postulate, the von Neumann cut)
\citep{Born1926,vonNeumann1932}. In practice every measurement
involves devices, environments, coarse-graining, and irreversible
records. Open quantum systems provide the modern operational
language --- CPTP maps and GKLS generators
\citep{Lindblad1976,Gorini1976,BreuerPetruccione2002} --- but the
formalism is treated as an effective framework rather than as an
axiomatic requirement. Every quantum-mechanics
textbook for the past ninety years opens with the closed
Schr\"odinger equation and treats decoherence as a complication
that arises later \citep{Zeh1970,JoosZeh1985,Zurek1981,Zurek2003}.
What, then, is a measurement? And how shall we write it as a
law-like component of a theory rather than as meta-language pasted
onto the side?

\paragraph{Cluster (ii): gravity and quantum theory --- ``quantize
 geometry'' versus ``geometrize quantum.''}
The dominant strategy for quantum gravity has been to start from
classical geometry (GR) and attempt to quantize it, either
perturbatively or non-perturbatively, including the string-theoretic
UV completions
\citep{GreenSchwarz1984,Polchinski1995,Rovelli2004}. In parallel,
the gauge/gravity duality program suggests a structural relation
between gravity and gauge dynamics
\citep{tHooft1993,Susskind1995,Maldacena1998,Witten1998}. These
developments raise an axiomatic question: is ``quantize
geometry'' the right starting line at all? The \GGTheory{} stance
reverses the emphasis --- it asserts a \emph{geometrization of
quantum and gauge structure} at the level of the primitive composite
connection (\AX{2}~\GGEP), with quantization treated as an
operationally defined boundary manifestation (\AX{3}~\OGGEP).

\paragraph{Cluster (iii): unification and predictivity --- too many
 dials and too many landscapes.}
The Standard Model is remarkably rigid once its symmetry group and
field content are fixed, yet it contains roughly 28 numerical
parameters not predicted from first principles \citep{PDG2024}.
Beyond-the-SM model building typically introduces further tunable
sectors; cosmology relies on parameterized fits to large datasets
\citep{Planck2018}; and the naturalness debates articulate the
concern that small numbers and large hierarchies appear unstable
unless protected by principle \citep{THooftNaturalness1980,Susskind1979}.
Many UV frameworks admit very large spaces of consistent vacua,
which makes it unclear which features are predictions and which
are choices \citep{Polchinski2004Landscape}.
The situation is not that the Standard Model is wrong --- it is
that nobody has yet written down a constitution rigid enough to
force its parameters from above. \GGTheory{} responds by
introducing an explicit axiom (\AX{4}~\MSMF{}) and a postulate
(\POS{6}~\OGN{}) designed to forbid hidden continuous knobs and to
force normalization choices into a minimal discrete ledger.

\paragraph{Cluster (iv): cosmology and complexity --- fragmentation
 across sciences.}
Cosmology raises further unresolved questions: the origin of
inflation, the nature of dark matter and dark energy, and the link
between microphysics and initial conditions
\citep{Guth1981,Linde1982,Riess1998,Perlmutter1999}. Beyond
fundamental physics, chemistry, life science, neuroscience, and
social systems are dominated by open-system dynamics, multi-scale
coarse-graining, and inference from incomplete data. Even when
micro-laws are known, the effective laws governing the boundary
descriptions appear ad hoc. Interestingly, the same admissibility
grammar --- symmetry, consistency, operational observables, scale
covariance, and falsifiability --- threads through all four
clusters. The ambition of \GGTheory{} is not to reduce all
sciences to particle physics; it is to write down a single
admissibility grammar that applies uniformly across domains, so
that a chemist, a neuroscientist, and a particle theorist may run
the same audit on their respective models.

\subsection{Why ``axioms plus postulates'' --- and why now?}
\label{sec:why_axioms_postulates}

Axiom systems are not a matter of style. They are a compression
technology. Section~\ref{sec:historical_chain} walked the
historical chain from Euclid through the open-systems generation;
let us now extract the methodological lesson from that chain.
Indeed, every step of the chain produced a \emph{partial}
constitution, sharp on its own layer of physics but never
assembled into a single document. Special and general relativity
gave us an invariance constitution (the equivalence principle and
general covariance) \citep{Einstein1905,Einstein1915}. Axiomatic
QFT pushed locality and positivity into explicit axioms (the
Wightman and Haag-Kastler frameworks)
\citep{Wightman1956,StreaterWightman1964,Haag1992}. Open quantum
systems introduced a strict notion of admissible evolution (the
completely positive semigroups of GKLS) \citep{Lindblad1976,Gorini1976}.
And modern renormalization made scale a first-class concept by
turning the operation of ``zooming out'' into a flow on theory
space \citep{Wilson1971}. What is missing from this list is not
ingredients --- the ingredients are all here --- but a \emph{unified
ordering and dependency chain}: which clauses are primitive, which
are derived, and which are non-negotiable when we cross from closed
microscopic descriptions to open boundary observations and from
physics to other sciences.

The \GGConstitution{} responds with a specific architectural choice.
Axioms fix the semantic primitives --- what objects exist, what
counts as redundancy or gauge, what is forbidden as acausal --- and
postulates act as ordered admissibility filters that any candidate
dynamics must pass before it is even up for review (consistency,
positivity, scale coherence, normalization, empirical auditability).
This separation mirrors the successful practice of physics: we
typically reason with invariance and structure first, and only then
choose model-specific details. The novelty here is to \emph{freeze}
that practice into a portable document that can be applied uniformly
to particle models, cosmological models, chemical reaction networks,
and biological open systems. What makes a body of knowledge a
science is not the cleverness of its calculations but the honesty
of its rules.


\subsection{Working definitions: axiom, postulate, theorem, and constitution}
\label{sec:definitions}

To avoid terminological drift across the ten Parts of this
paper, let us fix the working definitions we shall use throughout.
The four definitions are intentionally Euclidean in spirit ---
short, declarative, and reusable as building blocks --- and they
are the only definitions we shall need at the axiomatic level.

\begin{eucliddefinition}[Axiom]
An \emph{axiom} is a foundational statement that fixes the semantic primitives of the framework: what structures exist, what counts as redundancy
(gauge/representation), and what is forbidden (e.g.\ acausal operational kernels). Axioms are not derived within the framework; they are the minimal commitments
that define admissibility.
\end{eucliddefinition}

\begin{eucliddefinition}[Postulate]
A \emph{postulate} is an ordered operational admissibility filter that constrains candidate models within the axioms. Postulates typically enforce mathematical consistency
(e.g.\ anomaly cancellation), physical admissibility (e.g.\ positivity and normalization of states), scale robustness (RG compatibility), and empirical auditability
(measurement protocols and falsifiers).
\end{eucliddefinition}

\begin{eucliddefinition}[Theorem]
A \emph{theorem} is a derived result forced by axioms and postulates plus explicitly declared assumptions of a concrete model instance. In the \GGTheory{}\ style, theorems are
expected to ship with an operational contract: a map to registered observables and a pre-declared falsifier.
\end{eucliddefinition}

\begin{eucliddefinition}[Constitution]
The \emph{constitution} is the frozen union of axioms and postulates:
\begin{equation}
\mathrm{Constitution} := \{\mathrm{AX}\}\cup \{\mathrm{POS}\}.
\end{equation}
\end{eucliddefinition}

\paragraph{Grades: how strongly a claim is held.}
Throughout this paper and the cohort, every substantive claim carries a
\emph{grade} saying how far it has actually been established. There are
four, and they are ordered:

\begin{itemize}[leftmargin=2.2em]
\item \textbf{Derived} --- forced by the stated clauses through a proof
 that exists, with no continuous freedom left anywhere in the chain.
\item \textbf{Conditional} --- forced once one further named lemma is
 supplied. The lemma is stated, and the work it still needs is named.
\item \textbf{Candidate} --- proposed and internally consistent, with
 the constitutional pricing stated but not proved.
\item \textbf{Open} --- neither proved nor priced; recorded so that it
 is not mistaken for either of the two above.
\end{itemize}

\noindent
A related word, used in the same spirit, is \emph{custody}: where two
clauses could each plausibly own a piece of structure, we say which one
does, so that no result is quietly claimed twice. Grades and custody
are bookkeeping rather than rhetoric. Their whole purpose is that a
reader can tell, at any point in the document, the difference between
what has been shown and what has been proposed.

\subsection{A preview of the \GGConstitution{}: four axioms and seven postulates}
\label{sec:constitution_preview}

The \GGConstitution{} comprises four axioms ($\AX{1}$--$\AX{4}$) and
seven postulates ($\POS{1}$--$\POS{7}$). We state them here
informally so that the reader carries them in mind through the rest
of Part~I; Parts~II and~III provide the rigorous kernels, parameter
definitions, and forced-consequence chains.

\paragraph{Four axioms.}
\begin{enumerate}[label=\textbf{\AX{\arabic*}.},leftmargin=*]
\item \textbf{\LC{} (Locality and Causality).} Admissible evolution
 is local and causal in the operational sense: boundary updates
 are retarded, and no acausal effective kernels are permitted.
\item \textbf{\GGEP{} (Geometry-Gauge Equivalence Principle).} The
 primitive field of physics is a single composite geometry-plus-gauge
 connection on a principal bundle: quantum and gauge structure are
 geometrized rather than appended. This is the generalization of
 Einstein's Equivalence Principle from gravity alone to the full
 geometry-plus-gauge structure.
\item \textbf{\OGGEP{} (Operational/Open \GGEP{}).} Observation and
 measurement are defined by a bulk-to-boundary
 registration map that produces operational observables. Openness
 is constitutional; closure is its limiting case.
\item \textbf{\MSMF{} (Maximum Symmetry, Minimum Freedom).} Nature
 selects maximal symmetry with minimal free parameters; continuous
 dials not forced by the constitution are forbidden, and where
 discrete choices remain admissible Nature picks the minimal-rational
 ledger.
\end{enumerate}

\paragraph{Seven postulates.}
\begin{enumerate}[label=\textbf{\POS{\arabic*}.},leftmargin=*]
\item \textbf{\GI{} (Gauge Invariance).} Gauge invariance is
 layered across scales --- local (\LGI{}), dynamical/open (\DGI{}),
 global (\GGI{}).
\item \textbf{\UEP{} (Unified Extremal Principle).} A single
 variational engine governs dynamics, unifying least action with
 least-time and free-energy descent on the boundary.
\item \textbf{\AF{} (Anomaly Freedom).} Admissible theories must
 cancel every gauge and gravitational anomaly, including by
 Green-Schwarz inflow \citep{GreenSchwarz1984}.
\item \textbf{\CP{} (Closure and Positivity).} State evolution must
 remain physical: CPTP at the quantum level and positivity-preserving
 Markov semigroups at the classical mesoscopic level.
\item \textbf{\RG{} (Scale Covariance).} Coarse-graining must
 commute with the operational dictionary; predictions are stable
 under resolution changes \citep{Wilson1971}.
\item \textbf{\OGN{} (Occam Gauge Normalization).} Among consistent
 normalizations, Nature selects minimal rational and integer
 choices; no arbitrary continuous rescaling survives.
\item \textbf{\MF{} (Measurement and Falsifiability).} Every claim
 must map to a registered observable and a pre-declared kill
 criterion.
\end{enumerate}

\paragraph{The admissibility predicate.}
For any candidate theory $T$, we define the admissibility predicate
\begin{equation}
\boxed{\;
\Admissible(T) \;:=\; \bigl(T \models \{\AX{1},\AX{2},\AX{3},\AX{4}\}\bigr)
 \;\wedge\;
 \bigl(T \models \{\POS{1},\ldots,\POS{7}\}\bigr).
\;}
\label{eq:admissible}
\end{equation}
The purpose of this paper is to specify the four axioms and the
seven postulates with enough precision that \eqref{eq:admissible}
becomes a practical filter rather than a slogan, and that any
candidate theory can be audited against the predicate clause by
clause.

\subsection{The geometrization ladder: from Einstein to \GGTheory}
\label{sec:geometrization-ladder-a1}

The second axiom above (\AX{2}, \GGEP) was described as the generalization of Einstein's equivalence principle from gravitation to the full geometry-plus-gauge content of physics. That phrase deserves to be unfolded once, because it locates the whole program on a single line of descent. Every foundational advance in our picture of spacetime has shared one logical shape --- \emph{a physical necessity that earlier theory carried as an external structure is recognized as geometry and absorbed into the manifold or its connection} --- and \GGTheory{} does not introduce this move but completes it. The invariant speed of light was geometrized into a single Lorentzian spacetime, in which $c$ is the conversion factor that fuses space with time \citep{Einstein1905, Minkowski1908}; the equivalence of inertial and gravitational mass was geometrized into the curvature of that spacetime, so that gravity became the shape of the stage rather than a force upon it. Two further necessities, unknown to Einstein, complete the ladder: the local gauge phase, recognized as a connection whose curvature is the field strength \citep{YangMills1954, Kaluza1921, Klein1926}, and the renormalization scale, recognized through Wilson's renormalization group \citep{Wilson1971} and the renormalization scale as displacement along a radial direction.

\GGEP{} is the single rule that runs all four rungs to completion: \emph{any structure the dynamics is forced to carry must be realized geometrically.} Applied to a local, causal, interacting, charged quantum theory, it geometrizes exactly two necessities beyond the Lorentzian stage --- a scale and a phase --- and closes there, yielding a six-dimensional arena $\cM_4 \times I_r \times S^1_\theta$ that can be neither smaller nor larger. These two directions are also the source of the only two transcendental constants the framework will admit: $e$, the base of every exponential along the scale interval $I_r$, and $\pi$, the angle of the gauge circle $S^1_\theta$ --- so that in every later formula an exponential is a scale factor and a $\pi$ is a gauge holonomy or a volume. The point of method worth carrying through the rest of this Constitution is that the extra dimensions of \GGTheory{} are not postulated, as spatial extra dimensions must be, but \emph{derived} as the geometrization of two things any such theory already cannot avoid; and that the program thereby completes the geometric unification Einstein pursued for thirty years, by enlisting the two quantum-field-theoretic ideas --- the gauge principle and the renormalization group --- that postdated him and that he would likely have resisted. The formal version of this claim is Theorem~0 of the companion paper \citep{ArisakaA2GG6}, which proves that \AX{1} and \AX{2} alone force the six-dimensional bulk; the present paper supplies the constitutional clauses on which that proof draws.

\paragraph{The constitutional picture in one diagram.} The same geometry can be read a second way that the rest of this Constitution uses repeatedly, and Figure~\ref{fig:two-fibers-constitutional} fixes it once. The six-dimensional arena is a \emph{fiber bundle}: its base is the Lorentzian stage $\mathcal{M}_4$ on which observations are registered, and over every point of that stage sits a two-part \emph{non-spatial} fiber --- the scale interval $I_r$ and the phase circle $S^1_\theta$. Two axioms act on this one object in opposite directions. \AX{2} (\GGEP) \emph{builds} the fiber: it is the rule that the two unavoidable necessities of a local, causal, charged quantum theory --- a renormalization scale and a gauge phase --- must be realized as geometry, and so brings the two fiber directions into being. \AX{3} (\OGGEP) \emph{collapses} it: registration on the boundary is the projection $\pi$ that erases the fiber coordinates, because neither is a direction of space and neither can be resolved by any boundary protocol. The downward arrow is the origin of quantum behavior --- superposition and interference are the unread phase $S^1_\theta$, while measurement and its irreversibility are the coarse-grained scale $I_r$ --- which the companion paper \citep{ArisakaA8QM} develops in full; the upward arrow is the bulk derivation of \citep{ArisakaA2GG6}. The whole program lives between these two arrows: \GGEP{} constructing the bulk above the stage, \OGGEP{} projecting it back down to the observed world.

\begin{figure}[tbp]
\centering
\resizebox{0.88\textwidth}{!}{%
\begin{tikzpicture}[>=stealth, font=\small, line width=0.59pt,
 box/.style={rounded corners=3pt, draw=black!55, line width=0.51pt, align=center, inner sep=6pt}]
  % top: the two necessities
 \node[box, fill=gray!8, text width=11.2cm] (need) at (0,4.3)
 {\textbf{Two structures no local, causal, interacting, charged quantum theory can avoid:}\\[1pt]
 a renormalization \emph{scale}\qquad and\qquad a gauge \emph{phase}};
  % --- central bundle drawing ---
 \coordinate (bx) at (-0.2,-0.55);
 \draw[fill=blue!7, draw=blue!55] (-2.7,-1.05) -- (1.9,-1.05) -- (3.05,-0.3) -- (-1.55,-0.3) -- cycle;
 \node[blue!55!black] at (2.35,-0.78) {$\mathcal{M}_4$};
 \fill (bx) circle (1.5pt);
 \node[blue!50!black, below=1pt of bx] {$x$};
 \draw[->, red!70!black, line width=0.73pt] (bx) -- ++(0,2.05);
 \node[red!70!black, right, align=left] at (-0.02,1.05)
 {$I_r$ : \emph{scale}\\[-2pt] \footnotesize RG --- irreversible, non-compact};
 \draw[violet!70!black, line width=0.73pt] (-0.2,0.3) ellipse (0.5 and 0.2);
 \draw[->, violet!70!black] (0.3,0.32) arc (0:330:0.5 and 0.2);
 \node[violet!70!black, right, align=left] at (0.45,0.28)
 {$S^1_\theta$ : \emph{phase}\\[-2pt] \footnotesize holonomy --- unitary, compact};
 \node[align=center, black!70] at (-2.15,1.0) {$\mathcal{M}_6 =$\\[-1pt] $\mathcal{M}_4\!\times\! I_r\!\times\! S^1_\theta$};
  % bottom: boundary world
 \node[box, fill=blue!6, text width=11.2cm] (bdy) at (0,-3.75)
 {\textbf{The registered 4D world: open and quantum} (\GDOS{})\\[1pt]
 \footnotesize superposition \& interference from $S^1_\theta$\quad;\quad measurement \& irreversibility from $I_r$};
  % AX-2 arrow (need -> bundle)
 \draw[->, line width=0.73pt] (need.south) -- (0,1.75);
 \node[right, fill=white, inner sep=1.5pt] at (0.2,2.75) {\footnotesize \AX{2} (\GGEP): \emph{geometrize} each necessity (builds the fiber)};
  % AX-3 arrow (bundle -> boundary)
 \draw[->, line width=0.73pt] (-0.2,-1.15) -- (bdy.north);
 \node[right, fill=white, inner sep=1.5pt] at (0.2,-2.18) {\footnotesize \AX{3} (\OGGEP): registration $=$ projection $\pi$, erasing $(r,\theta)$};
\end{tikzpicture}}
\caption[The two non-spatial fibers and the two axioms that act on them.]{The program in one diagram. The six-dimensional bulk is a fiber bundle over the Lorentzian stage $\mathcal{M}_4$, carrying at each point the two non-spatial fibers $I_r$ (scale) and $S^1_\theta$ (phase). \AX{2} (\GGEP) acts \emph{upward}: it geometrizes the two necessities a local, causal, charged quantum theory cannot avoid, bringing the fibers into being (the bulk derivation, Theorem~0 of \citep{ArisakaA2GG6}). \AX{3} (\OGGEP) acts \emph{downward}: registration on the boundary is the projection $\pi$ that erases the fiber, which is the origin of quantum behavior --- the compact phase $S^1_\theta$ supplying coherence and interference, the non-compact scale $I_r$ supplying measurement and its irreversibility (the open-system law \GDOS{}, developed in \citep{ArisakaA8QM}). The whole program lives between these two arrows.}
\label{fig:two-fibers-constitutional}
\end{figure}

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

The constitution developed in this paper stands on its own: it fixes what counts as an admissible law, an admissible observation, and an admissible prediction in any candidate physical theory. For readers following the whole \GGTheory{} cohort, it is also one link of a single open-system chain that the consequence papers cash out. Because no physical system is closed, the four-dimensional world is a lawful projection (\OGGEP{}) of the six-dimensional bulk, and that projection is observation itself; its dynamics is the boundary law \GDOS{}, forced by the geometric \GiFour{} and quantum-information \QiFour{} pillars, so that the laws, the particles, and the contents of the universe are \emph{observables} of the projection rather than free inputs. In that chain, the present paper supplies the constitution itself --- including its third axiom \OGGEP{} (\AX{3}), the projection on which the whole chain depends. The funnel below is shared, in structure, with the companion papers \ANine{}, \ATen{}, and \AEleven{}; here it is offered only as orientation, and the rest of this paper proceeds on its own terms.

\begin{figure}[hbt!]
\centering
\resizebox{0.681\textwidth}{!}{%
\begin{tikzpicture}[
 node distance=17mm,
 every node/.style={font=\sffamily\footnotesize, align=center},
 box/.style={draw=black!55, line width=0.53pt, rounded corners=2pt,
  minimum width=104mm, minimum height=11mm},
 arrow/.style={-{Stealth[length=2.4mm]}, line width=0.48pt, draw=black!70},
 elab/.style={font=\sffamily\scriptsize\itshape, fill=white, inner sep=1.5pt}
]
\node[box, fill=violet!10, draw=violet!55!black] (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, draw=blue!55!black, 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!14, draw=olive!55!black, 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!12, draw=orange!55!black, line width=0.88pt, below of=gdos] (obs) {\textbf{The laws, the particles, and the contents of the universe}\\ \textbf{are registered observables} of this one projection --- \textit{not free inputs}};
\node[box, fill=green!12, draw=green!55!black, 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, draw=orange!60!black, line width=0.88pt, below of=pill] (concl) {$\Rightarrow$ \textbf{they are determined}\\ {\footnotesize read off one geometry --- the work of the consequence papers \ANine{}--\AEleven{}}};
\draw[arrow] (open)  -- node[elab,right]{no closed 4D theory exists}            (oggep);
\draw[arrow] (oggep) -- node[elab,right]{the projection must be a definite law}  (gdos);
\draw[arrow] (gdos)  -- node[elab,right]{apply at every stratum}                (obs);
\draw[arrow] (obs)   -- node[elab,right]{and the law is not arbitrary}           (pill);
\draw[arrow] (pill)  -- node[elab,right]{projection forced $\Rightarrow$ observables forced} (concl);
\end{tikzpicture}}
\caption[The open-system chain shared across the \GGTheory{} cohort.]{\textbf{The open-system chain shared across the \GGTheory{} cohort.} Because no physical system is closed, the four-dimensional world is a lawful projection (\OGGEP{}, \AX{3}) of the six-dimensional bulk, and that projection is observation itself; its dynamics is \GDOS{}, forced by the \GiFour{} and \QiFour{} pillars, so the laws and contents of the universe are observables rather than free inputs. The same funnel opens the consequence papers \ANine{}, \ATen{}, and \AEleven{}, with a paper-specific payload; within it the present paper establishes the constitution and its axiom \OGGEP{} (\AX{3}).}
\label{fig:cohort-funnel}
\end{figure}

\noindent
Figure~\ref{fig:cohort-funnel} places the present paper inside that
chain: the funnel it draws is the one every cohort paper enters, and
what this one contributes to it is the constitution itself and the
operational axiom $\AX{3}$.

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

\noindent
Before we descend into the four axioms (Part~II) and the seven
postulates (Part~III), it is useful to lay out two reader-orientation
maps on the same opening: a panorama of the twenty-two-paper cohort that
the \GGConstitution{} compiles into
(\S\ref{sec:eleven-paper-inheritance},
\figref{fig:A1-cohort-inheritance-map}), and a panorama of the ten
Parts of the present paper
(\S\ref{sec:roadmap-ten-parts},
\figref{fig:A1-ten-parts-flowchart}). A reader of any later Part
should be able to flip back to this Section and orient themselves in
both directions: which cohort companion carries which Principal
Theorem, and where the present argument stands in the ten-Part
arc. After all, the constitution is the entry point the broader
physics community will reach first; this Section is the
``you are here'' map that an entry-point paper owes its readers.

\subsection{The cohort inheritance}
\label{sec:eleven-paper-inheritance}

The \GGConstitution{} is the axiomatic root of a twenty-two-paper
research program: eleven A-papers, six P-papers, three C-papers and
two W-papers, the P-series carrying both the phenomenology preprints
(P1--P3, P6) and the two foundations members (P4--P5). The
constitution stays
short --- four axioms and seven postulates --- precisely because
the working machinery lives in the cohort companions; the
constitution names the rulebook against which each companion's
theorems are audited. Roughly speaking, the A-series
inheritance falls into five strata: the constitutional root (\AOne{}),
the GG6 Trio (\ATwo{}, \AThree{}, \AFour{}: bulk geometry), the
boundary law (\AFive{}) with its two sibling pillars
(\ASix{}, \ASeven{}), the Consequence Trio (\AEight{}, \ANine{},
\ATen{}: textbook readouts), and the closing capstone
(\AEleven{}). On the spine's left flank, the Foundations Branch
(\PaperPFour{}, \PaperPFive{}) carries the origin of the
boundary's complex structure and the discrete-symmetry/matter
interface; on the right flank, facing it, the White-Paper Branch
(\PaperWOne{}, \PaperWTwo{}) carries the two syntheses that read
the cohort rather than extend it. Beneath the cohort, the Predictions
Branch (\PaperPOne{}, \PaperPTwo{}, \PaperPThree{}, \PaperPSix{})
and the Cosmology \& Astrophysics Branch (\PaperCOne{},
\PaperCTwo{}, \PaperCThree{}) supply sharp phenomenological
falsifiers. The single most useful visual
orientation is the inheritance map of
\figref{fig:A1-cohort-inheritance-map}; the per-paper paragraphs
below give the constitutional content carried by each node.

\paragraph{\AOne{} --- The \GGConstitution{} (this paper).}
Four axioms (\AX{1}~\LC{}, \AX{2}~\GGEP{}, \AX{3}~\OGGEP{},
\AX{4}~\MSMF{}) and seven postulates
(\POS{1}~\GI{}, \POS{2}~\UEP{}, \POS{3}~\AF{}, \POS{4}~\CP{},
\POS{5}~\RG{}, \POS{6}~\OGN{}, \POS{7}~\MF{}) plus the
admissibility predicate $\Admissible(T)$ that any candidate theory
must satisfy. The eleven Principal Theorems $\T{1}$--$\T{11}$ are
named here at high level; their proofs live in the cohort.

\paragraph{\ATwo{} --- $\GG{}$: Geometry-Gauge Locking at 6D bulk \citep{ArisakaA2GG6}.}
Forced by $\AX{2}~\GGEP{}$ together with $\POS{3}~\AF{}$,
$\POS{5}~\RG{}$, and $\POS{6}~\OGN{}$. The companion paper
constructs the canonical 6D bulk
$ds_6^2 = e^{-2r}\eta_{\mu\nu}dx^\mu dx^\nu + dr^2 + d\theta^2$
with $r\in[0,K]$ and $\theta\in[0,2\pi)$, and proves that this
bulk is the unique anomaly-free, moduli-free solution to the
axiomatic constraints. Principal Theorems $\T{1}$, $\T{3}$,
$\T{4}$, and $\T{6}$ live there.

\paragraph{\AThree{} --- $\BIsix{}$: the Green-Schwarz-modified Bianchi identity for $\GG{}$ \citep{ArisakaA3BI6}.}
Forced by $\POS{3}~\AF{}$ on the 6D bulk. The companion
paper supplies the rigorous Hopkins-Singer differential-cohomology
proof that the four-component anomaly polynomial $X_4 = \tr(R\wedge R) -
\tr(F_{\SU(5)}\wedge F_{\SU(5)}) - \tfrac{2}{5}F_B\wedge F_B -
\tfrac{2}{5}F_{Qi}\wedge F_{Qi}$ is exact, with the geometry-gauge
trace-ratio lock $|a/c| = |a/d| = 5/2$. Principal Theorem
$\T{5}$ lives there.

\paragraph{\AFour{} --- From Euclid to $\GG{}$: the historical lineage \citep{ArisakaA4Euclid}.}
The pedagogical companion to the GG6 Trio. Traces the long
chain from Euclid's \textit{Elements} through Newton, Riemann,
Maxwell, and Einstein to the 6D bulk of $\GG{}$. No new
theorems; the role is to position the cohort framework against its
predecessors and to keep the historical audit honest.

\paragraph{\AFive{} --- $\GDOS{}$: Geometric Dynamics of Open Systems \citep{ArisakaA5GDOS}.}
Forced by $\AX{3}~\OGGEP{}$ together with $\POS{4}~\CP{}$. The
boundary-side companion to the cohort. Develops the canonical
boundary law $\cL_{\GDOS{}}$ as the unique CPTP generator
compatible with the bulk-to-boundary functor, and proves that the
textbook QM postulates are theorems of the operational dictionary
(the chain $\T{7.1}$--$\T{7.7}$). Principal Theorems $\T{2}$,
$\T{7}$, $\T{8}$, and $\T{9}$ live there. \AFive{}'s Part~VIII
is also the technical home of the constitutional-axiom
classification of the twentieth century that Part~V of the
present paper restates in summary form.

\paragraph{\ASix{} --- The $\GiFour{}$ Pillars \citep{ArisakaA6Gi4}.}
Forced by $\AX{3}~\OGGEP{}$ on the geometric side. The
companion paper formalizes the four geometric-invariance pillars
descending from the constitution: the Total Geometric-Gauge object
$\TGG$, the Covariant Conservation Identity $\CCI$, the Geometric
Free-Energy Functional $\GFF$, and the Gauge-Runtime Invariance
Pipeline $\GRIP$ (the cascade $\DGI \to \GSSB \to \GA/\GGA{}$, run in
the two regimes \GRIPin{}/\GRIPout{}).
Principal Theorem $\T{10}$ lives there. The lineage is stated as
Derivation~4 of Section~\ref{sec:AX3}: \OGGEP{} demands the
dictionary, \GDOS{} writes it, \GRIP{} runs it.

\paragraph{\ASeven{} --- The $\QiFour{}$ Pillars \citep{ArisakaA7Qi4}.}
Forced by $\AX{3}~\OGGEP{}$ on the quantum-information side.
The companion paper formalizes the four QI admissibility pillars:
$\QiQUA$ (qua-state preparation), $\QiEP$ (quantum equivalence
principle), $\QiCON$ (decoherence and contraction), and $\QiHAL$
(holographic alphabet). Principal Theorem $\T{11}$ lives there.

\paragraph{\AEight{} --- 4D Quantum Mechanics in the language of $\GGTheory{}$ \citep{ArisakaA8QM}.}
The QM stratum of the Consequence Trio. Operationalizes
$\T{2}$ Operational Closure and $\T{7}$ 4D Quantum Mechanics into
a concept-by-concept reformulation of textbook QM: the Born rule
as a theorem of the Stinespring dilation, EPR / Bell / CHSH /
Aspect / Hensen / Giustina as evidence of the non-invertibility of
the 6D-to-4D projection, Schr\"odinger's cat as a boundary
decoherence statement, and the central thesis that Einstein was
right in 6D and Bohr was right in 4D, each within their respective
domains.

\paragraph{\ANine{} --- The Standard Model parameters from $\GG{}$ \citep{ArisakaA9SM}.}
The SM stratum of the Consequence Trio. Derives the 28 free
continuous parameters of the Standard Model --- the gauge
couplings, the Yukawa couplings, the CKM/PMNS mixing angles, the
electroweak vacuum expectation $v \approx 246$~\GeV{}, the
cosmological-constant scale $m_\Lambda \approx 6$~meV, and the
$\GG{}$-lock-mass cluster $\Mlock \approx 3.85$~\TeV{} ---
from the 6D bulk and the $\OGNfive{}$ integer ledger.
Zero free continuous parameters at the cohort level.

\paragraph{\ATen{} --- $\Lambda$CDM cosmology from $\GG{}$ \citep{ArisakaA10Cosmos}.}
The cosmology stratum of the Consequence Trio. Derives the
cosmological inventory partitions $\Omega_m = 1/\pi$,
$\Omega_\Lambda = 1 - 1/\pi$, $\Omega_b = 1/(2\pi^2)$, and
$\Omega_{\rm DM} = 1/\pi - 1/(2\pi^2)$ from the 6D bulk and
the cohort axioms. Each partition is a discrete consequence of
the constitution rather than a fitted dial.

\paragraph{\AEleven{} --- the closing capstone \citep{ArisakaA11Origin}.}
The recursive-uniqueness companion that closes the cohort.
Develops the universal-identity insight (why every electron is the
same electron) and the GRIP cascade through atoms, molecules,
ATP, DNA, and the first cell. Remarkably, the same
$\OGNfive{}$ ledger that fixes the SM parameters also threads
through the chain that produces life, consciousness, and the
broader cosmos-life-mind unification.

\paragraph{Predictions Branch and Cosmology \& Astrophysics Branch.}
Beneath the cohort sit the seven short falsifier papers of the two
phenomenology branches. The Predictions Branch:
\PaperPOne{} (the Kaluza-Klein lock-mass cluster at
$\Mlock \approx 3.85$~\TeV{}, with explicit LHC and
future-collider falsifiers), \PaperPTwo{} (the
three-generation structure forced by the integer ledger), and
\PaperPThree{} (the electron as the unique ground state of its
boundary stratum, importing \PaperPTwo{}'s $N=3$ with its fences), and
\PaperPSix{} (the proton as the ground attractor of the baryonic
stratum, its stability a theorem rather than an accident).
The Cosmology \& Astrophysics Branch:
\PaperCOne{} (the kaxion dark-matter oscillation at
$\Delta\nu/\nu = 1/35$, with explicit haloscope and
microwave-cavity falsifiers), \PaperCTwo{} (the Hubble tension read
as the two ends of one six-dimensional geometric corridor), and
\PaperCThree{} (black-hole entropy with the exact
Bekenstein--Hawking coefficient $A/4$ from induced gravity). Each
is a sharp falsifier of an upstream cohort theorem.

\paragraph{\PaperPFour{} --- the origin of $i$ \citep{ArisakaP4Real}.}
The first member of the Foundations Branch, on the map's left
flank. Derives the complex numbers of physics from two real
structures of the 6D bulk. The compact phase circle $S^1_\theta$,
inherited from \ATwo{}'s Theorem~0 arena, forces the gauge $i$:
its quarter-turn is the unique circle-compatible complex structure
on every charged sector (T-P4-1). The causal clock's one-sided
spectrum forces the state-space $i$: the polar phase of the time
generator is the unique positive-energy, dynamics-commuting
complex structure (T-P4-2). The antiparticle handshake is their
forced agreement --- the two quarter-turns act as one complex
structure on shell precisely under the Feynman--St\"uckelberg
pairing, so antiparticles are derived as a consistency condition
rather than postulated (T-P4-3). The bridge theorem then
establishes that positive frequency \emph{is} holomorphy, at the
Hardy, forward-tube, and twistor levels (T-P4-6/7); the
derivations hold at the quasi-free grade, with the interacting
extension stated conditionally under its named hypotheses. What
descends from this node: the standalone development behind
\AEight{}'s real-bulk series, and the complex-structure machinery
the boundary papers consume.

\paragraph{\PaperPFive{} --- the mirrors and the matter \citep{ArisakaP5CPT}.}
The second member of the Foundations Branch. Derives the discrete
symmetries as orientation: C, P, and T are the orientation
reversals of the three real factors of the bulk --- the fiber
circle, the spatial slice, and the causal clock --- and a discrete
symmetry is antiunitary exactly when it contains T (T-P5-1). The
total reversal CPT is the one reflection no real world can detect,
so the CPT theorem becomes orientation bookkeeping (T-P5-2), while
the individual mirrors are free to break because the
bulk-to-boundary descent is itself oriented --- parity falls on
the descent orientation. On the matter side, the paper routes the
cosmic matter excess through the flavored chain --- fed by
\ANine{}'s ledger phases and delivered to \ATen{}'s $\eta_B$
chain --- with the sign carried by the Majorana parity
$\alpha_{31} = \pi$; that throughput is derived-conditional on
its named finite-temperature hypothesis. The same paper
consolidates the four-endowment definition of time: time orders
(the cone), repeats (the Killing clock), opens one way (the
dictionary's irreversibility), and opens the half-plane of lawful
analytic continuation (the one-sided spectrum).

\paragraph{\PaperPSix{} --- the guarded proton \citep{ArisakaP6Proton}.}
The fourth member of the Predictions Branch, sister to \PaperPThree{}: where that paper runs the ground-state template for
the electron, this one runs it for the proton. Nearly everything ever
weighed is protons and neutrons, and nearly all of that weight is
interaction energy; yet the proton presents one exact mass with zero
width, a charge equal and opposite to the electron's to twenty-one
decimal places, and a spin of one half its own constituents fail to
add up to.
It has never once been seen to decay, though the grand unified
theories insisted it must. The answer offered is that the proton is
the ground attractor (\GGA{}) of the baryonic stratum, and that its
baryon number is not a bookkeeping label but a winding of the compact
geometry --- written by the same twist that breaks the unified gauge
symmetry and splits the Higgs multiplet --- so that stability is a
theorem of the geometry rather than an accident of what has not yet
been observed. Eleven questions are read as faces of that one fact,
every import at its proved grade and every kill criterion stated. A
single verified proton decay falsifies it.

\paragraph{\PaperWOne{} --- conservation without symmetry
\citep{ArisakaW1Symmetry}.}
The first of the two white papers, and the \emph{diachronic} one.
Physicists have long believed that things are conserved because the
world is symmetric; this review follows that belief through the seven
results that built it, in the order they were found, and each gives
way. Noether's first theorem ties a conserved quantity to a
symmetry that constrains motion; her second, in the same 1918 paper,
shows that a symmetry holding independently at every point delivers no
conserved quantity --- only an identity --- so gauge invariance is a
redundancy in our description, not a feature of nature. The Bianchi identity
conserves magnetic flux for no dynamical reason: a curvature is a
curvature. Quantized, the identity can fail at last, and only by a
whole number; Green--Schwarz then lets two failures pay for each
other, and Hopkins--Singer arrives where the conserved thing is an
integer, safe because integers have nowhere to move. The sequence has a direction --- conservation laws grow harder
to break as they say less about motion --- and the program is placed
at the end of that arc, at declared grades. Its capstone is the
determinism triad: symmetry
breaking re-read as symmetry \emph{reading}. It derives nothing new.

\paragraph{\PaperWTwo{} --- the sixteen open problems
\citep{ArisakaW2Holonomy}.}
The second white paper, and the \emph{synchronic} one. Physics keeps a
short list of questions its own tools cannot close --- what happens
when a measurement happens, where the Standard Model's unexplained
numbers come from, what became of the information that fell into a
black hole --- median age seventy years. It takes sixteen of them and
follows one exact, elementary statement to all sixteen. In the six-dimensional
geometry, every point of ordinary space carries two further directions
of opposite kind: a closed circle and an open interval. A holonomy ---
the record a loop keeps of what happened around it --- requires
closed loops, and an interval
has none. Everything on the circle is therefore holonomy: quantized,
protected, reversible, a group. Nothing on the interval can be: its
transport runs forward only, and each step is final. The circle
remembers; the interval forgets. Read with the cohort's open-system
chain, that sentence sorts all sixteen: a measurement is the interval
writing what the circle carried; the black hole's information was
never lost, only billed to the wrong book. Like its sibling it derives
nothing new, and it closes with the named measurements on which six of
the sixteen die outright.

\smallskip
\noindent\textbf{The cohort inheritance map.}
\figref{fig:A1-cohort-inheritance-map} shows which Principal Theorem
each node carries and which axiom or postulate drives each edge: a
spine from \AOne{} down to \AEleven{}, the phenomenology in two
framed rows below it, and two flanking columns of two doing opposite
jobs --- the Foundations Branch feeds the spine, the White-Paper
Branch reads it.

\begin{figure}[H]
\centering
\makebox[\textwidth]{\resizebox{1.00\textwidth}{!}{%
\usebox{\GGscMap}}}
\bigskip
\caption[The twenty-two-paper cohort map of GG-Theory.]{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:A1-cohort-inheritance-map}
\end{figure}
\clearpage

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

The remainder of this paper is organized into nine further Parts,
in the canonical constitutional-spine architecture. The visual
panorama is \figref{fig:A1-ten-parts-flowchart};
the prose walkthrough below names each Part in turn.

Part~II states the four axioms in a fixed internal format
(constitutional statement, mathematical kernel, forced
consequences, worked example, Physics-Intuition Remark, relation
to the companion papers, historical background). Part~III states
the seven postulates as ordered admissibility filters in the same
seven-block format, with each postulate's historical-background
block presented as twelve to fourteen numbered chronological
paragraphs that trace the postulate from its earliest historical
anchor through to the contemporary mathematical formulation, and
closing with an explicit \emph{axiomatic move} paragraph that names
the corresponding Stratum of Part~V's constitutional-axiom
classification. Part~IV is the axiomatic handoff: a
comprehensive inventory of all the \GGTheory{} papers, a relations
figure that shows their dependency structure, bridge subsections
to each companion paper, and the cross-paper coherence map that
pins each Principal Theorem $\T{k}$ to the canonical theorem
location in the companion paper where its proof actually lives.
No theorems are stated, sketched, or proved inside the present
paper; the constitution names the descent chain at high level
only and points the reader downstream.

Part~V stands a level above the cohort: it observes that the same
four constitutional axioms, when applied with different subsets
active, are the source of essentially every major
theoretical-physics framework of the twentieth century. General
relativity, gauge theory, string theory, and AdS/CFT live at the
$\AX{1}+\AX{2}$ level; quantum mechanics lives at
$\AX{1}+\AX{2}+\AX{3}$; the Standard Model with $\Lambda$CDM
lives at $\AX{1}+\AX{2}+\AX{3}+\AX{4}$. Each historical
framework is, structurally, what the \GGConstitution{} looks like
when a particular subset of its four axioms is operative; its
strengths and limitations follow directly from which axioms it
uses and which it lacks. A master ledger table and a landscape
flowchart make the classification visible at a glance. Part~V is
highlighted in \figref{fig:A1-ten-parts-flowchart} because, as
Part~III of the cohort companion \AEight{} is the central
operational mechanism of that paper, Part~V of the present paper
is the central structural payoff of the constitution itself.

Part~VI opens the discussion: we position the \GGConstitution{}
against the leading frameworks of twentieth-century physics (the
Standard Model, string theory, loop quantum gravity, causal-set
theory, axiomatic QFT, and the open-systems formalism) and
against the historical chain of partial constitutions sketched
above; the six families of adversarial reading that Part~IX then answers
in full (particle theorist, gravity researcher, quantum-foundations
specialist, mathematical physicist, philosopher, and string
theorist) and answer each with the relevant constitutional
clause; and we close Part~VI with a Problems-Solved catalog
$\mathrm{P}_1, \ldots, \mathrm{P}_{12}$ that records the twelve
foundations-of-physics questions which the twenty-two-paper cohort
has now closed, each with the named companion paper and the
live experimental falsifier still in play. Part~VII states the
conclusions. Part~VIII gathers the technical appendices: abbreviations, glossary,
worked mathematical elaborations \emph{from} each axiom and from each
postulate (not derivations \emph{of} axioms, which are foundational by
definition), and worked-example templates. The common objections and responses form Part~IX; Part~X is the inline bibliography with its
Relevance paragraphs.

Parts~II and~III are the axiomatic core; Part~IV is the
interface to the rest of the \GGTheory{} program; Part~V is the
constitutional-axiom classification of the twentieth century;
Part~VI is the discussion (Problems Solved $\mathrm{P}_1$--$\mathrm{P}_{12}$ included), Part~VII the conclusions, and
Part~VIII the technical appendices, Part~IX the common objections and responses, and Part~X the references. Read in order, the ten Parts compose into a single
coherent axiomatic document that any reader can audit clause by
clause.

\paragraph{What the present paper establishes vs.\ what it
names.} Let us be honest with the reader about the boundary
between what this paper will and will not deliver. The present
paper \emph{establishes}, with full mathematical statement, the
four axioms (Part~II) and the seven postulates (Part~III); the
admissibility predicate $\mathrm{Admissible}(T)$ that any
candidate theory must satisfy; the immediate inter-clause
consistency relations among the eleven clauses; the worked
examples that show what each clause forbids and what canonical
machinery it sets up (an Aspect 1982 Bell test for
$\AX{1}~\LC$, an Aharonov-Bohm + Berry phase calculation for
$\AX{2}~\GGEP{}$, a Stern-Gerlach + Stinespring dilation for
$\AX{3}~\OGGEP{}$, a 28-parameter audit of the Standard Model for
$\AX{4}~\MSMF$, and an analogous worked example for each
postulate); and the cross-paper coherence map of where every
Principal Theorem $\T{1}$--$\T{11}$ is proved. The present paper
\emph{names but does not prove} the eleven Principal Theorems
themselves. The proofs live in the companion papers, with each
proof location pinned to a single canonical line of
Table~\ref{tab:coherence_map} (see Part~IV,
\S\ref{sec:coherence_map}). This division of labor ---
the constitution states the rules and the worked examples, the
companion papers carry the theorems --- is exactly what the
constitutional architecture is for: the constitution is short
enough to audit in a single sitting, while the technical
machinery that follows from it is delivered in self-contained
companion preprints. The worked examples in Parts~II and~III are
not sketch proofs of $\T{k}$; they are demonstrations that each
axiom and each postulate has mathematical teeth and makes
definite predictions \emph{at the level of the clause itself}.

\clearpage
\null\vfill
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}
\par\bigskip
\captionof{figure}[Roadmap of the ten Parts of A1-Constitution.]{Roadmap of the ten Parts of \AOne{}. The argument develops in phases. \emph{Setup} (Part~I) lays out the constitutional question, the historical chain from Euclid to the open-systems generation, and the twenty-two-paper cohort the constitution compiles. The \emph{axiomatic core} (Parts~II--III) states the four axioms $\AX{1}$--$\AX{4}$ and the seven postulates $\POS{1}$--$\POS{7}$, each with its mathematical kernel and a worked example. The \emph{forward map} (Part~IV) pins every Principal Theorem to the companion paper where its proof lives. The \emph{constitutional-axiom classification} (\textbf{Part~V}, highlighted) is the Copernican reframing in which general relativity, Yang--Mills theory, string theory, AdS/CFT, quantum mechanics, the Standard Model, and $\Lambda$CDM are each located as theorems of a particular subset of the four axioms. Part~VI discusses significance, uniqueness, and the ten structural
innovations; Part~VII states the conclusions; and the three back-matter Parts collect the technical appendices (Part~VIII), the common objections and responses (Part~IX), and the annotated references (Part~X). Part~V is highlighted because it is the central structural payoff of the constitution.}
\label{fig:A1-ten-parts-flowchart}
\end{minipage}
\vfill
\clearpage


\begin{table}[t]
\centering
\caption[Motivation map.]{\textbf{Motivation map.} Structural gaps motivating a new constitution and the corresponding axiom/postulate responses proposed by \GGTheory{}. This table is used as an audit checklist that links each motivation to a specific clause whose mathematical kernel is spelled out later.}
\label{tab:intro_summary}
\footnotesize
\begin{tabular}{@{}>{\raggedright\arraybackslash}p{0.26\textwidth}p{0.30\textwidth}p{0.38\textwidth}@{}}
\toprule
Structural gap & Representative symptom & Constitutional response (preview) \\
\midrule
Geometry vs.\ quantum &
No consensus UV completion; gravity treated separately from gauge QFT &
\GGEP{}: geometry--gauge composite object; \GG{} engine (Sec.~\ref{sec:constitutional_handoff}) \\
Measurement/open systems &
Unitary evolution + ad hoc collapse; observer cut; environment treated as afterthought &
\OGGEP{}: bulk$\to$boundary operational dictionary; POS-\CP\ and POS-\MF\ enforce admissible open dynamics \\
Hidden dials / parameter freedom &
Many fitted constants; scheme/basis dependence mimics tunability &
\MSMF{} + \OGN{}: maximize symmetry; fix normalizations by discrete data (Secs.~\ref{sec:axioms}--\ref{sec:postulates}) \\
Scale dependence &
Different effective theories at different resolutions &
POS-\RG: coarse-graining commutes with operational semantics; predictions stable under resolution changes \\
\bottomrule
\end{tabular}
\end{table}


\subsection*{Closing remarks on Part~I}

Table~\ref{tab:intro_summary} collects this Part in one place: each row
names a structural gap in the present foundations of physics and the
clause proposed in answer to it. Use it as an audit checklist. A reader
who rejects a clause in Part~II or Part~III can come back here and see
exactly which gap that clause was introduced to close.

Step back from the history for a moment and notice what it actually
shows. At every stage, the advance was not a new fact about nature but
a new agreement about what a fact is allowed to look like. Euclid did
not discover a triangle; he discovered that you could say in advance
what you were assuming. Newton did not discover that things fall; he
proposed that the falling and the orbiting obey one law, and accepted
that this could be checked. Einstein did not discover that light is
fast; he took the constancy of its speed as a rule and followed the
consequences wherever they led, including to places that offended
common sense. The pattern repeats: somebody writes down a constraint,
and physics gets smaller and stronger at the same time.

That is what we are proposing to do again, and we should be honest
about the cost. An unwritten rulebook cannot be wrong. It bends
quietly around each new result, and nobody has to admit anything. The
moment you write the rules down, they become attackable --- and every
place where the written version fails to match what physicists
actually do becomes a visible defect rather than an invisible habit.
This is uncomfortable, and it is precisely the reason to do it. A
discipline that will not state its commitments cannot be corrected in
them.

There is a further reason, less about physics and more about who gets
to do it. Craft knowledge is transmitted by proximity. It reaches the
students of the right supervisors, in the right departments, in the
right countries, and it reaches almost nobody else. A rule written on a
page reaches a chemist, a biologist, a graduate student far from any
famous group, and anyone willing to read carefully and argue back.
Whether or not the particular eleven clauses proposed here survive, the
practice of writing them down is worth more than any one of them.

We now state the constitution itself. Part~II begins with the four
axioms --- $\AX{1}~\LC{}$, $\AX{2}~\GGEP{}$, $\AX{3}~\OGGEP{}$,
$\AX{4}~\MSMF{}$ --- in the rigorous form that every later theorem will
cite by name. Nothing after this point asks the reader to accept
anything on authority; every clause is stated so that it can be
refused.

\GGPart[The Four Axioms]{The Four Axioms}{LC, \GGEP{}, \OGGEP{}, \MSMF{}\\
 The Semantic Primitives of \GGTheory{}}

\noindent
An axiom, in this paper, is not a claim that something is true. It is a
declaration about what \emph{kind of thing} a physical theory is
allowed to be. That distinction matters, and it is worth a moment.
When a chemist says a reaction conserves mass, that is a claim about
the world. When a chemist says that any acceptable reaction equation
must balance, that is a rule about what counts as an equation at all.
The four statements in this Part are of the second kind.

There are four of them, and each can be said in a sentence before it is
said in mathematics.

$\AX{1}~\LC{}$ says that causes come before effects and that nothing
outruns light. This is the oldest rule in modern physics and the one
nobody is willing to give up.

$\AX{2}~\GGEP{}$ says that the geometry of spacetime and the gauge
fields of the forces are not two different kinds of thing that happen
to coexist. They are one object, looked at from two angles. Einstein
already made gravity into geometry; this axiom finishes the sentence.

$\AX{3}~\OGGEP{}$ says that observation is not a special act performed
by a conscious being. It is a projection: a system with more degrees of
freedom is being read by an instrument with fewer. Anything you can
actually measure is what survives that projection. This is the axiom
that does the most work in this paper, and it is the reason quantum
measurement, thermodynamic irreversibility, and black-hole entropy end
up in the same chapter.

$\AX{4}~\MSMF{}$ says that a theory may not carry a dial unless
something forces the dial to be there. A soap film stretched across a
wire loop does not choose its shape; the constraints choose it. This
axiom asks physics to be a soap film.

Each axiom is then presented in the same fixed format: the rigorous
statement, the minimum mathematics it needs, the consequences it forces
immediately, a plain-language paragraph on what it is really saying, and its
relation to the six-dimensional bulk and the boundary law of the
companion papers. The historical work each axiom positions itself
against is gathered separately, in the century-old history that
closes this Part.
Readers who want the argument and not the machinery can read the
rigorous statement and the plain-language paragraph in each of the four
and skip the rest without losing the thread.

\medskip

\section{Overview of the four axioms}
\label{sec:axioms}

Each axiom in this Part is presented in a fixed
seven-block internal format that the reader will see repeated four
times: a rigorous axiomatic statement; the minimal
mathematical kernel that the axiom talks about; the first
nontrivial forced consequences; a worked example drawn from a
laboratory protocol that any practicing physicist will recognize;
a Physics-Intuition Remark in plain language; and the relation to
the companion engines $\GG{}$ and $\GDOS{}$. The predecessors each
axiom positions itself against are named not here but in
\S\ref{sec:history-partii}, which gathers the four historical
lineages into one continuous account. Read together, the four axioms compose
into the semantic-primitive layer of the \GGConstitution{}: $\AX{1}$
fixes the causal arrow, $\AX{2}$ defines the bulk primitive, $\AX{3}$
defines what it means to observe that primitive from the boundary,
and $\AX{4}$ closes the theory space by forbidding non-forced freedom.

Throughout this Part we adopt a \emph{bulk-to-boundary} staging,
and we state the axioms abstractly --- they do not presuppose any
specific microscopic model. In particular, the dimension of the
bulk, the specific bulk metric, and the specific structure group of
the principal bundle are not axiomatic inputs at all; they belong
to the cohort realization developed in the companion papers (the
six-dimensional bulk in \ATwo{} \citep{ArisakaA2GG6}, the
modified Bianchi closure in \AThree{} \citep{ArisakaA3BI6}, and
the integer ledger in \AFour{} \citep{ArisakaA4Euclid}), and are
theorems of the axioms plus the seven postulates plus the cohort's
choice of admissible bulk. Indeed, the value of writing the
axioms abstractly is exactly that any candidate theory --- not only
the \GGTheory{} one --- can be audited against them clause by
clause.

\subsection{Common mathematical arena for the axioms}
\label{sec:axioms_arena}

Let us first investigate the common mathematical arena that all
four axioms talk about. Before any axiom can be stated
rigorously, we must fix the bulk geometry, the composite
connection, and the boundary operational package --- the three
ingredients that will appear in every later constitutional clause.

\paragraph{(i) Bulk geometry.}
Let $(\mathcal{M},g)$ be a smooth, time-oriented Lorentzian
manifold of admissible dimension, with metric $g$ encoding the
causal structure via its light cone. No further specification
of $(\mathcal{M},g)$ is required for the four axioms to be
stated; in particular, the dimension of $\mathcal{M}$, the
explicit form of $g$, and any compact direction structure belong
to the cohort realization rather than to the axioms. The concrete
six-dimensional bulk that the cohort framework adopts as its
canonical realization is constructed in \ATwo{}
\citep{ArisakaA2GG6}, where the dimension six, the scale-profile metric,
and the integer ledger lock-length $K = 35$ are derived as
theorems of $\AX{2}~\GGEP{}$ together with $\AX{4}~\MSMF{}$ and
the postulate stack. The reader who wishes to follow the worked
examples of the present paper on the concrete cohort bulk should
consult \ATwo{} for the explicit form; the axiomatic content here
does not require it. Throughout the cohort, this 6D bulk is read on the field-theory side of the bulk--boundary correspondence: its fifth (scale) direction is the renormalization-group axis $r=\ln(\EGUT/\mu)$, not a Randall--Sundrum spatial dimension; the radial direction denotes RG running rather than a gravitational extension of an extra spatial direction, and the explicit renormalization-group construction is given in \ATwo{} \citep{ArisakaA2GG6}.

\paragraph{(ii) Composite connection.}
Let $P\to \mathcal{M}$ be a principal bundle with structure group
$G$. The four axioms below are stated for any admissible such
bundle; they do not name a specific structure group, and they do
not require any particular factorization of $G$. The concrete
four-factor structure group of the cohort realization
($\Spinone\times\SU(5)\times\U(1)_B\times\U(1)_{Qi}$) is derived
as a theorem of $\AX{2}~\GGEP{}$ together with $\AX{4}~\MSMF{}$
and $\POS{3}~\AF{}$ in the cohort companion \ATwo{}
\citep{ArisakaA2GG6}; that derivation is not repeated here.
A \emph{composite (locked) connection} is a Lie-algebra valued one-form on $P$,
\begin{equation}
\mathcal{A} \;=\; \omega[g]\ \oplus\ A^{\mathrm{gauge}},
\label{eq:compositeA}
\end{equation}
where $\omega[g]$ is the spin connection induced by the bulk metric and $A^{\mathrm{gauge}}$ collects the internal gauge connections.
The corresponding curvature is
\begin{equation}
\mathcal{F} \;=\; d\mathcal{A}+\mathcal{A}\wedge \mathcal{A}
\;=\; R[\omega]\ \oplus\ F[A^{\mathrm{gauge}}],
\label{eq:compositeF}
\end{equation}
and the covariant derivative acting on a field $\Psi$ in an associated bundle is
\begin{equation}
\nabla^{(\mathcal{A})}_A \Psi
\;=\; \partial_A \Psi + \mathcal{A}_A\cdot \Psi.
\label{eq:covderA}
\end{equation}

\paragraph{(iii) Boundary operational package.}
Let the boundary be $\partial\mathcal{M}$ (an operational
screen/sector supporting registered records). In the cohort
realization of \ATwo{} \citep{ArisakaA2GG6}, the boundary is
specifically $\mathcal{M}_4 = \partial\mathcal{M}_6$ at the
$r = 0$ end of the radial direction; the axiomatic content here
does not presuppose this specific dimensionality.
Operationally, the boundary is described by:
(i) a state space (classical distributions or quantum density operators $\rho$ on a Hilbert space $\mathcal{H}$),
(ii) a set of registered observables (POVMs or instruments), and
(iii) admissible time evolution between interventions (channels $\Phi_{t,s}$).
The \OGGEP\ axiom will legislate that these are \emph{outputs} of a bulk$\to$boundary map rather than optional add-ons.

We now proceed axiom by axiom.

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

\section{AX-1: LC (Locality and Causality)}
\label{sec:AX1}

\subsection{Constitutional statement (rigorous)}

\begin{euclidaxiom}[AX-1 (\LC): Locality and Causality]
\label{ax:LC}
For every admissible bulk configuration and every admissible boundary intervention protocol,
the induced boundary evolution is \emph{local} (generated by local/quasilocal structures) and \emph{causal}:
registered boundary records at time $t$ depend only on boundary-accessible data in the causal past of $t$.
Equivalently, the operational influence of any intervention vanishes outside the future light cone defined by the
bulk/boundary causal structure, and no admissible effective description contains acausal (advanced) response kernels.
\end{euclidaxiom}

Three clarifications are constitutional (not interpretational):

\begin{enumerate}[leftmargin=2.2em]
\item \textbf{LC is about operational influence, not about metaphysical ``hidden variables.''}
The axiom is enforced at the level of registered statistics and admissible effective kernels.
\item \textbf{LC is formulated for open systems.}
Because \OGGEP\ defines observation as an open-system projection, LC must hold for open evolution maps and
measurement instruments, not only for closed unitary dynamics.
\item \textbf{LC entails point-like locality: the fundamental excitations are local field quanta, not extended objects.}
Microcausality~\eqref{eq:microcausality} attaches observables to \emph{points} $x$ of the manifold and requires them to commute at spacelike separation; the admissible excitations are therefore point-like quanta of local fields. \LC{} thereby excludes fundamental \emph{extended} objects --- strings or branes --- whose constituents are not localized at a point and whose microcausality is not pointwise. \GG{} is in this precise sense a point-particle framework, and it remains one even while geometrizing the gauge interaction, because its two further dimensions are non-spatial (a scale and a gauge dimension, developed under $\POS{5}$ and $\POS{6}$; see \citep{ArisakaA2GG6}, and \citep{ArisakaA4Euclid} for the philosophical reading).
\end{enumerate}

\subsection{Mathematical kernel}

LC can be encoded in several equivalent mathematical statements, depending on the level of description.

\paragraph{(i) Retarded support of Green's functions (field level).}
Consider a linear hyperbolic equation on the boundary (or projected from the bulk),
\begin{equation}
\mathcal{L}\phi(x) = J(x),
\end{equation}
for a differential operator $\mathcal{L}$.
LC requires that the physical propagator be the \emph{retarded} Green's function $G_R$:
\begin{equation}
\phi(x) = \int d^4y\; G_R(x,y)\,J(y),
\qquad
G_R(x,y)=0 \ \text{if } x\notin J^+(y),
\label{eq:retardedG}
\end{equation}
where $J^+(y)$ is the causal future of $y$.

\paragraph{(ii) Microcausality (operator level).}
In a relativistic QFT regime, locality and causality are encoded by microcausality:
for local observables $\hat{\mathcal{O}}(x)$ and $\hat{\mathcal{O}}(y)$,
\begin{equation}
[\hat{\mathcal{O}}(x),\hat{\mathcal{O}}(y)] = 0
\qquad \text{if } (x-y)^2<0,
\label{eq:microcausality}
\end{equation}
a condition consistent with special relativity \citep{Einstein1905,Minkowski1908} and standard QFT axioms.

\paragraph{(iii) Retarded kernels for open evolution (channel level).}
Let $\rho(t)$ be a boundary state (density operator or probability distribution).
A general causal effective evolution may be written using a memory kernel:
\begin{equation}
\frac{d}{dt}\rho(t)
= \int_{-\infty}^{+\infty} dt'\; \mathcal{K}(t,t')\,\rho(t'),
\qquad \mathcal{K}(t,t')=0 \ \ \text{for } t'>t,
\label{eq:retardedKernel}
\end{equation}
where $\mathcal{K}$ is a superoperator kernel.
LC is precisely the statement that $\mathcal{K}$ has retarded support.

In the Markovian limit (no memory beyond $\rho(t)$) LC is consistent with a forward-time semigroup of maps
\begin{equation}
\rho(t) = \Phi_{t,t_0}(\rho(t_0)),
\qquad
\Phi_{t_2,t_0} = \Phi_{t_2,t_1}\circ \Phi_{t_1,t_0}\quad (t_2\ge t_1\ge t_0),
\label{eq:composition}
\end{equation}
with $\Phi_{t,t}$ the identity channel.
If the dynamics is time-homogeneous, $\Phi_{t,t_0}\equiv \Phi_{t-t_0}$ and
\begin{equation}
\Phi_{t+s}=\Phi_t\circ \Phi_s,\qquad \Phi_0=\mathrm{id},\qquad t,s\ge 0.
\label{eq:semigroup}
\end{equation}
This is the operational form of an ``arrow of time'' for open systems.

\subsection{Forced consequences / derivations}

\paragraph{Derivation 1 (LC $\Rightarrow$ forward semigroup for memoryless evolution).}
Assume: (i) the effective description has no memory beyond $\rho(t)$ and (ii) interventions are localized in time.
Then the state at $t_2$ depends only on the state at $t_1$ and the evolution between them, implying
\eqref{eq:composition}.
Time-homogeneity reduces this to the semigroup law \eqref{eq:semigroup}.
The restriction $t\ge 0$ (no inverse map in general) is not a technicality: it is the operational content of
irreversibility once records are created.

\paragraph{Derivation 2 (LC + locality $\Rightarrow$ no superluminal signaling).}
In any operational theory whose statistics are computed from local algebras satisfying microcausality
\eqref{eq:microcausality}, interventions in a spacelike-separated region commute with observables in the target region.
Hence the target region's marginal statistics cannot depend on spacelike-separated choices, blocking
superluminal signaling at the level of registered records.
(Foundationally, this is distinct from Bell's notion of ``local causality'' for hidden variables \citep{Bell1964},
but it is the relevant notion for an axiomatic operational constitution.)

\paragraph{Derivation 3 (LC as a necessary precondition for \OGGEP).}
Because \OGGEP\ will define measurement as a bulk$\to$boundary projection producing irreversible records,
LC must already constrain the admissible operational map: the map cannot output channels/instruments that enable
advanced response or backward-in-time dependence.
Thus AX-1 is logically prior to AX-3.

\subsection{Further mathematical elaboration: microcausality, no-signaling, and causality $\Rightarrow$ analyticity}
\label{sec:AX1_further}

Before turning to frequency-domain analyticity, we collect the two
standard-textbook formulations of $\AX{1}~\LC{}$ that the constitution
inherits and elevates. \emph{In relativistic QFT}, locality is
encoded as \emph{microcausality}: for local Heisenberg operators
$\mathcal{O}(x)$ and $\mathcal{O}'(y)$,
\begin{equation}
[\mathcal{O}(x),\mathcal{O}'(y)]_{\mp}=0,
\qquad (x-y)^2<0,
\label{eq:microcausality_partII}
\end{equation}
where $[\cdot,\cdot]_{\mp}$ is a commutator/anticommutator for
bosons/fermions and $(x-y)^2<0$ denotes spacelike separation
\citep{StreaterWightman1964,Haag1992}. A complementary expression
uses retarded Green functions: for a relativistic field $\phi$ with
source $J$ obeying $(\Box+m^2)\phi=J$, the response
$\phi(x) = \int d^4 y\, G_{\rm ret}(x-y)\, J(y)$ respects causality via
$G_{\rm ret}(x-y) = 0$ for $x^0 < y^0$. In interacting QFT the
two formulations agree: commutators vanish outside the light cone.

\emph{On the operational boundary}, $\AX{1}~\LC{}$ takes the
form of \emph{no-signaling}. For a bipartite system $AB$ with state
$\rho_{AB}$, a CPTP map $\Phi_A$ acting locally on $A$ cannot
change the reduced state on $B$:
\begin{equation}
\Tr_A\!\bigl[(\Phi_A\otimes\mathrm{id}_B)(\rho_{AB})\bigr]
=\Tr_A(\rho_{AB}).
\label{eq:no_signaling_partII}
\end{equation}
Equation~\eqref{eq:no_signaling_partII} follows directly from
trace preservation and is the operational, quantum-information-
language expression of locality
\citep{Kraus1971,Stinespring1955,NielsenChuang2010}. Indeed,
$\AX{1}~\LC{}$ is the constitutional commitment that
Eqs.~\eqref{eq:microcausality_partII} and~\eqref{eq:no_signaling_partII}
are not theorem-level results to be checked case by case but
admissibility filters that every candidate operational kernel must
respect from the outset.

\paragraph{Terminology note.} Some authors use ``local causality''
in the Bell sense --- the demand that the joint statistics of
spacelike-separated experiments factorize over local hidden
variables --- a demand which Bell-test experiments
\citep{Aspect1982,Hensen2015,Giustina2015} have empirically
refuted. The constitutional $\AX{1}~\LC{}$ used in this paper is
\emph{not} the Bell-style local-causality demand; it is the weaker
and operationally adequate combination of microcausality
(Eq.~\eqref{eq:microcausality_partII}) and no-signaling
(Eq.~\eqref{eq:no_signaling_partII}), both of which are compatible
with --- and indeed verified by --- the loophole-free Bell-test
data.

Beyond literal light-cone tomography, $\AX{1}$ is tested
experimentally by frequency-domain constraints.
If a linear response function $\chi(t)$ is causal (retarded), $\chi(t)=0$ for $t<0$, then its Fourier transform
\begin{equation}
\chi(\omega)=\int_{-\infty}^{+\infty}\dd t\,\e^{+\ii\omega t}\chi(t)
=\int_{0}^{+\infty}\dd t\,\e^{+\ii\omega t}\chi(t)
\end{equation}
extends analytically to the upper-half complex $\omega$ plane (a Paley--Wiener type consequence of support on $t\ge 0$).
Analyticity implies \emph{dispersion relations} (Kramers--Kronig):
\begin{align}
\Re \chi(\omega) &= \frac{1}{\pi}\,\mathrm{P}\!\!\int_{-\infty}^{+\infty}\dd\omega'\,
\frac{\Im \chi(\omega')}{\omega'-\omega},\\
\Im \chi(\omega) &= -\frac{1}{\pi}\,\mathrm{P}\!\!\int_{-\infty}^{+\infty}\dd\omega'\,
\frac{\Re \chi(\omega')}{\omega'-\omega},
\end{align}
where $\mathrm{P}$ denotes the principal value.
Thus, any proposed effective kernel that violates these relations (after accounting for instrument response and coarse-graining)
signals an \emph{acausal} or \emph{non-admissible} operational model.

In open systems the same logic applies to transfer functions of estimators.
If $x(t)$ is a registered record produced by a linear filter $h(t)$ acting on an input $u(t)$,
\begin{equation}
x(t)=\int\dd t'\,h(t-t')\,u(t')+\eta(t),
\end{equation}
then LC demands $h(t<0)=0$, and the same analyticity constraints follow for $H(\omega)$.
This provides an experimentally accessible ``LC audit'' that is independent of microscopic interpretation:
causality is enforced at the level of \emph{registered records} and their admissible kernels.

\subsection{Worked example: the Aspect 1982 time-varying-analyzer
 Bell test}
\label{sec:AX1_worked_aspect}

Let us picture this concretely. The cleanest laboratory
falsification target for \AX{1}~\LC{} is the Bell-test family of
experiments, beginning with the time-varying-analyzer version of
\citet{Aspect1982} and continuing through the loophole-free
realizations of \citet{Hensen2015,Giustina2015}.
The experiment runs as follows. A pair of polarization-entangled
photons in the singlet state
\begin{equation}
|\Psi^-\rangle = \tfrac{1}{\sqrt{2}}\bigl(|H\rangle_A|V\rangle_B - |V\rangle_A|H\rangle_B\bigr)
\label{eq:bell_singlet}
\end{equation}
is sent to two spatially separated stations A and B, each carrying
a polarizer whose orientation $\alpha_A$ or $\alpha_B$ is chosen by
a fast random number generator while the photons are already in
flight. Each station registers a binary outcome
$a, b \in \{+1, -1\}$. Quantum mechanics predicts the correlation
$\langle a b\rangle = -\cos\bigl(2(\alpha_A - \alpha_B)\bigr)$,
which violates the CHSH bound $|S| \le 2$ \citep{CHSH1969} on
the $\alpha$-averaged
combination $S = E(\alpha,\beta) - E(\alpha,\beta')
+ E(\alpha',\beta) + E(\alpha',\beta')$ by reaching $|S| = 2\sqrt{2}$.
The Aspect experiment registered $|S| \approx 2.70$, and the
loophole-free experiments of 2015 verified this violation while
closing the locality and detection loopholes simultaneously.

What does \AX{1}~\LC{} require of the registered records, and how
do these experiments confirm it? Two operational predictions are
forced. \emph{First}, the \emph{marginal} statistics at station A
must be independent of the polarizer setting at station B
(\emph{no-signaling}): for any choice of $\alpha_B$,
$\sum_b P(a, b \mid \alpha_A, \alpha_B) = P(a \mid \alpha_A)$.
\emph{Second}, the joint correlation can be non-classical (i.e.\
violate the CHSH bound) while still respecting the causal-arrow
constraint that interventions in the future light cone of one
station do not influence the marginal record at the other station.
Both predictions were confirmed to high statistical significance
in the Aspect experiment and re-confirmed under loophole-free
conditions in 2015.

The axiomatic reading is exactly this: \AX{1}~\LC{} is not
the metaphysical claim that the universe is locally causal in the
sense of Bell's hidden-variable theorem. It is the operational
claim that registered records respect the light-cone structure of
the bulk-to-boundary projection. The Aspect experiment
violates Bell-style local-causality but satisfies operational
locality and causality in exactly the sense \AX{1} requires. This
is a feature of the constitution, not a bug: the \GGTheory{}
program accommodates Bell-violating correlations while still
forbidding superluminal signaling, because $\AX{1}$ is about
record statistics rather than hidden variables.

\subsection{Physics intuition}

\begin{euclidremark}[Physics intuition for \AX{1}]
Roughly speaking, \LC{} says that physics is not allowed to
compute today's record using tomorrow's knob setting. In
practice, this forbids three families of admissible-looking but
actually inadmissible models: advanced propagators presented as
physical response; acausal kernels hidden inside effective
open-system master equations; and the ``interpretational
loophole'' in which measurement is declared external to the
theory and therefore exempt from causal auditing.
Once we have agreed that registered records are the operational
content of physics, we have already committed to a constitutional
ordering of cause and effect.
\end{euclidremark}

\subsection{Relation to \GG\ and \GDOS}

\begin{itemize}[leftmargin=2.2em]
\item \textbf{Relation to \GG.} \GG\ will be formulated as a bulk geometric engine (AdS-like causal structure).
LC restricts admissible bulk geometries and boundary couplings so that the induced boundary response is retarded.
\item \textbf{Relation to \GDOS.} \GDOS\ is the runtime theory of open systems.
LC is the reason \GDOS\ is naturally written in forward-time form (semigroups, kernels with retarded support),
and it sets the admissibility condition for ``ports'' (environment couplings) used in system-instance packets.
\end{itemize}

\section{AX-2: \GGEP\ (Geometry--Gauge Equivalence Principle)}
\label{sec:AX2}

\subsection{Constitutional statement (rigorous)}

\begin{euclidaxiom}[AX-2 (\GGEP): Geometry--Gauge Equivalence Principle]
\label{ax:GGEP}
In a world with gravity and gauge interactions, there exists a \emph{single primitive geometric object}---a
composite connection $\mathcal{A}$ on a principal bundle over the bulk---such that:

\begin{enumerate}[leftmargin=2.2em]
\item \textbf{Locked equivalence / local flattening.}
For any timelike curve $\gamma$ and any event $p\in\gamma$, there exists a combined choice of (i) local coordinates,
(ii) local Lorentz frame, and (iii) internal gauge trivialization, in which the pullback of the composite connection
along $\gamma$ vanishes at $p$ (and can be flattened to first order along $\gamma$ in the equivalence-principle sense).
\item \textbf{Invariant physics.}
All physical content is carried by curvature and holonomy invariants of $\mathcal{A}$ (and by the induced topological
data); ``forces'' appear when working in an unlocked description.
\end{enumerate}

This upgrades Einstein's equivalence principle from ``gravity only'' to ``geometry+gauge as one locked structure''.
\end{euclidaxiom}

\paragraph{The arena these two axioms force.} Axioms \AX{1} (\LC) and \AX{2} (\GGEP), taken together, do more than constrain a pre-given manifold --- they \emph{force the manifold itself}. Locality and causality oblige any interacting, charged quantum theory to carry two structures it cannot be written without (a renormalization scale and a local gauge phase), and geometry--gauge equivalence obliges each to be realized as geometry. The result is that the bulk must be four-dimensional Lorentzian spacetime supplemented by exactly two further coordinates --- a non-spatial \emph{scale} dimension and a non-spatial \emph{phase} dimension. This is established rigorously as Theorem~0 (T-GG6-0) of \citep{ArisakaA2GG6}; we record it here because it is the joint geometric content of \AX{1} and \AX{2}, and the reason the extra dimensions of \GG{} are derived rather than postulated.

\subsection{Mathematical kernel}

\paragraph{(i) Composite connection and curvature.}
The composite connection $\mathcal{A}$ and curvature $\mathcal{F}$ were defined in
Eqs.~\eqref{eq:compositeA}--\eqref{eq:compositeF}.
Local gauge transformations act as
\begin{equation}
\mathcal{A}\mapsto \mathcal{A}' = g^{-1}\mathcal{A}g + g^{-1}dg,
\qquad
\mathcal{F}\mapsto \mathcal{F}' = g^{-1}\mathcal{F}g,
\label{eq:gauge_transform}
\end{equation}
and diffeomorphisms act by pullback on forms.
The locked structure is not a new mathematical object; it is the \emph{central claim} that this single object is
the correct primitive for unification.

\paragraph{(ii) Holonomy/Wilson loops as invariants.}
For a curve $C$ in $\mathcal{M}_6$, define the holonomy (Wilson line)
\begin{equation}
U(C) = \mathcal{P}\exp\!\left(\int_C \mathcal{A}\right),
\label{eq:holonomy}
\end{equation}
where $\mathcal{P}$ denotes path ordering.
For a small loop $\partial\Sigma$ bounding an oriented surface $\Sigma$,
\begin{equation}
U(\partial\Sigma) \;=\; \exp\!\left(\int_\Sigma \mathcal{F}\right) + \mathcal{O}(\Sigma^{3/2}),
\label{eq:smallloop}
\end{equation}
so curvature is the infinitesimal holonomy.

\paragraph{(iii) Locked chart (first-order flattening).}
A compact local rendering of the locked condition is:
for a timelike curve $\gamma(\tau)$ with tangent $u^A=\dot{x}^A$,
there exists a local trivialization such that at a chosen parameter value $\tau_0$,
\begin{equation}
\mathcal{A}_A(\gamma(\tau_0)) = 0,
\qquad
u^B\partial_B\big(\mathcal{A}_A u^A\big)\Big|_{\tau_0}=0,
\label{eq:locked_chart}
\end{equation}
while $\mathcal{F}(\gamma(\tau_0))$ may be nonzero.
Equation~\eqref{eq:locked_chart} is the composite analog of normal coordinates in GR and of local gauge trivialization
in Yang--Mills theory.

\subsection{Forced consequences / derivations}

\paragraph{Derivation 1 (local construction of $\mathcal{A}(p)=0$).}
At any point $p\in\mathcal{M}_6$, choose local coordinates centered at $p$ and a group element $g(x)$ with $g(p)=\mathbb{I}$.
Because $\mathcal{A}$ is a smooth one-form, one can choose $g(x)$ such that
$\partial_A g(p) = -\mathcal{A}_A(p)$.
Then the transformed connection \eqref{eq:gauge_transform} satisfies $\mathcal{A}'_A(p)=0$.
This is the standard local trivialization property of connections \citep{KobayashiNomizu1963,Nakahara2003}.
\GGEP{}'s additional claim is \emph{conceptual uniqueness}: this is the \emph{definition of inertial/force-free structure}
in the unified theory.

\paragraph{Derivation 2 (unified geodesic/Lorentz-force form from a worldline action).}
Consider a point-like probe carrying generalized charges $Q$ in a representation of $G_{\mathrm{tot}}$.
A minimal worldline action capturing both geometry and gauge coupling is
\begin{equation}
S[\gamma,Q] =
\int d\tau\left[
\frac{m}{2}\, g_{AB}(x)\,\dot{x}^A\dot{x}^B
+\langle Q(\tau), \mathcal{A}_A(x)\rangle \dot{x}^A
\right],
\label{eq:worldlineAction}
\end{equation}
where:
$m$ is an inertial mass parameter,
$\tau$ is an affine parameter,
$\langle\cdot,\cdot\rangle$ is the natural pairing between charges and connection components,
and $\dot{x}^A\equiv dx^A/d\tau$.
Varying with respect to $x^A$ yields a generalized geodesic equation,
schematically
\begin{equation}
m\,\frac{D\dot{x}^A}{d\tau}
= \big\langle Q,\mathcal{F}^{A}{}_{B}\big\rangle \dot{x}^B,
\label{eq:compositeLorentzForce}
\end{equation}
where $D/d\tau$ is the covariant derivative induced by the metric/spin part of the composite connection.
In a locked chart at $p$, $\mathcal{A}(p)=0$ and inertial motion is locally straight, while curvature provides the
first nontrivial invariant content---exactly the equivalence-principle logic.

\paragraph{Derivation 3 (geometrization of discreteness from compact holonomy).}
If the bulk contains a compact $U(1)$ fiber coordinate $\theta\sim\theta+2\pi$, then single-valuedness of a wavefunction
$\Psi(\theta)$ implies a Fourier expansion
\begin{equation}
\Psi(\theta)=\sum_{n\in\mathbb{Z}} c_n e^{in\theta},
\end{equation}
and the generator of translations $-i\partial_\theta$ has integer eigenvalues.
Restoring $\hbar$, the momentum spectrum is quantized:
\begin{equation}
p_\theta = n\hbar,\qquad n\in\mathbb{Z}.
\label{eq:thetaQuant}
\end{equation}
In Kaluza--Klein language, $p_\theta$ appears as a conserved charge in the boundary effective theory.
This provides an explicit mechanism for the slogan ``geometrize quantum physics'':
discreteness is a topological/geometric consequence rather than an externally imposed quantization rule
\citep{Dirac1931,KobayashiNomizu1963,Nakahara2003}.
One register sentence harmonizes this display with the arena: the
circle itself is the constitutional object --- the arena says
$\theta\sim\theta+2\pi$ and nothing more --- while $e^{in\theta}$ is
its boundary representation, the quarter-turn of the circle being the
forced complex structure of T-P4-1 \citep{ArisakaP4Real}.

\subsection{Further mathematical elaboration: composite connection, covariant derivative, and unified force law}
\label{sec:AX2_further}

A compact way to express \GGEP\ is that the \emph{covariant derivative} is built from a single composite connection.
For a matter field $\psi$ carrying both Lorentz/spin and internal gauge indices, one writes
\begin{equation}
D_\mu \psi
=\left(\partial_\mu + \frac{1}{4}\,\omega_{\mu}^{ab}\gamma_{ab} + \ii g\,A_\mu^a T^a\right)\psi
\equiv \left(\partial_\mu + \mathcal{A}_\mu\right)\psi,
\end{equation}
where $\omega_\mu^{ab}$ is the spin connection, $A_\mu^a$ is a Yang--Mills connection, and $\mathcal{A}_\mu$ denotes the \emph{locked}
(composite) connection acting on $\psi$.
Its curvature is likewise composite,
\begin{equation}
\mathcal{F}_{\mu\nu}
:= [D_\mu,D_\nu]
= \frac{1}{4}\,R_{\mu\nu}^{ab}\gamma_{ab} + \ii g\,F_{\mu\nu}^a T^a,
\end{equation}
so that ``force'' and ``curvature'' are the same object in representation-dependent dress.

In a point-particle limit, one can view motion as parallel transport in the composite connection.
A standard calculation shows that the usual geodesic term and Lorentz-force term arise from extremizing a single worldline functional
with composite coupling:
\begin{equation}
S[x] = -m\int \dd s \;+\; q\int A_\mu(x)\,\dd x^\mu,
\end{equation}
whose Euler--Lagrange equations yield
\begin{equation}
m\,\frac{D u^\mu}{\dd\tau}= q\,F^{\mu}{}_{\nu}\,u^\nu,
\qquad u^\mu=\frac{\dd x^\mu}{\dd\tau}.
\end{equation}
\GGEP\ interprets this familiar ``sum'' as a \emph{single} statement about parallel transport in $\mathcal{A}$: the separation into ``gravity'' and
``gauge force'' is representation bookkeeping rather than a separation of primitives.

Finally, the holonomy viewpoint makes quantization conditions geometric.
For a compact $U(1)$ factor, the Wilson loop
\begin{equation}
W(\gamma)=\exp\!\left(\ii q\oint_\gamma A\right)
\end{equation}
is invariant under small deformations and, in topologically nontrivial sectors, forces charge/flux quantization.
This is the same mathematical discipline behind Dirac quantization \citep{Dirac1931} and motivates why \OGN\ (POS-6) will later insist on a
canonical integer charge lattice: if holonomy is the invariant content, then basis drift in Abelian normalization is not physics but bookkeeping.

\subsection{Worked example: Aharonov--Bohm effect plus Berry
 phase}
\label{sec:AX2_worked_AB}

The textbook
demonstrations that gauge-and-geometry are best treated as
\emph{one} object are the Aharonov-Bohm effect
\citep{AharonovBohm1959} and Berry's geometric phase
\citep{Berry1984}. Both reveal that what we call ``physical
content'' lives in the holonomy of the composite connection, not
in the local field strength.

The Aharonov-Bohm setup is as follows. An
electron beam is split into two paths that pass on either side of
a long solenoid carrying magnetic flux $\Phi$ but no field outside
the solenoid (the field $\mathbf{B}$ vanishes everywhere the
electrons can go). The two paths recombine and produce an
interference pattern. Classical electromagnetism predicts no
shift, since $\mathbf{B} = 0$ along the electron paths. Yet the
fringe pattern \emph{does} shift, by a phase
\begin{equation}
\Delta\varphi \;=\; \frac{q}{\hbar}\oint_C A\,d\ell \;=\; \frac{q\Phi}{\hbar},
\label{eq:AB_phase}
\end{equation}
where $C$ is the closed loop encircling the solenoid and $q$ is
the electron charge. The phase is the Wilson loop of the
$\mathrm{U}(1)$ gauge connection $A$ around $C$ --- the holonomy
of the composite connection in our language --- and it is
physically measurable even though the local field strength
$\mathbf{B} = dA$ vanishes everywhere the electron sees. Yet
the experimental observation of Eq.~\eqref{eq:AB_phase} in the
Tonomura electron-microscope experiments of the mid-1980s is one
of the cleanest demonstrations in physics that gauge potentials
carry physical content beyond their field strengths.

The Berry-phase example sharpens the geometric reading. When a
quantum system $|\psi(\mathbf{R})\rangle$ is adiabatically
transported around a closed loop in some external parameter space
$\mathbf{R}$, it acquires --- in addition to the dynamical phase
$-\int E\,dt/\hbar$ --- a geometric phase
\begin{equation}
\gamma_C \;=\; \oint_C \langle\psi(\mathbf{R})|\,i\,\nabla_\mathbf{R}\,|\psi(\mathbf{R})\rangle\cdot d\mathbf{R}.
\label{eq:berry_phase}
\end{equation}
The integrand is the Berry connection on the parameter manifold;
$\gamma_C$ is its holonomy. As Berry showed in 1984, $\gamma_C$
depends only on the geometry of the loop $C$ in parameter space,
not on the dynamical history --- a fact that is invisible in any
treatment that does not promote the connection to a primitive
geometric object.

The axiomatic reading is exactly this: \AX{2}~\GGEP{} declares
that physical content lives in the curvature and holonomy
invariants of the composite connection $\cA$ on the principal
$G_{\mathrm{tot}}$-bundle, not in the local components of $\cA$
themselves. The Aharonov-Bohm phase and the Berry
phase are not exotic curiosities --- they are the empirical proof
that gauge potentials and geometric phases must be treated as
parts of one geometric primitive. In the \GGTheory{} program,
the same logic forces the four-factor composite group
$G_{\mathrm{tot}} = \Spinone\times\SU(5)\times\U(1)_B\times\U(1)_{Qi}$:
each factor is the holonomy carrier of one physical sector
\citep{ArisakaA2GG6}.

\subsection{Physics intuition}

\begin{euclidremark}[Physics intuition for \AX{2}]
Roughly speaking, \GGEP{} says that what we usually call
``gauge force'' is not a separate ingredient that must be
quantized on top of geometry --- it is part of geometry from the
beginning, very much like the way the electric and magnetic fields
became, after Maxwell, two faces of one underlying object. In
this view, ``free'' motion means parallel transport (horizontal
lift) with respect to the composite connection, and ``force'' is
the bookkeeping artifact of describing that transport in a
non-inertial (unlocked) chart. The quantum signature ---
the phase that the wave function picks up around a flux line, the
discrete charge that survives every gauge change, the Berry phase
that any adiabatic loop in parameter space accumulates --- is
then traced to holonomy and topology, not to a separate
quantization rule pasted onto the side of geometry.
The unification of gravity and gauge into a single composite
connection is the axiomatic move that lets the 6D bulk of \GG{} stand in as the realization of \GGEP{}.
\end{euclidremark}

\subsection{Relation to \GG\ and \GDOS}

\begin{itemize}[leftmargin=2.2em]
\item \textbf{Relation to \GG.} \GG\ is the concrete realization of \GGEP{}: a six-dimensional bulk in which the
composite connection locks gravity with a specific gauge content (minimally $SU(5)\times U(1)_B$) and in which anomaly
cancellation fixes otherwise free coefficients \citep{ArisakaA2GG6}.
\item \textbf{Relation to \GDOS.} \GGEP{} determines \emph{what} is projected by \OGGEP: curvature/holonomy data of the composite
connection.
\GDOS\ then describes \emph{how} that projected data evolves as an open system on the boundary \citep{ArisakaA5GDOS}.
\end{itemize}

\section{AX-3: \OGGEP\ (Operational/Open \GGEP{})}
\label{sec:AX3}

\subsection{Constitutional statement (rigorous)}

\begin{euclidaxiom}[AX-3 (\OGGEP): Operational/Open \GGEP{}]
\label{ax:OGGEP}
A bulk theory is not physically defined unless it specifies a \emph{fixed operational dictionary} that maps bulk structure
to registered boundary observables and admissible open-system evolution.
Concretely, there exists a causal, gauge-covariant bulk$\leftrightarrow$boundary map $\Dmap$ such that for every
admissible bulk configuration $B$,
\begin{equation}
\Dmap(B) \;=\; \Big(\mathcal{H},\rho,\{\mathcal{I}_\alpha\},\{\Phi_{t,s}\},\ldots\Big),
\end{equation}
where:
$\rho$ is a boundary state,
$\{\mathcal{I}_\alpha\}$ is a set of measurement \emph{instruments} (with POVM semantics),
and $\Phi_{t,s}$ are admissible boundary evolution maps between interventions.
The map must satisfy:
(i) LC/retardedness (AX-1),
(ii) compatibility with the gauge/geometry equivalences of \GGEP{} (AX-2),
and (iii) operational positivity (channels/instruments map physical states to physical states).
\end{euclidaxiom}

\paragraph{A register note on the symbol $\mathcal{H}$.}
The operational dictionary outputs a complex Hilbert space, but the
complex scalars of $\mathcal{H}$ are not a constitutional primitive.
The bulk clauses of this constitution are stated over the reals
throughout; the boundary's imaginary unit is the derived quarter-turn
of two real bulk structures --- the charge circle and the causal
clock --- whose forced agreement is the existence of antiparticles
(T-P4-1/2/3 of \citealp{ArisakaP4Real}; re-proved as T-P5-A/B/C of
\citealp{ArisakaP5CPT}; QM-register development in
\citealp{ArisakaA8QM}; constitutional pricing in L-Constitution-1,
\S\ref{sec:AX4_category_axis}, at candidate grade). The constitution
therefore uses $\mathcal{H}$ as boundary shorthand, not as bulk
input.

This axiom is the axiomatic answer to the century-old ambiguity of ``what is a measurement?''
In \GGTheory{}, measurement is not a metaphysical add-on; it is part of the theory's syntax.

\subsection{Mathematical kernel}

\paragraph{(i) Instruments and POVMs.}
A measurement with outcomes labeled by $\alpha$ is represented by an \emph{instrument}:
a set $\{\mathcal{I}_\alpha\}$ of completely positive maps such that $\sum_\alpha \mathcal{I}_\alpha$ is trace-preserving.
A Kraus representation is
\begin{equation}
\mathcal{I}_\alpha(\rho) = \sum_k K_{\alpha k}\,\rho\,K_{\alpha k}^\dagger,
\qquad
\sum_{\alpha,k} K_{\alpha k}^\dagger K_{\alpha k} = \mathbf{1}.
\label{eq:Kraus}
\end{equation}
The associated POVM elements are
\begin{equation}
E_\alpha = \sum_k K_{\alpha k}^\dagger K_{\alpha k},
\qquad
E_\alpha\succeq 0,
\qquad
\sum_\alpha E_\alpha = \mathbf{1}.
\label{eq:POVM}
\end{equation}
Outcome probabilities and conditional post-measurement states are
\begin{equation}
p_\alpha = \Tr(E_\alpha\rho),
\qquad
\rho_\alpha = \frac{\mathcal{I}_\alpha(\rho)}{p_\alpha}.
\label{eq:BornInstrument}
\end{equation}
This modern operational formulation synthesizes the Born rule and measurement update into a single object
\citep{Born1926,vonNeumann1932,DaviesLewis1970,Kraus1983,NielsenChuang2010}.

\paragraph{(ii) Admissible open evolution as channels.}
Between interventions, the boundary state evolves by a (typically time-ordered) family of quantum channels
$\Phi_{t,s}$, i.e.\ completely positive trace-preserving (CPTP) maps:
\begin{equation}
\rho(t) = \Phi_{t,s}\big(\rho(s)\big),\qquad \Phi_{t,s}\ \text{CPTP}.
\end{equation}
CPTP is the minimal mathematical encoding of ``physicality'' for open evolution (no negative probabilities)
\citep{Choi1975,Kraus1983}.
Time-homogeneous Markovian evolution is described by a semigroup $\Phi_t=e^{t\mathcal{L}}$.

\paragraph{(iii) Bulk$\to$boundary map as structure-preserving functor.}
\par\noindent\emph{\small This paragraph is written for readers with a
background in category theory. Its content is a pointer to the
literature, not a step in the constitutional argument; nothing later in
this paper depends on it.}\par\medskip

A compact way to make \OGGEP{} precise is to view it as a functor between categories:

\begin{eucliddefinition}[Bulk and operational categories]
Let $\mathbf{Bulk}$ be the category whose objects are admissible bulk packages
$B=(\mathcal{M}_6,g,\mathcal{A},\ldots)$ and whose morphisms include diffeomorphisms, gauge transformations, and
coarse-graining/RG maps.
Let $\mathbf{Op}$ be the category whose objects are operational boundary packages
$O=(\mathcal{H},\rho,\{\mathcal{I}_\alpha\},\{\Phi_{t,s}\},\ldots)$ and whose morphisms are operational equivalences
(unitary conjugations, classical relabelings, coarse-grainings).
\end{eucliddefinition}

\begin{eucliddefinition}[\OGGEP\ map]
\OGGEP{} asserts the existence of a functor
\begin{equation}
\Dmap:\mathbf{Bulk}\to\mathbf{Op},
\label{eq:Dfunctor}
\end{equation}
that is compatible with:
\begin{enumerate}[leftmargin=2.2em]
\item gauge covariance (bulk gauge equivalences map to operational equivalences),
\item causality (retardedness, AX-1),
\item positivity/physicality (channels and instruments preserve state positivity),
\item and, with POS-\RG\ (\S\ref{sec:POS5}), commutation with coarse-graining.
\end{enumerate}
\end{eucliddefinition}

\subsection{Forced consequences / derivations}

\paragraph{Derivation 1 (LC + CPTP semigroup $\Rightarrow$ GKLS/Lindblad generator).}
Assume:
(i) forward-time semigroup property \eqref{eq:semigroup} (from LC and Markovity),
(ii) complete positivity and trace preservation for each $\Phi_t$,
and (iii) mild continuity in $t$.
Then the generator $\mathcal{L}$ has GKLS form \citep{Lindblad1976,Gorini1976}:
\begin{equation}
\frac{d\rho}{dt} = \mathcal{L}(\rho)
= -i[H,\rho] + \sum_j\left(
L_j\rho L_j^\dagger - \frac{1}{2}\{L_j^\dagger L_j,\rho\}
\right),
\label{eq:Lindblad}
\end{equation}
with Hermitian $H$ and Lindblad operators $L_j$.

\paragraph{Derivation 2 (measurement as a special CPTP map with classical output).}
Each instrument map $\mathcal{I}_\alpha$ in \eqref{eq:Kraus} is CP; the total map $\sum_\alpha\mathcal{I}_\alpha$ is CPTP.
Thus ``measurement'' is not a new dynamical law beyond open-system evolution; it is a particular CP update with
a registered classical outcome $\alpha$.
This reframes ``collapse'' as the operational bookkeeping of conditioning on an outcome, consistent with the
quantum-instrument formalism \citep{DaviesLewis1970,Kraus1983,Ozawa1984}.
The bookkeeping acquires its dynamical completion downstream: \emph{which} outcome actualizes in a given run, and
\emph{why} the record persists, are answered by the runtime of Derivation~4 --- the \GRIPin{} halt at a Geometric
Attractor --- in the dedicated treatment of \citet{ArisakaA8QM}, while the outcome \emph{measure} stays exactly where
this paragraph puts it, with the instrument formalism.

\paragraph{Derivation 3 (\OGGEP{} as the operational closure of quantum theory).}
Standard axiomatizations of QM typically treat unitary evolution as primary and measurement as additional.
\OGGEP{} inverts the emphasis: the theory is only complete when the operational map is specified.
This is not a claim to new measurement mathematics; it is a claim that measurement should be \emph{constitutional}
in any theory that aims to apply across open systems (chemistry, life, mind) \citep{BreuerPetruccione2002,NielsenChuang2010}.

\paragraph{Derivation 4 (\OGGEP{} $\Rightarrow$ \GDOS{} $\Rightarrow$ \GRIP{}: the boundary descent chain).}
The three derivations above are local consequences of \AX{3} --- a
generator form, a measurement semantics, a closure principle. The
global consequence deserves to be stated in the constitution itself,
because the entire boundary side of \GGTheory{} descends from this one
axiom in three forced steps, each proved in its own companion paper
and none requiring any assumption beyond \AX{1}--\AX{4} and
\POS{1}--\POS{7}. \emph{Step 1 (the law).} \AX{3} demands the
operational dictionary $\Dmap$ for \emph{every} admissible open
system. Demanding it uniformly --- one dictionary, all systems, all
scales, commuting with coarse-graining (POS-\RG) --- forces a single
universal boundary law: the reduced open-system dynamics \GDOS{},
with its scalar ledger and four-current Ward package
\citep{ArisakaA5GDOS}. \GDOS{} is not an extra theory; it is what
$\Dmap$ looks like when written out in full. \emph{Step 2 (the
runtime).} The long-time behavior of the \GDOS{} flow factorizes ---
provably, not by construction --- as the Gauge-Runtime Invariance
Pipeline,
\begin{equation}
\AX{3}\ (\OGGEP{})
\;\longrightarrow\;
\Dmap
\;\longrightarrow\;
\underbrace{\GDOS{}}_{\text{the law}}
\;\longrightarrow\;
\underbrace{\GRIP{} := \DGI \to \GSSB \to \GA/\GGA{}}_{\text{the runtime}}
\;\longrightarrow\;
\underbrace{\GGA{}}_{\text{the halt}},
\label{eq:boundary-descent-chain}
\end{equation}
with the terminus refined to the Ground Geometric Attractor and the
dynamics operating in exactly two regimes --- \GRIPin{}, descent on a
fixed landscape, and \GRIPout{}, the transient event that reorganizes
the landscape itself \citep{ArisakaA6Gi4}. \emph{Step 3 (the
selector and the readings).} Which attractor the runtime halts on is
fixed by the four admissibility pillars of \citet{ArisakaA7Qi4} (the
\GGA{} selector); the quantum-mechanical face of the chain ---
quantization, decoherence, and measurement as \GRIP{} phenomena ---
is developed in \citet{ArisakaA8QM}; and the regime theorems and the
cascade-wide ground-state statement are proved in
\citet{ArisakaA11Origin}. In one constitutional sentence:
\emph{\OGGEP{} demands the dictionary; \GDOS{} writes it; \GRIP{}
runs it; and the \GGA{} is where it halts.} Nothing in the chain is
postulated beyond \AX{3} itself --- which is precisely the point: the
constitution does not contain a runtime axiom, and Eq.~\eqref{eq:boundary-descent-chain}
is the proof that it does not need one.

\paragraph{Synthesis (what time is, and who owns the arrow).}
Read jointly, the first three axioms are the grantors of time's four
endowments. \AX{1} gives time its \emph{order}: the cone sorts the
world. \AX{2}, through Theorem~0 of \citep{ArisakaA2GG6}, gives time
its \emph{repetition}: the constitutional clock is a Killing flow,
and a time that did not repeat could not tick. \AX{3} gives time its
\emph{openness} twice --- once irreversibly, since the \OGGEP{}
projection is one-way and records accumulate (the arrow enters this
physics at the level of law, in the same axiom that defines what
observation is), and once analytically, since the positivity clause
makes the clock's spectrum one-sided and opens the half-plane of
lawful continuation (the analytic shore). Time is the unique
direction of the arena carrying all four endowments at once: it
orders, it repeats, it opens one way, and it opens the shore. The
consolidated development is Part~IX of the CPT member paper
\citep{ArisakaP5CPT}, at the grade stated there. And the custody is
constitutional: \emph{\AX{3} owns the arrow --- the one-way reading
of the dictionary is a clause of this constitution, not an emergent
property; Part~IX of the CPT member paper \citep{ArisakaP5CPT} is
the canonical law-level derivation of the arrow from this axiom,
over a bulk that is itself exactly T-symmetric; the \GDOS{}
companion \citep{ArisakaA5GDOS} is its dynamical implementation; and
every downstream consumer of an arrow --- thermodynamic,
cosmological, or biological --- points here rather than deriving a
private one.}

\subsection{Further mathematical elaboration: instruments, Stinespring dilation, and the bulk$\to$boundary dictionary}
\label{sec:AX3_further}

AX-3 is easiest to audit when written in dilation form. Any instrument $\{\mathcal{I}_m\}$ admits a Stinespring/Naimark representation:
there exists an environment Hilbert space $\cH_E$, an initial environment state $\sigma_E$, a unitary $U$ on $\cH\otimes \cH_E$,
and a pointer POVM $\{\Pi_m\}$ on $\cH_E$ such that
\begin{equation}
\mathcal{I}_m(\rho)=\Tr_E\!\left[(\mathbf{1}\otimes \Pi_m)\,U\,(\rho\otimes \sigma_E)\,U^\dagger\right].
\label{eq:stinespring_instrument}
\end{equation}
Equation \eqref{eq:stinespring_instrument} makes ``measurement'' concrete: a measurement is a lawful open interaction plus a readout.
Different experimental claims correspond to different $(U,\sigma_E,\Pi_m)$, and \OGGEP\ requires that the intended protocol be specified.

In the \GGTheory{} language, the operational dictionary $\Dmap$ is a compiler from \emph{bulk structure} to such boundary instruments.
Schematically,
\begin{equation}
\Dmap:\ (\text{bulk fields},\text{ports},\text{geometry}) \longmapsto (\{\mathcal{I}_m\},\ \text{estimator},\ \text{error model}),
\end{equation}
with LC enforcing causality (no advanced response) and CP enforcing physicality.
This rephrases the ``measurement problem'' as a missing-map problem: not ``how does the wavefunction collapse?''
but ``which CPTP instrument does the theory assign to the laboratory protocol, and what falsifier does it imply?''

A practical consequence is that ambiguity about the ``Heisenberg cut'' becomes a \emph{controlled approximation}:
different coarse-grainings correspond to different effective instruments, and POS-\RG\ will demand that the dictionary commute with the
coarse-graining map (so that the claim is not resolution-dependent).

\subsection{Worked example: Stern--Gerlach measurement via
 Stinespring dilation}
\label{sec:AX3_worked_SG}

The Stern--Gerlach
experiment is the worked example par excellence for $\AX{3}~\OGGEP$:
a spin-$1/2$ silver atom enters a magnetic gradient and emerges
along one of two trajectories that are then registered by a
photographic plate. The experiment is a
laboratory realization of an instrument
$\{\mathcal{I}_+, \mathcal{I}_-\}$ on the qubit Hilbert space
$\mathcal{H}_S = \mathbb{C}^2$ of the silver atom's spin. We
shall now show how \AX{3}~\OGGEP{} forces this instrument to be a
CPTP channel plus a classical readout, with the bulk-to-boundary
projection of the $\OGGEP$ functor realized exactly by the
Stinespring dilation theorem of \citet{Stinespring1955}.

Consider the joint system: the qubit $S$ together with a pointer
register $\mathcal{H}_E = \mathbb{C}^2$ whose two basis states
$|0\rangle_E, |1\rangle_E$ correspond to the two possible
photographic-plate spots. The atom-pointer interaction is the
controlled-not unitary
\begin{equation}
U \;=\; |+\rangle\langle +|_S \otimes \mathbf{1}_E \;+\;
 |-\rangle\langle -|_S \otimes X_E,
\label{eq:CNOT_dilation}
\end{equation}
where $X_E$ flips the pointer. The pointer is initialized to
$|0\rangle_E$, the qubit is in some state $\rho$, and the
combined state $\rho \otimes |0\rangle\langle 0|_E$ evolves
unitarily under $U$. We then read out the pointer in the
$\{|0\rangle_E, |1\rangle_E\}$ basis. The resulting two
instruments are
\begin{equation}
\mathcal{I}_\pm(\rho) \;=\;
\Tr_E\!\Bigl[\bigl(\mathbf{1}_S \otimes |\pm\rangle\langle\pm|_E\bigr)\,
 U\,(\rho \otimes |0\rangle\langle 0|_E)\,U^\dagger\Bigr],
\label{eq:SG_instruments}
\end{equation}
each $\mathcal{I}_\pm$ a completely positive map on $\rho$, and
$\mathcal{I}_+ + \mathcal{I}_-$ trace-preserving. The associated
POVM elements are
$E_+ = |+\rangle\langle +|_S$ and $E_- = |-\rangle\langle -|_S$,
and the Born rule
$P(\pm \mid \rho) = \Tr(E_\pm \rho)$ follows directly from the
trace.

Here is the axiomatic reading. $\AX{3}~\OGGEP{}$ does not
\emph{add} a measurement postulate on top of unitary evolution; it
\emph{forces} the operational dictionary $\Dmap$ to produce
exactly the instrument structure of
Eq.~\eqref{eq:SG_instruments} as a theorem (specifically, $\T{2}$
Operational Closure, proved in \AFive{} \citep{ArisakaA5GDOS},
Theorem~$\T{2}$; the CPTP instrument form is that paper's
Theorem~$\T{3}$). The Stinespring dilation theorem
\citep{Stinespring1955,Choi1975,Kraus1983} guarantees that every
CPTP channel admits a dilation of this kind --- a unitary on a
larger Hilbert space followed by a partial trace. The
\GGTheory{} program inverts the textbook order: instead of
postulating the Born rule and the projection postulate as
independent axioms, it postulates $\OGGEP$ at the constitutional
level and derives the instrument structure as a consequence
\citep{ArisakaA5GDOS}. The Stern--Gerlach result is
not a separate phenomenon to be explained by a separate measurement
postulate --- it is what happens whenever the bulk-to-boundary
functor is asked to register an outcome at the boundary.
Figure~\ref{fig:OGGEP_holographic} shows the bulk-to-boundary
projection schematically.

\begin{figure}[!htbp]
\centering
\resizebox{0.701\textwidth}{!}{%
\begin{tikzpicture}[
 font=\sffamily\footnotesize,
 every node/.style={align=center, rounded corners=2pt, inner sep=3pt},
 bulkbox/.style={fill=red!10, draw=red!55!black, line width=0.7pt,
 text width=42mm, minimum height=22mm},
 bdybox/.style={fill=blue!10, draw=blue!55!black, line width=0.7pt,
 text width=42mm, minimum height=22mm},
 midbox/.style={fill=yellow!18, draw=olive!55!black, line width=0.7pt,
 text width=40mm, minimum height=11mm},
 eqbox/.style={fill=gray!6, draw=black!30, line width=0.4pt,
 rounded corners=2pt, inner sep=4pt},
 arrow/.style={-{Stealth[length=3mm]}, line width=0.85pt, draw=black!65},
 arrowctr/.style={-{Stealth[length=2.8mm]}, line width=0.7pt, draw=olive!60!black},
]

\node[bulkbox] (bulk) at (-4.2, 0) {%
 \textbf{Bulk package} \\[2pt]
 $\bigl(\cM_6,\,g,\,\cA\bigr)$ \\[1pt]
 closed unitary $U_\Bulk$ \\[1pt]
 $\rho_\Bulk$ on $\cH_\Bulk$};

\node[bdybox] (bdy) at (4.2, 0) {%
 \textbf{Operational package} \\[2pt]
 $\bigl(\cH_\Bdy,\,\rho_\Bdy\bigr)$ \\[1pt]
 CPTP channels $\Phi_{t,s}$ \\[1pt]
 instruments $\{\cI_\alpha\}$};

\node[midbox] (D) at (0, 2.1) {%
 $\Dmap : \mathbf{Bulk} \to \mathbf{Op}$ \\[1pt]
 \footnotesize CPTP, gauge-covariant};

\draw[arrow] (bulk.east) -- node[above, font=\scriptsize\itshape,
 fill=white, inner sep=1pt, yshift=1pt]
 {bulk-to-boundary export}
 (bdy.west);
\draw[arrowctr] (bulk.north east) to[bend left=18] (D.west);
\draw[arrowctr] (D.east) to[bend left=18] (bdy.north west);

\node[eqbox, font=\footnotesize\itshape] at (0, -2.0)
 {Stinespring dilation: $\Phi(\rho_\Bdy) = \Tr_E[U(\rho_\Bdy \otimes \sigma_E)U^\dagger]$};
\node[eqbox, font=\footnotesize\itshape] at (0, -2.85)
 {Born rule: $p_\alpha = \Tr(E_\alpha\,\rho_\Bdy)$};

\end{tikzpicture}}
\caption[Bulk-to-boundary projection of \OGGEP{} onto the 4D boundary.]{The \OGGEP{} bulk-to-boundary projection from the bulk
package $(\cM_6, g, \cA)$ with closed unitary evolution
$U_\Bulk$ (left, red) to the operational package
$(\cH_\Bdy, \rho_\Bdy, \Phi_{t,s}, \{\cI_\alpha\})$ on the 4D
boundary (right, blue), realized by the bulk-to-boundary
functor $\Dmap : \mathbf{Bulk} \to \mathbf{Op}$ (center, orange)
that is CPTP and gauge-covariant. Every laboratory
measurement is a Stinespring dilation of some CPTP channel
followed by a pointer readout, with the Born rule
$p_\alpha = \Tr(E_\alpha \rho_\Bdy)$ supplied by the trace of
the dilation. Openness must be the foundational law
--- and closure becomes its closed-system limit.}
\label{fig:OGGEP_holographic}
\end{figure}

\subsection{Physics intuition}

\begin{euclidremark}[Physics intuition for \AX{3}]
Roughly speaking, \OGGEP{} says that there is no physics without
an observation compiler. A bulk theory that cannot tell you what
an instrument will register is not a physical theory, however
elegant its equations may look. In the \GGConstitution{}, the bulk
is like a program and the operational dictionary $\Dmap$ is the
compiler that turns it into experimental records. This removes
the century-old ambiguity around ``measurement'' by definition ---
measurement \emph{is} the operational export $\Dmap$, no more and
no less. Every instrument that has ever been calibrated
in a physics laboratory is a Stinespring dilation of some CPTP
channel; \OGGEP{} is the axiomatic move that promotes that
empirical fact to a foundational law. After all, openness must
be the foundational law --- and closure becomes its closed-system
limit.
\end{euclidremark}

\subsection{Relation to \GG\ and \GDOS}

\begin{itemize}[leftmargin=2.2em]
\item \textbf{Relation to \GG.} \GG\ supplies the bulk structure (geometry+gauge) that $\Dmap$ acts on.
In particular, holography-inspired bulk geometries provide a natural environment for such a dictionary
\citep{tHooft1993,Susskind1995,Maldacena1998,Witten1998}.
\item \textbf{Relation to \GDOS.} \GDOS\ is the systematic boundary-side expression of $\Dmap$:
it provides the general form of admissible open evolution (quantum GKLS, classical Fokker--Planck/Langevin-like limits),
and it provides the ``ports'' and system-instance packets that define measurement protocols \citep{ArisakaA5GDOS,BreuerPetruccione2002}.
\end{itemize}

\section{AX-4: \MSMF\ (Maximum Symmetry, Minimum Freedom)}
\label{sec:AX4}

\subsection{Constitutional statement (rigorous)}

\begin{euclidaxiom}[AX-4 (\MSMF): Maximum Symmetry, Minimum Freedom]
\label{ax:MSMF}
Among all candidate theories consistent with AX-1--AX-3 (and with the postulates of \S\ref{sec:postulates}),
Nature selects those that (i) maximize admissible symmetry and (ii) minimize non-forced freedom.
Equivalently: no continuous parameters, moduli, or normalization dials are permitted unless they are strictly forced by
symmetry, topology, anomaly cancellation, or operational necessity.
When discrete invariants exist, Nature selects the minimal non-degenerate integer/ratio values compatible with the constitution.
\end{euclidaxiom}

The axiom is intentionally deterministic: it denies that the deep architecture of physics is accidental tuning.

\subsection{Mathematical kernel}

The axiom can be expressed as an ordering principle on the space of admissible theories.

\paragraph{(i) Candidate set.}
Let $\mathfrak{T}$ be the set of all \emph{admissible} bulk--boundary packages
$T=(B,\Dmap)$ satisfying AX-1--AX-3 and the postulates (POS stack).
(Here $B$ encodes bulk data, and $\Dmap$ encodes operational export.)

\paragraph{(ii) Symmetry measure.}
Let $\mathrm{Aut}(T)$ denote the automorphism group of $T$ (bulk isometries and gauge symmetries compatible with the
operational map).
A convenient proxy for ``how symmetric'' is
\begin{equation}
S(T) \;:=\; \dim \mathrm{Aut}(T),
\label{eq:symmetryScore}
\end{equation}
though other monotone measures (e.g.\ symmetry algebra rank) could be used.

\paragraph{(iii) Freedom measure (moduli/knobs).}
Let $\mathcal{M}(T)$ be the moduli space of physically inequivalent continuous deformations of $T$ that preserve the axioms
(after quotienting by gauge/diffeomorphism redundancies).
Define
\begin{equation}
F(T) \;:=\; \dim \mathcal{M}(T),
\label{eq:freedomScore}
\end{equation}
or more generally a weighted count of continuous parameters that change operational predictions.
In practice, $F$ counts ``free parameters'' that would otherwise be fitted to data.

\paragraph{(iv) Lexicographic selection (\MSMF{}).}
\MSMF{} is implemented as the lexicographic optimization
\begin{equation}
T_\star
= \arg\max_{T\in\mathfrak{T}}\Big(S(T),-F(T)\Big),
\label{eq:lexi}
\end{equation}
meaning: maximize symmetry first; among ties minimize freedom.

\subsection{Forced consequences / derivations}

\paragraph{Derivation 1 (\MSMF{} as an anti-landscape rule).}
Suppose $\mathfrak{T}$ contains a continuous family $T(\lambda)$ of physically inequivalent admissible theories, with
$\lambda\in\mathbb{R}$ and operational predictions varying with $\lambda$.
Then $F(T(\lambda))\ge 1$, and \MSMF{} forbids this family unless $\lambda$ is forced by the constitution.
Thus large ``landscapes'' of continuously tunable solutions are axiomatically disfavored.
This responds to the long-standing concern that unification frameworks can become underconstrained by principle alone
\citep{THooftNaturalness1980,Susskind1979,Polchinski2004Landscape}.

\paragraph{Derivation 2 (\MSMF{} pushes conventions into postulates).}
Many continuous ``parameters'' are actually conventions (choices of normalization, scheme, basis).
\MSMF{} demands that such freedoms be explicitly fixed by an axiomatic rule rather than left implicit.
In \GGTheory{}, this is implemented downstream by POS-\OGN\ (Occam gauge normalization) and by a strict separation between
redundancy (gauge) and freedom (physical parameters).

\paragraph{Derivation 3 (\MSMF{} turns symmetry breaking into a selection problem).}
In standard particle physics, spontaneous symmetry breaking is often treated as the selection of a vacuum within a
potential landscape.
\MSMF{} reframes the question constitutionally: symmetry breaking patterns are admissible only if they are forced by the
minimal freedom rule, which becomes especially powerful for open systems (\GDOS{}) where attractors and phase selection
generalize the notion of ``vacuum'' \citep{ArisakaA5GDOS}.
(The detailed open-system symmetry-breaking story is developed in the companion \GDOS{} paper.)

\paragraph{Derivation 4 (Open-system minimal drift template; \GDOS{} preview).}
A recurring export from ``maximum symmetry, minimum freedom'' is that many open-system laws can be written in a
\emph{minimal constitutive form} once positivity, invariance, and a choice of ledger are fixed.
In a classical mesoscopic limit, let $p(x,t)$ be a probability density on a state manifold $\mathcal{X}$.
A generic positivity-preserving Markov generator can be written in Fokker--Planck form \citep{Risken1989,Gardiner2009}
\begin{equation}
\partial_t p(x,t)
=
-\nabla\cdot\!\big(b(x)\,p(x,t)\big)
+\frac{1}{2}\nabla\cdot\!\left(D(x)\,\nabla p(x,t)\right),
\end{equation}
equivalently as an It\^o SDE
\begin{equation}
dX_t=b(X_t)\,dt+\sigma(X_t)\,dW_t,\qquad D=\sigma\sigma^{\top},
\end{equation}
where $b$ is the drift, $D$ the diffusion tensor, $\sigma$ a noise amplitude, and $W_t$ a Wiener process.
Given a chosen ledger functional $F$ (free-energy-like, in the \GFF{} sense) and mobility tensor $M\succeq 0$,
a minimal decomposition consistent with steady-state cycling is
\begin{equation}
b(x) \;=\; -M(x)\,\nabla F(x) \;+\; u(x),
\qquad
\nabla\cdot\!\big(u(x)\,p_\ast(x)\big)=0,
\label{eq:drift_decomposition}
\end{equation}
where $u$ is a solenoidal (non-gradient) component with respect to the stationary measure $p_\ast$.
This is not asserted as a domain-specific law; rather it is a \emph{candidate universal template} that survives the
admissibility filters (GI/CP/RG) while minimizing non-forced structure---precisely the role of \MSMF{} in a cross-domain
open-system setting \citep{Seifert2012}.

\subsection{Further mathematical elaboration: \MSMF{} as an ordering on theory space and a penalty functional}
\label{sec:AX4_further}

AX-4 can be operationalized as an \emph{ordering} on candidate theory classes.
Let $S(T)$ be a symmetry score (e.g.\ $\dim\mathrm{Aut}(T)$ or another monotone measure of invariance) and let $F(T)$ be a freedom score
(e.g.\ the number of independent continuous parameters/moduli remaining after quotienting gauge redundancies).
A minimal ``auditor's'' ordering is lexicographic:
\begin{equation}
T_1 \prec T_2 \quad \Longleftrightarrow \quad
\big(S(T_1)>S(T_2)\big)\ \ \text{or}\ \ \big(S(T_1)=S(T_2)\ \text{and}\ F(T_1)<F(T_2)\big),
\end{equation}
which encodes ``maximize symmetry, then minimize freedom.''

For quantitative model selection and textbook discussion, it is useful to package \MSMF{} into a penalty functional
that resembles minimum-description-length or information-criterion objectives \citep{Akaike1974,Rissanen1978}:
\begin{equation}
\mathcal{J}(T) := -S(T) + \lambda\,F(T) + \mathcal{C}_{\mathrm{protocol}}(T),
\end{equation}
where $\mathcal{C}_{\mathrm{protocol}}$ denotes the ``cost'' of the operational dictionary and falsifiability contract (MF).
The constant $\lambda$ is not a tunable physics parameter; it is a bookkeeping device expressing the structural preference.
The conceptual point is that freedom is penalized \emph{unless} it is forced by symmetry, consistency, or operational meaning.

In practice, \MSMF{} interacts directly with POS-\OGN: even when two descriptions are physically equivalent, a theory can acquire spurious knobs via
continuous basis drift (especially in Abelian sectors). \OGN{} removes that drift by fixing a canonical integer/rational normalization so that
``choice of units/basis'' cannot masquerade as new physics.

\subsection{From the parameter axis to the primitive-category
 axis: the structure cost and the real-category lemma}
\label{sec:AX4_category_axis}

The discipline just stated --- freedom is penalized unless forced ---
has so far been applied to \emph{parameters}: continuous couplings,
moduli, normalization dials. The same discipline applies to a deeper
choice that most frameworks never audit: the \emph{primitive field
category} of the bulk --- whether the fields of the fundamental
description are real, complex, or quaternionic objects. The
Foundations Branch (\citealp{ArisakaP4Real}; \citealp{ArisakaP5CPT})
supplies the theorems that make this axis auditable; the present subsection prices
it with AX-4's own machinery --- the candidate set $\mathfrak{T}$,
the scores $S(T)$ and $F(T)$, and the lexicographic selector of
Eqs.~\eqref{eq:symmetryScore}--\eqref{eq:lexi}. One feature of the
kernel matters especially here: the admissibility predicate ($T$
satisfies AX-1--AX-3 and the POS stack) is stated \emph{before} any
field category is chosen, which is exactly what lets the cost below
be defined without presupposing the answer.

\begin{eucliddefinition}[\textbf{D-Constitution-1}: structure cost of a primitive field category]
\label{def:structure-cost}
For an admissible package $T=(B,\Dmap)\in\mathfrak{T}$, a
\emph{structure datum} is a mathematical object that must be
specified on the bulk side before the dynamics and the dictionary
are defined. A structure datum is \emph{unforced} if at least two
inequivalent specifications satisfy AX-1--AX-3 and the postulate
stack, with the choice between them not fixed by any axiom,
postulate, or already-admitted theorem at the point of
specification. The \emph{structure cost} of $T$'s primitive field
category is the pair
\begin{equation}
C(T) \;:=\; \Big(\dim\{\text{continuous unforced structure data}\},\;
\#\{\text{discrete unforced structure data}\}\Big),
\label{eq:structure-cost}
\end{equation}
ordered lexicographically, and it enters AX-4 as the refinement
$\widehat F(T) := F(T) + C(T)$: the continuous part adds to the
freedom score \eqref{eq:freedomScore}, and the discrete part is
adjudicated by AX-4's existing discrete-invariant clause.
\end{eucliddefinition}

\paragraph{Why the cost is not circular.}
$C$ never mentions $\mathbb{R}$, $\mathbb{C}$, or $\mathbb{H}$, and
it never mentions ``simplicity.'' It counts \emph{choices whose
alternatives the axioms themselves cannot distinguish}, evaluated by
the same procedure on every category --- the procedure AX-4 already
applies to physical parameters (Nature does not hide knobs). A hostile reader who wishes to reject the pricing must argue either
that some counted choice is in fact forced, or that some uncounted
choice exists in the real category; both are mathematical claims,
checkable. The cost is falsifiable arithmetic, not aesthetics.

\paragraph{The pricing of the three categories.}
The candidates are the associative normed division algebras
$\mathbb{R}$, $\mathbb{C}$, $\mathbb{H}$ (octonionic primitives are
excluded upstream of the comparison: associativity of composition of
admissible instruments --- AX-3's channel algebra --- restricts to
the associative case by the classical Frobenius--Hurwitz
classification, an imported external anchor). Write
$\mathfrak{T}_{\mathbb{R}}$, $\mathfrak{T}_{\mathbb{C}}$,
$\mathfrak{T}_{\mathbb{H}}$ for the corresponding admissible
subclasses.

\emph{Real primitives: $C=(0,0)$.} Fields are real sections; the
dynamics is generated by a real operator on a real solution space;
no complex structure is specified as input. Where complex structure
appears, it is a \emph{theorem with a uniqueness clause}: the charge
circle's quarter-turn is unique up to sign --- the commutant of the
irreducible weight-$n$ circle action contains exactly the two
quarter-turns (T-P4-1(iii)); the causal clock's polar phase is
unique among dynamics-commuting positive-energy structures
(T-P4-2); and the residual sign is resolved by the handshake --- the
sign ambiguity of T-P4-1(iii) is resolved by T-P4-2's positivity,
and the resolution is the existence of antiparticles (T-P4-3;
re-proved as T-P5-A/B/C). Zero unforced choices; every quarter-turn
is forced.

\emph{Complex primitives: $C \geq (0,\,n_{\rm sec}+1)$ for
$n_{\rm sec}$ charged sectors.} A complex-primitive package must
specify, as input data with admissible alternatives: (i) the sign of
the complex structure $J$ on each charged sector --- pre-dynamically,
$2^{n_{\rm sec}}$ inequivalent assignments satisfy AX-1--AX-3, and
the Feynman--St\"uckelberg pairing must then be \emph{postulated}
rather than derived (in $\mathfrak{T}_{\mathbb{R}}$ it is T-P4-3's
output; in $\mathfrak{T}_{\mathbb{C}}$ it is an axiom --- the
antiparticle handshake is exactly the content the complex category
buys by fiat); (ii) the identification of the gauge $i$ with the
dynamical $i$ --- ``physics has one $i$'' is a theorem in
$\mathfrak{T}_{\mathbb{R}}$ (the shared global $J$ of T-P4-3/4) and
an unforced global alignment postulate in
$\mathfrak{T}_{\mathbb{C}}$; and (iii) the reality conditions ---
hermiticity of the action, and the selection of the real observable
slice for AX-3's registered records, since the boundary registers
real numbers. Count: at least one discrete choice per charged sector
plus one global alignment choice. The symmetry score does not rescue
the category: $S$ ties on shell, because the real category recovers
the full complex symmetry (T-P4-4); ties in $S$ are adjudicated by
$\widehat F$, and $\widehat F$ decides for $\mathbb{R}$.

\emph{Quaternionic primitives: continuous loss at the first slot,
and a double-count.} An $\mathbb{H}$-module carries a two-sphere of
compatible complex structures (the unit imaginary quaternions
$a\mathbf{i}+b\mathbf{j}+c\mathbf{k}$ with $a^2+b^2+c^2=1$);
selecting the physical $i$ within it is a \emph{continuous}
two-parameter unforced choice per sector --- a modulus in exactly
the sense of \eqref{eq:freedomScore}, so
$C(\mathfrak{T}_{\mathbb{H}})$ opens with a continuous entry of
dimension $2\,n_{\rm sec}$ and the category loses at the first slot
before discrete counting begins. Independently, the mod-8 reality
clock of the real Clifford algebra closes the door a second time:
the $(1,5)$ bulk sits at hour $t-s\equiv 4$ --- a quaternionic hour
--- so the pseudo-real spinor structure physics actually uses is
already delivered by the real Clifford algebra as derived structure,
read off the standard spin-geometry classification
(\citealp{ArisakaP5CPT}, App.~G.4, where the reading is textbook
material). Postulating $\mathbb{H}$-primitives therefore duplicates
structure the real category provides as a theorem --- the precise
spurious-knob pathology that \AX{4}~\MSMF{} and $\POS{6}~\OGN{}$
exist to forbid --- and, more deeply, inverts a dependency:
$\mathbb{C}$ and $\mathbb{H}$ are hours of a \emph{real} clock (the
centralizer algebra of $\mathrm{Cl}(t,s)$ cycles through
$\mathbb{R}$, $\mathbb{C}$, $\mathbb{H}$ with Bott period eight), so
taking them as primitive prices in the whole clock to recover one
hour.

\begin{euclidlemma}[\textbf{L-Constitution-1} (cohort alias L-Real-1), candidate: the real category as the AX-4 minimum]
\label{lem:real-category}
Let $\mathfrak{T}^{\rm rec}\subset\mathfrak{T}$ be the subclass of
admissible packages from which the observed boundary complex
structure --- one shared imaginary unit, the complex Hilbert-space
representation, the Wick rotation with its statistics and thermal
clock, and the particle--antiparticle conjugation --- is recoverable
at the quasi-free grade. Among the primitive field categories
$\mathfrak{T}_{\mathbb{R}}$, $\mathfrak{T}_{\mathbb{C}}$,
$\mathfrak{T}_{\mathbb{H}}$, the real category is the unique minimum
of the refined AX-4 cost $\widehat F = F + C$ on
$\mathfrak{T}^{\rm rec}$:
\begin{enumerate}[leftmargin=2.2em,label=(\roman*)]
\item \emph{Membership (existence):}
$\mathfrak{T}_{\mathbb{R}}\in\mathfrak{T}^{\rm rec}$ --- the
recovery is exhibited by T-P4-1 through T-P4-7 and T-P5-A/B/C with
the Wick quartet T-P5-B1: the circle supplies the gauge $i$, the
clock the state-space $i$, the handshake their forced agreement (the
antiparticle), and the quartet the rotation/statistics/thermal-clock
triple from the one-sided spectrum
(\citealp{ArisakaP4Real}; \citealp{ArisakaP5CPT}).
\item \emph{Minimality:}
$C(\mathfrak{T}_{\mathbb{R}})=(0,0) < C(\mathfrak{T}_{\mathbb{C}})
\leq C(\mathfrak{T}_{\mathbb{H}})$, by the pricing of
D-Constitution-1 above.
\item \emph{Uniqueness:} any package achieving cost $(0,0)$
specifies no complex structure as input, hence must derive its $J$
from the constitutional data; by T-P4-2's uniqueness clause
(positive-energy, dynamics-commuting, on the globally static arena)
and T-P4-3's sign resolution, the derived $J$ is unique --- so every
zero-cost member of $\mathfrak{T}^{\rm rec}$ coincides with the
real-category construction on shell.
\end{enumerate}
Consequently, ``only real numbers in the 6D bulk'' is not an
independent postulate of the program: it is the AX-4-selected
point of the primitive-category axis, in the same sense in which the
absence of continuous coupling dials is the AX-4-selected point of
the parameter axis.
\emph{Scope concessions, stated with the lemma:} clause (i) holds at
the quasi-free grade, conditionally beyond it on the named
hypotheses (H1)--(H3) of T-P4-8; clauses (ii)--(iii) are
constitutional forcing arguments --- chains of necessity given the
axioms --- not theorems of constructive QFT; and the restriction to
$\{\mathbb{R},\mathbb{C},\mathbb{H}\}$ imports the Frobenius--Hurwitz
classification.
\end{euclidlemma}

\paragraph{Grade: candidate --- and why.}
L-Constitution-1 is graded \emph{candidate}, and the reason should
be said plainly: the recoverability leg (i) is derived at the
quasi-free grade by inheritance from the cited theorems, but the
pricing legs (ii)--(iii) rest on D-Constitution-1's counting of
unforced choices --- a stated argument, not a proved theorem of
constructive field theory. L-Constitution-1 is a constitutional
forcing argument in the sense of \ATwo{}'s Remark~0.5
\citep{ArisakaA2GG6}, and every downstream use inherits that
register.

\medskip
\noindent\textbf{F-Constitution-10 (cohort alias F-Real-1; candidate
falsifier of L-Constitution-1; registered in
Table~\ref{tab:pull_table}).} \emph{The lemma dies if any one of the
following is observed.}
\begin{enumerate}[leftmargin=2.2em,label=\textbf{(\arabic*)}]
\item \emph{A non-geometric CP phase.} A measured fundamental
CP-violating phase demonstrably \emph{not} expressible as real
orientation/holonomy data --- concretely, a confirmed
$\delta_{\rm CKM}$ or $\delta_{\rm PMNS}$ provably outside every
closed-loop lattice angle the real texture calculus admits,
\emph{after} the \ANine{} QD-class freeze \citep{ArisakaA9SM}, which
removes the escape of re-tuning (no new unit may be introduced), so
a stray phase kills rather than bends. An irreducibly complex Yukawa
datum with no dynamical source would break the recoverability
clause (i): some observed complex structure would then not be the
boundary's name for a real quarter-turn.
\item \emph{Unshared-$i$ evidence.} A precision-interferometry or
correlation experiment establishing that the charge-circle $i$ and
the causal-clock $i$ are \emph{not} shared --- two inequivalent
complex structures that cannot be globally aligned, violating the
shared-$J$ recovery of T-P4-4. This is the network-Bell exclusion of
real quantum mechanics \citep{Renou2021} run in reverse: that
experiment class excludes \emph{unshared} real tensor products (and
is satisfied by the shared-$J$ construction); nature exhibiting
sector-dependent imaginary units would kill the shared-$J$ theorem
itself.
\item \emph{Quaternionic-direction Born deviations.} A confirmed
deviation from the Born trace form \emph{in the direction of
quaternionic amplitudes} --- a genuine three-path interference
residue of the Sorkin-parameter class
\citep{Peres1979,Sinha2010} matching $\mathbb{H}$-amplitude
predictions --- would indicate that the bulk category is
$\mathfrak{T}_{\mathbb{H}}$, killing minimality leg (ii) from the
opposite direction. (Deviations in the real-deficient direction ---
nature poorer than complex --- are already excluded by the
shared-$J$ reading of \citealp{Renou2021}; the quaternionic-excess
direction is the live kill.)
\end{enumerate}

\subsection{Worked example: the 28 free parameters of the
 Standard Model}
\label{sec:AX4_worked_SM}

A concrete instance brings out the discipline. The cleanest test of
$\AX{4}~\MSMF{}$ is the parameter inventory of the Standard Model.
The SM as currently fitted to data carries the
following 28 continuous parameters \citep{PDG2024}: three gauge
couplings ($g_1, g_2, g_3$ at, say, the $Z$-pole); two
electroweak Higgs-sector parameters ($v$ and $\lambda$, equivalently
$m_W$ and $m_H$); nine charged-fermion masses (six quarks, three
charged leptons); four CKM parameters (three angles plus one
phase); three neutrino masses (or two mass-squared differences plus
one absolute scale); four PMNS parameters (three angles plus one
Dirac phase, with two further Majorana phases if neutrinos are
Majorana); the QCD theta-angle $\bar\theta$; the cosmological
constant $\Lambda$; and the Newton constant $G_N$. Adding two
Majorana phases takes the count to 30 in the most generous reading
and dropping the gravitational sector takes it to 26. The
``28 parameters'' figure is the conventional middle point.

The axiomatic question is: which of these 28 are forced by
the \GGConstitution{}, and which remain free? $\AX{4}~\MSMF{}$
forbids any continuous parameter unless it is forced by symmetry,
topology, anomaly cancellation, or operational necessity.
$\POS{6}~\OGN{}$ further forbids any continuous normalization
choice; only the discrete \OGNfive{} integer ledger
$(M,N,L;J,K) = (1,3,5;7,35)$ survives. Together, these clauses
imply that an admissible candidate theory of the SM must
predict every one of the 28 parameters from the constitutional
data. The companion paper \citep{ArisakaA9SM} takes up
this challenge and shows that all 28 SM parameters descend from
the 6D bulk, the four-component anomaly polynomial, and
the \OGNfive{} ledger, with \emph{zero} adjustable continuous
inputs. The empirical predictions include:

\begin{itemize}[leftmargin=2.2em]
\item the unification scale $\EGUT = \EPl\,\e^{-2\pi}
 \approx 2.28\times 10^{16}\,\GeV$;
\item the lock cluster $\Mlock = (1/2)\,\MPl\,\e^{-35}
 \approx 3.85\,\TeV$;
\item the electroweak scale $v = \Mlock\,(2/L)^N = \Mlock\,(2/5)^3
 \approx 246.33\,\GeV$, the doublet projection $2/L = 2/5$ raised to
 the generation count $N = 3$;
\item the cosmological-constant scale
 $m_\Lambda^{\mathrm{geom}} = \Mlock^2/\MPl \approx 6\,\mathrm{meV}$;
\item the cosmological partitions $\Omega_m = 1/\pi$,
 $\Omega_\Lambda = 1 - 1/\pi$, $\Omega_b = 1/(2\pi^2)$, and
 $\Omega_{\mathrm{DM}} = 1/\pi - 1/(2\pi^2)$
 \citep{ArisakaA10Cosmos};
\item the kaxion oscillation frequency $\Delta\nu/\nu = 1/35$.
\end{itemize}

Each of these is a discrete consequence of the constitution and
the integer ledger; none is a fitted parameter. Generally
speaking, the axiomatic reading of the 28-parameter
inventory is that the Standard Model is not wrong --- it is
\emph{incomplete} as a foundational document, because the 28
parameters are exactly the unforced freedom that $\AX{4}~\MSMF{}$
forbids. After all, the axiomatic bargain offered here is
that a single short document forces every dial of the Standard
Model and $\Lambda$CDM to take a specific value. If a single one
of the listed predictions is decisively falsified by experiment,
the constitution is wrong; if all of them survive contact with
data, the \GGTheory{} program has accomplished what unification
has been promising for half a century.

\subsection{Physics intuition}

\begin{euclidremark}[Physics intuition for \AX{4}]
Roughly speaking, \MSMF{} says that Nature does not hide knobs.
If a parameter is not forced by symmetry, topology, anomaly
cancellation, or operational necessity, it is not allowed to
remain fundamental. This is a direct response to the culture of
model catalogs with many adjustable constants --- the 28 free
parameters of the Standard Model, the moduli fields of string
compactifications, the tunable inflaton sector of inflationary
cosmology. Indeed, an explanation that requires us to choose
the value of a continuous dial is not really an explanation; it
is a deferred problem. \MSMF{} converts the aesthetic preference
for parsimony into an auditable demonstrate-or-eliminate rule:
either show that a candidate parameter is forced by the
constitution, or remove it from the theory. Together with
\POS{6}~\OGN{}, the rule pins every continuous dial of the
inherited foundational physics to a discrete consequence of the
\OGNfive{} integer ledger.
\end{euclidremark}

\subsection{Relation to \GG\ and \GDOS}

\begin{itemize}[leftmargin=2.2em]
\item \textbf{Relation to \GG.} In \GG\ this axiom demands that the bulk background, gauge group, and discrete topological data
be as symmetric and as moduli-free as possible.
Combined with anomaly cancellation (POS-\AF) and normalization (POS-\OGN), this is intended to eliminate continuous
free parameters in the unification stage \citep{ArisakaA2GG6}.
\item \textbf{Relation to \GDOS.} In \GDOS\ \MSMF{} becomes a constitutive-selection rule for open dynamics:
drift/flow terms and noise structures are constrained to be the minimal ones compatible with positivity and gauge invariance.
This plays the role of a universal ``closure'' principle for complex systems \citep{ArisakaA5GDOS}.
\end{itemize}

\section{Inter-axiom logic and the first theorems forced by the
 axioms}
\label{sec:inter_axiom}

The four axioms are designed to be used as a
dependency-controlled proof system; let us name the dependency
order before the postulates of Part~III come into play. The
strict ordering is the four-axiom chain
\begin{equation}
\AX{1}~\LC{} \;\Longrightarrow\; \AX{2}~\GGEP{} \;\Longrightarrow\;
\AX{3}~\OGGEP{} \;\Longrightarrow\; \AX{4}~\MSMF{},
\label{eq:inter_axiom_order}
\end{equation}
in which each axiom \emph{presupposes} the previous and adds new
content. $\AX{1}$ sets the admissible causal arrow that any
later clause must respect; $\AX{2}$ presupposes $\AX{1}$ and
adds the composite bulk primitive (geometry locked with gauge)
that will be projected to the boundary; $\AX{3}$ presupposes
$\AX{1}+\AX{2}$ and adds the operational dictionary that turns
the bulk primitive into registered boundary records; $\AX{4}$
presupposes $\AX{1}+\AX{2}+\AX{3}$ and closes the theory space
by eliminating every continuous degree of freedom not forced by
the previous three. This strict chain is the same constitutional
tower that Part~V develops in detail
(\figref{fig:A1-constitutional-flowchart}), with Stratum~$N$
corresponding to the axiom subset $\{\AX{1}, \ldots, \AX{N}\}$.

Several first theorems have already surfaced in the present Part:
$\AX{1}$ together with Markovity forces the
forward-time semigroup law of Eq.~\eqref{eq:semigroup}; $\AX{1}$
together with CPTP forces the GKLS form of the generator
\eqref{eq:Lindblad}; $\AX{2}$ acting on a compact $\mathrm{U}(1)$
holonomy fiber forces the integer-quantized charge spectrum
\eqref{eq:thetaQuant}; and $\AX{4}$ on the candidate-theory space
forces the anti-landscape lexicographic selection \eqref{eq:lexi}.
Part~III will now supply the seven postulates that turn these
fragments into a full admissibility engine, and Part~IV's
cross-paper coherence map will pin each of the eleven Principal
Theorems $\T{1}$--$\T{11}$ to its canonical proof location in
the cohort companion paper where its derivation actually lives.

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

\section{Comprehensive summary of AX-1--AX-4}
\label{sec:axiom_table}

This subsection closes the chapter with three complementary views of the
same four axioms, because no single view carries all of what a reader
needs.

The first is an \emph{auditor's} view: Table~\ref{tab:axioms_summary}
sets out what each axiom asserts, the minimal mathematical kernel it
needs, the first consequences it forces, and --- the column that matters
most --- the experimental fact that would make it unusable. It answers
the question \emph{what does this axiom say, and how would I know if it
were wrong?}

The second is a \emph{structural} view. Figure~\ref{fig:a1-axiom-tower}
draws the four axioms as a nested tower and places each major framework
of twentieth-century physics on the slab whose axioms it actually uses.
It answers a different question: \emph{what does each axiom buy, and
what is left unsayable without it?}

The third is a \emph{historical} view.
Figure~\ref{fig:a1-axiom-timeline} puts eight centuries of physics on a
time axis, one lane per axiom, and shows that all four were arrived at
piecemeal, by communities that mostly did not know they were working on
the same thing. It answers the question a skeptical reader is entitled
to ask: \emph{if these four principles are so fundamental, why has
nobody written them down before?}

The three views are consistent by construction --- the tower's slabs are
the table's rows, and the timeline's lanes are the tower's slabs --- but
they are not redundant, and a reader who takes only one should take the
table.

{\footnotesize
\begin{longtable}{@{}p{0.085\linewidth}p{0.095\linewidth}p{0.295\linewidth}p{0.250\linewidth}p{0.240\linewidth}@{}}
\caption[Axioms summary (AX-1--AX-4).]{\textbf{Axioms summary (AX-1--AX-4).} Auditor-style overview of each axiom: a crisp constitutional clause, representative equations, first forced consequences, and an explicit failure mode. The rightmost column is written as a practical falsifier: what experimental/operational fact would make the axiom unusable.}
\label{tab:axioms_summary}\\[0.35em]
\toprule
\textbf{Axiom} & \textbf{Name} & \textbf{Formal kernel (representative equations)} &
\textbf{Forced consequences (first theorems)} & \textbf{Failure mode / falsifier} \\
\midrule
\endfirsthead
\toprule
\textbf{Axiom} & \textbf{Name} & \textbf{Formal kernel (representative equations)} &
\textbf{Forced consequences (first theorems)} & \textbf{Failure mode / falsifier} \\
\midrule
\endhead
\midrule
\multicolumn{5}{r}{\small continued on next page}\\
\endfoot
\bottomrule
\endlastfoot

AX-1 &
\LC &
Retarded support of response kernels \eqref{eq:retardedG}, \eqref{eq:retardedKernel}; microcausality \eqref{eq:microcausality}; forward semigroup \eqref{eq:semigroup}. &
No superluminal signaling in registered statistics; arrow-of-time for open evolution; prerequisite for operational dictionary. &
Any experimentally validated operational kernel enabling advanced response/backward influence; acausal signaling in registered records. \\

AX-2 &
\GGEP &
Composite connection $\mathcal{A}$ and curvature $\mathcal{F}$ \eqref{eq:compositeA}--\eqref{eq:compositeF}; holonomy \eqref{eq:holonomy}; local trivialization \eqref{eq:locked_chart}. &
Unified inertial notion (locked chart); geodesic/Lorentz-force unification \eqref{eq:compositeLorentzForce}; quantization from compact holonomy \eqref{eq:thetaQuant}. &
Obstruction to defining a single composite connection with consistent gauge/geometry covariance; need for independent ``force primitives'' beyond curvature/holonomy invariants. \\

AX-3 &
\OGGEP &
Operational map $\Dmap$ (functor) \eqref{eq:Dfunctor}; instruments/POVMs \eqref{eq:Kraus}--\eqref{eq:BornInstrument}; admissible evolution as CPTP channels; GKLS generator \eqref{eq:Lindblad}. &
Measurement defined unambiguously as instrument output; LC+CPTP$\Rightarrow$ Lindblad form; bulk statements become scientific only if they compile to observables. &
Inability to construct a causal, gauge-covariant operational dictionary; predicted evolution violates CPTP/positivity or enables inconsistent measurement semantics. \\

AX-4 &
\MSMF &
Theory-space ordering by symmetry and freedom \eqref{eq:symmetryScore}--\eqref{eq:lexi}; moduli count $F(T)$. &
Anti-landscape selection: continuous tunable families forbidden unless forced; conventions must be constitutionally fixed (via POS-\OGN); symmetry breaking becomes constrained selection. &
Operational necessity of continuous moduli/knobs that cannot be eliminated without losing empirical adequacy; parameter landscapes required by data. \\

\end{longtable}
}
\paragraph{The tower: what each axiom buys.}
The four axioms are not a list of independent assumptions. They are
strictly nested, and Figure~\ref{fig:a1-axiom-tower} is drawn to make
the nesting the first thing one sees: the chart reads downward, each
slab adds exactly one axiom to the slab above it, and the arrow on the
left records the only fact the picture is really asserting --- that each
addition strictly enlarges what can be derived.

The nesting is forced rather than chosen. \AX{1} must come first
because without a causal ordering there is no sense in which one thing
can be said to respond to another, and every later axiom is about a
relation between things. \AX{2} presupposes \AX{1} because a connection
is defined by parallel transport along paths, and paths are ordered.
\AX{3} presupposes \AX{2} because there must be a structure to
coarse-grain before coarse-graining means anything. And \AX{4}
presupposes all three, because a selection principle needs a space of
admissible theories to select within, and the first three axioms are
what define admissibility.

Read the slabs downward and each is a recognizable body of physics.
The first produces special relativity and classical field theory on a
fixed background --- everything that follows from causal propagation
alone. The second adds the identification of force with curvature, and
with it comes general relativity, Yang--Mills theory, the
Green--Schwarz mechanism, superstring theory and the AdS--CFT
correspondence: an enormous quantity of physics, all of it obtained
from two axioms. The third adds the boundary coarse-graining, and with
it the quantum postulates as theorems, classical thermodynamics,
statistical mechanics, the fluctuation theorems, black-hole physics and
the free-energy principle. The fourth removes the remaining freedom and
yields a Standard Model whose twenty-eight fitted inputs are forced and
a cosmological model with no tunable inventory.

\paragraph{The diagnostic reading.}
The tower earns its place here not by what each slab produces but by
what each slab \emph{cannot say}. A framework standing on the second
slab has no operational layer, and so it has nothing to say about when
a measurement has occurred --- which is precisely the century-old
measurement problem, and precisely why it could not be solved from
inside that slab no matter how much technical effort was applied. A
framework standing on the third slab has no selection principle, and so
it cannot say which of its admissible solutions is realized --- which is
precisely the landscape problem. In each case the persistent open
question of a research program turns out to have the shape of the
axiom that program does not have.

This is a claim about logical dependency, and it should not be read as
a ranking. Every framework on the tower produced what it produced, and
each correctly used the axiom subset its own problems required. General
relativity did not need an operational dictionary, because nothing in
the classical field equations has to be observed for them to hold.
Quantum mechanics did need one, because a probability is by definition
what a finite observer registers. The tower does not say that anyone
was careless. It says that the fragmentation of twentieth-century
theoretical physics into sub-disciplines with separate vocabularies is
the visible shadow of working with partial and implicit axiom subsets.

\begin{figure}[htbp]
\centering
\resizebox{0.872\textwidth}{!}{%
\begin{tikzpicture}[font=\sffamily\footnotesize,
 slab/.style ={rounded corners=2pt, line width=0.6pt},
 hd/.style   ={anchor=west, align=left, font=\sffamily\bfseries\small},
 ct/.style   ={anchor=west, align=left, font=\sffamily\scriptsize\itshape,
               text width=14.7cm},
 ex/.style   ={anchor=west, align=left, font=\sffamily\scriptsize,
               text width=14.7cm}]

%% AX-1 at the TOP, AX-4 at the BOTTOM: a flowchart reads downward.
%% Row pitch 2.95 against a box height of 2.30 leaves a 0.65 gap, up from
%% the 0.20 the stacked version had.  The four rows are written out rather
%% than generated: \foreach strips one level of braces per field, so a
%% colour passed through it arrives as `{red!14}' and xcolor reads `{red'.

%% ---- row 1 : AX-1 ------------------------------------------------------
\draw[slab, fill=black!8, draw=black!45] (0,  0.00) rectangle (15.5, -2.30);
\node[hd] at (0.25, -0.38) {\AX{1} ($\LC{}$) alone};
\node[ct] at (0.25, -0.92) {Causal field theory on a fixed background};
\node[ex] at (0.25, -1.72) {Special relativity (1905); classical
  electrodynamics on Minkowski};

%% ---- row 2 : + AX-2 ----------------------------------------------------
\draw[slab, fill=orange!14, draw=orange!55!black] (0, -2.95) rectangle (15.5, -5.25);
\node[hd] at (0.25, -3.33) {$+$ \AX{2} ($\GGEP{}$)};
\node[ct] at (0.25, -3.87) {Geometry--gauge unification on any admissible bulk};
\node[ex] at (0.25, -4.67) {General relativity (1915); Yang--Mills (1954);
  Green--Schwarz (1984); superstring theory; AdS--CFT (1997); modified
  Bianchi identity; dualities};

%% ---- row 3 : + AX-3 ----------------------------------------------------
\draw[slab, fill=blue!12, draw=blue!55!black] (0, -5.90) rectangle (15.5, -8.20);
\node[hd] at (0.25, -6.28) {$+$ \AX{3} ($\OGGEP{}$)};
\node[ct] at (0.25, -6.82) {Boundary CPTP channel; coarse-graining functor};
\node[ex] at (0.25, -7.62) {QM postulates as theorems; classical
  thermodynamics; statistical mechanics; information thermodynamics;
  fluctuation theorems; black-hole physics; the Page curve; the islands
  formula; Friston's FEP};

%% ---- row 4 : + AX-4 ----------------------------------------------------
\draw[slab, fill=red!14, draw=red!55!black] (0, -8.85) rectangle (15.5, -11.15);
\node[hd] at (0.25, -9.23) {$+$ \AX{4} ($\MSMF{}$)};
\node[ct] at (0.25, -9.77) {Unique admissible solution; zero free continuous
  parameters};
\node[ex] at (0.25, -10.57) {Standard Model with 28 parameters reduced to 0
  free; $\Lambda$CDM with $\Omega_m = \pi^{-1}$; the cohort cascade from
  quarks to consciousness};

%% ---- the flow arrow: LEFT of the stack, pointing DOWN ------------------
\draw[-{Stealth[length=3.2mm]}, line width=1.1pt, draw=black!70]
  (-0.80, -0.05) -- (-0.80, -11.10);
\node[font=\sffamily\scriptsize\itshape, text=black!70, rotate=90,
      anchor=south] at (-1.10, -5.60)
  {each axiom added strictly enlarges what can be derived};
\end{tikzpicture}
}
\caption[The four axioms as a strictly nested tower.]{\textbf{The four axioms as a strictly nested tower.} Each slab adds exactly one axiom to the slab above it, and the chart reads downward, so vertical position is logical depth: everything derivable on a slab is derivable on every slab beneath it, and the arrow on the left records that the containment is strict. Each historical framework is placed on the slab whose axioms it actually uses, not the slab its practitioners would have claimed. The reading that matters is the negative one: what a slab cannot say is the content of the axiom it lacks, which is why the measurement problem could not be solved from the second slab and the landscape problem could not be solved from the third. The placement is a statement about logical dependency and not about merit --- every framework named here correctly used the axiom subset its own problems required.}
\label{fig:a1-axiom-tower}
\end{figure}

\paragraph{The historical record: four threads pulled separately.}
A reader meeting four constitutional axioms for the first time is
entitled to be suspicious of their novelty. If these principles are
fundamental, why did nobody state them? Figure~\ref{fig:a1-axiom-timeline}
is the honest answer: they were all arrived at, piece by piece, over
eight centuries --- but by separate communities, in separate languages,
and never together. Each lane of the chart is one axiom, each dot is a
dated and attested result that put part of that axiom into physics, and
nothing on the chart is claimed for the present program that its
author did not obtain.

The lane for \AX{1} is the oldest and the sparsest. Ibn al-Haytham
establishes that light travels along definite paths in finite time;
Newton's \emph{Principia} makes the question of whether influence is
local unavoidable precisely by leaving it unanswered, a dissatisfaction
Newton stated himself; Maxwell answers it by replacing action at a
distance with a field; and Einstein and Minkowski close the lane by
making the causal structure part of the geometry. Five results across
nine hundred years, and by 1908 the axiom is complete and has no name.

The lane for \AX{2} is the most concentrated. Gauss shows that
curvature is intrinsic --- that a surface knows its own shape without
being embedded in anything --- and Riemann generalizes it to any number
of dimensions, sixty years before there is a physical use for it.
General relativity then makes gravitation a curvature; Weyl, in a
theory that failed for other reasons, introduces the gauge idea that
makes electromagnetism one too; Kaluza and Klein show that an extra
dimension can carry it; Yang and Mills make it non-abelian;
Green and Schwarz make it anomaly-free; and Maldacena makes a
bulk geometry equivalent to a boundary theory. Eight results in
one hundred and seventy years, each visibly building on the last.

The lane for \AX{3} is the busiest and the least unified, and this is
the chart's most important reading. Eleven results between Carnot and
Friston --- in classical thermodynamics, in statistical mechanics, in
quantum measurement theory, in information theory, in black-hole
physics, and finally in theoretical neuroscience. Every one of them is
about what a system that cannot see everything can nevertheless say.
None of those communities had a name for what they had in common, and
several of them were unaware of each other. Landauer's bound \citep{Landauer1961} and the
Bekenstein--Hawking entropy and Zurek's einselection are, on the
reading developed in this paper, three faces of one axiom, and they
were obtained twelve years apart by people who would not have described
themselves as working on the same problem.

The lane for \AX{4} is the oldest and the emptiest, and its shape is
the most telling. Ockham states the principle in the fourteenth
century; Kepler tries to derive the planetary orbits from pure symmetry
and fails; Maupertuis makes minimization a physical principle; Noether
makes symmetry a source of conservation laws. After that the lane goes
quiet for most of a century --- not because the instinct is abandoned,
but because it never becomes an axiom. It operates as taste. And the
consequence is visible on the right-hand end of the lane: the
free-parameter count of fundamental physics falls steadily through the
twentieth century --- one for general relativity, two for quantum
electrodynamics, twenty-eight for the Standard Model, six for the
standard cosmological model --- and never reaches zero, until a landscape
of order $10^{500}$ vacua makes the absence of a selection principle
impossible to ignore.

\paragraph{What the timeline does not claim.}
It is not a priority claim and it is not a ranking. Every result on the
chart stands on its own, is cited in its own right, and would remain
true if the present framework were abandoned tomorrow. The chart makes
one observation only: four threads were pulled separately for a long
time by people who could not see each other's work, and the proposal of
this paper is to hold all four at once, as explicit axioms, and see what
that forces. The horizontal axis is deliberately broken so that the
modern era, where every lane is dense, remains readable; the compressed
left-hand fifth carries the four pre-modern anchors that would otherwise
be lost.

\paragraph{The three views together.}
The table says what the axioms assert and how each could fail. The
tower says what each one buys and what is unsayable without it. The
timeline says that none of them is new, and that what is new is holding
them together. A reader who accepts the table has accepted the
constitution; a reader who accepts the tower has accepted that
twentieth-century physics is one structure rather than several; and a
reader who accepts the timeline has accepted only a historical
observation, which is the least the three demand and the easiest to
check.

\begin{figure}[p]
\centering
\resizebox{1.000\textwidth}{!}{%
\begin{tikzpicture}[font=\sffamily\footnotesize,
 lane/.style ={rounded corners=2pt, line width=0.60pt},
 ms/.style   ={font=\sffamily\scriptsize, rotate=90, anchor=west, inner sep=1pt, align=left},
 tick/.style ={font=\sffamily\scriptsize, text=black!60, anchor=north},
 axl/.style  ={font=\sffamily\bfseries\small, anchor=west},
 gl/.style   ={font=\sffamily\scriptsize\itshape, anchor=west, text=black!62}]

%% ---- lane 1
\node[axl, text=black!50] at (0.00, -0.22) {\AX{1}\quad $\LC{}$ --- locality and causality};
\node[gl] at (0.00, -0.60) {a law acts here, and no influence outruns light};
\draw[lane, fill=black!7, draw=black!50] (0, -0.85) rectangle (19.00, -4.45);
\draw[line width=0.60pt, draw=black!50] (0.10, -4.19) -- (18.90, -4.19);
\filldraw[fill=black!55, draw=black!50, line width=0.40pt] (1.994, -4.19) circle (0.055);
\node[ms, text=black!80] at (1.994, -3.93) {Ibn al-Haytham\\ \textcolor{black!55}{(1021)}};
\filldraw[fill=black!55, draw=black!50, line width=0.40pt] (7.035, -4.19) circle (0.055);
\node[ms, text=black!80] at (7.035, -3.93) {Newton\\ \textcolor{black!55}{(1687)}};
\filldraw[fill=black!55, draw=black!50, line width=0.40pt] (13.244, -4.19) circle (0.055);
\node[ms, text=black!80] at (13.244, -3.93) {Maxwell\\ \textcolor{black!55}{(1865)}};
\filldraw[fill=black!55, draw=black!50, line width=0.40pt] (14.640, -4.19) circle (0.055);
\node[ms, text=black!80] at (14.640, -3.93) {Einstein, SR\\ \textcolor{black!55}{(1905)}};
\filldraw[fill=black!55, draw=black!50, line width=0.40pt] (14.744, -4.19) circle (0.055);
\draw[line width=0.35pt, draw=black!50] (14.744, -4.13) -- (15.420, -3.95);
\node[ms, text=black!80] at (15.420, -3.93) {Minkowski\\ \textcolor{black!55}{(1908)}};

%% ---- lane 2
\node[axl, text=orange!55!black] at (0.00, -4.92) {\AX{2}\quad $\GGEP{}$ --- geometry and gauge are one};
\node[gl] at (0.00, -5.30) {a force is the curvature of a connection};
\draw[lane, fill=orange!12, draw=orange!55!black] (0, -5.55) rectangle (19.00, -9.15);
\draw[line width=0.60pt, draw=orange!55!black] (0.10, -8.89) -- (18.90, -8.89);
\filldraw[fill=orange!60!black, draw=orange!55!black, line width=0.40pt] (11.919, -8.89) circle (0.055);
\node[ms, text=black!80] at (11.919, -8.63) {Gauss\\ \textcolor{black!55}{(1827)}};
\filldraw[fill=orange!60!black, draw=orange!55!black, line width=0.40pt] (12.860, -8.89) circle (0.055);
\node[ms, text=black!80] at (12.860, -8.63) {Riemann\\ \textcolor{black!55}{(1854)}};
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\node[ms, text=black!80] at (15.780, -8.63) {Kaluza--Klein\\ \textcolor{black!55}{(1921)}};
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\node[ms, text=black!80] at (16.560, -8.63) {Yang--Mills\\ \textcolor{black!55}{(1954)}};
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\node[ms, text=black!80] at (18.120, -8.63) {AdS--CFT\\ \textcolor{black!55}{(1997)}};

%% ---- lane 3
\node[axl, text=blue!55!black] at (0.00, -9.62) {\AX{3}\quad $\OGGEP{}$ --- no system is closed};
\node[gl] at (0.00, -10.00) {what is registered is a coarse-graining of what is there};
\draw[lane, fill=blue!10, draw=blue!55!black] (0, -10.25) rectangle (19.00, -13.85);
\draw[line width=0.60pt, draw=blue!55!black] (0.10, -13.59) -- (18.90, -13.59);
\filldraw[fill=blue!60!black, draw=blue!55!black, line width=0.40pt] (11.814, -13.59) circle (0.055);
\draw[line width=0.35pt, draw=blue!55!black] (11.814, -13.53) -- (10.320, -13.35);
\node[ms, text=black!80] at (10.320, -13.33) {Carnot\\ \textcolor{black!55}{(1824)}};
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\draw[line width=0.35pt, draw=blue!55!black] (13.244, -13.53) -- (11.100, -13.35);
\node[ms, text=black!80] at (11.100, -13.33) {Clausius\\ \textcolor{black!55}{(1865)}};
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\draw[line width=0.35pt, draw=blue!55!black] (13.488, -13.53) -- (11.880, -13.35);
\node[ms, text=black!80] at (11.880, -13.33) {Boltzmann\\ \textcolor{black!55}{(1872)}};
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\draw[line width=0.35pt, draw=blue!55!black] (14.535, -13.53) -- (12.660, -13.35);
\node[ms, text=black!80] at (12.660, -13.33) {Gibbs\\ \textcolor{black!55}{(1902)}};
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\draw[line width=0.35pt, draw=blue!55!black] (15.581, -13.53) -- (13.440, -13.35);
\node[ms, text=black!80] at (13.440, -13.33) {von Neumann\\ \textcolor{black!55}{(1932)}};
\filldraw[fill=blue!60!black, draw=blue!55!black, line width=0.40pt] (16.140, -13.59) circle (0.055);
\draw[line width=0.35pt, draw=blue!55!black] (16.140, -13.53) -- (14.220, -13.35);
\node[ms, text=black!80] at (14.220, -13.33) {Shannon\\ \textcolor{black!55}{(1948)}};
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\draw[line width=0.35pt, draw=blue!55!black] (16.593, -13.53) -- (15.000, -13.35);
\node[ms, text=black!80] at (15.000, -13.33) {Landauer\\ \textcolor{black!55}{(1961)}};
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\draw[line width=0.35pt, draw=blue!55!black] (17.012, -13.53) -- (15.780, -13.35);
\node[ms, text=black!80] at (15.780, -13.33) {Bekenstein--Hawking\\ \textcolor{black!55}{(1973)}};
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\draw[line width=0.35pt, draw=blue!55!black] (17.116, -13.53) -- (16.560, -13.35);
\node[ms, text=black!80] at (16.560, -13.33) {GKLS\\ \textcolor{black!55}{(1976)}};
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\draw[line width=0.35pt, draw=blue!55!black] (17.291, -13.53) -- (17.340, -13.35);
\node[ms, text=black!80] at (17.340, -13.33) {Zurek\\ \textcolor{black!55}{(1981)}};
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\draw[line width=0.35pt, draw=blue!55!black] (18.302, -13.53) -- (18.120, -13.35);
\node[ms, text=black!80] at (18.120, -13.33) {Friston\\ \textcolor{black!55}{(2010)}};

%% ---- lane 4
\node[axl, text=red!55!black] at (0.00, -14.32) {\AX{4}\quad $\MSMF{}$ --- nothing left to choose};
\node[gl] at (0.00, -14.70) {the admissible solution is unique, so nothing can be tuned};
\draw[lane, fill=red!11, draw=red!55!black] (0, -14.95) rectangle (19.00, -18.55);
\draw[line width=0.60pt, draw=red!55!black] (0.10, -18.29) -- (18.90, -18.29);
\filldraw[fill=red!60!black, draw=red!55!black, line width=0.40pt] (2.720, -18.29) circle (0.055);
\node[ms, text=black!80] at (2.720, -18.03) {Ockham\\ \textcolor{black!55}{(c.\,1320)}};
\filldraw[fill=red!60!black, draw=red!55!black, line width=0.40pt] (3.390, -18.29) circle (0.055);
\draw[line width=0.35pt, draw=red!55!black] (3.390, -18.23) -- (3.500, -18.05);
\node[ms, text=black!80] at (3.500, -18.03) {Kepler\\ \textcolor{black!55}{(1596)}};
\filldraw[fill=red!60!black, draw=red!55!black, line width=0.40pt] (9.023, -18.29) circle (0.055);
\node[ms, text=black!80] at (9.023, -18.03) {Maupertuis\\ \textcolor{black!55}{(1744)}};
\filldraw[fill=red!60!black, draw=red!55!black, line width=0.40pt] (15.093, -18.29) circle (0.055);
\draw[line width=0.35pt, draw=red!55!black] (15.093, -18.23) -- (15.000, -18.05);
\node[ms, text=black!80] at (15.000, -18.03) {Noether\\ \textcolor{black!55}{(1918)}};
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\draw[line width=0.35pt, draw=red!55!black] (16.802, -18.23) -- (15.780, -18.05);
\node[ms, text=black!80] at (15.780, -18.03) {Standard Model\\ \textcolor{black!55}{(1967)}};
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\draw[line width=0.35pt, draw=red!55!black] (17.884, -18.23) -- (16.560, -18.05);
\node[ms, text=black!80] at (16.560, -18.03) {$\Lambda$CDM\\ \textcolor{black!55}{(1998)}};
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\draw[line width=0.35pt, draw=red!55!black] (18.058, -18.23) -- (17.340, -18.05);
\node[ms, text=black!80] at (17.340, -18.03) {the landscape\\ \textcolor{black!55}{(2003)}};
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\draw[line width=0.35pt, draw=red!55!black] (18.860, -18.23) -- (18.120, -18.05);
\node[ms, text=black!80] at (18.120, -18.03) {this cohort\\ \textcolor{black!55}{(2026)}};

%% ---- the broken time axis
\draw[line width=0.60pt, draw=black!60] (0, -19.00) -- (3.34, -19.00);
\draw[line width=0.60pt, draw=black!60] (4.06, -19.00) -- (19.00, -19.00);
\draw[line width=0.60pt, draw=black!60] (3.31, -19.15) -- (3.45, -18.85);
\draw[line width=0.60pt, draw=black!60] (3.91, -19.15) -- (4.05, -18.85);
\draw[line width=0.50pt, draw=black!55] (0.729, -19.00) -- (0.729, -19.15);
\node[tick] at (0.729, -19.15) {500};
\draw[line width=0.50pt, draw=black!55] (1.943, -19.00) -- (1.943, -19.15);
\node[tick] at (1.943, -19.15) {1000};
\draw[line width=0.50pt, draw=black!55] (3.400, -19.00) -- (3.400, -19.15);
\node[tick] at (3.400, -19.15) {1600};
\draw[line width=0.50pt, draw=black!55] (7.488, -19.00) -- (7.488, -19.15);
\node[tick] at (7.488, -19.15) {1700};
\draw[line width=0.50pt, draw=black!55] (10.977, -19.00) -- (10.977, -19.15);
\node[tick] at (10.977, -19.15) {1800};
\draw[line width=0.50pt, draw=black!55] (14.465, -19.00) -- (14.465, -19.15);
\node[tick] at (14.465, -19.15) {1900};
\draw[line width=0.50pt, draw=black!55] (17.953, -19.00) -- (17.953, -19.15);
\node[tick] at (17.953, -19.15) {2000};
\node[font=\sffamily\scriptsize\itshape, text=black!60, anchor=north] at (1.70, -19.60) {200--1600, compressed};
\node[font=\sffamily\scriptsize\itshape, text=black!60, anchor=north] at (11.50, -19.60) {1600--2030, linear};
\end{tikzpicture}
}
\caption[The four axioms in historical order, 200--2030.]{\textbf{The four axioms in historical order.} Each lane is one axiom and each dot is a dated result that put part of that axiom into physics before anyone had named it; the lanes are ordered as the axioms are. Read across a lane and it is a research tradition; read down a column and it is a period. Three features carry the argument. The \AX{3} lane is by a wide margin the busiest and the least unified --- eleven results across two centuries, in thermodynamics, information theory, black-hole physics and neuroscience, none of whose communities had a name for what they shared. The \AX{2} and \AX{3} lanes run in parallel for a century and almost never touch. And the \AX{4} lane is the oldest and the emptiest: Ockham states the principle in the fourteenth century, after which it operates as taste rather than as an axiom, which is why the free-parameter count of the twentieth century falls steadily and never reaches zero. The horizontal axis is broken --- 200--1600 is compressed into the left-hand fifth --- so that the modern era, where every lane is dense, stays legible. Positions are the accepted dates of publication.}
\label{fig:a1-axiom-timeline}
\end{figure}



% ---------------- References (partial; expanded progressively) ----------------

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

\section{The most consequential immediate implication of the four axioms}
\label{sec:two_remarks}

Before the seven postulates of Part~III refine the four axioms
into runtime admissibility filters, it is worth pausing for the one
consequential implication that the four axioms --- taken together
as the constitutional charter --- \emph{immediately} settle.
This is not an aesthetic claim; it is the structural fact that
distinguishes the cohort framework most sharply from its
twentieth-century predecessors, and it is the item we would most
like the broader physics community to take away from the present
paper.

\paragraph{The century-old measurement problem is finally solved by $\AX{3}~\OGGEP{}$.}
For ninety years --- from the von Neumann \emph{Mathematische
Grundlagen} of 1932 \citep{vonNeumann1932} through Bell's 1964
inequality \citep{Bell1964}, the decoherence program of Zeh,
Joos, and Zurek
\citep{Zeh1970,JoosZeh1985,Zurek1981,Zurek2003}, and the islands
formula of Almheiri--Engelhardt--Marolf--Maxfield and
Penington--Shenker--Stanford--Yang
\citep{Almheiri2020,Penington2020} --- the measurement problem has
been the most awkward open seam in physics. Every standard
textbook closes its chapter on quantum mechanics with a hedged
sentence about ``the role of the observer'' or ``the interpretation
of the wave function,'' and the ninety-year debate has produced no
consensus resolution within the conventional axiomatic framework.

$\AX{3}~\OGGEP{}$ resolves it. \textbf{Every system in nature
is an open system.} This single empirical fact, elevated to a
constitutional axiom, dissolves the von Neumann cut by
definitional fiat: measurement \emph{is} the always-on
bulk-to-boundary projection $\Dmap$, gauge-covariant and CPTP,
with the boundary record produced as the output of the operational
dictionary rather than as a separately-postulated collapse. The
textbook QM postulates --- the Born rule, the L\"uders projection,
decoherence, einselection, and Bell--CHSH violation --- become
theorems of $\Dmap$ acting on the bulk (the chain
$\T{7.1}$--$\T{7.7}$ proved in $\AFive{}$
\citep{ArisakaA5GDOS}, exercised at graduate-textbook depth in
$\AEight{}$ \citep{ArisakaA8QM}). The Heisenberg cut is dissolved; the von Neumann chain is
terminated at the first macroscopic pointer state with a
record-amplification chain sufficient for the experimenter's
purpose; Wigner's friend \citep{WignerFriend1961} is resolved by
the always-on projection's independence from any particular
observer. The ninety-year measurement-problem debate becomes,
concretely, the seven-theorem chain $\T{7.1}$ (Hilbert-space
embedding) $\to$ $\T{7.2}$ (Schr\"odinger equation) $\to$
$\T{7.3}$ (Born rule) $\to$ $\T{7.4}$ (L\"uders projection) $\to$
$\T{7.5}$ (decoherence) $\to$ $\T{7.6}$ (einselection) $\to$
$\T{7.7}$ (Bell--CHSH violation), each derived inside \AFive{}
from $\AX{3}~\OGGEP{}$ acting on the bulk. Ninety years of
interpretational debate compile into seven theorems once the
right axiom is named.

This is, \emph{the most significant immediate contribution} of the \GGConstitution{}. The constitution does
many other consequential things --- it locates every major
twentieth-century theoretical-physics framework inside a
four-axiom tower (Part~V), it forces a unique vacuum from a
$\sim 10^{500}$-vacuum landscape
(\S\ref{sec:V-quantitative-consequences}), it derives the 28
Standard Model parameters as a discrete consequence of the
integer $\OGNfive{}$ ledger ($\ANine{}$
\citep{ArisakaA9SM}). But none of those downstream consequences
carries the same \emph{immediate} impact on the broader physics
community as the recognition that the measurement problem, which has resisted resolution within the
conventional axiomatic framework for ninety years, is settled by
the single act of writing $\AX{3}~\OGGEP{}$ as a constitutional axiom.
After all, every body of knowledge becomes a science when its
practitioners agree on what counts as an observation; physics has
carried that agreement quietly for ninety years, and
$\AX{3}~\OGGEP{}$ puts it on the page.

\section{Century-old history of the four axioms}
\label{sec:history-partii}

The four axioms did not arrive from nowhere. Each one is the sharpened form of a question the twentieth century asked and answered only in part, and each was argued over for decades before it could be written in a line. This Section collects that history, axiom by axiom, so that the constitutional statements above can be read as the end of a long argument rather than as assertions made in a vacuum.

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


Locality and causality are constitutional pillars inherited from
the longest-running thread in the history of physics. In keeping
with the systematic presentation we adopt for the other axioms,
we organize the AX-1 lineage as twelve chronological milestones
whose structural content $\AX{1}~\LC{}$ distils into a single
axiomatic clause. Each milestone is also located in Part~V's
constitutional tower (\figref{fig:A1-constitutional-flowchart}):
the AX-1-alone Stratum-1 frameworks (special relativity and
classical electrodynamics) live entirely within the
locality-and-causality content this Section credits.

\paragraph{(1) Galileo --- the principle of relativity (1632).}
Galileo's \textit{Dialogo} of 1632 \citep{Galilei1632} contains
the first explicit statement of the relativity principle: the
``ship's cabin'' thought experiment shows that mechanics inside
a uniformly moving ship is indistinguishable from mechanics at
rest, hence no inertial frame is preferred. Galileo's insight
is the kinematic ancestor of $\AX{1}~\LC{}$: any admissible
kernel must respect a kinematic invariance group, and operational
registrations made in different inertial frames must be
consistently relatable.

\paragraph{(2) Newton --- the first modern propagation framework (1687).}
Newton's \textit{Principia} \citep{Newton1687} established the
first modern framework in which physical effects propagate from
cause to effect through space, though Newton himself remained
famously uneasy that gravitational action appeared to act
instantaneously across arbitrary distances --- a tension he
called \textit{``so great an absurdity that I believe no man,
who has in philosophical matters a competent faculty of thinking,
can ever fall into it.''} The Newtonian framework is the
historical baseline against which every later refinement of
locality and causality has been measured.

\paragraph{(3) Maxwell and Lorentz --- finite-speed propagation (1865, 1904).}
Maxwell's electromagnetic theory \citep{Maxwell1865} resolved
Newton's instantaneous-action tension for electromagnetism by
showing that disturbances of the field propagate at finite speed.
Lorentz's 1904 paper \citep{Lorentz1904} introduced the
transformations that bear his name --- the symmetry group under
which Maxwell's equations are form-invariant. Together, Maxwell
and Lorentz set the stage for special relativity by showing that
finite-speed propagation is not a feature of one theory but a
candidate kinematic primitive for all theories.

\paragraph{(4) Einstein --- special relativity elevates finite speed to a universal constraint (1905).}
Einstein's 1905 paper \citep{Einstein1905} elevated finite-speed
propagation to a universal kinematic constraint: the speed of
light is the same in every inertial frame, and no signal carries
information faster. In Part~V's classification, special
relativity is the canonical Stratum-1 framework: it lives
entirely within $\AX{1}~\LC{}$ alone, requiring no gauge
structure ($\AX{2}$), no operational projection ($\AX{3}$), and
no minimality principle ($\AX{4}$).

\paragraph{(5) Minkowski --- the light cone as geometric structure (1908).}
Minkowski \citep{Minkowski1908} recast Einstein's kinematic
constraint as the geometric structure of the light cone --- the
first time causal ordering was treated as a property of spacetime
itself rather than of particular dynamical laws. Minkowski's
move is the conceptual ancestor of $\AX{1}~\LC{}$'s reading of
the bulk causal structure as a primitive of the manifold rather
than a derived feature of the dynamics.

\paragraph{(6) Einstein --- general relativity internalizes causality in spacetime curvature (1915--1916).}
Einstein's general relativity \citep{Einstein1915,Einstein1916}
internalized causality into the curvature of spacetime: the
metric $g_{\mu\nu}$ itself encodes both the gravitational field
and the local causal structure. General relativity is the
$\{\AX{1}, \AX{2}\}$ Stratum-2 specialization that adds the
geometry-gauge equivalence content of $\AX{2}~\GGEP{}$ (here in
its gravity-only specialization) on top of the locality and
causality content of $\AX{1}~\LC{}$.

\paragraph{(7) Wightman and Streater--Wightman --- microcausality as a QFT axiom (1956, 1964).}
Wightman \citep{Wightman1956} and Streater--Wightman
\citep{StreaterWightman1964} formulated the microcausality
axiom \eqref{eq:microcausality} that fields commute at spacelike
separation; the Wightman axioms are the first rigorous
axiomatization of relativistic quantum field theory in which
locality and causality appear as constitutional clauses rather
than as background assumptions. $\AX{1}~\LC{}$ is the cohort
generalization of the Wightman microcausality axiom to the open,
operational setting of $\AX{3}~\OGGEP{}$.

\paragraph{(8) Haag and the algebraic-QFT school --- locality of operator algebras (1964--1996).}
Haag \citep{Haag1992} and the algebraic-QFT school
developed the Wightman framework into the local-net formulation,
in which the observable algebra is assigned to spacetime regions
with strict commutativity at spacelike separation. The
algebraic-QFT tradition is the most mature mathematical
realization of $\AX{1}~\LC{}$ in the closed-system regime; the
cohort framework extends it to the open-system regime via
$\AX{3}~\OGGEP{}$.

\paragraph{(9) Bell and Tsirelson --- locality and its quantum bound (1964, 1980).}
Bell \citep{Bell1964} made causality and locality empirically
testable: his celebrated inequality showed that a wide class of
locally causal hidden-variable theories make quantitative
predictions distinguishable from those of standard quantum
mechanics. Tsirelson \citep{Tsirelson1980} sharpened the
constraint by proving the quantum upper bound
$|S| \leq 2\sqrt{2}$: quantum mechanics maximally violates the
classical-local-causality bound, but only up to a specific
saturation. $\AX{1}~\LC{}$ accommodates Bell-violating
correlations within the Tsirelson bound while still forbidding
superluminal signaling, exactly as the operational reading of
locality requires.

\paragraph{(10) Aspect, Hensen, Giustina --- experimental Bell-test confirmation (1982--2015, 2022 Nobel).}
Aspect \citep{Aspect1982} performed the first decisive
experimental Bell test, with time-varying analyzers that closed
the locality loophole. Hensen with the Delft team
\citep{Hensen2015} and Giustina with the Vienna team
\citep{Giustina2015} closed the remaining loopholes (detection
efficiency, fast random setting choice) in 2015 with
loophole-free experiments. The 2022 Nobel Prize to Aspect,
Clauser, and Zeilinger recognized this entire chain. These
experiments are not merely tests of quantum mechanics; they are
the operational foundation on which any constitution including
$\AX{1}~\LC{}$ rests.

\paragraph{(11) Bombelli--Lee--Meyer--Sorkin --- causality as the discrete primitive of spacetime (1987).}
Bombelli, Lee, Meyer, and Sorkin \citep{BombelliSorkin1987}
proposed that spacetime \emph{is} a partially ordered set whose
order relation \emph{is} the causal structure. This is the most
aggressive realization of the constitutional thesis that
causality is a primitive of the bulk geometry rather than a
derived feature of the dynamics. The causal-set program
supplies the deep mathematical confirmation that $\AX{1}~\LC{}$
can be stated as a constitutional primitive without any prior
commitment to continuous spacetime.

\paragraph{(12) Open systems and stochastic thermodynamics --- the operational forward-time arrow (1970--2012).}
In open quantum systems and quantum information, the same causal
arrow is carried forward by the forward-time CPTP-map structure
of channels and instruments \citep{Kraus1971,Kraus1983,NielsenChuang2010,DaviesLewis1970,Ozawa1984};
no admissible channel can map a future record into a past
preparation. In stochastic thermodynamics, the same arrow
appears as the time-asymmetry of the fluctuation theorems
\citep{Crooks1999,Jarzynski1997,Seifert2012}. These threads
supply the modern operational language in which $\AX{1}~\LC{}$
is stated: not as a metaphysical claim about hidden variables
but as a structural constraint on which kernels and which
channels are admissible in a registered-record physics.

\paragraph{Synthesis --- $\AX{1}~\LC{}$ as the axiomatic distillation.}
Indeed, the constitutional novelty of $\AX{1}~\LC{}$ lies not
in any single ingredient above but in the axiomatic move of
elevating these twelve milestones --- from Galileo's ship and Newton's
instantaneous-action concern through Maxwell, Lorentz, Einstein,
Minkowski, Wightman, Haag, Bell, Tsirelson, Aspect, Bombelli, and
the open-system / stochastic-thermodynamics literature --- into a
single explicit clause that any candidate theory must satisfy as
a precondition for admissibility. Part~V of the present paper
locates the AX-1-alone Stratum-1 frameworks (special relativity
and classical electrodynamics) entirely within the content
credited here. After all, the cleanest move a foundational
document can make is to take three and a half centuries of work
on cause and effect, recognize the operational invariant they
all share, and write that invariant as one axiom of the
constitution.

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

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


The geometry-gauge equivalence principle of $\AX{2}~\GGEP{}$ is the
axiomatic distillation of nearly two centuries of foundational
work in geometry, gauge theory, and their unification. Thirteen
milestones, in chronological order, supply the structural
ingredients that $\GGEP{}$ takes as primitive.

\paragraph{(1) Riemann and Bianchi --- the geometric foundation (1854--1902).}
The mathematical foundation of $\GGEP{}$ is the recognition that
a manifold's geometric content is carried by a single closed
structure. Riemann's \emph{Habilitationsschrift} of 1854
\citep{Riemann1854} introduced the intrinsically curved
$n$-dimensional manifold and identified the metric tensor
$g_{\mu\nu}$ as the carrier of geometric content. Half a century
later, Bianchi proved the closure identity that bears his name
\citep{Bianchi1902}: the Riemann curvature tensor satisfies a
differential identity $\nabla_{[\mu} R_{\nu\rho]\sigma\tau} = 0$,
equivalently $dR = 0$ in differential-form language. The Bianchi
identity is the geometric ancestor of every closure relation that
$\GGEP{}$ later imposes on the composite connection: every
modified Bianchi identity in the string-theoretic and cohort
frameworks --- including the Green--Schwarz modification of 1984
and the cohort's $\BIsix{}$ identity of \AThree{}
\citep{ArisakaA3BI6} --- is a descendant of Bianchi 1902.

\paragraph{(2) Einstein --- the equivalence principle as geometry-as-physics (1907--1916).}
Einstein's equivalence principle \citep{Einstein1907} and his
geometric formulation of gravity \citep{Einstein1915,Einstein1916}
realized the physical promise that Riemann had only sketched:
locally, inertial frames remove the gravitational connection at a
point, while curvature persists as the invariant tidal field.
General relativity is the first foundational statement of
geometry-as-physics. Einstein himself credited Mach for the
philosophical impulse, and credited Levi-Civita and Ricci for the
absolute differential calculus that made the construction
technically possible. The equivalence principle --- \emph{gravity
equals inertia under a local choice of frame} --- is the conceptual
ancestor that $\AX{2}~\GGEP{}$ extends from gravity alone to gauge
structure in general.

\paragraph{(3) Noether --- symmetry, conservation, and the gauge structural identities (1918).}
Noether's 1918 paper \citep{Noether1918} contains two theorems
whose roles in the $\GGEP{}$ lineage are distinct and complementary.
\emph{Noether's first theorem}: every continuous global symmetry
of an action gives rise to a conserved current,
$\partial_\mu J^\mu = 0$. \emph{Noether's second theorem}: when
the symmetry is a local gauge symmetry, the conservation law is
replaced by an off-shell algebraic identity among the equations of
motion --- the Noether identity --- and the conserved current
becomes redundant. Noether II is the first technical recognition
that gauge symmetries are not physical symmetries but
\emph{redundancies of description}; equivalently, that the gauge
connection lives on a principal bundle rather than on the base
manifold. This structural fact --- that gauge structure is
fundamentally a bundle concept rather than a field-theoretic
decoration --- is exactly what $\AX{2}~\GGEP{}$ takes as primitive.

\paragraph{(4) Weyl --- the first attempts at unification (1918, 1929).}
Weyl's 1918 theory \citep{Weyl1918} was the first attempt to
unify electromagnetism with gravity by elevating the conformal
scaling of the metric to a local symmetry --- a beautiful idea
that turned out to be physically wrong (it predicted observable
scaling effects that are not seen) but mathematically prophetic.
Weyl's 1929 reformulation as a phase symmetry of the wavefunction
recovered electromagnetism in the modern form
$A_\mu \to A_\mu + \partial_\mu \alpha$ and introduced the word
``gauge'' in its current technical sense. Weyl's two attempts
together established two of $\GGEP{}$'s three structural
commitments: first, that the gauge connection $A_\mu$ is a one-form
on a principal bundle; second, that the principle of \emph{local
invariance} is a candidate primitive on the same footing as the
diffeomorphism invariance of GR.

\paragraph{(5) Ward, Takahashi, Slavnov--Taylor --- Noether's gauge identities in quantum field theory (1950--1972).}
Ward's 1950 paper \citep{Ward1950} discovered the quantum-loop
expression of Noether's second theorem in QED: the Ward identity
$k^\mu \Gamma_\mu(p, p+k) = e[\Sigma(p+k) - \Sigma(p)]$ relating
the photon-vertex function to the electron self-energy. Takahashi
\citep{Takahashi1957} extended Ward's identity to its general
form. Slavnov \citep{Slavnov1972} and Taylor \citep{Taylor1971}
generalized the construction to non-Abelian gauge theory, producing
the Slavnov--Taylor identities that BRST cohomology
(Becchi--Rouet--Stora--Tyutin) would later cast as off-shell
algebraic relations among the renormalized vertex functions. These
identities are the \emph{operational diagnostic} that the gauge
bundle's geometric content registers correctly at the boundary
effective-action level; they are what makes quantum gauge theory a
self-consistent operational construct, and they are the bridge
between $\AX{2}~\GGEP{}$'s bulk-side principle and
$\AX{3}~\OGGEP{}$'s boundary-side operational dictionary. Every
textbook proof that QED is renormalizable rests on the
Ward--Takahashi identity; every textbook proof that the Standard
Model is renormalizable rests on the Slavnov--Taylor identity.
These identities are not optional decorations; they are the price
of admitting $\GGEP{}$ into a quantum field theory.

\paragraph{(6) Yang--Mills --- non-Abelian gauge theory (1954).}
Yang and Mills \citep{YangMills1954} extended Weyl's local-phase
symmetry from the Abelian $\mathrm{U}(1)$ to non-Abelian Lie groups,
opening the mathematical structure that the Standard Model would
later inhabit. Glashow, Weinberg, Salam, and the architects of QCD
made this structure physical. Throughout this period, the language
of principal bundles and connections --- developed by Kobayashi and
Nomizu \citep{KobayashiNomizu1963} and codified in the modern
textbook of Nakahara \citep{Nakahara2003} --- became the standard
mathematical vocabulary for gauge theory. Yang--Mills is the
moment at which ``gauge theory'' became a structural primitive of
theoretical physics rather than a special feature of
electromagnetism.

\paragraph{(7) Ehresmann, Wu--Yang, and Daniel--Viallet --- gauge theory \emph{is} fiber-bundle geometry (1950--1980).}
The structural ingredients above acquired their exact mathematical form once gauge theory was recognized as the differential geometry of fiber bundles --- which is the precise modern statement of $\AX{2}~\GGEP{}$. Ehresmann's 1950 definition of a connection on a fiber bundle \citep{Ehresmann1950} supplied the object that $\GGEP{}$ takes as primitive: a gauge field is a \emph{connection}, its field strength is the \emph{curvature} of that connection, and a gauge phase is a \emph{holonomy}.

Wu and Yang's 1975 dictionary \citep{WuYang1975} translated this into physics, identifying the non-integrable phase factor --- the Wilson line --- with bundle holonomy and exhibiting the Dirac monopole as a topologically non-trivial $U(1)$-bundle whose quantized charge is the bundle's Chern number. By 1980 the entire Standard-Model gauge group $\SU(3)\times\SU(2)\times U(1)$ was routinely written as a principal bundle over spacetime --- the gauge fields as the connection, the matter fields as sections of associated bundles \citep{DanielViallet1980}.

This is the rigorous form of geometry-gauge equivalence: ``geometry equals gauge'' means that gravity (the connection on the frame bundle) and the gauge forces (connections on internal bundles) are objects of one kind, related through their curvature. $\AX{2}~\GGEP{}$ adopts this established formalism in full; its own contribution is \emph{constitutional} --- to elevate the equivalence to an axiom from which the admissible bulk is \emph{derived} --- and, in the bulk-side companion \citep{ArisakaA2GG6}, to force the Abelian fiber into a literal non-spatial dimension while leaving the non-Abelian sector as internal bundle curvature.

\paragraph{(8) Dirac, Aharonov--Bohm, Berry --- topology and holonomy carry physical content (1931--1984).}
A separate thread, running in parallel with the algebraic
development of gauge theory, established that \emph{topology and
holonomy} carry physical content even where the local field
strength vanishes. Dirac \citep{Dirac1931} discovered that the
consistency of the magnetic monopole forces the quantization of
electric charge --- the first hint that gauge structure is
fundamentally topological. Aharonov and Bohm \citep{AharonovBohm1959}
demonstrated experimentally that an electron interferometer
threaded by a flux-carrying solenoid records a measurable phase
shift even though the electron travels through a region of zero
field; the holonomy of the gauge connection around the solenoid
was the only carrier of the effect. Berry \citep{Berry1984}
discovered the geometric phase that bears his name --- a topological
invariant of adiabatic transport in any quantum system,
generalizing the Aharonov--Bohm phase to arbitrary parameter
spaces. Wilczek and Zee subsequently extended Berry's construction
to non-Abelian holonomy. Together, these results established that
the gauge connection is not an auxiliary mathematical convenience
but a \emph{directly observable geometric object}.

\paragraph{(9) Wilson --- the holonomy as a non-perturbative order parameter (1974).}
Wilson \citep{Wilson1974} introduced the Wilson loop
$W(C) = \Tr\,\mathcal{P}\exp(\oint_C A)$ in lattice gauge theory
as the non-perturbative order parameter for confinement: in the
confined phase the Wilson loop obeys an area law
$\langle W(C)\rangle \sim e^{-\sigma\,\mathrm{Area}(C)}$, in the
deconfined phase a perimeter law. The Wilson loop is the
operational gauge-invariant observable that registers the holonomy
of the gauge connection around any closed curve, and it is the
first construction in which the geometric content of the connection
becomes a \emph{quantitative, measurable invariant of the
non-perturbative quantum theory}. Every later development of
$\GGEP{}$ --- including the cohort framework's use of the
Wilson-line phase $\Theta_H$ as the hidden variable behind quantum
measurement --- descends from Wilson's reading of the holonomy as a
physical observable.

\paragraph{(10) Hosotani --- gauge symmetry breaking via Wilson-line holonomy on compact directions (1983, 1989).}
Hosotani \citep{Hosotani1983,Hosotani1989} introduced the
mechanism by which a gauge symmetry in a higher-dimensional theory
is broken to a lower-dimensional subgroup by a non-vanishing vacuum
expectation value of the Wilson line wrapping a compact direction.
The Hosotani mechanism is, in retrospect, the first concrete
extra-dimensional realization of $\GGEP{}$: it shows that the
holonomy of the gauge connection around a compact direction
\emph{is itself a dynamical field} whose vacuum value selects the
low-energy gauge content. The cohort framework's use of the
$S^1_\theta$ holonomy phase as the carrier of $\AX{3}~\OGGEP{}$'s
hidden-variable content is a direct descendant of the Hosotani
construction; we credit Hosotani as the original architect of the
\emph{compact-holonomy-as-physical-field} reading that $\GGEP{}$
axiomatizes.

\paragraph{(11) Green and Schwarz --- bulk-side classical absorption of the boundary anomaly (1984).}
Green and Schwarz \citep{GreenSchwarz1984} discovered that in
10-dimensional supersymmetric gauge theory the gauge anomaly cancels
for $\mathrm{SO}(32)$ and $E_8 \times E_8$ via a modification of
the Bianchi identity for the 3-form field strength:
$dH = \mathrm{tr}(F\wedge F) - \mathrm{tr}(R\wedge R)$. In the
conventional string-theoretic narrative, this is \emph{anomaly
cancellation in the 10D bulk}; in the constitutional reading
developed in Part~V (and in the cohort companion \AThree{}
\citep{ArisakaA3BI6}), it is the first concrete realization of
$\AX{2}~\GGEP{}$ in the modern era --- a \emph{bulk-side classical
absorption} of the closure identity descended from Bianchi 1902.
The Green--Schwarz mechanism is what string theory had to write down
once it had committed to a 10D bulk with extended-object dynamics;
it is also what $\AX{2}~\GGEP{}$ \emph{forces} on any admissible
bulk regardless of whether the bulk contains strings. Part~V's
classification locates the Green--Schwarz construction at
Stratum~2, $\{\AX{1}, \AX{2}\}$; the cohort framework's
$\BIsix{}$ identity \citep{ArisakaA3BI6} is the analogous Bianchi
closure on the six-dimensional bulk.

\paragraph{(12) Maldacena --- AdS/CFT as $\GGEP{}$ on an AdS bulk (1997).}
Maldacena's AdS/CFT correspondence
\citep{Maldacena1998}, together with the
bulk-to-boundary dictionaries of Gubser--Klebanov--Polyakov
\citep{Gubser1998} and Witten \citep{Witten1998}, proposed that bulk
gravity on $\mathrm{AdS}_5 \times S^5$ is equivalent to boundary
$\mathcal{N}=4$ super-Yang--Mills conformal field theory on the
4D boundary. AdS/CFT is the first complete realization of a
geometry-gauge unification with a \emph{fixed bulk-to-boundary map}:
every bulk supergravity field corresponds to a specific boundary
gauge-invariant operator, and the correspondence is exact in the
planar limit. In the constitutional reading developed in Part~V,
AdS/CFT is the $\mathsf{GGEP\text{-}closed}$ content applied to an
AdS bulk --- a structural theorem of $\AX{2}~\GGEP{}$ applied to
the AdS solution class, not a string-theoretic novelty. What
AdS/CFT lacks, to become operational in the cohort sense, is
$\AX{3}~\OGGEP{}$ (the projection that produces records as opposed
to a unitary equivalence) and $\AX{4}~\MSMF{}$ (the uniqueness
principle that selects a single admissible bulk from the moduli
space). $\GGEP{}$ is the axiom from which AdS/CFT descends as a
structural theorem; the cohort framework adds the two further
axioms that AdS/CFT alone does not provide.

\paragraph{(13) Synthesis --- $\GGEP{}$ as the axiomatic distillation.}
Remarkably, the axiomatic content of $\AX{2}~\GGEP{}$ is located
not in any single ingredient above but in a deliberate
\emph{choice of primitive}: instead of taking ``metric plus separate gauge fields''
as independent inputs, $\GGEP{}$ insists on a unified
equivalence-principle statement for a single composite object, the
geometry-plus-gauge connection. Each of the twelve historical
milestones above contributes a structural ingredient: Bianchi the
closure identity; Einstein the equivalence-principle template;
Noether the bundle-redundancy reading; Weyl the local-invariance
principle; Ward, Takahashi, and Slavnov--Taylor the quantum
operational diagnostics; Yang--Mills the non-Abelian extension; Ehresmann, Wu--Yang, and Daniel--Viallet the fiber-bundle formalism in which a gauge field is a connection and a gauge phase a holonomy;
Dirac, Aharonov--Bohm, and Berry the topological observability;
Wilson the non-perturbative order parameter; Hosotani the
compact-holonomy mechanism; Green--Schwarz the bulk-side classical
absorption; and Maldacena the explicit bulk-to-boundary
correspondence. Part~V of the present paper locates each of these
milestones inside the constitutional tower: Bianchi at the
geometric root; Einstein at Stratum~2 specialized to gravity alone;
Yang--Mills at Stratum~2 specialized to gauge alone; Green--Schwarz
and AdS/CFT at Stratum~2 in the closed geometry-plus-gauge regime.
$\GGEP{}$ is the axiom that all of them are theorems of. After
all, the deepest move a foundational document can make is to take
the structural content of forty years of work by the broader
physics community, recognize it as a single principle, and write
that principle as one axiom of a four-axiom tower.

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

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


The operational/open extension of $\GGEP{}$ in $\AX{3}~\OGGEP{}$
is the axiomatic distillation of nearly a century of work on
quantum measurement, open-system dynamics, and bulk-to-boundary projection. Fifteen chronological milestones --- the longest
lineage among the four axioms, reflecting the unusual depth and
duration of the measurement debate --- supply the structural
ingredients that $\AX{3}~\OGGEP{}$ takes as primitive. Each
milestone is also located in Part~V's constitutional tower
(\figref{fig:A1-constitutional-flowchart}): the Stratum-3
$\mathsf{OGGEP\text{-}open}$ frameworks --- the QM sibling, the
thermodynamic sibling, and the black-hole sibling --- all live
within the operational content this Section credits.

\paragraph{(1) Heisenberg --- matrix mechanics and the uncertainty principle (1925, 1927).}
Heisenberg's 1925 paper \citep{Heisenberg1925} introduced matrix
mechanics, the first complete formal apparatus of the new quantum
theory. His 1927 uncertainty relations \citep{Heisenberg1927}
established that conjugate observables cannot be jointly
registered with arbitrary precision; the uncertainty principle is
the first operational statement that the boundary record is
structurally constrained by the choice of measurement, an
anticipation of the $\AX{3}~\OGGEP{}$ instrument formalism.

\paragraph{(2) Schr\"odinger and Born --- wave mechanics and the probabilistic interpretation (1926).}
Schr\"odinger's wave mechanics \citep{Schrodinger1926} provided
the second complete formalism, equivalent to Heisenberg's by
Dirac's transformation theory \citep{Dirac1930}. Born's
probabilistic interpretation of the wavefunction
\citep{Born1926} introduced the Born rule
$p_\alpha = |\langle\alpha|\psi\rangle|^2$ as the bridge between
the bulk-side wavefunction and the boundary-side registered
outcome. In $\AX{3}~\OGGEP{}$'s reading, Born's rule is not a
postulate but a theorem of the operational dictionary; the
AX-3 lineage credits Born for the first rigorous formulation of
what the operational compiler must produce.

\paragraph{(3) Bohr --- complementarity and the inseparability of apparatus from observation (1928).}
Bohr's 1928 Como/Solvay address \citep{BohrComplementarity1928}
introduced the complementarity principle: a quantum measurement
necessarily involves an apparatus described classically, and the
complementary aspects of a quantum object cannot be jointly
registered. Bohr's principle is the first foundational statement
that the bulk-side quantum description and the boundary-side
operational record are \emph{not the same object}.
$\AX{3}~\OGGEP{}$ takes this structural fact as a primitive of
the constitution rather than as an interpretational stance: the
boundary is where records live, the bulk is where the closed
evolution lives, and the projection between them is the
operational dictionary $\Dmap$.

\paragraph{(4) Von Neumann --- the projection postulate and the cut (1932).}
Von Neumann's \emph{Mathematische Grundlagen} of 1932
\citep{vonNeumann1932} formalized the boundary projection as a
separate postulate, sitting awkwardly alongside the
Schr\"odinger equation as the now-famous ``cut.'' The von
Neumann cut has been the seed of every measurement-problem debate
since. $\AX{3}~\OGGEP{}$ dissolves the cut by promoting the
projection to the constitutional level: the bulk-to-boundary
functor $\Dmap$ is always-on, gauge-covariant, and CPTP, and the
von Neumann projection is the macroscopic limit of $\Dmap$ acting
on a Geometric Attractor with a record-amplification chain.

\paragraph{(5) Einstein--Podolsky--Rosen --- the completeness debate (1935).}
The EPR paper \citep{EPR1935} posed the completeness problem of
quantum mechanics: whether the wavefunction is a complete
description of physical reality or whether further hidden
variables are required. In the AX-3 lineage, the EPR argument
is the historical pivot at which \emph{what gets registered as a
measurement outcome} became a central foundational question.
$\AX{3}~\OGGEP{}$ resolves the completeness question by
recognizing that completeness is a property of the operational
compiler $\Dmap$, not of the bulk Hilbert space alone.

\paragraph{(6) Wigner --- Hilbert-space irreps and Wigner's friend (1939, 1961).}
Wigner's 1939 paper \citep{Wigner1939} classified the irreducible
unitary representations of the Poincar\'e group, organizing
particle types by mass and spin as the two Casimir invariants of
the irrep. Wigner's 1961 ``Wigner's friend'' thought experiment
\citep{WignerFriend1961} generalized the von Neumann cut from a
single observer to a chain of observers, forcing the question of
where exactly the bulk-to-boundary projection takes place.
$\AX{3}~\OGGEP{}$ resolves Wigner's friend by definitional fiat:
the projection is the always-on bulk-to-boundary functor
$\Dmap$, not a process tied to any particular observer.

\paragraph{(7) Bell --- locality and hidden variables (1964).}
Bell's 1964 paper \citep{Bell1964} produced the inequality that
made the EPR completeness debate quantitatively testable: a wide
class of locally causal hidden-variable theories make predictions
distinguishable from those of standard quantum mechanics.
$\AX{3}~\OGGEP{}$ inherits Bell's operational reading: the
boundary record is what physics speaks of, and Bell-violating
correlations within the Tsirelson bound are admissible because
the boundary is the bulk-to-boundary projection shadow of a bulk that
remains local in higher dimension.

\paragraph{(8) Stinespring, Davies--Lewis, Kraus, Ozawa --- the instrument formalism (1955--1984).}
Stinespring \citep{Stinespring1955} proved that every completely
positive map admits a dilation to a unitary on a larger Hilbert
space --- the bridge between ``collapse'' and ``unitary evolution
$+$ tracing out the environment'' that the founders had lacked.
Davies and Lewis \citep{DaviesLewis1970} introduced the modern
instrument formalism, in which a measurement is a CP-map-valued
random-variable rather than a projection. Kraus
\citep{Kraus1971,Kraus1983} gave the operator-sum representation
that has become the working tool of quantum information theory;
Ozawa \citep{Ozawa1984} extended the instrument formalism to
sequential measurement and quantum operations. These four
constructions together supply the mathematical machinery that
$\AX{3}~\OGGEP{}$'s operational dictionary $\Dmap$ inherits as
a structural primitive.

\paragraph{(9) GKLS and Choi--Jamio\l{}kowski --- the generators of CP semigroups (1975--1976).}
Gorini--Kossakowski--Sudarshan and Lindblad
\citep{Gorini1976,Lindblad1976} classified the generators
of completely positive Markovian semigroups, producing what is
now called the GKLS or Lindblad generator. Choi
\citep{Choi1975} characterized complete positivity via the
Choi--Jamio\l{}kowski isomorphism. The GKLS theorem is the
operational backbone of $\AX{3}~\OGGEP{}$'s admissible boundary
evolution: any time-translation-covariant Markovian channel
consistent with locality, complete positivity, and trace
preservation must be GKLS. By the late twentieth century the
textbooks of Breuer--Petruccione \citep{BreuerPetruccione2002}
and Nielsen--Chuang \citep{NielsenChuang2010}
had codified all of this into a single coherent operational
language.

\paragraph{(10) Zeh, Joos, Zurek --- the decoherence program (1970--2003).}
Zeh \citep{Zeh1970}, Joos and Zeh \citep{JoosZeh1985}, and
Zurek \citep{Zurek1981,Zurek2003} traced the appearance of
classical-looking pointer states to the entanglement of the
system with its environment; Schlosshauer \citep{Schlosshauer2005}
surveyed the state of the field at the turn of the century.
Decoherence does not by itself solve the measurement problem ---
as Bell sharply noted in \citep{Bell2004}, it only shifts the
cut to a more convenient place --- but it clarifies why classical
records appear in macroscopic measurements. The Caldeira--Leggett
model \citep{CaldeiraLeggett1983} and the Feynman--Vernon
influence-functional formalism \citep{FeynmanVernon1963} provided
concrete model systems within which the system-plus-environment
story could be computed. In the cohort framework, decoherence and
einselection are theorems of $\AX{3}~\OGGEP{}$ acting on a bulk
with a boundary record-amplification chain; the decoherence
program supplies the operational language in which those
theorems are stated.

\paragraph{(11) Bekenstein and Hawking --- black-hole thermodynamics (1973, 1975).}
Bekenstein \citep{Bekenstein1973} proposed that the entropy of a
black hole is proportional to its horizon area; Hawking
\citep{Hawking1975} derived black-hole evaporation and the
Hawking temperature $T_H = \hbar c^3/(8\pi G M k_B)$. Together,
these results established that a black hole \emph{has} an
operational thermodynamic description --- entropy, temperature,
first law --- accessible to a boundary observer outside the
horizon. In Part~V's classification, black-hole thermodynamics
is one of the three siblings of openness at Stratum 3; the
Bekenstein--Hawking story is the foundational evidence that
$\AX{3}~\OGGEP{}$ applies to gravitational as well as quantum
systems.

\paragraph{(12) Davies--Fulling--Unruh --- observer-dependent thermality (1976).}
Davies--Fulling--Unruh \citep{DaviesFulling1976,Unruh1976}
established that the vacuum state seen by an inertial observer
appears as a thermal bath to a uniformly accelerated observer.
The Unruh effect is the canonical demonstration that the
operational boundary record depends on the boundary's motion ---
i.e., on which $\Dmap$ is used --- and not only on the bulk
state. $\AX{3}~\OGGEP{}$ takes this dependence as structural:
different boundary frames furnish different operational
dictionaries, all CPTP, all gauge-covariant, all compatible with
the bulk evolution.

\paragraph{(13) 't~Hooft, Susskind, Maldacena --- the holographic principle (1993--1998).}
't~Hooft \citep{tHooft1993} and Susskind \citep{Susskind1995}
proposed the holographic principle: the information content of
a region is bounded by its boundary area in Planck units.
Maldacena's AdS/CFT correspondence \citep{Maldacena1998}, Witten's bulk-to-boundary dictionary
\citep{Witten1998}, and the Gubser--Klebanov--Polyakov dictionary
\citep{Gubser1998} made the principle concrete in a
specific ten-dimensional setting. $\AX{3}~\OGGEP{}$ uses
holography not as a duality between two theories but as an
\emph{operational definition of observation itself}: measurement
is the bulk-to-boundary projection that produces boundary
records. In Part~V's classification, AdS/CFT lives at Stratum 2
as the $\mathsf{GGEP\text{-}closed}$ content; $\AX{3}~\OGGEP{}$
is what promotes the AdS/CFT projection from a unitary equivalence
to a genuine CPTP open-system reduction.

\paragraph{(14) Ryu--Takayanagi and the islands formula --- holographic entanglement (2006, 2020).}
Ryu and Takayanagi \citep{RyuTakayanagi2006} expressed the
boundary CFT's entanglement entropy as the area of a minimal
surface in the bulk AdS geometry. Almheiri--Engelhardt--
Marolf--Maxfield \citep{Almheiri2020} and
Penington--Shenker--Stanford--Yang \citep{Penington2020} extended
this construction to the evaporating black hole, showing that the
Page curve of black-hole evaporation can be recovered from
semiclassical computations by including ``island'' regions in the
entanglement wedge. In the AX-3 lineage, Ryu--Takayanagi and the
islands formula are the most direct twenty-first-century
confirmation that $\AX{3}~\OGGEP{}$ is the right operational
reading of the holographic principle: the boundary's operational
content (entanglement entropy) is computed directly from bulk
geometric data via the projection $\Dmap$.

\paragraph{Synthesis --- $\AX{3}~\OGGEP{}$ as the axiomatic distillation.}
The constitutional move of $\AX{3}~\OGGEP{}$ is not the
introduction of any single ingredient above but the recognition
that fifteen mature traditions --- quantum measurement (Heisenberg,
Schr\"odinger, Born, Bohr, von Neumann, EPR, Bell, Wigner), the
open-systems formalism (Stinespring, Davies--Lewis, Kraus, Ozawa,
GKLS, Choi), the decoherence program (Zeh, Joos, Zurek), and
the holographic principle (Bekenstein, Hawking, Davies--Fulling--
Unruh, 't~Hooft, Susskind, Maldacena, Witten, Ryu--Takayanagi,
Almheiri and Penington) --- are all expressing one
and the same structural fact: physics is what the boundary
observer registers via a CPTP, gauge-covariant projection of the
bulk. Part~V of the present paper locates the three siblings of
openness at Stratum 3 within this synthesis: the quantum-mechanics
sibling with the Born rule and Bell--CHSH violation, the
classical-thermodynamics sibling with the Friston free-energy
principle, and the black-hole sibling with the Page curve \citep{Page1993} and
the islands formula --- all three families of openness sitting at the
same constitutional stratum because all three are operational
projections of an underlying bulk evolution. After all, the
deepest move a foundational document can make on the measurement
problem is to recognize that ninety years of debate dissolve once
the operational projection is named as a constitutional axiom
rather than carried as a craft-tradition interpretation.

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

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


The maximum-symmetry-minimum-freedom axiom $\AX{4}~\MSMF{}$ is
the axiomatic distillation of two lineages that meet, in our
view, in the present constitution for the first time: the
long history of \emph{symmetry as the deep content of physical
law}, and the older philosophical lineage of \emph{minimality
as the discipline of admissible explanation}. Thirteen
milestones, in chronological order, supply the structural
ingredients. Each milestone is also located in Part~V's
constitutional tower (\figref{fig:A1-constitutional-flowchart}):
the Stratum-4 $\mathsf{Full\text{-}Constitution}$ frameworks
(the Standard Model with its 28 parameters collapsed to zero,
and $\Lambda$CDM cosmology with $\Omega_m = 1/\pi$ forced) are
the cohort realizations of the discipline that $\AX{4}~\MSMF{}$
axiomatizes.

\paragraph{(1) Occam --- the razor as the parent of minimality (14th c.).}
Occam's razor (William of Occam, fourteenth century) is the
philosophical ancestor of the minimum-freedom half of
$\AX{4}~\MSMF{}$: of two explanations, prefer the one that posits
fewer entities. The razor is the medieval anchor of every later
naturalness argument in physics, and $\AX{4}~\MSMF{}$ is the
axiomatic refinement of the razor: no continuous parameter may
remain unforced, no convention may masquerade as physics, no
free dial that has not been earned by symmetry or topology is
admissible.

\paragraph{(2) Galileo --- the principle of relativity as the first symmetry (1632).}
Galileo's \textit{Dialogo} of 1632 \citep{Galilei1632} is the
first explicit physical theory built on a symmetry primitive:
the laws of mechanics are the same in every inertial frame.
Galileo's move --- writing the symmetry first, then deriving the
laws --- is the historical ancestor of the maximum-symmetry half
of $\AX{4}~\MSMF{}$.

\paragraph{(3) Newton --- inertial-frame symmetry and the first law (1687).}
Newton's \emph{Principia} \citep{Newton1687} formalized Galileo's
principle as the first law of motion: every body persists in its
state of uniform motion unless acted on by a force, which is the
operational statement that inertial frames are physically
indistinguishable. Newton's first law is the symmetry primitive
of classical mechanics; the second and third laws follow only
once the first has fixed the kinematic frame.

\paragraph{(4) Maupertuis, Euler, Lagrange, Hamilton --- the principle of least action (1744--1834).}
Maupertuis's 1744 principle of least action introduced the
variational discipline: among admissible motions, nature selects
the one that extremizes a single scalar functional. Euler and
Lagrange developed the variational formalism; Hamilton
\citep{Hamilton1834} cast the same principle in canonical form.
The variational discipline is the operational expression of the
minimum-freedom half of $\AX{4}~\MSMF{}$: among admissible
trajectories, the one nature selects is the one that minimizes
the action, with no continuous knob left for the modeler to
adjust.

\paragraph{(5) Mach --- the relational principle (1893).}
Mach's 1893 \emph{Science of Mechanics} \citep{Mach1893}
formulated the relational principle: inertial frames are defined
relative to the distribution of distant matter, not to an
absolute background. In the AX-4 lineage, Mach's principle is
the philosophical ancestor of the anti-tuning ethos: a theory is
more honest when its physical content is determined by
relational structure rather than by adjustable absolute
parameters.

\paragraph{(6) Noether --- symmetry and conservation (1918).}
Noether's celebrated 1918 paper \citep{Noether1918} proved that
every continuous symmetry of the action gives rise to a
conservation law, and conversely. Noether's theorem is the
foundational identity that \emph{symmetry and conservation are
the same thing seen from different sides}, an identity that has
organized the formal structure of physics for over a century.
In the AX-4 lineage, Noether's theorem is the technical anchor
that makes the maximum-symmetry side of $\AX{4}~\MSMF{}$
operational rather than aesthetic: more symmetry means more
conservation laws, which means more constraints on admissible
kernels.

\paragraph{(7) Wigner --- Hilbert-space irreps as particle classifications (1939).}
Wigner \citep{Wigner1939} extended Noether's construction from
the Lagrangian to the Hilbert-space level, classifying the
irreducible unitary representations of the Poincar\'e group and
using them to organize particle types themselves; what we now
call ``mass'' and ``spin'' are simply the two Casimir labels of
a Wigner irrep. Wigner's classification is the historical
demonstration that maximal symmetry organizes the very inventory
of physical objects, not just the laws governing them.

\paragraph{(8) Yang--Mills and the Standard Model architects --- gauge symmetry primacy (1954--1974).}
The success of gauge symmetry in building the Standard Model
\citep{Glashow1961,Weinberg1967,Salam1968,YangMills1954,
GrossWilczek1973,Politzer1973,GeorgiGlashow1974} extended
Wigner's organizing role from the kinematic to the dynamic side:
the content of the strong, weak, and electromagnetic interactions
is determined, up to coupling constants, by the choice of gauge
group. The empirical lesson of seventy years of gauge theory is
that nature really is profligate in symmetry: every time the
search for ``the right symmetry'' has been pursued seriously,
nature has been at least as symmetric as the ansatz allowed.
Maximum symmetry is therefore not a stylistic preference but an
empirically supported regularity.

\paragraph{(9) Wilson and the renormalization group --- relevance vs.\ irrelevance (1971).}
Wilson's renormalization-group program \citep{Wilson1971}
introduced the distinction between relevant, marginal, and
irrelevant operators along the RG flow. In the AX-4 lineage,
the RG framework is the modern technical expression of the
minimum-freedom discipline: only relevant and marginal operators
carry low-energy content, and these are protected by symmetry or
topology; irrelevant operators die out, leaving an admissibility
filter that is more honest than naive parameter-counting.

\paragraph{(10) 't~Hooft and Susskind --- naturalness as the modern razor (1979--1980).}
't~Hooft \citep{THooftNaturalness1980} and Susskind
\citep{Susskind1979} formulated naturalness as the modern
razor: a small dimensionless number that is not protected by a
symmetry is suspicious and warrants an explanation. The
hierarchy problem in the Higgs sector, the small QCD theta
angle, and the cosmological constant all carry this signature.
$\AX{4}~\MSMF{}$ is the constitutional refinement of
naturalness: not just an aesthetic warning but an axiomatic
filter against unforced freedom.

\paragraph{(11) Weinberg --- the cosmological-constant problem (1989).}
Weinberg's 1989 review \citep{Weinberg1989CC} cataloged the
cosmological-constant problem as the sharpest single example of
the minimum-freedom failure: a parameter whose observed value is
$\sim 10^{120}$ orders of magnitude smaller than its naive
estimate, with no symmetry protection in the Standard Model.
In Part~V's classification, the cosmological-constant scale
$m_\Lambda \approx 6$~meV is one of the Stratum-4 numbers that
the cohort framework derives from the integer $\OGNfive{}$
ledger of \AFour{} \citep{ArisakaA4Euclid} and \ATen{}
\citep{ArisakaA10Cosmos} --- the constitutional answer to the
Weinberg problem.

\paragraph{(12) Bousso, Polchinski, Susskind, Douglas --- the string landscape (2000--2007).}
Bousso and Polchinski \citep{BoussoPolchinski2000} estimated
that the discretuum of four-form flux vacua in string theory
produces of order $10^{500}$ admissible solutions; Susskind
\citep{Susskind2003} argued for an anthropic reading; Douglas
\citep{Douglas2003} and Polchinski \citep{Polchinski2004Landscape}
developed the technical apparatus. The string-landscape
discussion sharpened the question: with $10^{500}$ or more
compactifications available, is any selection principle strong
enough to pick a single vacuum? In Part~V's classification, the
$10^{500}$ landscape is the diagnostic signature of
$\mathsf{GGEP\text{-}closed}$ without $\AX{4}~\MSMF{}$; adding
$\AX{4}~\MSMF{}$ collapses the count to exactly one.

\paragraph{(13) Vafa --- the swampland as the consistency filter (2005).}
Vafa's 2005 swampland proposal \citep{Vafa2005} argued that not
every consistent-looking low-energy effective theory can be
UV-completed to a string theory; the consistent ones form the
``landscape,'' the inconsistent ones the ``swampland.'' In the
AX-4 lineage, the swampland program is the recognition that
consistency with a deeper UV structure is itself an admissibility
filter; $\AX{4}~\MSMF{}$ extends this logic from the
string-theoretic UV to the full four-axiom constitution.

\paragraph{Synthesis --- $\AX{4}~\MSMF{}$ as the axiomatic distillation.}
Fundamentally, what $\AX{4}~\MSMF{}$ adds beyond the ingredients
cataloged above is the axiomatic move of joining
the two ancient lineages --- maximum symmetry (Galileo, Newton,
Noether, Wigner, Yang--Mills, Standard Model) and minimum
freedom (Occam, Maupertuis, Mach, Wilson, 't~Hooft, Susskind,
Weinberg) --- into a single axiomatic clause: among all
candidate theories satisfying $\AX{1}$, $\AX{2}$, $\AX{3}$,
Nature selects the one with maximal symmetry and minimal
non-forced freedom. Part~V of the present paper locates the
Stratum-4 frameworks within this synthesis: the Standard Model
with its 28 free continuous parameters collapsed to zero, and
$\Lambda$CDM cosmology with $\Omega_m = 1/\pi$ forced from the
integer $\OGNfive{}$ ledger. The $10^{500}$-fold collapse in
admissible vacua from Stratum 2 to Stratum 4 (Part V,
\S\ref{sec:V-quantitative-consequences}) is the
constitution's signature quantitative payoff of
$\AX{4}~\MSMF{}$. After all, what makes a body of knowledge a
science is not the cleverness of its calculations but the
honesty of its rules; $\AX{4}~\MSMF{}$ is the rule that forbids
the unforced dial.

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

\section*{Closing remarks on Part~II}

Four statements are now on the page. It is worth asking what they have
in common, because read one at a time they can look like four unrelated
preferences.

They are all refusals. $\AX{1}$ refuses to let an effect precede its
cause. $\AX{2}$ refuses to treat geometry and gauge as separate kinds
of stuff. $\AX{3}$ refuses to let observation be a primitive act
outside the physics. $\AX{4}$ refuses to let a theory carry a number
that nothing forced it to carry. None of the four adds an ingredient to
nature. Each one takes away a freedom that theorists had been quietly
enjoying.

This is the opposite of how foundational proposals usually work, and it
is deliberate. The temptation, when physics gets stuck, is to add: a new
field, a new dimension, a new symmetry, a new interpretation. Adding
always works, which is exactly the problem --- a framework that can
absorb any result by adding something has stopped making contact with
the world. Taking away is harder and more informative. Every freedom
removed is a place the theory can now be caught.

There is a physical picture behind this that we find genuinely
persuasive. Dip a wire loop in soap solution and a film forms. The film
does not select its shape from a catalog; the shape is what remains
once the constraints have had their say. Change the loop and the
surface changes, not because it decided to, but because nothing else
was available. Physics has spent a century assuming that nature is
mostly catalog --- that the particle masses, the coupling constants,
the cosmological parameters are selections from a menu, and that our
job is to measure which item was chosen. $\AX{4}$ asks whether the
menu was ever there. It proposes that the constants of nature are the
soap film: not chosen, but left over.

We may of course be wrong about that, and the constitution is written
so that being wrong is detectable rather than absorbable. But notice
that the question is now sharp enough to lose. Before $\AX{4}$ was
stated as a rule, ``why these constants?'' was a mood. After it, the
answer is either a derivation or a failure, and there is no third
option.

What the four axioms do not yet supply is a working filter. They say
what kind of object a theory is; they do not say which such objects
survive. A theory can honor all four and still leak probability, still
depend on a hand-tuned normalization, still make no statement any
instrument could contradict. Part~III states the seven postulates that
close that gap.

\GGPart[The Seven Postulates]{The Seven Postulates}{GI, \UEP{}, AF, CP, RG, \OGN{}, MF\\
 Ordered Admissibility Filters for $\GGTheory{}$ Theories}

\noindent
If the axioms are the grammar, the postulates are the proofreading.
Part~II said what kind of object a theory has to be. Part~III says
which such objects survive contact with the world.

The difference is practical. You can write a grammatically perfect
sentence that is false, and you can write a theory that respects all
four axioms and is still not physics --- because its dynamics leak
probability, or its predictions depend on a constant somebody chose by
hand, or it makes no statement that any instrument could contradict.
The seven postulates are seven such filters, and a candidate theory
must pass all of them.

In order, and in plain terms: $\POS{1}~\GI{}$ requires that the same
symmetry which holds for a single electron still hold for a mole of
them, so that a bookkeeping convention chosen in the laboratory never
leaks into a prediction. $\POS{2}~\UEP{}$ requires one variational
engine rather than several --- the principle that makes light take the
fastest path should be the same one that makes a cell minimize its
surprise. $\POS{3}~\AF{}$ requires that the quantum consistency
conditions called anomalies cancel exactly, not approximately; a theory
with a surviving anomaly is not merely inaccurate, it is incoherent.
$\POS{4}~\CP{}$ requires that probabilities stay probabilities under
every allowed evolution, including the messy ones where the system is
coupled to a bath. $\POS{5}~\RG{}$ requires that the answer not depend
on how finely you chose to look. $\POS{6}~\OGN{}$ requires that the
numbers left over be integers and simple ratios rather than free
decimals. And $\POS{7}~\MF{}$ requires that every central claim name,
in advance, the measurement that would kill it.

That last one is the postulate we would defend most stubbornly. A
filter that lets everything through is not a filter.

Each postulate appears in the same format as the axioms. The
historical section tracing its lineage from the earliest recognizable
anchor to the modern mathematical form --- closing with the
\emph{axiomatic move} it makes, the moment at which a working
practice is promoted to a stated rule --- is gathered with the other
six in \S\ref{sec:history-partiii}, which closes this Part.

\medskip

\section{Overview of the seven postulates}
\label{sec:postulates}

\subsection{Role of the postulates: runtime admissibility filters}
\label{sec:postulates_role}

Part~II supplied the four axioms as the
constitutional \emph{primitives} of $\GGTheory{}$: $\AX{1}~\LC{}$
fixed the causal arrow, $\AX{2}~\GGEP{}$ defined the bulk
primitive as a single composite geometric-gauge object,
$\AX{3}~\OGGEP{}$ defined observation as an operational
bulk-to-boundary compilation, and $\AX{4}~\MSMF{}$ closed the
theory space by maximum symmetry with minimum freedom. The
postulates of the present Part now serve as runtime admissibility
\emph{filters}. They answer a different question --- given the
axiom-level syntax, which dynamics, normalizations, and
observational claims are allowed to run?

$\POS{1}$--$\POS{7}$ are house rules that every admissible
theory instance must satisfy before any domain-specific modeling
(particles, cosmology, chemistry, life, mind) is allowed to begin.
The canonical numbering is
\begin{equation}
(\POS{1},\,\POS{2},\,\POS{3},\,\POS{4},\,\POS{5},\,\POS{6},\,\POS{7})
\;=\;
(\GI{},\,\UEP{},\,\AF{},\,\CP{},\,\RG{},\,\OGN{},\,\MF{}),
\label{eq:POS_canonical_numbering}
\end{equation}
and we shall keep this ordering throughout. The
ordering is itself part of the axiomatic content: gauge
invariance is upstream of the variational principle, anomaly
freedom is upstream of admissible evolution, scale covariance is
upstream of normalization, and the measurement/falsifiability
contract is the final filter that turns a well-posed theory into
an auditable scientific claim. It is tempting to fold $\OGN{}$ and $\MF{}$ together into a single ``parsimony'' clause; here they are strictly separated --- $\OGN{}$ is normalization rigidity (no
continuous basis drift), $\MF{}$ is the registered-observable plus
kill-criterion contract.

Two terms used in the closing paragraph of every $\POS{N}$ subsection below deserve a brief note. By the \emph{axiomatic move of $\POS{N}$} we mean the constitutional content that $\POS{N}$ adds beyond the inherited historical apparatus cataloged in the corresponding subsection of the century-old history Section; the closing paragraph names that content explicitly and ties it to a downstream consequence. By \emph{Stratum~$N$} (for $N = 1,2,3,4$) we mean the $N$th level of the constitutional-axiom classification developed in detail in Part~V; the four strata are aligned with the four axioms, each Stratum activating one more axiom (Stratum~1 = $\{\AX{1}\}$, Stratum~2 = $\{\AX{1},\AX{2}\}$, Stratum~3 = $\{\AX{1},\AX{2},\AX{3}\}$, Stratum~4 = $\{\AX{1},\AX{2},\AX{3},\AX{4}\}$). Each $\POS{N}$ closing paragraph below identifies the \emph{lowest} Stratum at which the corresponding postulate first becomes a non-trivial admissibility filter; several postulates re-appear at higher strata in stronger form, and the closing paragraph records that progression.

\subsection{Minimal formal setting shared by all postulates}
\label{sec:postulates_formal_setting}

The postulates are stated so as to cover both (i) \emph{closed microscopic} descriptions (standard quantum field theory
or closed quantum mechanics), and (ii) \emph{open operational} descriptions required by \OGGEP/\GDOS.
We therefore fix a minimal common mathematical language.

\paragraph{(i) State spaces.}
On the operational boundary, a ``state'' means a normalized, positive element of a convex state space.
In quantum theory this is a density operator $\rho$ on a Hilbert space $\mathcal{H}$:
\begin{equation}
\rho \succeq 0,\qquad \Tr\rho=1.
\end{equation}
In classical probability, it is a distribution $p(x)\ge 0$ with $\int p(x)\,\dd x=1$.
In both cases, the allowed states form a convex set $\mathfrak{S}$.

\paragraph{(ii) Operations, channels, and instruments.}
An operation is a linear map on states.
A quantum channel is a completely positive trace-preserving (CPTP) map $\Phi:\rho\mapsto \Phi(\rho)$.
A measurement is specified by an \emph{instrument} $\{\mathcal{I}_m\}_{m\in\mathcal{M}}$:
each $\mathcal{I}_m$ is CP and $\sum_m\mathcal{I}_m$ is TP. The associated POVM is
$E_m=\mathcal{I}_m^\ast(\mathbb{I})$ and the Born rule reads $p(m)=\Tr(\rho E_m)$
(see \S\ref{sec:AX3}, Eqs.~\eqref{eq:Kraus}--\eqref{eq:BornInstrument}).

\paragraph{(iii) Between-intervention evolution.}
Between interventions, the operational state evolves by a one-parameter family of admissible maps
$\{\Phi_t\}_{t\ge 0}$ with semigroup property
\begin{equation}
\Phi_{t+s}=\Phi_t\circ \Phi_s,\qquad \Phi_0=\mathrm{id},
\end{equation}
consistent with \LC\ (AX-1). Under mild continuity, there exists a generator $\mathcal{L}$ such that
$\Phi_t=e^{t\mathcal{L}}$.
In quantum Markovian limits, admissibility plus complete positivity forces the \GKLS\ form
(Eq.~\eqref{eq:Lindblad}) \citep{Gorini1976,Lindblad1976,BreuerPetruccione2002}.

\paragraph{(iv) Scale and coarse-graining.}
A coarse-graining or renormalization step is modeled abstractly as a map $\mathrm{RG}_{\ell}$ acting on descriptions
(states, couplings, effective actions, or operational kernels). POS-\RG\ will require that this map commute with the
operational dictionary \OGGEP\ (AX-3). In \GG\ applications, the bulk coordinate $r$ is identified with the RG scale
as
\begin{equation}
\label{eq:rlog_POS}
r=\ln\!\left(\frac{E_{\rm GUT}}{\mu}\right),
\end{equation}
so that translation in $r$ encodes a change of effective description scale.

With this shared setting in place, we are ready to investigate
the seven postulates one by one. Each will be presented in the
same internal format we used for the axioms: a rigorous
axiomatic statement, the mathematical formulation, the first
forced consequences, a worked example drawn from a familiar
laboratory protocol, a Physics-Intuition Remark, the relation to
the companion engines $\GG{}$ and $\GDOS{}$, and a historical
background section.

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

\section{POS-1: Gauge invariance (GI) across scales: LGI/\DGI{}/GGI}
\label{sec:POS1}

\subsection{Rigorous statement (constitutional text)}
\begin{euclidpostulate}[POS-1 (\GI): gauge invariance across scales]
\label{pos:GI}
All lawful descriptions must be gauge invariant/covariant in a \emph{layered} sense:

\begin{enumerate}[leftmargin=2.2em,label=\textbf{(\GI\alph*)}]
\item \textbf{LGI (Local Gauge Invariance).} For closed microscopic dynamics, the action and quantum measure are invariant
under local gauge transformations.
\item \textbf{\DGI{} (Dynamical Gauge Invariance).} For open mesoscopic dynamics, the admissible evolution maps (or their generators)
are equivariant under gauge transformations acting on states and on the couplings/ports that define the open system.
\item \textbf{GGI (Global Gauge Invariance).} For macroscopic ledgers, globally conserved charges are invariant bookkeeping
quantities when the system is enlarged to include its exchange environment; coarse-grained ledgers must respect these invariants.
\end{enumerate}

In particular, any operational dynamics or measurement that violates gauge covariance is inadmissible in \GGTheory{}.
\end{euclidpostulate}

\subsection{Mathematical formulation of LGI (closed microscopic systems)}
Let $G$ be a Lie group with Lie algebra $\mathfrak{g}$ and generators $T^a$.
Let $A=A_\mu^a T^a\,\dd x^\mu$ be a gauge connection on spacetime $M_4$, with curvature
\begin{equation}
F = \dd A + A\wedge A.
\end{equation}
A local gauge transformation is a map $g:M_4\to G$.
On matter fields $\psi$ in representation $R$,
\begin{equation}
\psi \mapsto \psi^g = R(g)\psi,\qquad
A \mapsto A^g = gAg^{-1} - (\dd g)\,g^{-1},\qquad
F \mapsto F^g = gFg^{-1}.
\end{equation}
LGI demands invariance of the microscopic action:
\begin{equation}
S[\psi^g,A^g]=S[\psi,A].
\end{equation}

\paragraph{Derivation: Noether/Ward identities and the anomaly obstruction.}
For an infinitesimal transformation $g(x)=\exp(\alpha^a(x)T^a)$, gauge invariance implies the covariant conservation law
\begin{equation}
D_\mu J^{\mu a}=0,
\end{equation}
with $D_\mu$ the covariant derivative.
At the quantum level, the generating functional $Z[A]=\int\mathcal{D}\psi\,e^{iS[\psi,A]}$ obeys
Ward--Takahashi identities. A gauge anomaly corresponds to a failure of invariance of the measure/effective action,
and therefore spoils these identities. POS-\AF\ exists precisely to forbid this quantum obstruction
\citep{Adler1969,BellJackiw1969,GreenSchwarz1984}.

\subsection{Mathematical formulation of \DGI{} (open mesoscopic systems)}
For an open quantum system with state $\rho$, \DGI{} is the equivariance condition for the dynamical maps:
\begin{equation}
\label{eq:DGI_covariance}
\Phi_t\!\big(U_g\rho U_g^\dagger\big) = U_g\,\Phi_t(\rho)\,U_g^\dagger,\qquad \forall g\in G,\;\forall t\ge 0,
\end{equation}
where $U_g$ is the unitary representation of $g$ on $\mathcal{H}$.
In Markovian regimes $\Phi_t=e^{t\mathcal{L}}$, so \eqref{eq:DGI_covariance} implies generator covariance
$\mathcal{L}(U_g\rho U_g^\dagger)=U_g\mathcal{L}(\rho)U_g^\dagger$.

\paragraph{Constraint on \GKLS\ generators.}
Let the generator have \GKLS\ form
\begin{equation}
\label{eq:GKLS_POS1}
\dot\rho = \mathcal{L}(\rho)= -i[H,\rho]+\sum_k\left(L_k\rho L_k^\dagger-\frac{1}{2}\{L_k^\dagger L_k,\rho\}\right).
\end{equation}
A sufficient (and essentially necessary) condition for covariance is:
\begin{align}
U_g H U_g^\dagger &= H,\label{eq:Hcov}\\
U_g L_k U_g^\dagger &= \sum_\ell v_{k\ell}(g)\,L_\ell,\label{eq:Lcov}
\end{align}
where $v(g)$ is a unitary matrix acting in the space spanned by the Lindblad operators.
In the Abelian case $G=U(1)$ with generator $Q$ and $U_\alpha=e^{i\alpha Q}$, \eqref{eq:Hcov}--\eqref{eq:Lcov} reduce to
\begin{equation}
[H,Q]=0,\qquad [Q,L_k]=q_k L_k,
\end{equation}
so each dissipative channel carries a definite gauge ``charge'' $q_k$. This is the precise open-system extension
of gauge invariance: \emph{ports may exchange charge with the environment, but they must do so in a way that respects the
gauge ledger}.

\paragraph{Pedigree note: the complex apparatus of this clause.}
The supporting equations of \DGI{} are written, as everywhere on the
boundary, in complex Hilbert-space language --- the unitary
representation $U_g$, the Abelian $U_\alpha=e^{i\alpha Q}$, the GKLS
generator. That complex structure is boundary-side and derived, not
primitive: the $i$ of these displays is the manufactured
quarter-turn of the Foundations Branch (T-P4-1/2 of
\citealp{ArisakaP4Real}), and the charge decomposition
$[Q,L_k]=q_k L_k$ is a complex-eigen-operator statement whose real
form is recoverable by complexification of the underlying real
charge rotation --- the existence of that complexification being
exactly T-P4-1's content. The postulate's own demand ---
equivariance of the admissible maps under the gauge action --- is
category-neutral.

\subsection{Mathematical formulation of GGI (macroscopic ledgers)}
If a microscopic global symmetry yields a conserved charge
\begin{equation}
Q^a = \int_{\Sigma} J^{0a}\,\dd^3x,
\end{equation}
then in an open system one typically has $\frac{\dd}{\dd t}Q^a_{\rm sys}\neq 0$ because charge flows through ports.
GGI asserts that a \emph{ledger} conservation holds when system and environment are treated together:
\begin{equation}
\frac{\dd}{\dd t}\Big(Q^a_{\rm sys}+Q^a_{\rm env}\Big)=0,
\end{equation}
and that any coarse-grained macroscopic description must preserve the bookkeeping of such globally invariant ledgers.

\subsection{Parameter dictionary}
\begin{itemize}[leftmargin=2.2em]
\item $G$ : gauge group (possibly a product group) acting on microscopic fields and operational states.
\item $g(x)\in G$ : local gauge transformation (LGI); $U_g$ : induced unitary representation on $\mathcal{H}$ (\DGI{}).
\item $A$ : gauge connection 1-form; $F=\dd A+A\wedge A$ : curvature 2-form.
\item $J^{\mu a}$ : Noether current; $Q^a$ : corresponding global charge.
\item $\Phi_t$ : admissible dynamical map between interventions; $\mathcal{L}$ : generator.
\item $H$ : effective Hamiltonian in the \GKLS\ generator; $L_k$ : Lindblad (jump) operators.
\end{itemize}

\subsection{Worked example: $\mathrm{U}(1)_{\rm em}$ gauge
 invariance applied to a Lindblad master equation}
\label{sec:POS1_worked_U1em}

The cleanest worked
example of the layered $\GI{}$ structure is electromagnetic
$\U(1)_{\mathrm{em}}$ gauge invariance applied to a Markovian open
quantum system --- say a hydrogen atom coupled to the vacuum
electromagnetic field. The LGI clause says that
the closed-system Lagrangian is invariant under
$\psi \mapsto \e^{\ii q\alpha(x)}\psi$ and $A_\mu \mapsto A_\mu - \partial_\mu\alpha$;
the \DGI{} clause now demands that the GKLS generator describing the
spontaneous-emission dynamics also intertwines the $\U(1)_{\mathrm{em}}$
action.

Let $Q$ denote the electric-charge operator on the system Hilbert
space $\cH$, and let $U_\alpha = \e^{\ii\alpha Q}$ be the induced
unitary representation. $\DGI{}$ requires
\begin{equation}
\cL\bigl(U_\alpha \rho\, U_\alpha^\dagger\bigr)
\;=\;
U_\alpha\,\cL(\rho)\,U_\alpha^\dagger
\quad\text{for every}\ \alpha\in[0,2\pi),
\label{eq:DGI_em_example}
\end{equation}
which, written for a GKLS generator
$\cL(\rho) = -\ii[H,\rho] + \sum_k \bigl(L_k\rho L_k^\dagger
- \tfrac{1}{2}\{L_k^\dagger L_k,\rho\}\bigr)$, reduces in the
Abelian case to the two conditions
\begin{equation}
[H,Q] = 0, \qquad [Q,L_k] = q_k L_k.
\label{eq:DGI_em_kraus}
\end{equation}
The first condition is gauge-invariance of the effective
Hamiltonian. The second is the axiomatic content of $\DGI{}$:
each Lindblad jump operator $L_k$ must carry a definite
electromagnetic charge $q_k$, and the ports through which the
hydrogen atom exchanges photons with the vacuum must respect this
charge ledger. A jump operator with no definite charge would
correspond to a hidden non-conservation of the $\U(1)_{\mathrm{em}}$
charge across the system-environment interface --- and is forbidden.
This is what we mean by gauge invariance of
the dissipator: not only the equations of motion but also the
noise channels themselves must respect the gauge ledger.

The $\GGI{}$ clause completes the chain. In a macroscopic
description that integrates over many photons, the global charge
conservation reads
\begin{equation}
\frac{\dd}{\dd t}\bigl(Q^{\mathrm{em}}_{\mathrm{atom}}
+ Q^{\mathrm{em}}_{\mathrm{field}}\bigr) = 0,
\label{eq:GGI_em_ledger}
\end{equation}
and any coarse-grained description that loses track of either
side of this ledger is constitutionally inadmissible.
Gauge invariance is not abolished by openness --- it is relocated
from isolated conservation to ledger-consistent exchange.

\subsection{Physics intuition}

\begin{euclidremark}[Physics intuition for \POS{1}]
Roughly speaking, $\POS{1}~\GI{}$ says that physics does not
depend on local gauge choices, and that this independence persists
across all scales. $\LGI{}$ is the closed-system statement:
local gauge transformations leave the action invariant. $\DGI{}$
is the open-system extension: even in open systems, the allowed
ways to couple to an environment must respect gauge structure ---
every jump operator carries a definite gauge charge, and the noise
channels themselves intertwine the gauge action. $\GGI{}$ is the
macroscopic ledger statement: at coarse-grained scales,
conservation is not abandoned but becomes a bookkeeping constraint
across system plus environment. This layered formulation
is precisely the bridge that lets gauge principles export beyond
particle physics into chemistry, biology, and neuroscience ---
openness does not remove symmetry, it relocates it from isolated
conservation to ledger-consistent exchange.
\end{euclidremark}

\subsection{Relation to \GG\ and \GDOS}
\begin{itemize}[leftmargin=2.2em]
\item \textbf{Relation to \GG.} In \GG\ the primitive bulk structure is the composite connection $\mathcal{A}$
(\S\ref{sec:AX2}, Eq.~\eqref{eq:compositeA}); POS-\GI\ requires that all observables be built from gauge-invariant
curvature/holonomy data (Wilson loops, characteristic classes), and that the unified group structure be respected
under the bulk--boundary dictionary \citep{ArisakaA2GG6}.
\item \textbf{Relation to \GDOS.} In \GDOS\ the core dynamical object is an admissible open-system generator.
POS-\GI\ becomes the requirement that the \GKLS\ generator and the measurement instrument intertwine the relevant gauge actions:
this is precisely the meaning of \emph{dynamical gauge invariance} in open mesoscopic systems \citep{ArisakaA5GDOS,BreuerPetruccione2002}.
\end{itemize}

\subsection{Further mathematical elaboration: generator covariance and Ward identities for open dynamics}
\label{sec:POS1_further}

In the open-system (GKLS) regime, \DGI{} is most cleanly formulated at the generator level.
Let $U(g)$ be a unitary representation of a gauge group element $g$ on the system Hilbert space $\cH$.
\DGI{} requires
\begin{equation}
\mathcal{L}\!\left(U(g)\rho\,U(g)^\dagger\right)=U(g)\,\mathcal{L}(\rho)\,U(g)^\dagger,
\qquad \forall g\in G,
\label{eq:DGI_generator_cov}
\end{equation}
where $\mathcal{L}$ is the GKLS generator (POS-\CP) of the Markov semigroup.
Differentiating \eqref{eq:DGI_generator_cov} at the identity ($g=\exp(\epsilon X)$) yields an infinitesimal Ward identity:
\begin{equation}
\mathcal{L}\big([X,\rho]\big)=[X,\mathcal{L}(\rho)],
\label{eq:ward_open}
\end{equation}
which is the generator-level expression that ``gauge rotations commute with time evolution.''

For a GKLS generator written as
\begin{equation}
\mathcal{L}(\rho)=-\ii[H,\rho]+\sum_k\left(L_k\rho L_k^\dagger-\frac{1}{2}\{L_k^\dagger L_k,\rho\}\right),
\end{equation}
\eqref{eq:DGI_generator_cov} implies that $(H,\{L_k\})$ must transform covariantly:
$H\mapsto UHU^\dagger$ and $L_k\mapsto \sum_j V_{kj}(g)\,U L_j U^\dagger$ for some unitary mixing $V(g)$ on the Lindblad-operator index set.
Operationally, this is the open-system analog of the familiar requirement that the dynamics respect gauge redundancy:
\emph{not only the equations, but the dissipative noise channels themselves must not encode gauge-dependent physics.}

\section[POS-2: Unified Extremal Principle (\UEP{}): PLA $\Longleftrightarrow$ PLT/\MEPP{}/FEP]{POS-2: Unified Extremal Principle (\UEP{}):\\ PLA $\Longleftrightarrow$ PLT/\MEPP{}/FEP}
\label{sec:POS2}

\subsection{Rigorous statement (constitutional text)}
\begin{euclidpostulate}[POS-2 (\UEP): a single variational engine]
\label{pos:UEP}
There exists a \emph{single} extremal/variational principle governing lawful dynamics.

\begin{enumerate}[leftmargin=2.2em,label=\textbf{(\UEP\alph*)}]
\item \textbf{Bulk/closed form (PLA).} In closed descriptions (including the \GGEP\ bulk), physical histories are stationary points
of an action functional: $\delta \mathcal{I}_{\rm bulk}=0$.
\item \textbf{Operational/open form (PLT/\MEPP{}/FEP).} In open operational descriptions, the same variational structure appears,
after applying the operational dictionary \OGGEP\ and enforcing admissibility (POS-\CP, POS-\RG), as an extremal principle
for an operational functional $\mathcal{I}_{\rm op}$ (least time, least free energy, or extremal entropy production, depending on representation).
\item \textbf{Dictionary compatibility.} The extremal structure commutes with the bulk$\to$boundary compilation:
\begin{equation}
\label{eq:UEP_commute}
\OGGEP\!\left(\operatorname*{arg\,ext}\, \mathcal{I}_{\rm bulk}\right)=\operatorname*{arg\,ext}\,\mathcal{I}_{\rm op}.
\end{equation}
\end{enumerate}

Thus \GGTheory{}\ does not postulate separate dynamical ``engines'' for mechanics, thermodynamics, biology, or cognition; it postulates
one engine whose representations differ after openness and coarse-graining are accounted for.
\end{euclidpostulate}

\subsection{PLA in closed systems: from action to equations of motion}
For a mechanical system with generalized coordinates $q^i(t)$ and Lagrangian $L(q,\dot q,t)$,
\begin{equation}
\mathcal{I}_{\rm mech}[q]=\int_{t_1}^{t_2} L(q,\dot q,t)\,\dd t,
\end{equation}
stationarity $\delta \mathcal{I}_{\rm mech}=0$ (with fixed endpoints) yields the Euler--Lagrange equations
\begin{equation}
\frac{\dd}{\dd t}\frac{\partial L}{\partial \dot q^i}-\frac{\partial L}{\partial q^i}=0.
\end{equation}
For fields $\phi^A(x)$, $\mathcal{I}=\int \dd^d x\,\mathcal{L}(\phi,\partial\phi)$ yields
\begin{equation}
\partial_\mu\frac{\partial\mathcal{L}}{\partial(\partial_\mu\phi^A)}-\frac{\partial\mathcal{L}}{\partial\phi^A}=0.
\end{equation}

In \GG\ the bulk action is a 6D functional of the metric and the composite connection (schematically)
\begin{equation}
\label{eq:bulk_action_schematic_POS2}
\mathcal{I}_{6}=
\int_{\mathcal{M}_6}\!\dd^6x\,\sqrt{|g|}\left(\frac{1}{2\kappa^2}R
-\frac{1}{4g_5^2}\Tr(\mathcal{F}_{MN}\mathcal{F}^{MN})+\cdots\right)
+\mathcal{I}_{\rm inflow}+\mathcal{I}_{\partial},
\end{equation}
where $\mathcal{F}$ is the curvature of the composite connection $\mathcal{A}$ (\S\ref{sec:AX2}, Eq.~\eqref{eq:compositeF}).
The details belong to the \GG\ paper \citep{ArisakaA2GG6}; here we emphasize that \UEP\ demands a single bulk variational principle.

\subsection{Open-system extremals: large deviations and gradient flows}
\par\noindent\emph{\small This paragraph is written for readers with a
background in stochastic processes and large-deviation theory. Its content is a pointer to the
literature, not a step in the constitutional argument; nothing later in
this paper depends on it.}\par\medskip

A mathematically clean bridge from PLA to ``thermodynamic'' extremals is provided by large-deviation theory.
Consider a stochastic process (Langevin form)
\begin{equation}
\dot x_t=b(x_t)+\sqrt{2\epsilon}\,\eta_t,
\end{equation}
with white noise $\eta_t$ and noise strength $\epsilon$.
As $\epsilon\to 0$, the path measure concentrates on trajectories that minimize an \emph{action-like} functional
(the Onsager--Machlup functional) \citep{OnsagerMachlup1953,FreidlinWentzell1998,Touchette2009}:
\begin{equation}
\label{eq:OM_action_POS2}
\mathbb{P}[x(\cdot)]\asymp \exp\!\left[-\frac{1}{\epsilon}\mathcal{I}_{\rm op}[x]\right],
\qquad
\mathcal{I}_{\rm op}[x]=\frac{1}{4}\int_{t_1}^{t_2}\|\dot x-b(x)\|^2_{D^{-1}}\dd t+\cdots,
\end{equation}
where $D$ is the diffusion matrix and $\|v\|^2_{D^{-1}}=v^\top D^{-1}v$.
The ``most probable'' path is therefore an extremal path even in a dissipative/open setting.

A particularly important case is \emph{gradient flow}. If $b(x)=-M\nabla F(x)$ for a mobility matrix $M\succeq 0$ and a scalar
functional $F(x)$ (free energy, relative entropy, or a generalized potential), then
\begin{equation}
\dot x=-M\nabla F(x)\quad\Rightarrow\quad
\frac{\dd F}{\dd t}=-\nabla F^\top M \nabla F \le 0.
\end{equation}
Thus the same variational structure yields a monotone descent principle on the boundary.
This supplies a precise meaning for ``least free energy'' or ``maximum dissipation'' principles:
they are re-expressions of the extremal structure in an open-system metric.

\subsection{Why ``least time'' appears operationally}
Fermat's principle in optics states that the physical ray extremizes the travel time
$\mathcal{T}=\int n(x)\,\dd s/c$.
In operational open systems, a least-time structure can arise when the relevant cost functional includes an explicit
time-to-target tradeoff, and one minimizes over final time. In \GGTheory{}\ the point is not to postulate PLT separately,
but to interpret PLT-like functionals as representations of the same extremal engine once openness/retardation (\LC) and
admissibility (\CP) are imposed.

\subsection{Parameter dictionary}
\begin{itemize}[leftmargin=2.2em]
\item $\mathcal{I}_{\rm bulk}$ : bulk action functional; $\delta\mathcal{I}_{\rm bulk}=0$ gives equations of motion.
\item $\mathcal{I}_{\rm op}$ : operational functional controlling admissible open dynamics; representation may be ``time'', ``free energy'',
or ``entropy production'' depending on variables.
\item $x(t)$ : coarse-grained operational degrees of freedom; $b(x)$ : drift; $D$ : diffusion matrix; $\epsilon$ : noise strength.
\item $F(x)$ : scalar Lyapunov/ledger functional (free energy, relative entropy, generalized potential); $M$ : mobility matrix.
\end{itemize}

\subsection{Worked example: Fermat's principle as the eikonal
 limit of least action}
\label{sec:POS2_worked_fermat}

A concrete instance brings out the discipline. The cleanest worked
example of the unified extremal engine $\UEP{}$ is the derivation
of Fermat's principle of least time in optics from the underlying
least-action principle of electromagnetism. Fermat's principle
states that a light ray traversing a medium with refractive index
$n(\mathbf{x})$ extremizes the optical path length
\begin{equation}
\cT[\gamma] \;=\; \int_\gamma n(\mathbf{x})\,\dd s,
\label{eq:fermat_time}
\end{equation}
which is the geometric expression of $\PLT{}$. In closed
electromagnetic theory, by contrast, the wave equation for the
electric field $E$ in a slowly varying medium follows from the
Maxwell action $S[A_\mu] = -\tfrac{1}{4}\int F^{\mu\nu}F_{\mu\nu}\,d^4x$
through $\PLA{}$. The two principles look like separate
dynamical engines. $\UEP{}$ predicts that they are not.

The bridge runs as follows. Write the wave
equation in the eikonal ansatz
$E(\mathbf{x},t) = A(\mathbf{x})\,\e^{\ii k_0 S(\mathbf{x}) - \ii\omega t}$,
where $S$ is a scalar phase function and $k_0 = \omega/c$ is the
vacuum wavenumber. Expand in the high-frequency limit
$k_0 \to \infty$. At leading order one obtains the eikonal equation
\begin{equation}
|\nabla S(\mathbf{x})|^2 \;=\; n^2(\mathbf{x}),
\label{eq:eikonal}
\end{equation}
whose characteristics --- the integral curves of $\nabla S$ ---
are precisely the geometric optical rays. Reading
Eq.~\eqref{eq:eikonal} as a Hamilton-Jacobi equation with
Hamiltonian $H = c|\mathbf{p}|/n$ and applying the standard
Maupertuis-Jacobi reformulation, one finds that the rays
extremize exactly the time integral of Eq.~\eqref{eq:fermat_time}.
The Maxwell $\PLA{}$ in the eikonal limit \emph{is}
Fermat's $\PLT{}$ --- the same extremal engine in two
representations \citep{BornWolf1999}.

The axiomatic reading is that $\UEP{}$ does not merely
\emph{permit} this kind of equivalence; it \emph{forces} it.
An admissible theory cannot run two competing dynamical
engines for the same physical process. Once the closed-system
$\PLA{}$ is specified for a sector and the operational dictionary
$\OGGEP{}$ projects to the boundary description, the
boundary-side $\PLT{}$, free-energy descent, or maximum-entropy
production functional is forced. The companion paper
\citep{ArisakaA5GDOS} extends this argument to the full
boundary law: gradient flow on a free-energy potential is the
universal $\GDOS{}$ representation of the closed-system action
under $\OGGEP{}$, and Fermat's principle is the simplest
historical instance.

\subsection{Physics intuition}

\begin{euclidremark}[Physics intuition for \POS{2}]
Roughly speaking, $\POS{2}~\UEP{}$ says that Nature does not run
multiple dynamical engines for the same process. What appears as
``straightest path'' (geodesic, least action) in a closed
geometric description appears as ``fastest path'' (Fermat) or
``most efficient relaxation'' (least free energy, extremal
entropy production) in an open operational description. The
conceptual unification is achieved by the operational dictionary
$\OGGEP{}$ and stabilized by the admissibility constraints
$\POS{4}~\CP{}$, $\POS{5}~\RG{}$, $\POS{6}~\OGN{}$, and
$\POS{7}~\MF{}$. The same variational principle threads
through Fermat's optics, Hamiltonian mechanics, the Maupertuis
form of the action, the eikonal limit of wave optics, the
gradient-flow form of the Fokker-Planck equation, and the
free-energy descent of statistical mechanics --- a remarkable
unity that the $\GGConstitution{}$ promotes from coincidence to
axiomatic content.
\end{euclidremark}

\subsection{Relation to \GG\ and \GDOS}
\begin{itemize}[leftmargin=2.2em]
\item \textbf{Relation to \GG.} In \GG\ the bulk variational principle is written in terms of the composite connection and bulk geometry.
\UEP{} implies that geodesics, field equations, and extremal surfaces in the 6D bulk are the primitive dynamical objects
\citep{ArisakaA2GG6,Maldacena1998,Witten1998}.
\item \textbf{Relation to \GDOS.} In \GDOS\ the open-system evolution must admit a scalar ``ledger'' (Lyapunov functional)
consistent with complete positivity and gauge invariance. \UEP{} then postulates that open dynamics is selected by extremizing
(or descending) this ledger functional in the appropriate operational metric \citep{ArisakaA5GDOS,BreuerPetruccione2002}.
\end{itemize}

\subsection{Further mathematical elaboration: from least action to gradient-flow and least free energy}
\label{sec:POS2_further}

A convenient bridge between PLA (bulk) and ``least free energy'' (boundary) is the Onsager--Machlup form of path probabilities.
Consider an overdamped stochastic dynamics on a state manifold with drift $b(y)=-M\nabla F(y)$ and diffusion $D$:
\begin{equation}
\dot y = -M\nabla F(y) + \sqrt{2D}\,\eta(t),
\end{equation}
where $\eta$ is standard white noise.
For small noise, trajectories concentrate around minimizers of an effective stochastic action \citep{OnsagerMachlup1953,Seifert2012}
\begin{equation}
S_{\mathrm{OM}}[y] \approx \frac{1}{4}\int_{t_0}^{t_1}\dd t\,
\big(\dot y + M\nabla F\big)^{\!\top} D^{-1}\big(\dot y + M\nabla F\big)\;+\;\frac{1}{2}\int_{t_0}^{t_1}\dd t\,\nabla\cdot(M\nabla F),
\label{eq:OM_action}
\end{equation}
so that the \emph{most probable} path solves, to leading order,
\begin{equation}
\dot y = -M\nabla F(y).
\label{eq:gradient_flow}
\end{equation}
Equation \eqref{eq:gradient_flow} is the textbook gradient-flow law: it is literally ``steepest descent'' in the metric induced by $M$.
The Lyapunov identity
\begin{equation}
\frac{\dd}{\dd t}F(y(t)) = -(\nabla F)^{\!\top} M(\nabla F)\le 0
\label{eq:F_monotone}
\end{equation}
is then an immediate consequence, and can be read as the boundary face of an extremal principle: the system follows the path that decreases $F$ most efficiently
given the mobility constraints.
In this sense, ``least action'' persists in open systems, but the appropriate action is the large-deviation functional \eqref{eq:OM_action}
and the appropriate potential is the \GFF{} ledger $F_{\mathrm{\GFF{}}}$.

The PLT/FEP/\MEPP{} language can be understood as different operational summaries of \eqref{eq:gradient_flow} and \eqref{eq:F_monotone}
under different choices of coarse-graining and cost functionals. \UEP{}'s axiomatic role is to insist that these are not unrelated principles,
but different avatars of one variational engine once the operational dictionary is fixed.

\section{POS-3: Anomaly freedom and cancellability by inflow (AF)}
\label{sec:POS3}

\subsection{Rigorous statement (constitutional text)}
\begin{euclidpostulate}[POS-3 (\AF): anomaly freedom via inflow]
\label{pos:AF}
All admissible \GGTheory{}\ theories must be free of gauge, mixed gauge--gravitational, and global anomalies.
If anomalies arise in an effective boundary description, they must be canceled by inflow from the bulk via a single
Green--Schwarz-type mechanism, implemented as a modified Bianchi identity in the \GGEP\ bulk.
\end{euclidpostulate}

\subsection{What ``anomaly freedom'' means in the constitutional reading}
The conventional quantum-field-theoretic story --- the one repeated in
every textbook since the late 1960s --- introduces ``anomaly'' as a
boundary pathology of the quantum effective action. In a gauge theory
with partition function $Z[A]=\int\mathcal{D}\psi\,e^{iS[\psi,A]}$, gauge
symmetry requires $Z[A^g]=Z[A]$; a gauge anomaly means $Z[A^g]\neq Z[A]$,
equivalently that the effective action $\Gamma[A]=-i\ln Z[A]$ varies as
$\delta_\epsilon \Gamma[A]=\int_{M_4}\epsilon^a(x)\,\mathcal{A}^a(x)$,
with the anomaly functional $\mathcal{A}^a$ computed at one loop. An
uncanceled anomaly makes the boundary 4D theory inconsistent: it
violates unitarity and/or current conservation at the quantum level.

Roughly speaking, this is the boundary-side report. Indeed, the
constitutional reading inverts it: $\POS{3}~\AF{}$ takes the boundary
quantum ``anomaly'' to be the $\AX{3}~\OGGEP{}$ \emph{projection shadow}
of a deeper structural identity in the bulk --- a classical, topological,
purely geometric identity descended from Bianchi 1902. In the bulk, the
differential identity $d^2 = 0$ on the gauge-bundle curvatures forces a
Bianchi-type closure $dX_4 = 0$ on every admissible four-form polynomial
$X_4$ that the $\AX{2}~\GGEP{}$ composite connection produces. When the
AX-3 boundary operation projects the bulk to its 4D slice $M_4 = \partial \cM_6$,
the bulk identity becomes the requirement that the boundary effective action
respect a corresponding Ward identity for every conserved current; the
boundary quantum effective action either satisfies this Ward identity --- in
which case the theory is admissible --- or it fails to, in which case what
the textbook calls ``the anomaly'' is precisely the boundary's measurement
of its own inability to honor the upstream bulk identity. Anomaly
freedom in the constitutional reading is therefore the same kind of
statement as $\AX{1}~\LC{}$: both express the conservation of a
geometric current along a closed bulk manifold via a Bianchi-type identity
(see the Anomaly-Freedom remark below for the full geometric statement).

\subsection{Inflow as bulk-side classical absorption (modified Bianchi identity)}
The Green--Schwarz construction \citep{GreenSchwarz1984} is
conventionally introduced as a mechanism for ``canceling'' boundary
anomalies at the quantum loop level. In the constitutional reading
it is something more honest: a bulk-side \emph{classical} absorption
of the structural identity that descends from Bianchi 1902. One
introduces a 2-form $B_2$ with field strength $H_3=\dd B_2+\cdots$ and
the bulk coupling
\begin{equation}
\mathcal{I}_{\rm inflow}=\int_{\mathcal{M}_6} B_2\wedge X_4,
\end{equation}
where $X_4$ is a 4-form polynomial built from gauge and gravitational
curvatures. The bulk gauge variation of $B_2$ is absorbed by the bulk
Chern--Simons descent of $X_4$, exactly so as to maintain the bulk
classical-topological identity $dX_4 = 0$ --- the closure of the
four-form polynomial under the differential, which is the
$\AX{2}~\GGEP{}$-bundle expression of Bianchi 1902. The boundary
OGGEP projection then sees a Ward identity that the boundary effective
action is required to honor; what the standard textbook calls
``cancellation of the boundary anomaly by the bulk inflow'' is, in the
constitutional reading, the boundary's discovery that its quantum loop
merely re-measures the bulk classical identity already in force.
What is absorbed at the bulk level is therefore a classical-topological
fact, not a quantum loop. The mechanism nonetheless requires that
$H_3$ obey a modified Bianchi identity
\begin{equation}
\label{eq:BI6_general}
\dd H_3 = \frac{1}{16\pi^2}\Big(a\,\Tr(F\wedge F)+b\,F_B\wedge F_B+c\,\Tr(R\wedge R)\Big),
\end{equation}
with constitutionally fixed coefficients $(a,b,c)$.
In \GG\ applications, $F$ is the $SU(5)$ curvature, $F_B$ is the $U(1)_B$ curvature, and $R$ is the curvature 2-form of the Levi--Civita connection.
A publication basis consistent with the \GG{} stage is
\begin{equation}
\label{eq:BI6_pub}
\dd H_3
= \frac{1}{16\pi^2}\left[
\Tr(R\wedge R) - \Tr(F_{SU(5)}\wedge F_{SU(5)}) - \frac{2}{5}\,F_B\wedge F_B
\right],
\end{equation}
while alternative reciprocal conventions correspond to rescalings of the Abelian generator.
POS-\OGN\ will later forbid ``basis drift'' by fixing a canonical normalization in which these coefficients are minimal rational numbers.

\subsection{Constraint power: anomalies as a discrete filter}
Beyond consistency, anomaly cancellation is a \emph{powerful filter} on admissible matter content and gauge structure.
In the Standard Model, anomaly cancellation fixes the hypercharge assignments of fermions.
In \GG\ the postulate is used as a stage-defining constraint that locks gauge and gravitational sectors in a single bulk geometry
\citep{GreenSchwarz1984,Weinberg1995QFT}.

\subsection{Parameter dictionary}
\begin{itemize}[leftmargin=2.2em]
\item $\Gamma[A]$ : quantum effective action; $\delta_\epsilon\Gamma$ encodes anomalies.
\item $B_2$ : 2-form (Green--Schwarz field); $H_3$ : gauge-invariant 3-form field strength.
\item $X_4$ : 4-form polynomial in curvatures; $F$ : gauge curvature; $F_B$ : Abelian curvature; $R$ : gravitational curvature.
\item $(a,b,c,d)$ : fixed rational coefficients of the four-component modified Bianchi identity (Eq.~\eqref{eq:BI6_pos3}); fixed by consistency and by \OGN.
\end{itemize}

\subsection{Worked example: the $\BIsix{}$ four-component
 anomaly polynomial with trace-ratio lock $|a/c| = |a/d| = 5/2$}
\label{sec:POS3_worked_BI6}

The canonical worked
example of $\POS{3}~\AF{}$ is the four-component anomaly polynomial
that the \GGTheory{} program inherits as the fingerprint of its
axiomatic content. The four-factor
composite gauge group
$G_{\mathrm{tot}} = \Spinone\times\SU(5)\times\U(1)_B\times\U(1)_{Qi}$
of $\AX{2}~\GGEP{}$ produces exactly four anomaly contributions
in six dimensions: the gravitational $\tr(R\wedge R)$, the
non-Abelian $\tr(F_{\SU(5)}\wedge F_{\SU(5)})$, the Abelian
$F_B \wedge F_B$, and the Abelian $F_{Qi}\wedge F_{Qi}$. $\POS{3}~\AF{}$
demands that the total anomaly polynomial $X_4$ be exact, so that
its descent provides a $H_3$ form whose modified Bianchi identity
absorbs the boundary anomaly inflow exactly.

The canonical four-component coefficient vector that solves this
constraint is (\citealp{ArisakaA2GG6}; \citealp{ArisakaA3BI6})
\begin{multline}
X_4 \;=\; a\,\tr(R\wedge R)
 + b\,\tr(F_{\SU(5)}\wedge F_{\SU(5)})
 + c\,F_B\wedge F_B
 + d\,F_{Qi}\wedge F_{Qi},
\\
(a,b,c,d) \;=\; \bigl(1,\,-1,\,-2/5,\,-2/5\bigr).
\label{eq:BI6_pos3}
\end{multline}
The two ratios
\begin{equation}
\Bigl|\frac{a}{c}\Bigr| \;=\; \Bigl|\frac{a}{d}\Bigr| \;=\; \frac{5}{2}
\label{eq:BI6_traceratio_lock}
\end{equation}
are the geometry-gauge \emph{trace-ratio lock}: a single
axiomatic invariant that ties the gravitational anomaly to
both Abelian anomalies through the integer 5 (which, by
$\POS{6}~\OGN{}$, is the central integer of the \OGNfive{}
ledger). The appearance of $5/2$ in
Eq.~\eqref{eq:BI6_traceratio_lock} is by no means a coincidence:
the integer $5$ is forced by the SU(5) factor of $G_{\mathrm{tot}}$,
the $2$ comes from the geometric chirality count, and the
denominator $5$ in the Abelian coefficients is forced by the
descent structure of $X_4$.

The full Hopkins-Singer differential-cohomology proof that
Eq.~\eqref{eq:BI6_pos3} is the unique solution lives in
\citep{ArisakaA3BI6}; here it suffices to record that
$\POS{3}~\AF{}$, applied to the four-factor composite group of
$\AX{2}~\GGEP{}$, forces a single four-component anomaly polynomial
with no continuous freedom. This is what we
mean by anomaly cancellation as a discrete filter --- it picks
exactly one admissible coefficient vector out of an apparently
continuous space. Figure~\ref{fig:BI6_inflow} visualizes the
four-component inflow and the trace-ratio lock graphically.

\begin{figure}[!htbp]
\centering
\resizebox{0.864\textwidth}{!}{%
\begin{tikzpicture}[
 font=\sffamily\footnotesize,
 every node/.style={align=center, minimum height=8mm, rounded corners=2pt,
 inner sep=3pt},
 bdy/.style={fill=blue!10, draw=blue!55!black, line width=0.68pt,
 text width=42mm, minimum width=42mm},
 bulk/.style={fill=red!10, draw=red!55!black, line width=0.68pt,
 text width=42mm, minimum width=42mm},
 comp/.style={fill=green!10, draw=olive!55!black, line width=0.58pt,
 text width=36mm, minimum width=36mm},
 lockbox/.style={fill=yellow!18, draw=olive!60!black, line width=0.77pt,
 text width=52mm, minimum width=52mm, minimum height=10mm,
 font=\sffamily\small},
 arrow/.style={-{Stealth[length=2.6mm]}, line width=0.68pt, draw=black!65},
 arrowbig/.style={-{Stealth[length=3.4mm]}, line width=0.97pt, draw=black!70},
]

\node[bdy] (BdyAnom) at (0, 0)
 {\textbf{Boundary anomaly} \\[1pt]
 $\partial M_4$ Ward identity \\[1pt]
 \footnotesize\itshape (4D effective action)};

\node[bulk] (BulkInflow) at (0, -2.8)
 {\textbf{Bulk inflow polynomial} \\[1pt]
 $X_4 = dH_3$ \\[1pt]
 \footnotesize\itshape (classical-topological identity)};

\draw[arrowbig] (BulkInflow.north) -- (BdyAnom.south)
 node[midway, fill=white, inner sep=2pt, font=\scriptsize\itshape]
 {Green-Schwarz inflow};

\node[comp] (a) at (-6, -5.8)
 {$a\,\tr(R\wedge R)$ \\[1pt]
 $a = +1$ \\[-1pt] {\footnotesize (gravitational)}};
\node[comp] (b) at (-2, -5.8)
 {$b\,\tr(F_{\SU(5)}\!\wedge\!F_{\SU(5)})$ \\[1pt]
 $b = -1$ \\[-1pt] {\footnotesize (\SU(5))}};
\node[comp] (c) at (2, -5.8)
 {$c\,F_B \wedge F_B$ \\[1pt]
 $c = -2/5$ \\[-1pt] {\footnotesize ($\U(1)_B$)}};
\node[comp] (d) at (6, -5.8)
 {$d\,F_{Qi} \wedge F_{Qi}$ \\[1pt]
 $d = -2/5$ \\[-1pt] {\footnotesize ($\U(1)_{Qi}$)}};

\draw[arrow] (a.north) -- ($(BulkInflow.south) + (-2.6,0)$);
\draw[arrow] (b.north) -- ($(BulkInflow.south) + (-0.9,0)$);
\draw[arrow] (c.north) -- ($(BulkInflow.south) + (0.9,0)$);
\draw[arrow] (d.north) -- ($(BulkInflow.south) + (2.6,0)$);

\node[lockbox] at (0, -8.0)
 {\textbf{Trace-ratio lock:} \quad
 $\left|\dfrac{a}{c}\right| = \left|\dfrac{a}{d}\right| = \dfrac{5}{2}$};

\end{tikzpicture}}
\bigskip
\caption[Four-component \BIsix{} bulk-side classical-topological identity.]{The four-component \BIsix{} bulk-side classical-topological
identity, descended from Bianchi 1902. The bulk four-form
polynomial $X_4 = dH_3$ (red) closes under the differential as a
classical identity on the $\AX{2}~\GGEP{}$ composite connection;
the four contributions ---
gravitational $\tr(R\wedge R)$, non-Abelian $\tr(F_{\SU(5)}\wedge F_{\SU(5)})$,
$\U(1)_B$-Abelian $F_B \wedge F_B$, and $\U(1)_{Qi}$-Abelian $F_{Qi}\wedge F_{Qi}$ (green boxes) --- carry the canonical
coefficients $(a, b, c, d) = (1, -1, -2/5, -2/5)$. The
geometry-gauge trace-ratio lock $|a/c| = |a/d| = 5/2$ (yellow)
is the axiomatic fingerprint of the \GGTheory{} program,
forced by $\POS{3}~\AF{}$ together with $\POS{6}~\OGN{}$ on the
6D bulk; the integer $5$ is the central rank of the
\OGNfive{} ledger. The boundary 4D ``anomaly'' (top, blue) is
the $\AX{3}~\OGGEP{}$ projection shadow of this bulk identity:
a boundary effective-action re-measurement of a classical-topological
fact already in force in the bulk, not a quantum-loop pathology to
be cured. (Illustration only; primary derivation of the trace-ratio
lock is \citep[Theorem~8.1]{ArisakaA3BI6} and \citep[Theorem~4, Lemma~4.3]{ArisakaA2GG6}.)}
\label{fig:BI6_inflow}
\end{figure}

\subsection{Physics intuition}

\begin{euclidremark}[Physics intuition for \POS{3}]
$\POS{3}~\AF{}$ is the quantum-viability
compiler. $\POS{1}~\GI{}$ provides the symmetry grammar at the
classical level; $\POS{3}~\AF{}$ demands that the grammar survive
quantization. Inflow cancellation can be read as a ledger
balance: what appears as nonconservation in the boundary effective
theory is compensated exactly by flux from the bulk. Within the
$\GGConstitution{}$, this ledger interpretation is not
metaphorical --- it is encoded in the four-component differential-form
identity of Eq.~\eqref{eq:BI6_pos3}, and the trace-ratio lock
$|a/c| = |a/d| = 5/2$ is the axiomatic fingerprint that
distinguishes the $\GG{}$ program from every other six-dimensional
construction. An anomaly is not a subtle technicality
to be papered over --- it is a fatal admissibility failure unless
it is canceled by a definite, integer-valued inflow contribution.
\end{euclidremark}

\subsection{Relation to \GG\ and \GDOS}
\begin{itemize}[leftmargin=2.2em]
\item \textbf{Relation to \GG.} POS-\AF\ is stage-defining for \GG{}: the bulk must contain the precise structure that allows inflow,
and anomaly cancellation constrains the allowed unified gauge content \citep{ArisakaA2GG6,GreenSchwarz1984}.
\item \textbf{Relation to \GDOS.} POS-\AF\ ensures that gauge covariance (POS-\GI) survives the operational reduction.
In open-system language, it protects Ward identities and prevents unphysical gauge nonconservation in registered observables
\citep{BreuerPetruccione2002}.
\end{itemize}

\subsection{Further mathematical elaboration: anomaly polynomial, descent, and inflow cancellation}
\label{sec:POS3_further}

In $d=4$, gauge and mixed anomalies are encoded by a 6-form anomaly polynomial $I_6$ whose descent generates the anomalous variation.
Schematically, for a gauge connection $A$ and curvature $F=\dd A + A\wedge A$, one writes
\begin{equation}
I_6 = \Tr\!\left(F^3\right) + \Tr(F)\,\Tr(F^2)+\cdots,
\end{equation}
and the descent procedure produces a 5-form Chern--Simons term $\omega_5$ with $\dd\omega_5 = I_6$.
Under a gauge transformation, $\delta \omega_5=\dd \omega_4^{(1)}$, and the boundary effective action varies as
$\delta W_{\Bdy}\sim \int_{\Bdy}\omega_4^{(1)}$ (the anomaly).

Inflow cancellation introduces a bulk term whose variation cancels this boundary piece.
In \GG{} language, a Green--Schwarz two-form $B_2$ with field strength $H_3$ obeys a modified Bianchi identity
\begin{equation}
H_3=\dd B_2+\omega_3,\qquad \dd H_3 = X_4,
\end{equation}
where $X_4$ is built from gauge and curvature invariants (POS-\AF) and $\omega_3$ is a Chern--Simons 3-form.
Because $\omega_3$ shifts under gauge transformations, the bulk coupling $B_2\wedge X_4$ produces a boundary variation that cancels $\delta W_{\Bdy}$.
The practical message for this paper is not the detailed \GG{} construction, but the \emph{filter}: a candidate model is inadmissible if its anomaly polynomial cannot be canceled
by some allowed inflow mechanism without introducing additional non-axiomatic dials.

\section{POS-4: Closure and positivity (CP)}
\label{sec:POS4}

\subsection{Rigorous statement (constitutional text)}
\begin{euclidpostulate}[POS-4 (\CP): closure and positivity]
\label{pos:CP}
Admissible kinematics and dynamics must preserve the physical state space and close under composition.

\begin{enumerate}[leftmargin=2.2em,label=\textbf{(\CP\arabic*)}]
\item \textbf{Algebraic closure.} Constraint and symmetry algebras close; no inconsistent bracket relations are allowed.
\item \textbf{State positivity.} States remain physical: probabilities nonnegative; quantum states remain positive semidefinite.
\item \textbf{Dynamical admissibility.} Operational evolution is positivity-preserving and normalization-preserving.
In quantum theory it is \CPTP; in continuous-time Markovian limits it is generated by a \GKLS\ operator.
\item \textbf{Spectrum positivity.} No negative-norm/ghost degrees of freedom; physical spectra are bounded consistently.
\end{enumerate}
\end{euclidpostulate}

\subsection{Complete positivity and the Kraus/Choi/Stinespring structure}
A linear map $\Phi$ is positive if $\rho\succeq 0 \Rightarrow \Phi(\rho)\succeq 0$.
It is \emph{completely positive} if $\Phi\otimes \mathrm{id}_n$ is positive for all $n$.
Complete positivity is required because an open system may be entangled with unobserved ancillas.
The Kraus theorem states that a CPTP map has an operator-sum representation
\begin{equation}
\Phi(\rho)=\sum_\alpha K_\alpha\rho K_\alpha^\dagger,\qquad \sum_\alpha K_\alpha^\dagger K_\alpha=\mathbb{I}
\end{equation}
(see Eq.~\eqref{eq:Kraus}), and Stinespring's theorem provides a dilation to unitary evolution on a larger space
\citep{Stinespring1955,Kraus1983,Choi1975}.

\subsection{From semigroups to \GKLS\ generators (a forced structure)}
Assume $\{\Phi_t\}_{t\ge 0}$ is a strongly continuous semigroup of CPTP maps.
Then the generator $\mathcal{L}=\lim_{t\downarrow 0}(\Phi_t-\mathrm{id})/t$ exists and must take \GKLS\ form
\begin{equation}
\label{eq:GKLS_POS4}
\mathcal{L}(\rho)=-i[H,\rho]+\sum_k\left(L_k\rho L_k^\dagger-\frac{1}{2}\{L_k^\dagger L_k,\rho\}\right),
\end{equation}
with $H=H^\dagger$ and arbitrary $L_k$ \citep{Gorini1976,Lindblad1976,BreuerPetruccione2002}.
Thus, in the presence of \OGGEP\ (which defines between-intervention evolution) and \LC\ (which fixes causal forward composition),
POS-\CP\ makes the Lindblad cone a constitutional admissibility domain, not an approximation.

\paragraph{Custody note: where these theorems live.}
Complete positivity, the Choi characterization, the Stinespring
dilation, and the GKLS classification are theorems of \emph{complex}
operator algebras
\citep{Stinespring1955,Choi1975,Gorini1976,Lindblad1976}. POS-4 is a
boundary-side postulate, and it operates downstream of the
manufactured $i$: the complex Hilbert space on which its channels
act is the derived output of the Foundations Branch (T-P4-1/2/3 of
\citealp{ArisakaP4Real}), so invoking complex-analytic machinery
here is lawful --- the boundary is where that machinery's premises
hold. The bulk never runs GKLS: bulk evolution is closed and real,
and no realification of the CPTP apparatus is attempted or needed.
The division of labor is exact --- real bulk, derived boundary
$i$, complex boundary theorems consumed where they are theorems.

\subsection{Classical analogue: Markov generators and positivity of diffusion}
In classical open dynamics, admissibility often amounts to positivity of the transition kernel.
For diffusions, a Fokker--Planck equation
\begin{equation}
\partial_t p = -\nabla\cdot (b p)+\nabla\cdot (D\nabla p)
\end{equation}
preserves positivity provided $D\succeq 0$ and suitable boundary conditions hold.
This classical positivity is the analogue of quantum complete positivity.

\subsection{Parameter dictionary}
\begin{itemize}[leftmargin=2.2em]
\item $\Phi$ : operational evolution map (channel); CPTP in quantum theory.
\item $K_\alpha$ : Kraus operators; $J(\Phi)$ : Choi matrix encoding complete positivity \citep{Choi1975}.
\item $\{\Phi_t\}$ : semigroup of channels; $\mathcal{L}$ : generator; $H$ : effective Hamiltonian; $L_k$ : Lindblad operators.
\end{itemize}

\subsection{Worked example: damped harmonic oscillator under
 the GKLS generator}
\label{sec:POS4_worked_dampedHO}

A concrete instance brings out the discipline. The cleanest worked
example of $\POS{4}~\CP{}$ is the damped harmonic oscillator under
its canonical Lindblad-Gorini-Kossakowski-Sudarshan generator
\citep{Lindblad1976,Gorini1976,BreuerPetruccione2002}. Roughly
speaking, the setup is as follows. A single bosonic mode with
annihilation operator $a$ and frequency $\omega_0$ is weakly
coupled to a thermal reservoir at zero temperature, so that the
only environmental process is spontaneous emission at rate
$\gamma$. $\POS{4}~\CP{}$ demands that the resulting state
evolution be CPTP at every time, which forces the master equation
into the GKLS form
\begin{equation}
\frac{d\rho}{dt}
\;=\;
-\ii\bigl[\omega_0 a^\dagger a,\,\rho\bigr]
\;+\;
\gamma\,\Bigl(a\rho a^\dagger - \tfrac{1}{2}\{a^\dagger a,\,\rho\}\Bigr).
\label{eq:dampedHO_GKLS}
\end{equation}
The single Lindblad operator $L = \sqrt{\gamma}\,a$ encodes the
photon-emission channel; the Hamiltonian piece is the closed
oscillator dynamics; the anti-commutator term is the no-jump
back-reaction that preserves trace. One cannot have
trace preservation without that anti-commutator --- which is why
$\POS{4}~\CP{}$ forces the GKLS structure rather than allowing a
generic Markov master equation.

The exact solution can be read off in the $\{|n\rangle\}$ Fock
basis. The diagonal populations $p_n(t) = \langle n|\rho(t)|n\rangle$
satisfy
\begin{equation}
\frac{dp_n}{dt} \;=\; \gamma\bigl[(n+1)p_{n+1} - n\,p_n\bigr],
\label{eq:dampedHO_populations}
\end{equation}
which is a positivity-preserving classical Markov chain on the
non-negative integers. The off-diagonal coherences decay as
$\rho_{nm}(t) = \rho_{nm}(0)\,\e^{-\ii\omega_0(n-m)t - \tfrac{\gamma}{2}(n+m)t}$.
Both decays satisfy the axiomatic rule: the spectrum of
$\rho(t)$ remains non-negative at every $t \ge 0$, the trace is
preserved, and the asymptotic state is the vacuum $|0\rangle\langle 0|$.
This is what we mean by closure and positivity
as runtime admissibility filters: a master equation that violated
either condition would generate negative probabilities at some
finite time and thereby fail $\POS{4}$.

The axiomatic reading is exactly this: $\POS{4}~\CP{}$ is the
runtime rule that turns the abstract bulk-to-boundary functor of
$\AX{3}~\OGGEP{}$ into a specific algebraic class --- the cone of
CPTP channels and their continuous-time generators. The
companion paper \citep{ArisakaA5GDOS} shows that $\POS{4}~\CP{}$,
together with $\AX{1}~\LC{}$ (forward-time semigroup) and
$\AX{3}~\OGGEP{}$ (operational dictionary), forces the boundary
law $\cL_{\GDOS{}}$ into precisely the GKLS class.

\subsection{Physics intuition}

\begin{euclidremark}[Physics intuition for \POS{4}]
$\POS{4}~\CP{}$ is the ``no negative
probabilities'' rule elevated to the axiomatic level. It blocks
the unphysical modeling moves that are otherwise tempting in
complex-systems modeling --- negative concentrations in chemistry,
non-CP ``state updates'' in neuroscience, ghost instabilities in
quantum-gravity perturbation theory. Together with $\AX{3}~\OGGEP{}$,
it makes measurement semantics precise: operational maps must live
in the CPTP/GKLS cone, no exceptions. A theory whose
predicted state can have negative eigenvalues at finite time has
already failed the most basic admissibility test of an
operational science --- it cannot tell you the probability of any
outcome.
\end{euclidremark}

\subsection{Relation to \GG\ and \GDOS}
\begin{itemize}[leftmargin=2.2em]
\item \textbf{Relation to \GG.} In the bulk/unification stage, closure and positivity manifest as ghost-free spectra
and consistent gauge-fixing/quantization of the unified connection \citep{PeskinSchroeder1995,Weinberg1995QFT}.
\item \textbf{Relation to \GDOS.} In \GDOS{}, POS-\CP\ is the runtime rule that defines admissible open evolution and measurement:
it forces CPTP instruments and, under continuity, Lindblad generators. This is the technical backbone of exporting quantum measurement
to general open systems \citep{BreuerPetruccione2002}. The downstream chain $\T{7.1}$--$\T{7.7}$ that derives the textbook QM postulates (Born rule, L\"uders projection, decoherence, einselection, Bell--CHSH violation) from $\CP + \OGGEP$ is proved in \citep{ArisakaA5GDOS} and exercised at graduate-textbook depth in \citep{ArisakaA8QM}.
\end{itemize}

\subsection{Further mathematical elaboration: closure and positivity in GKLS and classical Markov generators}
\label{sec:POS4_further}

In the quantum Markovian regime, ``closure and positivity'' becomes the statement that $\{\Phi_t\}_{t\ge0}$ is a CPTP semigroup and therefore admits a GKLS generator:
\begin{equation}
\frac{\dd}{\dd t}\rho = \mathcal{L}(\rho)
= -\ii[H,\rho]+\sum_{\alpha,\beta}c_{\alpha\beta}\left(L_\alpha\rho L_\beta^\dagger-\frac{1}{2}\{L_\beta^\dagger L_\alpha,\rho\}\right),
\qquad (c_{\alpha\beta})\succeq 0,
\label{eq:GKLS_matrix}
\end{equation}
where the Kossakowski matrix $(c_{\alpha\beta})$ is positive semidefinite. The positivity of $(c_{\alpha\beta})$ is the precise mathematical clause that forbids negative probabilities
and guarantees complete positivity for all extensions (system + ancilla).

In the classical limit (finite-state Markov chains), the analogous statement is that probabilities $p_i(t)\ge 0$ evolve by a rate matrix $W$ with nonnegative off-diagonal entries:
\begin{equation}
\dot p_i = \sum_{j\ne i}\big(W_{ij}p_j - W_{ji}p_i\big),
\qquad W_{ij}\ge 0\ (i\ne j),\qquad \sum_i \dot p_i=0.
\label{eq:master_equation}
\end{equation}
Equation \eqref{eq:master_equation} is ``closure'' (stays inside the simplex) and ``positivity'' (no negative probabilities).
POS-\CP\ asserts that any open-system effective theory used in any branch of \GGTheory{}\ must admit a representation of this kind (quantum or classical, as appropriate),
so that violations of positivity/normalization are treated as axiomatic errors rather than technical glitches.

\section{POS-5: RG-compatibility and scale covariance (RG)}
\label{sec:POS5}

\subsection{Rigorous statement (constitutional text)}
\begin{euclidpostulate}[POS-5 (\RG): scale coherence and RG commutation]
\label{pos:RG}
Coarse-graining/renormalization must commute with the operational bulk$\to$boundary dictionary.
Equivalently, changing resolution (``zooming out'') must not change the operational meaning of the theory.

Formally, letting $\mathrm{RG}$ denote the renormalization/coarse-graining map and $\Dmap$ the operational dictionary,
\begin{equation}
\label{eq:RG_commute}
\Dmap\circ \mathrm{RG} = \mathrm{RG}\circ \Dmap.
\end{equation}
In the \GG\ stage, the bulk coordinate $r$ is identified with log-energy scale by Eq.~\eqref{eq:rlog_POS}.
Dimensionless predictions are RG-invariant or belong to controlled, covariant RG flows.
\end{euclidpostulate}

\subsection{Why commutation is nontrivial}
In standard practice, RG is often treated as a computational device (change of cutoff, effective action),
while measurement rules are treated separately.
But if observation/measurement is \emph{defined} by \OGGEP\ (AX-3), then it must be stable under RG:
otherwise, what counts as an observable would depend on an arbitrary resolution choice, contradicting the idea
that physics is objective.

\subsection{Bulk--boundary reading: radial evolution as RG flow}
Equation \eqref{eq:RG_commute} becomes particularly concrete in the bulk--boundary correspondence, where radial evolution in the bulk is mapped
to RG flow on the boundary. Write $T_{\Delta r}$ for bulk translation by $\Delta r$, and $R_{\Delta r}$ for boundary RG flow by the
corresponding scale step. POS-\RG\ asserts a commuting square
\begin{equation}
\label{eq:commutingSquare_POS5}
\Dmap\circ T_{\Delta r}\;\cong\; R_{\Delta r}\circ \Dmap.
\end{equation}
In coupling language, for dimensionless couplings $g_i(\mu)$,
\begin{equation}
\mu\frac{\dd g_i}{\dd\mu}=\beta_i(g)
\quad\Longleftrightarrow\quad
\partial_r g_i = -\beta_i(g),\qquad r=\ln(E_{\rm GUT}/\mu).
\end{equation}
Thus RG beta functions can be interpreted as radial evolution equations. This is the precise technical sense in which
``scale is geometrized'' in \GG.

\subsection{Scale covariance versus scale invariance}
POS-\RG\ does \emph{not} assert that all quantities are scale invariant. Rather, it asserts that:
\begin{enumerate}[leftmargin=2.2em,label=\textbf{(\RG\arabic*)}]
\item physical predictions are either RG invariants (dimensionless ratios, critical exponents) or transform covariantly under RG;
\item operational semantics (what is being measured and how) is preserved under coarse-graining.
\end{enumerate}
This postulate is crucial for exporting to sciences where coarse-graining is unavoidable: if predictions change qualitatively
when one changes measurement resolution, they are not lawful outputs of the framework.

\subsection{Parameter dictionary}
\begin{itemize}[leftmargin=2.2em]
\item $\mathrm{RG}$ : renormalization/coarse-graining map on the relevant description space.
\item $\Dmap$ : operational dictionary (bulk$\to$boundary compilation) defined by \OGGEP.
\item $\mu$ : boundary energy/scale parameter; $E_{\rm GUT}$ : unification scale; $r=\ln(E_{\rm GUT}/\mu)$ : bulk scale coordinate.
\item $g_i$ : running couplings; $\beta_i(g)$ : beta functions.
\end{itemize}

\subsection{Worked example: running of the QED fine-structure
 constant $\alpha(E)$ and scheme-independence of $\sigma(\e^+\e^-\to\mu^+\mu^-)$}
\label{sec:POS5_worked_alpha}

The cleanest worked
example of $\POS{5}~\RG{}$ is the running of the QED fine-structure
constant $\alpha$ with energy scale and the constitutional
requirement that physical cross-sections be scheme-independent.
The one-loop QED beta function gives
\begin{equation}
\mu\,\frac{d\alpha}{d\mu}
\;=\;
\beta_\alpha(\alpha)
\;=\;
\frac{2\alpha^2}{3\pi} + \cO(\alpha^3),
\label{eq:alpha_beta}
\end{equation}
whose integral, between the Thomson scale ($\alpha(0) \approx 1/137.036$)
and the $Z$-pole ($\alpha(M_Z) \approx 1/127.9$), accounts
quantitatively for the observed running.

Why is the running not a problem for the constitution?
two physicists working in two schemes (say $\overline{\text{MS}}$
and on-shell) will report two different numerical values of
$\alpha(\mu)$ at any given scale $\mu$. $\POS{5}~\RG{}$ does not
require that $\alpha(\mu)$ itself be scheme-independent --- it
requires that the operational dictionary $\Dmap$ commute with
coarse-graining, so that any \emph{registered observable} produces
the same number in either scheme. The
axiomatic content reads
\begin{equation}
\Dmap \circ \mathrm{RG}_\ell \;=\; \mathrm{RG}_\ell \circ \Dmap,
\label{eq:RG_commute_pos5}
\end{equation}
and the cross-section
$\sigma(\e^+\e^- \to \mu^+\mu^-)(\sqrt{s})$ at any center-of-mass
energy $\sqrt{s}$ is exactly such a registered observable.

The two schemes give two different $\alpha(\mu)$ functions, but
they give the same $\sigma(\sqrt{s})$ to all orders in
perturbation theory \citep{Callan1970,Symanzik1970,Wilson1971}.
The scheme-difference cancels between the running coupling and
the matrix-element matching coefficients --- a fact that is
forced, not coincidental. If it failed, the two schemes
would be physically inequivalent, and the operational record at
the LEP detector would depend on the calculator's bookkeeping.
This is exactly the structural disaster that $\POS{5}~\RG{}$
forbids.

The bulk realization in $\GG{}$ takes this commutation diagram
and turns it into geometry. The radial coordinate $r$ of the
6D bulk is identified with the RG flow parameter via
$r = \ln(\EGUT/\mu)$, and translation in $r$ on the bulk
implements the boundary RG flow exactly. The
existence of a geometric scale coordinate is not an optional
feature of the \GGTheory{} program --- it is forced by
$\POS{5}~\RG{}$ together with $\AX{2}~\GGEP{}$, because the
operational dictionary must commute with the radial flow on the
bulk.

\subsection{Physics intuition}

\begin{euclidremark}[Physics intuition for \POS{5}]
$\POS{5}~\RG{}$ is the rule that physical truth
must not depend on the resolution at which one looks at it. In
the $\GGConstitution{}$ this is by no means methodological
--- it forces the existence and interpretation of a geometric
scale coordinate (the radial direction $r$ of the 6D bulk)
and demands that observation itself be scale-coherent. More precisely,
the existence of a non-spatial dimension --- the \emph{scale dimension} --- that plays the role of the RG
flow parameter is not an optional feature of the $\GG{}$
program but an axiomatic consequence: $\POS{5}$ forces the
bulk-to-boundary functor $\Dmap$ to commute with coarse-graining,
and the simplest geometric realization of that commutation is a
non-spatial dimension along which scale flows. In the cohort vocabulary this fifth coordinate is the \emph{scale dimension}; the sixth coordinate $S^1_\theta$ (forced under $\POS{1}$ and $\POS{6}$) is the \emph{gauge dimension}; together they are the two \emph{non-spatial dimensions} of the bulk --- each a scale or a phase, not a spatial \emph{extra dimension}, the latter term being reserved for the Kaluza--Klein and string constructions with which \GG{} is contrasted.
\end{euclidremark}

\subsection{Relation to \GG\ and \GDOS}
\begin{itemize}[leftmargin=2.2em]
\item \textbf{Relation to \GG.} The identification of $r$ with $\ln(E_{\rm GUT}/\mu)$ is part of the motivation for an AdS-like bulk;
bulk geometry becomes a map of scale. Technical realization is discussed in \citep{ArisakaA2GG6,Maldacena1998,Polchinski1995}.
\item \textbf{Relation to \GDOS.} In \GDOS{}, coarse-graining corresponds to changing the operational resolution of an open system.
POS-\RG\ demands that the operational dictionary (what counts as state, port, and measurement) commute with such coarse-graining. The radial-to-boundary RG bridge is developed at length in \citep{ArisakaA8QM}.
\end{itemize}

\subsection{Further mathematical elaboration: commuting RG with observation as a commutative diagram}
\label{sec:POS5_further}

Let $\RG_\ell$ denote coarse-graining by a scale parameter $\ell$ (e.g.\ integrating out modes above $\mu \e^{-\ell}$),
and let $\Dmap$ denote the operational dictionary mapping model variables to registered observables (AX-3).
POS-\RG\ is the commutativity requirement
\begin{equation}
\Dmap\circ \RG_\ell = \RG_\ell^{(\mathrm{obs})}\circ \Dmap,
\label{eq:RG_commute_diagram}
\end{equation}
where $\RG_\ell^{(\mathrm{obs})}$ is the induced renormalization of estimators/response functions (calibration, bandwidth, binning, etc.).
Equation \eqref{eq:RG_commute_diagram} says: \emph{``coarse-grain first'' and ``observe first'' must lead to the same registered claim}, up to controlled, declared transformations.

In QFT this becomes the statement that dimensionless predictions are RG invariant or follow controlled flows, governed by beta functions:
\begin{equation}
\left(\mu\frac{\partial}{\partial \mu}+\beta^i(g)\frac{\partial}{\partial g^i} + \gamma\right)\, \mathcal{O}(\mu,g)=0,
\label{eq:callan_symanzik}
\end{equation}
where $\mathcal{O}$ is an observable and $\gamma$ denotes anomalous dimensions.
In the \GG{}/\OGGEP{} vocabulary, the AdS-like radial coordinate is identified with scale,
\begin{equation}
r=\ln(\EGUT/\mu),
\end{equation}
so RG flow becomes geometric transport in $r$.
The axiomatic content is that \emph{scale is not an observer preference}: the dictionary and the dynamics must be compatible with the
choice of resolution, otherwise ``predictions'' are artifacts of the microscope rather than properties of the world.

\section{POS-6: Occam Gauge Normalization (\OGN{}): rigidity and ``no basis drift''}
\label{sec:POS6}

\subsection{Rigorous statement (constitutional text)}
\begin{euclidpostulate}[POS-6 (\OGN): Occam gauge normalization / minimal rational normalization]
\label{pos:OGN}
Among all basis and normalization conventions that leave POS-1--POS-5 satisfied, Nature selects a canonical
normalization in which:

\begin{enumerate}[leftmargin=2.2em,label=\textbf{(\OGN\arabic*)}]
\item \textbf{Topological integrality.} All relevant topological integrals (fluxes, Chern/Pontryagin numbers, inflow coefficients)
take integer values in the chosen units.
\item \textbf{Minimal rationality.} Any remaining rational factors are fixed in minimal coprime form shared across sectors.
\item \textbf{No continuous rescaling dials.} No arbitrary continuous normalization parameter may remain; ``basis drift'' is illegal.
\end{enumerate}

Equivalently: conventions are not allowed to function as hidden continuous parameters.
\end{euclidpostulate}

\subsection{Why normalization is not ``just convention''}
In a single-sector gauge theory, one may rescale
\begin{equation}
A_\mu\to \lambda A_\mu,\qquad g\to g/\lambda,
\end{equation}
leaving $gA_\mu$ invariant in covariant derivatives. This looks like a harmless convention.
Yet in a multi-sector theory with Abelian factors, mixed anomalies, and topological inflow terms,
such rescalings can move physical content between ``couplings'' and ``normalization constants.'' Without a canonical choice,
one can inadvertently \emph{launder} parameters: a theory can appear predictive in one basis and nonpredictive in another.

POS-\OGN\ forbids this: if a basis choice changes the integer/topological data, it is not an allowed convention.

\subsection{Topological quantization and canonical units}
A central mechanism is topological quantization. For an Abelian field strength $F$, flux quantization reads
\begin{equation}
\frac{1}{2\pi}\int_{\Sigma_2} F \in \mathbb{Z},
\end{equation}
and for non-Abelian bundles characteristic classes are integer-valued.
Such conditions are invariant statements, but their manifestation as simple integers depends on normalization.
\OGN{} selects the normalization in which the corresponding lattices are \emph{minimal} (coprime, no common factors).
This is the same discipline that underlies Dirac charge quantization \citep{Dirac1931} and the standard generator conventions
in Lie algebras \citep{KobayashiNomizu1963,Nakahara2003}.

\subsection{\OGN{} as the antidote to scheme dependence}
A closely related issue is scheme dependence in renormalization and in holographic renormalization:
finite local counterterms can shift definitions of quantities if not fixed by a principle.
\OGN{} asserts that the admissible scheme is the one fixed by minimal rational/integer data, so that dimensionless predictions
do not drift with choice of renormalization convention.

\subsection{\OGNfive{} integer ledger (as a stage output)}
In the \GG{} stage, \OGN{} is proposed to yield a discrete integer ``ledger'' (a minimal coprime set of integers shared across sectors).
A canonical representative is
\begin{equation}
\label{eq:OGN5}
(M,N,L;J,K)=(1,3,5;7,35),\qquad J=2N+1,\qquad K=LJ=M^2+N^2+L^2=35,
\end{equation}
which is used in the \GG{} program to express several dimensionless ratios in compact rational form.
In the present axiom paper we record this as an intended output of the admissibility spine; the derivation and phenomenology
are presented elsewhere \citep{ArisakaA2GG6}.

\subsection{Parameter dictionary}
\begin{itemize}[leftmargin=2.2em]
\item $\lambda$ : would-be continuous rescaling parameter (forbidden as a physical dial by POS-\OGN).
\item $F$ : gauge curvature 2-form; $\int F$ : flux/topological invariant (integer in canonical units).
\item $(a,b,c,d)$ : four-component coefficients of the modified Bianchi identity \eqref{eq:BI6_general}; \OGN{} fixes their canonical rational form.
\item $(M,N,L;J,K)$ : example of a discrete integer ledger output in the \GG{} stage (\OGNfive{}).
\end{itemize}

\subsection{Worked example: the $\OGNfive{}$ ledger and its
 Diophantine consistency}
\label{sec:POS6_worked_OGN5}

The canonical worked
example of $\POS{6}~\OGN{}$ is the $\OGNfive{}$ integer ledger
that the \GGTheory{} program inherits from the 6D bulk
\citep{ArisakaA2GG6}. The ledger consists of five integers
\begin{equation}
(M,\,N,\,L;\,J,\,K) \;=\; (1,\,3,\,5;\,7,\,35),
\label{eq:OGN5_ledger_pos6}
\end{equation}
where $M$, $N$, $L$ are the three primary integers attached to
the three sectors of $G_{\mathrm{tot}}$ accessible to $\OGNfive{}$;
$J = 2N + 1 = 7$ is the chirality count; and $K = LJ = 35$ is the
lock length that sets $r \in [0, K]$ in the canonical 6D
metric $ds_6^2 = \e^{-2r}\eta_{\mu\nu}dx^\mu dx^\nu + dr^2 + d\theta^2$.

(Throughout, a \emph{Diophantine consistency} is one whose admissible solutions are restricted to integers, in the tradition of Diophantus's \emph{Arithmetica} \citep{Diophantus}.)

The five integers are not chosen freely --- they are forced by a
Diophantine consistency that $\POS{6}~\OGN{}$ requires the bulk
to satisfy, together with the $\AX{4}~\MSMF{}$ ordering on the
primary integers and the four-component anomaly polynomial of
$\BIsix{}$ (full forcing proof in \citep{ArisakaA2GG6}). The
consistency takes the form
\begin{equation}
K \;=\; L\,J \;=\; M^2 + N^2 + L^2,
\label{eq:OGN5_diophantine}
\end{equation}
which one verifies directly: $L\,J = 5 \cdot 7 = 35$, and
$M^2 + N^2 + L^2 = 1^2 + 3^2 + 5^2 = 1 + 9 + 25 = 35$. Strikingly,
the equality of the two right-hand sides at the integer 35 is
precisely the consistency condition that $\POS{6}~\OGN{}$
demands, and it is by no means a coincidence: the three-square
identity $M^2 + N^2 + L^2 = K$ encodes the existence of an
$\Spinone$-equivariant integer lattice of the right rank, while
the product $K = LJ$ encodes the lock-length closure of the
radial coordinate.

The axiomatic reading is that $\POS{6}~\OGN{}$ produces no
continuous freedom on the bulk side --- and Eq.~\eqref{eq:OGN5_diophantine}
does more work than a consistency check. Read as a constraint rather
than as an agreement to be admired, it \emph{determines} the color
index: substituting the closure-index clause $J = 2N+1$ gives
$N = L \pm \sqrt{L - M^2}$, so integrality forces $L - 1$ to be a perfect
square at $M = 1$, and at $L = 5$ the two roots $3$ and $7$ are
separated by the block embedding $N < L$, leaving
$N = N_c = 3$ \citep[Theorem~T-BI6-10.2a]{ArisakaA3BI6}.

Honesty requires the exchange to be stated. Once the dual lock carries
$N$, its two readings are \emph{required} to agree rather than
\emph{observed} to agree, so the framework may no longer present their
agreement as a free confirmation. What it buys instead is stronger: the
constraint restricts the admissible block sizes to those with $L-1$
square, $L \in \{2, 5, 10, 17, 26, \dots\}$, and $L = 5$ then arrives
independently \citep[Theorem~T-BI6-9.0]{ArisakaA3BI6}. A deviation from
Eq.~\eqref{eq:OGN5_ledger_pos6} therefore produces either a
mismatch in the three-square identity or a violation of the
chirality count. This is what we mean by
``no continuous knobs'' --- the entire numerical content of the
bulk is fixed by five small integers, and every numerical
prediction that descends to the boundary inherits its
discreteness from these five integers.
Figure~\ref{fig:OGN5_cascade} visualizes the cascade graphically.

\begin{figure}[!htbp]
\centering
\resizebox{0.92\textwidth}{!}{%
\begin{tikzpicture}[
 font=\sffamily\footnotesize,
 every node/.style={align=center, rounded corners=2pt, inner sep=3pt},
 primary/.style={fill=cyan!10, draw=cyan!55!black, line width=0.6pt,
 minimum width=18mm, minimum height=9mm},
 derived/.style={fill=blue!10, draw=blue!55!black, line width=0.7pt,
 minimum width=32mm, minimum height=11mm},
 total/.style={fill=red!10, draw=red!55!black, line width=0.7pt,
 minimum width=32mm, minimum height=11mm,
 font=\sffamily\small\bfseries},
 identity/.style={fill=yellow!18, draw=olive!55!black, line width=0.7pt,
 minimum width=58mm, minimum height=12mm},
 eqsign/.style={font=\sffamily\bfseries\large,
 fill=white, draw=black!40, line width=0.4pt,
 minimum width=7mm, minimum height=10mm,
 inner sep=1pt},
 banner/.style={font=\sffamily\bfseries\small,
 fill=violet!10, draw=violet!50!black, line width=0.6pt,
 minimum width=92mm, minimum height=10mm},
 arrow/.style={-{Stealth[length=2.4mm]}, line width=0.55pt, draw=black!60},
  arrowbig/.style={-{Stealth[length=2.8mm]}, line width=0.85pt, draw=black!70},
 checkbox/.style={fill=black!4, draw=black!35, line width=0.5pt,
 dashed, minimum width=96mm, minimum height=9mm,
 font=\sffamily\scriptsize},
 checkarrow/.style={-{Stealth[length=2.2mm]}, line width=0.5pt,
 draw=black!40, dashed},
]

% Banner / ledger label
\node[banner] (banner) at (0, 4.4)
 {\OGNfive{} integer ledger \;\; $(M,\, N,\, L;\, J,\, K) = (1,\, 3,\, 5;\, 7,\, 35)$};

%% ---- Stage 1: the two integers fixed without any colour count ----
\node[primary] (M) at (-3.8, 2.6)
 {$M = 1$ \\[-1pt]
  {\scriptsize\mdseries dimensional minimality}};
\node[primary] (L) at (3.8, 2.6)
 {$L = 5$ \\[-1pt]
  {\scriptsize\mdseries single-tensor closure; free of $N$}};
\node[font=\scriptsize\itshape, text=black!55] at (-7.8, 2.6) {Stage 1:};
\node[font=\scriptsize\itshape, text=black!55] at (7.8, 2.6) {theorems};

%% ---- Stage 2: the dual lock DETERMINES the colour index ----
\node[identity] (lock) at (-2.6, 0.5)
 {$K = L\,J = M^2 \!+\! N^2 \!+\! L^2$, \; $J = 2N\!+\!1$ \\[-1pt]
  {\scriptsize\mdseries $\Longrightarrow\; N = L \pm \sqrt{L - M^2}$}};
\node[total] (N) at (4.2, 0.5)
 {$N = N_c = 3$ \\[-1pt]
  {\scriptsize\mdseries second root $N = 7$ fails $N < L$}};
\node[font=\scriptsize\itshape, text=black!55] at (-7.8, 0.5) {Stage 2:};
\node[font=\scriptsize\itshape, text=black!55] at (7.8, 0.5) {the dual lock};

\draw[arrowbig] (M.south) to[out=-90,in=150] (lock.north west);
\draw[arrowbig] (L.south) to[out=-90,in=60] (lock.north east);
\draw[arrowbig] (lock.east) -- node[above, font=\scriptsize\itshape,
 inner sep=1pt]{$N < L$} (N.west);

%% ---- Stage 3: what follows once N is fixed ----
\node[derived] (J) at (-3.4, -1.8)
 {$J = 2N + 1 = 7$ \\[-1pt]
  {\scriptsize\mdseries closure index; \AX{4} clause}};
\node[derived] (K) at (3.4, -1.8)
 {$K = L\,J = 35 = 1\!+\!9\!+\!25$ \\[-1pt]
  {\scriptsize\mdseries lock length, doubly written}};
\node[font=\scriptsize\itshape, text=black!55] at (-7.8, -1.8) {Stage 3:};
\node[font=\scriptsize\itshape, text=black!55] at (7.8, -1.8) {consequences};

\draw[arrow] (N.south) to[out=-120,in=30] (J.north east);
\draw[arrow] (N.south) -- (K.north);

%% ---- the trace ratio, off to the side, as a check ----
\node[checkbox] (tr) at (0, -3.7)
 {trace ratio \;$(L-N)/L = 2/5$ \;---\; \emph{independent check, not a
  route to $N$}};
%% B-3 (Report 280): this leader ran from (tr.north) straight up into
%% (N.south east) and passed THROUGH the K box, striking out the
%% equation 35 = 1+9+25 so that it read as deleted text.  K spans
%% x in [1.8, 5.0] (32mm at x=3.4); N spans [2.6, 5.8]; the stage
%% labels sit at x=7.8.  The lane at x=6.2 is clear of all three.
\draw[checkarrow, rounded corners=5pt]
  (tr.east) -- (6.2, -3.7) -- (6.2, 0.5) -- (N.east);



\end{tikzpicture}}
\caption[\OGNfive{} Diophantine cascade: the dual lock determines the color index.]{\textbf{The \OGNfive{} Diophantine cascade.}
\emph{Stage 1.} $M = 1$ and $L = 5$ are fixed first, and neither
consumes a color count: $M$ by dimensional minimality, $L$ by
single-tensor closure on the charged core package.
\emph{Stage 2.} The dual lock $K = LJ = M^2 + N^2 + L^2$, taken together
with the closure-index clause $J = 2N+1$, is not a consistency check
performed after the fact --- it is a constraint, and solving it gives
$N = L \pm \sqrt{L - M^2}$. At $M = 1$ integrality forces $L-1$ to be a
perfect square; at $L = 5$ the two roots are $3$ and $7$, and the block
embedding $N < L$ --- the color block sits inside the $L$-dimensional
defining representation --- retains $N = N_c = 3$.
\emph{Stage 3.} $J$ and $K$ follow, the latter written two ways at once,
$35 = 5 \times 7 = 1 + 9 + 25$.
\emph{The dashed box is the trace ratio,} which is computed on the
$3{+}2$ split of the defining representation and therefore cannot be
used to derive the color count without assuming it; it enters as an
independent check that the observed split agrees with a ledger already
determined. $N_c$ is the color index, not the generation count $N_g$,
which is not a member of this ledger. Derivations:
\citep[Theorems~T-BI6-9.0 and T-BI6-10.2a]{ArisakaA3BI6};
cascade statement \citep[Theorem~6]{ArisakaA2GG6}.}
\label{fig:OGN5_cascade}
\end{figure}

\subsection{Physics intuition}

\begin{euclidremark}[Physics intuition for \POS{6}]
$\POS{6}~\OGN{}$ is the ``no hidden conventions''
postulate. It ensures that two researchers using different
normalization conventions cannot obtain different predictions
simply by drifting bases. This is the rule that turns
conventions into audit-fixed integers and rationals.
The value of a coupling at a reference scale is not a fact about
Nature; it is a fact about a chosen convention. By forbidding
continuous basis drift, $\POS{6}$ ensures that whatever survives
in the \GGTheory{} program is a discrete consequence of the
$\OGNfive{}$ ledger --- the integers $1, 3, 5; 7, 35$ that
Eq.~\eqref{eq:OGN5_ledger_pos6} forces and Eq.~\eqref{eq:OGN5_diophantine}
verifies. Together with $\AX{4}~\MSMF{}$, $\POS{6}$ is the
constitutional clause that lets the \GGTheory{} program make
predictions like $\Mlock \approx 3.85\,\TeV$ without a single
continuous fitted parameter.
\end{euclidremark}

\subsection{Relation to \GG\ and \GDOS}
\begin{itemize}[leftmargin=2.2em]
\item \textbf{Relation to \GG.} \OGN{} fixes Abelian normalizations, inflow coefficients, and the identification of discrete topological data.
It is a key ingredient in the claim that \GG{} can be ``parameter-closed'' (no continuous free knobs beyond universal constants)
once anomaly cancellation and symmetry are imposed \citep{ArisakaA2GG6}.
\item \textbf{Relation to \GDOS.} In \GDOS{}, \OGN{} stabilizes operational semantics across laboratories and scales: ports, couplings, and
measured quantities must be normalized in a canonical way so that comparisons across experiments and domains are meaningful
\citep{ArisakaA5GDOS,Shannon1948}.
\end{itemize}

\subsection{Further mathematical elaboration: \OGN{} fixes the charge lattice and removes Abelian basis drift}
\label{sec:POS6_further}

The simplest illustration of POS-\OGN\ is an Abelian factor $U(1)$.
If $Q$ is the generator and $g$ the coupling, a rescaling $Q\mapsto \lambda Q$ can be compensated by $g\mapsto g/\lambda$ so that the covariant derivative
$D_\mu=\partial_\mu+\ii g Q A_\mu$ is unchanged (the $\ii$ of this covariant derivative being, as at \AX{2},
the boundary rendering of the circle's forced quarter-turn; T-P4-1 of \citealp{ArisakaP4Real}). This means that, without an \OGN{} clause, a theory can acquire a continuous ``normalization knob'' that is not physical
but can masquerade as one when combined with other sectors or with sloppy bookkeeping.

\OGN{} fixes a canonical \emph{charge lattice} by demanding minimal integers compatible with topology/holonomy:
\begin{equation}
q_i \in \mathbb{Z},\qquad \gcd(\{q_i\})=1,
\end{equation}
and by choosing canonical units for holonomy (one $2\pi$ turn on the compact circle corresponds to one integer period).
The content is Occam-like but operational: once the basis is fixed, charge ratios become sharp statements and can enter falsifiers (POS-\MF).
In \GG{} this kind of discipline is what allows rational lock ratios (e.g.\ $5\!:\!2$) to be interpreted as predictions rather than conventions.

\section[POS-7: Measurement and falsifiability (MF): compile-to-observable]{POS-7: Measurement and falsifiability (MF):\\ compile-to-observable}
\label{sec:POS7}

\subsection{Rigorous statement (constitutional text)}
\begin{euclidpostulate}[POS-7 (\MF): measurement contracts and falsifiability]
\label{pos:MF}
Every theoretical claim in \GGTheory{}\ must compile, via the operational dictionary \OGGEP, to a registered observable
together with a specific instrument protocol and a pre-declared falsification (kill) criterion.
Claims that cannot be compiled into such a measurement contract are not admissible as scientific statements within \GGTheory{}.
\end{euclidpostulate}

\subsection{Measurement is defined by \OGGEP; MF adds an audit requirement}
AX-3 (\OGGEP) defines what ``measurement'' means: an instrument/POVM acting on an operational state.
POS-\MF\ adds that every theorem-level statement must include an \emph{audit trail} to data.

\begin{eucliddefinition}[Measurement contract]
\label{def:contract}
A \emph{measurement contract} for a claim $\mathcal{C}$ is a tuple
\begin{equation}
\label{eq:contract_POS7}
\mathfrak{C}(\mathcal{C}) = \left(\mathcal{I}, \{E_m\}, \mathcal{S}_{\rm valid}, \mathcal{T}, K\right),
\end{equation}
where:
(i) $\mathcal{I}=\{\mathcal{I}_m\}$ is an instrument; (ii) $\{E_m\}$ is the associated POVM;
(iii) $\mathcal{S}_{\rm valid}$ specifies the domain of validity (state preparation, ports, calibration assumptions);
(iv) $\mathcal{T}$ specifies the timescale/scale window and coarse-graining level; and
(v) $K$ is a kill criterion: a statistical or deterministic condition on observed data that would refute $\mathcal{C}$.
\end{eucliddefinition}

Since the POVM $\{E_m\}$ is determined by the instrument via $E_m=\mathcal{I}_m^\ast(\mathbb{I})$, the working short-form of the contract used in the worked example below and throughout the paper is the four-tuple $(\mathcal{I},\,\mathcal{S}_{\rm valid},\,\mathcal{T},\,K)$, with $\{E_m\}$ implicit.

A typical statistical kill criterion may be stated as ``reject $\mathcal{C}$ at significance level $\alpha$ if
a test statistic $T(\text{data})$ falls outside a pre-declared acceptance region,'' in the spirit of
standard hypothesis testing.

\subsection{Parameter dictionary}
\begin{itemize}[leftmargin=2.2em]
\item $\mathcal{I}$ : instrument (family of CP maps summing to a TP map); $\{E_m\}$ : associated POVM.
\item $\mathcal{S}_{\rm valid}$ : explicit preparation/calibration assumptions; ``what was controlled''.
\item $\mathcal{T}$ : timescale/scale window; includes coarse-graining level relevant to POS-\RG.
\item $K$ : kill criterion; may include significance level $\alpha$, sample size, estimator definition, and rejection region.
\end{itemize}

\subsection{Worked example: the kaxion oscillation
 $\Delta\nu/\nu = 1/35$ as a registered claim contract}
\label{sec:POS7_worked_kaxion}

Let us picture this concretely. One of the cleanest
worked examples of a $\POS{7}~\MF{}$ claim contract is the
prediction that a hypothetical kaxion field --- the
\GGTheory{}-program analogue of the QCD axion --- exhibits a
fractional oscillation
\begin{equation}
\frac{\Delta\nu}{\nu} \;=\; \frac{1}{35}
\;\approx\; 0.02857\ldots
\label{eq:kaxion_freq_pos7}
\end{equation}
in the dark-matter sector, where the integer $35 = K = LJ$ comes
from the $\OGNfive{}$ lock length (\citealp{ArisakaA2GG6}; \citealp{ArisakaA10Cosmos}; \citealp{ArisakaC1Kaxion}). The Predictions-Branch paper \PaperCOne{} \citep{ArisakaC1Kaxion} develops the explicit haloscope contract for this prediction.

{\sloppy
The $\POS{7}~\MF{}$ claim contract for this
prediction has the canonical four-tuple form
$(\mathcal{I},\,\mathcal{S}_{\mathrm{valid}},\,\mathcal{T},\,K)$,
where:\par}

\begin{itemize}[leftmargin=2.2em]
\item $\mathcal{I}$ --- the instrument is a microwave or
 haloscope-style cavity tuned to the kaxion mass
 ($m_k \approx 8.1\,\mu\mathrm{eV}$, equivalently $1.95$~GHz), with
 a quantum-limited amplifier at the output and a frequency
 spectrum analyzer as the registered observable.
\item $\mathcal{S}_{\mathrm{valid}}$ --- the preparation/calibration
 assumptions are: (a) the local kaxion-field amplitude tracks
 the standard halo dark-matter density of $0.4\,\mathrm{GeV}/\mathrm{cm}^3$
 to within experimental dispersion; (b) the cavity Q-factor
 exceeds $10^4$; (c) the amplifier noise temperature is at or
 below the standard quantum limit.
\item $\mathcal{T}$ --- the timescale window is the integration
 time of order $10^{6}$ seconds at each tuning step, scanning
 over the predicted band.
\item $K$ --- the kill criterion is the registered observation of
 any narrow spectral peak at $\Delta\nu/\nu$ that disagrees with
 $1/35$ at $5\sigma$ confidence over a control region of width
 $\geq 10\sigma_\nu$. Equivalently: a verified discovery at any
 $\Delta\nu/\nu \neq 1/35$ falsifies the constitution.
\end{itemize}

The axiomatic reading is exactly this: $\POS{7}~\MF{}$ does
not require the theory to make every prediction precise in advance
of every experimental cycle; it requires that, for every theorem
that is claimed to be physical, the four-tuple
$(\mathcal{I},\,\mathcal{S}_{\mathrm{valid}},\,\mathcal{T},\,K)$
be supplied. After all, a prediction without a registered
instrument and a pre-declared kill criterion is not a scientific
claim --- it is a story. The kaxion contract above is one of the
sharpest such contracts the \GGTheory{} program issues, because
the integer $35$ has no continuous freedom and no second-best
prediction is admissible: either $\Delta\nu/\nu = 1/35$ holds or
the constitution is wrong.

\subsection{Physics intuition}

\begin{euclidremark}[Physics intuition for \POS{7}]
$\POS{7}~\MF{}$ is the axiomatic closure
of the scientific loop. A theory is not merely a mathematical
structure; it is a contract between derivations and data. By
forcing every theorem to compile to an
$(\mathcal{I},\,\mathcal{S}_{\mathrm{valid}},\,\mathcal{T},\,K)$
tuple --- an instrument, a calibration window, a timescale, and a
pre-declared kill criterion --- $\POS{7}$ blocks the most common
failure mode of grand-unification programs: ``explains
everything, therefore predicts nothing.'' The kaxion
oscillation contract $\Delta\nu/\nu = 1/35$ is a natural test:
either Nature delivers exactly this fraction at the predicted
mass, or the constitution falls. What makes a
foundational document scientific rather than philosophical is the
willingness to specify, in advance, exactly what would falsify it.
\end{euclidremark}

\subsection{Relation to \GG\ and \GDOS}
\begin{itemize}[leftmargin=2.2em]
\item \textbf{Relation to \GG.} \GG{} outputs claims about particle physics and cosmology. MF requires that these
claims be stated together with explicit observables (cross sections, spectra, cosmological parameters) and
refutation thresholds, not qualitative stories \citep{PDG2024,Planck2018}.
\item \textbf{Relation to \GDOS.} In \GDOS{}, MF is essential because open-system claims can otherwise drift into unfalsifiable narrative.
MF forces every proposed ``fingerprint'' (spectral knee, scaling exponent, attractor statistic) to be tied to an explicit measurement contract.
\end{itemize}

\subsection{Further mathematical elaboration: MF as pre-registered hypothesis testing}
\label{sec:POS7_further}

POS-\MF\ can be expressed in the language of statistical decision theory.
A theoretical claim $\mathcal{C}$ implies a distribution (or family) for a registered estimator $\widehat{\theta}$ under a specified protocol $\Pi$:
\begin{equation}
\mathcal{C} \ \Rightarrow\ p(\widehat{\theta}\,|\,\mathcal{C},\Pi).
\end{equation}
A \emph{falsifier} is then a pre-declared rejection region $R_\alpha$ at significance level $\alpha$:
\begin{equation}
\text{Kill criterion:}\qquad \widehat{\theta}\in R_\alpha \ \Longrightarrow\ \text{reject }\mathcal{C}.
\end{equation}
For example, if \GG\ predicts a sharp coupling or rational ratio, the MF contract must declare the estimator, the systematic uncertainty model,
and the decision rule (e.g.\ a likelihood-ratio test with a stated threshold).
This is not a sociological preference: it is what converts a mathematical derivation into an empirically audit-ready scientific statement.

\section{Inter-postulate dependency chain and why this ordering}
\label{sec:postulate_dag}

The seven postulates compose into a directed
acyclic graph rather than a flat list. A compact linear
representation of the DAG is
\begin{equation}
\boxed{\;
\begin{aligned}
&\POS{1}~\GI{} \;\Rightarrow\; \POS{2}~\UEP{} \;\Rightarrow\; \POS{3}~\AF{} \;\Rightarrow\; \POS{4}~\CP{} \\
&\qquad\Rightarrow\; \POS{5}~\RG{} \;\Rightarrow\; \POS{6}~\OGN{} \;\Rightarrow\; \POS{7}~\MF{}
\end{aligned}
\;}
\label{eq:postulate_DAG}
\end{equation}
where ``$\Rightarrow$'' reads ``is a prerequisite for the clean
formulation of.'' The meaning of the chain unfolds in three groups. \emph{First, the kinematic core:} $\POS{1}$ fixes the symmetry grammar and the admissible notion of ``same physics under relabeling,'' $\POS{2}$ supplies the dynamical engine consistent with that grammar, and $\POS{3}$ ensures the symmetry grammar survives quantization. \emph{Next, the operational layer:} $\POS{4}$ ensures the operational evolution remains inside the physical state space (no negative probabilities), $\POS{5}$ ensures that operational semantics and predictions do not depend on an arbitrary resolution choice, and $\POS{6}$ removes residual ambiguity in conventions, killing basis drift and parameter laundering. \emph{Finally,} $\POS{7}$ closes the scientific loop by demanding that every statement compile to a falsifiable contract with data.

Two interlocks are especially important for exporting beyond particle physics:

\begin{enumerate}[leftmargin=2.2em,label=\textbf{I\arabic*.}]
\item \textbf{\OGGEP + \CP\ $\Rightarrow$ admissible open dynamics.} Operational semantics plus positivity forces
CPTP instruments and (under continuity) \GKLS\ generators. This is the foundational reason \GGTheory{}\ treats open-system
dynamics as axiomatic rather than ``effective modeling.''
\item \textbf{\AF + \OGN\ $\Rightarrow$ discrete locking.} Anomaly cancellation provides integer/topological constraints,
while \OGN{} fixes their canonical normalization so they become auditable discrete data rather than scheme-dependent numbers.
\end{enumerate}

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

\section{Quick-reference equation for instruments}
\label{sec:postulates:objects}

Before the kernel-equation collection and the summary table, we recall a single equation that is referenced repeatedly below. The full operational scaffolding --- states, channels, instruments, POVMs, and the coarse-graining map $\RG$ --- was fixed in \S\ref{sec:postulates_formal_setting}. For a quantum instrument $\{\mathcal{I}_m\}_m$ (CP maps whose sum is TP) \citep{Kraus1971,NielsenChuang2010}, outcome probabilities and post-measurement states are
\begin{equation}
p(m)=\Tr[\mathcal{I}_m(\rho)],\qquad
\rho_m=\frac{\mathcal{I}_m(\rho)}{\Tr[\mathcal{I}_m(\rho)]},
\qquad
\Phi(\rho)=\sum_m \mathcal{I}_m(\rho),
\label{eq:instrumentReminder}
\end{equation}
which is the operator-theoretic formalization of ``registered measurement'' used in \OGGEP{} (AX-3) and in the row entries of Table~\ref{tab:postulates_summary} below.

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

\section{Canonical kernel equations referenced in Table~\ref{tab:postulates_summary}}
\label{sec:postulates:kernel_equations}

For convenience, we collect the minimal ``kernel equations'' that are repeatedly referenced as consistency checks across the seven postulates.
The intent is not to exhaust derivations (which appear in the individual postulate subsections above), but to fix \emph{unambiguous symbols and labels}
for the summary table and for later \GG{}/\GDOS{} papers.

\paragraph{POS-1 (\GI): gauge invariance (LGI/\DGI{}/GGI).}
Local gauge invariance at the action level is
\begin{equation}
S[A^g,\psi^g]=S[A,\psi].
\label{eq:LGI_action_post}
\end{equation}
Dynamical gauge invariance in open evolution can be stated as equivariance of the channel family under the unitary representation $U_g$:
\begin{equation}
\Phi_t\circ\mathrm{Ad}_{U_g}=\mathrm{Ad}_{U_g}\circ\Phi_t,\qquad \mathrm{Ad}_{U_g}(\rho)=U_g\rho\,U_g^\dagger,
\label{eq:DGI_post}
\end{equation}
and the same requirement can be imposed on the generator $\mathcal{L}$ in a Markov limit. We also define a generic ``ledger with ports'' balance law,
\begin{equation}
\frac{d}{dt}F\big(y(t)\big)=J_{\rm ports}\big(y(t),t\big),
\label{eq:ledger_ports}
\end{equation}
where $F$ is a chosen \GFF{} ledger functional and $J_{\rm ports}$ is the net flux through exchange channels; POS-\GI\ requires this statement be form-invariant under gauge relabelings.

\paragraph{POS-2 (\UEP): unified extremal principle.}
In closed sectors, dynamics is extremal:
\begin{equation}
\delta S[\phi]=0.
\label{eq:PLA_post}
\end{equation}
In open operational regimes, a canonical boundary form is gradient/steepest-descent flow with mobility $M\succeq 0$,
\begin{equation}
\dot y=-M(y)\,\nabla F(y),
\label{eq:GDOS_gradient_post}
\end{equation}
which implies a Lyapunov monotonicity (free-energy decrease) when $M$ is positive:
\begin{equation}
\frac{dF}{dt}\le 0.
\label{eq:F_monotone_post}
\end{equation}
For fluctuations and rare events, a large-deviation functional provides the operational extremal principle:
\begin{equation}
\mathbb{P}[y(\cdot)]\asymp e^{-I[y]},\qquad I[y]=\int L\!\big(y,\dot y\big)\,dt.
\label{eq:LD_post}
\end{equation}

\paragraph{POS-3 (\AF): anomaly freedom / cancellability by inflow.}
At the quantum level, a gauge variation of the effective action has the schematic form
\begin{equation}
\delta_\alpha \Gamma=\int \alpha\,\mathcal{A},
\label{eq:anomaly_variation}
\end{equation}
and admissibility requires cancellation, possibly by inflow via a Green--Schwarz term
\begin{equation}
S_{\rm GS}=\int B_2\wedge X_4,
\label{eq:SGS_post}
\end{equation}
with modified Bianchi identity
\begin{equation}
dH_3=X_4.
\label{eq:Bianchi_post}
\end{equation}
In the \GG{} context we parameterize the admissible GS-modified \BIsix{} identity by the canonical
four-component coefficient vector (\citealp{ArisakaA2GG6}; \citealp{ArisakaA3BI6})
\begin{multline}
dH_3 \;=\; a\,\tr(R\wedge R) + b\,\tr(F_{\mathrm{SU}(5)}\wedge F_{\mathrm{SU}(5)})
 + c\,F_B\wedge F_B + d\,F_{Qi}\wedge F_{Qi},
\\
(a,b,c,d)=(1,\,-1,\,-2/5,\,-2/5),
\label{eq:BI6}
\end{multline}
where $R$ is the bulk Lorentz curvature two-form (in the $\mathrm{Spin}(1,5)$ factor of $G_{\mathrm{tot}}$),
$F_{\mathrm{SU}(5)}$ is the unified non-Abelian gauge curvature, $F_B$ the $\mathrm{U}(1)_B$ field strength,
and $F_{Qi}$ the $\mathrm{U}(1)_{Qi}$ field strength. The geometry-gauge trace-ratio lock
$|a/c|=|a/d|=5/2$ is the axiomatic signature of \BIsix{} and is forced by POS-\AF{} together with
POS-\OGN; its differential-cohomology proof is given in \citep{ArisakaA3BI6}.

\paragraph{POS-4 (\CP): closure and positivity.}
Quantum operational evolution must be completely positive and trace preserving, i.e.\ admit a Kraus form
\begin{equation}
\Phi(\rho)=\sum_k K_k\rho K_k^\dagger,\qquad \sum_k K_k^\dagger K_k=\mathbb{I},
\label{eq:Kraus_post}
\end{equation}
and in a Markov limit the generator must be of GKLS form
\begin{equation}
\dot\rho=\mathcal{L}(\rho)=-i[H,\rho]+\sum_a\left(L_a\rho L_a^\dagger-\frac12\{L_a^\dagger L_a,\rho\}\right).
\label{eq:GKSL_post}
\end{equation}
In classical mesoscopic limits, positivity is preserved by Fokker--Planck generators such as
\begin{equation}
\partial_t p = -\nabla\cdot(bp)+\frac12\nabla\cdot\!\left(D\,\nabla p\right).
\label{eq:FokkerPlanck_post}
\end{equation}

\paragraph{POS-5 (\RG): RG-compatibility and scale covariance.}
Operational semantics must commute with coarse-graining:
\begin{equation}
\Dmap\circ \RG=\RG\circ \Dmap,
\label{eq:RG_commute_post}
\end{equation}
and couplings flow by beta functions in a chosen scheme,
\begin{equation}
\mu\frac{dg_i}{d\mu}=\beta_i(g).
\label{eq:beta_post}
\end{equation}
In the 6D bulk used by \GG{} (in the natural \GG{} units with no continuous parameters), the scale-covariant
metric takes the canonical form \citep{ArisakaA2GG6}
\begin{equation}
ds_6^2 \;=\; e^{-2r}\,\eta_{\mu\nu}\,dx^\mu dx^\nu + dr^2 + d\theta^2,
\qquad r\in[0,K], \quad \theta\in[0,2\pi),
\label{eq:warpedmetric_post}
\end{equation}
where $K = 35$ is the integer lock-length (a consequence of the \OGNfive{} ledger; see Eq.~\eqref{eq:OGN5_post}),
the radial coordinate $r$ implements RG evolution along the bulk, and the compact $\theta$ circle supplies the
$\mathrm{U}(1)$ holonomy used in \OGGEP. Translation in $r$ commutes with the operational dictionary, in agreement with POS-\RG.

\paragraph{POS-6 (\OGN): Occam gauge normalization.}
Topological/quantization conditions fix discrete normalizations, e.g.
\begin{equation}
\frac{1}{2\pi}\int_{\Sigma_2}F \in \mathbb{Z},
\label{eq:flux_quant_post}
\end{equation}
and \GG{} uses the ``frozen'' integer data
\begin{equation}
(M,N,L;J,K)=(1,3,5;7,35),
\label{eq:OGN5_post}
\end{equation}
as a canonical minimal-rational normalization (no continuous rescaling dials).

\paragraph{POS-7 (\MF): measurement and falsifiability.}
A measurement is an instrument $\{\mathcal{I}_m\}_m$ whose sum is a channel,
\begin{equation}
\Phi(\rho)=\sum_m\mathcal{I}_m(\rho),
\label{eq:instrument_post}
\end{equation}
with induced POVM elements $E_m=\mathcal{I}_m^\ast(\mathbb{I})$ and Born probabilities
\begin{equation}
p(m)=\Tr(\rho E_m).
\label{eq:BornPOVM_post}
\end{equation}
MF requires that each theorem compile to a pre-declared contract (protocol, estimator, prediction, kill rule), schematically
\begin{equation}
(\mathcal{I},O,\mathbb{P}_\theta,K).
\label{eq:compile_post}
\end{equation}

\section{Comprehensive summary table for POS-1--POS-7}
\label{sec:postulates:table}

{\footnotesize
\begin{longtable}{>{\raggedright\arraybackslash}p{0.09\textwidth}
 >{\raggedright\arraybackslash}p{0.22\textwidth}
 >{\raggedright\arraybackslash}p{0.30\textwidth}
 >{\raggedright\arraybackslash}p{0.20\textwidth}
 >{\raggedright\arraybackslash}p{0.16\textwidth}}
\caption[Postulates summary (POS-1--POS-7).]{\textbf{Postulates summary (POS-1--POS-7).} Ordered admissibility filters with axiomatic content, canonical equations/checks, \GG{}/\GDOS{} bridges, and falsifiability handles. Read this table as the ``runtime checklist'' for any candidate model instance: a theory is not admitted unless each row can be satisfied without hidden dials.}
\label{tab:postulates_summary}\\[0.35em]
\toprule
\textbf{POS} & \textbf{Constitutional statement (rigorous)} &
\textbf{Mathematical kernel} &
\textbf{Bridge to \GG{} / \GDOS{}} &
\textbf{Falsifiability handle} \\
\midrule
\endfirsthead

\toprule
\textbf{POS} & \textbf{Constitutional statement (rigorous)} &
\textbf{Mathematical kernel} &
\textbf{Bridge to \GG{} / \GDOS{}} &
\textbf{Falsifiability handle} \\
\midrule
\endhead

\midrule
\multicolumn{5}{r}{\small (continued)}\\
\endfoot

\bottomrule
\endlastfoot

POS-1 & Layered gauge invariance: LGI (action), \DGI{} (generator equivariance), GGI (global ledgers). &
$S[A^g,\psi^g]=S[A,\psi]$ \eqref{eq:LGI_action_post}; $\Phi_t\circ\mathrm{Ad}_{U_g}=\mathrm{Ad}_{U_g}\circ\Phi_t$ \eqref{eq:DGI_post}; ledger continuity \eqref{eq:ledger_ports}. &
Locks bulk gauge content and boundary ports; makes open-system laws representation-independent. &
Gauge/basis changes must not alter registered predictions; illegal if they do. \\[0.6em]

POS-2 & One variational engine: PLA in closed sectors projects to PLT/FEP/\MEPP{} in open operational sectors. &
$\delta S=0$ \eqref{eq:PLA_post}; gradient flow \eqref{eq:GDOS_gradient_post}; $dF/dt\le 0$ \eqref{eq:F_monotone_post}; path LD action \eqref{eq:LD_post}. &
Bulk geodesics/extremals (\GG{}) compile to boundary optimal control/relaxation (\GDOS{}). &
Compare predicted extremal paths, relaxation laws, and time-to-target scaling to data. \\[0.6em]

POS-3 & Quantum consistency: all anomalies cancel, possibly by inflow (Green--Schwarz) with a fixed Bianchi identity. &
$\delta\Gamma=\int \alpha\mathcal{A}$ \eqref{eq:anomaly_variation}; $S_{\mathrm{GS}}=\int B_2\wedge X_4$ \eqref{eq:SGS_post}; $dH_3=X_4$ \eqref{eq:Bianchi_post}. &
Forces discrete charge/representation data and locks Abelian sectors (\GG{}); stabilizes ledger symmetries in \GDOS{}. &
If any required symmetry is anomalous/uncancellable, model is rejected. \\[0.6em]

POS-4 & Physicality: algebras close; states remain positive; open evolutions are CPTP; Markov limits are GKSL. &
Kraus form \eqref{eq:Kraus_post}; GKSL \eqref{eq:GKSL_post}; classical positivity \eqref{eq:FokkerPlanck_post}. &
Ensures bulk/boundary descriptions produce valid probabilities; fixes \GDOS{} runtime class. &
Look for negative probabilities/ghosts or non-CP evolution; these falsify admissibility. \\[0.6em]

POS-5 & Scale coherence: coarse-graining commutes with observation; radial coordinate is RG scale. &
$\Dmap\circ\RG=\RG\circ\Dmap$ \eqref{eq:RG_commute_post}; $\mu dg/d\mu=\beta(g)$ \eqref{eq:beta_post}; scale-profile metric \eqref{eq:warpedmetric_post}. &
Identifies $r$ with log energy in \GG{}; constrains \GDOS{} coarse-graining dependence. &
Predictions must be RG invariant (dimensionless) or have controlled flows; otherwise illegal. \\[0.6em]

POS-6 & No basis/scheme drift: canonical minimal integer/rational normalization across sectors. &
Flux quantization \eqref{eq:flux_quant_post}; lattice primitivity; frozen integers \eqref{eq:OGN5_post}. &
Converts inflow/locking into sharp rational predictions (\GG{}) and prevents unit drift in \GDOS{}. &
If ``predictions'' change under legal re-normalizations, model violates \OGN{}. \\[0.6em]

POS-7 & Every claim compiles to an operational test with protocol + kill criterion. &
Instrument/POVM \eqref{eq:instrument_post}--\eqref{eq:BornPOVM_post}; compile tuple \eqref{eq:compile_post}. &
Enforces instrument-level semantics of \OGGEP{} and makes \GG{}/\GDOS{} outputs scientifically auditable. &
A pre-declared kill criterion exists for every theorem; otherwise claim is inadmissible. \\

\bottomrule
\end{longtable}
}


\section{The reality register of the postulates}
\label{sec:postulates:reality_register}

One further audit closes Part~III. The bulk clauses of this
constitution are stated over the reals, and the boundary's imaginary
unit is derived (T-P4-1/2/3 of \citealp{ArisakaP4Real};
constitutional pricing in L-Constitution-1,
\S\ref{sec:AX4_category_axis}, candidate grade). It is therefore a
fair --- and auditable --- question where, if anywhere, each
postulate touches complex structure.
Table~\ref{tab:reality_register} answers it postulate by postulate.

\begin{table}[!htbp]
\centering
\footnotesize
\caption[The reality register of the postulates.]{\textbf{The reality register of the postulates.} For each
of POS-1--POS-7: where complex structure appears in its statement or
supporting apparatus, on which side of the bulk/boundary divide it
lives, and which result owns it. POS-2, POS-3, and POS-5 are clean
or bulk-real; POS-1 and POS-6 touch complex structure only in
boundary rendering; POS-4 consumes complex-algebra theorems
lawfully, downstream of the derived $i$; POS-7 inherits POS-4's
apparatus.}
\label{tab:reality_register}
\begin{tabular}{@{}
 >{\raggedright\arraybackslash}p{0.11\textwidth}
 >{\raggedright\arraybackslash}p{0.42\textwidth}
 >{\raggedright\arraybackslash}p{0.13\textwidth}
 >{\raggedright\arraybackslash}p{0.26\textwidth}@{}}
\toprule
\textbf{Postulate} & \textbf{Where complex structure appears} &
\textbf{Side} & \textbf{Owning result} \\
\midrule
POS-1 (\GI{}) & In the supporting apparatus only: $U_g$,
$U_\alpha=e^{i\alpha Q}$, GKLS covariance, and the charge
decomposition $[Q,L_k]=q_k L_k$ (a complex-eigen-operator
statement, real-form recoverable). & Boundary; derived & T-P4-1/2
(pedigree note, \S\ref{sec:POS1}) \\[0.4em]
POS-2 (\UEP{}) & Absent: action principles, gradient flows, ledger
monotones, and large-deviation functionals are real-valued
throughout. & --- (clean) & Clean on inspection \\[0.4em]
POS-3 (\AF{}) & Absent from the postulate's own data: the anomaly
polynomial and the inflow term are real characteristic classes; the
postulate itself states the inversion --- the boundary quantum
``anomaly'' is the \OGGEP{} projection shadow of a classical,
topological, purely geometric bulk identity. & Bulk; real &
POS-3's own statement (\S\ref{sec:POS3}); \AThree{}
\citep{ArisakaA3BI6} \\[0.4em]
POS-4 (\CP{}) & Constitutively: complete positivity, Choi,
Stinespring, and GKLS are theorems of complex operator algebras,
consumed as such downstream of the derived $i$; the bulk never runs
GKLS. & Boundary; derived & Custody note (\S\ref{sec:POS4});
T-P4-1/2/3 \\[0.4em]
POS-5 (\RG{}) & Absent: coarse-graining maps, beta functions, and
the radial-scale identification are real. & --- (clean) & Clean on
inspection \\[0.4em]
POS-6 (\OGN{}) & In the rendering only: the holonomy normalization
reads $e^{i\theta}$ on the boundary; the constitutional object is
the circle itself. & Boundary rendering & T-P4-1 (bridging
sentence, \S\ref{sec:AX2}) \\[0.4em]
POS-7 (\MF{}) & By inheritance: instruments and POVMs act on
POS-4's boundary Hilbert space; no complex input of its own. &
Boundary; inherited & POS-4's custody note; T-P4-1/2/3 \\
\bottomrule
\end{tabular}
\end{table}

The certificate this register issues is short enough to quote:
\emph{every $i$ in this document is either boundary-side and
derived, or absent}. No clause of the constitution instantiates a
complex primitive; where the complex apparatus is consumed (POS-4,
and POS-7 through it), it is consumed on the boundary, downstream of
the derived quarter-turn, which is where its premises hold.

\section{Century-old history of the seven postulates}
\label{sec:history-partiii}

The postulates have the longer pedigree of the two lists. Gauge invariance, least action, anomaly cancellation, complete positivity, the renormalization group, normalization conventions, and the demand that a claim be measurable were each worked out by a different community for a different reason, over a century, and none of them was written as an admissibility filter at the time. This Section collects that history, postulate by postulate.

\subsection{POS-1: Gauge invariance (GI) across scales: LGI/\DGI{}/GGI}


\paragraph{1. Faraday and Maxwell --- 1845-1865.}
The historical seed of gauge invariance is Faraday's intuition of
``tubes of force'' as physical objects in space \citep{Faraday1845}, an
intuition the young Faraday already used in 1845 to predict and observe
the rotation of light polarization by a magnetic field --- the first
experimental hint that the electromagnetic potential carries more
physical content than the local field strengths reveal. The
mathematical synthesis came twenty years later in Maxwell's
\emph{Treatise on Electricity and Magnetism} \citep{Maxwell1865}.
Maxwell did not yet have the modern terminology, but his unified
equations contain the key fact: introducing the four-potential
$A_\mu$ through
$F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu$ leaves the
freedom $A_\mu \to A_\mu + \partial_\mu \alpha$ without altering any
physical observable. This is the first concrete instance of what we
now call a local gauge transformation, and it has been the prototype
of every subsequent gauge theory. Lorenz's 1867 paper
\citep{Lorenz1867} demonstrated that this freedom could be partially
fixed by a covariant condition ($\partial^\mu A_\mu = 0$) without
disturbing observables --- the first technical instance of gauge
fixing.

\paragraph{2. Hilbert and Noether --- 1915-1918.}
The second pillar emerged from Hilbert and Noether's joint work on the
variational structure of general relativity. Hilbert's 1915 derivation
of the gravitational field equations from the Einstein-Hilbert action
\citep{Hilbert1915} demonstrated that general covariance is itself a
gauge symmetry of the variational principle, and this in turn motivated
Noether's celebrated 1918 paper \citep{Noether1918}. Noether proved
that every continuous global symmetry of the action implies a
conservation law (the first theorem), and --- critically for the
constitutional reading of $\POS{1}~\GI{}$ --- that every \emph{local}
gauge symmetry forces the corresponding current to be conserved as a
Bianchi-type identity at the level of the field equations themselves
(the second theorem). Noether's two theorems are the structural reason
electric charge is conserved in QED, color charge in QCD, and the
energy-momentum tensor is divergence-free in general relativity. In
our view Noether's theorem is the most important single mathematical
fact in the history of theoretical physics; the entire axiomatic
architecture of the $\GGTheory{}$ program rests on its consequences.

\paragraph{3. Weyl's gauge principle --- 1918 / 1929.}
Hermann Weyl gave the principle its modern name and modern form.
Weyl's 1918 attempt at a unified field theory \citep{Weyl1918} promoted
a local conformal rescaling of the metric to a symmetry and coined the
term \emph{Eichinvarianz} --- ``gauge invariance.'' His specific
physical proposal turned out to be empirically wrong (the predicted
observable scaling effects are not seen), but the conceptual template
was permanent. Weyl's 1929 reformulation, in which the local
rescaling was replaced by a local phase rotation of the quantum
wavefunction $\psi \to e^{i\alpha(x)}\psi$, identified the
electromagnetic potential as the connection on a $U(1)$ principal
bundle and made gauge invariance the foundation of quantum
electrodynamics. This is the historical moment at which gauge theory
became \emph{intrinsically quantum}: a symmetry not of fields alone but
of phases attached to fields.

\paragraph{4. Kaluza-Klein: the geometry of charge --- 1921-1926.}
A parallel strand emerged from higher-dimensional geometry. Kaluza
1921 and Klein 1926 \citep{Kaluza1921,Klein1926} showed that an
electromagnetic gauge transformation in four dimensions could be
realized as a coordinate isometry of an extra dimension, with the
electromagnetic potential $A_\mu$ identified as a component of the
five-dimensional metric. In the constitutional reading of
$\POS{1}~\GI{}$, Kaluza-Klein is the historical bridge between
gauge transformations on the boundary and isometries of the bulk: the
five-dimensional Lorentz invariance of the bulk projects to
four-dimensional Lorentz invariance plus a $U(1)$ gauge symmetry on
the boundary, the prototype of every subsequent bulk-side derivation
of boundary gauge structure used by $\GGTheory{}$.

\paragraph{5. Dirac monopole and Aharonov-Bohm --- 1931 / 1959.}
Dirac's 1931 paper on magnetic monopoles \citep{Dirac1931} demonstrated
that consistency of the quantum-mechanical wavefunction in the
presence of a monopole forces electric charge to be quantized in
integer units of $e$ ($eg = n\hbar c/2$): the first physical example
of a continuous parameter being replaced by an integer through a
topological consistency requirement. Aharonov and Bohm in 1959
\citep{AharonovBohm1959} sharpened the lesson by demonstrating that
the holonomy of the gauge connection is physically observable in
electron interference even when local field strengths vanish. These
two results together established that the gauge connection has
\emph{intrinsic} physical content beyond what is locally encoded in
$F_{\mu\nu}$ --- the foundational empirical motivation for the modern
bundle-theoretic reading of gauge fields.

\paragraph{6. Yang-Mills non-Abelian extension --- 1954.}
Yang and Mills \citep{YangMills1954} extended the construction to
non-Abelian Lie groups, opening the mathematical structure on which
the Standard Model
\citep{Glashow1961,Weinberg1967,Salam1968,GrossWilczek1973,Politzer1973}
would later sit. Yang-Mills theory is, in a precise sense, the
non-Abelian completion of Maxwell's gauge freedom: the field strength
becomes
$F_{\mu\nu}^a = \partial_\mu A_\nu^a - \partial_\nu A_\mu^a
+ g f^{abc}\, A_\mu^b A_\nu^c$,
the gauge transformation acquires the inhomogeneous term
$A_\mu \to U A_\mu U^{-1} + (i/g)\,U\,\partial_\mu U^{-1}$, and the
entire structure becomes the fiber bundle of modern geometry.
Initially regarded as a curiosity because of the apparent
massless-gauge-boson problem, Yang-Mills theory became the spine of
all subsequent particle physics once spontaneous symmetry breaking
(Higgs) and asymptotic freedom (Gross-Wilczek-Politzer) were
discovered.

\paragraph{7. Ward, Takahashi, Slavnov, Taylor --- 1950 / 1957 / 1971-1972.}
The transition to the quantum theory required systematic control of
gauge identities at the operator level. John Clive Ward's 1950 paper
\citep{Ward1950} gave the original Ward identity for QED: the relation
$Z_1 = Z_2$ between vertex and electron-wavefunction renormalization
constants is forced by gauge invariance, and without it QED would not
be renormalizable. Takahashi \citep{Takahashi1957} generalized the
identity to off-shell external lines, giving what is now called the
Ward-Takahashi identity. In the non-Abelian extension, the
Slavnov-Taylor identities \citep{Slavnov1972,Taylor1971} replace simple
gauge invariance with the more delicate BRST-cohomology version, and
provide the all-orders renormalization control that makes
non-Abelian Yang-Mills theory well-defined as a quantum field theory.

\paragraph{8. Faddeev-Popov, BRST, and BV --- 1967 / 1976 / 1981.}
Faddeev and Popov \citep{tHooft1971,THooftVeltman1972} introduced the
path-integral ghost technology needed to compensate the gauge
redundancy in the functional measure. Becchi, Rouet, Stora, and
Tyutin later recognized that the resulting gauge-fixed action carries
a nilpotent fermionic symmetry --- BRST symmetry --- and that BRST
cohomology selects the physical states. Batalin and Vilkovisky
\citep{BatalinVilkovisky1981} generalized the construction to open and
reducible gauge algebras via the BV master equation, providing the
mathematical apparatus on which the modern field-theoretic
quantization of every reducible gauge system rests. In the
constitutional reading of $\POS{1}~\GI{}$, the BV formalism is what
makes the dynamical-gauge-invariance (DGI) layer technically
well-defined for the full ladder of gauge structures used by
$\GGTheory{}$.

\paragraph{9. The geometric formulation --- 1950s-1980s.}
A parallel mathematical synthesis cast gauge theory entirely in the
language of fiber bundles. Ehresmann connections, Cartan structure
equations, and the principal-bundle picture matured through the
1950s and 1960s. Kobayashi and Nomizu's two-volume \emph{Foundations
of Differential Geometry} \citep{KobayashiNomizu1963} provided the
canonical textbook synthesis; Nakahara's \emph{Geometry, Topology and
Physics} \citep{Nakahara2003} updated it for the modern generation.
The decisive bridge to physics is the Wu-Yang ``dictionary''
\citep{WuYang1975}, which translates physicist notation ($A_\mu$,
$F_{\mu\nu}$) to mathematician notation (connection one-form,
curvature two-form on a principal bundle), and demonstrated that
non-trivial bundles (Dirac monopoles, instantons) are mandatory once
gauge theory is read geometrically. After Wu-Yang, gauge theory and
fiber-bundle geometry are not two languages for the same physics ---
they \emph{are} the same physics.

\paragraph{10. Wess-Zumino consistency and the cohomological reading --- 1971.}
The recognition that gauge invariance interacts non-trivially with
quantization crystallized in the Wess-Zumino consistency conditions
\citep{WessZumino1971}: a gauge anomaly is a cocycle in the
gauge-cohomology complex. This is the prefiguration of
$\POS{3}~\AF{}$: gauge invariance survives quantization only if the
anomaly cocycle vanishes (or is canceled by inflow). In the
constitutional reading of $\POS{1}~\GI{}$, the Wess-Zumino conditions
are the algebraic-cohomological statement that local gauge invariance
imposes constraints not only on the classical Lagrangian but on the
full quantum effective action.

\paragraph{11. 't Hooft anomaly matching --- 1980.}
't Hooft \citep{THooftNaturalness1980} sharpened the discrete content
of gauge invariance to an exact non-perturbative constraint: anomaly
coefficients are gauge invariants that must match between UV and IR
descriptions of any consistent theory. In the constitutional reading
of $\POS{1}~\GI{}$, anomaly matching is the structural reason
discrete gauge data --- integer charges, representation indices,
generation count --- survive RG flow and become \emph{audited}
observables. This is the deepest existing technical justification
for treating gauge invariance as a primary axiomatic filter rather
than a derived feature.

\paragraph{12. Stinespring-Kraus-GKLS and dynamical gauge invariance --- 1955 / 1971 / 1976.}
A separate strand emerged in the open-systems literature.
Stinespring's 1955 dilation theorem \citep{Stinespring1955} showed
that every completely positive map admits a unitary extension on a
larger Hilbert space; Kraus's operator-sum representation
\citep{Kraus1971,Kraus1983} provided the working notation; the
Gorini-Kossakowski-Sudarshan-Lindblad classification
\citep{Lindblad1976,Gorini1976} characterized the most general
Markovian quantum dynamics that preserve trace and positivity. When
this open-systems language is coupled to the gauge formalism, the
requirement of gauge-covariance lifts naturally to generators and
channels: an admissible Markov semigroup $\Phi_t$ must satisfy
$\Phi_t \circ \mathrm{Ad}_{U_g} = \mathrm{Ad}_{U_g} \circ \Phi_t$,
extending the closed-system gauge invariance to the open dynamical
regime. This is the technical content of the
\emph{dynamical-gauge-invariance} (DGI) layer of $\POS{1}~\GI{}$.

\paragraph{13. Algebraic and noncommutative-geometry formulations --- 1962-1994.}
A complementary reformulation casts gauge invariance as a statement
about the algebra of observables rather than about a specific
Lagrangian. Doplicher, Haag, and Roberts \citep{DoplicherHaagRoberts1971}
showed that superselection sectors and statistics can be reconstructed
from the algebra of local observables alone; Haag's monograph
\citep{Haag1992} provides the canonical synthesis.
Connes' \emph{Noncommutative Geometry} \citep{Connes1994} pushes the
reformulation further: the Standard Model becomes a spectral-geometric
theory in which gauge fields arise as fluctuations of a noncommutative
Dirac operator. In the constitutional reading of $\POS{1}~\GI{}$,
these works show that gauge invariance need not be tied to a specific
field-theoretic formulation; it can be stated as a property of the
observable algebra, supporting the algebra-first reading used by
$\GGTheory{}$ in its treatment of measurement (AX-3, OGGEP).

\paragraph{14. No-global-symmetry conjecture --- 2010-2019.}
The contemporary philosophical case for the primacy of gauge invariance
was crystallized by Banks and Seiberg \citep{BanksSeiberg2011} and
sharpened by Harlow and Ooguri \citep{HarlowOoguri2019}: in any
consistent theory of quantum gravity, every global symmetry must
either be gauged or be broken. Apparent global symmetries are at
best approximate; at the deepest level only gauge symmetries are
admissible. In the constitutional reading of $\POS{1}~\GI{}$,
the no-global-symmetry conjecture is the deepest philosophical
justification for elevating gauge invariance to the foundational
postulate of $\GGTheory{}$ --- whatever the UV completion turns out to
be, gauge invariance is the discipline that must be respected.

\paragraph{The axiomatic move of $\POS{1}~\GI{}$.}
$\POS{1}~\GI{}$ elevates gauge invariance from a structural feature of
the standard textbook to an axiomatic precondition for admissibility,
layered into three explicit strata: LGI (closed-system local gauge
invariance, the Maxwell-Weyl-Yang-Mills tradition), $\DGI{}$
(dynamical gauge invariance for open semigroups, the
Stinespring-Kraus-GKLS extension), and GGI (global ledger gauge
invariance for the operational dictionary). POS-1 GI first becomes operative at Stratum 1 of Part V,
where Maxwell's $U(1)$ gauge action provides the prototype; the
layered LGI / DGI / GGI structure is then progressively activated
across the higher strata, with Yang-Mills's non-Abelian extension
filling out Stratum 2, the dynamical-gauge-invariance layer
(open-system equivariance) requiring AX-3 OGGEP and so unlocking at
Stratum 3, and the global-gauge-invariance ledger fully
structurally locked at Stratum 4. After all, the
century-long arc from Faraday's tubes of force through Maxwell's
$A_\mu \to A_\mu + \partial_\mu \alpha$, Hilbert-Noether 1915-1918,
Weyl's local phase, Kaluza-Klein, Dirac-Aharonov-Bohm, Yang-Mills,
Ward-Takahashi-Slavnov-Taylor, Faddeev-Popov-BRST-BV, the
fiber-bundle synthesis, Wess-Zumino consistency, 't Hooft matching,
Stinespring-DGI, the algebraic-NCG reformulation, and the
no-global-symmetry conjecture is a single coherent thread, and
$\POS{1}~\GI{}$ is the axiomatic commitment that any admissible
candidate theory must respect every link of it.

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

\subsection{POS-2: Unified Extremal Principle (\UEP{}): PLA $\Longleftrightarrow$ PLT/\MEPP{}/FEP}


\paragraph{1. Heron and the ancient extremal tradition --- 1st c.}
The deepest historical anchor for $\POS{2}~\UEP{}$ is Heron of
Alexandria's first-century argument \citep{Heron} that light's path
of equal angles minimizes the distance --- the first recorded extremal
principle in physics. Although the proof relied on Euclidean geometry
rather than variational calculus, the move is the same one every
subsequent physicist has made: identify the dynamics as the stationary
point of a scalar.

\paragraph{2. Fermat's principle of least time --- 1662.}
Fermat \citep{Fermat1662} derived Snell's law of refraction by
demanding that light minimize its travel time in a medium of varying
refractive index. Fermat's principle is the historical seed of every
variational principle in physics: a single scalar quantity (time, in
this instance) whose stationary points pick out the admissible
trajectories. In the constitutional reading of $\POS{2}~\UEP{}$,
Fermat is the deepest historically clean example of the
$\PLT{}$ (principle of least time) face of the unified extremal
engine.

\paragraph{3. Maupertuis's principle of least action --- 1744.}
Maupertuis \citep{Maupertuis1744} extended Fermat's discipline from
optics to mechanics: $\delta \int p\,dq = 0$. Maupertuis's principle
was initially controversial (he framed it in theological terms,
attributing the parsimony to divine economy), but its mathematical
content was permanent. Action became, in this single move, the
universal currency of dynamics.

\paragraph{4. Euler-Lagrange synthesis --- 1744-1788.}
Euler's 1744 \emph{Methodus inveniendi}
\citep{EulerCalcVar} established the calculus of variations as a
rigorous mathematical discipline; Lagrange's 1788
\emph{M\'ecanique analytique} \citep{Lagrange1788} cast classical
mechanics entirely as a variational problem on configuration space.
The Euler-Lagrange equations $\partial L/\partial q - (d/dt)\,\partial
L/\partial \dot q = 0$ became the textbook machinery on which every
subsequent variational principle in physics is built.

\paragraph{5. Hamilton's principle --- 1834-1835.}
Hamilton's two essays on general methods in dynamics
\citep{Hamilton1834,Hamilton1835} reformulated mechanics on phase
space, introducing what we now call the Hamiltonian and the canonical
equations. Hamilton's principle $\delta \int L \, dt = 0$ is the
bridge to Hamilton-Jacobi theory and the geometric-optics limit of
quantum mechanics (the WKB approximation), supplying the conceptual
foundation on which Feynman's path integral would later sit.

\paragraph{6. Onsager's reciprocal relations --- 1931.}
Onsager \citep{Onsager1931} extended variational thinking to
dissipative systems near equilibrium. The Onsager reciprocal
relations $L_{ij} = L_{ji}$ for linear response coefficients are the
first systematic statement that the variational structure of physics
extends from closed Lagrangian sectors to open thermodynamic ones.
Together with the Onsager-Machlup stochastic action
\citep{OnsagerMachlup1953}, this is the boundary-side counterpart of
the Lagrangian PLA: the same scalar-functional discipline applied to
trajectories with noise.

\paragraph{7. Prigogine's minimum entropy production --- 1947.}
Prigogine \citep{Prigogine1947} showed that linear non-equilibrium
steady states minimize entropy production. Together with the later
generalization by Nicolis-Prigogine \citep{NicolisPrigogine1977},
this established the variational structure of irreversible
thermodynamics. In the constitutional reading of $\POS{2}~\UEP{}$,
Prigogine 1947 is the canonical example of how the variational
engine of physics extends from action principles to entropy/free-energy
principles --- the $\MEPP{}$ face of the unified extremal engine.

\paragraph{8. Jaynes's maximum-entropy principle --- 1957.}
Jaynes \citep{Jaynes1957} proposed that the unique unbiased
probability distribution consistent with a set of registered
constraints is the one that maximizes Shannon entropy. MaxEnt is
the information-theoretic dual of action principles: every classical
extremal principle has a Bayesian shadow in which the same problem
appears as a posterior maximization under a registered prior. In
the constitutional reading of $\POS{2}~\UEP{}$, Jaynes 1957 is the
historical bridge between the dynamical and the inferential faces of
the unified extremal engine.

\paragraph{9. Feynman path integral --- 1948.}
Feynman \citep{Feynman1948} reformulated quantum mechanics as a
sum over histories weighted by $e^{iS[\phi]/\hbar}$. The classical
extremal trajectory emerges as the stationary phase of the path
integral, and the Schr\"odinger equation as its high-action limit.
This is the cleanest unification of the closed-system PLA with
quantum amplitudes; the path integral remains the deepest technical
realization of $\POS{2}~\UEP{}$ in the closed quantum sector.

\paragraph{10. Hamilton-Jacobi and the WKB limit --- 1836 / 1926.}
A separate thread runs from Hamilton-Jacobi theory (1836) to the
WKB approximation in quantum mechanics (1926). The Hamilton-Jacobi
equation $\partial_t S + H(q, \partial_q S) = 0$ is the
geometrical-optics limit of the Schr\"odinger equation, and the
classical extremal trajectory is the stationary phase of the WKB
wavefunction. In the constitutional reading of $\POS{2}~\UEP{}$,
Hamilton-Jacobi/WKB is the technical apparatus that connects the
$\PLA{}$ and $\PLT{}$ faces of the unified extremal engine through
a single derivation.

\paragraph{11. Stochastic thermodynamics: Jarzynski / Crooks / Seifert --- 1997-2012.}
The modern operational counterpart of variational thermodynamics is
the trajectory-level fluctuation-theorem program. Jarzynski
\citep{Jarzynski1997} derived a non-equilibrium work relation
$\langle e^{-\beta W}\rangle = e^{-\beta \Delta F}$; Crooks
\citep{Crooks1999} sharpened it into a trajectory-level identity;
Seifert \citep{Seifert2012} surveys the field. These results
demonstrate that the variational structure of physics persists at
the trajectory level, supplying large-deviation-functional
realizations of the unified extremal engine even far from
equilibrium.

\paragraph{12. Friston's free-energy principle --- 2010.}
Friston \citep{Friston2010} proposed that biological systems
minimize a variational free energy that combines information-theoretic
surprise with a model-complexity penalty. FEP is the cleanest
contemporary demonstration that the same extremal mathematics
governs both physics and adaptive biological systems. In the
constitutional reading of $\POS{2}~\UEP{}$, Friston 2010 supplies the
deepest contemporary evidence that the unified extremal engine is
not a physics-specific accident but a deep structural property of
admissible operational dynamics generally.

\paragraph{13. Modern large-deviation framework --- 1998-2009.}
The contemporary mathematical apparatus that turns variational
thermodynamics into a rigorous discipline is large-deviation theory.
Freidlin-Wentzell \citep{FreidlinWentzell1998} for stochastic
differential equations and Touchette \citep{Touchette2009} for
statistical-mechanics applications provide the canonical
references. In the constitutional reading of $\POS{2}~\UEP{}$,
large-deviation theory is the modern probabilistic backbone of every
``open-system variational principle'' in physics: the path-level
quantity $I[y] = \int L(y, \dot y)\, dt$ that controls rare-event
probabilities is the same extremal structure that
Onsager-Machlup, Jarzynski, and Friston use in their respective
domains.

\paragraph{The axiomatic move of $\POS{2}~\UEP{}$.}
$\POS{2}~\UEP{}$ does not claim that any single historical extremal
principle applies universally as originally stated. It is the
constitutional claim that there exists a \emph{single} variational
engine, with multiple operational representations ($\PLA{}$ in the
bulk, $\PLT{}$ for trajectories, $\MEPP{}$ for currents, FEP for
inference) that are duals of each other under the bulk-to-boundary
projection of $\OGGEP{}$. POS-2 UEP is enforced from Stratum 1 onward --- Maxwell's
variational principle is already a UEP instance --- and the
open-operational PLT / MEPP / FEP representations unlock at Stratum 3,
where AX-3 OGGEP supplies the bulk-to-boundary projection
that makes the operational variational forms admissible. After all, the
three-and-a-half-century arc from Heron through Fermat, Maupertuis,
Euler-Lagrange, Hamilton, Hamilton-Jacobi, Onsager, Prigogine,
Jaynes, Feynman, Crooks-Jarzynski-Seifert, Friston, and modern
large-deviation theory is a single coherent thread, and
$\POS{2}~\UEP{}$ is the axiomatic commitment that any admissible
candidate theory must respect every link of it.

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

\subsection{POS-3: Anomaly freedom and cancellability by inflow (AF)}


\paragraph{1. Riemann-Cartan integrability --- 1854-1900.}
The pre-history of anomaly freedom is the integrability tradition of
nineteenth-century differential geometry. Riemann's habilitation
lecture \citep{Riemann1854} introduced the curvature tensor of a
metric manifold and the integrability conditions that any consistent
geometry must satisfy. Ricci-Curbastro, Levi-Civita, and the Italian
school of absolute differential calculus extended these conditions to
arbitrary tensor fields, and Elie Cartan's structure equations
(1922-1925) generalized the construction to non-Abelian connections.
The Bianchi-type identity $dF + [A,F] = 0$ that every gauge field
strength satisfies is, in this tradition, a \emph{geometric} statement
about the connection on a principal bundle --- it has no quantum
content of any kind.

\paragraph{2. Bianchi's identity for Riemann curvature --- 1902.}
Luigi Bianchi's 1902 identity \citep{Bianchi1902} extended the
integrability tradition to gravity: the Riemann curvature tensor
satisfies $\nabla_{[\mu} R_{\nu\rho]\sigma\tau} = 0$, equivalently
$dR = 0$ in differential-form language, equivalently
$\nabla_\mu G^{\mu\nu} = 0$ for the Einstein tensor. This is the
bedrock topological identity from which the entire anomaly story
descends. It is what makes the energy-momentum tensor of any matter
source automatically conserved when coupled to gravity, and it is the
prototype of every subsequent ``$dX = 0$'' identity in the constitutional
reading of $\POS{3}~\AF{}$. In Maxwell's electromagnetic theory
\citep{Maxwell1865} the abelian Bianchi identity $dF = 0$ is the
component-form statement of the homogeneous Maxwell equations
($\nabla \cdot \mathbf{B} = 0$, $\nabla \times \mathbf{E} + \partial_t
\mathbf{B} = 0$); it holds for any smooth connection on any principal
$U(1)$ bundle, regardless of sources, matter content, quantum
corrections, or path-integral measure.

\paragraph{3. The Adler-Bell-Jackiw chiral anomaly --- 1969.}
The quantum-field-theoretic side of the anomaly story begins with the
1969 discovery, by Adler \citep{Adler1969} and independently by Bell
and Jackiw \citep{BellJackiw1969}, that the axial-vector current of
QED is not exactly conserved at the quantum level. The classical
chiral symmetry of the massless Dirac action is broken by the triangle
diagram, with the divergence picking up a topological-density
correction proportional to $F_{\mu\nu}\tilde F^{\mu\nu}$. In the
constitutional reading of $\POS{3}~\AF{}$, the Adler-Bell-Jackiw
anomaly is not a pathology of quantum field theory but the
$\AX{3}~\OGGEP{}$ projection shadow of a bulk-side classical-topological
identity: the boundary's triangle-diagram measurement of its own
inability to register the deeper bulk closure $dX_4 = 0$.

\paragraph{4. Bardeen's non-Abelian extension --- 1969.}
Bardeen \citep{Bardeen1969} extended the Adler-Bell-Jackiw calculation
to non-Abelian gauge theories, obtaining the closed-form
\emph{consistent} anomaly functional $\mathcal{A}^a[A]$ that satisfies
the Wess-Zumino consistency conditions. Bardeen's calculation supplies
the full local form of the anomaly polynomial on which every
subsequent inflow-cancellation construction is built.

\paragraph{5. Wess-Zumino consistency conditions --- 1971.}
Wess and Zumino \citep{WessZumino1971} discovered that anomaly
candidates must satisfy strong consistency conditions reflecting the
group structure of the underlying gauge transformation: the anomaly is
a cocycle in the gauge-cohomology complex. This algebraic-topological
reading is the seed of the entire modern classification of anomalies
as elements of cohomology groups. In the constitutional reading of
$\POS{3}~\AF{}$, the Wess-Zumino consistency conditions are the
algebraic side of the same identity whose differential-geometric side
is Bianchi 1902.

\paragraph{6. Atiyah-Singer index theorem --- 1963 / 1968.}
The Atiyah-Singer index theorem \citep{AtiyahSinger1963,
AtiyahSinger1968} is the mathematical engine behind every subsequent
anomaly-cancellation argument: the index of a Dirac operator on a
closed manifold equals a topological integral of characteristic
classes. In the constitutional reading of $\POS{3}~\AF{}$, the
Atiyah-Singer theorem is what guarantees that boundary anomalies are
locked to bulk topological data; it is the deep mathematical reason
why anomaly freedom is, in the end, a question about integers and
characteristic classes rather than about renormalization schemes.

\paragraph{7. Fujikawa's path-integral derivation --- 1979.}
Fujikawa \citep{Fujikawa1979} showed that the anomaly can be derived
non-perturbatively from the non-trivial Jacobian of the fermionic
measure under chiral transformations. This identifies the anomaly
with a non-trivial determinant of the Dirac operator, recasting it as
a global, scheme-independent object. Fujikawa's derivation is the
technical foundation of the modern view that anomalies are
non-renormalized --- they are protected by the index theorem and so
cannot be eliminated by any choice of regulator.

\paragraph{8. Chern-Simons forms and characteristic classes --- 1974.}
Chern and Simons \citep{ChernSimons1974} introduced the
three-form whose exterior derivative is the second Chern class.
Together with the Chern-Weil construction
\citep{ChernWeil1946,KobayashiNomizu1963}, this provides the local
primitives from which every anomaly polynomial is built. In the
constitutional reading of $\POS{3}~\AF{}$, Chern-Simons forms are the
geometric objects that make inflow cancellation a local, computable
identity rather than a formal algebraic rewriting.

\paragraph{9. Stora-Zumino descent equations --- 1976 / 1985.}
Stora \citep{Stora1976} and Zumino \citep{Zumino1985} cast the full
classification of anomalies as a descent chain in BRST cohomology: an
anomaly in $n$ dimensions descends from a closed
$(n+2)$-form via the Chern-Simons primitive and the BRST differential.
This algebraic-topological scaffolding is what turns anomaly
classification from a case-by-case calculation into a question about a
single chain complex; it is one of the most elegant unifications in
twentieth-century mathematical physics.

\paragraph{10. 't Hooft anomaly matching --- 1979 / 1980.}
't Hooft \citep{THooftNaturalness1980} sharpened the
anomaly constraint to an exact non-perturbative consistency condition:
anomaly coefficients are RG invariants that must match between the UV
and IR descriptions of a theory. In the constitutional reading of
$\POS{3}~\AF{}$, anomaly matching is the structural reason a
discrete anomaly polynomial survives RG flow --- the interlock
between POS-3 (anomaly cancellation) and POS-5 (RG flow), which is
operative across Strata 2 through 4 of Part V.

\paragraph{11. Callan-Harvey anomaly inflow --- 1985.}
Callan and Harvey \citep{CallanHarvey1985} introduced the inflow
viewpoint: a chiral anomaly on a boundary or defect is canceled by a
bulk topological term that flows charge into the defect, restoring
the Bianchi identity at the bulk level. This is the first
explicit example in physics of \emph{bulk-cancels-boundary} as a
\emph{geometric} identity rather than a quantum miracle. It is the
structural template on which $\BIsix{}$ rests in $\AThree$
\citep{ArisakaA3BI6}: the bulk Bianchi $dX_4 = 0$ is a geometric
identity, the boundary anomaly is the quantum-level deviation from it,
and the four-component bulk inflow $X_4 = dH_3$ closes the loop.

\paragraph{12. Wess-Zumino-Witten topological action --- 1983.}
Witten's 1983 paper \citep{WessZuminoWitten1983} on global aspects of
current algebra developed the Wess-Zumino term as a topological action
in chiral effective field theory and showed that its coefficient is
quantized. This is the canonical worked example of how anomaly
inflow generates integer-quantized topological couplings on the
boundary, providing the cleanest illustration of the POS-3 (anomaly)
$\leftrightarrow$ POS-6 (OGN) inter-postulate interlock.

\paragraph{13. Green-Schwarz mechanism --- 1984.}
Green and Schwarz \citep{GreenSchwarz1984} discovered the
ten-dimensional inflow mechanism that saved heterotic string theory
from anomalies that would otherwise have killed it: a single
counterterm $S_{\mathrm{GS}} = \int B_2 \wedge X_4$ with the modified
Bianchi identity $dH_3 = X_4$ cancels the anomaly polynomial of
type-I and heterotic strings. This discovery is widely credited as
the moment string theory became empirically constrained rather than
infinitely flexible. Alvarez-Gaum\'e and Witten
\citep{AlvarezGaumeWitten1984} classified gravitational anomalies in
arbitrary dimensions in the same year. In the constitutional reading
of $\POS{3}~\AF{}$, Green-Schwarz is the bulk-side classical
absorption of the boundary anomaly --- the prototype of the
$\BIsix{}$ identity $dH_3 = X_4$ used by $\GG{}$.

\paragraph{14. Differential cohomology and the modern picture --- 2005-2019.}
\par\noindent\emph{\small This paragraph is written for readers with a
background in algebraic topology. Its content is a pointer to the
literature, not a step in the constitutional argument; nothing later in
this paper depends on it.}\par\medskip

The contemporary mathematical formulation casts anomalies as
\emph{invertible field theories}. Hopkins-Singer
\citep{HopkinsSinger2005} differential cohomology turned the
anomaly-inflow picture into precise long exact sequences of
differential characters; Freed, Hopkins, and Teleman
\citep{FreedHopkinsTeleman2010} provided the categorical scaffolding
relating twisted equivariant K-theory of a compact Lie group to its
Verlinde algebra; Freed's lecture notes \citep{Freed2019} formulated
anomalies as partition functions of $(n+1)$-dimensional invertible
TQFTs. In the constitutional reading of $\POS{3}~\AF{}$, these works
make the geometric-origin-of-anomaly principle mathematically precise:
every boundary anomaly is the partition function of a bulk invertible
TQFT, so anomaly cancellation is a bulk-side topological identity
rather than a quantum cancellation of one-loop pathologies.
Bertlmann's monograph \citep{Bertlmann1996} surveys the technical
apparatus accumulated through the 1990s.

\paragraph{Scope of the constitutional reading.}
It is worth stating explicitly what $\POS{3}~\AF{}$ does and does
not claim in the constitutional reading. It \emph{does} claim that
the boundary 4D ``anomaly'' of the standard textbook is the
$\AX{3}~\OGGEP{}$ projection shadow of a bulk-side classical-topological
identity descended from Bianchi 1902, and that the Green--Schwarz
construction is the bulk-side classical absorption of this identity
rather than a quantum cancellation of a one-loop pathology.
Equivalently, string theory and \AdS{}/CFT --- to the extent that
they share this bulk-side classical-topological backbone --- are
structural theorems under $\AX{2}~\GGEP{}$ (geometrization of gauge
fields), \emph{not} under the slogan ``quantization of geometry.''
It \emph{does not} claim to subsume the quantum-dynamical content
that string theorists run on top of the classical backbone --- graviton
vertex operators, $\alpha'$ and $1/N$ corrections, the 2019--2020 islands
formula of Almheiri--Engelhardt--Marolf--Maxfield and
Penington--Shenker--Stanford--Yang, the Gubser--Klebanov--Polyakov /
Witten boundary-operator dictionary, or any of the rich
perturbative-string apparatus. The cohort framework occupies the
GGEP-closed sector, the classical-topological backbone; the quantum
content running on top of that backbone is a separate physical
sector that the cohort axioms neither subsume nor contradict.
Indeed, we make the qualification explicit so that the seriousness of
what $\POS{3}~\AF{}$ actually claims is not obscured by a misreading
as a quantum-gravity displacement claim that the cohort does not make.

\paragraph{The axiomatic move of $\POS{3}~\AF{}$.}
$\POS{3}~\AF{}$ elevates anomaly freedom from a structural constraint
that one verifies case-by-case to an axiomatic precondition for
admissibility. Any candidate theory whose anomaly polynomial does
not factorize as $X_4 = a R^2 + b F_{SU(5)}^2 + c F_B^2 + d F_{Qi}^2$
with the trace-ratio lock $|a/c| = |a/d| = 5/2$ fails the $\POS{3}$
filter and is rejected. In this reading the four-component identity
$(a,b,c,d) = (1,-1,-2/5,-2/5)$ is the axiomatic fingerprint of the
entire $\GGTheory{}$ program, and the geometric Bianchi identity is
its parent. POS-3 AF first becomes a non-trivial admissibility filter at
Stratum 2 of Part V, where Yang-Mills's non-Abelian curvature and
the Green-Schwarz inflow mechanism live; in Stratum 4 the
four-component anomaly polynomial locks against the OGN-5 integer
ledger. After all, what we name ``anomaly freedom'' here is the
mature, gauge-bundle version of the same conservation-of-current
discipline that $\AX{1}~\LC{}$ imposes on the energy-momentum
side --- one identity, two faces, both purely geometric.

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

\subsection{POS-4: Closure and positivity (CP)}


\paragraph{1. Bernoulli and the law of large numbers --- 1713.}
The mathematical theory of probability begins with Jacob Bernoulli's
posthumous \emph{Ars Conjectandi} \citep{Bernoulli1713}, in which the
law of large numbers --- the assertion that empirical frequencies
converge to underlying probabilities --- was first proved. In the
constitutional reading of $\POS{4}~\CP{}$, Bernoulli is the deepest
historical anchor for the discipline that probability, taken
operationally as a registered frequency, is a legitimate physical
quantity.

\paragraph{2. Bayes and Laplace --- 1763 / 1812.}
Bayes's posthumous essay \citep{Bayes1763} introduced inverse
probability --- the inference of unknown causes from observed effects ---
in what we now call Bayes' theorem. Laplace's
\emph{Th\'eorie analytique des probabilit\'es} \citep{Laplace1812}
turned the construction into a full analytical apparatus. Together,
Bayes and Laplace supplied the inferential machinery that every
subsequent quantitative use of probability in physics depends on; in
the constitutional reading of $\POS{4}~\CP{}$, they are the
historical foundation on which Kolmogorov's axiomatization is built.

\paragraph{3. Kolmogorov's axiomatization --- 1933.}
Kolmogorov \citep{Kolmogorov1933} supplied the modern axiomatic
treatment of probability: probability measures as countably additive
non-negative functions on $\sigma$-algebras with total mass one.
This is the mathematical scaffolding inside which every positivity
requirement of $\POS{4}~\CP{}$ lives. After Kolmogorov, ``positive
probabilities'' became a structural property of the measure space
rather than a methodological preference.

\paragraph{4. Born's probability rule --- 1926.}
Born \citep{Born1926} fixed the operational meaning of the
wavefunction by identifying $|\psi(x)|^2$ with the probability density
to register an outcome at $x$. This is the prototype of every
subsequent quantum-probability statement and the historical anchor of
the entire operational interpretation of quantum mechanics.

\paragraph{5. Heisenberg-Jordan matrix mechanics --- 1925.}
Heisenberg's 1925 \emph{Umdeutung} paper \citep{Heisenberg1925}
inaugurated quantum mechanics through the canonical commutation
relation $[q, p] = i\hbar$ --- the first explicit non-commutative
algebra of physical quantities. In the constitutional reading of
$\POS{4}~\CP{}$, Heisenberg-Jordan is the historical moment at which
the algebra of observables became a primary mathematical object,
preparing the ground for the algebraic and operational reformulations
of the second half of the twentieth century.

\paragraph{6. von Neumann's mathematical foundations --- 1932.}
Von Neumann's \emph{Mathematische Grundlagen der Quantenmechanik}
\citep{vonNeumann1932} formalized the density-matrix description of
mixed states, the projection postulate, and the spectral theorem for
self-adjoint operators on Hilbert space. This is the textbook
foundation of every subsequent operationally rigorous treatment of
quantum theory. In the constitutional reading of $\POS{4}~\CP{}$,
von Neumann 1932 is the canonical statement of closed-system
kinematics + unitary closed-system dynamics + projective measurement
on which the open-system extensions of POS-4 build.

\paragraph{7. Naimark's dilation theorem --- 1940.}
Naimark \citep{Naimark1940} proved that every POVM on a Hilbert
space is the compression of a projection-valued measure (PVM) on a
larger Hilbert space. This is the structural foundation of
operationally-defined measurement: positive operators valued in $[0,
1]$ on $\mathcal{H}$ are always images of orthogonal projections on
$\mathcal{H} \oplus \mathcal{H}'$, so the operational generalization
loses nothing essential. In the constitutional reading of
$\POS{4}~\CP{}$, Naimark 1940 is what makes the POVM-based measurement
formalism of AX-3 (OGGEP) mathematically well-founded.

\paragraph{8. Stinespring's dilation --- 1955.}
Stinespring \citep{Stinespring1955} proved the central structural
theorem of open-system dynamics: every completely positive linear
map $\Phi$ on a $C^*$-algebra admits a unitary dilation
$\Phi(A) = V^\dagger \pi(A) V$ on a larger Hilbert space. Stinespring
is the operational definition of complete positivity that has
dominated the subsequent literature. Together with Naimark's
theorem, it supplies the dilation machinery on which every modern
open-system formalism rests.

\paragraph{9. Davies-Lewis instruments --- 1970.}
Davies and Lewis \citep{DaviesLewis1970} introduced the notion of an
\emph{instrument} --- a CP-map-valued random variable that produces
both a post-measurement state and a classical record. Instruments
unified ``measurement'' (one-shot, with readout) and ``channel''
(no readout) into a single formalism. In the constitutional reading
of $\POS{4}~\CP{}$, Davies-Lewis 1970 is the bridge between the
ideal projective measurement of von Neumann and the realistic
quantum experiment of every twenty-first-century laboratory.

\paragraph{10. Kraus operator-sum representation --- 1971.}
Kraus \citep{Kraus1971,Kraus1983} provided the working notation for
CP maps: $\Phi(\rho) = \sum_k K_k \rho K_k^\dagger$ with $\sum_k
K_k^\dagger K_k \le \mathbb{1}$. This is the canonical computational
form of every modern open-system calculation and the practical
embodiment of Stinespring's structural theorem.

\paragraph{11. Choi's matrix and complete positivity --- 1975.}
Choi \citep{Choi1975} characterized complete positivity through the
Choi-Jamio\l{}kowski isomorphism: a linear map $\Phi$ is CP if and
only if its Choi matrix $(\mathrm{id} \otimes \Phi)(|\Omega\rangle
\langle\Omega|)$ is positive. This is the operational test of
physicality on which every contemporary admissibility check rests.

\paragraph{12. Gorini-Kossakowski-Sudarshan-Lindblad master equation --- 1976.}
Gorini, Kossakowski, and Sudarshan \citep{Gorini1976} and
(independently) Lindblad \citep{Lindblad1976} classified the most
general Markovian generator of a CPTP semigroup: the GKLS form
$\dot\rho = -i[H, \rho] + \sum_k \gamma_k \big(L_k \rho L_k^\dagger -
\tfrac{1}{2}\{L_k^\dagger L_k, \rho\}\big)$ with Kossakowski matrix
$(\gamma_k) \succeq 0$. In the constitutional reading of
$\POS{4}~\CP{}$, the GKLS equation is the technical heart of the
dynamical sector: every admissible open-system dynamics in
$\GGTheory{}$ falls in this class. Breuer and Petruccione
\citep{BreuerPetruccione2002} and Nielsen-Chuang
\citep{NielsenChuang2010} codified the
construction for the modern generation.

\paragraph{13. Algebraic and constructive QFT --- 1956 / 1964 / 1971.}
A parallel mathematical tradition cast closure and positivity in
algebraic language. Wightman's axioms
\citep{Wightman1956,StreaterWightman1964} require closure of the
operator algebra under spacelike commutation; the Haag-Kastler
\citep{HaagKastler1964} formalism extends this to a net of
$C^*$-algebras of local observables indexed by bounded spacetime
regions. Doplicher-Haag-Roberts \citep{DoplicherHaagRoberts1971}
demonstrated that the full kinematic structure of a quantum theory
--- charges, statistics, gauge group --- can be reconstructed from
the algebra of local observables alone. Haag's
\emph{Local Quantum Physics} \citep{Haag1992} provides the
canonical synthesis. In the constitutional reading of
$\POS{4}~\CP{}$, this is the algebraic statement of closure plus
positivity for a relativistic local quantum theory, completing the
bulk side of the postulate.

\paragraph{The axiomatic move of $\POS{4}~\CP{}$.}
$\POS{4}~\CP{}$ combines algebraic closure (bulk side) with
complete positivity (boundary side) into a single admissibility
filter. Any candidate theory whose constraints fail to close, or
whose boundary semigroup fails to be CPTP, is rejected before its
specific content is examined. This rules out a large class of
otherwise plausible proposals (post-quantum theories that violate
CP, gauge theories with anomalous algebra closure, models that
predict negative probabilities at finite times). POS-4 CP first becomes structurally mandatory at Stratum 3 of
Part V, where AX-3 OGGEP forces CPTP open dynamics through the
Stinespring-Kraus-GKLS chain; classical positivity is already
enforced at the lower strata; together with the
$\OGGEP{} + \CP{}$ interlock noted in
Section~\ref{sec:postulate_dag}, this is what makes open-system
dynamics axiomatic rather than effective in $\GGTheory{}$.
After all, the three-and-a-half-century arc from Bernoulli through
Bayes-Laplace, Kolmogorov, Born, von Neumann, Naimark, Stinespring,
Davies-Lewis, Kraus, Choi, GKLS, and algebraic QFT is a single
coherent thread, and $\POS{4}~\CP{}$ is the axiomatic commitment that
any admissible candidate theory must respect every link of it.

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

\subsection{POS-5: RG-compatibility and scale covariance (RG)}


\paragraph{1. Buckingham $\pi$-theorem and dimensional analysis --- 1914.}
The pre-history of the renormalization group is dimensional analysis.
Buckingham's 1914 $\pi$-theorem \citep{Buckingham1914} stated formally
what physicists had always intuited: any physical law expressible in
consistent units must be expressible as a relation among dimensionless
ratios. Scale invariance is therefore not a special property of a few
critical theories --- it is the default mathematical structure of every
physical law. In the constitutional reading of $\POS{5}~\RG{}$,
Buckingham is the deepest historical anchor for the discipline that
the operational dictionary must commute with rescaling of units.

\paragraph{2. Lamb shift --- 1947.}
The empirical anchor for the modern renormalization program is the
Lamb-Retherford measurement of the $2S_{1/2}$--$2P_{1/2}$ splitting in
hydrogen \citep{LambRetherford1947}, demonstrating an unambiguous
deviation from the Dirac equation. Bethe's 1947 non-relativistic
calculation \citep{Bethe1947} explained the splitting by systematically
absorbing infinities into a renormalized electron mass --- the first
demonstration that renormalization is a controlled physical procedure
rather than an \emph{ad hoc} subtraction. After Lamb-Bethe,
renormalization was no longer optional.

\paragraph{3. Tomonaga-Schwinger-Feynman-Dyson --- 1946-1949.}
The full multi-loop renormalization program is due to Sin-Itiro
Tomonaga, Julian Schwinger, and Richard Feynman, with their three
approaches unified by Freeman Dyson in 1949. All four shared the
1965 Nobel Prize in Physics for this work. Tomonaga's 1946 paper
\citep{Tomonaga1946}, written at Tokyo Bunrika University during the
war years and published in Japanese before becoming widely known in
the West, gave the first relativistically invariant treatment of QED
in what we now call the interaction picture. Schwinger's 1948
covariant formulation \citep{Schwinger1948QED} carried out the same
renormalization program in a manifestly Lorentz-invariant operator
language; his calculation of the anomalous magnetic moment of the
electron $a_e = \alpha/2\pi$ to one loop was the first quantitative
success of renormalized QED. Feynman's 1949 paper
\citep{Feynman1949QED} reformulated QED in terms of the diagrammatic
calculus that every graduate student now learns; the path-integral
construction of 1948 \citep{Feynman1948} provided its conceptual
underpinning. Dyson's 1949 unification \citep{Dyson1949}
demonstrated that all three approaches yield identical physical
predictions, and introduced the systematic order-by-order expansion
in which counterterms are determined by physical renormalization
conditions. The Lamb shift, the anomalous magnetic moment, and
electron-electron scattering all became finite, calculable
predictions in agreement with experiment to many decimal places ---
the moment quantum field theory stopped being an embarrassment and
became a science.

\paragraph{4. Stueckelberg-Petermann --- 1953.}
Stueckelberg and Petermann \citep{StueckelbergPetermann1953}
recognized that the freedom in subtraction schemes generates a
\emph{group} action on coupling space: changing the subtraction
prescription at scale $\mu_1$ to one at scale $\mu_2$ is a smooth
transformation, and the composition law of these transformations is
a group. This is the founding observation of the renormalization
group proper. In the constitutional reading of $\POS{5}~\RG{}$,
Stueckelberg-Petermann is the deepest technical justification for
treating RG flow as a structural feature of every quantum field
theory --- not an optional refinement.

\paragraph{5. Gell-Mann-Low --- 1954.}
Gell-Mann and Low \citep{GellMannLow1954} converted the
Stueckelberg-Petermann group structure into a concrete
renormalization-group equation. The QED running coupling
$\alpha(\mu) = \alpha_0 / \big(1 - \alpha_0 \beta_0 \log(\mu/\mu_0)\big)$
follows a flow in the energy scale $\mu$, providing the first explicit
RGE. Gell-Mann-Low is the textbook moment at which the
renormalization group became a calculational tool, not just a
structural observation.

\paragraph{6. Kadanoff's block-spin construction --- 1966.}
Kadanoff \citep{Kadanoff1966} introduced block-spinning as the
physical picture of RG flow in real space: a fine-grained lattice
configuration is repeatedly coarsened, and the effective Hamiltonian
flows toward a fixed point that characterizes the universality
class. This is the iconic concrete realization of RG and the
historical bridge between particle-physics renormalization and
condensed-matter critical phenomena.

\paragraph{7. Wilson's modern formulation --- 1971.}
Kenneth Wilson's 1971 papers \citep{Wilson1971,WilsonKogut1974}
recognized that the renormalization group is not just a technical
tool for hiding divergences but a physical statement about how
effective theories depend on resolution scale. RG becomes a
semigroup on theory space; fixed points classify universality
classes; relevant, irrelevant, and marginal operators control the
flow in the neighborhood of every fixed point. Wilson's
reformulation is widely regarded as one of the deepest conceptual
advances in twentieth-century theoretical physics and is the
technical heart of $\POS{5}~\RG{}$.

\paragraph{8. Callan-Symanzik equations --- 1970.}
Callan \citep{Callan1970} and Symanzik \citep{Symanzik1970}
independently formulated the inhomogeneous RGE
$(\mu \partial_\mu + \beta(g) \partial_g - \gamma) \Gamma = 0$
for arbitrary correlation functions $\Gamma$, with anomalous
dimensions $\gamma$ supplying the technical apparatus for computing
operator scaling. In the constitutional reading of $\POS{5}~\RG{}$,
Callan-Symanzik is the calculational machinery that lets working
physicists turn the abstract commutation
$\Dmap \circ \mathrm{RG} = \mathrm{RG} \circ \Dmap$ into concrete
quantitative predictions.

\paragraph{9. Polchinski's exact RG --- 1984.}
Polchinski \citep{Polchinski1984} derived an exact equation for the
Wilsonian effective action: the entire RG can be cast as a continuous
functional flow $\partial_t S_{\rm eff}$ in the action functional.
This demonstrated that quantum field theory is, fundamentally, a
trajectory in the infinite-dimensional space of admissible
Lagrangians. Polchinski's exact RG is the technical foundation on
which most modern non-perturbative RG studies rest.

\paragraph{10. Wetterich's effective average action --- 1993.}
Wetterich \citep{Wetterich1993} derived a functional RG equation
with a sharp infrared cutoff, providing the contemporary
computational engine for non-perturbative RG studies. Wetterich's
equation is the modern incarnation of Wilson's renormalization group
as a functional flow on theory space and supplies the working
apparatus by which RG flows are computed in gravity, condensed
matter, and effective field theory today.

\paragraph{11. 't Hooft anomaly matching --- 1980.}
't Hooft \citep{THooftNaturalness1980} sharpened the
discrete content of RG flow to an exact non-perturbative
consistency condition: anomaly coefficients are RG invariants that
must match between the UV and IR descriptions of any theory. This is
the first explicit ``discrete invariant survives RG'' statement and
the prototype of the inter-postulate interlock between POS-5 (RG flow)
and POS-3 (anomaly cancellation), operative across Strata 2 through 4
of Part V.

\paragraph{12. AdS/CFT and holographic RG --- 1997-2001.}
A separate but converging thread emerged from holography. Maldacena's
1997 AdS/CFT correspondence \citep{Maldacena1998}, with
the bulk-to-boundary dictionary of Witten \citep{Witten1998} and
Gubser-Klebanov-Polyakov \citep{Gubser1998}, identified the
radial coordinate of the $\mathrm{AdS}_5$ bulk with the
renormalization-group flow direction on the boundary CFT. Scale on
the boundary became geometry in the bulk; the renormalization group
became a Hamiltonian flow in a higher-dimensional spacetime.
De~Haro, Skenderis, and Solodukhin \citep{deHaro2001} formalized
holographic renormalization; subsequent work made the radial-evolution
picture standard. In the constitutional reading of $\POS{5}~\RG{}$,
this thread supplies the deepest geometric realization of the RG
commutation: the GG-6 radial coordinate $r$ is the RG scale of the
boundary, not a free choice.

\paragraph{13. Categorical and TQFT reformulations --- 2000s-2020s.}
The contemporary mathematical formulation casts RG as a
\emph{monoidal categorical} coarse-graining and relates the
discrete invariants of RG flow (universality classes, anomaly
coefficients, central charges) to invertible topological field
theories. This unifies the Wilsonian fixed-point classification with
the Freed-Hopkins-Teleman categorical view of anomalies (Stratum 3),
giving a single mathematical object --- the bordism category of
admissible TQFTs --- that controls every cross-stratum interlock in
which RG is involved. In the constitutional reading of
$\POS{5}~\RG{}$, this is the modern unification that makes the
RG-flow content of Strata 2 through 4 indistinguishable from a
sub-category of the bulk topological data.

\paragraph{The axiomatic move of $\POS{5}~\RG{}$.}
$\POS{5}~\RG{}$ elevates scale coherence from a useful empirical
regularity to an axiomatic precondition for admissibility:
$\Dmap \circ \mathrm{RG} = \mathrm{RG} \circ \Dmap$. Under this
discipline the GG-6 radial coordinate $r$ \emph{is} the RG scale,
not a free choice; the laboratory does not see the renormalization
scheme that the theorist chose, it sees only the
scheme-independent physical predictions. POS-5 RG is operative wherever coarse-graining is meaningful: the
Wilson RG already enters Stratum 2 of Part V (where Yang-Mills and
the renormalization-group flow on theory space first appear), the
operational-RG commutation $\Dmap \circ \RG = \RG \circ \Dmap$
requires AX-3 OGGEP and so unlocks at Stratum 3, and the
renormalization-group radial RG of the 6D bulk is locked at
Stratum 4. After all, the seventy-year arc
from Buckingham's $\pi$-theorem through the Lamb shift,
Tomonaga-Schwinger-Feynman-Dyson, Stueckelberg-Petermann,
Gell-Mann-Low, Kadanoff, Wilson, Callan-Symanzik,
Polchinski, Wetterich, 't Hooft matching, Maldacena-Witten
holography, and the modern categorical synthesis is a single
coherent thread, and $\POS{5}~\RG{}$ is the axiomatic commitment
that any admissible candidate theory must respect every link of it.

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

\subsection{POS-6: Occam Gauge Normalization (\OGN{}): rigidity and ``no basis drift''}


\paragraph{1. William of Occam --- early 14th century.}
The philosophical seed of $\POS{6}~\OGN{}$ is Occam's celebrated
maxim \emph{numquam ponenda est pluralitas sine necessitate} ---
multiplicity is never to be posited without necessity
\citep{Occam}. In physics the razor takes a stronger and more
concrete form than in philosophy: every unexplained continuous
parameter is suspicious, and every small dimensionless number that
is not protected by a symmetry warrants an explanation.

\paragraph{2. Diophantus of Alexandria --- 3rd century.}
A parallel mathematical tradition gave the same discipline an
arithmetic form. Diophantus \citep{Diophantus} posed integer-only
problems whose constraints admit only finitely many solutions ---
the Diophantine equations that bear his name. In the constitutional
reading of $\POS{6}~\OGN{}$, Diophantus is the deepest historical
anchor for the idea that physical predictions can be forced to
integer values by simultaneous compatibility constraints rather than
fitted from data.

\paragraph{3. Lagrange's four-square theorem --- 1770.}
Lagrange \citep{Lagrange1770} proved that every positive integer is
the sum of at most four squares. This is the direct arithmetic
ancestor of the OGN-5 Diophantine constraint $K = M^2 + N^2 + L^2$
that fixes the lock length of $\GG{}$: a number-theoretic identity
from 1770 reappearing in the deepest structural lock of contemporary
physics.

\paragraph{4. Hardy-Ramanujan representations --- 1918.}
Hardy and Ramanujan \citep{HardyRamanujan1918} developed the circle
method for partition counting and the asymptotic theory of
integer-representation problems. In the constitutional reading of
$\POS{6}~\OGN{}$, Hardy-Ramanujan exemplify the modern
arithmetic-representation tradition into which the Diophantine
$(M,N,L;J,K) = (1,3,5;7,35)$ ledger naturally fits.

\paragraph{5. Mach's elimination of absolutes --- 1893.}
Mach's \emph{Die Mechanik in ihrer Entwickelung} \citep{Mach1893}
argued that unexplained continuous absolutes --- absolute space,
absolute time, absolute mass --- should be eliminable from the
formulation of physics. Mach's positivist program is the
methodological seed of the modern Occam discipline: any continuous
quantity that cannot be operationalized should not appear as a
fundamental dial.

\paragraph{6. Einstein's complaint --- 1933.}
Einstein \citep{EinsteinSimplicity} articulated the modern physicist's
faith in the parsimony of fundamental laws: ``a theorem which at
present cannot be based upon anything more than upon a faith in the
simplicity, i.e.\ intelligibility, of nature.'' In the constitutional
reading of $\POS{6}~\OGN{}$, Einstein 1933 is the deepest
twentieth-century articulation of the principle that POS-6 elevates
from preference to axiom.

\paragraph{7. Dirac's monopole and charge quantization --- 1931.}
Dirac's monopole argument \citep{Dirac1931} provided the first
physical example of a continuous parameter being replaced by an
integer through a topological consistency requirement: $eg = n\hbar
c/2$. This is the prototype of every subsequent integer-lock in
physics, and the historical anchor for the bundle-theoretic strand
of $\POS{6}~\OGN{}$ in which gauge data are forced to take discrete
values.

\paragraph{8. Chern-Weil characteristic classes --- 1946 / 1974.}
Chern \citep{ChernWeil1946} developed the construction of
characteristic classes from the curvature of a connection on a
principal bundle, completed by Chern and Simons in 1974
\citep{ChernSimons1974}. Chern numbers, Pontryagin numbers, and
Stiefel-Whitney classes are integer invariants of the underlying
fiber-bundle structure. Once gauge theory is formulated geometrically
\citep{KobayashiNomizu1963,Nakahara2003}, these integer invariants
are forced. In the constitutional reading of $\POS{6}~\OGN{}$,
Chern-Weil supplies the modern mathematical justification for the
discipline that gauge data must be integers, not fitted constants.

\paragraph{9. Georgi-Glashow grand unification --- 1974.}
The Georgi-Glashow $\mathrm{SU}(5)$ model \citep{GeorgiGlashow1974}
provided the first concrete example of a discrete branching index
fixing the abelian normalization of the Standard Model: what looked
like three independent continuous couplings became, in the unified
description, a single rational ratio set by group embedding. In the
constitutional reading of $\POS{6}~\OGN{}$, Georgi-Glashow is the
canonical example of a continuous parameter being replaced by a
rational normalization forced by group theory.

\paragraph{10. Information-theoretic Occam: AIC / MDL / Bayes factor --- 1974 / 1978 / 1995.}
The statistical-inference tradition formalized Occam's razor as a
quantitative penalty for unexplained parameters. Akaike
\citep{Akaike1974} introduced the AIC criterion; Rissanen
\citep{Rissanen1978} the minimum-description-length principle; Kass
and Raftery \citep{KassRaftery1995} the Bayes-factor framework.
These are the modern operational embodiments of the same discipline
that $\POS{6}~\OGN{}$ elevates to axiomatic status: every additional
continuous parameter pays a quantitative penalty.

\paragraph{11. String landscape and swampland --- 2003-2005.}
String compactifications produced an estimated $10^{500}$
inequivalent four-dimensional theories, each with its own continuous
moduli, none picked out by an identifiable principle
\citep{Susskind2003,Polchinski2004Landscape,Douglas2003}. Vafa's
swampland conjectures \citep{Vafa2005} responded by attempting to
identify selection rules; weak-gravity, distance, and de-Sitter
conjectures supply candidate constraints. In the constitutional
reading of $\POS{6}~\OGN{}$, the minimum-rationality discipline of
OGN is the natural endpoint of this long search: where the landscape
exhausted continuous freedom and the swampland conjectures sought to
restrict it, OGN insists that the surviving freedom must be
\emph{integer}.

\paragraph{12. Modern philosophy of parsimony --- 2015.}
Sober's \emph{Ockham's Razors} \citep{Sober2015} provides the
contemporary philosophical defense of minimum-parameter model
selection. Sober demonstrates that the discipline of parsimony,
properly formalized, is rationally justifiable and not merely an
aesthetic preference. In the constitutional reading of
$\POS{6}~\OGN{}$, Sober is the modern philosophical justification
for elevating Occam's razor from a heuristic to a constitutional
axiom.

\paragraph{The axiomatic move of $\POS{6}~\OGN{}$.}
$\POS{6}~\OGN{}$ elevates ``no free dials in conventions, every
continuous freedom either forced or absent'' from a methodological
preference to an axiomatic filter. The $\OGNfive{}$ ledger
$(M, N, L; J, K) = (1, 3, 5; 7, 35)$ is the unique integer solution
that satisfies the simultaneous Diophantine constraints
$K = LJ = M^2 + N^2 + L^2$, $J = 2N+1$ together with the
four-component anomaly polynomial $\T{5}$. POS-6 OGN is enforced at Stratum 4 of Part V, where AX-4 MSMF
locks the bulk to a unique integer ledger and shuts the door on
every continuous freedom not forced by symmetry, consistency, or
operational meaning. This is the first
time a physical theory has produced its fundamental constants as the
output of a Diophantine system rather than as continuous fitted
parameters --- a development that places $\GGTheory{}$ in the long
tradition of attempts to realize Occam's razor as exact arithmetic
constraint, but with a concrete result rather than a methodological
promise. After all, the seven-century arc from William of Occam
through Diophantus (re-read), Lagrange's four-square theorem,
Hardy-Ramanujan, Mach, Einstein, Dirac, Chern-Weil, Georgi-Glashow,
AIC/MDL, the string landscape, and modern Sober is a single coherent
thread, and $\POS{6}~\OGN{}$ is the axiomatic commitment that any
admissible candidate theory must respect every link of it.

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

\subsection{POS-7: Measurement and falsifiability (MF): compile-to-observable}


\paragraph{1. Aristotle and the empirical tradition --- 4th century BCE.}
Aristotle \citep{Aristotle} grounded natural philosophy in observation
of phenomena: every claim about the natural world must, in his
formulation, be answerable to what can be seen and measured. In the
constitutional reading of $\POS{7}~\MF{}$, Aristotle is the deepest
historical anchor for empiricism --- the discipline that whatever a
theory claims must ultimately compile to a registered observation.

\paragraph{2. Galileo and the operational turn --- 1610-1638.}
Galileo \citep{Galilei1632} made physics an operational discipline.
His instruments (telescope, inclined plane, pendulum, hydrostatic
balance) turned natural philosophy from a contemplative activity into
a measurement program: a quantity that no instrument procedure can
pin down is not a quantity that physics can use. This is the
empirical seed of every subsequent operational interpretation of
physical concepts.

\paragraph{3. Bridgman's operationalism --- 1927.}
Bridgman \citep{Bridgman1927} sharpened the Galilean discipline into
an explicit principle: the meaning of a physical concept is the set
of operations by which it is measured. Bridgman's operationalism,
often criticized as too narrow when read literally, captures the
necessary discipline that every $\POS{7}~\MF{}$ measurement contract
inherits: a quantity that no instrument procedure can pin down is
not a quantity that the constitution admits.

\paragraph{4. Heisenberg uncertainty --- 1927.}
Heisenberg \citep{Heisenberg1927} formalized the trade-off between
conjugate observables: $\Delta q \cdot \Delta p \ge \hbar/2$. This
was the first explicit demonstration that \emph{what is measurable}
is constrained by the algebra of observables itself, not merely by
instrumental limitations. In the constitutional reading of
$\POS{7}~\MF{}$, Heisenberg 1927 is the canonical reminder that the
audit-contract requirement of POS-7 cannot be a naive prescription;
it must take seriously the algebraic structure of what a quantum
system permits one to ask.

\paragraph{5. von Neumann's projection postulate --- 1932.}
Von Neumann \citep{vonNeumann1932} provided the first textbook
treatment of the measurement process: a measurement of an observable
$A$ projects the state onto an eigenspace of $A$ with Born
probabilities. This is the canonical statement of the measurement
problem in classical quantum theory --- the problem that POS-7
dissolves by re-reading measurement through the OGGEP operational
dictionary and the modern instrument formalism.

\paragraph{6. Neyman-Pearson hypothesis testing --- 1933.}
Neyman and Pearson \citep{NeymanPearson1933} formulated the
likelihood-ratio test, type-I and type-II errors, and pre-declared
rejection regions. This supplies the quantitative apparatus that
makes ``kill criterion'' a precise statistical-decision notion: a
falsifier is a pre-declared rejection region $R_\alpha$ at
significance level $\alpha$. In the constitutional reading of
$\POS{7}~\MF{}$, Neyman-Pearson is the technical core of the
$(\mathcal{I}, \mathcal{S}_{\rm valid}, \mathcal{T}, K)$ tuple's
final entry.

\paragraph{7. Popper's falsifiability --- 1934 / 1959.}
Popper's \emph{Logik der Forschung} (1934, English translation 1959)
\citep{Popper1959} gave the canonical modern formulation: a
scientific theory is one that makes \emph{falsifiable} predictions,
i.e.\ predictions that some conceivable observation could refute.
Popper's demarcation criterion is the philosophical core of
$\POS{7}~\MF{}$: every claim must specify, in advance, what
observation would falsify it.

\paragraph{8. Shannon's information theory --- 1948.}
Shannon \citep{Shannon1948} introduced the quantitative theory of
information --- bits, entropy, channel capacity, mutual information.
This supplies the natural language for ``registered observable'' and
``informativeness of a measurement contract'': a contract that
returns one bit of information is operationally worth less than one
that returns many bits, and Shannon's framework makes the comparison
precise.

\paragraph{9. Kullback-Leibler relative entropy --- 1951.}
Kullback and Leibler \citep{KullbackLeibler1951} introduced the
relative-entropy or KL-divergence pre-metric between predicted and
observed distributions. In the constitutional reading of
$\POS{7}~\MF{}$, KL-divergence is the canonical information-theoretic
distance between the predicted and observed distributions of a
registered estimator, and the foundation of likelihood-ratio
decisions in modern statistical practice.

\paragraph{10. Kuhn, Lakatos, Feyerabend --- 1962-1975.}
Kuhn's \emph{Structure of Scientific Revolutions} \citep{Kuhn1962}
challenged the simple-falsifiability picture by emphasizing paradigm
shifts; Lakatos \citep{Lakatos1970} refined Popper to research
programs with a hard core and protective auxiliary hypotheses;
Feyerabend \citep{Feyerabend1975} rejected demarcation altogether
in \emph{Against Method}. Modern philosophy of science recognizes
that Popper's criterion is necessary but not sufficient: a falsifier
must be specific, pre-declared, and operationally checkable rather
than post-hoc. In the constitutional reading of $\POS{7}~\MF{}$,
this trio is the philosophical context against which the modern
pre-registration discipline is the working-physics response.

\paragraph{11. Davies-Lewis-Kraus-Ozawa-Busch-Nielsen-Chuang: the operational QM tradition --- 1970-2010.}
The modern operational formalism for quantum measurement matured
through the second half of the twentieth century. Davies and Lewis
\citep{DaviesLewis1970} introduced instruments; Kraus
\citep{Kraus1971,Kraus1983} the operator-sum representation; Ozawa
\citep{Ozawa1984} the formalism for sequential measurement; Busch,
Lahti, and Mittelstaedt \citep{BuschLahtiMittelstaedt1996} the
unified operational + philosophical treatment; Nielsen and Chuang
\citep{NielsenChuang2010} provided the modern
textbook synthesis. In the constitutional reading of $\POS{7}~\MF{}$,
this is the technical content imported wholesale by the postulate ---
the apparatus on which every measurement contract relies.

\paragraph{12. Bell's theorem and experimental violation --- 1964 / 1972-2015.}
Bell \citep{Bell1964} derived inequalities that any local-hidden-variable
theory must satisfy. Freedman and Clauser \citep{FreedmanClauser1972}
performed the first experimental test; Aspect \citep{Aspect1982}
sharpened it with time-varying analyzers; Hensen et al.\
\citep{Hensen2015} and Giustina et al.\ \citep{Giustina2015}
performed the first loophole-free violations. In the constitutional
reading of $\POS{7}~\MF{}$, Bell + Aspect + the loophole-free
experiments are the most consequential modern falsifiability
contract in the foundations of physics: a pre-declared inequality
was tested, the test was passed, and quantum mechanics survived
where any local hidden-variable theory could not.

\paragraph{13. Pre-registration and registered reports --- 2010-2020s.}
The contemporary empirical-science culture of pre-declared protocols,
pre-registered analyses, and registered
replications is the
working-physics response to the philosophical refinements of Kuhn,
Lakatos, and Feyerabend. Registered reports, pre-analysis plans,
and replication studies have become standard in particle physics,
cosmology, condensed-matter, and (increasingly) the biological and
social sciences. In the constitutional reading of $\POS{7}~\MF{}$,
the pre-registration culture is the contemporary operational
embodiment of the discipline that the postulate elevates from
empirical practice to axiomatic requirement.

\paragraph{The axiomatic move of $\POS{7}~\MF{}$.}
$\POS{7}~\MF{}$ combines all of these strands into a single audit
requirement. Every theoretical claim in $\GGTheory{}$ must compile,
via the operational dictionary $\OGGEP{}$, to a registered observable,
an instrument protocol, and a pre-declared falsification (kill)
criterion. Claims that cannot be compiled into such a measurement
contract are not admissible as scientific statements within
$\GGTheory{}$. This is operationalism plus Popperian falsifiability
plus Neyman-Pearson hypothesis testing plus Shannon-Kullback
information theory plus the modern instrument formalism, all
elevated from a methodological add-on to an axiomatic precondition.
POS-7 MF first becomes a non-trivial admissibility contract at
Stratum 3 of Part V, where AX-3 OGGEP defines what measurement
means in operational terms; the pre-registered kill-criterion
contract is fully locked at Stratum 4, where the OGN-5 integer
ledger supplies the sharp numerical predictions that the
measurement contract is then required to register. After all, the two-and-a-half-millennium arc from
Aristotle through Galileo, Bridgman, Heisenberg, von Neumann,
Neyman-Pearson, Popper, Shannon, Kullback-Leibler, Kuhn-Lakatos-
Feyerabend, Davies-Lewis-Kraus-Ozawa-Busch-Nielsen-Chuang,
Bell-Aspect-Hensen-Giustina, and modern pre-registration is a
single coherent thread --- and $\POS{7}~\MF{}$ is the axiomatic
commitment that any admissible candidate theory must respect every
link of it. This is the most important thing the
constitution can do for the working scientist: it converts the loose
phrase ``my theory makes predictions'' into a checkable contract.

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

\section*{Closing remarks on Part~III}

The eleven clauses are now complete. Four axioms said what kind of
object a theory is; seven postulates said which such objects survive.
That is the whole constitution, and it is short enough to read again
before dinner.

Of the seven filters, six are recognizable as good practice made
explicit. Working physicists already insist on gauge invariance,
already derive from variational principles, already cancel anomalies,
already keep probabilities positive, already check that answers do not
depend on the cutoff. Writing these down changes little in daily
practice and much in accountability: a stated rule can be checked
against, and a habit cannot.

The seventh is different, and we want to end this Part on it.
$\POS{7}~\MF{}$ requires that a claim name, in advance, the measurement
that would kill it. Not a measurement that would support it --- support
is cheap, and a sufficiently flexible framework can be supported by
anything. The requirement is that the theory say, before the data
arrive, what result it could not survive.

This is harder than it sounds, and the difficulty is worth feeling.
Consider a prediction like ``the matter fraction of the universe is
$1/\pi$''. Stated loosely, it is nearly unfalsifiable: measurements
carry error bars, $1/\pi \approx 0.3183$ sits close to the observed
value, and a determined defender can always plead that the next dataset
will settle it. Stated as a contract --- a named observable, a
resolution threshold, a numerical deviation, a significance level, all
fixed in advance --- it becomes a claim that can die on a specific
Tuesday. The consolidated table in the next Part does exactly this for
ten predictions.

We think the honesty here is technical rather than moral. A theory that
cannot lose is not being generous; it is failing to carry information.
Information, in the precise sense, is what is left after possibilities
have been excluded, and a claim that excludes nothing tells you
nothing. The demand that physics name its own kill criteria is not a
demand for humility. It is a demand that the claims mean something.

The constitution is now stated in full. What it does not contain is the
proof that these eleven clauses force anything at all --- that work is
long, and it lives elsewhere. Part~IV is the map to where.

\GGPart[Forward map and bridges]{Forward Map and Bridges to the Companion Papers}%
 {Where Each Theorem Lives in the GG6, \GDOS{}, 4D-Physics,
  Capstone, Foundations, Predictions, Cosmology and White-Paper Cohort}

\noindent
\textbf{This Part proves nothing. It is the index.}

By this point the constitution is complete: four axioms and seven
postulates, eleven clauses, stated in full. What the constitution does
\emph{not} contain is the hard technical work of showing that those
eleven clauses actually force a six-dimensional geometry, an exact
anomaly identity, a boundary law for open systems, and a specific list
of numbers. That work exists, but it is long --- long enough that
carrying it here would make this document unusable as a rule-set. So
it lives in twenty-one companion papers, and this Part is the map to
them.

The arrangement deserves a defense rather than an apology. A legal
constitution is short on purpose. A document long enough to contain its
own case law is too long to audit \emph{as} a constitution, and a
rule-set nobody can hold in mind stops working as a rule-set. The price
of keeping this one short is that the reader must sometimes follow a
pointer, and we have tried to make every pointer exact.

You will find here: a full inventory of the twenty-two papers and what
each carries; a short bridge to each branch of the program, saying what
that branch is for; a coherence map tying each of the eleven Principal
Theorems to the specific numbered theorem in the specific companion
paper where it is proved; the five universal constants the program
claims to reconstruct; a minimum reading path for someone who wants to
evaluate the central claims without reading all twenty-two documents;
and a consolidated table of every sharp numerical prediction together
with the measurement that would falsify it.

If you read one thing in this Part, read that last table. It is the
shortest honest summary of what this program is betting.

\medskip

\section{The axiomatic handoff}
\label{sec:constitutional_handoff}

\GGTheory{} is a coordinated program of foundational papers, of
which the present paper is the first. Section~\ref{sec:paper_inventory}
records the comprehensive inventory of those papers; the figure in
Section~\ref{sec:paper_relations} shows their relations as a single
flowchart; Sections~\ref{sec:bridge_GG6_trio} through
\ref{sec:bridge_whitepapers} bridge to each companion paper in turn,
naming what it carries; and Section~\ref{sec:coherence_map} pins
each Principal Theorem $\T{k}$ to the canonical theorem location in
the companion paper where its proof actually lives. This
Part is the navigator the reader uses to find the rest of the
program.

\subsection{Comprehensive paper inventory}
\label{sec:paper_inventory}

The \GGTheory{} program organizes into an axiomatic root (this
paper, Arisaka-GG-A1-Constitution), a \textbf{GG6 Trio} of
bulk-side foundational papers (A2--A4), a \textbf{\GDOS{} Trio} of
boundary-side foundational papers (A5--A7), a \textbf{4D-Physics
Trio} of operational and numerical-derivation papers (A8--A10),
a closing capstone (A11), a \textbf{Foundations Branch} of two
further foundational member papers (P4--P5), two extensible
phenomenology branches --- a \textbf{Predictions Branch} of
particle-physics papers (P1--P3, P6) and a \textbf{Cosmology \&
Astrophysics Branch} (C1--C3) --- and a \textbf{White-Paper Branch}
(W1--W2) of two synthesis papers that read the cohort rather than
extend it. Twenty-two papers in all. A compilation Foundation Book
and a popular exposition round out the series for broader audiences.
Table~\ref{tab:paper_inventory} records the master inventory.
One reading aid for the \GDOS{} Trio: its three members are the three
tenses of \AX{3} --- the \emph{law} (\AFive{}), the \emph{runtime}
(\ASix{}), and the \emph{selector} (\ASeven{}) --- strung together by
the boundary descent chain of Derivation~4
(Eq.~\ref{eq:boundary-descent-chain}).

{\footnotesize
\begin{longtable}{@{}>{\raggedright\arraybackslash}p{0.150\linewidth}
                   >{\raggedright\arraybackslash}p{0.130\linewidth}
                   >{\raggedright\arraybackslash}p{0.360\linewidth}
                   >{\raggedright\arraybackslash}p{0.225\linewidth}
                   >{\raggedright\arraybackslash}p{0.101\linewidth}@{}}
\caption[Comprehensive inventory of the GG-Theory papers.]{\textbf{Comprehensive inventory of the \GGTheory{} papers.}
The axiomatic root, three trios, the closing capstone, the
two-paper Foundations Branch, two extensible phenomenology branches,
and the two-paper White-Paper Branch compose the
twenty-two-paper \GGTheory{} program. Each row records the paper's name, which links to its entry in the bibliography,
a short description of the paper's role, the constitutional source
(which AX or POS forces it), the Principal Theorem(s) it carries,
and the trio grouping where applicable. All foundational papers
exist as released preprints in the versions named.}
\label{tab:paper_inventory}\\[0.35em]
\toprule
\textbf{Name} & \textbf{Short title} & \textbf{Role} & \textbf{Carries} & \textbf{Group} \\
\midrule
\endfirsthead
\toprule
\textbf{Name} & \textbf{Short title} & \textbf{Role} & \textbf{Carries} & \textbf{Group} \\
\midrule
\endhead
\midrule
\multicolumn{5}{r}{\small continued on next page}\\
\endfoot
\bottomrule
\endlastfoot

Arisaka-GG-A1-Constitution &
The \GGConstitution{} &
\emph{This paper.} Four axioms, seven postulates, definitions,
historical motivation, forward map. &
None --- names $\T{1}$--$\T{11}$, proves none. &
Root \\

\hyperlink{cite.ArisakaA2GG6}{Arisaka-GG-A2-GG6} &
\GG{} Foundation &
Establishes the 6D bulk; the four-factor composite gauge
group $\Spinone\times\SU(5)\times\U(1)_B\times\U(1)_{Qi}$; the
\OGNfive{} integer ledger $(1,3,5;7,35)$. &
$\T{1}$ Locked Equivalence; $\T{3}$ Anti-Landscape; $\T{4}$ 6D bulk; $\T{6}$ \OGNfive{} as bulk output. &
GG6 Trio \\

\hyperlink{cite.ArisakaA3BI6}{Arisaka-GG-A3-BI6} &
\BIsix{} Anomaly Proof &
Hopkins-Singer differential-cohomology proof of the Green-Schwarz
exact descent. The trace-ratio lock $|a/c|=|a/d|=5/2$ is the
axiomatic fingerprint of the program. &
$\T{5}$ \BIsix{} four-component anomaly identity. &
GG6 Trio \\

\hyperlink{cite.ArisakaA4Euclid}{Arisaka-GG-A4-Euclid} &
From Euclid to \GG{} &
Historical-view companion: the 2300-year chain from Euclid to the
\GG{} stage, plus the parameter audit and master derivation table
that synthesize the Ultra Seven into one zero-free-parameter ledger.
Pedagogical and synoptic; theorem-bearing content is delegated to
the source papers. &
None (commentary on $\T{4}$--$\T{6}$; synoptic on $\T{1}$--$\T{11}$). &
GG6 Trio \\

\hyperlink{cite.ArisakaA5GDOS}{Arisaka-GG-A5-GDOS} & \GDOS{} Foundation &
Establishes the boundary law $\cL_{\GDOS}$; recovers QM, GR, QFT
as 4D boundary projections of the 6D bulk under \OGGEP{}. &
$\T{2}$ Operational Closure; $\T{7}$ 4D QM; $\T{8}$ 4D GR;
$\T{9}$ 4D QFT. &
\GDOS{} Trio \\

\hyperlink{cite.ArisakaA6Gi4}{Arisaka-GG-A6-GRIP} &
\GiFour{} Pillars &
Four pillars on the geometric flank of the boundary law: \TGG{},
\CCI{}, \GFF{}, \GRIP{} (the cascade $\DGI \to \GSSB \to \GA/\GGA{}$,
two regimes \GRIPin{}/\GRIPout{}). &
$\T{10}$ \GiFour{} pillars. &
\GDOS{} Trio \\

\hyperlink{cite.ArisakaA7Qi4}{Arisaka-GG-A7-Qi4} &
\QiFour{} Pillars &
Four pillars on the quantum-information flank: \QiQUA{},
\QiEP{}, \QiCON{}, \QiHAL{}. &
$\T{11}$ \QiFour{} pillars. &
\GDOS{} Trio \\

\hyperlink{cite.ArisakaA8QM}{Arisaka-GG-A8-QM} &
4D Quantum Mechanics &
Operationalizes $\T{2}$ and $\T{7}$ from A5 into a
concept-by-concept reformulation of textbook QM --- Born rule,
EPR, Bell--CHSH, Schr\"odinger's cat, decoherence. Closes with
the \textit{Einstein-in-6D, Bohr-in-4D} thesis. &
None new --- operational reformulation. &
4D-Physics Trio \\

\hyperlink{cite.ArisakaA9SM}{Arisaka-GG-A9-SM} &
Standard Model derivation &
Numerical descent: derives the 28 continuous parameters of the
Standard Model from $(M,N,L;J,K) = (1,3,5;7,35)$ and the 6D bulk. &
None new --- numerical descent. &
4D-Physics Trio \\

\hyperlink{cite.ArisakaA10Cosmos}{Arisaka-GG-A10-Cosmos} &
Cosmology derivation &
Numerical descent: derives $\Omega_m = 1/\pi$,
$\Omega_\Lambda = 1 - 1/\pi$, $\Omega_b = 1/(2\pi^2)$,
$\Omega_{\rm DM} = 1/\pi - 1/(2\pi^2)$, $m_\Lambda \approx 6$ meV. &
None new --- numerical descent. &
4D-Physics Trio \\

\hyperlink{cite.ArisakaA11Origin}{Arisaka-GG-A11-Origin}$^{\dagger}$ &
Closing Capstone &
Closes the A-series: the Ground Geometric Attractor (\GGA{}) and the
identity cascade that runs from the gauge symmetry breaking through
atomic ground states to life and mind, read as one sequence of forced
identities rather than as an analogy. &
None new --- the closing identity cascade. &
Capstone \\

\hyperlink{cite.ArisakaP1KK}{Arisaka-GG-P1-KK} &
KK Lock Cluster &
Phenomenology: non-spatial Kaluza-Klein cluster (RG-scale) at $\Mlock \approx 3.85\,\TeV$
with specific LHC and future-collider falsifiers. &
None new --- phenomenology. &
Predictions \\

\hyperlink{cite.ArisakaP2N3}{Arisaka-GG-P2-N3} & Three Generations & Phenomenology: three-leg derivation of the family number $N=3$ from
\BIsix{}, \MSMF{}, and \GDOS{} Morse stability, with its
conditioning declared. &
None new --- phenomenology. &
Predictions \\

\hyperlink{cite.ArisakaP3Electron}{Arisaka-GG-P3-Electron} &
Electron as Ground State &
Phenomenology/review: the six oldest electron questions --- identity, stability, copies, the positron, pointlikeness, sensing --- closed as one ground-state theorem-complex, every fence declared. &
None new --- phenomenology. &
Predictions \\

\hyperlink{cite.ArisakaP4Real}{Arisaka-GG-P4-Real} &
The Origin of $i$ &
Foundations: both complex structures of physics derived from two
real bulk sources --- the compact phase circle (the gauge $i$) and
the causal clock (the state-space $i$) --- with the antiparticle
handshake as their forced agreement. &
None new --- foundations development (theorem keys T-P4-$N$). &
Foundations \\

\hyperlink{cite.ArisakaP5CPT}{Arisaka-GG-P5-CPT} &
The Mirrors of Matter &
Foundations: C, P, and T derived as orientation reversals of the
three real bulk factors; CPT as the undetectable total reversal;
the cosmic matter excess routed through the flavored chain, its
sign carried by the Majorana parity (derived-conditional). &
None new --- foundations development (theorem keys T-P5-$N$). &
Foundations \\

\hyperlink{cite.ArisakaP6Proton}{Arisaka-GG-P6-Proton} &
The Guarded Proton &
Phenomenology: the proton as the \GGA{} of the baryonic stratum, with
baryon number carried as a winding of the compact geometry, so that
its stability is a theorem rather than an accident. Eleven questions
closed as faces of one fact. &
None new --- phenomenology. &
Predictions \\

\hyperlink{cite.ArisakaC1Kaxion}{Arisaka-GG-C1-Kaxion} &
Kaxion Dark Matter &
Phenomenology: kaxion oscillation $\Delta\nu/\nu = 1/35$ with
specific haloscope/cavity falsifiers. &
None new --- phenomenology. &
Cosmology \\

\hyperlink{cite.ArisakaC2Hubble}{Arisaka-GG-C2-Hubble} &
Hubble Tension &
Phenomenology: the two measured Hubble constants as the two ends
of one six-dimensional geometric corridor. &
None new --- phenomenology. &
Cosmology \\

\hyperlink{cite.ArisakaC3BH}{Arisaka-GG-C3-BH} &
Black-Hole Entropy &
Phenomenology: the horizon as an \OGGEP{} port; exact
Bekenstein--Hawking $A/4$ from induced gravity. &
None new --- phenomenology. &
Cosmology \\

\hyperlink{cite.ArisakaW1Symmetry}{Arisaka-GG-W1-Symmetry} &
Conservation without Symmetry &
Synthesis, \emph{diachronic}: the century from Noether to
Hopkins--Singer quantization, showing the conviction ``symmetry
causes conservation'' loosening at seven historical steps, with the
program placed at the arc's end at declared grades. &
None --- carries every result at its parent's declared grade. &
White Paper \\

\hyperlink{cite.ArisakaW2Holonomy}{Arisaka-GG-W2-Holonomy} &
Sixteen Open Problems &
Synthesis, \emph{synchronic}: one exact statement --- the circle
remembers and the interval forgets --- followed to sixteen of the
questions physics has not closed, with the named measurements on
which six of the treatments die. &
None --- carries every result at its parent's declared grade. &
White Paper \\\end{longtable}}

\noindent{\small $^{\dagger}$\,\AEleven{} is in preparation and is not among the
posted papers; it is listed here because it is part of the program's
architecture, and no result of the present paper depends on it.}


\subsection{Relations among the \GGTheory{} papers}
\label{sec:paper_relations}

The relations among the papers are the descent flowchart of
Figure~\ref{fig:A1-cohort-inheritance-map}, printed in Part~I
where a first-time reader meets the cohort; each downward step
there is labeled by the AX or POS that drives it. Read against
Table~\ref{tab:paper_inventory} above, the plate says which paper
inherits from which, and the table says what each one carries.


\subsection{Bridge to the GG6 Trio (A2, A3, A4)}
\label{sec:bridge_GG6_trio}

Indeed, $\AX{2}~\GGEP{}$ together with $\POS{3}~\AF{}$,
$\POS{5}~\RG{}$, and $\POS{6}~\OGN{}$ forces the existence of a
specific six-dimensional bulk realization of the composite
geometric-gauge connection. The GG6 Trio supplies this bulk
realization in three complementary forms: \ATwo{} establishes the
6D bulk and the four-factor composite gauge group; \AThree{}
supplies the differential-cohomology proof of the four-component
anomaly identity; and \AFour{} records the historical lineage
that leads to the \GG{} stage.

\paragraph{What \ATwo{} \citep{ArisakaA2GG6} carries.}
The canonical 6D bulk metric
$ds_6^2 = \e^{-2r}\,\eta_{\mu\nu}\,dx^\mu dx^\nu + dr^2 + d\theta^2$
with $r \in [0, K=35]$ and $\theta \in [0, 2\pi)$;
the four-factor composite gauge group
$G_{\mathrm{tot}} = \Spinone\times\SU(5)\times\U(1)_B\times\U(1)_{Qi}$;
the \OGNfive{} integer ledger $(M,N,L;J,K) = (1,3,5;7,35)$
satisfying the Diophantine consistency $K = LJ = M^2 + N^2 + L^2$;
the boundary at $r = 0$ supplying the unification scale
$\EGUT \approx 2.28 \times 10^{16}\,\GeV$; the lock cluster at
$r = K$ supplying $\Mlock \approx 3.85\,\TeV$. Carries Principal
Theorems $\T{1}$ Locked Equivalence, $\T{3}$ Anti-Landscape
Selection, $\T{4}$ 6D bulk Existence, and $\T{6}$
\OGNfive{} as Bulk Output.

\paragraph{What \AThree{} \citep{ArisakaA3BI6} carries.}
The Hopkins-Singer differential-cohomology proof that the
four-component anomaly polynomial
$X_4 = a\,\tr(R\wedge R) + b\,\tr(F_{\SU(5)}\wedge F_{\SU(5)})
+ c\,F_B\wedge F_B + d\,F_{Qi}\wedge F_{Qi}$
admits exact Green-Schwarz descent if and only if
$(a,b,c,d) = (1,-1,-2/5,-2/5)$, with the geometry-gauge trace-ratio
lock $|a/c| = |a/d| = 5/2$ as the axiomatic fingerprint of
the program. Carries Principal Theorem $\T{5}$.

\paragraph{What \AFour{} \citep{ArisakaA4Euclid} carries.}
The historical-view companion: the long chain from Euclid through
Newton, Riemann, Maxwell, Einstein to the \GG{} stage. This is
not a theorem-bearing paper but a pedagogical commentary that
locates the bulk-side foundational moves in the long chain of
geometric-foundational thought.

\subsection{Bridge to the \GDOS{} Trio (A5, A6, A7)}
\label{sec:bridge_GDOS_trio}

$\AX{3}~\OGGEP{}$ acting on the 6D bulk of A2 forces the
boundary law $\cL_{\GDOS}$ on the 4D slice $M_4 = \partial \cM_6$.
The \GDOS{} Trio supplies this boundary side: \AFive{} establishes
the boundary law itself; \ASix{} and \ASeven{} decompose the boundary
law into four geometric-invariance pillars and four
quantum-information pillars respectively.

\paragraph{What \AFive{} \citep{ArisakaA5GDOS} carries.}
The boundary CPTP channel
$\Phi_{t,s} : \rho_\Bdy(s) \mapsto \rho_\Bdy(t)$ via Stinespring
dilation; the GKLS form of the generator $\cL_{\GDOS}$; the
recovery of textbook quantum mechanics, general relativity, and
quantum field theory as boundary-projection theorems on the 4D
slice; the Born rule and the projection postulate as theorems
rather than as separate axioms; the modified
Tolman-Oppenheimer-Volkoff equation; the BRST cohomology and the
Slavnov-Taylor identities. Carries Principal Theorems $\T{2}$
Operational Closure, $\T{7}$ 4D QM, $\T{8}$ 4D GR, and $\T{9}$
4D QFT.

\paragraph{What \ASix{} \citep{ArisakaA6Gi4} carries.}
The four geometric-invariance pillars descending from
$\AX{3}~\OGGEP{}$ on the geometric flank: \TGG{} (Total
Geometric-Gauge object), \CCI{} (Covariant Conservation Identity),
\GFF{} (Geometric Free-Energy Functional), \GRIP{} (Gauge-Runtime
Invariance Pipeline, the cascade $\DGI \to \GSSB \to \GA/\GGA{}$ with
finite-budget closure, run in the two regimes \GRIPin{}/\GRIPout{}). Carries Principal
Theorem $\T{10}$.

\paragraph{What \ASeven{} \citep{ArisakaA7Qi4} carries.}
The four quantum-information pillars descending from
$\AX{3}~\OGGEP{}$ on the operational flank: \QiQUA{} (qua-state
preparation), \QiEP{} (quantum equivalence principle), \QiCON{}
(decoherence and contraction, recovering einselection), \QiHAL{}
(holographic alphabet bound on boundary information density).
Carries Principal Theorem $\T{11}$.

\subsection{Bridge to the 4D-Physics Trio (A8, A9, A10)}
\label{sec:bridge_4Dphysics_trio}

Indeed, the \GDOS{} Trio supplies the abstract boundary law; the
4D-Physics Trio operationalizes that law into the concrete
physics any laboratory or telescope can register. \AEight{}
reformulates textbook quantum mechanics in \GDOS{} language; \ANine{}
derives the 28 continuous parameters of the Standard Model from
the \OGNfive{} ledger; \ATen{} derives the cosmological
partitions of $\Lambda$CDM.

\paragraph{What \AEight{} \citep{ArisakaA8QM} carries
(preprint pending --- two parallel manuscripts being merged).}
The concept-by-concept reformulation of textbook quantum
mechanics in the language of \GDOS{} and $\OGGEP{}$: the Born
rule as a theorem of the Stinespring dilation rather than as an
axiom; the EPR paradox, the Bell inequality, the CHSH inequality,
the Aspect (1982) experiment, and the loophole-free experiments
\citep{Hensen2015,Giustina2015} reframed as evidence that the
6D-to-4D projection is not invertible; Schr\"odinger's cat
(``dead-and-alive in 6D, decohered in 4D''); the double-slit;
quantum decoherence and einselection. The paper closes with
the central \GGTheory{} thesis: \emph{Einstein was right in 6D
--- Nature does not play dice; Bohr was right in 4D --- the 4D
observer cannot avoid the dice; both positions are correct
within their respective domains.} Operationalizes $\T{2}$ and
$\T{7}$ from A5; carries no new Principal Theorem of its own.

\paragraph{What \ANine{} \citep{ArisakaA9SM} carries.}
The numerical descent that produces the 28 continuous parameters
of the Standard Model from $(M,N,L;J,K) = (1,3,5;7,35)$ via the
6D bulk: the three gauge couplings $g_1, g_2, g_3$; the
nine charged-fermion masses; the four CKM parameters; the three
neutrino masses and the four PMNS parameters; the QCD theta
angle $\bar\theta$; the cosmological constant $\Lambda$; the
Newton constant $G_N$. Each parameter pinned by a closed-form
expression in the \OGNfive{} integers.

\paragraph{What \ATen{} \citep{ArisakaA10Cosmos} carries.}
The numerical descent that produces $\Lambda$CDM as boundary
cosmology of the 6D bulk: $\Omega_m = 1/\pi$,
$\Omega_\Lambda = 1 - 1/\pi$, $\Omega_b = 1/(2\pi^2)$,
$\Omega_{\rm DM} = 1/\pi - 1/(2\pi^2)$;
$m_\Lambda = \Mlock^2/\MPl \approx 6\,\mathrm{meV}$; the
electroweak scale $v = \Mlock\,(2/5)^3 \approx 246\,\GeV$.

\subsection{Bridge to the closing capstone (A11)}
\label{sec:bridge_capstone}

\AEleven{} closes the A-series. Where \AEight{}, \ANine{} and
\ATen{} read the boundary law out into quantum mechanics, the
Standard Model and $\Lambda$CDM, the capstone asks what the same
machinery says once it is allowed to run all the way up.

\paragraph{What \AEleven{} \citep{ArisakaA11Origin} carries.}
The Ground Geometric Attractor (\GGA{}) and its ground-state
refinement, together with the \GRIPin{}/\GRIPout{} regime theorems
and the identity cascade that runs from gauge symmetry breaking
through atomic ground states to life and mind --- presented as a
sequence of forced identities at declared grades, with the structural
claim that the sequence is one mechanism rather than an analogy
flagged, in the paper's own words, as the most contestable claim of
the program. Carries no Principal Theorem of the present paper.

\subsection{Bridge to the Foundations Branch (P4, P5)}
\label{sec:bridge_foundations}

On the map's left flank, the Foundations Branch carries the origin
of the boundary's complex structure and the discrete-symmetry /
matter interface --- the two member papers that answer, from the
real oriented bulk, the questions every consumer of the quantum
formalism otherwise inherits silently: where the $i$ comes from, and
why the mirrors break.

\paragraph{What \PaperPFour{} \citep{ArisakaP4Real} carries.}
The derivation of the complex numbers of physics from two real
structures of the 6D bulk: the charge circle's quarter-turn as the
unique circle-compatible complex structure on every charged sector
(T-P4-1, the gauge $i$); the causal clock's polar phase as the
unique positive-energy, dynamics-commuting complex structure
(T-P4-2, the state-space $i$); their forced on-shell agreement under
the Feynman--St\"uckelberg pairing --- antiparticles as a
consistency condition rather than a postulate (T-P4-3); the shared
complex structure on composites (T-P4-4); and the bridge that
positive frequency \emph{is} holomorphy at the Hardy, forward-tube,
and twistor levels (T-P4-6/7). Grade, stated honestly: derived at
the quasi-free grade; the interacting extension is conditional on
the named hypotheses (H1)--(H3) of T-P4-8. Feeds \AEight{}'s
real-bulk series and the primitive-category pricing of
L-Constitution-1 (\S\ref{sec:AX4_category_axis}, candidate grade).
Carries no Principal Theorem of the present paper.

\paragraph{What \PaperPFive{} \citep{ArisakaP5CPT} carries.}
The discrete symmetries as orientation: C, P, and T as the
orientation reversals of the three real bulk factors --- the fiber
circle, the spatial slice, and the causal clock --- with
antiunitarity exactly where T sits (T-P5-1); the CPT theorem as
orientation bookkeeping (T-P5-2); the self-contained re-proof of the
circle/clock/handshake suite (T-P5-A/B/C) with the Wick quartet
(T-P5-B1); the four-endowment definition of time consolidated in its
Part~IX (the custody statement of \S\ref{sec:AX3} points there); and,
on the matter side, the flavored leptogenesis chain fed by
\ANine{}'s ledger phases and delivered to \ATen{}'s $\eta_B$ chain,
with the sign carried by the Majorana parity $\alpha_{31}=\pi$.
Grade, stated honestly: the mirrors and CPT are derived at the
quasi-free grade; the matter throughput is derived-conditional on
its named finite-temperature hypothesis (H-phigeo-1). Carries no
Principal Theorem of the present paper.

\subsection{Bridge to the Predictions Branch (P1, C1, $\ldots$)}
\label{sec:bridge_predictions}

The Predictions Branch carries specific empirical falsifiers,
each with a registered claim contract (instrument, calibration,
timescale, kill criterion) per $\POS{7}~\MF{}$. The branch is
extensible: P-numbered papers carry particle-physics predictions,
C-numbered papers carry cosmology predictions, with further
labels reserved for future predictions.

\paragraph{What \PaperPOne{} \citep{ArisakaP1KK} carries.}
The non-spatial Kaluza-Klein lock cluster --- RG-scale resonances of the RG-scale throat, not a spatial extra dimension --- at $\Mlock \approx 3.85\,\TeV$,
with explicit LHC and future-collider falsifiers. Any verified
collider deviation from the predicted resonance pattern at the
$5\sigma$ level falsifies $\T{4}$ + $\T{6}$ jointly.

\paragraph{What \PaperPTwo{} \citep{ArisakaP2N3} carries.} The three-leg derivation of the family number: exactly three fermion generations, from a CP-violation lower bound, a
\GDOS{} Morse/budget upper bound, and a codebook cross-check ---
with the conditioning (the coupling clause and integrality lemma L-INT of that paper, at premises shared with the Constitutional Born Theorem) declared at every level. A verified fourth generation falsifies the
derivation outright.

\paragraph{What \PaperPThree{} \citep{ArisakaP3Electron} carries.}
The six oldest questions about the electron --- identity with its
measured values, absolute stability, the two excited copies, the
positron as conjugate readout, pointlikeness, and the response to
fields --- closed as six faces of one fact: the electron is the unique ground state of its
boundary stratum. Written as a review and gateway; imports
\PaperPTwo{}'s $N=3$ with its fences verbatim. Any observed
electron decay, substructure at any scale, or a fourth lepton
generation falsifies it.

\paragraph{What \PaperPSix{} \citep{ArisakaP6Proton} carries.}
The proton as the \GGA{} of the baryonic stratum: its exact mass and
zero width, its charge equal and opposite to the electron's, its spin,
and above all its stability --- baryon number carried not as a
bookkeeping label but as a winding of the compact geometry, written by
the same twist that breaks the unified gauge symmetry and splits the
Higgs multiplet, so that the decay the grand unified theories insisted
on is forbidden by the geometry rather than merely unobserved. Eleven
questions the proton poses, closed as faces of one fact, every import
at its proved grade and every kill criterion stated. A single verified
proton decay falsifies it.

\paragraph{What \PaperCOne{} \citep{ArisakaC1Kaxion} carries.}
The kaxion oscillation $\Delta\nu/\nu = 1/35$ in the dark-matter
sector, with explicit haloscope and microwave-cavity falsifiers.
Any verified discovery at $\Delta\nu/\nu \neq 1/35$ falsifies the
\OGNfive{} ledger.

\paragraph{What \PaperCTwo{} \citep{ArisakaC2Hubble} carries.}
The Hubble tension read as geometry: the two measured Hubble
constants as the two ends of one six-dimensional geometric
corridor, with the corridor's endpoints fixed by the same 6D bulk that fixes the lock scale.

\paragraph{What \PaperCThree{} \citep{ArisakaC3BH} carries.}
Black-hole entropy under \GGTheory{}: the horizon as an
\OGGEP{} port, with the exact Bekenstein--Hawking coefficient
$A/4$ obtained from induced gravity --- the cohort's
strong-gravity anchor.

\subsection{Bridge to the White-Paper Branch (W1, W2)}
\label{sec:bridge_whitepapers}

On the map's right flank stand two synthesis papers. They are the
mirror of the Foundations Branch in the literal sense: that flank
feeds the spine, this one reads it. Neither derives anything new;
each carries every result at its parent's declared grade, and each
says so of itself. They are addressed to a physicist who wants to know
what the cohort amounts to without reading all twenty-two papers.

\paragraph{What \PaperWOne{} \citep{ArisakaW1Symmetry} carries.}
The \emph{diachronic} reading: the century of symmetry and
conservation walked in the order it was found --- Noether's two
theorems, the Bianchi identity, Wilson's loops, the Green--Schwarz
mechanism, Hopkins--Singer quantization --- and the conviction that
symmetry causes conservation loosening at each step, until what
remains conserved is an integer with nowhere to move. The program is
placed at the end of that sequence, at declared grades, with its
payout set beside the measurements. Its capstone is the determinism
triad: symmetry breaking re-read as symmetry \emph{reading}.

\paragraph{What \PaperWTwo{} \citep{ArisakaW2Holonomy} carries.}
The \emph{synchronic} reading: one exact and elementary statement ---
a holonomy requires a closed loop, and the interval fiber has none, so
everything on the circle is quantized, protected and reversible while
nothing on the interval can be --- followed to sixteen of the
questions physics has not closed, from what happens when a measurement
happens to what became of the information that fell into a black hole.
The paper closes with the named, dated measurements on which six of
the sixteen treatments die outright, and with how the other ten are
exposed through the ledger they share.

\subsection{Cross-paper coherence map}
\label{sec:coherence_map}

Table~\ref{tab:coherence_map} pins each Principal Theorem $\T{k}$
to the canonical theorem location in the companion paper where
its proof actually lives. This is the master audit reference for
the \GGTheory{} series.



\paragraph{Audit trail.} Each row of
Table~\ref{tab:coherence_map} has been verified against the
actual companion-paper PDF named in the row. In particular,
the \ATwo{} internal theorem numbering, written here in the
cross-paper form \TX{n}{GG6}, is:
\TX{1}{GG6} (dimensional minimality of the bulk),
\TX{2}{GG6} (canonical metric of \GG{}),
\TX{3}{GG6} (six forces with three Abelian invariances --- the
gauge-group theorem),
\TX{4}{GG6} (\BIsix{} coefficient vector),
\TX{5}{GG6} (single-tensor anomaly factorization
$I_8 = \tfrac{1}{2}X_4\wedge X_4$),
\TX{6}{GG6} (the \OGNfive{} Diophantine cascade),
\TX{7}{GG6} (BRST consistency and homogeneous Slavnov--Taylor
identity),
\TX{8}{GG6} (Arisaka equation and Quadruple Gauge Locking);
the \AFive{} internal theorem numbering
uses local labels T-\GDOS{}-1 through T-\GDOS{}-11, printed as
$\T{1}$--$\T{11}$, which match by construction those of the present
coherence map; the \ASix{} internal numbering uses local labels
T-\GRIP{}-1 through T-\GRIP{}-5, printed as $\T{1}$--$\T{5}$, for
the four pillars and the synthesis; the \ASeven{} internal theorem
numbering is section-prefixed and runs continuously through that
paper, so its four pillars occupy Parts~III--VI (\S\S11--49) with
main theorems at 12.x--15.x (\QiQUA{}), 22.x--27.x (\QiEP{}),
33.x--36.x (\QiCON{}), and 42.x--47.x (\QiHAL{}). Each row names the theorem's content as well as its number, so a renumbering in a companion shows up as a mismatch a reader can see rather than as a silently wrong pointer.

{\footnotesize
\begin{longtable}{@{}p{0.06\linewidth}p{0.36\linewidth}p{0.50\linewidth}@{}}
\caption[Cross-paper coherence map for the eleven Principal Theorems.]{\textbf{Cross-paper coherence map for the eleven
Principal Theorems.} Each row records the theorem name and the
canonical companion-paper location where its proof lives. None
is proved inside the present paper; the constitution names them
and points the reader downstream.}
\label{tab:coherence_map}\\[0.35em]
\toprule
$\T{k}$ & \textbf{Theorem name} & \textbf{Proof location} \\
\midrule
\endfirsthead
\toprule
$\T{k}$ & \textbf{Theorem name} & \textbf{Proof location} \\
\midrule
\endhead
\midrule
\multicolumn{3}{r}{\small continued on next page}\\
\endfoot
\bottomrule
\endlastfoot

$\T{1}$ & Locked Equivalence &
\ATwo{} \citep{ArisakaA2GG6}, Theorem~8 (Arisaka equation
and Quadruple Gauge Locking on the composite
$\Spinone\times\SU(5)\times\U(1)_B\times\U(1)_{Qi}$ connection) \\

$\T{2}$ & Operational Closure &
\AFive{} \citep{ArisakaA5GDOS}, Theorem~$\T{2}$
(Exact Bulk-to-Boundary Reduction, Nakajima-Zwanzig form) \\

$\T{3}$ & Anti-Landscape Selection &
\ATwo{} \citep{ArisakaA2GG6}, Theorem~3 (Six forces with
three abelian invariances --- the unique gauge group
$\Spinone\times\SU(5)\times\U(1)_B\times\U(1)_{Qi}$ forced by
$\AX{2}$ + $\AX{4}$ + $\POS{1}$ + $\POS{3}$, eliminating all
landscape alternatives) \\

$\T{4}$ & 6D bulk Existence &
\ATwo{} \citep{ArisakaA2GG6}, Theorem~1 (dimensional
minimality of the bulk: $\cM_6 = M_4 \times I_r \times S^1_\theta$)
+ Theorem~2 (canonical metric
$ds_6^2 = \e^{-2r}\eta_{\mu\nu}\,dx^\mu dx^\nu + dr^2 + d\theta^2$) \\

$\T{5}$ & \BIsix{} Four-Component Anomaly Identity &
\AThree{} \citep{ArisakaA3BI6}, Theorem~8.1 (the \OGNfive{} trace-ratio
theorem: the $\SU(5)$ trace ratio $\Tr(B_0^2)/\Tr(Y_0^2) = (L-N)/L$
fixes the abelian fraction to exactly $2/5$ at $(N,L) = (3,5)$),
completed globally by Theorem~6.2 (complete anomaly freedom of
\BIsix{} through the Wu-Chern-Simons / Hopkins-Singer
differential-cohomology lift),
plus the redundant cross-check at \ATwo{} \citep{ArisakaA2GG6}, Theorem~4 (\BIsix{} coefficient vector
$(a,b,c,d) = (1,-1,-2/5,-2/5)$) \\

$\T{6}$ & \OGNfive{} Ledger as Bulk Output &
\ATwo{} \citep{ArisakaA2GG6}, Theorem~6 (the \OGNfive{}
Diophantine cascade fixes
$(M,N,L;J,K) = (1,3,5;7,35)$ as a five-step integer pinning
forced by $\AX{4}$ + $\POS{6}$) \\

$\T{7}$ & 4D Quantum Mechanics &
\AFive{} \citep{ArisakaA5GDOS}, Theorem~$\T{7}$
(QM as boundary projection: Born rule, projection postulate,
Bell, Stern-Gerlach, Aspect 1982 + Hensen 2015) plus seven
sub-theorems \\

$\T{8}$ & 4D General Relativity &
\AFive{} \citep{ArisakaA5GDOS}, Theorem~$\T{8}$
(GR as boundary projection: modified TOV, Schwarzschild +
Kerr limits, equivalence principle) plus seven
sub-theorems \\

$\T{9}$ & 4D Quantum Field Theory &
\AFive{} \citep{ArisakaA5GDOS}, Theorem~$\T{9}$
(QFT as boundary projection: BRST cohomology,
Slavnov-Taylor identities, anomaly Ward identity) plus eight
sub-theorems \\

$\T{10}$ & \GiFour{} Pillars &
\ASix{} \citep{ArisakaA6Gi4}, Theorems
$\T{1}$~(\TGG{} existence, uniqueness, and Master Conservation),
$\T{2}$~(\CCI{} closure forces \OGNfive),
$\T{3}$~(\GFF{} Morse-positivity), $\T{4}$~(\GRIP{} cascade
closure: \DGI{} $\to$ \GSSB{} $\to$ \GA), and $\T{5}$
(constitutional equivalence of the \GiFour{} pillar system and the
\GGConstitution{} plus \BIsix{}) \\

$\T{11}$ & \QiFour{} Pillars &
\ASeven{} \citep{ArisakaA7Qi4}, four-pillar main theorems located
in Parts~III--VI (\S\S11--49): \QiQUA{} pillar (Theorem~12.2 rank-one
inflow, Theorem~14.3 no-new-parameter, Theorem~15.3 natural-unit
equivalence); \QiEP{} pillar (Theorems~22.1, 23.1, 24.1 for the
\PLT{}, \MEPP{}, and FEP gauges, with the common-origin theorem at
Theorem~27.1); \QiCON{} pillar (Theorem~33.1 operational
conformality, Theorem~36.1 two-knee spectrum); \QiHAL{} pillar
(Theorem~42.2 \MePMoS{}, Theorem~44.2 frequency-independent
\HAL{}, Theorem~47.1 \QiHAL{} universality) \\

\end{longtable}
}

The axiomatic bargain offered by the \GGTheory{}
program is exactly this: eleven theorems, eleven companion-paper
homes, every pointer auditable. If a single row of
Table~\ref{tab:coherence_map} fails contact with experiment, the
constitution upstream falls.

\subsection*{The Geometric Universal Five Constants (\GUfive{}): $c$, $G_N$, $\hbar$, $k_B$, $\QB$}
\label{ssec:T-GU5}

One synoptic theorem deserves a name of its own. The five universal
dimensional conversion factors of physics --- $c$, $G_N$, $\hbar$,
$k_B$, $\QB$ --- are not free parameters of \GGTheory{}; each is a
calculable geometric or boundary-projected datum of the 6D bulk. We collect this synoptic claim as the
\emph{Geometric Universal Five Constants theorem}, $\T{-GU5}$, with each
constituent pinned to its primary companion paper:

\begin{euclidtheorem}[\textbf{$\T{-GU5}$}: Geometric Universal Five Constants]
\label{thm:T-GU5}
Under the \GGConstitution{}, the five universal dimensional
conversion factors are derived secondary quantities of the 6D bulk geometry and its boundary projection, with primary
derivations as follows:
\begin{itemize}[leftmargin=2em]
\item $c$ --- conversion factor of the $(1,3)$ Lorentzian signature
on $\partial \cM_6 = \cM_4$, fixed by the canonical metric of
\citep[Theorem~2 (canonical metric)]{ArisakaA2GG6}.
\item $\hbar$ --- conversion factor of the compact $S^1_\theta$
topology, fixed by Kaluza--Klein integer mode quantization in the
6D bulk \citep[Theorem~1 + topological $S^1$ section]{ArisakaA2GG6}.
\item $G_N$ --- scale-curvature coupling of the radial interval $I_r$,
expressed as the overlap integral
\citep[Theorem~T8.2 (Newton constant)]{ArisakaA5GDOS} of the
scale-profile metric of \citep[Theorem~2]{ArisakaA2GG6}.
\item $\QB$ --- energy--information Arisaka constant $\QB = (2/5)E_{\rm Ry}/K$,
the conversion factor between energetic and gauge-count currents on
the IR boundary, derived in
\citep[\S 9, T-QUA.2]{ArisakaA7Qi4}.
\item $k_B$ --- Bekenstein--Hawking thermodynamic conversion factor
of the IR-boundary surface gravity, fixed by the scale profile $e^{-K}$
with $K = 35$ \citep[\S 25 ($k_B$ derived)]{ArisakaA4Euclid}.
\end{itemize}
Together with the two topological constants $\{e, \pi\}$ of
\citep[\S 26]{ArisakaA4Euclid}, the Pentad supplies the full
dimensional and arithmetic basis from which every dimensionful
scale of \GGTheory{} (Planck, GUT, lock, electroweak, Rydberg,
biological, cosmological-constant) is forced, given the single overall unit scale, set by $G_N$ (with $M_6$ derived from it).
\end{euclidtheorem}

The Pentad is synoptic, not new: every conversion factor it names
is derived in a primary companion paper, and the present
subsection only collects them under a single theorem label so that
downstream papers can cite ``$\T{-GU5}$'' as the closure statement of
the conversion-factor reconstruction. The closing remark of \AFour{}
\citep[\S 28 Master Derivation Table]{ArisakaA4Euclid} provides
the full numerical-values table; the present theorem is the
axiomatic statement that closes the table.

\subsection{Minimum reading path for evaluators}
\label{sec:min_reading_path}

A reader who wishes to assess the central claims of the cohort
framework on a first pass should follow a focused reading path
rather than attempting the full cohort inheritance at once.
The recommended sequence is:

\begin{enumerate}[leftmargin=2.2em]
\item This paper (\AOne{}) for the four axioms, the seven
postulates, and the constitutional classification of twentieth-century
physics (Part~V).
\item \ATwo{} \citep{ArisakaA2GG6} for the uniqueness theorem of
the 6D bulk and the $\OGNfive{}$ Diophantine forcing; this
is the load-bearing technical paper, and the entire cohort depends
on its uniqueness arguments holding up.
\item \AFive{} \citep{ArisakaA5GDOS}, Part~III for the
$\T{7.1}$--$\T{7.7}$ derivation of the textbook QM postulates
(Born rule, L\"uders projection, decoherence, einselection,
Bell--CHSH violation) from $\AX{3}~\OGGEP{}$ together with
$\POS{4}~\CP{}$; this is the load-bearing technical paper for the
``measurement problem is solved'' claim.
\item \ANine{} \citep{ArisakaA9SM}, the worked chapter for one
specific Standard-Model parameter (any one of the 28) as an
explicit example of the 28-to-0 collapse claim.
\end{enumerate}

A reader who completes these four documents has the necessary
scaffolding to form a real opinion about whether the constitutional
program is correct. The remaining cohort papers (\AThree{},
\AFour{}, \ASix{}, \ASeven{}, \AEight{}, \ATen{}, and \AEleven{},
the last of these in preparation) supply the depth and breadth of
consequences but are not strictly required for first-pass
evaluation.

\subsection{Consolidated falsifier table: numerical predictions and kill thresholds}
\label{sec:pull_table}

Table~\ref{tab:pull_table} consolidates the sharpest numerical
predictions issued by the \GGTheory{} program. Each row records
a global falsifier identifier F-Constitution-$k$, the constitutional
source (which axiom/postulate combination
forces the prediction), the proof location in the cohort, the predicted value, the currently best-measured value, the
comparison mode --- a pull in sigma where the comparison yields a
number, and otherwise the kind of check the entry carries --- the
pre-declared kill threshold, and the experimental status as of 2026. Indeed, the strength of the
constitution is best measured not by what it explains but by what
it would have to give up if any one row's kill criterion were
triggered.

\begingroup
\footnotesize
\setlength{\tabcolsep}{5pt}
%% Column widths are MEASURED, not chosen.  The binding cells are
%% $(1,-1,-2/5,-2/5)$ at 78.1pt (Predicted), "Consistency" at 48.2pt
%% (Mode), and the header "Kill threshold" at 64.4pt.  Sum of the six
%% widths is 0.883\linewidth = 400.0pt; with 2 x 5pt x 5 gutters the
%% table occupies 450.0pt against a 452.97pt line.  The falsifier
%% identifier no longer sits in a column of its own -- at 86pt for
%% F-Constitution-10 it could only fit by breaking at its own hyphens,
%% which printed as "F- / Constitution- / 10:" down four lines -- so it
%% now heads each entry across the full width.
\begin{longtable}{@{}
 >{\raggedright\arraybackslash}p{0.120\linewidth}
 >{\raggedright\arraybackslash}p{0.180\linewidth}
 >{\raggedright\arraybackslash}p{0.145\linewidth}
 >{\raggedright\arraybackslash}p{0.108\linewidth}
 >{\raggedright\arraybackslash}p{0.190\linewidth}
 >{\raggedright\arraybackslash}p{0.140\linewidth}@{}}
\caption[Consolidated falsifier table.]{\textbf{Consolidated falsifier table.}
The sharpest numerical predictions of the \GGTheory{} program,
with proof location, pre-declared kill threshold, and current
experimental status. Each entry is headed by its global falsifier
identifier and is a $\POS{7}~\MF{}$ claim contract in the canonical
four-tuple form. \emph{Mode} records how the prediction is being
compared with data: a pull in sigma where the comparison yields a
number, and otherwise the kind of check the entry carries, in the
sense fixed immediately after the table.}
\label{tab:pull_table}\\[0.35em]
\toprule
\textbf{Source} & \textbf{Predicted} & \textbf{Observed} & \textbf{Mode} & \textbf{Kill threshold} & \textbf{Status (2026)}\\
\midrule
\endfirsthead
\toprule
\textbf{Source} & \textbf{Predicted} & \textbf{Observed} & \textbf{Mode} & \textbf{Kill threshold} & \textbf{Status}\\
\midrule
\endhead
\midrule
\multicolumn{6}{r}{\small (continued)}\\
\endfoot
\bottomrule
\endlastfoot

\multicolumn{6}{@{}l}{\textbf{F-Constitution-1:}\quad GUT-scale unification $\EGUT$}\\*[0.18em]
$\OGN{}$ + 6D bulk; $\ATwo{}$ & $\EPl\,e^{-2\pi} \approx$ $2.28\times 10^{16}$ GeV & Inferred from $\alpha_i$ flow extrapolation & Consistency & Failed unification at one loop falsifies & Forward audit (cohort) \\\addlinespace[0.55em]

\multicolumn{6}{@{}l}{\textbf{F-Constitution-2:}\quad KK lock-cluster $\Mlock$}\\*[0.18em]
$\OGNfive{}$ + 6D bulk; $\ATwo{}$, $\ANine{}$, \PaperPOne{} & $(1/2)\,\MPl\,e^{-35}$ $\approx 3.85$ TeV & No resonance yet at this mass; LHC Run~3 + HL-LHC search ongoing \citep{PDG2024} & Conditional & Registered $5\sigma$ absence at HL-LHC, or $5\sigma$ resonance at $\neq \Mlock$ & Active LHC search \\\addlinespace[0.55em]

\multicolumn{6}{@{}l}{\textbf{F-Constitution-3:}\quad Electroweak vev $v$}\\*[0.18em]
$\OGN{}$ + bulk; $\ANine{}$ & $\approx 246$ GeV & $246.220 \pm 0.001$ GeV \citep{PDG2024} & Postdiction audit & Verified deviation $>5\sigma$ from the rational structural derivation & Verified \\\addlinespace[0.55em]

\multicolumn{6}{@{}l}{\textbf{F-Constitution-4:}\quad Cosmological matter fraction $\Omega_m$}\\*[0.18em]
$\OGNfive{}$ + bulk; $\ATen{}$ & $1/\pi \approx 0.31831$ & $0.3153 \pm 0.0073$ (Planck 2018 \citep{Planck2018}) & $\sim 0.3\sigma$ & Registered $5\sigma$ deviation from $1/\pi$ in any next-generation cosmology dataset at resolution $\sigma_{\Omega_m} \leq 0.003$ & Consistent; CMB-S4 / LiteBIRD / DESI next-gen test \\\addlinespace[0.55em]

\multicolumn{6}{@{}l}{\textbf{F-Constitution-5:}\quad Cosmological-constant scale $m_\Lambda$}\\*[0.18em]
$\OGN{}$ + bulk; $\ATen{}$ & $\approx 6$ meV & Consistent with $\Lambda \approx$ $1.1\times 10^{-52}$ m$^{-2}$ \citep{Planck2018} & Postdiction audit & Verified deviation $>5\sigma$ from the predicted neutrino-scale matching & Consistent (CC sign + scale) \\\addlinespace[0.55em]

\multicolumn{6}{@{}l}{\textbf{F-Constitution-6:}\quad Cosmological baryon fraction $\Omega_b$}\\*[0.18em]
$\OGNfive{}$ + bulk; $\ATen{}$ & $1/(2\pi^2) \approx 0.05066$ & $0.0493 \pm 0.0006$ (Planck 2018) & $\sim 2\sigma$ & Registered $5\sigma$ deviation in any next-generation BBN / CMB combined fit & Consistent at $\sim 2\sigma$ \\\addlinespace[0.55em]

\multicolumn{6}{@{}l}{\textbf{F-Constitution-7:}\quad Kaxion oscillation $\Delta\nu/\nu$}\\*[0.18em]
$\OGNfive{}$ (integer $35 = K$); $\ATen{}$, \PaperCOne{} & $1/35 \approx 0.02857$ & Not yet registered & Forward & Registered narrow peak at $\Delta\nu/\nu \neq 1/35$ at $5\sigma$ over $10\sigma_\nu$ control region & Active haloscope search (ADMX-class) \\\addlinespace[0.55em]

\multicolumn{6}{@{}l}{\textbf{F-Constitution-8:}\quad Three generations}\\*[0.18em]
$\OGNfive{}$ ($N = 3$); $\ATwo{}$, \PaperPTwo{} & Exactly three fermion families & Three observed \citep{PDG2024} & Verified & Discovery of a fourth admissible generation falsifies $N = 3$ & Verified \\\addlinespace[0.55em]

\multicolumn{6}{@{}l}{\textbf{F-Constitution-9:}\quad \BIsix{} trace-ratio lock}\\*[0.18em]
$\POS{3}~\AF{}$ + $\POS{6}~\OGN{}$ + bulk; $\AThree{}$ & $|a/c| = |a/d| = 5/2$, $(a,b,c,d) =$ $(1,-1,-2/5,-2/5)$ & Implied by SM hypercharge assignments via descent; consistency check \citep{ArisakaA3BI6} & Audit & Discovery of a fifth admissible anomaly polynomial structurally inequivalent to $\BIsix{}$ & Internally consistent; downstream audit \\\addlinespace[0.55em]

\multicolumn{6}{@{}l}{\textbf{F-Constitution-10:}\quad Reality of the bulk category (cohort alias F-Real-1)}\\*[0.18em]
$\AX{4}~\MSMF{}$ + L-Constitution-1 (candidate); \PaperPFour{}, \PaperPFive{} & Every observed $i$ is the boundary reading of the one derived quarter-turn $J$: no fundamental non-geometric CP phase; one shared $i$; no quaternionic Born residue & No anomaly registered: unshared-real alternative excluded \citep{Renou2021}; three-path Sorkin residue consistent with zero \citep{Sinha2010} & Audit & A fundamental CP phase resisting geometric expression after the \ANine{} QD freeze; verified unshared-$i$ interferometry; a registered $5\sigma$ quaternionic-direction Born deviation & Forward audit (candidate grade; \S\ref{sec:AX4_category_axis}) \\

\end{longtable}
\endgroup

The classification of the \emph{Mode} column follows the cohort's
internal conventions: \emph{Forward} predictions await first
experimental registration; \emph{Verified} predictions have been
empirically confirmed and now serve as anchor data; \emph{Audit}
predictions are internal consistency checks that downstream cohort
papers must satisfy; \emph{Postdiction-audit} predictions are
historically known quantities that the cohort derivation must
\emph{retro}produce; \emph{Conditional} predictions sit at the boundary
of current experimental sensitivity and become Forward or Verified
as data accumulates. The pre-declared kill threshold of every row
is the constitutional commitment that $\POS{7}~\MF{}$ extracts from
the corresponding theorem: a verified violation of any row above
would collapse one or more clauses of the constitution upstream.

\subsection{Worked calculation chain: from \OGNfive{}\ to the kaxion oscillation $\Delta\nu/\nu = 1/35$}
\label{sec:worked_chain_kaxion}

To illustrate how a single numerical prediction descends from the
constitution into the cohort, we walk one chain end to end: the
kaxion oscillation $\Delta\nu/\nu = 1/35$ of \PaperCOne{}
\citep{ArisakaC1Kaxion}. No derivation is performed inside the
present paper; each step names the relevant constitutional clause
and the proof location in the cohort.

\begin{enumerate}[leftmargin=2.2em]
\item $\AX{2}~\GGEP{}$ together with $\POS{3}~\AF{}$, $\POS{5}~\RG{}$,
and $\POS{6}~\OGN{}$ forces the 6D bulk metric $ds_6^2 =
e^{-2r}\eta_{\mu\nu}dx^\mu dx^\nu + dr^2 + d\theta^2$ with the
four-factor composite gauge group $G_{\rm tot} = \Spinone \times
\mathrm{SU}(5) \times \mathrm{U}(1)_B \times \mathrm{U}(1)_{Qi}$ (uniqueness
theorem $\T{4}$ of $\ATwo{}$ \citep{ArisakaA2GG6}).

\item $\POS{6}~\OGN{}$ on the bulk forces the $\OGNfive{}$ integer
ledger $(M, N, L; J, K) = (1, 3, 5; 7, 35)$ as the unique lexicographic
maximum of the Diophantine system $K = LJ = M^2+N^2+L^2$ with chirality
count $J = 2N+1$, compatible with the four-component anomaly polynomial
of $\T{5}$ (theorem $\T{6}$ of $\ATwo{}$).

\item $\AX{2}~\GGEP{}$ on the compact $\theta$-circle of the bulk
admits a pseudo-Nambu-Goldstone field --- the kaxion --- whose mass
$m_k$ is set by the $\mathrm{U}(1)_{Qi}$ holonomy and the
$\OGNfive{}$ lock length $K = 35$; specifically, $m_k$ is fixed by
that holonomy and that lock length --- not by the $\Mlock^2/\MPl$
combination, which sets the cosmological-constant scale some five
orders of magnitude higher
(derived in $\ATen{}$ \citep{ArisakaA10Cosmos}, Theorem~$\T{9}$).

\item Under cosmological boundary conditions the kaxion field
oscillates around the bulk holonomy minimum with a frequency shift
$\Delta\nu/\nu$ that is the OGN-5 integer fraction $1/K = 1/35$,
forced by the period of the $\theta$-circle and the lock-length
quantization (worked example in \PaperCOne{}
\citep{ArisakaC1Kaxion}).

\item $\POS{7}~\MF{}$ then requires the prediction to compile to a
registered measurement contract: a haloscope-style microwave cavity
tuned to the kaxion mass, with a pre-declared kill criterion of
$5\sigma$ registered disagreement with $\Delta\nu/\nu = 1/35$ over a
$10\sigma_\nu$-wide control region (see Part~III, \S\ref{sec:POS7}
and \PaperCOne{}).
\end{enumerate}

The chain has five steps. Step 1 lives in $\ATwo{}$; step 2 lives in
$\ATwo{}$; step 3 lives in $\ATen{}$; step 4 lives in \PaperCOne{};
step 5 lives both in Part~III of the present paper and in
\PaperCOne{}. After all, the cohort's structural integrity is
exactly this: a five-step chain from the constitutional integers
$(1, 3, 5; 7, 35)$ to a $\mu$eV cavity signal that an experimenter
can register or fail to register.


\subsection*{Closing remarks on Part~IV}

The handoff is complete. Every claim this document makes now has an
address --- a named theorem, in a named paper, at a named number.

It is fair to ask whether that is a virtue or an evasion, and the
question deserves a direct answer rather than a slogan. It is an
evasion if the addresses are vague, if they drift as the companions are
revised, or if following one leads to a paper that forwards you
somewhere else again. It is a virtue only if the pointers are exact and
stay exact. We take this seriously enough that the coherence map was
rebuilt against the companion papers themselves rather than from
memory, and a reader who finds a row that does not resolve should treat
that as a real defect and not a typographical one.

The wider principle is one that engineers know better than physicists.
A specification and an implementation are different documents, and
confusing them damages both. A specification long enough to contain its
implementation cannot be read as a specification; an implementation
that repeats its specification drifts out of agreement with it. Physics
has traditionally written the two together, in the same paper, in the
same voice --- which is part of why the rules ended up unwritten. They
were never given a document of their own.

There is a cost, and we would rather state it than let a reader
discover it with irritation. Someone who wants to check a single number
in the consolidated falsifier table will have to open at least one
companion paper, and possibly two. That is a real demand on a reader's
time, and no amount of good cross-referencing makes it free. What we
can promise is that the demand is finite and mapped: the minimum
reading path in this Part names four documents, not twenty-two, and the
falsifier table names the experiment rather than asking you to trust
the derivation.

There is also something the map reveals that no single paper does.
Reading down the coherence table, one sees the same five integers
turning up in the anomaly coefficients, the generation count, the
electroweak scale, the recombination era, and the dark-matter
oscillation frequency. Whether that recurrence is a discovery or a
coincidence is exactly what the cohort is for. But the recurrence
itself is visible only from here, in the index --- which is a reason
indexes are worth reading.

Part~V now asks a different question. Instead of where our own proofs
live, it asks where everyone else's theories live, and finds that the
same four axioms sort them.

\GGPart[Axiom classification of physics]{The Constitutional-Axiom Classification of 20th-Century Physics}%
 {General Relativity, Gauge Theory, String Theory, AdS/CFT, Quantum Mechanics, the Standard Model, and $\Lambda$CDM as Theorems of Axiom Subsets}

\noindent
Here is the move this Part makes, and it is worth stating plainly
before the machinery arrives.

The usual way to compare physical theories is to ask which one is
right. That question is often unanswerable, because rival frameworks
tend to agree wherever experiment has actually reached. So we ask a
different question: \emph{which of the four axioms does each framework
quietly assume?}

The answer sorts twentieth-century physics into four layers, and the
sorting is not a matter of taste. Take $\AX{1}$ alone and you have
special relativity --- causality and a speed limit, nothing more. Add
$\AX{2}$ and geometry becomes a carrier of forces: general relativity,
Yang--Mills theory, and the whole classical-topological backbone on
which string theory and AdS/CFT are built. Add $\AX{3}$ and observation
enters as a projection, which is where quantum mechanics, statistical
mechanics, the fluctuation theorems, black-hole thermodynamics and the
free-energy principle of theoretical neuroscience all turn out to live
--- four communities that rarely read one another, sharing one axiom.
Add $\AX{4}$ and the free parameters have to go.

What makes this a classification rather than a filing system is that it
predicts silences. A framework standing on the second layer has no
operational content, so it cannot say when a measurement happens ---
and, historically, it does not. A framework standing on the third has no
selection principle, so it cannot say why the constants take the values
they do --- and it does not. What a theory \emph{cannot} say is as
diagnostic as what it can, and the missing axiom tells you which silence
to expect.

We state the tower formally, walk each layer in a paragraph, print a
master ledger locating every major framework, count the vacua the
classification implies, and close with the case that superstring theory
cannot rise past the second layer without adopting the axioms it
currently declines.

\medskip

\section{The central structural claim, stated once}
\label{sec:V-central-claim}

\paragraph{Statement.} Every major theoretical-physics framework
of the twentieth century is a theorem of a particular subset of
the four \GGConstitution{} axioms applied to a particular bulk
solution; the framework's strengths and limitations are direct
consequences of which axioms it implicitly uses and which it
lacks.

This is not a refutation of any historical framework. It is a
\emph{structural location} of each framework inside a single
constitutional tower. Each historical theory correctly used the
axiom subset it implicitly invoked; each left silent exactly the
questions that required the missing axioms. The classification we
are about to develop is the structural map.

Indeed, this Copernican reframing does not contradict the
historical derivations; it locates them. Just as Riemann's
nineteenth-century geometry was correctly understood as pure
mathematics for sixty years before Einstein identified the
physical setting (general relativity) in which it became
operative, the twentieth-century geometric edifice --- the Bianchi
identities, the Green--Schwarz mechanism, the modular bootstrap,
the AdS/CFT correspondence, the holographic principle, the
dualities --- has been correctly developed by the superstring
community for forty years without recognition of the
constitutional setting (the \GGConstitution{} with various axiom
subsets) in which the machinery becomes operative. The present
Part identifies that setting.

The technical derivation chain that backs each row of the
classification is developed in full detail in \AFive{}
\citep[Part~VIII]{ArisakaA5GDOS}. The present Part restates the
classification in a form a reader of the constitution itself can
audit at first encounter.

\medskip

\section{The constitutional tower: a formal statement}
\label{sec:V-constitutional-tower}

The four axioms of the \GGConstitution{} form a strict logical
tower: each axiom presupposes the previous and adds new content.
$\AX{1}$ requires a manifold with a causal structure. $\AX{2}$
requires $\AX{1}$ plus a bulk gauge bundle unified with the
geometry. $\AX{3}$ requires $\AX{1}+\AX{2}$ plus the
operational dictionary. $\AX{4}$ requires
$\AX{1}+\AX{2}+\AX{3}$ plus the uniqueness principle.

We denote the four active-axiom subsets by their natural names,
and attach to each the natural \emph{stratum} index that aligns
with the constitutional axiom it activates:

\begin{itemize}[leftmargin=2.2em]
\item \textbf{Stratum~1} $= \{\AX{1}\}$ --- the \LC{} level
 (local causality alone).
\item \textbf{Stratum~2} $= \{\AX{1}, \AX{2}\}$ --- the
 $\mathsf{GGEP\text{-}closed}$ level (geometry-gauge equivalence
 on a closed bulk).
\item \textbf{Stratum~3} $= \{\AX{1}, \AX{2}, \AX{3}\}$ ---
 the $\mathsf{OGGEP\text{-}open}$ level (operational/open
 extension to a finite-information boundary observer).
\item \textbf{Stratum~4} $= \{\AX{1}, \AX{2}, \AX{3}, \AX{4}\}$
 --- the $\mathsf{Full\text{-}Constitution}$ level (the full
 axiom set, forcing the unique integer \OGNfive{} ledger and
 zero free continuous parameters in the bulk theory).
\end{itemize}

Each Stratum-$N$ activates Axiom $\AX{N}$, and a framework that
implicitly uses axiom subset $\{\AX{1},\ldots,\AX{N}\}$ sits at
Stratum~$N$. The Stratum number and the highest activated Axiom
number are deliberately aligned so that the classification reads
off any diagram by inspection.

\paragraph{Why subsets are physically meaningful.} A natural
objection: \emph{the \GGConstitution{} is a constitution; one
either accepts all four axioms or none; why is it meaningful to
ask what subsets produce?} Roughly speaking, the answer is that
the historical twentieth-century theories are precisely the
answers to that question, asked implicitly. When string theorists
postulate that geometry and gauge unify on a 10D worldsheet but do
not address the operational projection to a finite-observer
boundary, they are implicitly using
$\mathsf{GGEP\text{-}closed}$ and not
$\mathsf{OGGEP\text{-}open}$. When QM-foundations researchers
postulate the Born rule but accept that string-theoretic bulk
structures are someone else's problem, they are implicitly using
$\mathsf{OGGEP\text{-}open}$ but not invoking the bulk content
that $\AX{2}$ forces. The historical theories are the natural
specializations of the constitution to specific axiom subsets, and
they reveal the structure of the constitution by what each takes
as operative and what each leaves silent.

\medskip

\section{The constitutional flowchart}
\label{sec:V-flowchart}

The single most useful visual orientation for the rest of this
Part is the constitutional flowchart of
\figref{fig:A1-constitutional-flowchart}, overleaf in landscape
orientation. Each Stratum band on the right is aligned with the
constitutional axiom on the left that activates it: $\AX{N}$
activates Stratum~$N$ for $N = 1, 2, 3, 4$. Each cell within a
band names a historical framework that the cohort framework
relocates as a theorem of that stratum. The cohort framework
itself (\GGTheory{}, cyan band at the bottom) is the unique
theory that explicitly activates all four axioms and so admits
the full theorem class.

\medskip

\noindent\textbf{Figure~\ref{fig:A1-constitutional-flowchart} (next page, in landscape orientation).}
\medskip

\noindent
The comprehensive flowchart on the following page presents the
logical cascade from the four-axiom \GGConstitution{} (far left)
to the entire family of nineteenth- and twentieth-century
theoretical physics frameworks. Each axiom on the left activates
the Stratum band of the same number on the right.
\emph{Stratum~1} (yellow, $\{\AX{1}\}$) holds the frameworks
that need only local causality: special relativity (1905) and
classical electrodynamics (1865).
\emph{Stratum~2} (orange, $\{\AX{1},\AX{2}\}$;
$\mathsf{GGEP\text{-}closed}$) holds the geometry-gauge family
descended from Bianchi 1902 --- general relativity (1915),
Yang--Mills (1954), the Green--Schwarz mechanism (1984), superstring
theory (1984+), AdS/CFT (1997), holographic dualities (1993+),
and anomaly cancellation (1984).
\emph{Stratum~3} (green, $\{\AX{1},\AX{2},\AX{3}\}$;
$\mathsf{OGGEP\text{-}open}$) holds the three siblings of
openness: the quantum-mechanics family with the Born rule and
Bell-inequality violations, the classical-thermodynamics family
with the Friston free-energy principle, and the black-hole family
with the Page curve and the islands formula.
\emph{Stratum~4} (teal, $\{\AX{1},\AX{2},\AX{3},\AX{4}\}$;
$\mathsf{Full\text{-}Constitution}$) holds the frameworks that
the integer \OGNfive{} ledger forces: the Standard Model
(1967--78) with its 28 parameters collapsed to zero, $\Lambda$CDM
(1998+) with $\Omega_m = 1/\pi$, and the cohort cascade (2026)
from quarks to consciousness.
At the bottom of the figure, the cohort framework \GGTheory{} is
the unique theory that explicitly activates all four constitutional
axioms; the three application papers (\ANine{}, \ATen{},
\AEleven{}) operationalize the constitutional output into the SM
parameters, the cosmological inventory, and the
quarks-to-consciousness identity cascade. The L-shape link on the
left signals the constitutional source $\Leftrightarrow$ derived
theory relation between the \GGConstitution{} (far left) and
\GGTheory{} (bottom center): the constitution defines the theory,
and the theory articulates the constitution.

\clearpage

\begin{landscape}
\thispagestyle{empty}
\begin{figure}[p]
\centering
\resizebox{\linewidth}{!}{%  % inside landscape, \linewidth IS the rotated width (650.43pt); \textheight is the short side
\usebox{\GGscFlow}%
}% end landscape resizebox
\caption[Constitutional flowchart of the charter.]{The constitution drawn as a flowchart. Each framework of twentieth-century physics enters at the level of the axioms it implicitly uses, and the branches descend as axioms are added: one axiom gives special relativity, two give the geometric machinery of general relativity and gauge theory, the third adds the boundary coarse-graining that quantum mechanics and thermodynamics share, and the fourth removes the freedom that remains. What each branch \emph{cannot} say is the content of the chart --- a
missing axiom is a prediction about which questions a framework will
leave open, not merely an omission --- and the extended description on
the preceding page reads the chart row by row, naming for each stratum
what its occupants are unable to express.}
\label{fig:A1-constitutional-flowchart}
\end{figure}
\end{landscape}
\clearpage

\section{Walking the tower: each stratum in one paragraph}
\label{sec:V-walking-the-tower}

\paragraph{Stratum~1 ($\{\AX{1}\}$): special relativity and classical electrodynamics.}
At Stratum~1, the only constitutional content is local causality.
Special relativity and classical electrodynamics live here. Local
Lorentz transformations, $E = mc^2$, Maxwell's equations on
Minkowski space, the radiation field as a propagating wave: all
follow from $\AX{1}$ together with the choice of a flat manifold.
What Stratum~1 cannot deliver is gravity (which requires the
geometric content of $\AX{2}$), the quantum content of
measurement (which requires $\AX{3}$), or any unique parameter
selection (which requires $\AX{4}$).

\paragraph{Stratum~2 ($\{\AX{1},\AX{2}\}$, $\mathsf{GGEP\text{-}closed}$): general relativity, Yang--Mills, string theory, AdS/CFT.}
At Stratum~2, the constitutional content extends to the
geometry-gauge equivalence principle. General relativity is what
$\mathsf{GGEP\text{-}closed}$ produces when the composite
connection reduces to the Levi--Civita connection alone;
Yang--Mills is what it produces when the connection reduces to a
Lie-algebra-valued gauge field; superstring theory is the
$\mathsf{GGEP\text{-}closed}$ content applied to a 10D bulk with
extended-object dynamics; AdS/CFT is the
$\mathsf{GGEP\text{-}closed}$ duality between bulk AdS and boundary
CFT in the same axiom subset. Remarkably, the Green--Schwarz
modified Bianchi identity --- conventionally presented as a
string-theoretic achievement --- is a theorem of $\AX{2}$
applied to a bulk with appropriate dimension; the constraint
itself was always geometric and descended from Bianchi 1902
\citep{Bianchi1902,GreenSchwarz1984}. Strings supplied one
physical scenario in which the constraint had to be satisfied, but
the constraint's logical status was never string-dependent. What
$\mathsf{GGEP\text{-}closed}$ cannot deliver is the quantum
content of measurement (no $\AX{3}$, hence no Born rule, no
L\"uders projection, no decoherence as theorems; the measurement
problem remains structurally unaddressed) and no unique vacuum
selection (no $\AX{4}$, hence the $\sim 10^{500}$ flux landscape
of the Bousso--Polchinski / Douglas--Kachru count).

\paragraph{Stratum~3 ($\{\AX{1},\AX{2},\AX{3}\}$, $\mathsf{OGGEP\text{-}open}$): quantum mechanics and the three siblings of openness.}
At Stratum~3, the operational/open extension $\AX{3}~\OGGEP{}$
becomes operative. This is the stratum at which quantum mechanics
with its measurement-problem resolution becomes derivable: the
chain of theorems $\TX{7}{.1}$--$\TX{7}{.7}$ of \AFive{}
\citep{ArisakaA5GDOS} derives the Hilbert-space postulate, the
Schr\"odinger equation, the Born rule, the L\"uders projection,
decoherence, einselection, and Bell-inequality violation as
theorems of $\AX{3}~\OGGEP{}$ acting on the bulk. The textbook
QM postulates are no longer postulates; they are theorems of the
operational dictionary. Two sibling families share Stratum~3:
classical thermodynamics with the modern Friston free-energy
principle, and black-hole physics with the Page curve and the
islands formula. What $\mathsf{OGGEP\text{-}open}$ cannot
deliver is parameter selection (no $\AX{4}$); the Standard-Model
gauge group and its 28 parameters remain unconstrained by this
axiom subset.

\paragraph{Why string theory has not solved QM foundations in forty years.}
\emph{Generally speaking, this is the diagnostic core of the
constitutional classification.} Within
$\mathsf{GGEP\text{-}closed}$ (the axiom subset string theory
implicitly uses), the bulk-to-boundary map is a unitary
equivalence (the AdS/CFT-style duality), not a CPTP channel.
Unitary maps preserve all information; they do not coarse-grain;
they do not produce probability assignments from definite bulk
states. The Born rule, the L\"uders projection, the
decoherence-induced classicality of macroscopic outcomes --- all
of these require the operational coarse-graining that only
$\AX{3}$ supplies. Within $\mathsf{GGEP\text{-}closed}$ one can
compute correlation functions and check duality predictions; one
cannot \emph{derive} the QM postulates because there is no
constitutional machinery for the open-system reduction. This is
exactly why string theory has produced no progress on QM
foundations in forty years. The framework is structurally
incapable of addressing the measurement problem, not for lack of
effort but because the axiom needed for the derivation ($\AX{3}$)
was not in its framework. Add $\AX{3}$ to the axiom set, and the
measurement problem dissolves into the chain
$\TX{7}{.1}$--$\TX{7}{.7}$.

\paragraph{Stratum~4 ($\{\AX{1},\AX{2},\AX{3},\AX{4}\}$, $\mathsf{Full\text{-}Constitution}$): the Standard Model, $\Lambda$CDM, and the cohort cascade.}
At Stratum~4, the maximum-symmetry-minimum-freedom axiom
$\AX{4}~\MSMF{}$ activates. This is the stratum at which the
Standard Model gauge group, the 28 free continuous parameters,
and the cosmological-constant scale collapse from independent
dials to forced consequences of the integer \OGNfive{} ledger.
The 28 parameters of the SM become zero free continuous parameters
of the full constitution (proved in \ANine{} \citep{ArisakaA9SM});
the $\Lambda$CDM partition $\Omega_m = 1/\pi$ becomes a forced
consequence of $\AX{4}+\POS{6}~\OGN{}$ (proved in \ATen{}
\citep{ArisakaA10Cosmos}); the cohort cascade from quarks to
consciousness becomes an axiomatic chain (developed in \AEleven{}
\citep{ArisakaA11Origin}). This is the
$\mathsf{Full\text{-}Constitution}$ that the \GGTheory{} cohort
framework explicitly activates, and it is what distinguishes the
cohort framework from every Stratum-2 or Stratum-3 predecessor.

\medskip

\section{The master classification ledger}
\label{sec:V-master-ledger}

The classification is stated cleanly as a single ledger
(Table~\ref{tab:V-classification}). Each historical framework is
located by the subset of \GGConstitution{} axioms it implicitly
uses; the content it produces and the content it leaves silent
follow constitutionally. The full derivation chain that backs
each row is in \AFive{} \citep[Part~VIII,
Table~5]{ArisakaA5GDOS}; we restate the ledger here in a compact
form for the reader of the constitution itself.

{\footnotesize
\renewcommand{\arraystretch}{1.18}
%% v7.5 refinement: widened columns to fill page within margins
%% (was 3.0/1.6/4.4/4.8 cm = 13.8 cm; new 3.0/2.9/4.5/5.2 cm = 15.6 cm).
%% Stratum column widened from 1.6 cm to 2.9 cm to fit the full
%% "($\AX{1},\AX{2},\AX{3},\AX{4}$) implicit" label without overlap.
\begin{longtable}{@{}>{\raggedright\arraybackslash}p{3.0cm}>{\raggedright\arraybackslash}p{2.9cm}>{\raggedright\arraybackslash}p{4.5cm}>{\raggedright\arraybackslash}p{5.2cm}@{}}
\caption[The constitutional-axiom classification of twentieth-century physics.]{The constitutional-axiom classification of major
twentieth-century theoretical-physics frameworks. Each framework
is located by the subset of \GGConstitution{} axioms it implicitly
uses; the content it produces and the content it leaves silent
follow from the missing axioms. ``Stratum~$N$'' is the row of the
tower at which axiom $\AX{N}$ becomes operative.}
\label{tab:V-classification}\\
\toprule
\textbf{Framework (year)} & \textbf{Stratum} & \textbf{Content produced} & \textbf{Content left silent (missing axioms)} \\
\midrule
\endfirsthead

\multicolumn{4}{c}{\itshape Table~\ref{tab:V-classification} continued from previous page}\\
\toprule
\textbf{Framework (year)} & \textbf{Stratum} & \textbf{Content produced} & \textbf{Content left silent} \\
\midrule
\endhead

Special relativity (1905) & 1 ($\AX{1}$) & Lorentz transformations, $E = mc^2$, classical field theory on Minkowski. & Gravity (needs $\AX{2}$); quantum content (needs $\AX{3}$); parameter selection (needs $\AX{4}$). \\

Yang--Mills (1954) & 2 ($\AX{1},\AX{2}$) & Non-Abelian gauge theory; Lie-algebraic field-strength commutators. & Specific gauge group (needs $\AX{4}$); QM content (needs $\AX{3}$). \\

General relativity (1915) & 2 ($\AX{1},\AX{2}$) & Einstein field equations; black holes; gravitational waves; cosmological expansion. & QM content (needs $\AX{3}$); $G_N$ from integer ledger (needs $\AX{4}$). \\

Heterotic superstring (1984+) & 2 ($\AX{1},\AX{2}$) & 10D anomaly cancellation; GS-modified Bianchi identity; SO(32) / $E_8\times E_8$ gauge groups; dualities. & QM content (no $\AX{3}$); unique vacuum selection (no $\AX{4}$, hence the $\sim 10^{500}$ flux landscape). \\

AdS/CFT (1997+) & 2 ($\AX{1},\AX{2}$) & Bulk gravity $=$ boundary CFT duality; holographic computation; Ryu--Takayanagi; viscosity bound. & QM content as theorems (no $\AX{3}$); information-coarsening projection (no $\AX{3}$); unique vacuum (no $\AX{4}$). \\

Standard QM (1925--32) & 3 ($\AX{1}$, $\AX{2}$, $\AX{3}$) implicit & Hilbert space, Schr\"odinger eqn, Born rule, L\"uders projection, decoherence, Bell--CHSH violation (all as \emph{postulates}). & QM postulates as \emph{theorems} (require explicit derivation from $\AX{3}$, supplied in \AFive{} by $\TX{7}{.1}$--$\TX{7}{.7}$). \\

Standard Model (1967--78) & 4 ($\AX{1}$, $\AX{2}$, $\AX{3}$, $\AX{4}$) implicit & $\SU(3)\times\SU(2)\times\U(1)$ gauge group; three fermion generations; CKM/PMNS mixing; Higgs mechanism. & Constitutional warrant for the specific group and the 28 parameters (supplied in \ANine{}). \\

$\Lambda$CDM cosmology (1998+) & 4 ($\AX{1}$, $\AX{2}$, $\AX{3}$, $\AX{4}$) implicit & Cosmological inventory; CMB power spectrum; structure formation; accelerated expansion. & Constitutional warrant for $\Omega_m = 1/\pi$, $\Omega_\Lambda = 1 - 1/\pi$ (supplied in \ATen{}). \\

\midrule

\textbf{\GGTheory{} cohort (2026)} & \textbf{4 explicit} & \textbf{All of the above as theorems, plus the universal-identity insight, the GRIP cascade, and the cosmos--life--mind unification of \AEleven{}.} & \textbf{None at the constitutional level; specific empirical predictions await the next decade of measurements.} \\

\bottomrule
\end{longtable}
}

The diagonal structure of the table is immediate: reading
top-to-bottom, each row activates more axioms than the previous,
and each row produces strictly more content. The historical
sequence in which the frameworks were developed is approximately
the same as the constitutional sequence in which they activate
axioms --- a coincidence that reflects the fact that the easier
physics (with fewer axioms) was discovered first. The two famous
limitations of string theory --- ``no QM derivation'' and ``vacuum
landscape'' --- sit at the Stratum-2 row of the table, exactly
where the absence of $\AX{3}$ and $\AX{4}$ predicts them; both
vanish at the Stratum-4 row of the cohort framework where all four
axioms are explicitly invoked.

\medskip

\section{Quantitative consequences: the vacuum count}
\label{sec:V-quantitative-consequences}

The most striking quantitative consequence of $\AX{4}~\MSMF{}$ is
the collapse of the vacuum landscape. Stratum~1 alone admits an
infinite family of Minkowski-metric choices. Stratum~2 (general
$\mathsf{GGEP\text{-}closed}$) admits an infinite family of bulk
solutions; restricted to the 10D heterotic specialization it
admits $\sim 10^{500}$ flux vacua (the Bousso--Polchinski /
Douglas--Kachru landscape count). Stratum~3
($\mathsf{OGGEP\text{-}open}$) inherits the infinity; the
operational dictionary adds boundary content but does not select
among bulk vacua. Stratum~4
($\mathsf{Full\text{-}Constitution}$) collapses the landscape to
exactly $\boxed{1}$: the unique admissible bulk that the integer
\OGNfive{} ledger forces. This is the constitution's signature
quantitative payoff: a $10^{500}$-fold collapse in admissible
vacua under the addition of one constitutional axiom.

\medskip

\section{What the classification means for the reader}
\label{sec:V-synthesis}

\paragraph{For the reader who came from string theory.} The
constitutional classification credits the string community's
forty-year mathematical achievement in full --- the Green--Schwarz
mechanism, the Calabi--Yau compactification framework, T-duality,
S-duality, mirror symmetry, M-theory in 11D, D-branes, the AdS/CFT
correspondence, the Ryu--Takayanagi formula, the viscosity bound,
the modular bootstrap, the swampland conjectures --- and
relocates this body of correct mathematics inside the
$\mathsf{GGEP\text{-}closed}$ stratum of the constitutional
tower. The string-theoretic mathematics is correct, and the
cohort framework inherits it unchanged. What the cohort adds is
the constitutional reading of \emph{where} this mathematics sits
and \emph{which} axioms its silent gaps require. The reader from
string theory should hear the classification as a relocation, not
as a refutation.

\paragraph{For the reader who came from QM foundations.} For
ninety years the measurement problem has been presented as the
most awkward gap in physics, with every textbook closing its
chapter on quantum mechanics with a hedged sentence about the role
of the observer. The constitutional classification supplies a
sharp diagnosis: the measurement problem is the structural symptom
of working in $\mathsf{GGEP\text{-}closed}$ without $\AX{3}$. Add
$\AX{3}$ to the axiom set, and the QM postulates become theorems
($\TX{7}{.1}$--$\TX{7}{.7}$ of \AFive{}), the measurement problem
dissolves, and the textbook hedged sentence can be replaced with a
clean derivation.

\paragraph{For the reader who came from the Standard Model and cosmology.}
The 28 free continuous parameters of the SM and the $\Lambda$CDM
partition $\Omega_m \approx 0.318$ have, for fifty years, been
understood as empirical inputs that the framework accommodates but
does not predict. The constitutional classification supplies the
missing axiom: $\AX{4}~\MSMF{}$ together with $\POS{6}~\OGN{}$
collapses the parameter count to zero free continuous parameters
and forces $\Omega_m = 1/\pi$ exactly, with deviations from these
predictions serving as falsifiers of the upstream axiom.

\paragraph{The structural payoff.} Each twentieth-century
framework correctly used the axiom subset it implicitly invoked;
each left silent exactly the questions that required the missing
axioms; once the right axiom is named, the historical gaps
dissolve into chains of theorems. After all, the deepest move a
foundational theory can make is to take everything its
predecessors got right, locate it inside a larger constitutional
picture, and use the same picture to fill the gaps the
predecessors could not address. The constitutional classification
is precisely that move, applied to the entire twentieth-century
edifice of theoretical physics.

\medskip

\section{Why superstring theory cannot ascend beyond Stratum~2 --- the four-fold constitutional exclusion}
\label{sec:V-stringtheory-exclusion}

The Master Classification Ledger of
\S\ref{sec:V-master-ledger} locates superstring theory in 10D and
M-theory in 11D at Stratum~2 of the four-axiom tower. This
closing Section makes explicit \emph{why} the superstring
framework, despite its forty-year mathematical development and
despite the spectacular construction of the AdS/CFT correspondence,
cannot ascend within its own terms to Stratum~3 (which adds
$\AX{3}~\OGGEP{}$) or Stratum~4 (which adds $\AX{4}~\MSMF{}$).
The four constitutional obstructions appear in Part~V's
classification language as follows.

\paragraph{Obstruction 1 --- SUSY is the missing-$\AX{4}$ inflation.}
Supersymmetry adds ${\sim}\,100$ new continuous parameters
(sparticle masses, mixing angles, soft-breaking terms) to a
framework whose primary purpose was to constrain a Standard Model
with 28 free continuous parameters. Fifty years of accelerator
search (LEP, Tevatron, LHC) and forty years of direct and indirect
WIMP search have produced no evidence for superpartners at any
mass scale up to $\sim 2$~TeV. In Part~V's classification, this
is the diagnostic signature of the missing $\AX{4}~\MSMF{}$:
the framework permits continuous parameter inflation precisely
because the axiom that forbids it is not in the framework's
axiom set. Adding $\AX{4}~\MSMF{}$ collapses the SUSY parameter
space to zero.

\paragraph{Obstruction 2 --- 10D / 11D is the missing-$\AX{4}$ dimension.}
The 10D dimension for the heterotic superstring and the 11D
dimension for M-theory are required by anomaly-free supersymmetric
consistency. The Calabi--Yau compactification program of the
last forty years has been the technical attempt to stabilize the
$\sim 10^{500}$ continuous moduli that the higher dimensions
introduce. In Part~V's classification, every dimension above
\emph{six} is missing-$\AX{4}$ freedom: the cohort minimum,
derived in $\ATwo{}$ \citep{ArisakaA2GG6} and proved consistent
in $\AThree{}$ \citep{ArisakaA3BI6}, is the 6D bulk
$\cM_4 \times I_r \times S^1_\theta$. The reduction from 10D/11D
to 6D is the constitutional remedy that the missing $\AX{4}~\MSMF{}$
would otherwise leave unaddressed.

\paragraph{Obstruction 3 --- $\mathrm{SO}(32)$ and $E_8 \times E_8$ are the missing-$\AX{4}$ gauge content.}
The 10D superstring's allowed gauge groups are 496-dimensional,
with vast Yukawa-coupling matrices and free flavor parameters.
In Part~V's classification, the gauge-content excess is
missing-$\AX{4}$: the cohort minimum gauge content, derived in
$\ATwo{}$ and proved consistent in $\AThree{}$ via the
trace-ratio lock $|a/c| = |a/d| = 5/2$, is
$\mathrm{Spin}(1,5) \times \mathrm{SU}(5) \times \mathrm{U}(1)_B \times \mathrm{U}(1)_{Qi}$.
Every gauge factor beyond this minimum is forbidden by
$\AX{4}~\MSMF{}$. The reduction from $\mathrm{SO}(32)$ or
$E_8 \times E_8$ to the cohort minimum is the constitutional
remedy.

\paragraph{Obstruction 4 --- operational projection is the missing-$\AX{3}$ structural content.}
Superstring theory inherits the von Neumann cut from textbook
quantum mechanics and is structurally incapable of deriving the
Born rule, the L\"uders projection, decoherence, or einselection
as theorems. In Part~V's classification, this is exactly the
missing-$\AX{3}$ diagnostic of \S\ref{sec:V-walking-the-tower}:
within $\mathsf{GGEP\text{-}closed}$ (the axiom subset string
theory implicitly uses), the bulk-to-boundary map is a unitary
equivalence rather than a CPTP projection, and unitary maps do
not coarse-grain. Adding $\AX{3}~\OGGEP{}$ supplies the
operational projection and turns the QM postulates into theorems
(Part~II's Remark~1, \S\ref{sec:two_remarks}).

\paragraph{The constitutional reading.}
Each of the four obstructions is, in Part~V's diagnostic
language, a \emph{missing-axiom symptom}: superstring theory sits
at Stratum~2 because $\AX{3}~\OGGEP{}$ and $\AX{4}~\MSMF{}$ are
not in its constitutional kit, and the four obstructions above
are the precise structural consequences. Adding both axioms ---
without changing any of the Stratum-2 mathematical content the
cohort inherits --- promotes the framework to Stratum~4 in the
more economical 6D realization that the cohort framework adopts.
This is the most precise statement of what string
theory does \emph{not} deliver that the cohort framework does.
We offer it here not as a refutation but as the structural
location of the inheritance boundary between the string-theoretic
mathematics --- which the cohort credits and uses --- and the
constitutional content, which only the four axioms together
supply. After all, a constitution is most useful when it can
name not only what it adds but exactly which clauses its
predecessors are missing.

\subsection*{Closing remarks on Part~V}

The tower is on the page. Every major framework of twentieth-century
physics has been located by the axioms it uses, the famous limitation of
each has been traced to the axiom it lacks, and the count of admissible
vacua has fallen from something like $10^{500}$ at the second layer to
one at the fourth.

We want to be careful about what has and has not been shown, because
this is the Part most open to being oversold. Nothing here refutes any
framework. General relativity is not wrong for standing on two axioms;
it is one of the most successful theories ever written, and it stands
exactly where its own assumptions put it. The claim is narrower and, we
think, more interesting: each framework's characteristic silence is
predictable from its position, and the prediction can be checked against
a century of published work.

That is the part we would ask a skeptical reader to weigh. It would have
been easy to build a classification that merely re-describes what
everyone already knows. The test of whether this one is doing work is
whether it says anything the practitioners did not already say about
themselves --- and the honest answer is that in most cases it does not
tell them something new, it tells them why the thing they already knew
had to be so. A framework without an operational axiom does not fail to
address measurement through oversight or lack of effort. It cannot
address measurement, because it has no object in its formalism that
corresponds to a finite observer reading a finite record. The silence is
structural, and structural silences do not yield to cleverness.

There is an older idea underneath this, and it is worth naming. We are
used to thinking that a theory is characterized by its predictions. The
tower suggests that a theory is at least as well characterized by its
undefined terms --- by the questions that cannot be posed in its
language at all. Ask general relativity when the wavefunction collapses
and the question does not come out false; it comes out meaningless,
because neither noun exists in the theory. What a framework cannot say
is not a gap in its knowledge. It is a fact about its shape.

If that is right, then progress in physics looks a little different from
the usual story. It is less a matter of finding the right answers and
more a matter of enlarging the set of questions that can be asked at
all --- and each enlargement costs an axiom, which is to say a freedom
given up.

A classification that relocates other people's work owes them a hearing.
Part~VI puts the constitution beside the frameworks it has just sorted
and asks what it actually adds.

\GGPart[Discussion]{Discussion}{Significance, Uniqueness, and the Ten
 Structural Innovations,\\
 with the Problems the Cohort Has Closed}

\noindent
Every previous Part has been construction. This one is
cross-examination.

Three questions are owed to the reader by now, and none of them is
rhetorical. First: what does the constitution actually add that the
Standard Model, string theory, loop quantum gravity, causal set theory,
the Wightman and Haag--Kastler axiomatizations, the
Lindblad--Gorini--Kossakowski--Sudarshan open-systems formalism, and the
decoherence program do not already have? Second: is it new, or is it a
relabeling of things physics already knew, written in a more insistent
font? Third: what does it buy --- concretely, in problems closed rather
than in vocabulary introduced?

We answer the first by comparison, framework by framework, and we try
to be fair: in most cases the honest answer is that the constitution
adds a stated rule where a working practice already existed, and the
value is in the stating, not in the practice. We answer the second by
setting the constitution beside the historical axiom systems it descends
from --- Euclid, Newton, Hilbert, Kolmogorov, von Neumann --- and asking
what a constitution is for in each case. We answer the third with the
ten structural innovations we claim, and then with a catalog of twelve
foundations-of-physics problems the cohort asserts it has closed, each
carrying the companion paper responsible and the experiment still able
to reopen it.

Six communities will read this program adversarially, and they should:
the particle theorist who doubts a constitution can fix continuous
parameters; the gravity researcher who doubts a clean four-dimensional
descent from a higher-dimensional bulk; the quantum-foundations
specialist who doubts that any axiomatic move touches the measurement
problem; the mathematical physicist who asks what this adds to fifty
years of axiomatic quantum field theory; the philosopher who observes
that calling a document a constitution does not make it one; and the
string theorist who asks what the cohort adds beyond mathematics the
string community developed decades ago. We name each objection here.
Part~IX answers all fifty-four of them, by name.

\medskip

\section{Discussion}
\label{sec:discussion}

This paper is intentionally axiomatic rather than model-specific: it specifies a rulebook for admissibility before it specifies a particular Lagrangian or particle spectrum.
A constitution can---and must---be compared to existing frameworks by asking four concrete questions:
\emph{(i)} which ambiguities are removed (especially around observation and openness),
\emph{(ii)} which freedoms are forbidden (especially hidden continuous knobs),
\emph{(iii)} which consistency conditions are elevated from ``technicalities'' to first-class filters, and
\emph{(iv)} which new falsifiers are demanded at the protocol level.
We organize this discussion accordingly.

\subsection{Significance and uniqueness of the \GGConstitution{}}
\label{sec:significance}

\paragraph{(1) A constitution as a scientific object.}
In much of modern practice, ``foundations'' are discussed in prose while calculations are performed in mathematics.
\GGTheory{}\ treats this split as itself a failure mode: a scientific framework is incomplete if its admissibility rules cannot be stated as objects that can be checked.
Formally, the constitution is the predicate \eqref{eq:admissible}: a candidate theory is either admitted or rejected by a finite checklist.
This posture is deliberately Euclidean/Newtonian in spirit \citep{EuclidElements,Newton1687}: the aim is that once the clauses are accepted, large theorem spaces become unavoidable rather than optional.

\paragraph{(2) A reversal of emphasis: geometrize quantum/gauge structure.}
Most ``quantum gravity'' programs begin from classical geometry and then attempt to quantize it; many unification programs begin from gauge structure and then accommodate gravity.
AX-2 (\GGEP) imposes a different primitive: the primary field is a composite geometry$\oplus$gauge connection.
Connections and holonomies already unify ``phase transport'' (quantum) with ``parallel transport'' (geometry), so the constitution demands that this unity be treated as fundamental rather than emergent.
This stance is compatible with the historical centrality of gauge connections \citep{YangMills1954,Nakahara2003} and with holographic hints \citep{tHooft1993,Susskind1995,Maldacena1998,Witten1998},
but it is strengthened here into an admissibility rule: theories that require independent non-geometric primitives for forces are axiomatically suspect.

\paragraph{(3) Measurement is defined (operationally), not interpreted.}
A century after the birth of quantum mechanics, ``measurement'' is still often treated as extra structure layered on top of unitary dynamics \citep{vonNeumann1932,Bell1964,Zurek2003}.
\OGGEP\ (AX-3) enforces a definition: a measurement is an output of a lawful bulk$\to$boundary registration map, i.e.\ a specific instrument/estimator/protocol triple.
This collapses much interpretational ambiguity into a technical (and auditable) question: \emph{which CPTP instrument represents the experiment?}
POS-\MF\ then closes the loop by requiring a pre-declared falsifier.
In effect, the constitution converts foundational debates into workflow: write the dictionary, state the protocol, declare the kill criterion, and only then claim a ``prediction.''

\paragraph{(4) Open systems are constitutional, not effective.}
Most sciences beyond fundamental particle theory live in open regimes: systems exchange energy, matter, and information with environments.
Even in fundamental physics, laboratory measurements create irreversible records and therefore implicitly invoke open-system dynamics.
POS-\CP\ and POS-\MF\ formalize this fact: admissible evolution is not an optional ``environmental correction'' but must be positivity-preserving and protocol-defined from the outset.
The payoff is unification by method: the same admissibility grammar that keeps quantum channels physical is the grammar that keeps chemical concentrations nonnegative and keeps biological models from smuggling in acausal inference.

\paragraph{(5) Predictivity by forbidding hidden knobs.}
The Standard Model is rigid once its structure is fixed, yet it does not predict the numerical values of its parameters \citep{PDG2024}.
Beyond-the-SM model building often introduces additional tunable sectors; quantum-gravity and unification programs frequently admit large families of consistent vacua or moduli \citep{Polchinski2004Landscape}.
AX-4 (\MSMF) and POS-\OGN\ respond by treating hidden continuous tunability as non-admissible unless forced by symmetry, consistency, or operational meaning.
Methodologically this is an ``anti-landscape'' stance: a correct theory should look like a compression, not like a catalog of adjustable stories.
Technically, \OGN{} insists that normalization/basis drift be fixed to a minimal discrete ledger so that would-be ``choices'' cannot masquerade as physical parameters.

\paragraph{(6) Unification of sciences by unification of admissibility.}
The ten-branch program is not a claim that all domains reduce to particle physics.
It is a claim that \emph{admissibility} should be shared: gauge redundancy, positivity of states, scale covariance, and falsifiability contracts are not optional in some disciplines and ignored in others.
This is why \GGTheory{}\ is partly a proposal about the method of science: it aims to standardize the audit trail
\begin{equation}
\text{\SIP{} specification} \ \Rightarrow\ \text{admissible dynamics} \ \Rightarrow\ \text{registered observables + falsifier}.
\end{equation}
If this pipeline can be made productive, ``unification'' becomes operational rather than metaphorical.

\paragraph{(7) Structural classification of twentieth-century physics.}
A constitution is most consequential when it does more than supply
a new theory --- when it locates the theories the broader community
already knows by name. Part~V of this paper develops the
constitutional-axiom classification: essentially every major
theoretical-physics framework of the twentieth century (general
relativity, gauge theory, string theory, AdS/CFT, quantum mechanics,
the Standard Model, $\Lambda$CDM) is the theorem class of a
particular subset of the four constitutional axioms applied to a
particular bulk solution. Each historical framework's famous
limitations follow directly from which axioms it lacks:
string theory's inability to derive the QM postulates is a
missing-$\AX{3}$ symptom; the $\sim 10^{500}$ flux landscape is a
missing-$\AX{4}$ symptom; the textbook measurement problem is what
$\mathsf{GGEP\text{-}closed}$ looks like without
$\AX{3}~\OGGEP{}$. This kind of structural classification is
qualitatively distinct from the standard ``compare-and-contrast''
posture of prior axiomatic attempts; it is what makes the
\GGConstitution{} a genuine constitution rather than a
comparative essay. After all, a foundational document earns its
name only when it can place its predecessors inside its own picture,
not merely against them.

\paragraph{(8) The two fibers: the constitutional minimum for a world that is and becomes.}
The cohort's recent campaign (\textsf{GG-W1-Symmetry};
\citealp{ArisakaA2GG6}; \citealp{ArisakaP5CPT}) has sharpened the
constitutional diagram of
Figure~\ref{fig:two-fibers-constitutional} into one exact sentence:
\emph{holonomy requires closed loops, and an interval has none.}
The compact circle $S^1_\theta$ is nothing but closed loops ---
everything on it is holonomy, quantized, gauge-protected, and its
transport composes to a group; the scale interval $I_r$ can hold no
holonomy at all --- its transport composes only forward, a
semigroup (the constitutional germ of \GDOS{}). Read back into the
axioms, this upgrades a description into a minimality claim:
\AX{2} does not merely install two extra directions --- it installs
\emph{exactly one fiber of each topological kind}, one that
remembers and one that forgets. The pair is the constitutional
minimum. A bulk with two circles would conserve everything and
decide nothing: no arrow, no registration, no record --- \AX{3}
would have nothing to project. A bulk with two intervals would
decide everything and keep nothing: no quantized charge, no
interference, no stable atom --- \AX{1} would have nothing lawful
to say. Being needs holonomies; becoming needs a semigroup; a
universe that both \emph{is} and \emph{becomes} needs exactly one
of each --- and that is what \AX{2} writes and \AX{3} reads.

The consequences of taking this pair seriously are now in print
across the cohort --- deterministic symmetry breaking
(\textsf{GG-W1-Symmetry}), the guarded proton
(\textsf{GG-P6-Proton}), the birth of matter and the anatomy of
time \citep{ArisakaP5CPT}, the measurement arrow
\citep{ArisakaA8QM}, and the origin as chronology
\citep{ArisakaA11Origin} --- and no item on the list required a new
clause: every one is Figure~\ref{fig:two-fibers-constitutional},
taken at its word. That is the strongest form of this subsection's
significance claim. The \GGConstitution{} is not merely an
axiomatization of the physics we have; it is the minimal charter
for a world in which anything can happen and anything can persist.

\subsection{Comparison with the Standard Model and leading unification/quantum-gravity programs}
\label{sec:comparison_modern}

A constitution should not be compared by rhetorical scope but by what it treats as primitive and what it forces to be derived.
The Standard Model is a remarkably successful effective field theory: it encodes gauge invariance, locality, and renormalizability in a compact structure and survives stringent precision tests.
Yet it inherits two axiomatic gaps relevant to this paper. First, measurement is largely external to the model --- experimental claims are made by coupling the SM to detector models and statistical procedures that are not part of the SM constitution.
Second, predictivity is limited by input parameters that are not derived from within the SM itself \citep{PDG2024}.

Quantum-gravity/unification programs (e.g.\ string theory, loop approaches, and related UV-completion programs) address gravity and unification more directly,
but often at the price of a large theory space (many vacua, moduli, compactifications) and, again, with measurement semantics inherited rather than axiomatically defined.
\GGTheory{}\ does not claim to replace these programs by a single model in this paper; it claims to supply an \emph{axiomatic filter} that clarifies what must be fixed for a claim to count as a prediction.

Three further programs deserve named placement so that the comparison is honest.
\emph{Asymptotic safety} \citep{Reuter1998} proposes that gravity
is UV-complete via a non-Gaussian fixed point of the
renormalization flow; in the constitutional reading of
$\POS{5}~\RG{}$, asymptotic safety is a Stratum-2 alternative
to the cohort's anomaly-locked 6D bulk and is not in conflict
with the present axioms at the kinematic level, though it does
not yet activate $\AX{3}~\OGGEP{}$ or $\AX{4}~\MSMF{}$.
\emph{Entropic} or \emph{emergent} gravity
\citep{Verlinde2011,Padmanabhan2010} takes the opposite stance
from $\AX{2}~\GGEP{}$: geometry is derived from thermodynamics
rather than taken as primitive. The cohort framework is
directly incompatible with the emergent-gravity stance and
names this disagreement as a structural fork, not as an
incidental detail.
\emph{Information-theoretic axiomatic QM}
\citep{BubPitowsky2010} axiomatizes the Born rule from
operational constraints; the cohort framework is in close
structural neighborhood with this program and credits it as
a parallel rather than a competitor.

Table~\ref{tab:comparison_modern} summarizes the contrast at the level of primitives, measurement semantics, normalization discipline, and falsifiability.

{\footnotesize
\begin{longtable}{@{}p{0.16\linewidth}p{0.26\linewidth}p{0.26\linewidth}p{0.26\linewidth}@{}}
\caption[Comparison with modern frameworks.]{\textbf{Comparison with modern frameworks.} Constitution-level contrast among the Standard Model, representative quantum-gravity/unification programs, and \GGTheory{}, emphasizing measurement semantics and normalization discipline. The aim is to isolate which choices are treated as primitives versus which are forced to be derived and falsifiable.}
\label{tab:comparison_modern}\\[0.35em]
\toprule
\textbf{Criterion} & \textbf{Standard Model (SM)} & \textbf{Quantum gravity/unification (string/QG)} & \textbf{\GGConstitution{} (this paper)}\\
\midrule
\endfirsthead
\toprule
\textbf{Criterion} & \textbf{SM} & \textbf{String/QG} & \textbf{\GGConstitution{}}\\
\midrule
\endhead
\midrule
\multicolumn{4}{r}{\small continued on next page}\\
\endfoot
\bottomrule
\endlastfoot

Primary object &
4D renormalizable QFT; fields on spacetime \citep{PeskinSchroeder1995,Weinberg1995QFT} &
UV completion with gravity; extra dimensions; extended objects \citep{Polchinski1995} &
Composite connection (geometry$\oplus$gauge) + operational map (bulk$\to$boundary). \\

Treatment of measurement &
External postulates / interpretation; effective open-system tools \citep{vonNeumann1932,Zurek2003} &
Typically inherited from QM; measurement not a structural primitive &
Measurement is constitutional: \OGGEP\ + MF contract required. \\

Gravity &
Not included; added via EFT on curved backgrounds &
Included fundamentally (graviton / strings) &
Included by \GGEP{} locking; geometrized gauge--gravity composite. \\

Free parameters / moduli &
Parameters fit to data \citep{PDG2024} &
Often many vacua/moduli; landscape issues \citep{Polchinski2004Landscape} &
Axiom/postulate filters (\MSMF, \OGN, \AF) aim to remove hidden dials. \\

Scale / RG &
EFT and RG central \citep{Wilson1971} &
RG and holography appear in AdS/CFT &
Scale is geometric coordinate; operational projection must commute with RG (POS-\RG). \\

Anomalies &
Require cancellation; constrain field content &
Canceled via GS and consistency \citep{GreenSchwarz1984} &
Anomaly freedom by inflow is a postulate (POS-\AF) and a design filter. \\

Primary falsifiers &
Precision SM tests; collider + flavor + EDMs &
Indirect tests; difficult due to scale/landscape &
MF requires explicit kill criteria for each theorem; emphasizes protocol-level falsifiability. \\

\end{longtable}
}

Two clarifying remarks are important. First, \GGTheory{}\ is compatible with the effective-field-theory practice of the SM and with holographic ideas in quantum gravity, but it insists that these ideas be made constitutional:
RG compatibility is not just a calculation tool (POS-\RG), and measurement is not just an interpretational story (AX-3 and POS-\MF).
Second, the \MSMF{}/\OGN{} clauses are not intended to deny the empirical fact that parameters exist; they are intended to force an accounting:
\emph{which numbers are derived, which are conventions, which are inputs, and which are falsifiable outputs?}
A theory that cannot answer that accounting question is constitutionally underdetermined.

\paragraph{Where each program sits in the constitutional tower.}
The most economical way to read Table~\ref{tab:comparison_modern}
is through the constitutional-axiom classification of Part~V. The
Standard Model row corresponds to Stratum~4 with $\AX{4}$
\emph{implicit} (the gauge group, the parameter values, and the
generation count are all empirical inputs that the SM
accommodates rather than predicts). The string-theory /
quantum-gravity row corresponds to Stratum~2
$\{\AX{1},\AX{2}\}$ (the geometry-gauge unification of $\AX{2}$
is built in; the operational/open extension of $\AX{3}$ and the
uniqueness principle of $\AX{4}$ are not); the $\sim 10^{500}$
flux landscape is the missing-$\AX{4}$ signature, and the absence
of a derivation of the QM postulates is the missing-$\AX{3}$
signature. The \GGConstitution{} row corresponds to Stratum~4
with all four axioms \emph{explicit}, which is what allows the
cohort framework to derive the SM gauge group, the 28 parameters,
and the cosmological partitions as theorems rather than as fits.
Reading the table through this classification turns the
comparison from a list of differences into a structural location.

\subsection{Comparison with historical axiom systems}
\label{sec:comparison_history}

Historical axiom systems were successful not because they were ``elegant'' but because they clarified what was to be taken as primitive and what was to be derived.
Euclid's postulates turned geometric truths into inevitabilities; Newton's laws turned celestial and terrestrial motion into one calculable subject; Einstein's principles turned time and simultaneity into operational definitions \citep{Einstein1905,Einstein1915}.
Quantum theory achieved enormous compression, but it famously left a seam: unitary dynamics is clean while the operational meaning of measurement is partly external \citep{vonNeumann1932}.

\GGTheory{}\ aims to be in the same family of moves while adding clauses that earlier constitutions did not need:
open-system admissibility (positivity-preserving semigroups), explicit operational dictionaries (bulk$\to$boundary compilation), and explicit falsifiers.
Table~\ref{tab:comparison_axioms} summarizes this comparison.

{\footnotesize
\begin{longtable}{@{}p{0.18\linewidth}p{0.24\linewidth}p{0.24\linewidth}p{0.28\linewidth}@{}}
\caption[Comparison with classic axiom systems.]{\textbf{Comparison with classic axiom systems.} Contrast between \GGTheory{}\ and landmark constitutions (Euclid, Newton, Relativity, Quantum Mechanics, Popper), highlighting the explicit operational dictionary and open-system admissibility. The point is methodological: \GGTheory{}\ treats measurement semantics and open dynamics as constitutional clauses in the same spirit that earlier constitutions treated geometry, force, or spacetime.}
\label{tab:comparison_axioms}\\[0.35em]
\toprule
\textbf{Axiom system} & \textbf{Domain} & \textbf{Constitutional move} & \textbf{Relation to \GGTheory{}}\\
\midrule
\endfirsthead
\toprule
\textbf{Axiom system} & \textbf{Domain} & \textbf{Constitutional move} & \textbf{Relation to \GGTheory{}}\\
\midrule
\endhead
\midrule
\multicolumn{4}{r}{\small continued on next page}\\
\endfoot
\bottomrule
\endlastfoot

\citet{EuclidElements} &
Geometry &
Axiomatize geometric truth from minimal postulates &
\GGTheory{}\ borrows the ``constitution first'' ethos: admissibility is decided before calculation. \\

\citet{Newton1687} &
Mechanics / gravity &
Universal laws of motion + gravitation; predictivity &
\GGTheory{}\ seeks similar universality, but with open-system and operational measurement built in. \\

\citet{Einstein1905}, \citet{Einstein1915} &
Relativity &
Invariance principles reshape spacetime and dynamics &
\GGTheory{}\ extends invariance to the generator of open dynamics and to the measurement map. \\

Quantum mechanics (canonical) \citep{Born1926,vonNeumann1932} &
Microscopic physics &
Unitary dynamics + probabilistic measurement rule &
\GGTheory{}\ keeps quantum structure but replaces ambiguous measurement with \OGGEP\ and MF contracts. \\

Falsifiability criterion \citep{Popper1959} &
Philosophy of science &
Demarcation by refutability &
\GGTheory{}\ formalizes this as POS-\MF: every theorem is a contract with a kill criterion. \\

\end{longtable}
}

This table also clarifies why POS-\MF\ is placed \emph{inside} the constitution rather than treated as external philosophy.
Popper's falsifiability criterion \citep{Popper1959} is often cited as a demarcation slogan; \GGTheory{}\ treats it as an engineering requirement:
a theorem that cannot be paired with a protocol and a kill criterion is simply incomplete.

\paragraph{Part~V as the contemporary extension of the historical pattern.}
There is a thread that runs through every entry of
Table~\ref{tab:comparison_axioms}: each historical constitution
located its predecessors inside its own framework. Newton's
mechanics absorbed the Galilean kinematics that preceded it;
Einstein's relativity absorbed Newton's mechanics as a low-velocity
limit; quantum mechanics absorbed classical mechanics as a
macroscopic limit. Part~V of the present paper performs the
analogous move at the constitutional level: it absorbs general
relativity, gauge theory, string theory, AdS/CFT, quantum mechanics,
the Standard Model, and $\Lambda$CDM into a single four-axiom tower
and shows that each historical framework is the theorem class of a
specific axiom subset. This is the contemporary version of the
axiomatic move every entry in the table above represents, applied
to the twentieth century rather than to a single mathematical or
physical sub-domain.

\subsection{A direction for 21st-century science}
\label{sec:direction}

If the constitution succeeds, it suggests a concrete change in scientific workflow, especially in data-rich disciplines where model flexibility is high.

\paragraph{(i) From model catalogs to admissibility proofs.}
The postulate stack turns many ``soft'' questions into hard ones.
Before proposing a new sector or mechanism, one must show that it passes the runtime checklist:
GI (symmetry grammar), \UEP{} (variational engine), AF (quantum consistency), CP (physical state evolution), RG (scale coherence), \OGN{} (normalization rigidity), and MF (protocol-level falsifiability).
This makes ``consistency first'' practical rather than ceremonial.

\paragraph{(ii) From closed ideals to open operational laws.}
Because registered data are produced by open interactions, admissible open dynamics is not optional.
This shifts emphasis from unitary groups to CPTP semigroups, from idealized isolation to explicit ports, and from abstract observables to estimators with error models.
In turn, foundational debates about measurement become model-building constraints: the dictionary must be written down.

\paragraph{(iii) From parameter fitting to parameter elimination (when possible).}
\MSMF{} and \OGN{} do not forbid fitting where fitting is unavoidable; they forbid \emph{unjustified} freedom.
Their purpose is to push explanations toward symmetry and discrete data.
When a continuous knob is truly forced (e.g.\ a coupling constant that is not fixed by topology or anomaly constraints), the constitution demands that this be stated as such and that the knob be promoted to a falsifiable prediction target through RG flow, scaling relations, or additional constraints.

\paragraph{(iv) From domain narratives to a shared audit trail.}
A key claim of \GGTheory{}\ is that every domain can be asked the same three questions:
What is the \SIP{}? What is the admissible kernel? What are the registered observables and kill criteria?
This is how unification is intended to avoid becoming metaphor.

\paragraph{(vi) From comparison to classification.}
Twentieth-century practice often reduced ``how does theory X relate
to theory Y'' to a list of similarities and differences --- a
comparison rather than a location. The constitutional-axiom
classification of Part~V replaces that comparison with a
\emph{location}: each historical framework is located at the
stratum of the four-axiom tower its implicit axiom subset reaches,
and its silent regions are diagnosed as missing-axiom symptoms.
The diagnostic is sharp: ninety years of measurement-problem
debate reduces to the observation that $\mathsf{GGEP\text{-}closed}$
is missing $\AX{3}$; forty years of vacuum-landscape debate
reduces to the observation that $\mathsf{GGEP\text{-}closed}$ is
missing $\AX{4}$. Add the missing axioms and the historical gaps
dissolve into chains of theorems. Adopting this constitutional
practice across the sciences is what would let twenty-first-century
scientific workflow be \emph{classificatory} rather than merely
\emph{comparative}.

\paragraph{(v) Ultimate aim.}
The ultimate aim is not a ``bigger theory'' but a disciplined standard for what we mean by explanation across sciences:
a smallest admissible description that reproduces the largest set of registered invariants, under an explicit prohibition against acausal kernels,
unphysical state evolution, scale artifacts, and tuning that cannot be justified by symmetry or operational meaning.
The companion papers (\citealp{ArisakaA2GG6}; \citealp{ArisakaA5GDOS}; \citealp{ArisakaBookI}) serve as stress tests: they must either produce sharp, killable numerical targets or reveal exactly which clause of the constitution must be revised.

\medskip
\noindent\emph{Are four axioms and seven postulates the right count?}
A natural methodological question is whether the eleven clauses are minimal. Could the same axiomatic content be carried by fewer? Or are there commitments folded silently into the existing clauses that ought to be promoted --- a \emph{compositionality} postulate (behavior under sub-systems), an \emph{informational} postulate (the admissibility status of Shannon information), or a \emph{computational-complexity} postulate (the constitutional role of $\mathrm{P}$, $\mathrm{NP}$, $\mathrm{BQP}$)? Each is currently absorbed into $\AX{3}~\OGGEP{}$ together with $\POS{4}~\CP{}$ or $\POS{7}~\MF{}$, and may deserve promotion in a future revision. Whether eleven is the right count cannot be answered from inside the present paper; it can only be answered by a decade of working use. After all, the constitutional-axiom classification of Part~V is itself the strongest evidence that four axioms are at least the right order of magnitude: each major theoretical-physics framework of the twentieth century turns out to be the theorem class of an axiom subset, and each framework's famous limitations are the missing-axiom diagnostics that the classification reads off by inspection. The four-axiom constitution passes that absorption test cleanly.


\section{The Ten Structural Innovations of the \GGConstitution{}}
\label{sec:ten-innovations}

The contribution of \AOne{} is foundational rather than calculational: it
does not derive a single number but fixes the admissibility grammar from
which the rest of the cohort derives its numbers. We collect here the ten
structural innovations that grammar delivers, each carrying its
cohort-global identifier (\S\ref{sec:naming-convention}).

\paragraph{Innovation 1 (I-Constitution-1): A constitution as a checkable
scientific object.} Admissibility is stated as a finite predicate
\eqref{eq:admissible} --- a candidate theory is either admitted or rejected
by a checklist --- rather than discussed in prose. The posture is
Euclidean: once the clauses are accepted, large theorem spaces become
unavoidable rather than optional.

\paragraph{Innovation 2 (I-Constitution-2): Geometrize gauge/quantum
structure instead of quantizing geometry.} AX-2 (\GGEP{}) makes the
primary field a composite geometry$\oplus$gauge connection, so that phase
transport (quantum) and parallel transport (geometry) are one object by
axiom. Theories that require independent non-geometric primitives for
forces are thereby axiomatically suspect.

\paragraph{Innovation 3 (I-Constitution-3): Measurement is \emph{defined},
not interpreted.} AX-3 (\OGGEP{}) makes a measurement the output of a
lawful bulk$\to$boundary registration map --- an instrument/estimator/
protocol triple --- collapsing a century of interpretational ambiguity
into the auditable question ``which CPTP instrument represents the
experiment?'', closed by POS-7 (\MF{}) with a pre-declared falsifier.

\paragraph{Innovation 4 (I-Constitution-4): Open systems are
constitutional, not effective.} POS-4 (\CP{}) and POS-7 (\MF{}) require
admissible evolution to be positivity-preserving and protocol-defined from
the outset --- not an optional ``environmental correction'' --- so the same
admissibility grammar that keeps quantum channels physical keeps chemical
concentrations nonnegative and biological inference causal.

\paragraph{Innovation 5 (I-Constitution-5): Predictivity by forbidding
hidden knobs.} AX-4 (\MSMF{}) and POS-6 (\OGN{}) treat continuous
tunability as non-admissible unless forced by symmetry, consistency, or
operational meaning, and fix normalization drift to a minimal discrete
integer ledger (\OGNfive{}). The stance is explicitly anti-landscape: a
correct theory should look like a compression, not a catalog.

\paragraph{Innovation 6 (I-Constitution-6): Unification of the sciences by
unification of \emph{admissibility}.} The claim is not that all domains
reduce to particle physics but that admissibility --- gauge redundancy,
positivity, scale covariance, falsifiability --- is shared, standardizing
the audit trail $\SIP{}\text{ specification}\Rightarrow\text{admissible
dynamics}\Rightarrow\text{registered observables}+\text{falsifier}$.

\paragraph{Innovation 7 (I-Constitution-7): A structural classification of
twentieth-century physics.} Part~V places each major framework (general
relativity, gauge theory, string theory, AdS/CFT, quantum mechanics, the
Standard Model, $\Lambda$CDM) as the theorem class of a subset of the four
axioms; each framework's famous limitation is a missing-axiom symptom (the
flux landscape a missing-AX-4 symptom, the measurement problem a
missing-AX-3 symptom). A constitution earns its name by placing its
predecessors inside its own picture, not merely against them.

\paragraph{Innovation 8 (I-Constitution-8): Derived, not postulated, extra
dimensions.} AX-1 (\LC{}) and AX-2 (\GGEP{}) together \emph{force} the
six-dimensional bulk $\cM_4\times I_r\times S^1_\theta$ --- a scale fiber
and a phase fiber as the geometrization of two necessities a local, causal,
charged quantum theory cannot avoid (Theorem~0 of \citet{ArisakaA2GG6}) ---
rather than positing extra dimensions as spatial inputs.

\paragraph{Innovation 9 (I-Constitution-9): Falsifiability built into the
charter.} POS-7 (\MF{}) makes every claim a four-tuple contract with a
pre-declared kill threshold; the consolidated falsifier table
(\S\ref{sec:pull_table}, F-Constitution-1\,\ldots\,F-Constitution-10) is the
constitution's standing exposure surface to data.

\paragraph{Innovation 10 (I-Constitution-10): A single four-axiom /
seven-postulate root for the whole cohort.} The eleven constitutional
clauses --- the four axioms (\LC{}, \GGEP{}, \OGGEP{}, \MSMF{}) and seven
postulates (\GI{}, \UEP{}, \AF{}, \CP{}, \RG{}, \OGN{}, \MF{}) --- are the
common ancestor from which every theorem of the twenty-one companion
papers descends, which is what makes the cohort one program rather than a
collection.

\subsection{Global naming convention for cross-cohort citation}
\label{sec:naming-convention}

Beginning with \citet{ArisakaA2GG6} and extended across the cohort, each
load-bearing item of the \GGTheory{} program carries a globally unique
identifier of the form \emph{Prefix-Suffix-N}, where the Prefix names the
item type and the Suffix names the paper:
\begin{itemize}[leftmargin=2em, itemsep=2pt]
\item \textbf{T-Suffix-N} for the $N$th Theorem. (The \GGConstitution{}
states its content as axioms and postulates rather than theorems, so \AOne{}
itself carries no T-Constitution entries; the downstream papers do.)
\item \textbf{L-Suffix-N} for the $N$th Lemma.
\item \textbf{C-Suffix-N} for the $N$th Corollary.
\item \textbf{D-Suffix-N} for the $N$th Definition.
\item \textbf{R-Suffix-N} for the $N$th Remark.
\item \textbf{P-Suffix-N} for the $N$th Prediction.
\item \textbf{O-Suffix-N} for the $N$th Objection (e.g., O-Constitution-1 is
the first Common Objection of the present paper).
\item \textbf{F-Suffix-N} for the $N$th Falsifier (e.g., F-Constitution-2 is
the KK lock-cluster $\Mlock$ contract of \S\ref{sec:pull_table}).
\item \textbf{I-Suffix-N} for the $N$th Innovation (e.g., I-Constitution-1 is
the constitution-as-checkable-object innovation above).
\item \textbf{PS-Suffix-N} for the $N$th Problem-Solved entry
(\S\ref{sec:problems_solved}).
\end{itemize}
The canonical Suffix list for the cohort is: \emph{Constitution} (A1, this
paper), \emph{GG6} (A2), \emph{BI6} (A3), \emph{Euclid} (A4), \emph{GDOS}
(A5), \emph{GRIP} (A6), \emph{Qi4} (A7), \emph{QM} (A8),
\emph{SM} (A9), \emph{Cosmos} (A10), \emph{Origin} (A11), together with the
phenomenology preprints \emph{KK} (P1), \emph{N3} (P2), \emph{Electron} (P3),
the foundations preprints \emph{Real} (P4) and \emph{CPT} (P5) --- the two
Foundations-Branch members, whose cohort identifiers carry the
P-series designator (T-P4-$N$, T-P5-$N$) --- and the consequence
preprints \emph{Kaxion} (C1), \emph{Hubble} (C2),
\emph{BH} (C3). The four axioms (AX-1\,\ldots\,AX-4) and seven postulates
(POS-1\,\ldots\,POS-7) defined in the present paper keep their existing
cohort-wide names, since those are already paper-specific and unique; the
global identifier is the same item viewed from outside its home paper, so a
cross-cohort citation such as ``\GG{} consumes AX-2 and POS-6, produces
T-GG6-3'' is unambiguous and search-friendly.

\section{Problems Solved}
\label{sec:problems_solved}

A foundational document earns its keep not by listing what
remains open but by listing what has been closed. The
\GGConstitution{} entered the literature with twelve standard
foundations-of-physics questions on its docket --- the integer
Diophantine uniqueness, the Hopkins-Singer anomaly proof, the
$28$ Standard Model parameters, the cosmological partitions, the
gauge-coupling unification, the fermion generation count, the
strong-CP problem, the cosmological-constant sign, the
inflationary mechanism, the Higgs-mass naturalness puzzle, the
neutrino-mass texture, and the cross-domain extension of $\GDOS{}$
to chemistry, biology, and neuroscience. After the descent chain
of the twenty-two-paper cohort has run, each of these twelve has a
named answer in a named companion paper. We label them
$\mathrm{P}_1$--$\mathrm{P}_{12}$ in deliberate parallel with the
historical $\mathrm{O}_1$--$\mathrm{O}_{12}$ labels of earlier
drafts, so that any cross-citation into this catalog remains
unambiguous; the difference is the status column, which now reads
\textbf{Resolved} for every entry. We list the twelve in the order
in which their corresponding clause appears in the constitution.

\paragraph{P1 (PS-Constitution-1) Diophantine uniqueness for the \OGNfive{} ledger
 --- \emph{Resolved.}}
Theorem $\T{6}$ asserts that $(M,N,L;J,K) = (1,3,5;7,35)$ is the
unique integer solution of the Diophantine system $K = LJ$,
$K = M^2 + N^2 + L^2$, $J = 2N+1$, compatible with the
four-component anomaly polynomial of $\T{5}$ and the chirality
count of the 6D bulk. The proof lives in \ATwo{}
\citep{ArisakaA2GG6}, Theorem~6 (the \OGNfive{} Diophantine
cascade), with the redundant cross-check at \AThree{}
\citep{ArisakaA3BI6}, Theorem~7.2 (the trace-ratio theorem).
The deeper philosophical question --- whether the five integers
themselves count as ``free parameters'' or as derived data of
the bulk geometry --- is addressed at length in \AFour{}
\citep{ArisakaA4Euclid}, which proves that the integers are
\emph{theorems} forced by a five-step Diophantine cascade in
exact parallel with the conversion-factor theorems for
$c$, $\hbar$, and $G_N$, and that the nine-element ledger
($\BIsix{}$ coefficients plus $\OGNfive{}$ integers) constitutes
the entire dimensionless arithmetic content of physics, leaving
no continuous knob in the gauge or cosmological sectors of
\GGTheory{}. \emph{Live falsifier:} an exhaustive integer search
to a much larger bound producing a second admissible solution
under the same axiomatic ordering would collapse $\T{6}$ and
force a revision of $\POS{6}~\OGN{}$.

\paragraph{P2 (PS-Constitution-2) Hopkins--Singer differential-cohomology proof of
 $\BIsix{}$ --- \emph{Resolved.}}
Theorem $\T{5}$ is established by the differential-cohomology
construction of \AThree{} \citep{ArisakaA3BI6}, Theorem~7.2 (the
\OGNfive{} trace-ratio theorem), with the redundant cross-check at
\ATwo{} \citep{ArisakaA2GG6} Theorem~4. \emph{Live falsifier:}
a published counter-derivation by an outside differential-cohomology
specialist exhibiting an admissible $X_4$ with $|a/c| \neq 5/2$
or $|a/d| \neq 5/2$ would collapse the trace-ratio lock and
revise $\POS{3}~\AF{}$.

\paragraph{P3 (PS-Constitution-3) Closed-form derivation of the 28 Standard Model
 parameters --- \emph{Resolved.}}
Theorems $\T{6}$ together with $\T{9}$ force every continuous
parameter of the Standard Model to descend from the five integers
$(1,3,5;7,35)$. \ANine{} \citep{ArisakaA9SM} carries out the
descent and produces explicit closed-form expressions
$X = X(M, N, L; J, K)$ for each of the 28 parameters, with six \emph{Derived} entries (including $\sin^2\theta_W = 25/108$,
$\alpha_s^{-1}(M_Z) = \pi e$, $\lambda_H = 1/8$, $y_t = 1$, and
the two Majorana phases), fourteen \emph{Derived (conditional on a
single named lemma)} entries, and eight \emph{Asserted} entries on an explicit
promotion roadmap. \emph{Live falsifier:} any precision SM
measurement deviating from the predicted closed-form value at
$> 5\sigma$ falsifies the parameter assignment in
\citep{ArisakaA9SM}.

\paragraph{P4 (PS-Constitution-4) Cosmological partitions and the $\pi$-locked
 fractions --- \emph{Resolved.}}
\ATen{} \citep{ArisakaA10Cosmos} carries $\Omega_m = 1/\pi$,
$\Omega_\Lambda = 1 - 1/\pi$, $\Omega_b = 1/(2\pi^2)$, and
$\Omega_{\rm DM} = 1/\pi - 1/(2\pi^2) \approx 0.2676$ as theorems
forced by the 6D bulk on its 4D boundary, agreeing with
Planck at the 0.27-$\sigma$ level. The appearance of $\pi$ is
not a free continuous parameter: it is the topological volume of
the compact $S^1_\theta$ fiber and is therefore admissible under
$\POS{6}~\OGN{}$. The corridor-switch redshift of the recombination era descends
from the integer $J = 7$ as $z_* = e^J \approx 1096.6$ --- a
topological marker distinct from the measured decoupling redshift,
which \ATen{}'s own atomic chain reproduces at the $0.1\%$ level
with zero fitted parameters, the switch sitting $0.4\%$ (in
e-folds) above it. \emph{Live falsifier:} a future
high-precision survey establishing $\Omega_m$ outside the
$1/\pi$ corridor at $> 5\sigma$ collapses the partition theorem.

\paragraph{P5 (PS-Constitution-5) One-loop gauge-coupling unification at $\EGUT$
 --- \emph{Resolved.}}
\PaperPOne{} \citep{ArisakaP1KK} computes the Kaluza--Klein
threshold corrections from the 6D bulk and demonstrates
that the three Standard Model couplings $g_1, g_2, g_3$ unify
cleanly at $\EGUT = \EPl\,e^{-2\pi} \approx 2.28 \times
10^{16}\,\GeV$ without invoking ad hoc supersymmetric content;
the lock cluster at $\Mlock \approx 3.85\,\TeV$ supplies the
required threshold corrections. \emph{Live falsifier:} an LHC
or future-collider exclusion of the predicted KK resonance
pattern at $\Mlock \approx 3.85\,\TeV$ at $5\sigma$ collapses
$\T{4}$ jointly with $\T{6}$.

\paragraph{P6 (PS-Constitution-6) The fermion generation count of three
 --- \emph{Resolved.}}
The Kaluza--Klein reduction on the 6D bulk produces
exactly three chiral generations on the 4D boundary from the
integer $J = 2N+1 = 7$ of the \OGNfive{} ledger together with
the chirality counter on $S^1_\theta$; the closure of the cascade
$S_{\rm tot}(3) = 3\pi$ exactly saturates the $\BIsix{}$-permitted
total instanton action. The derivation lives in \ANine{}
\citep{ArisakaA9SM}. \emph{Live falsifier:} the discovery of a
fourth chiral generation collapses both the KK reduction and the
$\OGNfive{}$ ledger.

\paragraph{P7 (PS-Constitution-7) The strong-CP problem and the kaxion mechanism
 --- \emph{Resolved.}}
\PaperCOne{} \citep{ArisakaC1Kaxion} establishes the kaxion field
with mass and decay constant both pinned by $\OGNfive{}$, and
shows $\bar\theta \to 0$ via a Peccei--Quinn-style mechanism whose
oscillation signature is the axiomatic invariant
$\Delta\nu/\nu = 1/35$ in the microelectronvolt range.
\emph{Live falsifier:} a haloscope or microwave-cavity search
that excludes $\Delta\nu/\nu = 1/35$ at the predicted
parameter point at $5\sigma$ collapses the strong-CP closure.

\paragraph{P8 (PS-Constitution-8) The cosmological-constant scale and sign
 --- \emph{Resolved.}}
The constitution forces the magnitude $m_\Lambda^{\mathrm{geom}}
= \Mlock^2/\MPl$, of order a few~meV with zero free continuous
parameters --- a result no other framework has produced ---
and \ATen{} \citep{ArisakaA10Cosmos} shows that the radial
orientation of the 6D bulk (boundary at $r = 0$, lock
cluster at $r = K = 35$) selects positive $\Lambda$ (de~Sitter)
on the 4D boundary. The residual factor-of-three between the
constitutional estimate and the empirical value is a falsifier on
the lock-mass prediction, not an unresolved foundational
question. \emph{Live falsifier:} a precision-cosmology
measurement establishing $m_\Lambda$ outside the
$\Mlock^2/\MPl$ corridor at $5\sigma$ collapses the projection
chain in \citep{ArisakaA10Cosmos}.

\paragraph{P9 (PS-Constitution-9) The inflationary mechanism --- \emph{Resolved.}}
\ATen{} \citep{ArisakaA10Cosmos} carries TeV-scale slow-smooth
inflation with a graceful exit, sourced by the inherited
inflationary-modulus sub-stratum of the 6D bulk rather than
by an external scalar field. The horizon, flatness, and
unwanted-relic problems of standard $\Lambda$CDM are solved by
this single epoch; the inflationary energy scale is locked to
the same lock cluster that produces $\Mlock \approx 3.85\,\TeV$,
and the graceful exit is structural rather than fine-tuned.
The relict primordial gravitational-wave amplitude
$r = 12/N^{*2} = 0.0043$ follows from the Starobinsky channel on the
derived GUT-adjacent plateau. \emph{Live falsifier:} a CMB B-mode
measurement inconsistent with $r \simeq 0.004$ \emph{in either
direction} --- a detection well above it, or a null pushing $r$ below
$\sim10^{-3}$ --- at $5\sigma$ collapses it
\citep{ArisakaA10Cosmos}.

\paragraph{P10 (PS-Constitution-10) Higgs-mass radiative stability and the hierarchy
 --- \emph{Resolved.}}
The electroweak vacuum expectation value
$v = \Mlock\,(2/L)^N = \Mlock\,(2/5)^3 \approx 246.33\,\GeV$ is a forced descendant of
$\Mlock$ and the \OGNfive{} ledger, not a free input. The scale profile $e^{-r}$ of the \GG{} bulk supplies the geometric
hierarchy explicitly: the Planck--electroweak ratio of $10^{17}$
is the integer $K = 35$ exponentiated, and the standard
naturalness puzzle does not appear because the Higgs mass has
nowhere to drift to. This is the structural answer to the
hierarchy problem: the Higgs sector inherits its scale from the
same lock that fixes every other scale in the descent chain.
The derivation lives in \citep{ArisakaA9SM} with the explicit
KK-threshold cross-check in \citep{ArisakaP1KK}. \emph{Live
falsifier:} a precision Higgs measurement establishing $v$
outside the predicted corridor at $5\sigma$, or a discovery of
unprotected radiative drift of the Higgs mass, collapses the
hierarchy resolution.

\paragraph{P11 (PS-Constitution-11) Neutrino-mass texture --- \emph{Resolved.}}
\ANine{} \citep{ArisakaA9SM} carries the descent that produces
the three neutrino masses, the three PMNS mixing angles, the
Dirac CP phase (resolved to a two-fold discrete ambiguity
between $\delta = \pi$ and $\delta = 3\pi/2$, pending tighter
experimental input), and the two Majorana phases (both
\emph{Derived} as cost-minimum winners under $\MSMF{}/\OGN{}$).
The full texture --- mass hierarchy (normal vs.\ inverted),
absolute mass scale within $\sum m_\nu \lesssim 0.1$~eV, and
Dirac vs.\ Majorana character --- is fixed by the cascade.
\emph{Live falsifier:} a KATRIN-style absolute-mass measurement
outside the predicted window, or a neutrinoless-double-beta-decay
detection incompatible with the predicted Majorana character, at
$5\sigma$, collapses \citep{ArisakaA9SM}.

\paragraph{P12 (PS-Constitution-12) Biological, chemical, and neuroscientific
 extension of $\GDOS{}$ --- \emph{Resolved.}}
\AEleven{} \citep{ArisakaA11Origin} carries the
eighteen-layer $\GRIP{}$ cascade from gauge structure
through the periodic table, ATP, RNA, DNA, the first cell
(LUCA), eukaryogenesis, brain organization, and the conscious
moment, each layer a $\GRIP{}$ event on the same lawful state
manifold. The hard problem of consciousness is dissolved by
structural stipulation rather than left as a free metaphysical
postulate. The energy--information Arisaka constant
$Q_{\rm B} \approx 0.156$~eV per qubit is calibrated against the
biological energy currency (ATP) --- a quantitative bridge from
the 6D bulk to the universal currency of biological energy that no
prior framework has supplied; the per-qubit calibration is
developed in (\citealp{ArisakaA7Qi4}; \citealp{ArisakaA11Origin}). \emph{Live falsifier:} an
experimental measurement of $Q_{\rm B}$ outside the predicted
corridor, or a structural counter-example showing that one of
the eighteen $\GRIP{}$ events fails to descend from the same
constitutional axioms, collapses
\citep{ArisakaA11Origin}.

\paragraph{Triage summary.}
All twelve entries are \textbf{Resolved}. This is not the same
as saying the program is finished: each entry carries a
specific live laboratory or precision-cosmology falsifier in its
companion paper, and the test of the constitution over the
coming decade is whether those falsifiers collapse one or more
clauses upstream. The honest reading is that the constitution
has been written, the cohort has done the technical work, and
the discipline now has a falsifiable, auditable, parameter-free
proposal on the table. After all, a constitution shows it is
serious by collapsing under a single registered measurement, not
by surviving every measurement; the live falsifiers above are the
program's standing offer to be wrong, in detail, in twelve
specific ways.


\subsection*{Closing remarks on Part~VI}

Having compared the constitution with everything we could think to
compare it against, we should say what we think the comparison actually
shows.

On most points, the constitution does not out-compute its neighbors.
The Standard Model predicts collider cross-sections to many decimal
places, and nothing here improves on that. String theory has developed
mathematics of a depth this document does not approach. The
open-systems formalism was worked out by others and is imported whole.
What the constitution offers is not a better calculation but a
different kind of object: a statement of the commitments under which
those calculations are legitimate, short enough to be argued with as a
whole.

Whether that is worth having is a fair question, and we would answer it
this way. Consider how the free-parameter count of fundamental physics
has moved. One for general relativity. Two for quantum electrodynamics.
Twenty-eight for the Standard Model. Six more for cosmology. And then,
when the search for a selection principle came up empty, an estimated
$10^{500}$ vacua and a serious argument that the constants might have no
explanation at all. That trajectory is not a series of technical
setbacks. It is what happens to a discipline that never wrote down
whether unforced freedom is allowed. $\AX{4}$ is a one-sentence answer
to a question the field spent fifty years not asking out loud.

Two things follow that we would ask a reader to keep separate. The
first is the constitutional move itself --- stating the rules,
classifying by axiom subset, requiring a named kill criterion. The
second is the particular descent chain this cohort proposes: a
six-dimensional bulk, an anomaly identity, five integers. The second
could fail entirely while the first survives, and we would count that a
good outcome rather than a wasted decade. It is a bad sign when a program is built so that nothing inside it can
be given up.

The twelve problems cataloged above are all marked resolved, and that
word should be read narrowly. Resolved means: a mechanism is proposed,
the proposal is written down where it can be checked, and a measurement
is named that would undo it. It does not mean settled, and it does not
mean believed. Each entry carries a live falsifier for exactly this
reason.

Part~VII draws the conclusions --- including what we think should
survive if the descent chain does not.


\GGPart[Conclusions]{Conclusions}%
 {The Constitutional Shift toward a Fourth Scientific Revolution}

\noindent
This Part is short, and it is where we say what we think we have done
and what we know we have not.

The proposal is one sentence: the next foundational step in physics is
not a new particle, a new symmetry, or a new dimension, but the writing
down of the rulebook physics has been using without writing down. Four
axioms and seven postulates, eleven clauses, short enough to read in an
afternoon. The claim is that those eleven clauses are rigid enough to
force --- as theorems proved elsewhere, not as assertions made here ---
a six-dimensional bulk geometry, an exact four-component anomaly
identity, a five-integer ledger, a boundary law for open systems, and a
list of numbers sharp enough to be wrong.

We also say what a reader should take away if the descent chain
eventually fails. The classification of Part~V, the operational reading
of measurement in $\AX{3}$, and the discipline of $\POS{7}$ --- name your
kill criterion before you look --- do not depend on the six-dimensional
geometry being correct. They are worth something on their own, and we
would rather they survive us than that they be bundled with a
compactification and discarded together.

The wider view comes at the end: what twenty-two papers attempt
together, what the project is in one sentence, why any of this should
matter to someone who is not a physicist, and what we hope a careful
reader carries away.

\medskip

\section{Conclusions}
\label{sec:conclusion}

This paper introduced the \GGConstitution{}: four axioms ($\AX{1}$--$\AX{4}$) and seven postulates ($\POS{1}$--$\POS{7}$) intended to function as an admissibility grammar
for theories across ten scientific branches. The core claim is constitutional: several persistent bottlenecks of modern science arise from missing or implicit
rulebook clauses---especially around measurement, open systems, and unconstrained modeling freedom. The present work therefore aims to provide a stable public
citation anchor (Arisaka-GG-A1-Constitution) for subsequent technical papers and for textbook-scale expansion.

\paragraph{What the axioms do.}
AX-1 (\LC) is the operational expression of ``cause precedes effect'': registered boundary updates must be local and retarded, and effective kernels are inadmissible
if they silently use information that could not have arrived through allowed channels. This is familiar as microcausality in relativistic QFT, but it becomes nontrivial
as a constitutional clause precisely because much of modern science is dominated by open systems and coarse-grained records.

AX-2 (\GGEP) asserts a deliberate reversal of emphasis: rather than attempting to quantize classical geometry and then append internal gauge structure, the primitive object
is declared to be a composite geometry$\oplus$gauge connection. In this view, quantum/gauge structure is already geometric when expressed in the correct bundle language
(holonomy, parallel transport, and consistency constraints). The novelty in this paper is not a particular Lagrangian but the central claim that unification must be written
as geometry-first at the primitive level.

AX-3 (\OGGEP) addresses a century-old ambiguity by making measurement part of the definition of a theory. An observation is not a metaphysical ``collapse'' statement;
it is the output of a lawful bulk$\to$boundary registration map that produces operational observables (POVMs, instruments, estimators) together with declared protocols and error models.
In this constitution, it is not enough to write down dynamics: a theory is incomplete if it does not specify how its claims become registered records.

AX-4 (\MSMF) enforces predictivity as an axiomatic demand. Nature selects maximal symmetry with minimal continuous freedom; continuous knobs that are not compelled by deeper symmetry,
consistency, or operational meaning are treated as inadmissible. This axiom is motivated by the long history of successful symmetry-based compression, and by the modern experience that
when model spaces become too flexible (many parameters, many vacua, many conventions), the distinction between prediction and choice becomes ambiguous.

\paragraph{What the postulates do.}
The seven postulates turn the axioms into a theorem machine. POS-1 (GI) generalizes gauge invariance to a layered principle across scales: local gauge redundancy in closed microscopic systems,
generator-level covariance for open mesoscopic dynamics (\DGI{}), and ledger-level invariance in macroscopic accounting (GGI). POS-2 (\UEP{}) asserts that a single variational ``engine'' underlies
dynamics: least action in closed bulk limits and least-time/least-free-energy avatars in open operational limits. POS-3 (AF) elevates anomaly freedom to an admissibility filter:
consistency obstructions are not optional subtleties but architecture constraints, including cancellability by inflow where required. POS-4 (CP) enforces physicality:
states must remain positive and normalized under evolution (quantum: CPTP semigroups; classical: positivity-preserving Markov dynamics). POS-5 (RG) insists on scale coherence:
coarse-graining must commute with the operational dictionary; predictions cannot depend on arbitrary resolution choices. POS-6 (\OGN{}) fixes normalization discipline:
among consistent conventions, nature selects minimal rational/integer ledgers, forbidding hidden continuous rescalings. Finally, POS-7 (MF) closes the scientific loop:
every claim must map to a registered observable and a pre-declared falsifier (kill criterion).

\paragraph{The constitutional classification of the twentieth century.}
Beyond compiling into the cohort's own technical machinery, the
\GGConstitution{} furnishes a structural classification of
essentially every major theoretical-physics framework of the
twentieth century. Part~V develops this classification at length:
general relativity, gauge theory, string theory, and AdS/CFT live
at the $\mathsf{GGEP\text{-}closed}$ stratum (Stratum-2,
$\{\AX{1},\AX{2}\}$); quantum mechanics with its measurement
resolution lives at the $\mathsf{OGGEP\text{-}open}$ stratum
(Stratum-3, $\{\AX{1},\AX{2},\AX{3}\}$); the Standard Model and
$\Lambda$CDM cosmology live at the
$\mathsf{Full\text{-}Constitution}$ stratum (Stratum-4, full set).
Each historical framework correctly used the axiom subset it
implicitly invoked; each left silent exactly the questions that
required the missing axioms. The diagnostic value of the
classification is sharp: ninety years of debate about the
measurement problem reduces to the observation that
$\mathsf{GGEP\text{-}closed}$ is missing $\AX{3}$, and forty
years of the string-theory landscape reduces to the observation
that $\mathsf{GGEP\text{-}closed}$ is missing $\AX{4}$. Add the
missing axioms and both historical gaps dissolve into chains of
theorems. This is the most consequential structural
payoff of the four axioms: not just a new theory, but a structural
map of the theories the broader physics community already knows by
name.

\paragraph{Two parental engines and the ten-branch export.}
With the constitution in place, \GGTheory{}\ is designed to compile into two complementary parental engines.
The first is \GG\ (Geometry--Gauge Locking in six dimensions), a bulk stage intended to realize \GGEP{} with anomaly inflow, RG geometry, and normalization discipline.
The second is \GDOS\ (Geometric Dynamics of Open Systems), a boundary runtime in which open-system admissibility and observation are first-class.
These engines are developed in the companion preprints \citet{ArisakaA2GG6} and \citet{ArisakaA5GDOS}; the closing capstone \citet{ArisakaA11Origin} carries the GRIP cascade from quarks through atoms, molecules, ATP, DNA, and the first cell to the universal-identity insight; the ten-branch export map is developed in \citet{ArisakaBookI}.
The ten branches are not claimed to be identical in microscopic degrees of freedom. Rather, they are required to share a single audit trail:
\emph{declare a \SIP{}} (state variables, symmetries, ports, ledger), \emph{evolve by an admissible kernel}, and \emph{export observables with falsifiers}.
This is how unification is intended to avoid becoming metaphor: cross-domain claims are constrained to be operationally testable and scale-coherent.

\paragraph{A philosophical closing (and a falsifiability opening).}
Historically, scientific revolutions are as much about rules of explanation as about new equations. Euclid did not merely list geometric facts; he made them inevitable by a constitution.
Newton did not merely fit planetary data; he provided a rulebook that made celestial and terrestrial dynamics one subject. Einstein did not just change formulas; he redefined the operational
meaning of time and simultaneity. In the twentieth century, quantum theory and relativity achieved enormous explanatory compression, but they also inherited open axiomatic seams:
measurement remained partially informal, and the status of ``what is input vs.\ what is derived'' became increasingly blurred as theories grew in complexity.

The present paper proposes a specific answer to that blur: treat the constitution as an explicit object. A scientific claim is not only an equation; it is a contract that (i) specifies its operational
observables, (ii) preserves physical state space under admissible dynamics, and (iii) forbids hidden knobs that would convert prediction into choice. In a data-rich century, the danger is not lack of
models but excess of flexibility. A constitution is a discipline for saying ``no''---no to acausal kernels, no to unphysical evolution, no to scale artifacts, and no to tuning that cannot be justified
by symmetry or operational meaning. Under this discipline, explanatory progress becomes a compression game played under audit: the goal is the smallest admissible description that reproduces the
largest set of registered invariants.

This conclusion is therefore also a falsifiability opening. The
constitution will be judged by whether its downstream engines can
generate sharp, killable targets without reintroducing hidden
dials --- quantitative coupling and normalization relations,
anomaly-locked integer ledgers, cosmological budget constraints,
spectral fingerprints, and protocol-level measurement contracts
that survive contact with data. If the program succeeds, the
standard of what we mean by explanation across the sciences
ought to shift: not toward more detailed models but toward a shared
grammar for admissibility and auditability. If it fails, the
failure modes are themselves informative --- they identify which
constitutional clauses must be revised and which parts of the
open-system and measurement semantics resist being made
universal. In either outcome, the value of writing the
constitution down is not in being right; it is in being
auditable.

After all, what makes a body of knowledge a science is not the
cleverness of its calculations but the honesty of its rules. The
historical chain cataloged in Part~I --- from Euclid through Newton
and the nineteenth-century unifications to the open-systems generation
--- has now reached the moment
where the axiomatic move is to write the rulebook itself.
The four axioms and seven postulates of the \GGConstitution{} are
that rulebook. They are short enough to be read in an afternoon,
explicit enough to be audited line by line, and rigid enough to
force the eleven Principal Theorems and the sharp numerical
predictions of the \GGTheory{} program. Remarkably, the same
four axioms also locate every major theoretical-physics framework
of the twentieth century inside a single constitutional tower
(Part~V): each predecessor is what the constitution looks like when
a particular subset of its axioms is operative, and each
predecessor's famous limitations are missing-axiom diagnostics that
the classification reads off by inspection. Whether the
constitution's downstream predictions survive contact with
experiment over the coming decade will determine whether the
eighth step in the long historical chain has been taken or whether
we must wait for a ninth. The constitution invites both outcomes;
that is precisely what makes it scientific.


\subsection*{The wider view: what the twenty-two papers attempt together}

\noindent
A constitution is only as serious as the country it governs.
The \GGConstitution{} is the founding document of a twenty-two-paper research program --- eleven foundational A-papers, two Foundations-Branch papers, four Predictions-Branch papers, three Cosmology \& Astrophysics papers, and two White-Paper-Branch syntheses --- and a few words about what those papers together attempt may be useful to a reader who has not yet looked
beyond the present volume.

The eleven A-papers fall into a single descending arc, the two
Foundations-Branch papers carry its deepest structural
derivations, and the six phenomenology papers carry its
predictions to experiment.
\AOne{} (the present paper) writes the rulebook ---
four axioms and seven postulates, no more. \ATwo{}, \AThree{},
and \AFour{} build the geometric stage on which the rulebook acts:
a six-dimensional bulk, a rigorous proof that its anomaly
content closes without remainder, and a historical accounting of
the long road from Euclid to the present. \AFive{} crosses from
the bulk to the boundary --- from what nature does to what an
observer in a laboratory can register --- and \ASix{} and \ASeven{}
work out the two halves of that boundary law in detail: the
geometric runtime that delivers conservation laws and the
quantum-information runtime that delivers measurement.
\AEight{}, \ANine{}, and \ATen{} then walk back through textbook
physics --- ordinary quantum mechanics, the Standard Model of
particle physics, and the cosmology of the visible universe ---
and show that each of these familiar frameworks is what the
constitution looks like when its axioms are read on the
appropriate stratum. \AEleven{} closes the cohort with a single
audacious claim: that the same constitutional ladder that explains
the fine structure of the proton also explains the formation of
the first cell, of conscious experience, and of the long sequence
of \emph{identities} --- atoms, molecules, organisms, minds ---
that the universe has built out of its own ingredients. The
Foundations Branch (\PaperPFour{}, \PaperPFive{}) stands at the
arc's flank and derives what the textbook chapters assume first:
the origin of the complex numbers of quantum mechanics, and the
mirrors C, P, and T together with the origin of the cosmic matter
excess. The
Predictions Branch (\PaperPOne{}, \PaperPTwo{}, \PaperCOne{})
hangs from that ladder as a set of sharp, falsifiable numerical
targets: a non-spatial Kaluza-Klein lock cluster near $3.85$~TeV, a
three-generation count forced by the integer ledger, and a kaxion
oscillation at fractional frequency $1/35$ in the microelectronvolt range.

\subsection*{What the project is, in one sentence}

\noindent
Stripped of all technical apparatus, the twenty-two-paper program
is an attempt to do for physics what the founders did for political
philosophy: to ask whether the discipline can be governed by a
short, written constitution rather than by accumulated custom.

A short constitution does not replace the working scientist's
laboratory or the theorist's calculation. It asks them to declare,
before the work begins, which clauses they have implicitly agreed to.
A theory that quietly turns off causality, or that admits hidden
continuous parameters, or that treats measurement as something the
mathematics may ignore, is not thereby wrong. But it is now
\emph{visible} as a theory of a particular kind: a theory whose
constitutional stratum can be named and whose limitations can be
diagnosed by inspection rather than by decades of post-hoc
analysis. The cost of writing the constitution down is small.
The benefit is that the work of the discipline becomes auditable
across centuries instead of across careers.

\subsection*{Why this matters beyond physics}

\noindent
The constitutional move is not a piece of mathematical
housekeeping. It is a stance about what science is for.

After Newton, the dominant model of explanation in physics was the
\emph{successful prediction}: a theory was judged by what it could
forecast. This served the field well for three centuries. But by
the closing decades of the twentieth century the cost of that
model had become visible: predictions multiplied, theories
multiplied with them, and the field accumulated frameworks faster
than it could test them. The string-theory landscape, the
multiplication of inflationary scenarios, and the proliferation of
interpretations of quantum measurement are three faces of the same
underlying problem. The problem is not that the predictions
became uninformative; it is that the rulebook for what counts as
an admissible theory was never written down. Without an
admissibility rulebook, the discipline cannot tell, at the level
of \emph{principle}, which of its competing frameworks are
candidates and which are not.

The twenty-two-paper program replaces \emph{successful prediction}
with \emph{auditable admissibility} as the primary scientific
discipline. Predictions still matter --- the constitution forces
sharp, killable ones --- but they are no longer the only test.
A theory must \emph{also} declare its constitutional axioms and
postulates, accept that those clauses can be falsified independently
of any specific prediction, and accept the bookkeeping that goes
with such a declaration. This second discipline is the older one.
It is the discipline of Euclid, of Newton's \emph{Principia}, of
the framers who wrote down the political constitutions of the
modern world. It is the discipline of saying out loud what the
rules of the game are before the game begins.

\subsection*{What we hope a careful reader will take away}

\noindent
We hope a reader who has worked through the present volume comes
away with three convictions. The first is that the central
question of foundations of physics is no longer \emph{what is the
correct theory} but \emph{which axioms is the correct theory a
theorem of}. The second is that ``measurement,'' ``openness,''
``locality,'' and ``parsimony'' are not interpretational add-ons
but constitutional primitives, on the same footing as ``space,''
``time,'' and ``symmetry.'' The third is that the same four
axioms that organize twentieth-century physics also organize the
sequence of \emph{identities} that the universe has built --- from
the electron through the atom and the molecule to the cell, the
brain, and whatever comes next --- and that the right scientific
attitude toward that long sequence is not awe, but careful
constitutional accounting.

After all, the most important sentence in the twenty-two-paper
program is not any of its theorems or numerical predictions.
It is the recognition that the same four clauses constrain the
proton, the photon, the genome, and the conscious moment, and that
this is the deepest empirical regularity that physics has yet
written down. If the constitution is right, the next century of
physics will not be the discovery of new physics so much as the
\emph{readout} of the constitution on a longer and longer chain of
admissible identities. If the constitution is wrong, the failure
will be specific, locatable, and instructive. Either way, the
discipline will be the better for having written the rulebook
down.

The constitution is short; the cohort is the long argument that
the constitution is enough. We invite the reader to test both.


\subsection*{Closing remarks on Part~VII}

A last word about what kind of document this has been. We have not
reported a discovery. No new particle is announced, no new equation
solved, no measurement made. What is proposed is a change in
bookkeeping: that physics state, in one short document, the commitments
it has been making for three and a half centuries without recording
them. The eleven clauses offered here are a first draft, and first
drafts of constitutions are usually wrong in ways their authors cannot
see. What we would defend is not the particular clauses but the
practice.

There is a reason this matters beyond physics, and it is the thing we
would most like a reader to carry away. The rules proposed here are not
about particles. $\AX{3}$ says that observation is a projection from
more degrees of freedom to fewer --- as true of a cell reading its
environment through a handful of receptors as of a detector reading a
collision from a shower of tracks. If that clause is right, the
boundary between physics and the sciences of the living is not a
boundary in nature. It is a boundary in our bookkeeping, and
bookkeeping can be changed.

We do not know whether the descent chain will survive the decade. But
the question the constitution raises does not depend on the answer.
Either the constants of nature are chosen, or they are left over ---
and physics never had to decide, because it never wrote down whether
choosing was allowed. It is written down now.


\GGAcknowledgments

\GGAppendixPart[Technical Appendices]{Technical Appendices}{Detailed mathematical machinery: abbreviations, the glossary of key symbols, mathematical bridges, and worked-example templates}

\noindent
Part~VIII gathers the technical machinery
of the constitution. This is where the reader who
wants to verify a specific axiom step, run a specific worked example,
or trace a specific reference will spend time --- while the reader
who wants only the central claim and its forward consequences
can stop after Part~VII and treat Part~VIII as a self-contained
companion volume. Appendix~A is the abbreviations table, and it is the first place to
look when an unfamiliar name appears anywhere in the volume.
Appendix~B is the glossary of symbols. Appendix~C works out, from each
of the four axioms, what that clause forbids and what standard
machinery it sets up --- these are elaborations \emph{from} the axioms,
not derivations \emph{of} them, since an axiom by definition is stated
rather than proved. Appendix~D holds worked examples and reusable
templates: a driven oscillator, a two-state gene switch, Bayesian
updating, and two templates a reader can copy when writing a falsifiable
claim of their own.

Those last two are worth singling out. The gene-regulation example and
the free-energy monotonicity calculation are there because we mean the
cross-domain claim literally: the same boundary law that governs a
damped oscillator governs a cell deciding whether to transcribe. A
reader from the life sciences who wants to check that claim rather than
take it on faith should start in Appendix~D.

The ten Parts together are meant to be self-checking. The first three
state the constitution; Part~IV hands it forward and says exactly where
each proof lives; Part~V classifies twentieth-century physics by the
axioms each framework uses; Parts~VI and~VII weigh what the constitution
does and does not buy; and Parts~VIII to~X supply the technical,
adversarial and bibliographic apparatus that lets a reader audit the
rest without trusting us.

\clearpage
\section{Summary of Abbreviations}
\label{app:abbrev}

This appendix collects every abbreviation used in the present paper.
All program-defined names are rendered in \emph{sans-serif type}
(\GGTheory{}, \GGConstitution{}, \GG{}, \GDOS{}, \BIsix{}, \OGNfive{},
\QGL{}, \GiFour{}, \QiFour{}, \GRIP{}, \BFFT{}, etc.) so the reader
can see at a glance which terms belong to the \GGTheory{} program
proper. Standard physics abbreviations imported from the wider
literature (KK, RS, QFT, TOV, etc.) are rendered in plain uppercase
Roman, following the convention of the major physics journals.
Mathematical objects are in roman or italic in the usual way. The
third column gives the \emph{primary link}: the paper in the
Arisaka-GG series and the section where each program-defined name
is first introduced in detail.

{\footnotesize
\begin{longtable}{@{}>{\raggedright\arraybackslash}p{0.16\linewidth} >{\raggedright\arraybackslash}p{0.56\linewidth} >{\raggedright\arraybackslash}p{0.24\linewidth}@{}}
\caption[Summary of abbreviations used in the present paper.]{Every abbreviation used in this paper, with its meaning and the place it is introduced in full. Typography carries information here: program-defined names are set in sans serif, standard physics abbreviations in plain uppercase Roman, and mathematical objects in roman or italic as usual, so a reader can tell at a glance whether a name belongs to this framework or to the general literature. The third column links each entry to its primary source --- the present paper for names defined here, the companion paper otherwise. A reader who meets an unfamiliar name anywhere in the volume should come here first.}
\label{tab:appA-abbrev}\\
\toprule
\textbf{Symbol} & \textbf{Meaning} & \textbf{Primary link} \\
\midrule
\endfirsthead

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

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

\bottomrule
\endlastfoot

\multicolumn{3}{l}{\itshape Framework name and core program objects} \\
\midrule
\GGTheory{} & Grand Geometric Theory (also: Geometry-Gauge Equivalence Theory) & \AOne{}, Preface \\
\GGConstitution{} & The constitutional charter of the \GGTheory{} program: four axioms (\AX{1}--\AX{4}) and seven postulates (\POS{1}--\POS{7}) & \AOne{}, Parts~II--III \\
\GG{} & Geometry-Gauge locking in six dimensions: the 6D bulk & \ATwo{}, \S 8 (\TX{1}{GG6}) \\
\GDOS{} & Geometric Dynamics of Open Systems: the boundary law implied by \OGGEP{} & \AFive{} \\
\BIsix{} & Green--Schwarz-modified Bianchi identity at 6D & \AThree{}, \S 3 \\
\OGNfive{} & Occam-Gauge-Normalization 5-integer ledger $(M, N, L; J, K) = (1, 3, 5; 7, 35)$ & \AThree{}, \S 7; \TX{6}{GG6} \\
\QGL{} & Quadruple Gauge Locking: the four-factor bulk gauge group $\Spinone \times \SU(5) \times \U(1)_B \times \U(1)_{Qi}$ & \ATwo{}, \S 10 (\TX{3}{GG6}) \\
\GiFour{} & Geometric-Invariance four-pillar structure (\TGG{}, \CCI{}, \GFF{}, \GRIP{}) & \ASix{} \\
\QiFour{} & Quantum-Information four-pillar structure (\QiQUA{}, \QiEP{}, \QiCON{}, \QiHAL{}) & \ASeven{} \\
\midrule

\multicolumn{3}{l}{\itshape Four axioms (\AX{1}--\AX{4})} \\
\midrule
\AXkw{} & Axiom (axiomatic primitive; non-negotiable) & \AOne{}, \S\ref{sec:definitions} \\
\LC{} & \AX{1} Locality and Causality & \AOne{}, \S\ref{sec:AX1} \\
\GGEP{} & \AX{2} Geometry--Gauge Equivalence Principle & \AOne{}, \S\ref{sec:AX2} \\
\OGGEP{} & \AX{3} Operational/Open Geometry--Gauge Equivalence Principle (boundary observation dictionary) & \AOne{}, \S\ref{sec:AX3} \\
\MSMF{} & \AX{4} Maximum Symmetry, Minimum Freedom (no hidden continuous knobs) & \AOne{}, \S\ref{sec:AX4} \\
\midrule


\multicolumn{3}{l}{\itshape Seven postulates (\POS{1}--\POS{7})} \\
\midrule
\POSkw{} & Postulate (admissibility filter; testable in principle) & \AOne{}, \S\ref{sec:definitions} \\
\GI{} & \POS{1} Gauge invariance across scales (\LGI{} / \DGI{} / \GGI{}) & \AOne{}, \S\ref{sec:POS1} \\
\UEP{} & \POS{2} Unified Extremal Principle (PLA $\Leftrightarrow$ PLT / \MEPP{} / FEP) & \AOne{}, \S\ref{sec:POS2} \\
\AF{} & \POS{3} Anomaly Freedom and cancellability by inflow & \AOne{}, \S\ref{sec:POS3} \\
\CP{} & \POS{4} Closure and Positivity (CPTP-admissible open dynamics) & \AOne{}, \S\ref{sec:POS4} \\
\RG{} & \POS{5} Renormalization-Group covariance (scale covariance) & \AOne{}, \S\ref{sec:POS5} \\
\OGN{} & \POS{6} Occam Gauge Normalization (canonical integer/rational normalization; \emph{formerly} MRHN) & \AOne{}, \S\ref{sec:POS6} \\
\MF{} & \POS{7} Measurement and Falsifiability (operational contract + kill criteria) & \AOne{}, \S\ref{sec:POS7} \\
\LGI{} & Local Gauge Invariance (microscopic / closed-system layer of \GI{}) & \AOne{}, \S\ref{sec:POS1} \\
\GGI{} & Global Gauge Invariance (macroscopic / ledger layer of \GI{}) & \AOne{}, \S\ref{sec:POS1} \\
\midrule

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


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

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

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

\multicolumn{3}{l}{\itshape Program-defined mathematical symbols} \\
\midrule
$\QB$ & Quantum-information Bit -- the energy--information Arisaka constant $\QB \approx 0.1555$~eV per qubit; calibrated against biological energy currents ((\citealp{ArisakaA7Qi4}; \citealp{ArisakaA11Origin})) & \ATwo{}, \S 15 (\TX{8}{GG6}) \\

\\


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

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

\multicolumn{3}{l}{\itshape \quad Quantum and classical foundational theories} \\
\midrule
BRST & Becchi--Rouet--Stora--Tyutin (cohomology) & --- \\
GR & General Relativity & --- \\
QCD & Quantum Chromodynamics & --- \\
QED & Quantum Electrodynamics & --- \\
QFT & Quantum Field Theory & --- \\
QM & Quantum Mechanics & --- \\
\midrule

\multicolumn{3}{l}{\itshape \quad Standard Model and beyond} \\
\midrule
EFT & Effective Field Theory & --- \\
EW & Electroweak & --- \\
GUT & Grand Unified Theory & --- \\
SM & Standard Model of particle physics & --- \\
SSB & Spontaneous Symmetry Breaking (closed-system; generalized by \GSSB{}) & --- \\
SUSY & Supersymmetry & --- \\
\midrule

\multicolumn{3}{l}{\itshape \quad Higher-dimensional constructions} \\
\midrule
AdS & Anti--de Sitter spacetime & --- \\
AdS/CFT & Anti--de Sitter / Conformal Field Theory correspondence & --- \\
CFT & Conformal Field Theory & --- \\
KK & Kaluza--Klein (mode expansion / dimensional reduction) & --- \\
RS & Randall--Sundrum (warped extra-dimension model) & --- \\
\midrule

\multicolumn{3}{l}{\itshape \quad Quantum gravity and cosmology} \\
\midrule
$\Lambda$CDM & Standard cosmological model with cosmological constant + cold dark matter & --- \\
LQG & Loop Quantum Gravity & --- \\
TOV & Tolman--Oppenheimer--Volkoff (equation) & --- \\
\midrule

\multicolumn{3}{l}{\itshape \quad Foundations of quantum mechanics} \\
\midrule
CHSH & Clauser--Horne--Shimony--Holt (inequality) & --- \\
EPR & Einstein--Podolsky--Rosen (paradox / argument) & --- \\
\midrule


\multicolumn{3}{l}{\itshape \quad Open-system / extremal-principle sub-acronyms} \\
\midrule
FEP & Free Energy Principle (boundary inference avatar of \UEP{}) & \ASeven{}, \QiEP{} \\
\MEPP{} & Maximum Entropy Production Principle (heuristic boundary avatar of \UEP{}) & \ASeven{}, \QiEP{} \\
PLA & Principle of Least Action (closed/bulk avatar of \UEP{}; standard variational dynamics) & \ASeven{}, \QiEP{} \\
PLT & Principle of Least Time (operational/boundary avatar of \UEP{}; fastest lawful path) & \ASeven{}, \QiEP{} \\
\end{longtable}
}




\bigskip

\noindent\textbf{Extended description of
Figure~\ref{fig:abbrev-flow} --- the hierarchy of program-defined
acronyms.} Top: the framework name (\GGTheory{}) and its constitutional charter (\GGConstitution{}). Tier~2: the four axioms (\AX{1}--\AX{4}) and seven postulates (\POS{1}--\POS{7}) that together define the admissibility predicate. Tier~3: three bulk objects forced by the constitution --- \GG{} (6D bulk), \BIsix{} (modified Bianchi identity), \OGNfive{} (integer ledger). Tier~4: the boundary law \GDOS{} that the bulk projects to via \OGGEP{}, with input acronym \SIP{}. Tier~5: the two four-pillar structures, \GiFour{} (geometric flank, with sub-acronyms \GEP{}/\CEP{} blocks of \GFF{} and \DGI{}/\GSSB{}/\GA{} stages of \GRIP{}) and \QiFour{} (quantum-information flank, with \QiEP{} sub-acronyms PLT/\MEPP{}/FEP, \QiCON{} sub-concepts Conformal/L\'evy/Fractal/$1/f$, \QiHAL{} sub-acronyms \MePMoS{}/\NHT{}/\HAL{}/\BFFT{}, the boundary completion \OH{}${+}$\IH{}${=}$\EIC{}, and the runtime carrier \SeOS{} (Selfish Open System) underneath). Bottom: the energy--information Arisaka constant $\QB$ (derived secondary quantity), with the Arisaka equation $J_E / \QB = J_e = J_B = J_{Qi}$ that locks the four boundary currents through one conversion factor.

\begin{figure}[H]
\centering
\resizebox{0.824\textwidth}{!}{%
\begin{tikzpicture}[
 font=\sffamily,
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 line width=0.80pt,
 every node/.style={align=center},
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 arrow/.style ={->, line width=0.7pt, draw=black!65},
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]

% --- Top tier: programme name and charter -----------------
\node[top] (GGT) at (0, 0) {\GGTheory{}};
\node[charter] (GGC) at (0, -16mm) {\GGConstitution{}};
\draw[arrow] (GGT.south) -- (GGC.north);

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

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

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

% --- Tier 3: bulk theorems (3 boxes: GG-6, BI-6, OGN-5) ---
\node[font=\sffamily\bfseries\large] (BTL) at (0, -108mm) {Bulk theorems forced by \textbf{AX} + \textbf{POS}};
\node[bulk] (GG6) at (-46mm, -120mm) {\textbf{GG-6}};
\node[bulk] (BI6) at (0mm, -120mm) {\textbf{BI-6}};
\node[bulk] (OGN5) at (46mm, -120mm) {\textbf{OGN-5}};

\draw[arrow] (AX4.south) -- ([yshift=-2mm]BTL.north -| AX4.south);
\draw[arrow] (P7.south) -- ([yshift=-2mm]BTL.north -| P7.south);

% --- Tier 4: GDOS boundary law (only SIP under it) --------
\node[gdos] (GDOS) at (0, -136mm) {\textbf{GDOS} \;\; (Geometric Dynamics of Open Systems on $\partial\mathcal{M}_6$) \;\; \textbf{SIP}};

\draw[arrow] (GG6.south) -- (GDOS.north -| GG6.south);
\draw[arrow] (BI6.south) -- (GDOS.north -| BI6.south);
\draw[arrow] (OGN5.south) -- (GDOS.north -| OGN5.south);

% --- Tier 5a: Pillars headers ----------------------------
\node[gihead] (GiH) at (-46mm, -152mm) {\textbf{Gi-4 Pillars}};
\node[qihead] (QiH) at (46mm, -152mm) {\textbf{Qi-4 Pillars}};

\draw[arrow] (GDOS.south) -| (GiH.north);
\draw[arrow] (GDOS.south) -| (QiH.north);

% --- Tier 5b: Pillars content -----------------------
% Gi-4 box (left) -- 4 pillars + sub-acronyms
\node[pill] (TGG) at (-82mm, -170mm) {\textbf{TGG} \quad Total Geometric--Gauge object};
\node[pill] (CCI) at (-82mm, -179mm) {\textbf{CCI} \quad Covariant Conservation Identity};
\node[pill] (GFF) at (-82mm, -188mm) {\textbf{GFF} \quad Geometric Free-Energy Functional};
\node[sub] at (-82mm, -195mm) {\hphantom{XX}$\downarrow$ \;\textbf{GEP} (energetic) \,$\bullet$ \,\textbf{CEP} (entropic)};
\node[pill] (GRIP) at (-82mm, -204mm) {\textbf{GRIP} \quad Gauge-Runtime Invariance Pipeline};
\node[sub] at (-82mm, -211mm) {\hphantom{XX}$\downarrow$ \;\textbf{DGI} (initial) \,$\bullet$ \,\textbf{GSSB} (middle) \,$\bullet$ \,\textbf{GA} (attractor)};

% Qi-4 box (right) -- 4 pillars + sub-acronyms + EIC
\node[pillq] (QQUA) at (10mm, -170mm) {\textbf{Qi-QUA} \quad Quantum information for \textbf{QGL}};
\node[pillq] (QEP) at (10mm, -179mm) {\textbf{Qi-EP} \quad Quantum information for \textbf{UEP}};
\node[sub] at (10mm, -186mm) {\hphantom{XX}$\downarrow$ \;\textbf{PLT} \,$\bullet$ \,\textbf{MEPP} \,$\bullet$ \,\textbf{FEP}};
\node[pillq] (QCON) at (10mm, -195mm) {\textbf{Qi-CON} \quad Quantum information for Conformal Inv.};
\node[sub] at (10mm, -202mm) {\hphantom{XX}$\downarrow$ \;Conformal \,$\bullet$ \,L\'evy \,$\bullet$ \,Fractal \,$\bullet$ \,$1/f$};
\node[pillq] (QHAL) at (10mm, -211mm) {\textbf{Qi-HAL} \quad Quantum information for the \textbf{HAL}};
\node[sub] at (10mm, -218mm) {\hphantom{XX}$\downarrow$ \;\textbf{MePMoS} \,$\bullet$ \,\textbf{NHT} \,$\bullet$ \,\textbf{HAL} \,$\bullet$ \,\textbf{BFFT}};
\node[pillq] (EIC) at (10mm, -227mm) {\textbf{OH} \,$+$\, \textbf{IH} \,$=$\, \textbf{EIC} \quad (Evolving Inner Cosmos)};
\node[sub] at (10mm, -234mm) {\hphantom{XX}$\downarrow$ \;\textbf{SeOS} (Selfish Open System)};

% --- Tier 6: Q_B + Arisaka equation -----------------------
\node[qb] (QB) at (0, -250mm) {%
$\boldsymbol{\mathsf{Q}_{\mathsf{B}}}$ \;\; --- \;\; Quantum-information Bit \;\; (energy--information Arisaka constant) \\[1.5mm]
\textit{Arisaka equation:} \;\; $\boldsymbol{J_E / \mathsf{Q}_{\mathsf{B}} \,=\, J_e \,=\, J_B \,=\, J_{Qi}}$
};

\end{tikzpicture}
}
\caption[Hierarchy of program-defined acronyms in the \GGTheory{} program.]{How the program's own names are related to each other. The chart is a hierarchy, not a list: the charter sits at the top, the geometry it forces below it, the boundary law below that, and the derived pillars at the foot, so the position of a name on the chart tells the reader what it depends on. Names at the same level are siblings introduced by the same construction. The purpose is orientation rather than definition --- the meanings are in the abbreviations table --- and a reader who has lost track of which name belongs to which layer can recover it here in one glance.}
\label{fig:abbrev-flow}
\end{figure}

\clearpage


\clearpage
\section{Glossary of Key Symbols}
\label{app:symbols}

Table~\ref{tab:symbols} collects principal symbols and their meanings. We emphasize both bulk (\GGEP{}/\GG{}) objects and boundary operational (\OGGEP{}/\GDOS{}) objects.

\begin{longtable}{@{}>{\raggedright\arraybackslash}p{0.18\linewidth} >{\raggedright\arraybackslash}p{0.50\linewidth} >{\raggedright\arraybackslash}p{0.26\linewidth}@{}}
\caption[Key symbols, with meaning and typical context.]{Every symbol this paper uses frequently, with its meaning and the setting in which it normally appears. The third column is the working one: the same letter can carry a bulk meaning and a boundary meaning, and the context column is what distinguishes them. Because this is the charter paper, the ledger fixed here is the one the companion papers refer back to, so a symbol defined in this table should not be redefined downstream. A reader meeting an unfamiliar letter anywhere in the cohort should come here first.}
\label{tab:symbols}\\[0.35em]
\toprule
\textbf{Symbol} & \textbf{Meaning} & \textbf{Typical context} \\
\midrule
\endfirsthead
\toprule
\textbf{Symbol} & \textbf{Meaning} & \textbf{Typical context} \\
\midrule
\endhead
\midrule
\multicolumn{3}{r}{\small (continued)}\\
\endfoot
\bottomrule
\endlastfoot
$G_{\rm tot}$ & Total bulk gauge group, $\Spinone \times \SU(5) \times \U(1)_B \times \U(1)_{Qi}$; the single composite object $\AX{2}$ takes as primitive & Bulk (\GGEP{}) \\
$\cL_{\GDOS}$ & The \GDOS{} boundary law: the equation of motion obeyed by an open system read from outside itself & Boundary (\OGGEP{}) \\
$E_{\rm Ry}$ & Rydberg energy, $13.6$~eV; the atomic reference scale entering $\QB$ & Boundary, atomic \\
$\QB$ & Energy--information Arisaka constant, $\QB = (2/5)E_{\rm Ry}/K \approx 0.1555$~eV per qubit & Boundary ledger \\
$z^{*}$ & Corridor-switch redshift, $z^{*} = e^{J} = e^{7} \approx 1096.6$; a topological marker fixed by the integer $J$, and \emph{not} the same quantity as the measured decoupling redshift $z_{\rm dec} \approx 1090$ & Cosmology (\ATen{}, \PaperCTwo{}) \\


$\Mbulk$ & 6D bulk manifold (\GG{} stage) & Applications preview / \GG{} companion.\\
$\Mbdy$ & 4D operational boundary (registered observables) & \OGGEP{}, \GDOS{}.\\
$X^A=(x^\mu,r,\theta)$ & Bulk coordinates: boundary $x^\mu$, scale $r$, compact holonomy $\theta$ & \GG{} preview.\\
$g_{AB}$ & Bulk metric on $\Mbulk$; signature implicit & \GG{}.\\
$\omega$ & Spin connection (orthonormal-frame connection) & \GGEP{} primitive.\\
$A$ & Gauge connection one-form (Yang--Mills) & \GGEP{} primitive.\\
$\mathcal{A}_{\mathrm{lock}}=\omega\oplus A$ & Composite (locked) connection & AX-2, \GG{}.\\
$\mathcal{F}_{\mathrm{lock}}$ & Composite curvature $(\mathcal{R}\oplus F)$ & AX-2.\\
$F=\dd A + A\wedge A$ & Gauge field strength 2-form & AX-2.\\
$\Dmap$ & Operational dictionary (bulk$\to$boundary map) & AX-3.\\
$\cH$ & Hilbert space of boundary degrees of freedom & Quantum \GDOS{} limits.\\
$\rho$ & Density operator (quantum) or density on state manifold (classical) & POS-4.\\
$\Phi_t$ & Evolution map/semigroup (channel) & POS-4.\\
$\mathcal{L}$ & Generator of $\Phi_t$ (GKLS in quantum) & POS-4.\\
$\{\mathcal{I}_m\}$ & Quantum instrument (CP maps for outcomes) & AX-3/POS-7.\\
$\{E_m\}$ & POVM effects with $\sum_m E_m=\mathbb{I}$ & AX-3.\\
$\M$ & State manifold of a \SIP{} instance & \GDOS{} kernel.\\
$y^A$ & Coordinates/state variables on $\M$ & \GDOS{}/\UEP{}.\\
$G_{AB}(y)$ & Metric on $\M$ (e.g.\ Fisher metric) & POS-2/\UEP{}; information geometry.\\
$F_{\mathrm{\GFF{}}}$ & \GFF{} scalar ledger functional & POS-2; \GDOS{}.\\
$M^{AB}$ & Mobility/dissipation tensor ($\succeq 0$) & \GDOS{} drift.\\
$D^{AB}$ & Diffusion/noise tensor ($\succeq 0$) & \GDOS{} diffusion.\\
$u^A$ & Solenoidal/cycle drift term & \GDOS{} non-gradient flow.\\
$r$ & Scale/radial coordinate, $r=\ln(\EGUT/\mu)$ & POS-5 / applications.\\
$\mu$ & Coarse-graining (energy/resolution) scale & POS-5.\\
$\EGUT$ & Unification/locking scale & \GG{} preview.\\
$B_2,\,H_3$ & GS two-form and its (modified) field strength & POS-3; anomaly inflow.\\
$(M,N,L)$ & Primary integers of the \OGNfive{} ledger & POS-6; \GG{} bulk.\\
$J = 2N+1$ & Chirality count derived from $N$ ($J = 7$ when $N = 3$) & POS-6; \OGNfive{}.\\
$K = LJ$ & Lock length / radial range of 6D bulk ($K = 35$ when $L = 5, J = 7$) & POS-5/6; \GG{} bulk.\\
$\Mlock$ & Lock-cluster mass scale, $\Mlock = (1/2)\,\MPl\,\e^{-35}\approx 3.85\,\TeV{}$ & \PaperPOne{}; SM-scale predictions.\\
$\MPl,\,\EPl$ & Planck mass / Planck energy & Bulk reference scale.\\

\end{longtable}

\clearpage
\section{Mathematical equivalences and standard-textbook bridges}
\label{app:derivations}
\label{app:axioms-expanded}

This appendix provides a compact translation table between the four
constitutional axioms of the \GGConstitution{} and the standard-textbook
mathematical objects that a working physicist will recognize. Each
entry is deliberately brief: the full derivations live either in the
main text (Part~II for the axioms, Part~III for the postulates) or in
the cited companion papers. The point of the table is to allow a
reader who knows the standard machinery to identify, at first
encounter, which constitutional clause corresponds to which familiar
piece of mathematical apparatus. We close with one short worked
derivation that is genuinely useful as a stand-alone illustration:
how $\AX{1}~\LC{}$ together with $\POS{4}~\CP{}$ \emph{forces} the
Lindblad / GKLS form of the open-system generator.

\subsection{$\AX{1}~\LC{}$: locality, causality, and no-signaling}
\label{app:bridge_LC}

The microcausality formulation
$[\mathcal{O}(x),\mathcal{O}'(y)]_{\mp} = 0$ for spacelike $(x-y)^2 < 0$,
the retarded-Green-function formulation $G_{\rm ret}(x-y) = 0$ for
$x^0 < y^0$, and the operational no-signaling formulation
$\Tr_A[(\Phi_A \otimes \mathrm{id}_B)(\rho_{AB})] = \Tr_A(\rho_{AB})$
are all admissibility expressions of the same constitutional content.
Full derivations are in Part~II~\S\ref{sec:AX1_further};
$\AX{1}~\LC{}$ is the constitutional commitment to all three.

\subsection{$\AX{2}~\GGEP{}$: principal bundles and the composite locked connection}
\label{app:bridge_GGEP}

A gauge field is a connection one-form $A \in \Omega^1(M, \mathfrak{g})$
on a principal $G$-bundle $P(M, G)$, with curvature
$F = dA + A \wedge A$ and gauge transformation
$A \to A^g = g^{-1} A g + g^{-1} dg$ \citep{KobayashiNomizu1963,Nakahara2003}.
The gravitational analogue uses the spin connection $\omega$ on the
frame bundle, with curvature $\mathcal{R} = d\omega + \omega \wedge \omega$.
$\AX{2}~\GGEP{}$ elevates the \emph{composite locked connection}
\begin{equation}
\mathcal{A}_{\rm lock} \equiv \omega \oplus A,
\qquad
\mathcal{F}_{\rm lock} \equiv \mathcal{R} \oplus F,
\label{eq:locked_connection_app}
\end{equation}
to the primitive object: geometric and gauge data are not allowed to
vary independently. The Aharonov-Bohm experiment
\citep{AharonovBohm1959} confirms operationally that quantum phase is
holonomy of $\mathcal{A}_{\rm lock}$. The full bulk geometry is
constructed in \citep{ArisakaA2GG6}.

\subsection{$\AX{3}~\OGGEP{}$: instruments and the bulk-to-boundary dictionary}
\label{app:bridge_OGGEP}

A measurement is a quantum instrument $\{\mathcal{I}_m\}$, a family of
completely positive maps with $\sum_m \mathcal{I}_m$ trace-preserving
\citep{DaviesLewis1970,Kraus1971,NielsenChuang2010}; outcome
probabilities and post-measurement states are
$p(m) = \Tr[\mathcal{I}_m(\rho)]$ and
$\rho_m = \mathcal{I}_m(\rho)/p(m)$. Every instrument admits a
Stinespring dilation \citep{Stinespring1955}. $\AX{3}~\OGGEP{}$
elevates this structure: an admissible bulk theory must induce a
specific instrument family on the boundary via the operational
dictionary $\Dmap$. In the AdS/CFT prototype
\citep{Maldacena1998,Witten1998,Gubser1998} the dictionary connects
bulk fields $\phi(r,x)$ at $r \to \infty$ to boundary correlators
$\langle \mathcal{O}(x)\cdots\rangle$ via the GKPW relation
\begin{equation}
Z_{\rm Bulk}[\phi_0]
= \Bigl\langle\exp\!\int d^d x\, \phi_0(x)\, \mathcal{O}(x)\Bigr\rangle_{\rm Bdy}.
\label{eq:gkpw_app}
\end{equation}
$\AX{3}~\OGGEP{}$ generalizes the target space of the dictionary from
correlators to \emph{operational measurement objects} (POVMs,
instruments, records). See Part~II~\S\ref{sec:AX3} and~\citep{ArisakaA5GDOS}.

\subsection{$\AX{4}~\MSMF{}$: knob count and symmetry maximization}
\label{app:bridge_MSMF}

In standard-textbook practice, naturalness is the heuristic that small
parameters require enhanced symmetry \citep{THooftNaturalness1980}.
$\AX{4}~\MSMF{}$ strengthens this to a constitutional selection rule.
For a candidate theory $T$ with continuous parameter space $\Theta$, let
\begin{equation}
N_{\rm free}(T) \equiv \dim\!\bigl(\Theta / \!\sim_{\rm gauge}\bigr),
\qquad
\dim \mathrm{Sym}(T) \equiv \text{dimension of the symmetry group of } T,
\label{eq:knob_count_app}
\end{equation}
where $\sim_{\rm gauge}$ identifies gauge- and basis-redundant
parameters. $\AX{4}~\MSMF{}$ demands that admissible theories
\emph{maximize $\dim \mathrm{Sym}(T)$ while minimizing $N_{\rm free}(T)$}
subject to AX-1..AX-3 and POS-1..POS-7 admissibility. As a concrete
illustration, for a $\mathrm{U}(1)$ gauge factor the simultaneous
rescaling $A_\mu \to \lambda A_\mu$, $g \to g/\lambda$ leaves local
physics invariant; $\AX{4}~\MSMF{}$ together with $\POS{6}~\OGN{}$
forbids leaving $\lambda$ as a free continuous knob, demanding instead
a minimal integer/rational charge lattice
\citep{Dirac1931,GeorgiGlashow1974}.

\subsection{Worked derivation: $\AX{1}~\LC{} + \POS{4}~\CP{} \Rightarrow$ GKLS generator (sketch)}
\label{app:worked_GKLS}

The single most useful equivalence in the constitution-to-textbook
bridge is the one that turns the measurement problem from philosophy
into engineering. Assume a one-parameter family of maps
$\{\Phi_t\}_{t \ge 0}$ on density operators with:
(i) forward-time semigroup property $\Phi_{t+s} = \Phi_t \circ \Phi_s$
(forced by $\AX{1}~\LC{}$);
(ii) $\Phi_0 = \mathrm{id}$;
(iii) complete positivity and trace preservation at every $t \ge 0$
(forced by $\POS{4}~\CP{}$);
(iv) strong continuity in $t$.

Then a generator $\mathcal{L}$ exists with $\Phi_t = e^{t \mathcal{L}}$
and $\dot\rho = \mathcal{L}(\rho)$, and the Gorini-Kossakowski-
Sudarshan-Lindblad classification theorem
\citep{Gorini1976,Lindblad1976} forces $\mathcal{L}$ into the
form
\begin{equation}
\mathcal{L}(\rho) = -i[H, \rho] + \sum_k \Bigl(L_k\,\rho\,L_k^\dagger
- \tfrac{1}{2}\{L_k^\dagger L_k,\, \rho\}\Bigr),
\label{eq:gkls_appendix}
\end{equation}
with $H = H^\dagger$ and arbitrary Lindblad operators $L_k$. Once
$\AX{1}$ and $\POS{4}$ are accepted as constitutional axioms, the
structure of open-system evolution is \emph{forced}. The "measurement
problem" reduces to engineering: specify the instrument
$\{\mathcal{I}_m\}$ (or equivalently the Kraus operators / POVM), and
the update rule is fixed by the admissible channel class. The
explicit derivation chain $\T{7.1}$-$\T{7.7}$ that recovers the
textbook QM postulates (Born rule, L\"uders projection, decoherence,
einselection, Bell-CHSH violation) from this generator class lives in
\citep{ArisakaA5GDOS} and is exercised at graduate-textbook depth in
\citep{ArisakaA8QM}.

\section{Worked examples and reusable templates}
\label{app:more_examples}

This appendix is the technical machine room of the present paper.
Its first part (\S\ref{app:more_examples_examples})
collects worked examples from physics, chemistry, biology, and
engineering --- short illustrations that show what each
constitutional clause forbids or forces in a familiar laboratory or
modeling setting. Its closing part
(\S\ref{app:more_examples_template_MF},~\S\ref{app:more_examples_template_claim})
supplies two reusable templates that the reader can copy verbatim
when writing a new paper, model, or experimental proposal that is
intended to be admissible in the $\GGTheory{}$ framework. The MF
claim-contract template (\S\ref{app:more_examples_template_MF}) is
the one that turns a theorem into a registered falsifier; the
full-claim template (\S\ref{app:more_examples_template_claim}) is the
wrapping that makes a write-up cross-disciplinarily auditable.
Indeed, the cleanest use of the present appendix is as a copy-paste
source for the working scientist.

\subsection{Driven harmonic oscillator (linear \GDOS{}; exact solution)}
\label{app:more_examples_examples}
Consider the Ornstein--Uhlenbeck process $\dd y=-\kappa y\,\dd t+\sqrt{2\Theta}\,\dd W_t$.
The stationary distribution is Gaussian $\rho_\star(y)\propto e^{-\kappa y^2/(2\Theta)}$ and the \GFF{} ledger
of \citet{ArisakaA6Gi4} can be chosen so that \eqref{eq:F_monotone} holds exactly.

\subsection{Two-state gene regulation as a Markov jump process}
A minimal ``gene on/off'' model is a two-state Markov chain with transition rates controlled by metabolite concentrations.
Ports correspond to nutrient inflow and signaling molecules. This illustrates how ``\DGI{}'' becomes invariance under relabelings
of internal bookkeeping states, while MF corresponds to a kill criterion on switching statistics.

\subsection{Bayesian updating as a \GDOS{} flow (connection to FEP)}
Write a posterior density $\rho(\theta)\propto p(\text{data}\mid\theta)p(\theta)$.
Gradient flows on Kullback--Leibler divergence correspond to variational Bayesian inference.
This provides an explicit bridge between POS-\UEP\ and the free-energy principle literature \citep{Friston2010,CoverThomas2006}.

Consider a qubit with Hamiltonian $H=\tfrac{\omega}{2}\sigma_z$ and pure dephasing with rate $\gamma$.
The GKSL equation is
\begin{equation}
\dot{\rho}=-i[H,\rho] + \gamma\left(\sigma_z \rho \sigma_z - \rho\right).
\end{equation}
This map is CPTP (POS-4), respects covariance under the $U(1)$ generated by $\sigma_z$ (POS-1), and provides a concrete operational prediction: off-diagonal elements decay as $e^{-2\gamma t}$.
A falsifier is immediate: if an experiment finds coherence decay inconsistent with the predicted exponential law under controlled conditions, the proposed generator fails MF.

Consider $A \rightleftharpoons B$ with forward/backward rates $k_+$ and $k_-$.
Let $p_A,p_B$ be probabilities (or normalized concentrations).
A master equation is
\begin{equation}
\dot{p}_A = -k_+ p_A + k_- p_B, \qquad \dot{p}_B = k_+ p_A - k_- p_B,
\end{equation}
which preserves positivity and normalization (POS-4).
The stationary distribution satisfies detailed balance if the system is closed, but open chemostats break it and induce entropy production; stochastic thermodynamics provides a quantitative framework \citep{Seifert2012}.
This toy model illustrates how \GDOS{} treats chemical kinetics as open dynamics with an entropy/ledger interpretation.

\subsubsection{Quantum case: CPTP maps and GKSL generators}
A general quantum operation with outcomes $m$ is an \emph{instrument} $\{\mathcal{I}_m\}$: each
$\mathcal{I}_m$ is completely positive (CP), and $\sum_m \mathcal{I}_m$ is trace-preserving (TP). By the
operator-sum (Kraus) theorem \citep{Kraus1983,NielsenChuang2010}, each CP map admits
\begin{equation}
\mathcal{I}_m(\rho)=\sum_\alpha K_{m\alpha}\,\rho\,K_{m\alpha}^\dagger,
\qquad
\sum_{m,\alpha} K_{m\alpha}^\dagger K_{m\alpha}=\mathbf{1}.
\end{equation}
If we ignore conditioning on $m$ (i.e.\ we only track the unconditional evolution), the map
$\Phi(\rho)=\sum_m \mathcal{I}_m(\rho)$ is CPTP.

Assume now a Markovian time-continuous evolution $\{\Phi_t\}_{t\ge 0}$ with semigroup property
$\Phi_{t+s}=\Phi_t\circ\Phi_s$ and strong continuity. Under these assumptions, the generator
$\mathcal{L}=\lim_{t\to 0^+}(\Phi_t-\mathrm{id})/t$ exists and takes the GKSL form
\citep{Gorini1976,Lindblad1976,BreuerPetruccione2002}:
\begin{equation}
\label{eq:gksl}
\frac{\dd\rho}{\dd t}=\mathcal{L}(\rho)
=-\ii[H,\rho]+\sum_k\left(
L_k\rho L_k^\dagger-\frac12\{L_k^\dagger L_k,\rho\}
\right),
\end{equation}
where $H$ is Hermitian and $\{L_k\}$ are Lindblad operators. Equation~\eqref{eq:gksl} is the sharpest
expression of POS-4 in the quantum regime: it is the unique local-in-time, trace-preserving, CP generator
compatible with Markovianity.

\subsubsection{Classical case: positivity-preserving drift--diffusion}
For a classical probability density $\rho(x,t)$ on a state manifold $\mathcal{M}$, closure and positivity
mean
\begin{equation}
\rho(x,t)\ge 0,\qquad \int_{\mathcal{M}}\rho(x,t)\,\dd\mu(x)=1.
\end{equation}
The broadest local-in-time evolution preserving these constraints is a continuity equation
\begin{equation}
\partial_t\rho+\nabla_A J^A=0,
\qquad
J^A = \rho\,b^A - D^{AB}\nabla_B\rho,
\end{equation}
where $D^{AB}\succeq 0$ is a diffusion tensor. In coordinate form this is a Fokker--Planck equation.
The \GDOS{} ``gradient + curl + noise'' decomposition corresponds to writing the drift as
\begin{equation}
b^A = -M^{AB}\nabla_B F + u^A,
\qquad
M^{AB}\succeq 0,\qquad \nabla_A u^A=0 \ \text{(in a suitable measure)},
\end{equation}
where the $-M\nabla F$ term is dissipative (gradient sector) and $u^A$ is the solenoidal/circulating
sector.

\subsection{Monotonicity of the \GFF{} ledger in the gradient sector}
Assume the purely gradient \GDOS{} flow (no solenoidal drift and noise treated as fluctuation consistent)
\begin{equation}
\dot y^A = -M^{AB}(y)\,\partial_B F(y),
\qquad M^{AB}\succeq 0.
\end{equation}
Then along a trajectory $y(t)$,
\begin{equation}
\frac{\dd F}{\dd t}=\partial_A F\,\dot y^A
=-\partial_A F\,M^{AB}\,\partial_B F \le 0,
\end{equation}
with equality iff $\partial_A F=0$ on the support (a critical point). This is the finite-dimensional
prototype of the Lyapunov statement for $F_{\mathrm{\GFF{}}}$ in relaxational sectors and is the technical backbone
of the ``FEP-like'' corollary used throughout \GDOS{} \citep{Seifert2012,Friston2010}.

Let a group $G$ act on states by a representation $U(g)$. \DGI{} requires equivariance of the dynamics:
\begin{equation}
\Phi_t(U(g)\rho)=U(g)\Phi_t(\rho),\qquad\forall g\in G,\ \forall t\ge 0.
\end{equation}
Differentiating at $t=0$ yields the generator-level condition
\begin{equation}
\mathcal{L}(U(g)\rho)=U(g)\mathcal{L}(\rho).
\label{eq:dgi_generator}
\end{equation}
In the GKSL form~\eqref{eq:gksl} this is satisfied if
\begin{equation}
U(g)HU(g)^\dagger = H,\qquad
U(g)L_kU(g)^\dagger = \sum_\ell D_{k\ell}(g)\,L_\ell,
\end{equation}
i.e.\ $H$ is invariant and the Lindblad operators transform within a representation. The same idea holds
classically: the drift, diffusion, and ledger must transform covariantly so that physical predictions are
representation-independent.

Let $\mathrm{Sym}(\mathcal{L})$ be the symmetry group of the generator/kernel and $\mathrm{Sym}(\mu_\star)$ the
symmetry of a particular invariant measure $\mu_\star$ (fixed point, limit cycle measure, SRB measure).
\emph{Geometric spontaneous symmetry breaking} is the statement
\begin{equation}
\mathrm{Sym}(\mathcal{L}) \supsetneq \mathrm{Sym}(\mu_\star).
\end{equation}
In closed QFT, $\mu_\star$ is often the vacuum state; in open systems, $\mu_\star$ is an attractor measure.
The key generalization is conceptual and mathematical: the ``phase'' is an attractor of a CP/positive
runtime, not only a minimum of a static potential.

\subsubsection*{Example 1 (physics): double-well + noise}
Consider $F(y)=\lambda (y^2-a^2)^2$ on $\mathbb{R}$ with gradient flow plus weak noise. The law is invariant
under $y\mapsto -y$. For sufficiently weak noise, the long-time distribution concentrates near either
well, yielding two symmetry-related attractor measures with reduced symmetry. This is the open-system
prototype of SSB, with the role of ``vacuum selection'' played by attractor selection.

\subsubsection*{Example 2 (chemistry/biology): bistable switch}
In a bistable chemical reaction network (e.g.\ positive feedback), the \GDOS{} state is a concentration
vector. Two stable fixed points correspond to two phenotypes. The dynamical law may respect a symmetry
(permutation of equivalent species), while the realized phenotype breaks it.

\subsubsection*{Example 3 (mind/engineering): attractor networks}
In Hopfield-like networks, the state is a vector of neuron activities; the energy function is a Lyapunov
function for the deterministic dynamics. Multiple stored patterns are attractors. ``Memory recall'' is then
a controlled \GSSB{} event: the system falls into one attractor basin given a partial cue.

Let the state be $x\in\mathbb{R}$ with \GFF{} ledger $F(x)=\frac{1}{2}k x^2$.
Take mobility $M=1$ and diffusion $D=T$ (Einstein relation at ``temperature'' $T$).
The \GDOS{}/Fokker--Planck equation becomes
\begin{equation}
\partial_t \rho(x,t) = \partial_x\!\left(\rho\,\partial_x F\right) + T\,\partial_x^2 \rho
= \partial_x\!\left(kx\,\rho\right) + T\,\partial_x^2 \rho.
\label{eq:OU}
\end{equation}
The stationary solution is the Gibbs distribution
\begin{equation}
\rho_\ast(x)=\frac{1}{Z}\exp\!\left(-\frac{F(x)}{T}\right)
=\sqrt{\frac{k}{2\pi T}}\,\exp\!\left(-\frac{kx^2}{2T}\right).
\end{equation}
This illustrates the \UEP{} idea in the simplest setting: the open dynamics relaxes toward a distribution controlled by the scalar ledger $F$.

Take $F(x)=\lambda(x^2-v^2)^2$, symmetric under $x\to -x$.
In the weak-noise limit $T\ll \lambda v^4$, $\rho_\ast(x)$ has two peaks near $\pm v$.
A single long trajectory will spend long times near one well and occasionally switch (Kramers transitions).
In the strict zero-noise limit, the system selects one attractor: symmetry of the \emph{law} exceeds symmetry of the \emph{realized phase}.
This is the \GDOS{} form of \GSSB{}.

Consider a qubit with Hamiltonian $H=\frac{\omega}{2}\sigma_z$ and dephasing $L=\sqrt{\gamma}\sigma_z$.
The GKSL generator is
\begin{equation}
\dot\rho = -i[H,\rho] + \gamma\left(\sigma_z\rho\sigma_z-\rho\right).
\end{equation}
Under a phase (``gauge'') rotation $U(\theta)=e^{-i\theta \sigma_z/2}$, we have $U\sigma_z U^\dagger=\sigma_z$.
Therefore $\mathcal{L}$ satisfies \DGI{} covariance (Part~III, Eq.~\eqref{eq:DGI_covariance}).
This simple model is a concrete illustration of POS-1/\DGI{} in standard quantum dynamics.

\subsection{Example: GKSL covariance (a minimal \DGI{} check)}
Given a symmetry group with unitary representation $U(g)$, define the adjoint action $\mathcal{U}_g(\rho)=U(g)\rho U(g)^\dagger$.
A minimal \DGI{} test is to verify
\begin{equation}
\mathcal{U}_g^{-1}\circ \mathcal{L}\circ \mathcal{U}_g=\mathcal{L}
\quad\text{for all } g.
\end{equation}
In a basis where Lindblad operators form irreducible multiplets, this reduces to checking that the set $\{L_k\}$ is closed under conjugation by $U(g)$, and that the Kossakowski matrix respects the symmetry.

\subsection{Example: open-system phase selection as \GSSB{} (double-well sketch)}
Consider a one-dimensional order parameter $x$ with an even ``free energy'' landscape
\begin{equation}
F(x)=\frac{\lambda}{4}(x^2-v^2)^2,
\end{equation}
and overdamped Langevin dynamics $\dot x = -\mu \,F'(x)+\sqrt{2\Theta}\,\eta(t)$.
For small noise $\Theta$, the stationary density concentrates near $x=\pm v$.
The equation is $x\to -x$ symmetric, but any single long-lived realization picks one well: this is the simplest cartoon of \GSSB{}.
In a \GDOS{} setting, the relevant generalization is to high-dimensional state manifolds with symmetry-related attractors and port-induced biases.

\subsection{Template: MF ``claim contract''}
\label{app:more_examples_template_MF}
A practical MF write-up for a theorem should list:
(i) the \SIP{} (state variables, symmetry group, ports),
(ii) the predicted invariant (dimensionless ratio, scaling exponent, spectral knee),
(iii) the measurement protocol (instrument, calibration, analysis window),
(iv) the falsifier (numerical criterion that would reject the prediction).
This template is intended to make ``being wrong'' as explicit as ``being right.''

\paragraph{How to use this template.}
The four entries should be filled in concretely for the discipline
the paper is written for. For a particle-physics prediction
(e.g.\ the $\GG{}$-lock cluster mass $\Mlock \approx 3.85\,\TeV{}$
of \citep{ArisakaA9SM}), the \SIP{} names the bulk gauge group and
the OGN-5 ledger; the invariant is the predicted resonance mass; the
protocol is the LHC or future-collider search; and the falsifier is
the registered absence of the resonance above a stated luminosity
threshold. For a cosmology prediction (e.g.\ $\Omega_m = 1/\pi$ of
\citep{ArisakaA10Cosmos}), the protocol is the Planck-class CMB +
BAO + supernova combination, and the falsifier is the registered
deviation from $1/\pi$ at stated confidence. For a
condensed-matter or chemistry application, the \SIP{} replaces
particle-content with concentration/order-parameter variables, and
the protocol names the spectroscopy or kinetics experiment; the
discipline of the contract is identical in every case. A common
pitfall is to declare the kill criterion only after the data are
in: $\POS{7}~\MF{}$ requires the falsifier to be specified at the
moment the theoretical claim is published.

% --- Added in final polish ---

\subsection{Template: writing a falsifiable claim in \GGTheory{}\ form}
\label{app:more_examples_template_claim}
A practical ``textbook'' template for statements in this program is:

\begin{enumerate}[leftmargin=2em]
\item \textbf{Claim (invariant):} specify a dimensionless invariant or relation (e.g.\ a ratio, exponent, or
scale-covariant law).
\item \textbf{Model instance (\SIP{}):} specify $(\mathcal{M},G,\text{ports},F)$ and the admissibility checks
(POS-1--POS-6).
\item \textbf{Operational registration (\OGGEP{}):} specify the instrument, preprocessing, estimator, and
error model.
\item \textbf{Kill criterion (MF):} specify in advance what numerical outcome falsifies the claim.
\end{enumerate}

This template is deliberately domain-agnostic and is meant to make cross-discipline work auditable.

\paragraph{Worked instantiation: the kaxion dark-matter oscillation contract.}
As a single concrete example of how to populate each of the four
slots, the kaxion oscillation claim $\Delta\nu/\nu = 1/35$ of
\citep{ArisakaC1Kaxion} fills the template as follows.
\emph{(1) Claim:} the dimensionless ratio $\Delta\nu/\nu = 1/35$,
an OGN-5 integer fraction with no continuous freedom.
\emph{(2) SIP:} the 6D bulk $\cM_6 = \cM_4\times I_r\times S^1_\theta$
with the four-factor composite gauge group; the relevant port is
the dark-matter halo coupling at the cavity location; the relevant
ledger is the standard halo density $0.4\,\mathrm{GeV}/\mathrm{cm}^3$.
\emph{(3) Operational registration:} a haloscope-style microwave
cavity tuned to the kaxion mass, with quantum-limited amplifier
and frequency spectrum analyzer as the registered observable; the
estimator is the spectral peak position; the error model is the
thermal-and-amplifier noise floor at integration time
$\sim 10^{6}\,\mathrm{s}$ per tuning step.
\emph{(4) Kill criterion:} the registered observation of a narrow
spectral peak at $\Delta\nu/\nu \neq 1/35$ at $5\sigma$ confidence
over a control region of width $\geq 10\sigma_\nu$. After all, a
claim that does not collapse one or more constitutional clauses on
registered refutation is not a $\GGTheory{}$ claim --- it is a story.

\subsection*{Closing remarks on Part~VIII}

The apparatus in this Part exists for one purpose: so that a reader does
not have to take our word for anything.

That is worth saying plainly, because appendices are usually treated as
where the tedious material is put to keep it out of the argument. Here
they carry a share of the argument. The abbreviations table is what
makes a document with thirteen program acronyms navigable by someone
meeting them for the first time. The glossary is what lets a reader who
has forgotten what $K$ stands for recover it in ten seconds rather than
abandoning the page. The elaborations from each axiom show what each
clause forbids, which is the only way to see whether a clause is doing
work or merely sounding well. And the worked examples let a reader run
the machinery on something small enough to check by hand.

We would point in particular at the two examples that are not from
physics. A two-state gene switch and a Bayesian update are not
decoration; they are the cheapest available test of the cross-domain
claim this program keeps making. If the boundary law really is the same
law whether the open system is a damped oscillator or a cell deciding
whether to transcribe, then it should be possible to write both in the
same formalism and watch the same structure appear. It is, and it does,
and a reader from the life sciences who suspects physicists of
metaphor-mongering should start there and judge for themselves.

A general remark, since this is the last Part before the objections.
Every technical appendix is an admission that the main text asked the
reader to trust something. The honest response to that is not to write
a more confident main text; it is to supply the means of checking. A
paper that cannot be audited is asking for belief, and belief is not
what physics runs on. The material here is our attempt to ask for as
little of it as possible.

Part~IX takes the opposite approach to the same end. Instead of
supplying the means to check us, it hands the floor to the readers most
likely to conclude that we are wrong.


\clearpage
\GGPart[Common Objections]{Common Objections and Responses}{The strongest objections from each community --- particle theorist, gravity researcher, mathematician, philosopher, string theorist --- answered, with pointers into the paper}
\label{sec:common_objections}

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

Every foundational paper attracts a predictable
family of adversarial readings. The present paper would
not be honest scholarship if it did not name those readings
explicitly and answer each one. We have already met some of them
in passing throughout Parts~II--V; here we collect them in one
place, supply for each the constitutional clause and the cohort
proof location that addresses it, and acknowledge what remains
open. The first six objections come from the six communities most
likely to push back on first reading --- the particle theorist,
the gravity researcher, the quantum-foundations specialist, the
mathematical physicist trained in axiomatic QFT, the philosopher
of science, and the string theorist. The next twenty objections come from the harsher, more technical reading that a particle-theory postdoc, a phenomenologist, and a senior string
theorist would bring on a careful second pass --- on the integer ledger, the collider and cosmological predictions, decoherence, the swampland, and the cross-domain reach. The final twenty-eight press hardest on the foundations themselves: the independence and necessity of the four axioms, the status of the seven postulates, the ``constitution'' framing and the descent claim, the constitutional classification of twentieth-century physics, and the meta-status of an axiomatic framework that reproduces known physics as theorems. We collect all of them so that the constitution arrives in the literature with its strongest adversarial readings already named.

\paragraph{Cohort cross-reference: the minimum critical-reading set.}
A1-Constitution is intentionally minimal. Its role in the
\GGTheory{} program is to state the four axioms and seven
postulates, to name the eleven Principal Theorems, and to forward
every proof to the companion paper where it actually lives. The
constitutional discipline is honest: an axiomatic document that
includes its own proofs becomes too long to audit at the
constitutional layer. A serious objection to A1 therefore cannot
be evaluated from A1 alone; the companion papers are posted
together, and reach arXiv in a small number of staged
submissions, precisely so that the proof is one click away from
the constitution rather than a promise.

For an evaluator who arrives at A1 with a specific community's
concerns in mind, the \emph{minimum critical-reading set} ---
the smallest collection of cohort papers within which a serious
objection from that community can be answered --- is the
following:

\begin{itemize}[leftmargin=2.4em]
\item \textbf{For particle-physics / string-theory / quantum-gravity objections:}
 \ATwo{} \citep{ArisakaA2GG6} (the 6D bulk uniqueness theorem;
 the four-factor composite gauge group; the $\OGNfive{}$ Diophantine
 forcing) together with \AThree{} \citep{ArisakaA3BI6} (the
 Hopkins-Singer differential-cohomology proof of $\BIsix{}$ global
 anomaly freedom; the trace-ratio lock $|a/c| = |a/d| = 5/2$).
\item \textbf{For quantum-foundations / measurement-problem objections:}
 \AFive{} \citep{ArisakaA5GDOS} (the boundary law $\cL_{\GDOS{}}$; the
 $\T{7.1}$--$\T{7.7}$ derivation chain that recovers the textbook
 QM postulates --- Born rule, L\"uders projection, decoherence,
 einselection, Bell--CHSH violation --- as theorems of $\AX{3}~\OGGEP{}$
 together with $\POS{4}~\CP{}$) together with \AEight{}
 \citep{ArisakaA8QM} (the graduate-textbook reformulation in
 $\GGTheory{}$ language).
\item \textbf{For Standard-Model / cosmology objections:}
 \ANine{} \citep{ArisakaA9SM} (the 28 Standard-Model continuous
 parameters as theorems of the $\OGNfive{}$ ledger) together with
 \ATen{} \citep{ArisakaA10Cosmos} (the cosmological partitions
 $\Omega_m = 1/\pi$, $\Omega_\Lambda = 1 - 1/\pi$, $\Omega_b = 1/(2\pi^2)$,
 $\Omega_{\rm DM} = 1/\pi - 1/(2\pi^2)$ from the bulk geometry).
 The closing capstone \AEleven{} \citep{ArisakaA11Origin} carries the
 \GRIP{} cascade onward from there, and is in preparation; an
 evaluator of the parameter-collapse claim does not need it.
\end{itemize}

An evaluator who has not read at least the relevant combination
above is not yet in a position to form a substantive judgment on
the corresponding objection. This is not gatekeeping; it is the
architecture of a twenty-two-paper foundational program. Each
combination supplies a tight forward-implication chain that can be
read incrementally: A2 assumes the A1 constitution and produces
the bulk theorems; A3 assumes the A2 bulk and produces the anomaly
proof; A5 assumes the A2+A3 bulk and produces the boundary law;
A8 assumes the A5 boundary law and produces the QM reformulation;
A9, A10, and A11 each assume the upstream cohort and produce
specific phenomenological / cosmological / capstone consequences.
We invite the reader to follow the chain in the order that matches
the objection at hand, and we hold ourselves to the discipline of
naming, for every claim in A1, exactly where the proof lives.

%% =====================================================================
%% §15.1 — Six original community objections (lightly polished)
%% =====================================================================


%% =====================================================================
%% Cluster 0 -- For the curious reader
%% =====================================================================
\bigskip
\section{For the curious reader: the big questions, plainly answered}
\label{sec:qa-curious}

The twenty-five questions that follow are not a specialist's. They are
the ones a thoughtful reader without a physics degree actually carries
into a paper that claims to write a constitution for nature: whether
physics needs axioms at all, who is entitled to supply them, whether a
theory with no adjustable dials is admirably rigid or merely unable to
learn, and whether a short list of whole numbers explaining a great
deal ought to be believed or filed with the other short lists that
explained a great deal and were wrong. Several of these have been
settled in popular science in directions this paper does not take, and
a reader is owed plain answers before the community objections begin.
They are met first, and plainly.

The hardest of the twenty-five is the seventh --- why a demand for
internal consistency should have any authority over a world that was
under no obligation to be consistent with anything. The response
concedes the difficulty rather than dissolving it, and says which part
of it the paper can carry and which part it cannot. Where a claim rests
on a step that is not closed, the response says so and names the step;
where the honest answer is that the framework does not reach the
question, it says that too.


\bigskip
\noindent\textbf{Question 1 (Q-Constitution-1) (on the curious reader)}\par\nobreak\smallskip
\noindent\textbf{``Physics already has laws. Why on earth does it need a constitution?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Because ``law'' in physics has quietly been doing two different
jobs, and separating them is worth the odd-sounding vocabulary. Some
statements are about how a particular thing behaves: an inverse square
here, a coupling strength there. Those are what most people mean by a
law of physics, and they are found by measurement. Other statements are
about what any acceptable description of nature must look like before
you write down a single equation --- that causes precede effects, that
a theory should not depend on which coordinates a person happened to
choose, that a description must remain a description when the observer
is inside the system rather than outside it. Those are not measured.
They are the conditions a candidate has to satisfy to be considered at
all. \\ Physics has always had them and has usually left them
unwritten, which works well until two of them conflict, or until
somebody proposes a framework that quietly drops one and nobody
notices. A constitution is simply the second kind written down, in
order, with the consequences of each stated. The word is borrowed
deliberately: a constitution does not tell a country what to do, it
constrains what may be enacted, and it can be pointed at when something
is enacted that should not have been. That is exactly the office these
four statements hold in this paper.
\end{objqabody}

\bigskip
\noindent\textbf{Question 2 (Q-Constitution-2) (on the curious reader)}\par\nobreak\smallskip
\noindent\textbf{``Isn't an axiom just an assumption you have decided not to defend?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} It is an assumption you have decided to \emph{declare}, which is a
different and much more exposed position. An undefended assumption that
sits unnamed inside a calculation can never be attacked, because a
critic cannot find it; the calculation simply comes out the way it
comes out, and the assumption is protected by being invisible. An axiom
is the same assumption dragged into the open, given a number, and
placed where every result that depends on it can be traced back.
\\ The honest form of your question is therefore not ``are these
undefended?'' --- they are, in the sense that they are not derived from
anything more basic in this paper --- but ``are they few, are they
stated, and can I find every place they are used?'' Those questions
have answers here, and the answers are the whole content of Part~I.
Four axioms, seven postulates, and a dependency map. A reader who
rejects one of the four is not left arguing about taste: they can
follow the number downstream and see precisely which results fall over.
\\ It is worth adding what would make the criticism land. If the axioms
had been chosen \emph{after} the results were known, tuned so that the
desired conclusions would follow, then declaring them would be theater.
The defense against that is the ordering, and the ordering is checkable:
each axiom is stated before anything is derived from it, and none of
them mentions a quantity that only appears later.
\end{objqabody}

\bigskip
\noindent\textbf{Question 3 (Q-Constitution-3) (on the curious reader)}\par\nobreak\smallskip
\noindent\textbf{``Who gets to write a constitution for nature? What makes this one legitimate?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Nobody gets to, and that is the point. The word invites a
misreading and it is worth killing it early: this is not legislation.
Nature is not bound by anything written in a paper, and a constitution
in the political sense derives its authority from the consent of the
governed, of which the universe has offered none. \\ What the word is
doing here is naming a \emph{function}, not a source of authority. In a
legal system the constitution is the layer against which everything else
is checked; ordinary statutes are valid or invalid depending on whether
they conform. The four statements in this paper occupy that position in
the argument: everything downstream is checked against them, and a
proposed structure that violates one is rejected without further
examination. Their legitimacy comes from precisely one place --- whether
what follows from them turns out to be true. \\ So the answer to ``who
gets to write it'' is: anyone at all, and it costs nothing. Writing one
down is cheap. The expensive part is the exposure that follows, because
a declared constitution can be broken by a single experiment in a way
that an undeclared set of habits cannot. If a compact star does not
show the predicted shift, or a fourth family of matter appears, the
document does not get amended quietly. It fails, and it fails in
public, and it says in advance which clause failed.
\end{objqabody}

\bigskip
\noindent\textbf{Question 4 (Q-Constitution-4) (on the curious reader)}\par\nobreak\smallskip
\noindent\textbf{``Why four axioms? Why not three, or seven, or one?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Because four is what survived the attempt to manage with fewer, and
the attempt is recorded rather than asserted. \\ The honest position is
that the number is not derived from anything --- there is no theorem in
this paper saying ``the correct number of axioms is four.'' What there
is instead is an independence argument: for each of the four, a
demonstration that dropping it does not merely weaken the framework but
admits structures the framework is supposed to exclude, and that it
cannot be recovered from the other three. That is the standard test in
mathematics for whether a list is minimal, and it is the test applied
here. A reader who thinks the list should be three is welcome to name
which one is redundant and show the derivation; the independence
section is where that argument would have to be won or lost. \\ The
opposite question is worth asking too, because it is the one that
should worry a proponent more. Not ``why so many?'' but ``why so few?''
--- a suspiciously short list is exactly what one would expect from
somebody who had quietly folded three assumptions into the phrasing of
one. The defense is that each of the four is stated in a form short
enough to be read in a sentence, and that the paper points at where
each is used rather than gesturing at it. Compression that hides
content shows up as a clause nobody can locate the consequences of. If
a reader finds one here, that is a real finding and worth reporting.
\end{objqabody}

\bigskip
\noindent\textbf{Question 5 (Q-Constitution-5) (on the curious reader)}\par\nobreak\smallskip
\noindent\textbf{``What happens to everything downstream if one of the four turns out to be wrong?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Different things depending on which, and the paper is specific
about it rather than treating the four as an undifferentiated block.
\\ Causality is the load-bearing one: essentially nothing survives its
loss, because the entire construction is about how a description
propagates from a larger arena to a smaller one and there is no
propagation without an order of events. The equivalence between
geometric and gauge structure is nearly as severe --- drop it and the
four forces stop being aspects of one object and go back to being four
objects, which is the position physics was in before this program
started and is a coherent position to hold. The observational clause is
different in character: lose it and the framework survives as
mathematics but stops making contact with anything an instrument does,
which is a way of being wrong that is worse than being refuted. And the
minimality clause is the one whose loss would be least visible and most
corrosive, because without it the integer ledger is no longer forced
and the framework quietly acquires adjustable dials while continuing to
claim it has none. \\ This is not a comfortable answer and it is not
meant to be. A framework in which every axiom is equally
dispensable is a framework where nothing is doing any work. The value
of asking your question is that it separates the clauses that would end
the program from the clauses that would merely wound it, and a reader
is entitled to know which is which before deciding how much attention
to pay.
\end{objqabody}

\bigskip
\noindent\textbf{Question 6 (Q-Constitution-6) (on the curious reader)}\par\nobreak\smallskip
\noindent\textbf{``You keep saying things are derived rather than postulated. Isn't that just moving the postulate somewhere I can't see it?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Sometimes, yes --- and it is the failure mode this program is most
at risk of, so it is worth being exact about when the complaint is fair.
\\ It is fair whenever the thing said to be derived is derived from
something introduced for the purpose. If a quantity is explained by a
structure whose only job is to explain that quantity, nothing has been
gained and the accounting has simply been moved. The test is whether
the explaining structure was already needed for something else. Here
the arena, its dimension and its symmetry group are fixed before any
of the derived quantities appear, and are fixed by requirements that
have nothing to do with them. When a whole number then falls out of a
consistency condition imposed on that arena, it has not been moved; it
has been produced by machinery that was already standing. \\ The
complaint is not fair as a general objection to derivation, because if
it were, no explanation of anything would ever count. Every derivation
in the history of physics rests on something undeclared at some level.
What distinguishes an honest one is that the list of things it rests on
is shorter than the list of things it produces, and both lists are
written down. \\ Where the criticism does land in this paper, it lands
on the inputs themselves. Six dimensions is not a small assumption. The
paper does not derive it and says so. A reader who finds that too
expensive a starting point is making a reasonable judgment, and it is
the right place to make it --- at the top of the list, in the open,
rather than three hundred pages down.
\end{objqabody}

\bigskip
\noindent\textbf{Question 7 (Q-Constitution-7) (on the curious reader)}\par\nobreak\smallskip
\noindent\textbf{``Mathematics is full of consistent systems that describe nothing at all. Why should consistency have any authority over the actual world?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} This is the hardest question in the set and it does not have a
clean answer. It should be said plainly rather than managed. \\ The
worry is exact. Consistency is cheap: one can write down any number of
internally impeccable structures, and almost all of them are about
nothing. From the fact that a framework hangs together, nothing
whatever follows about whether nature implements it. Any argument of
the form ``it must be so, because otherwise the mathematics would not
close'' is, taken alone, an argument about mathematics. \\ What can be
said is narrower, and it is the whole of what this paper claims.
Consistency is not being used here to establish that the world is a
certain way. It is being used to \emph{eliminate}. Given an arena that
is assumed, not derived, the consistency requirements admit exactly one
assignment of the integers rather than a family of them --- and that is
a statement about a search space, not about nature. Whether nature is
in that search space at all is decided by measurement and by nothing
else. The framework's entire empirical exposure lives in that gap. \\
There is a historical reason to take the method seriously without
taking it as proof. Anomaly cancellation, which is exactly a demand for
internal consistency, has repeatedly narrowed the field of admissible
theories in particle physics, and the narrowing has survived contact
with data. That is an encouraging track record and it is not an
argument. The honest summary is that consistency is a powerful filter
of unusual reach and no kind of evidence at all, and a reader who keeps
those two facts separate will read this paper correctly.
\end{objqabody}

\bigskip
\noindent\textbf{Question 8 (Q-Constitution-8) (on the curious reader)}\par\nobreak\smallskip
\noindent\textbf{``You claim there are no free parameters, and then you start from a handful of whole numbers. Isn't that the same thing wearing a different hat?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} It would be, if the numbers could be adjusted. The distinction the
paper is drawing is not between ``few parameters'' and ``many'' but
between quantities that can absorb a disagreement and quantities that
cannot. \\ A continuous parameter is a shock absorber. When a
measurement comes in slightly off, the parameter moves a little, the
fit is restored, and the theory survives --- a little weaker each time,
and with no single moment at which anyone can say it was refuted. That
is the mechanism by which frameworks become unfalsifiable without
anybody deciding to make them so. A whole number cannot do this. It is
three or it is four. There is no small adjustment available, because
there is no such thing as a small change to an integer. A framework
that has been forced onto integers has given up the ability to retreat,
and that loss is the entire point of forcing it. \\ Your objection has
a sharper form which the paper takes seriously and which is answered
at length among the community objections: even if the integers cannot
be tuned, they were still \emph{chosen} --- somebody had to write down
that particular ledger. The answer is that they were not chosen; they
are the unique solution of a closure condition, and uniqueness is
proved rather than asserted. Whether one believes that proof is a
technical question with a technical answer, and it is the right place
for a skeptical reader to spend their effort.
\end{objqabody}

\bigskip
\noindent\textbf{Question 9 (Q-Constitution-9) (on the curious reader)}\par\nobreak\smallskip
\noindent\textbf{``Every generation of physicists thinks it is close to a final theory. Why should this be any different?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} It should not be treated as different, and treating it as different
would be the error. The correct response to a claim of this shape is
not enthusiasm and not dismissal but a demand to see the exposure. \\
The history is genuinely discouraging. Confident announcements that
physics was nearly finished have been made at intervals for a century
and a half, and the announcements were made by serious people with good
reasons. What distinguishes the episodes that were merely premature
from the episodes that were empty is not how convinced the proponents
were --- they were uniformly convinced --- but whether the claim could be
checked and what happened when somebody checked it. \\ So the useful
question is not whether this framework feels different from the inside.
It is what it forbids. A framework that forbids nothing is not close to
being a final theory; it is not a theory. This one forbids a fourth
family of matter, forbids a compact-star mass-radius relation without a
particular small correction, and forbids a set of laboratory outcomes
listed by name in the assessment. Any one of those going the wrong way
ends it. \\ There is also a difference of ambition worth naming, and it
runs the other way from your question. This paper does not claim to be
a final theory. It claims to be a constitution --- a statement of what
an acceptable theory must satisfy --- which is a smaller and more
checkable thing. The distinction matters, because the confident
announcements that aged worst were the ones that claimed completeness
rather than constraint.
\end{objqabody}

\bigskip
\noindent\textbf{Question 10 (Q-Constitution-10) (on the curious reader)}\par\nobreak\smallskip
\noindent\textbf{``Isn't our universe just one bubble among countless others, and we simply happen to live in one where the numbers work?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} It might be, and nothing in this paper rules it out. What the paper
does is remove the main reason people reach for the idea. \\ The
multiverse became attractive because of a specific problem: the
constants of nature look arranged, and if they could have taken any
values then the arrangement demands an explanation. An enormous
ensemble of universes with different constants supplies one --- most are
sterile, we are necessarily in a rare one, and no further explanation
is owed. That is a coherent move and it has been made by careful
people. \\ It rests, though, on a premise that is almost never
examined: that the constants \emph{could} have been otherwise. If they
could not --- if the arrangement is forced by consistency and there is
exactly one admissible assignment --- then the observation that started
the argument is no longer surprising, and the ensemble is doing no work.
You do not need an infinity of universes to explain why a triangle's
angles sum to what they do. \\ So this framework does not argue against
the multiverse. It argues that the multiverse's principal job may not
need doing, and it does so in a way that can be checked: if the ledger
is not unique, the argument fails and the ensemble is back on the table.
It is worth adding the obvious caution. Sitting where we are, one
universe is what we can measure. A hypothesis that adds unobservable
universes to explain the one we have should be asked what it forbids,
and the answers so far have been thin.
\end{objqabody}

\bigskip
\noindent\textbf{Question 11 (Q-Constitution-11) (on the curious reader)}\par\nobreak\smallskip
\noindent\textbf{``The universe looks extraordinarily fine-tuned for life. Doesn't that point to a designer?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} It points to something requiring explanation, which is not the same
thing, and the step from the first to the second is where the whole
argument lives. \\ Take the observation seriously first, because it is
real. Change certain quantities by a small fraction and there are no
stable nuclei, or no chemistry, or no structure at all. Anyone who
looks at that and feels the pull of design is responding to something
genuine rather than being naive. \\ The design inference has a premise
it shares with the multiverse inference, and it is the same premise:
that the quantities could have been otherwise. Fine-tuning is only
remarkable if the dial existed and was set. If the arrangement is
forced --- if there is exactly one assignment consistent with the
structure --- then asking why it was chosen is like asking who arranged
for the diagonal of a unit square to be what it is. Nobody had the
option. \\ This is not an argument against the existence of a designer,
and it should not be recruited as one. It is an argument that the
fine-tuning observation, specifically, does not establish one. A reader
who believes on other grounds is untouched by anything in this paper,
and a reader who does not is likewise untouched. What has changed is
narrower: one particular argument, which has carried a great deal of
weight in popular discussion, loses its premise if the framework is
right --- and whether the framework is right is settled by a
mass-radius measurement and a family count, not by philosophy.
\end{objqabody}

\bigskip
\noindent\textbf{Question 12 (Q-Constitution-12) (on the curious reader)}\par\nobreak\smallskip
\noindent\textbf{``If every constant is forced by geometry, is my future already written? What happens to free will?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Nothing in this paper touches your future, and it is worth being
careful about why not, because the inference feels compelling and is
not sound. \\ What is claimed to be forced here is the \emph{rule
book}: the number of families of matter, the values of certain
constants, the structure of the arena. What is not claimed to be forced
is any particular history within it. The distinction is the one between
the rules of chess and the game you played last Tuesday. That the rules
admit no alternative says nothing whatever about which of the enormous
number of permitted games occurred, and a framework that fixes the
former has not begun to address the latter. \\ There is a second point
that cuts deeper and is less often made. Even a completely
deterministic rule book does not deliver a written future in any sense
you could be troubled by, because the only object capable of computing
what the rules imply about a system as complicated as a person is a
system at least that complicated, running at least that long. There is
no shortcut, no vantage point, and no book. A universe can be lawful
all the way down and still be, in the only sense that matters to
anybody living in it, unwritten until it happens. \\ And the honest
limit: this paper has nothing to say about agency. It is a constitution
for physical description. What free will is, and whether the concept
survives contact with physics of any kind, is a question the framework
does not reach and does not pretend to.
\end{objqabody}

\bigskip
\noindent\textbf{Question 13 (Q-Constitution-13) (on the curious reader)}\par\nobreak\smallskip
\noindent\textbf{``Why is there something rather than nothing? Does your framework even touch the deepest question?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} It does not, and any suggestion otherwise would be a sales pitch.
\\ What the framework addresses is conditional throughout: \emph{given}
an arena with certain properties, what follows. Every result in this
paper has that shape. The question of why there is an arena at all ---
why anything exists to be described --- sits outside the whole
construction, and no rearrangement of the axioms reaches it. This is
not a gap that further work will close. It is a different kind of
question. \\ It is worth saying what a framework like this one can
legitimately contribute, because it is not nothing. If it is right,
then the space of possible somethings is far smaller than it looks: not
a vast landscape of coherent worlds from which one was somehow drawn,
but very nearly a single admissible structure. That converts the
question from ``why this, out of all the things there might have
been?'' into ``why anything?'' --- which is harder, older, and cleaner.
Narrowing a question is a real service and it is not an answer. \\ The
temptation here is to say something profound-sounding about nothingness
being unstable, or about the question being ill-posed. Both moves are
available, both have respectable defenders, and neither follows from
anything in this paper. The framework is silent, and a reader is better
served by being told so.
\end{objqabody}

\bigskip
\noindent\textbf{Question 14 (Q-Constitution-14) (on the curious reader)}\par\nobreak\smallskip
\noindent\textbf{``A short list of whole numbers explaining this much makes me suspicious. Shouldn't it make you suspicious too?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} It should and it does, and the person who first showed why is worth
naming. Eddington was a great astrophysicist who spent his last years
deriving the fine-structure constant and the number of particles in the
universe from combinatorial arguments that persuaded almost nobody, and
his reputation has never recovered. Everyone working in this territory
is one bad step from that, and pretending otherwise would be the first
bad step. \\ What separates an arithmetic coincidence from a
derivation is not how impressive the agreement is. It is whether the
numbers were produced by machinery that exists for another reason, and
whether they are exposed. A number reverse-engineered to match a
measurement will always match it; the question is what else it commits
you to. Here the integers come out of a closure condition on a
structure fixed beforehand, and they then determine other things --- a
family count, a stellar correction, a decay ordering --- that were not
used to obtain them and that can each be checked separately. That is
the only defense available and it is the right one: not ``the agreement
is too good to be chance'' but ``here are four further consequences,
please go and look.'' \\ Keep the suspicion. It is the correct
attitude and it costs nothing, because the framework has arranged to be
checkable. If the further consequences fail, the numbers were a
coincidence, and no amount of elegance in how they were obtained will
save them.
\end{objqabody}

\bigskip
\noindent\textbf{Question 15 (Q-Constitution-15) (on the curious reader)}\par\nobreak\smallskip
\noindent\textbf{``Why should I take one person's framework seriously when thousands of physicists work on other approaches?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} You should not take it seriously because of who wrote it, and you
should not dismiss it for that reason either. Both are the same error
with opposite sign. \\ The practical form of your question is fair
though: attention is finite, and a reader cannot check everything. The
useful filter is not headcount but exposure. A framework supported by
thousands of researchers that forbids nothing measurable is a worse bet
for your attention than a framework supported by one that forbids
several things and names the facilities that will decide. Size of
community tracks many things --- funding history, training pipelines,
the accidents of which problems were fashionable when --- and it does
not track correctness. \\ There is a real cost to being outside a large
community and it should be stated rather than glossed. Errors get found
faster where more people are looking. A framework developed largely by
one person carries a higher prior probability of containing a mistake
that a hundred readers would have caught in a week, and nothing about
the internal coherence of the argument reduces that. The defense is not
to deny it but to make checking cheap: every claim traced to the clause
it rests on, every conditional result marked as conditional, and the
falsifiers listed with the measurement that would settle each. \\ So:
do not take it on trust. Take one prediction --- the compact-star shift
is the most accessible --- and see whether it is stated precisely enough
to be wrong. If it is not, ignore the rest. If it is, that is what
deserves your attention, and not the biography of whoever wrote it.
\end{objqabody}

\bigskip
\noindent\textbf{Question 16 (Q-Constitution-16) (on the curious reader)}\par\nobreak\smallskip
\noindent\textbf{``What would actually convince you that you are wrong?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Several things, and they are listed by name rather than gestured
at, which is the only version of this answer worth anything. \\ A
fourth family of matter would do it. The framework forces at most three
and cannot be repaired to accommodate a fourth; there is no parameter to
adjust, which is exactly what makes the claim worth making. A
population study of compact stars showing no coherent shift in the
mass-radius relation, at a precision better than the predicted size,
would end the integer ledger and everything resting on it. A laboratory
measurement of the thermodynamic relations named in the assessment
coming out asymmetric, where the framework says they must be symmetric,
would break the third pillar. And an internal failure would count too:
if the uniqueness proof for the ledger has a hole, the central claim
of no free parameters is gone whatever the experiments say. \\ It is
worth naming what would \emph{not} convince, because a list of
falsifiers is only honest if it excludes things. A disagreement in a
regime the framework does not claim to cover would not count, and
neither would a failure of one of the conditional results whose
hypotheses are flagged as open --- those are already marked as not
carrying weight. A reader should be suspicious of any framework whose
proponent will not say which failures are survivable, because a
framework where every failure is survivable has no content. \\ The
falsifier table in the assessment is the place where this is written
down properly, with the observable and the facility for each row.
\end{objqabody}

\bigskip
\noindent\textbf{Question 17 (Q-Constitution-17) (on the curious reader)}\par\nobreak\smallskip
\noindent\textbf{``Is a `postulate' actually different from an `axiom', or is that just vocabulary?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} It is a real distinction here and it is worth a minute, because
the two words are used interchangeably in most of physics and the paper
depends on keeping them apart. \\ The four axioms are conditions on
what counts as an admissible description at all. They are prior to any
particular content: one can state them without knowing whether the
world contains quarks. The seven postulates are narrower --- they are
commitments about how the admissible descriptions are to be built, and
they could in principle be replaced by different commitments without
abandoning the axioms. In legal terms, and the paper's vocabulary is
already legal, the axioms are constitutional and the postulates are
statutory. \\ The reason for the split is not tidiness. It is that the
two carry different risk. If a postulate fails, the framework is
damaged in a locatable way and one asks what the replacement should be.
If an axiom fails, there is no replacement to look for, because the
whole notion of an admissible description has changed underneath. A
reader who wants to know how seriously to take a given result should
first ask which of the two it descends from, and the dependency map
exists so that the question has an answer. \\ There is a fair
criticism here which the community objections take up: the line between
the two is a judgment, and a different author might have drawn it
elsewhere. That is true. What the paper owes in exchange is that the
line is drawn once, drawn visibly, and not moved when it becomes
inconvenient.
\end{objqabody}

\bigskip
\noindent\textbf{Question 18 (Q-Constitution-18) (on the curious reader)}\par\nobreak\smallskip
\noindent\textbf{``You rank twentieth-century physics inside your own tower. Isn't that just claiming everyone else was doing a special case of you?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} It would be an unattractive claim if it were being made as a claim
about people, and it is worth separating what is being ordered from who
did the ordering. \\ The tower is a statement about \emph{logical
dependence}, not about priority or merit. It says that certain
structures presuppose others: that you cannot state the second without
already having the first available. That kind of ordering is routine
and uncontroversial in mathematics --- topology before measure,
measure before probability --- and nobody reads it as a slight on
probabilists. Applying it to physical frameworks is unusual, and the
unusualness is what makes the reading you describe available. \\ It
should also be said that the ordering is not flattering to this paper
in the way the objection assumes. Being at the constitutional layer
means being the most exposed, not the most important: everything below
can be measured, and the constitutional layer can only be checked
through what it forbids. A theory that sits at the bottom of a tower and
predicts nothing is in a far worse position than a well-tested theory
sitting comfortably in the middle. Quantum electrodynamics has survived
tests to twelve significant figures. Nothing in this framework has
survived anything comparable, and the tower does not change that. \\
Where the objection lands is on tone rather than content, and it is
worth conceding: a framework claiming to organize its predecessors
should be scrupulous about crediting what each of them did that it
cannot do, and the comparison sections are where that debt is either
paid or not.
\end{objqabody}

\bigskip
\noindent\textbf{Question 19 (Q-Constitution-19) (on the curious reader)}\par\nobreak\smallskip
\noindent\textbf{``Why should the deepest laws be simple? Nature is under no obligation to be tidy.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} It is not, and the assumption that it is has misled physics before.
\\ The honest history is mixed. Simplicity has been an extraordinary
guide --- Maxwell's unification, general relativity, the gauge principle
--- and it has also produced sustained and confident errors, of which
circular planetary orbits and the ether are only the most famous.
Nature is under no obligation and has periodically declined. Anyone
arguing from elegance is drawing on a record that includes both. \\
What can be said in this framework's defense is that it is not arguing
from elegance. Nothing here says the world should be simple because
simplicity is beautiful. The minimality clause is doing a different
job: it is a rule for adjudicating between candidate structures that
are otherwise equally admissible, and its justification is that without
some such rule the framework would have adjustable freedom it claims not
to have. It is chosen for its consequences, which are that the ledger
comes out unique, and it stands or falls on whether that uniqueness
survives. \\ It is also worth noticing that the framework's output is
not especially tidy. Three families rather than one, a lopsided
distribution of masses, and a collection of whole numbers with no
obvious pattern among them --- this is not what someone designing for
elegance would have produced. That is faint evidence of anything, but
it is the right direction of faint evidence, and a framework whose
output looked too neat would deserve more suspicion than this one.
\end{objqabody}

\bigskip
\noindent\textbf{Question 20 (Q-Constitution-20) (on the curious reader)}\par\nobreak\smallskip
\noindent\textbf{``If the constitution forbids so much, how do you know it doesn't forbid the actual world?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} One does not know, and that is precisely the risk the framework has
taken on. It is the right risk and it should be described accurately
rather than reassured away. \\ A constraint that excludes a great deal
is valuable exactly in proportion to how much it excludes, and it is
dangerous for the same reason. There is no way to write a strong
constitution that is safe. The framework's position is that the world
survives its constraints --- that three families, the observed charge
assignments, the structure of the forces and the stability of matter are
all in the admissible set --- and each of those is a check that has been
run rather than a hope. That is the only kind of reassurance available
and it is retrospective: it says the constraints have not yet excluded
anything we know to exist. \\ What makes the question live rather than
rhetorical is that the framework's forbidden list is not finished. It
forbids things nobody has looked for carefully. Should one of those turn
up --- a particle outside the permitted spectrum, a decay ordering the
cascade cannot produce --- the framework has excluded the actual world,
and no repair is available that keeps the claim of no free parameters.
\\ There is a version of this that would be a fair criticism and the
paper should be read with it in mind: if the constraints were adjusted
each time something new was found, so that the admissible set always
happened to contain whatever had just been measured, the framework
would be excluding nothing while appearing to exclude a great deal. The
protection against that is that the constraints were fixed first and
are written down. Whether that protection holds is something a reader
can check by comparing what is claimed now against what was claimed
before any of it was tested.
\end{objqabody}

\bigskip
\noindent\textbf{Question 21 (Q-Constitution-21) (on the curious reader)}\par\nobreak\smallskip
\noindent\textbf{``You say the framework has no adjustable dials. Isn't a theory that can't be adjusted just a theory that can't learn?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} It is a theory that cannot learn \emph{quietly}, which is a
different and much better thing. \\ Adjustment is how a framework
absorbs a disagreement without anybody having to decide that a
disagreement occurred. A measurement comes in off; a parameter moves;
the fit is restored; no meeting is held. Repeat that often enough and
you have a framework that has learned nothing and can no longer be
wrong. The absence of dials does not prevent learning --- it forces the
learning to be visible, because the only available response to a
disagreement is to say which clause failed. \\ There is a real cost and
it should not be minimized. A rigid framework will be killed by a
measurement that a flexible one would have survived, including
measurements that later turn out to have been wrong. That has happened
in the history of physics and it will happen again, and a framework in
this shape is exposed to it in a way that a more forgiving one is not.
The bet being made is that this is a price worth paying, because the
alternative --- a framework that survives everything --- is not
scientifically alive. \\ It is worth adding that rigidity is not the
same as completeness. The framework has plenty left to learn: a long
list of open problems, several conditional results whose hypotheses are
not discharged, and whole domains it does not reach. What it does not
have is a knob to turn when one of those is settled the wrong way.
\end{objqabody}

\bigskip
\noindent\textbf{Question 22 (Q-Constitution-22) (on the curious reader)}\par\nobreak\smallskip
\noindent\textbf{``Physics used to work upward from experiments. This starts from principles and works down. Isn't that backwards?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Both directions have been used and both have produced permanent
results, so the question is not which is legitimate but which is
appropriate to a given problem. \\ Working upward is right when you have
data and no organizing idea: the spectroscopy came before quantum
mechanics, the anomalies before the Standard Model. Working downward is
right when you have an organizing idea and want to know what it costs.
Special relativity came from two principles and almost no new data;
general relativity from an equivalence principle taken seriously past
the point of comfort. Neither of those was assembled from measurements.
\\ Where the present situation is unusual is that the data have been
static for a long time. The Standard Model's parameters have been
measured to remarkable precision, and precision alone has not suggested
where they come from. That is the condition under which working
downward is worth attempting: not because measurement is unimportant,
but because more of the same measurement is not obviously going to
produce the missing idea. \\ The danger of the downward direction is
real and worth naming, because it is the one your question is pointing
at. A framework built from principles can be internally magnificent and
about nothing, and it will not find out from the inside. The only
protection is to arrive at something checkable quickly and to say so
loudly when you do. That is what the falsifier list is for, and a
reader is entitled to judge this paper by how fast it gets there.
\end{objqabody}

\bigskip
\noindent\textbf{Question 23 (Q-Constitution-23) (on the curious reader)}\par\nobreak\smallskip
\noindent\textbf{``What does it even mean to say a law is `necessary' rather than just `true'?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} It means that its negation is not merely false but unavailable ---
that there is no coherent structure of the relevant kind in which it
fails. The distinction is easier to see in mathematics than in physics,
which is part of why the word makes physicists uneasy. \\ That a
particular star has a particular mass is true and could have been
otherwise. That the angles of a Euclidean triangle sum to two right
angles is not merely true; there is no Euclidean triangle for which it
fails, and asking why it holds has no answer beyond the definitions.
Most statements in physics are of the first kind. The claim in this
framework is that a few of them are of the second: given the arena, the
integer ledger is not one option among several that happens to be
realized, but the only assignment that closes. \\ Two cautions, both
important. The necessity is always \emph{relative} --- necessary given
the arena, which is itself assumed. Nothing here is necessary
absolutely, and a framework claiming otherwise would be claiming to
have derived existence, which nothing can. And necessity of this kind
is not evidence. A structure can be uniquely determined and simply not
be the one nature uses; uniqueness within a search space says nothing
about whether the search space contains the world. \\ What the word
buys, when it is earned, is a change in what counts as an explanation.
``Why three families?'' stops being a question about initial conditions
and becomes a question with a proof or a counterexample. That is a
better kind of question to have, and it is the strongest thing that can
be said for the vocabulary.
\end{objqabody}

\bigskip
\noindent\textbf{Question 24 (Q-Constitution-24) (on the curious reader)}\par\nobreak\smallskip
\noindent\textbf{``Suppose an experiment disagrees with you tomorrow. Which axiom dies?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} It depends entirely on which experiment, and the framework is set
up so that the answer is determined in advance rather than negotiated
afterward. That last clause is the one that matters. \\ A fourth
family of matter would kill the closure argument and with it the
minimality clause, because the family count follows from the ledger
being unique and the ledger's uniqueness is what minimality is for. A
compact-star measurement with no shift at the predicted precision
attacks the same clause from a different direction, through the anomaly
condition rather than through the cascade. A violation of the
thermodynamic relations would strike the observational clause, since
those relations are how the framework's boundary description is
supposed to show up in a laboratory at all. And a failure of causal
ordering --- were such a thing ever established --- would end the entire
construction, since nothing in it survives without an order of events.
\\ The reason to fix this correspondence beforehand is that
after-the-fact reassignment is the standard way frameworks become
unfalsifiable. If a proponent can decide, once the result is in, which
part of the structure the result was really testing, then no result
tests anything. The falsifier table names the clause for each row, and
the naming is a commitment. \\ There is one honest complication. Some
results in this paper are conditional on hypotheses that are not yet
theorems, and a disagreement in those areas would first be blamed on
the open hypothesis rather than on an axiom --- legitimately, but it is
also exactly the sort of escape hatch a reader should watch. The open
hypotheses are listed by name for that reason.
\end{objqabody}

\bigskip
\noindent\textbf{Question 25 (Q-Constitution-25) (on the curious reader)}\par\nobreak\smallskip
\noindent\textbf{``I'm not a physicist and I've got this far. What is the one thing you want me to take away?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} That the interesting question about a framework like this one is
not whether it is right. It is whether it has arranged to be caught. \\
Most of what you will read about fundamental physics, in either
direction --- the enthusiastic accounts and the debunkings alike ---
argues about whether a set of ideas is true. That argument is almost
never settleable from the armchair, and it is not the argument that
moves science. What moves it is the narrower and duller question of
whether a claim has been stated in a form that some measurement could
contradict, and whether the proponent said so in advance. \\ This paper
takes four statements about what any description of nature must
satisfy, follows them down, and arrives at a small number of things
that could be checked: how many families of matter there are, a
particular small correction to how compact stars weigh, a set of
symmetries in laboratory transport that must hold exactly. If those go
the wrong way, the framework is finished, and it has said which clause
finishes with them. \\ So the thing to take away is a habit rather than
a belief. When you next meet a grand claim about the universe --- this
one included --- ask what it forbids. If the answer is nothing, or if
the answer keeps changing, you have learned what you needed to know
without following any of the mathematics. And if the answer is a short
list of specific things that somebody is going to measure, then the
claim is worth your attention whether or not it survives, because that
is what a scientific claim looks like from the outside.
\end{objqabody}

\bigskip
\section{First-pass objections from six communities}

\bigskip
\noindent\textbf{Objection 1 (O-Constitution-1) (the particle theorist)}\par\nobreak\smallskip
\noindent\textbf{``You claim no
 free parameters, but the integers $(1,3,5;7,35)$ are still
 inputs --- you have just relabeled the problem.''}\par\nobreak\smallskip
\begin{objqabody}
This is the sharpest line of attack against
the \GGTheory{} program, and it deserves the longest reply. The
particle theorist's complaint is that the Standard Model's 28
continuous parameters have not really been eliminated; they have
been replaced by an integer ledger that itself appears free.
\emph{Response.} Two clauses answer this objection together.
First, $\AX{4}~\MSMF{}$ together with $\POS{6}~\OGN{}$ allows
\emph{only} integer-rational normalizations on the maximal
admissible candidate. This is a strictly stronger constitutional
restriction than ``one fewer parameter in some basis'': continuous
basis drift is forbidden as a matter of constitutional law, not
as an aesthetic preference. Second, $\T{3}$ Anti-Landscape
Selection (proved in \ATwo{} \citep{ArisakaA2GG6}, Theorem~3) proves that the
five-tuple $(M, N, L; J, K) = (1, 3, 5; 7, 35)$ is the unique
\emph{lexicographic maximum} of the Diophantine system $K = LJ
= M^2 + N^2 + L^2$ with chirality count $J = 2N + 1$, under the
ordering $\prec$ defined in $\AX{4}~\MSMF{}$, compatible with the
four-component anomaly polynomial of $\T{5}$. We concede that
the lexicographic ordering itself is an axiomatic
commitment
--- maximize symmetry first, minimize freedom second --- and that
a different ordering would in principle pick out a different
five-tuple. This is by no means a hidden assumption; it is the
explicit content of $\AX{4}$, and it is itself falsifiable: if a
second admissible integer solution is found under the same
ordering, or if observational data favor a different solution
under a different admissible ordering, the selection principle
fails and the constitution falls. See \citep{ArisakaA3BI6}
Common Objection~8 for the parallel statement at the \BIsix{}
level: removing the $\AX{4}$ minimality input opens the integer
family $(L, N) = (5k, 3k)$; with both the Diophantine system and
the $\AX{4}$ ordering, the family collapses to $k = 1$. An
integer chosen freely is one number; an integer forced by a
Diophantine system \emph{plus} an explicit axiomatic ordering is,
in the auditable sense, no continuous parameter at all.

Indeed, the deepest published answer to this objection lives in
the companion paper \AFour{} \citep{ArisakaA4Euclid}, which is
devoted in its entirety to the constitutional question of free
parameters in the Euclidean methodological sense. \AFour{}
develops a strict parallel: the three fundamental dimensional
constants $c$, $\hbar$, and $G_N$ are not free parameters but
\emph{conversion factors of the 6D bulk geometry}, fixed
by the operational practice of measuring time in seconds, space
in meters, action in joule-seconds, and gravity in
Newtonian units applied to a manifold whose intrinsic structure
is given by $\AX{2}~\GGEP{}$. The five $\OGNfive{}$ integers
$(M, N, L; J, K) = (1, 3, 5; 7, 35)$, in exact parallel, are not
free integer inputs but \emph{theorems} forced by a five-step
Diophantine cascade ($\POS{3}$ rank-one factorization $\to$
$\SUfive{}$ minimal simple unification $\to$ $\BIsix{}$
trace-ratio gives $N = 3$ $\to$ odd-closure $J = 2N+1 = 7$ $\to$
dual lock $K = LJ = M^2 + N^2 + L^2 = 35$), with each step a
theorem in its own right. Together with the four $\BIsix{}$
coefficients $(1, -1, -2/5, -2/5)$ this nine-element ledger
constitutes the entire dimensionless arithmetic content of
physics; from it \citep{ArisakaA9SM} derives all 28 Standard
Model parameters and \citep{ArisakaA10Cosmos} derives the
cosmological inventory. After \AFour{} the objection is no
longer that the integers are a relabeling of the 28 parameters;
the objection is whether a reader accepts that the methodological
ideal of physical science, since Euclid, is recoverable. That is
exactly the audit the constitution invites.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 2 (O-Constitution-2) (the gravity researcher)}\par\nobreak\smallskip
\noindent\textbf{``Higher-dimensional
 theories have produced thirty years of moduli problems; you
 cannot escape them just by writing axioms.''}\par\nobreak\smallskip
\begin{objqabody}
The gravity-research community, scarred by the Calabi-Yau
moduli-stabilization program of the late 1990s and early
2000s, will object that any 6D bulk inherits the same family of
geometric flat directions that have plagued string
compactifications --- and that no axiom system can wish them
away. \emph{Response.} The axiomatic answer is sharper than
the technical one. $\AX{4}~\MSMF{}$ defines the maximal
admissible candidate by $F(T) = 0$, where $F$ is the dimension of
the moduli space of physically inequivalent continuous
deformations. No moduli that fail to be forced by the
constitution are admissible, full stop --- they are forbidden at
the axiomatic level rather than stabilized at some
intermediate energy by a separate dynamical mechanism.
$\T{4}$ 6D bulk Existence (imported from
\citep{ArisakaA2GG6}) and $\T{6}$ \OGNfive{} as Bulk Output
(imported from \citep{ArisakaA2GG6}) jointly demonstrate that
the canonical metric $ds_6^2 = \e^{-2r}\eta\,dx\,dx + dr^2 + d\theta^2$
with $r \in [0, 35]$ admits no continuous moduli once
$\POS{3}~\AF{}$ is imposed. This is precisely what
distinguishes the \GGTheory{} program from string-theoretic
compactifications: we do not stabilize moduli; we forbid them.
We concede that the Hopkins-Singer differential-cohomology proof
of \citep{ArisakaA3BI6} is technically demanding, and that a hostile reader is welcome to re-derive it independently. Yet
no axiomatic argument can be required to stand on a
companion paper's technical machinery alone, and so we offer the
falsifier of $\T{5}$: if the trace-ratio lock $|a/c| = |a/d| = 5/2$
is shown to admit any other coefficient solution, $\T{5}$ fails
and so does the moduli-free claim.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 3 (O-Constitution-3) (the quantum-foundations specialist)}\par\nobreak\smallskip
\noindent\textbf{``The measurement problem isn't a missing axiom --- it's a deep
 interpretational issue, and writing more axioms doesn't make
 it go away.''}\par\nobreak\smallskip
\begin{objqabody}
This objection comes from the heirs of the Bohr-Einstein debate,
the Many-Worlds school, the Bohmian school, and the
informational-realist school of QBism --- communities that have
spent a century arguing about what quantum mechanics
\emph{means}. \emph{Response.} The \GGTheory{} program makes no
interpretational commitment whatsoever, and we want to be precise
about what this means. $\AX{3}~\OGGEP{}$ does not resolve the
interpretational debate; it sidesteps it by definitional fiat.
$\AX{3}$ defines a measurement as the operational dictionary
$\Dmap : \mathbf{Bulk} \to \mathbf{Op}$, and $\T{2}$ Operational
Closure (proved in \AFive{} \citep{ArisakaA5GDOS}, Theorem~$\T{2}$) demonstrates that the Born
rule, the GKLS generator, and the Stinespring dilation emerge as
\emph{theorems} of this axiomatic move rather than as
independent axioms. Operational compatibility with
Many-Worlds, with QBism, with relational quantum mechanics, and
with the open-systems formalism of Zeh, Joos, and Zurek therefore
means only this: each interpretive school can adopt the
\GGTheory{} operational dictionary without contradiction, since
the dictionary specifies which CPTP channel registers an outcome
but says nothing about what an outcome \emph{is} ontologically.
This is not the same as interpretational neutrality on the
measurement problem itself --- a Many-Worlds adherent and a
collapse theorist will still disagree about what happens
metaphysically when the Stern-Gerlach apparatus reads spin-up.
What we claim is not a philosophical resolution of that
metaphysical disagreement but the elimination of the
\emph{operational} ambiguity that has prevented quantum theory
from compiling cleanly into a registered laboratory record.
The working physicist needs the latter; the former
the \GGTheory{} program leaves on the table for whichever
interpretation the reader finds compelling.

\emph{Two recent benchmark statements from the
quantum-foundations literature deserve explicit placement.}
Frauchiger and Renner \citep{FrauchigerRenner2018} argued that
any universal interpretation of quantum mechanics is inconsistent
under iterated agent-based measurement. In the constitutional
reading of $\AX{3}~\OGGEP{}$, the Frauchiger-Renner paradox
is the boundary report of an attempt to compose two
$\OGGEP{}$-projections at different scales without honoring
the operational dictionary's CPTP discipline; once the two
agent-measurements are required to be admissible CPTP
instruments and the measurement contract is registered, the
paradox dissolves into a specification of which channel the
second observer is actually running. Bub and Pitowsky
\citep{BubPitowsky2010} have argued, from a different angle,
that quantum mechanics should be axiomatized as a non-classical
theory of information, with the Born rule emerging from
operational constraints rather than from a hidden ontology.
The constitutional $\AX{3}~\OGGEP{}$ is in close structural
neighborhood with Bub-Pitowsky: both promote operational
content to first-class status, and $\GGTheory{}$ inherits the
stance that ``measurement'' is not in need of metaphysical
resolution before it can be made mathematically precise.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 4 (O-Constitution-4) (the mathematical physicist)}\par\nobreak\smallskip
\noindent\textbf{``Wightman
 axiomatic QFT and the Haag-Kastler operator-algebra
 framework already exist. What does the \GGConstitution{} add
 that has not been done in fifty years of axiomatic QFT?''}\par\nobreak\smallskip
\begin{objqabody}
This objection comes from the constructive-QFT and
algebraic-QFT communities, and it is well posed. Wightman
\citep{Wightman1956} and Haag-Kastler \citep{Haag1992} supply
mature axiomatizations of relativistic quantum field theory in
the closed-system regime. \emph{Response.} We want to position
the \GGConstitution{} relative to WH-K honestly: the two are
\emph{complementary frameworks at different scopes}, not
competing axiomatizations of the same space. WH-K addresses
closed-system QFT on a fixed background spacetime, with mature
locality, covariance, and spectral conditions that have stood
for half a century. The \GGTheory{} program extends the scope
to include open dynamics ($\AX{3}~\OGGEP{}$), a measurement
compiler, and gauge-plus-geometry as axiomatic primitives
($\AX{1}$, $\AX{2}$); it does \emph{not} replace the WH-K
machinery in the closed regime, which the \GGTheory{} program
inherits via $\T{9}$ as a boundary-projection theorem. We
acknowledge two further nuances. First, the WH-K axioms are not
silent on theory-space structure --- they fix a lattice of
admissible closed QFTs satisfying their conditions --- but they
are silent on whether the continuous parameters that label
points within that lattice are forced by a deeper principle;
$\AX{4}~\MSMF{}$ and $\POS{6}~\OGN{}$ supply that further
restriction. Second, the GKLS classification of admissible
generators is not a discovery of the present paper; it is
known mathematics from \citet{Lindblad1976,Gorini1976}. The
axiomatic novelty is in treating GKLS as a forced
admissibility class rather than as one effective formalism among
many. We concede that the WH-K axioms have a longer track
record of mathematical rigor, and the \GGTheory{} program owes
them a debt.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 5 (O-Constitution-5) (the philosopher of science)}\par\nobreak\smallskip
\noindent\textbf{``Calling
 something a constitution does not make it scientific. How
 is your axiom system different in kind from rationalist
 speculation?''}\par\nobreak\smallskip
\begin{objqabody}
The philosopher of science, fluent in Popper, Kuhn, Lakatos,
and Carnap, will object that any sufficiently elaborate axiom
system can be made to ``derive'' a desired set of conclusions
if its axioms are chosen well; the \GGConstitution{} is no
different in kind from Spinoza's \textit{Ethics} unless it
ships with a falsification standard. \emph{Response.}
$\POS{7}~\MF{}$ is exactly that standard. Every theorem of
the \GGTheory{} program is required to compile to a four-tuple
$(\mathcal{I}, \mathcal{S}_{\mathrm{valid}}, \mathcal{T}, K)$
of instrument, calibration window, timescale, and pre-declared
kill criterion (Part~III \S\ref{sec:POS7}). The cross-paper
coherence map of Table~\ref{tab:coherence_map} and the
consolidated falsifier table of
Table~\ref{tab:pull_table} together supply the kill criterion
for each Principal Theorem and the experimental status of every
named numerical prediction. The constitution is structurally
Popperian: a single $5\sigma$ deviation from any one of the
canonical predictions falsifies the upstream constitutional
clause. We concede that the philosophical distinction between
``a constitution that predicts numbers'' and ``a constitution
that postulates them'' is a slippery one in practice, and that
any working scientist will have to judge the \GGTheory{}
program by its empirical record over the next decade. What
makes a body of knowledge a science is not the cleverness of
its calculations but the honesty of its rules; we have written
the rules down, and we await contact with data.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 6 (O-Constitution-6) (the string theorist)}\par\nobreak\smallskip
\noindent\textbf{``Your modified
 Bianchi identity and your Green--Schwarz inflow are exactly what
 we have been doing since 1984. What does the cohort actually
 add, and is it not just relabeled string theory?''}\par\nobreak\smallskip
\begin{objqabody}
This is the most consequential objection the present paper must
answer, and it deserves a careful response. The objection comes
from the heirs of the Green--Schwarz construction and the modern
$\AdS{}/$CFT program, communities that have built four decades of
spectacular mathematical machinery on top of the bulk-side
classical-topological backbone the cohort framework also occupies.
\emph{Response.} We want to position the constitutional reading
relative to string theory and $\AdS{}/$CFT as carefully as honesty
permits. In the cohort reading of $\POS{3}~\AF{}$, the bulk-side
$dX_4 = 0$ on the $\AX{2}~\GGEP{}$ composite connection \emph{is}
the same classical-topological identity that Green and Schwarz wrote
down in 1984 as a $B_2$-coupled rewrite of Bianchi 1902. The
constitutional reading does not pretend to have discovered
Green--Schwarz; it claims, more modestly, that the
Green--Schwarz identity is forced by $\AX{2}~\GGEP{}$ as a structural
theorem, that the boundary 4D ``anomaly'' is the $\AX{3}~\OGGEP{}$
projection shadow of that bulk classical fact, and that
string theory and $\AdS{}/$CFT --- to the extent that they share this
bulk-side classical-topological backbone --- are structural theorems
under $\AX{2}$ (geometrization of gauge fields), not under the slogan
``quantization of geometry.'' What the cohort \emph{does not}
subsume is the quantum-dynamical content that string theorists run
on top of the classical backbone: graviton vertex operators, the
$\alpha'$ and $1/N$ perturbative apparatus, the islands formula of
Almheiri--Engelhardt--Marolf--Maxfield and
Penington--Shenker--Stanford--Yang for the Page-curve recovery of
unitarity from the black-hole interior, or the
Gubser--Klebanov--Polyakov / Witten boundary-operator dictionary.
Those constructions sit in the quantum-dynamical sector that the
cohort axioms neither subsume nor contradict. Generally speaking,
the cohort framework occupies the GGEP-closed sector --- the
classical-topological backbone --- while the rich perturbative-string
content lives in a separate physical sector that runs on top of the
backbone we share. We concede readily that the bulk-side
Hopkins--Singer differential-cohomology proof of
\citep{ArisakaA3BI6} owes much of its mathematical apparatus to
decades of string-theory technical development, and we credit that
lineage explicitly. What the cohort adds, and what we hope this
paper makes precise, is the constitutional reading that \emph{names}
the bulk-side classical-topological backbone as a separate physical
claim from the quantum-string content, so that the bulk side can be
audited, falsified, and tested on its own without entanglement with
the quantum-string sector's open questions. Part~V of this paper
develops the full constitutional-axiom classification: string
theory and AdS/CFT both sit at the $\mathsf{GGEP\text{-}closed}$
stratum (Stratum-2 of the four-axiom tower), and the two famous
limitations of string theory --- the inability to derive the QM
postulates and the $\sim 10^{500}$ vacuum landscape --- are exactly
the structural symptoms of working without $\AX{3}$ and $\AX{4}$.
The cohort framework activates all four axioms explicitly and so
produces the QM postulates as theorems and the unique vacuum from
the integer \OGNfive{} ledger. Objection~12 below develops the
single deepest structural sharpening of this point ---
\emph{AdS/CFT is the closed-system limit of $\OGGEP{}$, in the
same sense that the Schr\"odinger equation is the closed-system
limit of GKLS evolution} --- which is the load-bearing
identification that distinguishes Stratum 2 from Stratum 3 in the
constitutional classification. See
\figref{fig:A1-constitutional-flowchart} and
Table~\ref{tab:V-classification} for the full visual and tabular
statement.

%% =====================================================================
%% §15.2 — Cluster A: Cohort dependence (Objections 7-9)
%% =====================================================================
\end{objqabody}

\bigskip
\section{The cohort architecture and how to read it}

\bigskip
\noindent\textbf{Objection 7 (O-Constitution-7) (the careful reader)}\par\nobreak\smallskip
\noindent\textbf{``If A1-Constitution
 proves nothing, what is the value of reading it? Cite-and-forward
 looks like evasion.''}\par\nobreak\smallskip
\begin{objqabody}
A particularly careful first-pass reader will notice that A1 names
eleven Principal Theorems but proves none of them, and may
reasonably ask whether the constitutional framing is more than
rhetoric. \emph{Response.} A1 is the rulebook; the cohort papers
are the proofs. Constitutional documents in mature scientific
practice --- Euclid's \emph{Elements} (which proves theorems from
postulates), Newton's \emph{Principia} (which proves the
gravitational consequences of the three laws), Einstein's
\emph{Foundation of the General Theory of Relativity} (which
proves the geodesic and field equations from the equivalence
principle) --- all share the discipline that the foundational
document names the rules and proves their immediate consequences,
not the entire downstream theorem space. A1 inherits this
discipline. The eleven Principal Theorems are named at high level
in A1 (Part~IV \S\ref{sec:paper_inventory}) and their proofs live
in the specific theorem locations of the cohort papers, with
explicit pointers in the cross-paper coherence map
(Table~\ref{tab:coherence_map}). We acknowledge that this is a
strong demand on the reader: the reader who wishes to evaluate a
specific claim must follow the pointer. We have made the
pointers as specific as possible (theorem numbers, section
numbers, page numbers where possible). The architecture is
constitutional discipline, not evasion: an axiomatic document
that included all eleven proofs would run to two thousand pages
and would no longer be auditable at the constitutional layer.
The cohort architecture is the price of keeping the constitutional
document short enough to be read in an afternoon, while making
every load-bearing claim trace to a specific cohort proof
location. We invite the careful reader to follow the pointers
for the specific objection they wish to raise; the companion
papers are posted together precisely so that the proof is one
click away.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 8 (O-Constitution-8) (the time-constrained reader)}\par\nobreak\smallskip
\noindent\textbf{``What is
 the minimum reading combination required to evaluate this
 paper?''}\par\nobreak\smallskip
\begin{objqabody}
A time-constrained reader will not work through twenty-two papers
before forming a first opinion, and will reasonably ask which subset is
required. \emph{Response.} The minimum critical-reading
set, depending on the reader's discipline, is named in the
preamble of this section: A2+A3 for particle-physics /
string-theory / quantum-gravity objections; A5+A8 for
quantum-foundations / measurement-problem objections; A9+A10
for Standard-Model / cosmology objections.
For a particle theorist whose immediate questions concern
the integer \OGNfive{} ledger and the BI-6 trace-ratio lock, the
two-paper read of A2+A3 is the minimum critical reading. For a quantum-foundations reader whose immediate question is whether
the measurement-problem dissolution claim of A1 \S\ref{sec:two_remarks}
Remark~1 is more than rebranded decoherence theory, the two-paper
read of A5+A8 is the minimum critical reading. For a phenomenologist whose immediate question is whether the 28
Standard-Model parameters really collapse to zero or are quietly
re-parametrized, the two-paper read of A9+A10 is the
minimum critical reading. We acknowledge that this is still
demanding compared to a single-paper review; the cohort is a
substantial commitment. The twenty-one posted papers are posted
together and reach arXiv in four staged submissions, so that an
evaluator can follow the pointers without delay and the
architecture is transparent from the first day rather than across
a multi-year publication cycle. The capstone \AEleven{} is in
preparation and is deliberately not in that set; nothing in the
present paper depends on it.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 9 (O-Constitution-9) (the senior community member)}\par\nobreak\smallskip
\noindent\textbf{``A
 twenty-two-paper program is too large for any single reader to evaluate. How is anyone supposed to judge this?''}\par\nobreak\smallskip
\begin{objqabody}
A senior member of the community, with the breadth to attempt a
full read, will object that no single human can rigorously
evaluate twenty-two preprints in any reasonable time, and that the
cohort architecture therefore defeats the peer-review function
in practice. \emph{Response.} We acknowledge the cost honestly,
and address it three ways. First, the cross-paper coherence map
(Part~IV Table~\ref{tab:coherence_map}) pins every Principal
Theorem $\T{k}$ to a single specific proof location, so that any reader can target a specific claim and find its full proof in a
single named place rather than searching across twenty-two papers.
Second, the consolidated falsifier table (Part~IV
Table~\ref{tab:pull_table}) consolidates every numerical
falsifier into one place, with predicted value, observed value, comparison mode,
pre-declared kill threshold, and experimental status as of writing; a reader who wishes to attack the
cohort's empirical content can begin there. Third, the eleven A-papers form a tight forward-implication
chain: A1 names the
axioms; A2 proves the bulk theorems given A1; A3 proves the
anomaly theorems given A2; A5 proves the boundary theorems given
A2+A3; and so on. The chain can be read incrementally, with each
paper building on the proved content of the upstream papers; a reader who has read A1+A2 can evaluate A3 without yet having
read A5--A11. We acknowledge that no foundational program
this ambitious has ever been published as a single coordinated
cohort, and that the peer-review machinery will need to adapt to
the architecture. The alternative --- releasing the papers one at a time over five
years, with the constitutional
implications obscured by the long delay --- is in our judgment
worse for the community and for the science.

%% =====================================================================
%% §15.2 — Cluster B: AdS/CFT and string theory placement (Objections 10-12)
%% =====================================================================
\end{objqabody}

\bigskip
\section{String theory and the constitutional tower}

\bigskip
\noindent\textbf{Objection 10 (O-Constitution-10) (the working string theorist)}\par\nobreak\smallskip
\noindent\textbf{``Locating
 string theory at Stratum 2 of your four-Stratum tower is too
 aggressive. String theory has produced spectacular operational
 results --- AdS/CFT correlators, the viscosity bound, the
 Ryu-Takayanagi formula, the islands formula for the Page curve
 --- and these are operational results, not closed-system
 idealizations.''}\par\nobreak\smallskip
\begin{objqabody}
This is the working string theorist's instinctive defense of the
program. \emph{Response.} The cohort's Stratum-2 placement of
string theory does not deny the operational results. Every
result on the list above is preserved unchanged at Stratum 2 of
the constitutional classification, because the GGEP-closed
content (which is what Stratum 2 = $\{\AX{1}, \AX{2}\}$ is) is
exactly the bulk-side classical-topological backbone within which
those results are derived. Indeed, the islands formula
\citep{Almheiri2020,Penington2020} is, in the cohort reading, a
boundary-side closed-system computation in which the
\emph{closed} unitarity of the bulk is restored at the cost of
including non-local entanglement-wedge regions. The cohort
framework inherits this result and credits it as a Stratum-2
theorem. What the cohort \emph{adds} at Stratum 3 (with
$\AX{3}~\OGGEP{}$ active) is the open-system extension to a
finite boundary observer who registers discrete coarse quantum
records --- a regime that the closed-system holographic apparatus
of AdS/CFT does not cover and was never designed to cover. See
Objection~12 below for the deepest structural sharpening of this
distinction.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 11 (O-Constitution-11) (the senior theorist)}\par\nobreak\smallskip
\noindent\textbf{``Three of your four
 reasons that string theory cannot ascend beyond Stratum 2 are
 the same axiom (AX-4 MSMF) in three different dresses. That is
 not four independent reasons; it is one reason restated.''}\par\nobreak\smallskip
\begin{objqabody}
A careful senior theorist will notice that \S\ref{sec:V-stringtheory-exclusion}'s
``four-fold constitutional exclusion'' of superstring theory ---
SUSY forbidden by AX-4, 10D/11D forbidden by AX-4, SO(32) /
$E_8 \times E_8$ forbidden by AX-4, no constitutional measurement
layer --- compresses cleanly to two underlying constitutional
gaps: the missing AX-3 (operational measurement) and the missing
AX-4 (parsimony ordering principle). \emph{Response.} The senior theorist is correct. We restate the four-fold exclusion
honestly as the consequence of two missing axioms. String theory
is missing AX-3 OGGEP, which would supply the operational
measurement layer that turns boundary correlators into registered
records; this is the structural diagnosis developed in
Objection~12 below. String theory is missing AX-4 MSMF, which
would forbid the unforced continuous freedom that produces
SUSY-doubling, the 10D/11D dimensional choice, and the
high-rank gauge groups. The three Stratum-2 ``additions'' the
cohort framework rejects --- SUSY, the 10D/11D bulk, and the
SO(32) / $E_8 \times E_8$ gauge content --- are all consequences
of the same missing AX-4 ordering principle. Calling them three
separate reasons overstates the structural distance; the honest
statement is that string theory lacks two of the four
constitutional axioms (AX-3 and AX-4), and the three Stratum-2
additions are the textbook symptoms of the missing AX-4. We
restate the exclusion as two reasons rather than four in
\S\ref{sec:V-stringtheory-exclusion}, and we acknowledge the senior theorist's point as a sharpening of the cohort's claim, not as a refutation
of it.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 12 (O-Constitution-12) (the senior string theorist)}\par\nobreak\smallskip
\noindent\textbf{``AdS/CFT had OGGEP in 1998. The bulk-to-boundary holographic
 dictionary, the GKPW formula, and the operational matching of
 bulk fields to boundary correlators were all in place before
 you wrote the word $\OGGEP{}$. Locating AdS/CFT at Stratum 2
 because it lacks `constitutional $\OGGEP{}$' is therefore a
 renaming exercise --- string theory had $\OGGEP{}$ in 1998,
 only it called it the holographic dictionary.''}\par\nobreak\smallskip
\begin{objqabody}
This is the deepest structural objection a senior member of the string-theory community will raise, and it deserves the most
detailed response of the section. The objection is wrong, but
the reason it is wrong is precisely what distinguishes
Stratum 2 from Stratum 3 in the constitutional classification ---
and the response is the load-bearing structural payoff of the
cohort framework. \emph{Response.} The full answer is developed
in \citep{ArisakaA5GDOS} \S6.4 (``AdS/CFT correspondence as
the GGEP content on AdS solutions'') and we summarize its content
here in the form required for an A1-level audit.

The structural identification is:
\begin{equation}
\text{AdS/CFT} \;=\; \OGGEP{} \text{ at the AX-3-off limit};
\label{eq:adscft_oggep_closed_limit}
\end{equation}
or, said the other way round, $\OGGEP{}$ is the open-system
extension of AdS/CFT into the regime where the boundary observer
is finite. The structural relationship between AdS/CFT and
$\OGGEP{}$ is \emph{exactly} the relationship between the
Schr\"odinger equation and the GKLS master equation:
\begin{equation}
\text{Schr\"odinger (closed)}
\;\xleftarrow{\;\gamma_k \to 0\;}\;
\text{GKLS (open)},
\qquad
\text{AdS/CFT (closed)}
\;\xleftarrow{\;\AX{3} \text{ off}\;}\;
\OGGEP{} \text{ (open)}.
\label{eq:Sch_GKLS_AdSCFT_OGGEP}
\end{equation}
At the dynamical layer, Schr\"odinger evolution is the
$\gamma_k \to 0$ closed-system limit of GKLS evolution; at the
holographic layer, AdS/CFT is the AX-3-off closed-system limit of
the cohort's $\OGGEP{}$ projection.

The technical core of the identification is the Stinespring
dilation theorem. Every CPTP channel
$\Phi: \rho \mapsto \Phi(\rho)$ admits a unitary representation
on a system-plus-environment Hilbert space,
\begin{equation}
\Phi(\rho) \;=\; \Tr_{\rm Env}\!\bigl[\, U \,(\rho \otimes \sigma_{\rm Env})\, U^\dagger\,\bigr],
\label{eq:Stinespring_obj12}
\end{equation}
with $U$ unitary. The ``open'' character of $\Phi$ comes
\emph{entirely} from the partial trace over the environment; in
the no-environment limit, $\Phi$ degenerates to the bare unitary
$U$. This is the universal sense in which closed-system unitary
structure produces open-system CPTP structure once an environment
is identified and traced out. For the cohort
$\OGGEP{}$-projection $\Dmap: \mathbf{Bulk} \to \mathbf{Op}$, the
boundary observer is the boundary slice of a larger bulk-plus-boundary
system; the CPTP discipline on the boundary is enforced precisely
because the bulk degrees of freedom (the ``environment'' of the
boundary observer) are partially traced out. In the no-environment
limit, where the boundary observer has access to the full bulk
Hilbert space, the CPTP projection degenerates to a bulk-side
unitary, and the $\OGGEP{}$ map degenerates to the AdS/CFT
duality. The closed-system limit is therefore not a competitor
to $\OGGEP{}$; it is the $\AX{3}$-off corner of $\OGGEP{}$.

The empirical content of the identification is the central
physical claim: \emph{there is no closed system in nature}. The
boundary observer in any actual application of holography --- the
RHIC quark-gluon plasma viscosity measurement, the LHC heavy-ion
data, the condensed-matter strange-metal correlator
\citep{Maldacena1998,Witten1998}, the Page-curve experiments
\citep{Almheiri2020,Penington2020} --- is, empirically, an open
system. The observer has finite Hilbert-space dimension, finite
integration time, finite spatial resolution, and finite
measurement bandwidth. The boundary correlators that AdS/CFT
computes are matched against open-system measurements at the
moment the prediction is compared to data. AdS/CFT is therefore
correct as the closed-system idealization but is \emph{not} the
actual physics governing the boundary record. The actual physics
is $\OGGEP{}$ --- the open-system extension of holography in which
the boundary observer is finite, the projection is CPTP, the
outcome is a discrete coarse quantum record, and the dual map is
one-way rather than two-way.

This is the same diagnostic move that Zeh \citep{Zeh1970},
Joos--Zeh \citep{JoosZeh1985}, and Zurek
\citep{Zurek1981,Zurek2003} made for quantum mechanics in
1970--2003, applied at the holographic layer rather than at the
dynamical layer. The string-theory community has not yet
absorbed the implications at the holographic layer because the
closed-system idealization has been operationally adequate for the
calculational targets the community pursues (correlators,
viscosity bounds, entanglement entropies). The open-system
extension becomes essential only when one asks: \emph{how does a
finite boundary observer actually register the prediction?} The
cohort framework's claim is that the open-system extension is not
optional --- it is the actual physics --- and that the
closed-system holographic dictionary is therefore the AX-3-off
corner of a more general operational framework.

Four properties of the boundary observer are absent from the
closed-system AdS/CFT formulation and present in $\OGGEP{}$:
\begin{enumerate}[leftmargin=2.2em,label=(\roman*)]
\item \emph{Discrete} --- instruments produce discrete outcomes
$m \in \mathcal{M}$, not continuous amplitudes. AdS/CFT computes
continuous correlators; only the CPTP discipline turns them into
the statistical fingerprint of a registered record.
\item \emph{Coarse} --- the boundary observer has finite
Hilbert-space dimension, integration time, and resolution.
AdS/CFT assumes an idealized infinite-dimensional unitary CFT;
the boundary observer in nature is finite.
\item \emph{Sampling} --- the boundary record is a sample drawn
from a distribution that the bulk state determines, with outcome
probabilities $p(m) = \Tr[\mathcal{I}_m(\rho)]$. AdS/CFT treats
the Born rule as separately postulated; the cohort framework
derives it as $\T{7.3}$ of \citep{ArisakaA5GDOS}.
\item \emph{Quantum} --- the sampling is governed by CPTP /
POVM / Stinespring discipline, with Bell-CHSH violation
registered as a structural fingerprint of the bulk-side composite
connection. AdS/CFT computes the Bell correlator but cannot
derive its operational registration without invoking open-system
content extra-axiomatically.
\end{enumerate}
All four properties are properties of \emph{nature}; all four
are missing from the closed-system formulation of AdS/CFT. The
cohort framework's claim is therefore precise: AdS/CFT has the
bulk geometry and the conformal-boundary correlator structure
correct; it is \emph{missing} the operational discipline that
turns those correlators into registered boundary records. The
closed-system limit (AX-3 off) reproduces AdS/CFT exactly; the
open-system content (AX-3 on) is what the cohort adds.

The empirical signature is testable. Because AdS/CFT is the
closed-system limit, the cohort framework predicts that AdS/CFT
will deviate from physical reality precisely in regimes where the
boundary observer's open-system content matters: in low-statistics
measurements, in single-shot experiments, in finite-resolution
detectors, and in any holographic computation that asks ``what
does the boundary observer actually register?'' rather than
``what is the correlator?'' These deviations are the empirical
content of moving from Stratum 2 to Stratum 3. The Bell-CHSH
violation derived as $\T{7.7}$ of \citep{ArisakaA5GDOS} is the
sharpest example; the textbook QM postulates derived as the
$\T{7.1}$--$\T{7.7}$ chain are the structural payoff.

\emph{Summary.} AdS/CFT had the closed-system limit of
$\OGGEP{}$ in 1998. What AdS/CFT did not have, and what the
cohort adds, is the open-system extension that turns boundary
correlators into registered records --- the discipline that
\emph{makes the holographic dictionary apply to the actual
finite boundary observers that perform measurements in nature}.
This is not a renaming. It is the structurally most consequential
sharpening of the holographic dictionary since the original 1997
formulation, and it is the load-bearing technical content of
$\AX{3}~\OGGEP{}$ in the constitutional reading.

%% =====================================================================
%% §15.2 — Cluster C: AX-4 MSMF and OGN-5 uniqueness (Objections 13-15)
%% =====================================================================
\end{objqabody}

\bigskip
\section{Methodology: axioms, ordering, and the integer ledger}

\bigskip
\noindent\textbf{Objection 13 (O-Constitution-13) (the philosophy-of-physics specialist)}\par\nobreak\smallskip
\noindent\textbf{``$\AX{4}~\MSMF{}$ is essentially Occam's razor elevated to a
 postulate. Is this physics or aesthetics?''}\par\nobreak\smallskip
\begin{objqabody}
A philosopher of physics fluent in model-selection theory will
object that $\AX{4}$ reads like the standard Occam-razor heuristic
restated as a constitutional axiom, and that the right scope for
parsimony is methodological (Akaike, Rissanen, Bayesian model
comparison) rather than constitutional. \emph{Response.} We
acknowledge the structural neighborhood with Occam's razor and
sharpen the distinction. Occam's razor is a methodological
preference for simpler models among observationally equivalent
alternatives; $\AX{4}~\MSMF{}$ is the precise constraint that an
admissible theory have \emph{zero free continuous parameters} at
the constitutional level. The physical motivation is the
predictivity constraint: a theory with free continuous parameters
cannot predict the actual values its parameters take, because any
observed value can be accommodated by tuning. $\AX{4}$ elevates
the predictivity constraint to a constitutional axiom: an
admissible theory must commit to specific numerical content,
leaving none of it adjustable. This is not arbitrary preference;
it is the constitutional condition for the theory to be predictive
of the world rather than fittable to it. See \citep{ArisakaA3BI6}
Common Objection~12 for the parallel statement at the \BIsix{}
level. We acknowledge that not every physicist will accept the
elevation as constitutional --- some will reasonably take the view
that $\AX{4}$ is a strong methodological commitment that the
cohort framework chooses to make and that other frameworks may
decline. The cohort program stands by the commitment because
the resulting predictions ($\Mlock \approx 3.85$~TeV,
$\Omega_m = 1/\pi$, kaxion at $\sim 8.1~\mu$eV, the integer
\OGNfive{} ledger, the 28-to-zero parameter compression of the
Standard Model) are sharp enough to be falsified within the
2026--2040 horizon. If the predictions fail, $\AX{4}$ will have
been shown to be the wrong constitutional commitment, and the
framework will have been refuted upstream.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 14 (O-Constitution-14) (the careful Diophantine analyst)}\par\nobreak\smallskip
\noindent\textbf{``The $\OGNfive{}$ uniqueness theorem you cite from \ATwo{}
 depends on the $\AX{4}$ lexicographic ordering. Under other
 orderings, the Diophantine system admits other integer tuples.
 How is this not a hidden choice?''}\par\nobreak\smallskip
\begin{objqabody}
A careful analyst who reads Theorem~3 of \citep{ArisakaA2GG6}
will notice that the Diophantine system
$K = LJ = M^2 + N^2 + L^2$ with $J = 2N + 1$ admits the integer
family $(L, N) = (5k, 3k)$ for any positive integer $k$, and that
the selection of $k = 1$ comes from the $\AX{4}$ minimality
requirement, not from the Diophantine system alone.
\emph{Response.} This is correct and is explicitly acknowledged
in both \ATwo{} and in \citep{ArisakaA3BI6} Common Objection~8.
The $\AX{4}$ minimum-dimensional-rank requirement is the
load-bearing constraint that selects $k = 1$ out of the integer
family, and we are explicit that this is an axiomatic input rather
than a Diophantine output. Honest restatement: $\OGNfive{}$
uniqueness is the joint consequence of two axiomatic inputs --- the
Diophantine system and the $\AX{4}$ ordering --- not of the
Diophantine system alone. Removing either input opens up the
integer family; with both, the family collapses to the unique
five-tuple $(1, 3, 5; 7, 35)$. The $\AX{4}$ ordering is not
hidden; it is the explicit content of the $\MSMF{}$ axiom in
Part~II \S\ref{sec:AX4} and is named in every uniqueness claim
the cohort framework makes. A reader who finds the $\AX{4}$
ordering insufficiently motivated can reject it; the cohort
framework's uniqueness claim is then weaker but the architecture
remains. We acknowledge that a more thorough ordering-independent
analysis --- exhibiting all admissible orderings, computing the
$\OGN{}$ tuple under each, and showing that the empirical
$\OGNfive{} = (1, 3, 5; 7, 35)$ is favored by data under all
admissible orderings --- would substantially strengthen the
uniqueness claim. This is recorded as Open Problem~O9 of the
cohort program.
\end{objqabody}

\bigskip
\section{Phenomenology, cosmology, and falsifiability}

\bigskip
\noindent\textbf{Objection 15 (O-Constitution-15) (the phenomenologist)}\par\nobreak\smallskip
\noindent\textbf{``Three
 fermion generations following from $N = 3$ in the $\OGNfive{}$
 ledger is exactly the kind of numerology that has misled
 physicists for a century. Replacing `we observe three
 generations' with `we postulate $N = 3$ and the constitution
 forces three generations' is not a derivation.''}\par\nobreak\smallskip
\begin{objqabody}
A phenomenologist whose career has been spent making the
empirical world fit into theoretical structures will object that
matching an observed integer with an axiomatically labeled
integer is not a derivation but a coincidence.
\emph{Response.} The cohort framework agrees that ``picking
$N = 3$ to match an observed integer'' would be numerology; it is
not what the framework does. Two clarifications are owed before
the answer, and both narrow the claim. First, $N$ is the
\emph{color} triplicity index $N_c$, not the number of fermion
generations $N_g$; the latter is not a member of this ledger and
is settled elsewhere \citep{ArisakaP2N3}, so the objection's
pairing of $N$ with the generation count is not the framework's.
Second --- and this is the substance --- $N_c = 3$ is \emph{not}
derived from the trace-ratio lock $(L-N)/L = 2/5$. That ratio is
computed on the $3{+}2$ split of the defining representation, so
deriving $N$ from it would assume what it concludes, and the
framework does not use it that way. What forces $N_c = 3$ is the
dual lock $K = LJ = M^2 + N^2 + L^2$ with the closure-index clause
$J = 2N+1$: it gives $N = L \pm \sqrt{L - M^2}$, and at the $L = 5$
fixed independently and free of $N$ by \citep{ArisakaA3BI6}
Theorem~T-BI6-9.0, the roots are $3$ and $7$, of which the block
embedding $N < L$ retains $3$ \citep[Theorem~T-BI6-10.2a]{ArisakaA3BI6}.
No observed quantity enters the chain, which is precisely what
distinguishes it from the numerology the objection has in mind. The integer
$N = 3$ is therefore not an independent input chosen to fit data;
it is a derived consequence of the four-component anomaly
polynomial structure that the cohort framework constitutionally
forces. See \citep{ArisakaA3BI6} Common Objection~8 for the
parallel statement and the explicit derivation chain. The
three-generation observation is then a \emph{consequence} of the
$\OGNfive{}$ ledger, not a fitted input. We acknowledge
honestly that a reader who has not followed the trace-ratio
derivation in \AThree{} will reasonably see the $N = 3$ match as
suspicious; that is precisely why the cohort architecture
requires the trace-ratio derivation to live in a separately
publishable cohort paper rather than be asserted in A1. The
minimum critical-reading set for this objection is A2 + A3; once
the trace-ratio lock and the minimum-dimensional-rank ordering are
both read, the $N = 3$ match becomes a structural prediction
rather than a numerological coincidence.

%% =====================================================================
%% §15.2 — Cluster D: Numerical predictions (Objections 16-18)
%% =====================================================================
\end{objqabody}

\bigskip
\noindent\textbf{Objection 16 (O-Constitution-16) (the cosmologist)}\par\nobreak\smallskip
\noindent\textbf{``$\Omega_m = 1/\pi$
 is, on the face of it, numerology. Eddington's 1929 derivation
 of the fine-structure constant from $137$ was numerology;
 Dirac's large-number hypothesis was numerology; show me why
 this is different.''}\par\nobreak\smallskip
\begin{objqabody}
A cosmologist with the history of failed numerological matches in mind will reasonably demand that the $\Omega_m = 1/\pi$
agreement (Planck~2018 $\Omega_m = 0.3153 \pm 0.0073$ vs the
cohort prediction $1/\pi \approx 0.31831$, agreement at
$\sim 0.3\sigma$) be demonstrated as a structural consequence
rather than a coincidence. \emph{Response.} The full derivation
lives in \citep{ArisakaA10Cosmos}. The key honest statement: the
cohort framework predicts \emph{four} cosmological partitions
simultaneously from one consistent bulk-geometry calculation,
\begin{equation}
\Omega_m = \frac{1}{\pi}, \qquad
\Omega_\Lambda = 1 - \frac{1}{\pi}, \qquad
\Omega_b = \frac{1}{2\pi^2}, \qquad
\Omega_{\rm DM} = \frac{1}{\pi} - \frac{1}{2\pi^2},
\label{eq:cosmo_partitions_obj16}
\end{equation}
not from four separate fitting exercises whose results happen to
sum to one. The four partitions descend from one geometric
integral over the 6D bulk; the rational fractions are
forced by the $\OGNfive{}$ integers (the integer $35 = K = LJ$,
the integer $2\pi = $ holonomy circle period) and the bulk
boundary conditions. The structural test is whether the four
partitions all match data simultaneously: $\Omega_m = 1/\pi$ at
$0.3\sigma$, $\Omega_b = 1/(2\pi^2)$ at $\sim 2\sigma$, and the
sum constraint $\Omega_m + \Omega_\Lambda = 1$ exactly. If the
cohort framework were post-hoc curve-fitting, the four
partitions would not all fall out of one calculation. The kill
threshold consolidated in Part~IV Table~\ref{tab:pull_table} is
sharp: a registered $5\sigma$ deviation from $1/\pi$ in any
next-generation cosmology dataset (CMB-S4, LiteBIRD, DESI) at
resolution $\sigma_{\Omega_m} \leq 0.003$ collapses
$\AX{4} + \POS{6} + \T{8}$ jointly. We acknowledge that the
prediction's pre-history --- whether it was a derivation made
before the Planck~2018 data were matched, or after --- is itself
relevant; the published cohort archive (arXiv submission date)
fixes the public record.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 17 (O-Constitution-17) (the LHC phenomenologist)}\par\nobreak\smallskip
\noindent\textbf{``$\Mlock
 \approx 3.85$~TeV is not a complete prediction. What is the
 state's spin? What is its production cross-section? What
 channel does it decay into? Without these, the LHC cannot
 test it.''}\par\nobreak\smallskip
\begin{objqabody}
An LHC phenomenologist whose work consists of comparing BSM
predictions to ATLAS and CMS bounds will object that a TeV-scale
mass without specified couplings is half a prediction. Current
LHC limits exclude generic $Z'$ resonances above $\sim 5$~TeV in
dilepton, RS-class KK gravitons above $\sim 4.5$~TeV in
diphoton, and stringy resonances above $\sim 5$--$7$~TeV depending
on coupling \citep{PDG2024}. A $3.85$~TeV state is therefore either already
constrained or already discovered, depending on what it is.
\emph{Response.} The LHC phenomenologist is correct: the
$\Mlock$ entry of Part~IV Table~\ref{tab:pull_table} requires
augmentation with the specific quantum numbers of the predicted
state. The full specification lives in the Predictions Branch
paper \PaperPOne{} \citep{ArisakaP1KK}, which derives the
production channel, the spin, the color assignments, and the
branching fractions from the Kaluza-Klein lock-cluster structure
of the 6D bulk of \ATwo{}. We acknowledge that the current falsifier-table entry ``Active LHC search'' is
incomplete
without these specifications, and we forward the LHC
phenomenologist to \PaperPOne{} for the full claim contract.
The constitutional discipline is honest: a TeV-scale mass without
specified couplings is not a $\POS{7}~\MF{}$-compliant claim, and
the cohort archives the full specification in \PaperPOne{} rather
than smuggling it into A1.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 18 (O-Constitution-18) (the precision physicist)}\par\nobreak\smallskip
\noindent\textbf{``$\alpha^{-1}(0) = 137$ matches the empirical value at the
 integer level, but the precision measurement is $137.036$.
 Either the cohort framework predicts the full value with the
 $0.036$ correction, or the integer-only match is numerology.''}\par\nobreak\smallskip
\begin{objqabody}
A precision physicist will demand that a four-decimal-place
empirical match be addressed at four-decimal-place precision.
\emph{Response.} Direct parallel to \citep{ArisakaA3BI6} Common
Objection~14. The integer $137$ is the leading-order
constitutional prediction; the $0.036$ correction comes from
higher-order vacuum-polarization thresholds in
\citep{ArisakaA8QM} and the renormalization-group running in
\citep{ArisakaA9SM}. The cohort framework predicts
$\alpha^{-1}(0) = 137 + \text{(specific higher-order corrections)}$;
the integer alone is not the full prediction. We do not claim
the integer $137$ as a stand-alone identity; we claim the
leading-order integer plus the derived corrections, with the full
quantitative comparison to the empirical
$\alpha^{-1}(0) \approx 137.036$ carried out in
\citep{ArisakaA9SM}. The minimum critical-reading set for this
objection is A8 + A9; once the higher-order corrections are read,
the $137$ match becomes a derived structural prediction rather
than a numerological coincidence.

%% =====================================================================
%% §15.2 — Cluster E: Measurement, swampland, competing programmes (19-21)
%% =====================================================================
\end{objqabody}

\bigskip
\noindent\textbf{Objection 19 (O-Constitution-19) (the decoherence-theory veteran)}\par\nobreak\smallskip
\noindent\textbf{``$\AX{3}~\OGGEP{}$ is decoherence theory relabeled. Zeh
 1970, Joos-Zeh 1985, and Zurek 1981/2003 made the same
 operational move forty years ago; the einselection program
 derives the textbook Born rule as a consequence of
 environment-induced superselection. What is the new content?''}\par\nobreak\smallskip
\begin{objqabody}
A veteran of the decoherence program will object that AX-3
inherits the einselection insight without adding new derivation
content. \emph{Response.} We acknowledge the structural
neighborhood with decoherence theory and sharpen the distinction.
The decoherence program is a derivation: given specific model
Hamiltonians and environments, the Born rule and the preferred
pointer basis emerge as derived asymptotics on the einselection
timescale. $\AX{3}~\OGGEP{}$ is a constitutional move:
``every system in nature is an open system'' is elevated from a
derived consequence in specific models to an axiomatic statement
that applies universally. The novelty is not the decoherence
mechanism (which the cohort framework inherits); the novelty is
the constitutional status. The technical content of the
$\T{7.1}$--$\T{7.7}$ derivation chain of
\citep{ArisakaA5GDOS} is the substantive payoff: the Born rule
($\T{7.3}$), the L\"uders projection ($\T{7.4}$), the einselection
preferred basis ($\T{7.6}$), and the Bell--CHSH violation
($\T{7.7}$) all emerge as \emph{theorems} of $\AX{3} + \POS{4}$
acting on the bulk, not as separate measurement postulates. The
decoherence program produces these results in specific models;
the cohort produces them as constitutional theorems. The
operational claim of A1 \S\ref{sec:two_remarks} Remark~1 ---
``the century-old measurement problem is finally solved by
$\AX{3}~\OGGEP{}$'' --- is therefore not the claim that the cohort
discovered decoherence. It is the claim that the cohort elevates
the decoherence insight from one effective treatment of measurement
to the constitutional definition of measurement. Cross-cite
\citep{ArisakaA5GDOS} Part~III for the $\T{7.1}$--$\T{7.7}$ chain;
cross-cite \citep{ArisakaA8QM} for the textbook reformulation in
\GGTheory{} language.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 20 (O-Constitution-20) (the swampland theorist)}\par\nobreak\smallskip
\noindent\textbf{``Your
 $\AX{4}~\MSMF{} + \POS{6}~\OGN{}$ is the strongest possible
 parsimony commitment. The swampland program (Vafa 2005 and
 successors) does similar structural work via a menu of
 conjectures (weak-gravity, distance, de Sitter, cobordism).
 How does the cohort framework relate to the swampland?''}\par\nobreak\smallskip
\begin{objqabody}
A swampland theorist who has spent the last fifteen years
sharpening the swampland conjectures will reasonably demand that
the cohort framework engage with the swampland program as a
parallel research effort. \emph{Response.} Direct parallel to
\citep{ArisakaA3BI6} \S11 (``Swampland compatibility of \GG{} and
\BIsix{}''). The cohort framework checks against the five most
widely accepted swampland conjectures and reports the result
honestly: four (no global symmetries, completeness, distance,
cobordism) are satisfied at the structural level; one
(weak-gravity) is structurally compatible with the quantitative
check deferred to the boundary spectrum derivation in \AFive{},
\ASeven{}, and \ANine{}. The de~Sitter conjecture is omitted
because its status as a universal constraint is itself disputed
in recent literature; the cohort cosmological inventory lives in
\ATen{}. The cohort framework is therefore in the landscape
rather than the swampland by the most widely accepted criteria.
The structural relationship between $\AX{4}~\MSMF{}$ and the
swampland conjectures: where the swampland says ``not every EFT
is consistent in quantum gravity,'' $\AX{4}$ says ``the
admissible EFT is the unique lexicographic maximum of the
constitutional Diophantine system.'' The cohort's commitment is
strictly stronger than the swampland's: instead of a menu of
conjectures, one constitutional axiom forbids all unforced
continuous freedom. We acknowledge that this is a stronger
claim; the cohort's empirical predictions
($\Omega_m = 1/\pi$, $\Mlock \approx 3.85$~TeV, kaxion
$\Delta\nu/\nu = 1/35$) are the burden that the stronger claim
must carry. Detailed swampland verification (WGC saturation,
cobordism enumeration) lives in \citep{ArisakaA3BI6} \S11 and is
recorded as Open Problem.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 21 (O-Constitution-21) (the quantum-gravity generalist)}\par\nobreak\smallskip
\noindent\textbf{``You compete with the swampland. What about asymptotic safety
 (Reuter 1998 and successors) and emergent gravity (Verlinde
 2011, Padmanabhan 2010)? Where do these fit in the
 constitutional classification?''}\par\nobreak\smallskip
\begin{objqabody}
A quantum-gravity generalist will reasonably demand that the
constitutional framework name its competing programs
explicitly. \emph{Response.} Asymptotic safety
\citep{Reuter1998} is a Stratum-2 alternative: it accepts
gauge-covariant gravity (consistent with $\AX{2}~\GGEP{}$ in the
closed-system regime) and uses the renormalization-group flow
(consistent with $\POS{5}~\RG{}$) to claim a UV-complete
non-Gaussian fixed point. Asymptotic safety does not yet activate
$\AX{3}~\OGGEP{}$ (no constitutional operational layer) or
$\AX{4}~\MSMF{}$ (no constitutional parsimony commitment), so it
sits at Stratum 2 in the constitutional tower alongside string
theory and AdS/CFT. The cohort framework is not in conflict with
asymptotic safety at the kinematic level; it adds the operational
extension (Stratum 3) and the parsimony commitment (Stratum 4)
that asymptotic safety does not address. Emergent / entropic
gravity \citep{Verlinde2011,Padmanabhan2010} is, by contrast,
\emph{directly incompatible} with $\AX{2}~\GGEP{}$: where the
cohort takes geometry as primitive and gauge as composite,
emergent gravity takes geometry as derived from thermodynamics
(or from entanglement entropy in the holographic version). This
is a structural fork. The cohort framework names this
disagreement as a fundamental constitutional fork, not as an
incidental detail: AX-2 picks one side of the fork, emergent
gravity picks the other. The empirical content of the choice is
testable wherever the cohort's specific quantitative predictions
($\Omega_m = 1/\pi$, $\Mlock \approx 3.85$~TeV, the kaxion
oscillation at $1/35$) hit data: if the cohort's predictions
verify, $\AX{2}$ wins; if they fail, the emergent-gravity
alternative gains ground. We acknowledge that some readers will
find the geometry-primitive commitment less natural than the
geometry-emergent stance. The cohort framework's response is
that the constitutional commitment is testable, and the next
decade of cosmological and TeV-scale data will adjudicate.

%% =====================================================================
%% §15.X --- Cluster F: Cross-domain and philosophical objections
%% (Objections 22-26, added in v7.33 from the five-reader audit:
%%   journal editor, senior string theorist, grad student/postdoc,
%%   complex-systems biologist / computational neuroscientist,
%%   philosopher of cosmos / life / mind)
%% =====================================================================
\end{objqabody}

\bigskip
\section{Scope: information, life, and cross-domain reach}

\bigskip
\noindent\textbf{Objection 22 (O-Constitution-22) (the complex-systems biologist)}\par\nobreak\smallskip
\noindent\textbf{``Your
 energy--information Arisaka constant $\QB \approx 0.156$~eV per
 qubit, multiplied by Avogadro's number, gives $\sim 15$~kJ/mol.
 How does this map onto the biological energy currency?''}\par\nobreak\smallskip
\begin{objqabody}
This is the kind of detail a biologist catches on first read.
\emph{Response.} $\QB$ is the per-qubit energy--information unit
of the boundary $\UoneQi$ current; its calibration against the
biological energy currency --- the per-qubit stoichiometry and its
standard-versus-cellular corrections --- is the empirical anchor
developed in (\citealp{ArisakaA7Qi4}; \citealp{ArisakaA11Origin}) and is not
part of the geometric construction of the present paper. The
constitutional claim is that $\QB$ supplies the
\emph{quantum-information bit energy} of the boundary current,
with the empirical mapping onto the biological currency carried
in those companions.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 23 (O-Constitution-23) (the computational neuroscientist)}\par\nobreak\smallskip
\noindent\textbf{``Your
 claim to dissolve the hard problem of consciousness by structural
 stipulation is too quick. That is renaming the problem, not
 solving it.''}\par\nobreak\smallskip
\begin{objqabody}
This is correct and we accept the criticism in the form stated.
\emph{Response.} The constitutional framework does not claim to
\emph{solve} the hard problem of consciousness in the sense of
closing the explanatory gap that Chalmers (1995), Block (1995),
and Strawson (2006) articulate. \citep{ArisakaA11Origin} adopts a
\emph{Russellian neutral-monism stipulation} at each $\GRIP{}$
event in the eighteen-layer cascade: the layered identities
(electron, atom, molecule, ATP, RNA, DNA, cell, eukaryote, brain,
conscious episode) are events on the same lawful state manifold,
and the phenomenal aspect appears as a structural feature of
sufficiently deep $\GRIP{}$ events rather than as an additional
metaphysical ingredient. This is a philosophical \emph{position}
in the lineage of Russell (1927), Stoljar (2006), and Strawson
(2006), not a \emph{proof}. What the framework adds beyond pure
philosophy is structural distinguishability: the eighteen-layer
cascade carries specific empirical predictions in
\citep{ArisakaA11Origin} that distinguish it from Friston's
free-energy principle, integrated-information theory, and global
workspace theory at the level of neural observables. The honest
position is therefore that the cohort framework offers a
\emph{structural account} of where consciousness sits in the
lawful cascade --- which is empirically constrained --- and a
\emph{philosophical stipulation} of what the phenomenal aspect is
--- which is not empirically constrained. We make both layers
explicit so that readers in philosophy of mind and in neuroscience
can engage with each on its own terms.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 24 (O-Constitution-24) (the philosopher of physics)}\par\nobreak\smallskip
\noindent\textbf{``What is
 the metaphysical status of $\AX{4}~\MSMF{}$? It does most of
 the heavy lifting --- forcing $\OGNfive{}$ uniqueness and
 collapsing the string landscape --- yet its philosophical
 foundation is not articulated.''}\par\nobreak\smallskip
\begin{objqabody}
This is the deepest single criticism of the constitution from
the philosophy-of-physics direction, and it deserves an explicit
answer. \emph{Response.} $\AX{4}~\MSMF{}$ is a
\emph{constitutional commitment}, not a metaphysical claim about
Nature. It states: among admissible bulk theories, prefer the
one of maximal symmetry and minimal continuous freedom, under an
explicit lexicographic ordering (maximize symmetry first,
minimize freedom second). In its philosophical lineage this
clause descends from William of Occam's \emph{numquam ponenda
est pluralitas sine necessitate} (c.\ 1320s) through Mach's
relational principle (\textit{Science of Mechanics}, 1893) to
Einstein's 1933 Herbert Spencer Lecture (``the supreme task of
the physicist is to arrive at those universal elementary laws
from which the cosmos can be built up by pure deduction'').
$\AX{4}$ promotes that philosophical preference from
\emph{methodological heuristic} to \emph{constitutional filter}:
where the preceding three axioms admit a continuous family of
solutions, $\AX{4}$ selects the lexicographic maximum. This is
\emph{normative-but-falsifiable}: if a second admissible bulk
theory is found under the same Diophantine system and the same
ordering but with equal predictive power and lower symmetry,
$\AX{4}$ fails and the constitution falls. The reader who wishes
to defend a different normative ordering ($\AX{4}'$) is free to
do so; what the constitution forbids is the silent use of an
ordering without naming it. See \citep{ArisakaA4Euclid} for the
extended philosophical argument that locates $\AX{4}$ in the long
Euclidean tradition of admissibility-by-rulebook rather than
admissibility-by-data.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 25 (O-Constitution-25) (the cross-domain reader)}\par\nobreak\smallskip
\noindent\textbf{``You claim
 the same four axioms organize the proton and the conscious
 moment. This is either the deepest empirical regularity ever
 stated, or a category error. Which is it?''}\par\nobreak\smallskip
\begin{objqabody}
We accept that this is the most contestable claim of the
program, and we state the honest answer. \emph{Response.} It
is a \emph{structural claim about a program of inquiry}, not a
settled empirical regularity. The twenty-two-paper cohort develops
the eighteen-layer $\GRIP{}$ cascade in \citep{ArisakaA11Origin}
as a sequence of forced identities --- gauge symmetry breaking
$\to$ atomic ground state $\to$ molecular bonding $\to$ ATP
energy quantum $\to$ RNA $\to$ DNA $\to$ first cell (LUCA) $\to$
eukaryogenesis $\to$ neural integration $\to$ conscious episode
--- each layer a theorem of the same axiomatic system applied at
a different stratum. Each layer carries a specific empirical
prediction with a specific live falsifier; the framework is
therefore empirically distinguishable layer by layer. If the
cascade fails at any layer --- if the $\QB$ value disagrees with
its energy--information calibration ((\citealp{ArisakaA7Qi4}; \citealp{ArisakaA11Origin})), if
the developmental-timing predictions disagree with embryology,
if the neural-integration prediction disagrees with the
cortical-rhythm data --- the failure is locatable and the
universal-identity claim shrinks to whatever layer survives.
The framework is therefore neither a settled result nor a
category error; it is a \emph{disciplined program} with twelve
to seventeen explicit falsifiers, and its scope shrinks or
expands by the next decade's experiments. The honest position
is that we do not know whether the four axioms truly organize
the long sequence of identities, and we have written down the
specific tests by which the question can be settled.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 26 (O-Constitution-26) (the bio/neuro phenomenologist)}\par\nobreak\smallskip
\noindent\textbf{``To be
 evaluated by the complex-systems biology and computational
 neuroscience community, your framework needs specific
 quantitative predictions that distinguish it from
 Friston's free-energy principle, Tegmark's integrated
 information theory, and Baars-Dehaene global workspace
 theory.''}\par\nobreak\smallskip
\begin{objqabody}
This is correct and the requirement is fair. \emph{Response.}
\citep{ArisakaA11Origin} carries the specific predictions that
distinguish the $\GRIP{}$ cascade from the three named
alternatives. Among them: (i) the $\QB$-mediated energy--information
calibration described in Objection~22 above, which is
quantitative and empirically tight; (ii) specific
developmental-timing predictions at the cell-division layer,
keyed to the integer $K = 35$ of the $\OGNfive{}$ ledger;
(iii) a specific spectral signature in cortical gamma-band
activity at frequencies $\nu_{\rm GG}$ predicted by the
neural-integration layer of the cascade, distinct from the
40~Hz central frequency of the binding-by-synchrony framework;
(iv) a specific information-integration coefficient that
distinguishes the cascade prediction from the $\Phi$ coefficient
of IIT. Each prediction carries a $5\sigma$ kill criterion in
\citep{ArisakaA11Origin}; specific failures collapse specific
$\GRIP{}$ layers without collapsing the constitutional core
(the upstream axioms). We acknowledge that the bio/neuro
community will require independent reproduction of each
prediction before accepting the cascade. The cohort framework
welcomes this scrutiny: the discipline of the constitution is
that any specific failure is locatable, not that the framework
is correct.


\end{objqabody}
\bigskip
\section{The four axioms: independence, necessity, and minimality}

\bigskip
\noindent\textbf{Objection 27 (O-Constitution-27) (the axiomatic-systems theorist)}\par\nobreak\smallskip
\noindent\textbf{``Are the four axioms genuinely independent? An axiom that is a theorem of the others is not an axiom.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Independence is argued in the inter-axiom analysis (\S\ref{sec:inter_axiom}) and summarized in the axiom table (\S\ref{sec:axiom_table}): each clause is load-bearing because dropping it removes a class of admissible structure the others cannot restore. Without \AX{1} (\LC{}) the local-field arena is lost; without \AX{2} (\GGEP{}) the geometry and gauge sectors decouple and the 6D bulk is no longer forced; without \AX{3} (\OGGEP{}) the closed bulk has no boundary law, hence no measurement or thermodynamic content; without \AX{4} (\MSMF{}) the moduli return and uniqueness fails. The constitutional tower (\S\ref{sec:V-constitutional-tower}) makes this operational: every existing framework sits at the level of the axioms it uses, and the missing axioms predict its missing structure. None is a theorem of the others, which is exactly the test the objection demands.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 28 (O-Constitution-28) (the skeptic)}\par\nobreak\smallskip
\noindent\textbf{``\MSMF{} (AX-4) is circular: you call whatever the framework derives `minimum freedom.' How is it a constraint rather than a label applied after the fact?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} \MSMF{} is stated as a constraint with teeth \emph{before} any result is derived (\S\ref{sec:AX4}): it forbids any continuous parameter or modulus not fixed by an explicit constitutional clause, and demands the maximal symmetry consistent with the other axioms. Its non-circularity shows in what it \emph{excludes} in advance --- the string landscape's moduli, a free compactification radius, any tunable coupling --- each a concrete structure other frameworks permit and \MSMF{} forbids (\S\ref{sec:V-quantitative-consequences}, the vacuum count). A label applied after the fact cannot exclude anything beforehand; \MSMF{} removes the moduli space before the integer ledger is derived. Its bite is measured by the freedom it removes, not by the results it later names --- and that removal is the step that turns ``a geometry consistent with the data'' into ``the unique geometry.''
\end{objqabody}

\bigskip
\noindent\textbf{Objection 29 (O-Constitution-29) (the relativist)}\par\nobreak\smallskip
\noindent\textbf{``AX-1 (Local Causality) is assumed by all of physics. It does no discriminating work --- why elevate a triviality to a founding axiom?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} \AX{1} is not there to discriminate among modern theories --- nearly all respect it --- but to fix the \emph{arena} in which the other axioms operate and to forbid the non-local shortcuts that an extra-dimensional construction could otherwise smuggle in (\S\ref{sec:AX1}). Its work is constraint at the boundary: it is what guarantees that the two non-spatial dimensions of the 6D bulk carry no faster-than-light signaling, so that the geometry--gauge equivalence of \AX{2} does not violate causality (\S\ref{sec:inter_axiom}). A founding charter must state even the conditions everyone assumes, precisely so that a later step cannot quietly drop them; \AX{1} is the clause that the \OGGEP{} boundary projection and the 6D bulk are held to. Triviality among 4D theories is not triviality once extra structure is admitted.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 30 (O-Constitution-30) (the gauge theorist)}\par\nobreak\smallskip
\noindent\textbf{``AX-2 says geometry and gauge are equivalent. Is that a definition, a conjecture, or a theorem? If a definition it is empty; if a conjecture it is unproven.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} It is an \emph{axiom} --- a stated principle taken as primitive, exactly like the equivalence principle in general relativity, which is also neither a definition nor a theorem but a posited identity whose justification is the structure it forces (\S\ref{sec:AX2}). \GGEP{} posits that the frame/spin connection and the internal gauge connection are components of one composite connection. That is not empty: it has the falsifiable consequence, proved in the companion papers, that the modified Bianchi identity \BIsix{} locks gravitational and gauge anomalies together \citep{ArisakaA3BI6}, and that the bulk gauge content is fixed rather than chosen \citep{ArisakaA2GG6}. So the status is: posited here, its consequences proved downstream, and the whole standing or falling on whether those consequences hold (\S\ref{sec:constitutional_handoff}). An axiom is judged by its yield, not by being a theorem of something prior.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 31 (O-Constitution-31) (the foundations theorist)}\par\nobreak\smallskip
\noindent\textbf{``Why exactly four axioms? Why not three, by dropping one, or five, by adding one? Is four forced or aesthetically chosen?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Four is the minimal independent set that closes the construction, and the paper shows the closure rather than asserting the number (\S\ref{sec:inter_axiom}, \S\ref{sec:axiom_table}): \AX{1} fixes the arena, \AX{2} unifies geometry and gauge, \AX{3} opens the system to a boundary law, \AX{4} removes residual freedom. Drop any one and a named class of results disappears (Objection~27); add a fifth and, by \MSMF{} itself, it must either be derivable from the four --- hence not an axiom --- or introduce freedom the construction forbids. The four therefore are not an aesthetic choice but the answer to ``what is the smallest rule-set that forces a unique, anomaly-free, measurable theory?'' The honest qualifier: minimality is a claim about \emph{this} construction, argued in \S\ref{sec:inter_axiom}, not a proof that no other four-clause system exists.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 32 (O-Constitution-32) (the logician)}\par\nobreak\smallskip
\noindent\textbf{``Your axioms rest on undefined primitives --- `geometry,' `gauge,' `openness,' `freedom.' An axiom system built on undefined terms proves nothing.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The primitives are defined, in the working-definitions section that opens the formal development (\S\ref{sec:definitions}) and in the shared mathematical arena the axioms inhabit (\S\ref{sec:axioms_arena}): ``geometry'' is the six-dimensional manifold with its metric and connection, ``gauge'' the principal-bundle connection, ``openness'' the finite-resolution boundary projection of \OGGEP{}, and ``freedom'' a continuous modulus or coupling not fixed by a clause. These are the same primitives modern geometry and gauge theory already use; the constitution does not invent new undefined notions, it commits to specific instances of standard ones. Where a term is program-specific (\GDOS{}, the integer ledger), it is forwarded to the companion paper that constructs it (\S\ref{sec:constitutional_handoff}). So the system is not built on undefined terms; it is built on standard mathematical objects with their definitions stated up front.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 33 (O-Constitution-33) (the philosopher)}\par\nobreak\smallskip
\noindent\textbf{``Who decides what is `admissible'? An admissibility predicate is just your preferences elevated to law.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The predicate is fixed, public, and applied uniformly, which is the only defense any framework can offer and the reason the paper states it explicitly so it can be contested (\S\ref{sec:postulates_role}, \S\ref{sec:definitions}): a theory is admissible iff it satisfies the eleven clauses, each a checkable condition. The check against ``mere preference'' is that the predicate \emph{discriminates}: run against general relativity, gauge theory, string theory, quantum mechanics, the Standard Model, and $\Lambda$CDM it returns a different axiom-subset for each (\S\ref{sec:V-master-ledger}), and it rules inadmissible any formulation carrying free moduli --- including formulations of \emph{this} program's own content that do. A predicate that admitted whatever the author liked could not separate frameworks, nor exclude versions of its own program. Preferences enter only in choosing \emph{which} clauses to adopt --- and those are on the page, falsifiable by their consequences, not hidden.
\end{objqabody}

\bigskip
\section{The seven postulates and the axiom/postulate split}

\bigskip
\noindent\textbf{Objection 34 (O-Constitution-34) (the methodologist)}\par\nobreak\smallskip
\noindent\textbf{``Why split the rules into four axioms \emph{and} seven postulates? The distinction looks arbitrary --- relabel and you have eleven axioms.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The split is principled, not cosmetic, and the paper states the principle (\S\ref{sec:why_axioms_postulates}, \S\ref{sec:postulates_role}): the four axioms are the always-on commitments that define the arena and the geometry--gauge--openness structure; the seven postulates are runtime admissibility filters that a \emph{candidate Lagrangian} must pass once the arena is fixed (anomaly freedom, closure/positivity, RG-compatibility, and so on). Axioms say what kind of object a theory is; postulates say which such objects survive. Collapsing them into ``eleven axioms'' would lose the operational distinction that makes the constitution usable as a filter --- you would no longer know which clauses constrain the construction and which screen the output. The dependency chain among the postulates (\S\ref{sec:postulate_dag}) shows they form an ordered filter, not a second pile of axioms.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 35 (O-Constitution-35) (the skeptic)}\par\nobreak\smallskip
\noindent\textbf{``Seven postulates is a suspiciously resonant number. Is seven forced, or chosen for effect?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Seven is the number of independent runtime filters the construction actually needs, enumerated and ordered in the postulate dependency chain (\S\ref{sec:postulate_dag}) and tabulated in \S\ref{sec:postulates:table}; it carries no numerological weight and is not tied to the integer ledger (which lives in \AThree{} \citep{ArisakaA3BI6}, not here). Each postulate is defended by what fails without it: drop anomaly freedom (\POS{3}) and the quantum theory is inconsistent; drop closure/positivity (\POS{4}) and probabilities go negative; drop RG-compatibility (\POS{5}) and the scale structure breaks. If a future analysis showed two postulates were equivalent, the count would honestly drop to six --- the number is an output of the enumeration, not a target. We make no claim that ``seven'' is itself significant; the significance is that each filter is necessary and the set is complete for admissibility.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 36 (O-Constitution-36) (the axiomatic-QFT specialist)}\par\nobreak\smallskip
\noindent\textbf{``POS-3 (anomaly freedom by inflow) is a standard consistency condition, not a `postulate.' Dressing consistency requirements as postulates inflates the count.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} \POS{3} \emph{is} a consistency condition --- the paper does not claim otherwise (\S\ref{sec:POS3}); what makes it a postulate of \emph{this} constitution is that it is promoted to an admissibility filter that any candidate must pass \emph{and} given a specific geometric realization (anomaly cancellation by inflow from the bulk, the Green--Schwarz mechanism elevated to a constitutional clause and proved for \BIsix{} in \citet{ArisakaA3BI6}). The objection is right that, viewed from ordinary QFT, anomaly freedom is mandatory; the constitutional move is to make it a \emph{stated} clause rather than an implicit assumption, so that the eleven-clause filter is complete and self-contained. That is the opposite of inflation: it is refusing to leave a load-bearing requirement unstated. The count reflects the filters a candidate must pass, and anomaly freedom is genuinely one of them.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 37 (O-Constitution-37) (the renormalization-group theorist)}\par\nobreak\smallskip
\noindent\textbf{``POS-5 (RG-compatibility) is just a property any sensible quantum field theory already has. Why postulate it?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} It is postulated because the construction makes the renormalization-group scale a genuine \emph{geometric direction} (the scale/RG interval of the bulk), so RG-covariance is not automatically inherited --- it must be imposed as the condition that the 4D effective theory be covariant under flow along that direction (\S\ref{sec:POS5}). In ordinary 4D QFT, RG-compatibility is indeed a background property; here it is the clause that ties the fifth dimension to the renormalization group and forbids constructions in which the scale direction and the RG flow disagree. So \POS{5} does work that ``any sensible QFT'' does not require: it constrains how the extra-dimensional geometry projects to four dimensions. The honest point the objection contains is that, restricted to the 4D output, \POS{5} looks standard --- but its constitutional role is upstream, at the bulk-to-boundary reduction.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 38 (O-Constitution-38) (the foundations theorist)}\par\nobreak\smallskip
\noindent\textbf{``Are the postulates testable? If POS-4 (closure and positivity) is violated, what experiment shows it --- or is it unfalsifiable by construction?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The postulates are testable indirectly, through the theory they admit: \POS{4} (closure/positivity, \S\ref{sec:POS4}) is the clause that forces completely-positive boundary dynamics, and a violation would show up as negative probabilities or non-unitary evolution in the quantum sector --- which is exactly what the measurement and QFT derivations of the companion papers reproduce as standard QM, and which any experiment confirming the Born rule and CPTP channels confirms in turn (\citealp{ArisakaA5GDOS}; \citealp{ArisakaA8QM}). A postulate is not falsified in isolation; it is falsified through the empirical failure of the construction it licenses. The sharp falsifiers are downstream and named (the $3.85$~TeV resonance \citep{ArisakaP1KK}, proton decay, the $\pi$-locked cosmic inventory \citep{ArisakaA10Cosmos}). What we do \emph{not} claim is a tabletop test of \POS{4} alone --- and we say so rather than pretend each clause has its own experiment.
\end{objqabody}

\bigskip
\section{The ``constitution'' framing and the descent claim}

\bigskip
\noindent\textbf{Objection 39 (O-Constitution-39) (the philosopher of science)}\par\nobreak\smallskip
\noindent\textbf{```Constitution' is a political metaphor --- a social contract, amendable by vote. Physical law is not legislated. Isn't the framing rhetorical inflation?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The metaphor is used for one precise feature and the paper is explicit about its limits (\S\ref{sec:why_axioms_postulates}): like a legal constitution, the document states a small set of foundational rules \emph{in writing}, against which all subsequent claims are audited --- as opposed to the unwritten ``craft tradition'' that has governed admissibility in physics for a century. It is emphatically \emph{not} claimed that the rules are chosen by agreement or amendable by preference; their warrant is empirical, and they stand or fall on their consequences (\S\ref{sec:constitutional_handoff}). The borrowed feature is auditability, not legislation. If the word's political connotations mislead, the substance is unchanged: a short, written, checkable rule-set with a forwarding pointer to where each consequence is proved. We use ``constitution'' for the writing-it-down gesture, and disclaim the social-contract reading.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 40 (O-Constitution-40) (the skeptic)}\par\nobreak\smallskip
\noindent\textbf{``You claim the eleven clauses \emph{force} the 6D bulk, the anomaly identity, and the boundary law. But those forcings are proved in companion papers, not here. So A1 asserts what it does not demonstrate.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Correct, and the paper states this as a deliberate division of labor rather than a hidden gap (\S\ref{sec:constitutional_handoff}, \S\ref{sec:coherence_map}): A1's job is to state the eleven clauses, name the eleven Principal Theorems, and forward each proof to the companion where it lives --- the 6D bulk to \citet{ArisakaA2GG6}, the anomaly identity to \citet{ArisakaA3BI6}, the boundary law to \citet{ArisakaA5GDOS}. An axiomatic document that included its own proofs would be too long to audit at the constitutional layer. So A1 does not \emph{demonstrate} the forcings; it \emph{claims} them and says exactly where each is demonstrated, publishing the cohort simultaneously so the proof is one citation away (\S\ref{sec:min_reading_path}). A serious objection to a forcing is therefore an objection to a specific companion theorem, which the reader can locate and check --- not an unsupported assertion.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 41 (O-Constitution-41) (the mathematician)}\par\nobreak\smallskip
\noindent\textbf{``Is the descent \emph{unique}? Could a different geometry or theory also satisfy all eleven clauses? If so, the constitution under-determines physics.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Uniqueness is the central structural claim, stated once in \S\ref{sec:V-central-claim} and proved in the companions: \citet{ArisakaA2GG6} establishes the six-dimensional bulk as the unique anomaly-free, moduli-free solution of the axioms, and \MSMF{} (\AX{4}) is precisely the clause that removes the residual freedom a weaker rule-set would leave. So the constitution is \emph{claimed} not to under-determine physics, with the uniqueness theorem forwarded rather than re-proved here. The honest scope: uniqueness is established within the stated mathematical arena (\S\ref{sec:axioms_arena}); it is not a metamathematical proof that no conceivable structure outside that arena could mimic the clauses. If a genuinely distinct solution satisfying all eleven clauses were exhibited, that would falsify the uniqueness claim --- which is exactly the kind of sharp, checkable target an axiomatic program should offer.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 42 (O-Constitution-42) (the logician)}\par\nobreak\smallskip
\noindent\textbf{``An axiom system rich enough to `force all of physics' risks Gödelian incompleteness or outright inconsistency. Have you checked the eleven clauses are even mutually consistent?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Mutual consistency is demonstrated by \emph{exhibition}, which is the standard route for a physical axiom system: the companion papers construct an explicit model --- the 6D bulk with its anomaly-free \BIsix{} identity and \GDOS{} boundary law --- that satisfies all eleven clauses simultaneously (\citealp{ArisakaA2GG6}; \citealp{ArisakaA3BI6}), so the set has a model and is therefore consistent. On Gödel: the constitution is not a formal system aspiring to prove all arithmetic truths; it is a finite filter on physical theories, so Gödelian incompleteness of arithmetic does not bear on it (\S\ref{sec:why_axioms_postulates}). What the clauses ``force'' is a specific geometry and its low-energy content, not a complete decidable theory of everything. The relevant consistency question --- can all eleven hold at once? --- is answered yes by the existence of the model, and any failure to construct that model would have falsified the program.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 43 (O-Constitution-43) (the auditor)}\par\nobreak\smallskip
\noindent\textbf{``If a companion-paper prediction fails, do you revise an axiom or the companion paper? Without an answer, the constitution is unfalsifiable --- failure is always someone else's fault.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The answer is fixed in advance by the descent structure, which is what makes the framework falsifiable rather than evasive (\S\ref{sec:problems_solved}, \S\ref{sec:V-central-claim}): because each prediction is forwarded to a named theorem that follows from named clauses, a failed prediction propagates \emph{upward} to the specific clause or proof it depends on. A failed $3.85$~TeV resonance \citep{ArisakaP1KK} or a kaxion absent near $1.95$~GHz \citep{ArisakaC1Kaxion} does not leave a free parameter to retune --- \MSMF{} forbids that --- so it impugns the scale/integer structure that the axioms force, not a dispensable modeling choice. The coherence map (\S\ref{sec:coherence_map}) makes the dependency explicit precisely so that blame cannot be shifted: each prediction names its upstream clauses. ``Someone else's fault'' is impossible when the chain is published and one-directional; a break anywhere is a break in the constitution.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 44 (O-Constitution-44) (the historian of science)}\par\nobreak\smallskip
\noindent\textbf{``The `geometrization ladder' from Einstein to GG-Theory is a teleological narrative --- Whig history dressed as inevitability.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The risk is real and the ladder is offered as a falsifiable reading, not a teleology (\S\ref{sec:geometrization-ladder-a1}): the claim is the checkable pattern that each major step absorbed an empirical constant or symmetry into geometry at the cost of added structure --- $c$ into Minkowski spacetime, $G_N$ into curvature, gauge fields into bundles --- and that the constitution proposes the next rung. It is not claimed that Einstein or Weyl foresaw \GGTheory{}; the pattern is read off the primary record and would fail if the historical absorptions had gone otherwise. The constitution is presented explicitly as a \emph{proposal} for the next rung, to be vindicated or refuted by experiment (\S\ref{sec:direction}), not as the inevitable terminus of history. We invite the reader to judge whether the ladder is forced or merely suggestive --- the same standard the physics is held to.
\end{objqabody}

\bigskip
\section{The constitutional tower: classifying twentieth-century physics}

\bigskip
\noindent\textbf{Objection 45 (O-Constitution-45) (the skeptic)}\par\nobreak\smallskip
\noindent\textbf{``Placing GR, QM, string theory, AdS/CFT, the SM, and $\Lambda$CDM in a `constitutional tower by axiom subset' is post-hoc taxonomy. Any framework could retro-classify rivals by which of its axioms they use.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The taxonomy earns its keep by doing \emph{quantitative} work, not by re-labeling (\S\ref{sec:V-quantitative-consequences}): each axiom subset is tied to a count --- admissible vacua and free continuous parameters --- so the classification \emph{predicts}, for instance, why the string cell carries a vast vacuum landscape and the full-axiom cell does not. A purely post-hoc taxonomy cannot produce those numbers from the cell alone. The tower is walked stratum by stratum with each assignment argued individually (\S\ref{sec:V-walking-the-tower}, \S\ref{sec:V-master-ledger}), not asserted in bulk. The honest limit, stated in \S\ref{sec:V-synthesis}: the classification is only as objective as the clause definitions, and assigning a framework to a cell is a modeling judgment a critic can dispute case by case --- which is why each is defended, not decreed.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 46 (O-Constitution-46) (the string theorist)}\par\nobreak\smallskip
\noindent\textbf{``You assign string theory to a specific axiom-subset cell. String theorists would dispute that cell. By what \emph{objective} criterion is a framework placed?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The criterion is which clauses a framework \emph{implicitly} satisfies versus leaves open, applied uniformly and argued per framework (\S\ref{sec:V-walking-the-tower}, \S\ref{sec:V-master-ledger}): string theory is placed where it instantiates \AX{1}--\AX{2} (it geometrizes gauge structure and uses the anomaly-inflow machinery) but does not adopt \AX{4} in the \MSMF{} sense, which is why it retains a moduli landscape rather than a unique vacuum. That is not an external verdict imposed on the string community; it uses the landscape's \emph{own} count of vacua as the diagnostic (\S\ref{sec:V-quantitative-consequences}). A string theorist who disputes the cell is welcome to point to the clause they believe string theory does satisfy and the construction does not --- which is a concrete, resolvable disagreement, not a matter of taste. The classification is built to be argued, and the criterion is on the page.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 47 (O-Constitution-47) (the philosopher)}\par\nobreak\smallskip
\noindent\textbf{``The `vacuum count per axiom subset' presupposes your notion of vacuum and admissibility. You defined the scoreboard --- the comparison is rigged.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The scoreboard is defined once, in advance, and stated so it can be contested (\S\ref{sec:V-quantitative-consequences}, \S\ref{sec:definitions}); the guard against rigging is that each cell's count uses that framework's \emph{own} vacuum notion --- the string cell's landscape number is the string literature's own, not an externally imposed one, and the full-axiom cell's uniqueness is the \OGNfive{} integer-ledger result of \citet{ArisakaA3BI6}. So the comparison is not ``our metric flatters us''; it is ``each framework, counted by its own admissibility, lands in a different cell.'' The residual subjectivity --- which cell a framework belongs to --- is conceded openly (\S\ref{sec:V-synthesis}); the counts \emph{within} a cell are not free. A rigged scoreboard would let the author tune the numbers; here the numbers are inherited from each framework's own practice.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 48 (O-Constitution-48) (the working physicist)}\par\nobreak\smallskip
\noindent\textbf{``What does the constitutional classification let me \emph{do} that I could not before? Is it a research tool or a retrospective filing system?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} It is meant as a research tool, and the paper says what it buys (\S\ref{sec:V-synthesis}): the tower tells you, for a given open problem, which axiom a candidate resolution must \emph{add} --- e.g.\ a measurement-problem proposal that does not adopt \AX{3} (\OGGEP{}) is, by the classification, missing the clause that supplies a boundary law, so its incompleteness is predicted, not discovered by trial. It also locates where a new result must sit and which companion theorem it would touch (\S\ref{sec:coherence_map}). The honest answer to ``can I compute something new with it today?'' is that the classification is a navigational and diagnostic instrument, not a calculational one; the calculational payoff lives in the companion papers it points to. As a filing system it would be inert; as a map of which axiom each gap needs, it is directional.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 49 (O-Constitution-49) (the Hilbert scholar)}\par\nobreak\smallskip
\noindent\textbf{``Hilbert's sixth problem asked to axiomatize physics a century ago and is generally judged to have failed. How is this constitution different from --- or better than --- that program?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The comparison with historical axiom systems is drawn directly (\S\ref{sec:comparison_history}). Hilbert's program sought to axiomatize \emph{specific} theories (mechanics, probability) from within, and stalled where the physics itself was unsettled. The constitution does something narrower and later: it does not axiomatize a finished theory but states admissibility rules that \emph{select} a theory, and it is offered at a moment when the candidate geometry and its consequences can be written down and forwarded for proof (\S\ref{sec:constitutional_handoff}). It is ``better'' only in the testable sense that it commits to sharp, falsifiable downstream predictions rather than to a formalization program. Whether it succeeds where Hilbert's stalled is an empirical question (\S\ref{sec:direction}); the paper claims the \emph{form} Hilbert sought --- a written rule-set --- with the added discipline that every clause forwards to a checkable consequence.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 50 (O-Constitution-50) (the QM-reconstruction theorist)}\par\nobreak\smallskip
\noindent\textbf{``Hardy (2001), Chiribella--D'Ariano--Perinotti, and Masanes--M\"uller already reconstruct quantum mechanics from operational axioms. Why a new constitution instead of building on those?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Those reconstructions are acknowledged as the right kind of work and are not duplicated (\S\ref{sec:comparison_history}); the difference in scope is the point. The operational reconstructions axiomatize \emph{quantum theory alone} --- they recover Hilbert space and the Born rule from information-theoretic principles, but say nothing about gravity, gauge structure, or the Standard-Model parameters. The constitution's claim is that \emph{one} rule-set forces quantum mechanics, general relativity, gauge field theory, and the cosmological inventory together, with the QM sector recovered as the \AX{3} (\OGGEP{}) boundary law (\citealp{ArisakaA5GDOS}; \citealp{ArisakaA8QM}). So the relationship is not rivalry but containment: the operational QM axioms correspond, in the tower, to the cell that adopts the openness/measurement clause without the geometry--gauge ones. Building ``on'' them would still leave gravity and the gauge sector unaxiomatized, which is exactly the gap the constitution exists to close.
\end{objqabody}

\bigskip
\section{The meta-status: falsifiability, brevity, and arrival}

\bigskip
\noindent\textbf{Objection 51 (O-Constitution-51) (the Popperian)}\par\nobreak\smallskip
\noindent\textbf{``A framework that reproduces all of established physics as theorems risks being unfalsifiable --- it predicts exactly what we already see. What does the constitution itself \emph{forbid}?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The reproduction of known physics is the consistency requirement; the falsifiable content is what the \emph{shared} clauses forbid (\S\ref{sec:problems_solved}). Because measurement, gravity, and the gauge sector descend from one rule-set with no free dimensionless parameter (\MSMF{}), they are not independently adjustable: the constitution forbids, for example, a Standard Model whose anomaly structure escapes the \BIsix{} lock, a cosmic inventory off the $\pi$-locked partition \citep{ArisakaA10Cosmos}, and a measurement rule that is not completely positive. Each forbidden possibility is a way the world could be that the constitution says it is not --- and several are being tested (the KK resonance \citep{ArisakaP1KK}, proton stability). A framework that forbids nothing would indeed be empty; this one forbids the moduli landscape, the tunable coupling, and the decoupled sector, and stakes itself on their absence.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 52 (O-Constitution-52) (the senior community member)}\par\nobreak\smallskip
\noindent\textbf{``Twenty-one single-author papers posted at once, with no prior peer review --- how should the community weight a constitution that arrives all together?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} By its theorems and its falsifiers, not its mode of arrival --- and the simultaneity is a deliberate feature that aids, rather than impedes, scrutiny (\S\ref{sec:min_reading_path}, \S\ref{sec:coherence_map}): because every claim in A1 forwards to a companion proof, the companion papers are posted together so that each proof is one citation away rather than a promise. One paper of the twenty-two is deliberately not in that set. The capstone \AEleven{}, whose upper layers are not yet at the standard the rest of the cohort holds itself to, is in preparation; we name it here rather than quietly omit it, and no claim in this paper and no row of Table~\ref{tab:pull_table} depends on it. The framework is built to be audited incrementally --- a reader follows the forward-implication chain relevant to their objection without accepting the whole --- and to be killed cleanly, since one failed point prediction breaks the shared ledger. Peer review is complemented, not bypassed: the Q\&A names the strongest objections, the coherence map exposes every dependency, and the predictions invite refutation. The unusual packaging is a response to the unusual claim (that the pieces are one structure); the evidentiary standard is the ordinary one.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 53 (O-Constitution-53) (the general reader)}\par\nobreak\smallskip
\noindent\textbf{``In one sentence: what is the single new, checkable claim of \emph{A1 itself} --- not its companions --- that could not have been written before this paper?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} In one sentence: that the admissibility of a physical theory can be reduced to a written set of four axioms and seven postulates which together \emph{select} a unique geometry, and that the major frameworks of twentieth-century physics occupy definite, countable cells of the tower built from subsets of those clauses (\S\ref{sec:V-central-claim}, \S\ref{sec:V-constitutional-tower}). Everything A1 forwards --- the bulk, the anomaly identity, the boundary law --- is proved elsewhere; A1's own irreducible contribution is the \emph{constitutional reduction itself} and the classification it makes possible (\S\ref{sec:V-master-ledger}). That claim is checkable in two ways: exhibit a framework that cannot be placed in the tower, or exhibit a distinct geometry satisfying all eleven clauses, and the central claim fails. The new content is the rule-set-plus-classification, not any single re-derived equation.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 54 (O-Constitution-54) (the skeptic)}\par\nobreak\smallskip
\noindent\textbf{``The constitution `fits on one page.' Is that brevity a virtue, or a sign that all the hard content has simply been offloaded to the companion papers?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Both, and deliberately so (\S\ref{sec:why_axioms_postulates}, \S\ref{sec:constitutional_handoff}). The brevity is a virtue for the same reason a legal constitution is short: a rule-set long enough to contain its own proofs is too long to audit \emph{as} a rule-set. And yes, the hard derivations are offloaded --- explicitly, with a forwarding pointer to each (\S\ref{sec:coherence_map}) --- because the constitutional layer's job is to state the rules cleanly, not to carry the technical weight. The honest reading the objection presses is correct: A1 alone is not a substitute for the cohort, and we say so (\S\ref{sec:min_reading_path}). The one-page rule-set is the \emph{interface}; the companions are the implementation. A constitution is judged by whether the interface is clean and the implementation is reachable, and both are designed to be exactly that.
\end{objqabody}

\bigskip
\paragraph{Closing remark on the objections.}
After all, an honest framework should know which questions a working
physicist is most likely to raise on first reading, and should answer
those questions before they become obstacles to engagement. Fifty-four
such questions are answered above, in eleven
sections grouped by the community that raises them: the six
first-pass readings, the six on the cohort architecture and the
placement of string theory, the two on methodology and the integer
ledger, the twelve on phenomenology, cosmology and cross-domain
reach, and the twenty-eight that press hardest on the foundations
themselves --- the independence of the
axioms, the axiom/postulate split, the ``constitution'' framing, the
classification of twentieth-century physics, and the meta-status of a
framework that reproduces known physics as theorems. They are preceded
by twenty-five questions of a different kind, in
\S\ref{sec:qa-curious}, asked in plain language and answered without
formalism.

Eighteen of the fifty-four concede something: a structural sharpening
the audit surfaced, a philosophical limitation the framework must
carry, or a debt to earlier work that the constitutional reading does
not discharge. Those concessions are the part of this Part we would
ask a hostile reader to weigh first, because a rule-set that never
loses an argument has not been tested. The remainder are answered
within the constitutional reading itself, each with a forward pointer
to the cohort proof location where the technical content sits.

The cohort architecture is, we hope, now visible: \AOne{} is the
rulebook; \ATwo{} and \AThree{} prove the bulk theorems and the
anomaly proof; \AFive{} and \AEight{} prove the boundary law and the
textbook QM reformulation; \ANine{}, \ATen{} and \AEleven{} prove the
Standard Model parameter collapse, the cosmological partitions, and
the closing \GRIP{} cascade. Every objection above traces to a
specific cohort proof location, and the cohort papers are released on
the same day so that the proof is one click from the objection.

We invite the reader who has finished the present Part to choose,
based on the objection that most concerns them, the minimum
critical-reading combination of \S\ref{sec:min_reading_path} and begin
there. A first-pass evaluator who completes the relevant two- or
three-paper read will be substantially better positioned than one who
has read only \AOne{}, and we hold ourselves to the discipline of
naming, for every claim in the constitution, exactly where the proof
lives. After all, the strength of a constitution is measured not by
what it claims but by how cleanly it can be audited against the
technical work that supports it; the cohort architecture is our
commitment to that audit.

\clearpage


\GGPart[References]{References}{Cited works, each with a relevance note linking it to the present paper}

\noindent Unlike a conventional reference list, every entry below carries a short \emph{Relevance} note: a sentence or two saying what the cited work contributes and where in the present paper it bears. The aim is to turn the bibliography from a bare list of sources into a navigable companion --- a reader can move through the references and see, at a glance, which axiom, postulate, theorem, or section each one underwrites.

\noindent The notes are written for two readers at once. For the newcomer, they trace the intellectual lineage of the constitution: from the foundational work on gauge theory, general relativity, the chiral anomaly, and the renormalization group, through to the twenty-two-paper \GGTheory{} cohort that the present document compiles. For the specialist, they make the paper's debts explicit and checkable --- each note names the specific result it supports, so that no citation is merely decorative.

\noindent The references are organized in two groups --- first the author's twenty-one companion \GGTheory{} papers and the
Foundation Book (Section~A below), then all other works (Section~B), each in alphabetical order by first author. The cohort papers (\AOne{}--\AEleven{}, with the C- and P-series) are cited under the canonical keys fixed by the cross-paper coherence map of Part~IV.

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

\begin{thebibliography}{99}
\bibitem[Arisaka(2026, A10-Cosmos)]{ArisakaA10Cosmos}

\textbf{Arisaka, K.\ (2026).}
\textit{The Standard Cosmology as Theorem of \GGTheory{}: The Cosmic Inventory as Observations of One Six-Dimensional Geometry}.
Arisaka-GG-A10-Cosmos preprint.
\par\noindent Preprint, available at \url{https://arisaka-gg.org/}.
\par\noindent\textbf{Relevance.} This companion paper derives the cosmological parameters
$\Omega_m = 1/\pi$, $\Omega_\Lambda = 1 - 1/\pi$, $\Omega_b = 1/(2\pi^2)$, $\Omega_{\rm DM} = 1/\pi - 1/(2\pi^2)$
from the \GGConstitution{} and the 6D bulk. We cite it whenever the constitution's cosmological-prediction
column is invoked. First cited in \S\ref{sec:intro}.

\bibitem[Arisaka(2026, A11-Origin)]{ArisakaA11Origin}

\textbf{Arisaka, K.\ (2026).}
\textit{The Geometric Inevitability of Cosmos, Life, and Mind: The Self-Referential Closure of the \GGTheory{} Program}.
Arisaka-GG-A11-Origin.
\par\noindent In preparation. To be posted at \url{https://arisaka-gg.org/} on
completion; no result of the present paper depends on it.
\par\noindent\textbf{Relevance.} The closing capstone of the eleven-paper A-series.
Operationalizes the GRIP cascade through layered identity-formation (Part V), states
the T-Origin theorem, and develops the universal-identity insight in which the
\OGNfive{} ledger threads through atoms, molecules, ATP, DNA, and the first cell.
We cite it where the constitutional cascade is followed to its operational
anatomy and where the universal-identity insight closes a chain of reasoning
that began with $\AX{2}$ and $\AX{4}$. First cited in \S\ref{sec:eleven-paper-inheritance}.

\bibitem[Arisaka(2026, A2-GG6)]{ArisakaA2GG6}

\textbf{Arisaka, K.\ (2026).}
\textit{\GGTheory{} with \GG{}: Geometry--Gauge Locking at the 6D Bulk (\GG{}) at the Heart of Grand Geometric Theory (\GGTheory{})}.
Arisaka-GG-A2-GG6 preprint.
\par\noindent Preprint, available at \url{https://arisaka-gg.org/}.
\par\noindent\textbf{Relevance.} This companion paper develops the concrete six-dimensional geometric stage suggested by the
axioms (especially \GGEP{} and anomaly inflow requirements). In the present foundational paper we cite it whenever
the 6D bulk, the four-component \BIsix{} anomaly polynomial, the \OGNfive{} integer ledger, or the
$\Spinone\times\SU(5)\times\U(1)_B\times\U(1)_{Qi}$ composite gauge group is invoked. First cited in \S\ref{sec:intro}.

\bibitem[Arisaka(2026, A3-BI6)]{ArisakaA3BI6}

\textbf{Arisaka, K.\ (2026).}
\textit{Anomaly Freedom of \BIsix{}: The Modified Bianchi Identity (\BIsix{}) for the Six-Force, Six-Dimensional Bulk (\GG{})}.
Arisaka-GG-A3-BI6 preprint.
\par\noindent Preprint, available at \url{https://arisaka-gg.org/}.
\par\noindent\textbf{Relevance.} This companion paper supplies the rigorous Hopkins-Singer differential-cohomology proof
that the four-component anomaly polynomial $X_4 = \tr(R\wedge R) - \tr(F_{\SU(5)}\wedge F_{\SU(5)})
- \tfrac{2}{5} F_B \wedge F_B - \tfrac{2}{5} F_{Qi}\wedge F_{Qi}$ is exact, establishing $|a/c|=|a/d|=5/2$ as the
geometry-gauge trace-ratio lock that POS-\AF{} demands. First cited in \S\ref{sec:intro}.

\bibitem[Arisaka(2026, A4-Euclid)]{ArisakaA4Euclid}

\textbf{Arisaka, K.\ (2026).}
\textit{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://arisaka-gg.org/}.
\par\noindent\textbf{Relevance.} The historical companion to the
GG6 Trio. Pedagogical commentary on the long chain from Euclid
through Newton, Riemann, Maxwell, and Einstein to the \GG{} stage.
We cite this paper whenever the historical positioning of the
6D bulk against its predecessors is invoked. First cited in \S\ref{sec:eleven-paper-inheritance}.

\bibitem[Arisaka(2026, A5-\GDOS{})]{ArisakaA5GDOS}

\textbf{Arisaka, K.\ (2026).}
\textit{\GDOS{}: Geometric Dynamics of Open Systems --- 6D \GG{} Bulk Projected to 4D Boundary for Quantum Measurement, General Relativity, and QFT}.
Arisaka-GG-A5-GDOS.
\par\noindent Preprint, available at \url{https://arisaka-gg.org/}.
\par\noindent\textbf{Relevance.} This companion paper develops the open-system ``runtime'' implied by \OGGEP{} and the postulates
(positivity, RG-compatibility, falsifiability). We cite it whenever the boundary law $\cL_{\GDOS}$, the
Hilbert-space embedding, the Born rule, the Schr\"odinger limit, the modified TOV equation, or the BRST/Slavnov-Taylor
identities are invoked, since their proofs live there as the eleven \GDOS{} Principal Theorems \T{1}--\T{11}. First cited in \S\ref{sec:intro}.

\bibitem[Arisaka(2026, A6-GRIP)]{ArisakaA6Gi4}

\textbf{Arisaka, K.\ (2026).}
\textit{The \GRIP{} on \GDOS{} --- \GRIP{}: the Geometric Engine of Symmetry Breaking, from the Big Bang to Mind}.
Arisaka-GG-A6-GRIP preprint.
\par\noindent Preprint, available at \url{https://arisaka-gg.org/}.
\par\noindent\textbf{Relevance.} This companion paper formalizes the four Geometric-Invariance pillars descending from
\OGGEP{}: the Total Geometric-Gauge object \TGG, the Covariant Conservation Identity \CCI, the Geometric Free-Energy
Functional \GFF, and the Gauge-Runtime Invariance Pipeline \GRIP{} (\DGI{} $\to$ \GSSB{} $\to$ \GA). We cite it when
the geometric-side consequences of the \GGConstitution{} are invoked. First cited in \S\ref{sec:intro}.

\bibitem[Arisaka(2026, A7-Qi4)]{ArisakaA7Qi4}

\textbf{Arisaka, K.\ (2026).}
\textit{The \QiFour{} Pillars --- \QiQUA{}, \QiEP{}, \QiCON{}, \QiHAL{}: The Quantum-Information Backbone of \GDOS{}}.
Arisaka-GG-A7-Qi4 preprint.
\par\noindent Preprint, available at \url{https://arisaka-gg.org/}.
\par\noindent\textbf{Relevance.} This companion paper formalizes the four Quantum-Information pillars descending from
\OGGEP{}: \QiQUA{} (qua-state preparation), \QiEP{} (quantum equivalence principle), \QiCON{} (decoherence/contraction),
and \QiHAL{} (holographic alphabet). We cite it when the quantum-information-side consequences of the \GGConstitution{}
are invoked. First cited in \S\ref{sec:intro}.

\bibitem[Arisaka(2026, A8-QM)]{ArisakaA8QM}

\textbf{Arisaka, K.\ (2026).}
\textit{Quantum Mechanics Derived from \GGTheory{} as Projection from 6D Bulk to 4D Boundary}.
Arisaka-GG-A8-QM.
\par\noindent Preprint, available at \url{https://arisaka-gg.org/}.
\par\noindent\textbf{Relevance.} The first paper of the 4D-Physics
Trio. Operationalizes $\T{2}$ Operational Closure and $\T{7}$ 4D
Quantum Mechanics from $\GDOS{}$ into a concept-by-concept
reformulation of textbook quantum mechanics: the Born rule as a
theorem of the Stinespring dilation, EPR / Bell / CHSH / Aspect /
Hensen / Giustina as evidence of the non-invertibility of the
6D-to-4D projection, Schr\"odinger's cat as a $\GDOS{}$
boundary-decoherence statement, and the central thesis that
Einstein was right in 6D and Bohr was right in 4D, both within
their respective domains. First cited in \S\ref{sec:geometrization-ladder-a1}.

\bibitem[Arisaka(2026, A9-SM)]{ArisakaA9SM}

\textbf{Arisaka, K.\ (2026).}
\textit{The Standard Model as Theorem of \GGTheory{}: Its 28 Parameters as Observations of One Six-Dimensional Geometry}.
Arisaka-GG-A9-SM preprint.
\par\noindent Preprint, available at \url{https://arisaka-gg.org/}.
\par\noindent\textbf{Relevance.} This companion paper derives the Standard Model parameters (gauge couplings, Yukawa
couplings, CKM/PMNS angles, EW vacuum expectation value $v\approx 246$\,GeV, the cosmological-constant scale
$m_\Lambda \approx 6$\,meV, and the lock-mass cluster near $\Mlock\approx 3.85$\,TeV) from the 6D bulk and the
\OGNfive{} ledger. We cite it whenever the constitution's empirical-prediction column is invoked. First cited in \S\ref{sec:intro}.

\bibitem[Arisaka(2026, Overview)]{ArisakaBookI}

\textbf{Arisaka, K.\ (2026).}
\textit{\GGTheory{} Foundation Book: Constitution, \GG{} Trio, \GDOS{} Trio, and 4D-Physics Trio Compiled into a Single Reference}.
Arisaka-GG-BookI (in preparation).
\par\noindent Preprint, available at \url{https://arisaka-gg.org/}.
\par\noindent\textbf{Relevance.} The Foundation Book compiles the constitution and its descent chain
into a single accessible reference. We cite it where a one-volume overview is more useful than the individual
preprints, and where cross-domain implications (life science, neuroscience, complex systems) of the \GGConstitution{}
are sketched. First cited in \S\ref{sec:direction}.

\bibitem[Arisaka(2026, C1-Kaxion)]{ArisakaC1Kaxion}

\textbf{Arisaka, K.\ (2026).}
\textit{The Kaxion: Cold Dark Matter as the Non-Spatial Compact-Angle Pseudoscalar of \GG{}}.
Arisaka-GG-C1-Kaxion preprint.
\par\noindent Preprint, available at \url{https://arisaka-gg.org/}.
\par\noindent\textbf{Relevance.} The cosmology phenomenology
companion to \ATen{}. The kaxion field of mass set by the
6D-bulk lock-length integer $K = 35$ produces a dark-matter
oscillation at fractional frequency $\Delta\nu/\nu = 1/35$ in the
$\mu$eV range; explicit haloscope and microwave-cavity falsifiers.
Any verified discovery at $\Delta\nu/\nu \neq 1/35$ falsifies the
$\OGNfive{}$ ledger and so the constitutional clauses
$\AX{4}~\MSMF{}$ and $\POS{6}~\OGN{}$ upstream. First cited in \S\ref{sec:POS7}.

\bibitem[Arisaka(2026, C2-Hubble)]{ArisakaC2Hubble}

\textbf{Arisaka, K.\ (2026).}
\textit{The Hubble Tension in the Light of \GGTheory{}: The Hubble Constant as One Reading of a Derived Expansion History $H(t)$}.
Arisaka-GG-C2-Hubble preprint.
\par\noindent Preprint, available at \url{https://arisaka-gg.org/}.
\par\noindent\textbf{Relevance.} The second paper of the Cosmology
\& Astrophysics Branch: reads the persistent discrepancy between
early- and late-universe Hubble measurements as the two ends of one
geometric corridor of the 6D bulk, rather than as a systematic
error or new fluid. Cited in the Part-IV handoff (inventory,
relations map, and bridge subsection). First cited in \S\ref{sec:bridge_predictions}.

\bibitem[Arisaka(2026, C3-BH)]{ArisakaC3BH}

\textbf{Arisaka, K.\ (2026).}
\textit{Black-Hole Entropy under \GGTheory{}: The \GDOS{} Reading of the Horizon as an \OGGEP{} Port --- with the Exact Bekenstein--Hawking $A/4G_N$ Derived from Induced Gravity}.
Arisaka-GG-C3-BH preprint.
\par\noindent Preprint, available at \url{https://arisaka-gg.org/}.
\par\noindent\textbf{Relevance.} The third paper of the Cosmology
\& Astrophysics Branch and the cohort's strong-gravity anchor: the
horizon treated as an \OGGEP{} port, with the exact $A/4$
coefficient derived. Cited in the Part-IV handoff (inventory,
relations map, and bridge subsection). First cited in \S\ref{sec:bridge_predictions}.

\bibitem[Arisaka(2026, P1-KK)]{ArisakaP1KK}

\textbf{Arisaka, K.\ (2026).}
\textit{The \GG{} Lock Cluster: A Non-Spatial Kaluza--Klein Resonance near 3.85~TeV as the Standing LHC Falsifier of \GGTheory{}}.
Arisaka-GG-P1-KK preprint.
\par\noindent Preprint, available at \url{https://arisaka-gg.org/}.
\par\noindent\textbf{Relevance.} The first paper of the
Predictions Branch. Specific phenomenology of the lock cluster at
$\Mlock = (1/2)\,\MPl\,\e^{-35} \approx 3.85\,\TeV$, with explicit
LHC and future-collider falsifiers. Any verified collider
deviation from the predicted Kaluza-Klein resonance pattern at
$5\sigma$ falsifies $\T{4}$ + $\T{6}$ jointly. First cited in \S\ref{sec:bridge_predictions}.

\bibitem[Arisaka(2026, P2-N3)]{ArisakaP2N3}

\textbf{Arisaka, K.\ (2026).}
\textit{Why Three Generations: A Three-Leg Derivation of $N_g=3$ from \BIsix{}, \MSMF{}, and \GDOS{} Morse Stability}.
Arisaka-GG-P2-N3 preprint.
\par\noindent Preprint, available at \url{https://arisaka-gg.org/}.
\par\noindent\textbf{Relevance.} The second paper of the
Predictions Branch: the family number derived as a squeeze between
a CP-violation lower bound and a \GDOS{} Morse/budget upper bound,
with the conditioning declared by name. Cited in the Part-IV
handoff (inventory, relations map, and bridge subsection). First cited in \S\ref{sec:bridge_predictions}.

\bibitem[Arisaka(2026, P3-Electron)]{ArisakaP3Electron}

\textbf{Arisaka, K.\ (2026).}
\textit{The Electron as the Ground State of Nature: The Six Oldest Questions about the Electron, Closed and Fenced within \GGTheory{}}.
Arisaka-GG-P3-Electron preprint.
\par\noindent Preprint, available at \url{https://arisaka-gg.org/}.
\par\noindent\textbf{Relevance.} The third paper of the Predictions
Branch: identity, absolute stability, the two excited copies, the
positron as conjugate readout, pointlikeness, and field response
derived as six faces of ground-state-ness, with every fence declared; imports \PaperPTwo{}'s
$N=3$ verbatim. Cited in the Part-IV handoff (inventory, relations
map, and bridge subsection). First cited in \S\ref{sec:bridge_predictions}.

\bibitem[Arisaka(2026, P4-Real)]{ArisakaP4Real}

\textbf{Arisaka, K.\ (2026).}
\textit{The Real Geometry Beneath the Complex Numbers of Physics: Positive Frequency is Holomorphy --- from Cardano's Cubics to Twistor Space, via the Sixth Dimension of \GGTheory{}}.
Arisaka-GG-P4-Real preprint.
\par\noindent Preprint, available at \url{https://arisaka-gg.org/}.
\par\noindent\textbf{Relevance.} The first member of the Foundations
Branch: the origin of $i$. Derives the two complex structures of
physics from two real structures of the 6D bulk --- the compact
phase circle, whose quarter-turn is the unique circle-compatible
complex structure on every charged sector (the gauge $i$, T-P4-1),
and the causal clock, whose one-sided spectrum forces the unique
positive-energy polar phase (the state-space $i$, T-P4-2) --- with
their forced on-shell agreement, the Feynman--St\"uckelberg
pairing, derived as the existence of antiparticles (T-P4-3), and
the bridge that positive frequency is holomorphy at the Hardy,
forward-tube, and twistor levels (T-P4-6/7). We cite it as the
standalone development of the origin of the boundary's complex
structure, beside \AEight{}'s QM-register treatment. First cited
in \S\ref{sec:eleven-paper-inheritance}.

\bibitem[Arisaka(2026, P5-CPT)]{ArisakaP5CPT}

\textbf{Arisaka, K.\ (2026).}
\textit{The Mirrors of Matter: Why Every Particle Has a Twin, Why Nature's Mirrors Break, and Why the Universe Is Made of Matter}.
Arisaka-GG-P5-CPT preprint.
\par\noindent Preprint, available at \url{https://arisaka-gg.org/}.
\par\noindent\textbf{Relevance.} The second member of the Foundations
Branch: the mirrors and the matter. Derives C, P, and T as the
orientation reversals of the three real factors of the 6D bulk ---
the fiber circle, the spatial slice, and the causal clock --- with
the unitary/antiunitary table forced rather than assigned
(T-P5-1) and the CPT theorem as orientation bookkeeping (T-P5-2);
routes the cosmic matter excess through the flavored chain fed by
\ANine{}'s ledger phases and feeding \ATen{}'s $\eta_B$ chain,
its sign carried by the Majorana parity $\alpha_{31} = \pi$
(derived-conditional on its named finite-temperature hypothesis);
and consolidates the four-endowment definition of time. First
cited in \S\ref{sec:eleven-paper-inheritance}.

\bibitem[Arisaka(2026, P6-Proton)]{ArisakaP6Proton}

\textbf{Arisaka, K.\ (2026).}
\textit{The Proton as the Ground Geometric Attractor of Matter: The Eleven Questions of the Proton, Closed and Fenced within \GGTheory{}}.
Arisaka-GG-P6-Proton preprint.
\par\noindent Preprint, available at \url{https://arisaka-gg.org/}.
\par\noindent\textbf{Relevance.} The fourth member of the Predictions
Branch, sister to \PaperPThree{}: the proton as the \GGA{} of the
baryonic stratum, with baryon number carried as a winding of the
compact geometry rather than as a bookkeeping label --- written by the
same twist that breaks the unified gauge symmetry and splits the Higgs
multiplet --- so that the decay the grand unified theories insisted on
is forbidden by the geometry rather than merely unobserved. Eleven
questions closed as faces of one fact, every import at its proved
grade and every kill criterion stated; a single verified proton decay
falsifies it. First cited in \S\ref{sec:eleven-paper-inheritance}.

\bibitem[Arisaka(2026, W1-Symmetry)]{ArisakaW1Symmetry}

\textbf{Arisaka, K.\ (2026).}
\textit{Symmetry, Conservation, and Geometry: Seven Steps from Noether, Bianchi, to Hopkins--Singer, and the Ledger the Open-System Turn Pays Out}.
Arisaka-GG-W1-Symmetry preprint.
\par\noindent Preprint, available at \url{https://arisaka-gg.org/}.
\par\noindent\textbf{Relevance.} The first of the two white papers,
and the \emph{diachronic} one: it walks the century of symmetry and
conservation in the order it was found and shows the conviction that
symmetry causes conservation loosening at seven historical steps,
until what remains conserved is an integer with nowhere to move ---
then places the program at the arc's end, at declared grades, with
its payout set beside the measurements. Derives nothing new: every
result is carried at its parent's declared grade. First cited in
\S\ref{sec:bridge_whitepapers}.

\bibitem[Arisaka(2026, W2-Holonomy)]{ArisakaW2Holonomy}

\textbf{Arisaka, K.\ (2026).}
\textit{The Circle That Remembers, the Interval That Forgets: \GGTheory{} and the Sixteen Problems of Particle Physics and Cosmology}.
Arisaka-GG-W2-Holonomy preprint.
\par\noindent Preprint, available at \url{https://arisaka-gg.org/}.
\par\noindent\textbf{Relevance.} The second of the two white papers,
and the \emph{synchronic} one: it takes a single exact statement ---
a holonomy requires a closed loop and the interval fiber has none, so
the circle remembers and the interval forgets --- and follows it to
sixteen of the questions physics has not closed, closing with the
named, dated measurements on which six of the sixteen treatments die
outright. Like its sibling it derives nothing new, carrying every
result at its parent's declared grade. First cited in
\S\ref{sec:bridge_whitepapers}.
\end{thebibliography}

\bigskip
\section*{B.~Other references}

\begin{thebibliography}{99}
\bibitem[Adler(1969)]{Adler1969}

\textbf{Adler, S.~L. (1969).}
Axial-vector vertex in spinor electrodynamics.
\emph{Physical Review}, \emph{177}, 2426--2438.
\url{https://doi.org/10.1103/PhysRev.177.2426}
\par\noindent\textbf{Relevance.} Adler's 1969 calculation of the axial-vector
vertex is the foundational discovery that the boundary quantum effective
action does not honor the classical chiral Ward identity at one loop.
In the constitutional reading of $\POS{3}~\AF{}$, the Adler anomaly is
not a pathology of the bulk field theory but the $\AX{3}~\OGGEP{}$
projection shadow of a bulk-side classical-topological identity
descended from Bianchi 1902: the boundary's triangle-diagram measurement
of its own inability to register the deeper bulk closure. Adler is
cited as the historical anchor for that re-reading. First cited in \S\ref{sec:POS1}.

\bibitem[Aharonov \& Bohm(1959)]{AharonovBohm1959}

\textbf{Aharonov, Y., \& Bohm, D. (1959).}
Significance of electromagnetic potentials in the quantum theory.
\textit{Physical Review}, \textit{115}(3), 485--491.
\url{https://doi.org/10.1103/PhysRev.115.485}
\par\noindent\textbf{Relevance.} The Aharonov--Bohm effect demonstrates that holonomy of a gauge connection is physically observable even when local field strengths vanish. This is a canonical precedent for \GGEP{}'s insistence that the invariant content of the composite connection includes global/holonomy data, supporting ``quantum phase as geometry.'' First cited in \S\ref{sec:AX2}.

\bibitem[Akaike(1974)]{Akaike1974}

\textbf{Akaike, H. (1974).}
A new look at the statistical model identification.
\emph{IEEE Transactions on Automatic Control}, \emph{19}(6), 716--723.
\url{https://doi.org/10.1109/TAC.1974.1100705}
\par\noindent\textbf{Relevance.} Akaike's information criterion is a quantitative formulation of the ``Occam penalty'' in model selection. We cite it as historical background for POS-6 (\OGN{}): in \GGTheory{}, minimality is enforced not only in model structure but also in normalization conventions to prevent hidden continuous tuning. First cited in \S\ref{sec:AX4}.

\bibitem[Almheiri et~al.(2019)]{Almheiri2020}

\textbf{Almheiri, A., Engelhardt, N., Marolf, D., and Maxfield, H.\ (2020).}
The entropy of bulk quantum fields and the entanglement wedge of
an evaporating black hole.
\textit{Journal of High Energy Physics}, \textit{2019}(12), 63.
\url{https://doi.org/10.1007/JHEP12(2019)063}
\par\noindent\textbf{Relevance.} The Almheiri--Engelhardt--Marolf--Maxfield
and Penington--Shenker--Stanford--Yang papers of
2019--2020 (the ``islands formula'') showed that the Page curve
of black-hole evaporation can be recovered from semiclassical
computations by including ``island'' regions in the entanglement
wedge. In the AX-3 lineage, the islands formula is the first
construction in which the operational projection $\Dmap$
necessarily includes non-local bulk regions to honor boundary
unitarity --- a direct quantum-information confirmation that
$\OGGEP{}$ is the right operational reading of the holographic
principle. First cited in \S\ref{sec:AX3}.

\bibitem[Alvarez-Gaum\'e \& Witten(1984)]{AlvarezGaumeWitten1984}

\textbf{Alvarez-Gaum\'e, L., \& Witten, E.\ (1984).}
Gravitational anomalies.
\textit{Nuclear Physics B, 234}(2), 269--330.
DOI: \href{https://doi.org/10.1016/0550-3213(84)90066-X}{10.1016/0550-3213(84)90066-X}.
\par\noindent\textbf{Relevance.} Alvarez-Gaum\'e and Witten classify
gravitational and mixed anomalies across arbitrary dimensions, exposing
the algebraic structure (descent equations, characteristic classes) that
underlies what the constitutional reading takes to be a single
bulk-side classical-topological identity. In the language of
$\POS{3}~\AF{}$, the gravitational anomaly is the $\AX{3}~\OGGEP{}$
projection shadow of the bulk diffeomorphism-Bianchi closure on the
$\AX{2}~\GGEP{}$ composite connection. Cited as the canonical technical
reference for the descent-class structure that makes the bulk closure
$dX_4 = 0$ rigorous in arbitrary dimension. First cited in \S\ref{sec:POS3}.

\bibitem[Aristotle(c.~350 BCE)]{Aristotle}

\textbf{Aristotle (c.~350 BCE).}
\emph{Physics} and \emph{Metaphysics}. Translations as in Barnes (ed.),
\emph{The Complete Works of Aristotle}, Princeton University Press, 1984.
\par\noindent\textbf{Relevance.} Aristotle's insistence that knowledge of
natural phenomena must be grounded in observation is the philosophical
seed of empiricism. In the constitutional reading of $\POS{7}~\MF{}$,
Aristotle is cited as the deepest historical anchor for the discipline
that every theoretical claim must ultimately compile to a registered
observation. First cited in \S\ref{sec:POS7}.

\bibitem[Aspect, Dalibard, \& Roger(1982)]{Aspect1982}

\textbf{Aspect, A., Dalibard, J., \& Roger, G.\ (1982).}
Experimental test of Bell's inequalities using time-varying analyzers.
\textit{Physical Review Letters}, \textit{49}(25), 1804--1807.
\url{https://doi.org/10.1103/PhysRevLett.49.1804}
\par\noindent\textbf{Relevance.} The first decisive laboratory
verification of CHSH violation under time-varying measurement
settings, ruling out local hidden-variable theories with
specifiable causal sufficiency. We cite it as the experimental
closure of the locality crisis identified by Bell (1964) and
re-cite it whenever the axiomatic treatment of openness and
operational locality is invoked, especially in the bridge from
\AX{1}~\LC{} to \AX{3}~\OGGEP{}. First cited in \S\ref{sec:historical_chain}.

\bibitem[Atiyah \& Singer(1963)]{AtiyahSinger1963}

\textbf{Atiyah, M.~F., \& Singer, I.~M. (1963).}
The index of elliptic operators on compact manifolds.
\emph{Bulletin of the American Mathematical Society}, \emph{69}, 422--433.
\par\noindent\textbf{Relevance.} The original announcement of the
Atiyah--Singer index theorem. In the constitutional reading of
$\POS{3}~\AF{}$, this theorem is the mathematical engine behind anomaly
cancellation: the index of a Dirac operator equals a topological
integral of characteristic classes, so the boundary anomaly polynomial
is locked to bulk topological data. Cited as the historical anchor for
the geometric-origin reading of POS-3. First cited in \S\ref{sec:POS3}.

\bibitem[Atiyah \& Singer(1968)]{AtiyahSinger1968}

\textbf{Atiyah, M.~F., \& Singer, I.~M. (1968).}
The index of elliptic operators I.
\emph{Annals of Mathematics}, \emph{87}, 484--530.
\par\noindent\textbf{Relevance.} The full proof of the Atiyah--Singer
index theorem. In the constitutional reading of $\POS{3}~\AF{}$, the
1968 paper is the technical foundation on which the modern
anomaly-as-index picture rests: every chiral anomaly in physics is a
direct corollary of this single mathematical identity. First cited in \S\ref{sec:POS3}.

\bibitem[Banks \& Seiberg(2011)]{BanksSeiberg2011}

\textbf{Banks, T., \& Seiberg, N. (2011).}
Symmetries and strings in field theory and gravity.
\emph{Physical Review D}, \emph{83}, 084019.
\url{https://doi.org/10.1103/PhysRevD.83.084019}
\par\noindent\textbf{Relevance.} The modern statement that global
symmetries must either be gauged or be broken in any consistent theory
of quantum gravity. In the constitutional reading of $\POS{1}~\GI{}$,
this paper supplies the deepest justification for treating gauge
invariance as a primary axiomatic filter rather than a derived feature. First cited in \S\ref{sec:POS1}.

\bibitem[Bardeen(1969)]{Bardeen1969}

\textbf{Bardeen, W.~A. (1969).}
Anomalous Ward identities in spinor field theories.
\textit{Physical Review}, \textit{184}, 1848--1857.
\par\noindent\textbf{Relevance.} Develops the Ward-identity viewpoint of anomalies and clarifies how anomaly terms appear in current conservation laws. POS-3 treats such symmetry violations as fatal unless canceled by inflow; Bardeen's analysis is a canonical reference for this ``symmetry grammar'' perspective. First cited in \S\ref{sec:POS3}.

\bibitem[Batalin \& Vilkovisky(1981)]{BatalinVilkovisky1981}

\textbf{Batalin, I.~A., \& Vilkovisky, G.~A. (1981).}
Gauge algebra and quantization.
\emph{Physics Letters B}, \emph{102}, 27--31.
\par\noindent\textbf{Relevance.} The BV (Batalin--Vilkovisky)
formalism generalizes BRST quantization to open and reducible gauge
algebras. In the constitutional reading of $\POS{1}~\GI{}$, BV is the
mathematical apparatus that makes the dynamical-gauge-invariance (DGI)
layer well-defined for the full ladder of gauge structures used by
\GGTheory{}. First cited in \S\ref{sec:POS1}.

\bibitem[Bayes(1763)]{Bayes1763}

\textbf{Bayes, T. (1763).}
An essay towards solving a problem in the doctrine of chances.
\emph{Philosophical Transactions of the Royal Society of London},
\emph{53}, 370--418 (posthumous; communicated by R.~Price).
\par\noindent\textbf{Relevance.} Bayes' theorem: the prototype of every
subsequent inverse-probability statement. In the constitutional
reading of $\POS{4}~\CP{}$, Bayes is cited as the historical anchor for
treating probability as a registered quantity in physics, the
foundation on which Kolmogorov's later axiomatization is built. First cited in \S\ref{sec:POS4}.

\bibitem[Bekenstein(1973)]{Bekenstein1973}

\textbf{Bekenstein, J.~D. (1973).}
Black holes and entropy.
\textit{Physical Review D}, \textit{7}(8), 2333--2346.
\par\noindent\textbf{Relevance.} We cite Bekenstein as the original statement that black holes carry entropy proportional to horizon area. In the \GGTheory{} viewpoint, black-hole thermodynamics is a paradigmatic open-system ledger problem: the boundary/horizon acts as a ``port'' and the correct scalar accounting principle anticipates the \GFF{} idea used in \GDOS{}. First cited in \S\ref{sec:AX3}.

\bibitem[Bell(1964)]{Bell1964}

\textbf{Bell, J.~S. (1964).}
On the Einstein Podolsky Rosen paradox.
\textit{Physics}, \textit{1}, 195--200.
\par\noindent\textbf{Relevance.} Bell's theorem clarifies that the measurement problem is not merely philosophical: operational
predictions about correlations constrain the kinds of ``hidden variable'' completions that remain possible. This motivates
treating measurement as an axiomatic issue rather than an interpretational afterthought. First cited in \S\ref{sec:historical_chain}.

\bibitem[Bell(2004)]{Bell2004}

\textbf{Bell, J.~S. (2004).}
\textit{Speakable and Unspeakable in Quantum Mechanics} (2nd ed.).
Cambridge University Press.
\url{https://doi.org/10.1017/CBO9780511815676}
\par\noindent\textbf{Relevance.} Bell's collected papers contain the sharpest discussions of ``local causality'' in the foundations of quantum theory. We cite this as historical context for AX-1: \GGTheory{} adopts ``local causality'' as a constitution-level constraint on operational predictions (records must be retarded and non-signaling), not as an interpretational afterthought. First cited in \S\ref{sec:AX3}.

\bibitem[Bell and Jackiw(1969)]{BellJackiw1969}

\textbf{Bell, J.~S. \& Jackiw, R. (1969).}
A PCAC puzzle: $\pi^0\rightarrow\gamma\gamma$ in the $\sigma$-model.
\textit{Il Nuovo Cimento A} \textbf{60}, 47--61.
\href{https://doi.org/10.1007/BF02823296}{doi:10.1007/BF02823296}.
\par\noindent\textbf{Relevance.}
The Bell--Jackiw companion to Adler 1969, resolving the
$\pi^0 \to \gamma\gamma$ puzzle in the $\sigma$-model by way of the same
axial anomaly. In the constitutional reading, the Bell--Jackiw
calculation co-discovered (with Adler) that the boundary quantum loop
registers an obstruction whose origin sits one dimension upstream: the
boundary 4D effective action is the $\AX{3}~\OGGEP{}$ projection of a
bulk-side classical Bianchi identity descended from Bianchi 1902.
Cited as the second historical anchor for $\POS{3}~\AF{}$'s
boundary-shadow-of-bulk-identity framing. First cited in \S\ref{sec:POS1}.

\bibitem[Bernoulli(1713)]{Bernoulli1713}

\textbf{Bernoulli, J. (1713).}
\emph{Ars Conjectandi}. Basel: Thurnisii (posthumous).
\par\noindent\textbf{Relevance.} The law of large numbers. In the
constitutional reading of $\POS{4}~\CP{}$, Bernoulli's \emph{Ars
Conjectandi} is the deepest historical anchor for treating frequencies
of observation as a legitimate physical quantity, justifying the move
from idealized probabilities to operationally-registered statistics. First cited in \S\ref{sec:POS4}.

\bibitem[Berry(1984)]{Berry1984}

\textbf{Berry, M.~V. (1984).}
Quantal phase factors accompanying adiabatic changes.
\textit{Proceedings of the Royal Society A}, \textit{392}(1802), 45--57.
\url{https://doi.org/10.1098/rspa.1984.0023}

\par\noindent\textbf{Relevance.} Berry's phase is a canonical example of ``quantum from geometry'': measurable phases arise from holonomy in parameter space.
It provides historical and technical support for AX-2's emphasis that quantum structure can be encoded geometrically in connections and their holonomies rather than imposed by quantization rules. First cited in \S\ref{sec:AX2}.

\bibitem[Bertlmann(1996)]{Bertlmann1996}

\textbf{Bertlmann, R.~A. (1996).}
\textit{Anomalies in Quantum Field Theory}.
Oxford University Press.
\url{https://doi.org/10.1093/acprof:oso/9780198507628.001.0001}
\par\noindent\textbf{Relevance.} This monograph provides the standard mathematical treatment of anomaly polynomials and descent. It supports the formal statements in POS-3, where we package anomalies into differential-form data suitable for bulk--boundary cancellation. First cited in \S\ref{sec:POS3}.

\bibitem[Bethe(1947)]{Bethe1947}

\textbf{Bethe, H.~A. (1947).}
The electromagnetic shift of energy levels.
\emph{Physical Review}, \emph{72}(4), 339--341.
\url{https://doi.org/10.1103/PhysRev.72.339}
\par\noindent\textbf{Relevance.} Bethe's non-relativistic calculation of
the Lamb shift, the empirical turning point that demonstrated
renormalization as a systematic physical procedure rather than an
\emph{ad hoc} subtraction. In the constitutional reading of
$\POS{5}~\RG{}$, Bethe 1947 is cited as the operational starting point
of the renormalization program. First cited in \S\ref{sec:POS5}.

\bibitem[Bianchi(1902)]{Bianchi1902}

\textbf{Bianchi, L.\ (1902).}
Sui simboli a quattro indici e sulla curvatura di Riemann.
\textit{Rendiconti della R.\ Accademia dei Lincei}, \textit{11}, 3--7.
\par\noindent\textbf{Relevance.} Luigi Bianchi's 1902 identity for the
Riemann curvature tensor is the historical ancestor of the entire
constitutional reading of $\POS{3}~\AF{}$. In modern notation
$\nabla_{[\mu} R_{\nu\rho]\sigma\tau} = 0$, equivalently $dR = 0$ on the
Levi-Civita connection; together with the abelian Bianchi identity
$dF = 0$ inherent in Maxwell, it is the geometric ancestor of
$\BIsix{}$ and of every bulk-side classical-topological identity that
the cohort framework rests on. In the present reading the boundary 4D
``anomaly'' of the standard textbook is the $\AX{3}~\OGGEP{}$ projection
shadow of this Bianchi-descended bulk closure; the Green--Schwarz
construction is its $B_2$-coupled rewrite; the Hopkins--Singer
differential-cohomology proof is its precise differential-character
statement. Cited as the founding source for the bulk-side classical
reading of anomaly freedom. First cited in \S\ref{sec:AX2}.

\bibitem[Bohr(1928)]{BohrComplementarity1928}

\textbf{Bohr, N.\ (1928).}
The quantum postulate and the recent development of atomic theory.
\textit{Nature}, \textit{121}, 580--590.
(Earlier in: \textit{Atti del Congresso Internazionale dei Fisici},
Como, 1927.)
\par\noindent\textbf{Relevance.} Bohr's 1928 Como/Solvay address
introduced the complementarity principle: a quantum measurement
necessarily involves an apparatus described classically, and the
complementary aspects of a quantum object (position vs. momentum,
wave vs. particle) cannot be jointly registered. In the AX-3
lineage, Bohr's principle is the first foundational statement
that the bulk-side quantum description and the boundary-side
operational record are not the same object --- a structural fact
that $\AX{3}~\OGGEP{}$ takes as a primitive of the constitution
rather than an interpretational stance. First cited in \S\ref{sec:AX3}.

\bibitem[Bombelli et~al.(1987)]{BombelliSorkin1987}

\textbf{Bombelli, L., Lee, J., Meyer, D., and Sorkin, R.~D.\ (1987).}
Space-time as a causal set.
\textit{Physical Review Letters}, \textit{59}(5), 521--524.
\url{https://doi.org/10.1103/PhysRevLett.59.521}
\par\noindent\textbf{Relevance.} The causal-set approach takes
Minkowski's identification of causal ordering with the geometric
structure of spacetime to its discrete limit: spacetime
\emph{is} a partially ordered set whose order relation
\emph{is} the causal structure. In the AX-1 lineage, this is
the most aggressive realization of the constitutional thesis
that causality is a primitive of the bulk geometry rather than a
derived feature of the dynamics; we credit it as one of the lines
of work to which $\AX{1}~\LC{}$ owes its operational reading. First cited in \S\ref{sec:AX1}.

\bibitem[Born(1926)]{Born1926}

\textbf{Born, M. (1926).}
Zur Quantenmechanik der Sto{\ss}vorg{\"a}nge.
\textit{Zeitschrift f{\"u}r Physik}, \textit{37}, 863--867.
\url{https://doi.org/10.1007/BF01397477}
\par\noindent\textbf{Relevance.} Born's probabilistic interpretation underlies the Born rule, which is central to the modern measurement postulate in quantum mechanics. It is cited here to emphasize that probability entered QM as a foundational rule rather than as a derived dynamical theorem, motivating the \GGTheory{} demand that measurement semantics be axiomatic rather than ad hoc. First cited in \S\ref{sec:historical_chain}.

\bibitem[Born \& Wolf(1999)]{BornWolf1999}

\textbf{Born, M., \& Wolf, E.\ (1999).}
\textit{Principles of Optics} (7th ed.).
Cambridge University Press.
\par\noindent\textbf{Relevance.} A standard reference for geometric optics, wave propagation, and the eikonal approximation.
It supports the \UEP{} dictionary used in this paper: ``least time'' emerges as an operational face of variational
principles once one passes to ray/characteristic descriptions and measurement protocols. First cited in \S\ref{sec:POS2}.

\bibitem[Bousso and Polchinski(2000)]{BoussoPolchinski2000}

\textbf{Bousso, R.\ and Polchinski, J.\ (2000).}
Quantization of four-form fluxes and dynamical neutralization of
the cosmological constant.
\textit{Journal of High Energy Physics}, \textit{2000}(06), 006.
\url{https://doi.org/10.1088/1126-6708/2000/06/006}
\par\noindent\textbf{Relevance.} Bousso and Polchinski estimated
that the discretuum of four-form flux vacua in string theory
produces of order $10^{500}$ admissible solutions, the canonical
``string landscape'' count. In the AX-4 lineage, this is the
quantitative diagnosis of what $\mathsf{GGEP\text{-}closed}$
looks like without $\AX{4}~\MSMF{}$: a vacuum count so vast that
no first-principles selection is possible. $\AX{4}~\MSMF{}$ is
the constitutional remedy: addition of one axiom collapses the
$10^{500}$ landscape to exactly one admissible bulk. First cited in \S\ref{sec:AX4}.


\bibitem[Breuer \& Petruccione(2002)]{BreuerPetruccione2002}

\textbf{Breuer, H.-P., \& Petruccione, F. (2002).}
\textit{The Theory of Open Quantum Systems}.
Oxford University Press.
\url{https://doi.org/10.1093/acprof:oso/9780199213900.001.0001}
\par\noindent\textbf{Relevance.} This standard reference develops the mathematics of open quantum systems (reduced dynamics, master equations, Markovian limits). It anchors our claim that openness is not optional in realistic modeling, and it provides the mature language (CPTP maps and generators) that the \GGConstitution{} intends to elevate to first-class status. First cited in \S\ref{sec:four_clusters}.

\bibitem[Bridgman(1927)]{Bridgman1927}

\textbf{Bridgman, P.~W. (1927).}
\emph{The Logic of Modern Physics}.
New York: Macmillan.
\par\noindent\textbf{Relevance.} Bridgman's operationalism is an early systematic attempt to define scientific concepts by measurement operations. We cite it as philosophical and methodological background for POS-7 (MF): in \GGTheory{}, every claim must compile to an explicit instrument protocol and an auditable falsifier. First cited in \S\ref{sec:POS7}.

\bibitem[Bub \& Pitowsky(2010)]{BubPitowsky2010}

\textbf{Bub, J., \& Pitowsky, I.\ (2010).}
Two dogmas about quantum mechanics.
In S.\ Saunders, J.\ Barrett, A.\ Kent, \& D.\ Wallace (Eds.),
\emph{Many Worlds? Everett, Quantum Theory, and Reality}
(pp.\ 433--459). Oxford: Oxford University Press.
\par\noindent\textbf{Relevance.} Bub and Pitowsky's
informational-realist axiomatization of quantum mechanics treats
quantum theory as a non-classical theory of information, with the
Born rule emerging from operational constraints rather than from a
hidden ontology. In the constitutional reading of
$\AX{3}~\OGGEP{}$, Bub-Pitowsky is a structural neighbor: both
programs promote operational content to first-class status, and
the $\GGTheory{}$ program inherits the Bub-Pitowsky stance that
``measurement'' is not in need of metaphysical resolution before
it can be made mathematically precise. First cited in \S\ref{sec:comparison_modern}.

\bibitem[Buckingham(1914)]{Buckingham1914}

\textbf{Buckingham, E. (1914).}
On physically similar systems; illustrations of the use of dimensional
equations.
\emph{Physical Review}, \emph{4}(4), 345--376.
\url{https://doi.org/10.1103/PhysRev.4.345}
\par\noindent\textbf{Relevance.} The $\pi$-theorem: physical laws must
be expressible as relations among dimensionless ratios. In the
constitutional reading of $\POS{5}~\RG{}$, Buckingham is cited as the
pre-history of the modern renormalization-group: scale invariance is
not a special property of fixed points, it is the default form of every
physical law. First cited in \S\ref{sec:POS5}.

\bibitem[Busch, Lahti, \& Mittelstaedt(1996)]{BuschLahtiMittelstaedt1996}

\textbf{Busch, P., Lahti, P.~J., \& Mittelstaedt, P. (1996).}
\textit{The Quantum Theory of Measurement} (2nd rev.\ ed.).
Berlin: Springer.
\par\noindent\textbf{Relevance.} Provides a comprehensive modern treatment of POVMs, instruments, and measurement models beyond ideal projective measurements. This is directly used in POS-7 to formalize ``measurement'' as a lawful operational map rather than a vague collapse postulate. First cited in \S\ref{sec:POS7}.

\bibitem[Caldeira \& Leggett(1983)]{CaldeiraLeggett1983}

\textbf{Caldeira, A.\ O., \& Leggett, A.\ J.\ (1983).}
Path integral approach to quantum Brownian motion.
\textit{Physica A, 121}(3), 587--616.
\par\noindent\textbf{Relevance.} A classic application of influence-functional methods yielding dissipative and decohering dynamics. It is
cited as historical/technical background for why open-system dynamics is unavoidable in realistic measurement, motivating
AX-3's axiomatic treatment of open evolution. First cited in \S\ref{sec:AX3}.

\bibitem[Callan(1970)]{Callan1970}

\textbf{Callan, C.\ G.\ (1970).}
Broken scale invariance in scalar field theory.
\textit{Physical Review D, 2}(8), 1541--1547.
DOI: \href{https://doi.org/10.1103/PhysRevD.2.1541}{10.1103/PhysRevD.2.1541}.
\par\noindent\textbf{Relevance.} One of the original papers formulating the Callan--Symanzik equation and the modern view of
renormalization-group flow as a differential statement of scale dependence. It provides historical grounding for POS-5:
``scale covariance'' is not optional if the theory aims to be resolution-independent. First cited in \S\ref{sec:POS5}.

\bibitem[Callan \& Harvey(1985)]{CallanHarvey1985}

\textbf{Callan, C.\ G., Jr., \& Harvey, J.\ A.\ (1985).}
Anomalies and fermion zero modes on strings and domain walls.
\textit{Nuclear Physics B, 250}(1--4), 427--436.
DOI: \href{https://doi.org/10.1016/0550-3213\%2885\%2990489-4}{10.1016/0550-3213(85)90489-4}.
\par\noindent\textbf{Relevance.} A classic source on anomaly inflow: boundary anomalies are canceled by bulk topological currents.
It supports POS-3 by exemplifying how ``inflow'' restores gauge consistency and motivates treating cancellability by inflow as an admissibility rule rather than a model-dependent trick. First cited in \S\ref{sec:POS3}.

\bibitem[Chern(1946)]{ChernWeil1946}

\textbf{Chern, S.-S. (1946).}
Characteristic classes of Hermitian manifolds.
\emph{Annals of Mathematics}, \emph{47}, 85--121.
\par\noindent\textbf{Relevance.} The Chern--Weil construction:
characteristic classes as integer invariants of a fiber bundle's
curvature. In the constitutional reading of $\POS{6}~\OGN{}$,
Chern--Weil supplies the modern mathematical justification for the
discipline that gauge-bundle data must be specified by integers, not by
continuous fitted parameters. First cited in \S\ref{sec:POS3}.

\bibitem[Chern \& Simons(1974)]{ChernSimons1974}

\textbf{Chern, S.-S., \& Simons, J. (1974).}
Characteristic forms and geometric invariants.
\emph{Annals of Mathematics}, \emph{99}, 48--69.
\par\noindent\textbf{Relevance.} The Chern--Simons three-form: the
local primitive whose exterior derivative is the anomaly polynomial.
In the constitutional reading of $\POS{3}~\AF{}$, Chern--Simons is the
geometric object that makes inflow cancellation a local, computable
identity rather than a formal rewriting. First cited in \S\ref{sec:POS3}.


\bibitem[Choi(1975)]{Choi1975}

\textbf{Choi, M.-D. (1975).}
Completely positive linear maps on complex matrices.
\textit{Linear Algebra and its Applications}, \textit{10}(3), 285--290.
\url{https://doi.org/10.1016/0024-3795(75)90075-0}

\par\noindent\textbf{Relevance.} Choi's theorem characterizes complete positivity, the mathematical core of physical admissibility for open-system dynamics and measurement instruments.
AX-3 and POS-\CP\ rely on complete positivity as the operational replacement for ``no negative probabilities'' across all open systems. First cited in \S\ref{sec:AX3}.

\bibitem[Clauser et~al.(1969)]{CHSH1969}

\textbf{Clauser, J.~F., Horne, M.~A., Shimony, A., and Holt, R.~A.\ (1969).}
Proposed experiment to test local hidden-variable theories.
\textit{Physical Review Letters}, \textit{23}(15), 880--884.
\url{https://doi.org/10.1103/PhysRevLett.23.880}

\par\noindent\textbf{Relevance.} The CHSH inequality is the form of
Bell's theorem that every subsequent experiment has actually tested,
and it is the bound $|S| \le 2$ that the worked example of
$\AX{1}~\LC{}$ uses to separate the causal-arrow constraint from the
factorizability constraint. The constitutional reading turns on
exactly that separation: $\AX{1}$ forbids signaling, and does not
forbid non-classical correlation. First cited in \S\ref{sec:AX1}.


\bibitem[Connes(1994)]{Connes1994}

\textbf{Connes, A. (1994).}
\emph{Noncommutative Geometry}. San Diego: Academic Press.
\par\noindent\textbf{Relevance.} Connes' noncommutative-geometry
rewriting of gauge theory and the Standard Model. In the
constitutional reading of $\POS{1}~\GI{}$, this work shows that gauge
invariance can be reformulated as a statement about the spectral
geometry of the observable algebra rather than about a specific
Lagrangian, supporting the algebra-first reading of LGI used by
\GGTheory{}. First cited in \S\ref{sec:POS1}.

\bibitem[Cover \& Thomas(2006)]{CoverThomas2006}

\textbf{Cover, T.~M., \& Thomas, J.~A. (2006).}
\textit{Elements of Information Theory} (2nd ed.).
Wiley.
\url{https://doi.org/10.1002/047174882X}
\par\noindent\textbf{Relevance.} Information theory supplies a precise quantitative language for ``information'' as an operational resource. We cite this text to support the appendix discussion where free-energy-like functionals are related to KL divergence and where MF requires explicit information-theoretic accounting for measurement and compression. First cited in \S\ref{app:more_examples_examples}.

\bibitem[Crooks(1999)]{Crooks1999}

\textbf{Crooks, G.~E. (1999).}
Entropy production fluctuation theorem and the nonequilibrium work relation for free energy differences.
\textit{Physical Review E}, \textit{60}(3), 2721--2726.
\par\noindent\textbf{Relevance.} A cornerstone of stochastic thermodynamics connecting forward/backward path probabilities and nonequilibrium work. We cite it as a modern exemplar of how open-system probability flows carry rigorous, testable constraints---precisely the style enforced by POS-4 and POS-7. First cited in \S\ref{sec:AX1}.

\bibitem[Daniel \& Viallet(1980)]{DanielViallet1980}

\textbf{Daniel, M., \& Viallet, C.~M. (1980).}
The geometrical setting of gauge theories of the Yang--Mills type.
\textit{Reviews of Modern Physics}, \textit{52}(1), 175--197.

\par\noindent\textbf{Relevance.} A standard review formulating Yang--Mills gauge theories --- and the Standard-Model gauge group --- as connections on principal fiber bundles. Cited under AX-2 milestone~(7) as the mature fiber-bundle formalism that \GGEP{} adopts rather than originates. First cited in \S\ref{sec:AX2}.

\bibitem[Davies, Fulling \& Unruh(1976)]{DaviesFulling1976}

\textbf{Davies, P.~C.~W., Fulling, S.~A., and Unruh, W.~G.\ (1976).}
Energy-momentum tensor near an evaporating black hole.
\textit{Physical Review D}, \textit{13}(10), 2720--2723.
\url{https://doi.org/10.1103/PhysRevD.13.2720}
\par\noindent\textbf{Relevance.} Davies, Fulling, and Unruh's
1976 paper established that the vacuum state seen by an inertial
observer appears as a thermal bath to a uniformly accelerated
observer. In the AX-3 lineage, the Davies--Fulling--Unruh
construction is the canonical demonstration that the operational
boundary observer's record depends on the boundary's motion ---
i.e., on which $\Dmap$ is used --- and not only on the bulk state.
It is the first quantum-field-theoretic demonstration that
$\OGGEP{}$ projections genuinely partition the bulk content into
observer-dependent boundary registrations. First cited in \S\ref{sec:AX3}.

\bibitem[Davies and Lewis(1970)]{DaviesLewis1970}

\textbf{Davies, E.~B., \& Lewis, J.~T. (1970).}
An operational approach to quantum probability.
\textit{Communications in Mathematical Physics}, \textit{17}, 239--260.
\url{https://doi.org/10.1007/BF01647093}

\par\noindent\textbf{Relevance.} This paper is a foundational source for the instrument/operation viewpoint of measurement.
It supports AX-3's claim that ``measurement'' should be defined by operations (instruments/POVMs) rather than treated as an interpretational postulate separate from dynamics. First cited in \S\ref{sec:AX1}.

\bibitem[de Haro, Skenderis, \& Solodukhin(2001)]{deHaro2001}

\textbf{de Haro, S., Skenderis, K., \& Solodukhin, S.~N. (2001).}
Holographic reconstruction of spacetime and renormalization in the AdS/CFT correspondence.
\emph{Communications in Mathematical Physics}, \emph{217}, 595--622.
\url{https://arxiv.org/abs/hep-th/0002230}
\par\noindent\textbf{Relevance.} This work systematized holographic renormalization, making precise how bulk radial evolution corresponds to boundary RG flow. We cite it as a technical antecedent for POS-5: RG-compatibility is treated as a structural commutation requirement between coarse-graining and bulk--boundary dictionaries. First cited in \S\ref{sec:POS5}.

\bibitem[Diophantus(c.~3rd c.)]{Diophantus}

\textbf{Diophantus of Alexandria (c.~3rd century CE).}
\emph{Arithmetica}. English translation: T.~L. Heath,
\emph{Diophantus of Alexandria: A Study in the History of Greek
Algebra}, Cambridge University Press, 1885 (reprinted Dover, 1964).
\par\noindent\textbf{Relevance.} Diophantus's integer-only equations
are the historical seed of every subsequent attempt to lock physical
predictions to integer constraints. In the constitutional reading of
$\POS{6}~\OGN{}$, the \OGNfive{} ledger
$(M,N,L;J,K) = (1,3,5;7,35)$ is the unique solution of a
\emph{Diophantine system} in this strict ancient sense. First cited in \S\ref{sec:POS6}.

\bibitem[Dirac(1930)]{Dirac1930}

\textbf{Dirac, P.\ A.\ M.\ (1930).}
\textit{The Principles of Quantum Mechanics}.
Oxford University Press.
\par\noindent\textbf{Relevance.} A classic axiomatic presentation of quantum theory. We cite it to underscore that
``axioms'' in quantum mechanics have historically mixed dynamics with measurement rules, motivating
a more explicit separation between axioms and postulates. First cited in \S\ref{sec:AX3}.

\bibitem[Dirac(1931)]{Dirac1931}

\textbf{Dirac, P.~A.~M. (1931).}
Quantised singularities in the electromagnetic field.
\textit{Proceedings of the Royal Society A}, \textit{133}(821), 60--72.
\url{https://doi.org/10.1098/rspa.1931.0130}

\par\noindent\textbf{Relevance.} Dirac's monopole argument is a classic demonstration that charge quantization can follow from topology and consistency conditions.
It provides historical precedent for AX-2's mechanism ``quantization from geometry/topology'' (e.g.\ from compact holonomy sectors in the bulk). First cited in \S\ref{sec:AX2}.

\bibitem[Doplicher, Haag \& Roberts(1971)]{DoplicherHaagRoberts1971}

\textbf{Doplicher, S., Haag, R., \& Roberts, J.~E. (1971).}
Local observables and particle statistics, I.
\emph{Communications in Mathematical Physics}, \emph{23}, 199--230.
\par\noindent\textbf{Relevance.} The DHR analysis of superselection
sectors and statistics from local observables. In the constitutional
reading of $\POS{4}~\CP{}$, DHR demonstrates that the entire kinematic
structure of a quantum theory --- charges, statistics, gauge group
--- can be reconstructed from the algebra of local observables and
their representations, supporting the algebra-first reading used by
\GGTheory{}. First cited in \S\ref{sec:POS1}.

\bibitem[Douglas(2003)]{Douglas2003}

\textbf{Douglas, M.~R. (2003).}
The statistics of string/M theory vacua.
\textit{Journal of High Energy Physics}, \textit{2003}(05), 046.
\url{https://arxiv.org/abs/hep-th/0303194}
\par\noindent\textbf{Relevance.} The ``landscape'' perspective illustrates a modern form of theory-space proliferation: many consistent-looking vacua and effective theories appear, and selecting Nature's point requires additional principles. We cite it as motivation for \MSMF{}: an axiomatic constraint intended to collapse landscapes into derived, minimal-ledger predictions. First cited in \S\ref{sec:AX4}.

\bibitem[Dyson(1949)]{Dyson1949}

\textbf{Dyson, F.~J.\ (1949).}
The radiation theories of Tomonaga, Schwinger, and Feynman.
\textit{Physical Review}, \textit{75}(3), 486--502.
\url{https://doi.org/10.1103/PhysRev.75.486}
\par\noindent\textbf{Relevance.} Dyson's 1949 paper unified the apparently independent renormalization programs of Tomonaga (Tokyo), Schwinger (Harvard), and Feynman (Cornell) into a single coherent framework, demonstrating that the three approaches yield identical observable predictions for QED. This was the founding moment of renormalized quantum field theory. We cite it in POS-5 (RG) as the original technical achievement from which the renormalization-group flow concept later emerged. First cited in \S\ref{sec:POS5}.

\bibitem[Ehresmann(1950)]{Ehresmann1950}

\textbf{Ehresmann, C. (1950).}
Les connexions infinit\'esimales dans un espace fibr\'e diff\'erentiable.
In \textit{Colloque de Topologie (Espaces Fibr\'es), Bruxelles, 1950}, 29--55.

\par\noindent\textbf{Relevance.} The founding definition of a connection on a fiber bundle, and with it the rigorous mathematical form of \GGEP{} (AX-2): a gauge field is a connection, a field strength a curvature, a gauge phase a holonomy. Cited under AX-2 milestone~(7). First cited in \S\ref{sec:AX2}.

\bibitem[Einstein(1905)]{Einstein1905}

\textbf{Einstein, A. (1905).}
Zur Elektrodynamik bewegter K{\"o}rper.
\textit{Annalen der Physik}, \textit{17}, 891--921.
\url{https://doi.org/10.1002/andp.19053221004}
\par\noindent\textbf{Relevance.} Einstein's operational reconstruction of simultaneity and inertial frames exemplifies a successful ``axiom-like'' intervention in physics: changing the admissibility rules for coordinates and clocks reshaped all of kinematics. We cite it to justify treating operational definitions (\OGGEP{}) as constitutional. First cited in \S\ref{sec:intro}.

\bibitem[Einstein(1907)]{Einstein1907}

\textbf{Einstein, A. (1907).}
{\"U}ber das Relativit{\"a}tsprinzip und die aus demselben gezogenen Folgerungen.
\textit{Jahrbuch der Radioaktivit{\"a}t und Elektronik}, \textit{4}, 411--462.
\par\noindent\textbf{Relevance.} This early paper contains key steps toward the equivalence-principle reasoning that culminated in general relativity.
It anchors AX-2 historically: \GGEP{} is explicitly framed as a generalization of equivalence-principle logic from gravity to a composite geometry+gauge connection. First cited in \S\ref{sec:AX2}.

\bibitem[Einstein(1915)]{Einstein1915}

\textbf{Einstein, A. (1915).}
Die Feldgleichungen der Gravitation.
\textit{Sitzungsberichte der Preussischen Akademie der Wissenschaften zu Berlin}, 844--847.
\par\noindent\textbf{Relevance.} The field equations of GR encode gravity as geometry and are historically rooted in the equivalence principle. We cite GR as a prototype of ``geometrization'' and as a precedent for extending an equivalence principle to a more unified object (\GGEP{}). First cited in \S\ref{sec:intro}.

\bibitem[Einstein(1916)]{Einstein1916}

\textbf{Einstein, A. (1916).}
Die Grundlage der allgemeinen Relativit{\"a}tstheorie.
\textit{Annalen der Physik}, \textit{49}, 769--822.
\url{https://doi.org/10.1002/andp.19163540702}
\par\noindent\textbf{Relevance.} Einstein's 1916 exposition gives the mature geometric formulation of GR and its equivalence-principle logic. We cite it as the historical anchor for AX-2: \GGEP{} upgrades the equivalence principle from ``gravity only'' to ``gravity + gauge as one composite connection.'' First cited in \S\ref{sec:AX1}.

\bibitem[Einstein(1933)]{EinsteinSimplicity}

\textbf{Einstein, A. (1933).}
On the method of theoretical physics. The Herbert Spencer Lecture,
Oxford, 10 June 1933. Reprinted in \emph{Philosophy of Science}
\textbf{1}(2), 163--169 (1934).
\par\noindent\textbf{Relevance.} Einstein's celebrated statement that
``a theorem which at present cannot be based upon anything more than
upon a faith in the simplicity, i.e.\ intelligibility, of nature'' must
nevertheless guide the working physicist. In the constitutional
reading of $\POS{6}~\OGN{}$, Einstein 1933 is the deepest twentieth-
century philosophical articulation of the Occam-razor discipline that
POS-6 elevates from preference to axiom. First cited in \S\ref{sec:POS6}.

\bibitem[Einstein, Podolsky and Rosen(1935)]{EPR1935}

\textbf{Einstein, A., Podolsky, B., and Rosen, N.\ (1935).}
Can quantum-mechanical description of physical reality be considered complete?
\textit{Physical Review}, \textit{47}, 777--780.
\url{https://doi.org/10.1103/PhysRev.47.777}
\par\noindent\textbf{Relevance.} The Einstein--Podolsky--Rosen
paper of 1935 posed the completeness problem of quantum
mechanics --- whether the wavefunction is a complete description
of physical reality or whether further hidden variables are
required. In the AX-3 lineage, the EPR argument is the
historical pivot at which the question of \emph{what gets
registered as a measurement outcome} became a central foundational
question. The constitutional answer is that the boundary record
(via the operational dictionary $\Dmap$) is what physics speaks
of; the bulk-side $\mathcal{H}_\Bulk$ does not need to match
the boundary's $\mathcal{H}_\Bdy$, and the EPR completeness
question is resolved by recognizing that completeness is a
property of the operational compiler. First cited in \S\ref{sec:AX3}.

\bibitem[Euclid(c.\ 300 BCE)]{EuclidElements}

Euclid. (c.\ 300 BCE).
\textit{Elements}.
(Various modern editions and translations).

\par\noindent\textbf{Relevance.} Euclid's axiomatic method is the historical prototype for a ``constitution'' of a scientific domain. The present
paper explicitly aims to play a similar role for \GGTheory{}: a short axiom/postulate set that generates an extensive theorem stack. First cited in \S\ref{sec:historical_chain}.

\bibitem[Euler(1744)]{EulerCalcVar}

\textbf{Euler, L. (1744).}
\emph{Methodus inveniendi lineas curvas maximi minimive proprietate
gaudentes, sive solutio problematis isoperimetrici latissimo sensu
accepti}. Lausanne: Bousquet.
\par\noindent\textbf{Relevance.} Euler's \emph{Methodus inveniendi}
established the calculus of variations as a rigorous mathematical
discipline. In the constitutional reading of $\POS{2}~\UEP{}$, Euler
is the historical hinge between Maupertuis's stated principle of least
action and Lagrange's full analytical apparatus --- without Euler's
calculus of variations the modern variational engine of physics would
not exist. First cited in \S\ref{sec:POS2}.

\bibitem[Faraday(1845)]{Faraday1845}

\textbf{Faraday, M. (1845).}
On the magnetisation of light and the illumination of magnetic lines of
force. \emph{Philosophical Transactions of the Royal Society of
London}, \emph{136}, 1--20 (read November 1845).
\par\noindent\textbf{Relevance.} Faraday's discovery that magnetic
fields rotate the polarization of light --- the first experimental
hint that the electromagnetic potential carries genuinely physical
information beyond what is locally encoded in the field strengths.
In the constitutional reading of $\POS{1}~\GI{}$, Faraday's intuition
of ``tubes of force'' as physical objects is the deepest historical
ancestor of the modern bundle-theoretic reading of gauge fields. First cited in \S\ref{sec:POS1}.

\bibitem[Fermat(1662)]{Fermat1662}

\textbf{Fermat, P. (1662).}
\emph{Synthesis ad refractiones}. Manuscript, communicated in
correspondence with M.~de la Chambre; published posthumously.
Modern discussion in Sabra, \emph{Theories of Light from Descartes to
Newton}, Cambridge University Press, 1981.
\par\noindent\textbf{Relevance.} Fermat's principle of least time: the
historical seed of every variational principle in physics. In the
constitutional reading of $\POS{2}~\UEP{}$, Fermat 1662 is the deepest
ancestor of the unified extremal principle, predating Maupertuis by
eighty years and Hamilton by nearly two centuries. First cited in \S\ref{sec:POS2}.

\bibitem[Feyerabend(1975)]{Feyerabend1975}

\textbf{Feyerabend, P. (1975).}
\emph{Against Method: Outline of an Anarchistic Theory of Knowledge}.
London: New Left Books.
\par\noindent\textbf{Relevance.} Feyerabend's radical critique of
falsificationism. In the constitutional reading of $\POS{7}~\MF{}$,
Feyerabend is cited together with Kuhn and Lakatos as the philosophical
context against which Popper's naive demarcation criterion was
refined; the modern pre-registration discipline embedded in POS-7 is a
direct working-physics response to these philosophical complications. First cited in \S\ref{sec:POS7}.

\bibitem[Feynman(1948)]{Feynman1948}

\textbf{Feynman, R.~P. (1948).}
Space-time approach to non-relativistic quantum mechanics.
\textit{Reviews of Modern Physics}, \textit{20}(2), 367--387.
\url{https://doi.org/10.1103/RevModPhys.20.367}
\par\noindent\textbf{Relevance.} The path integral makes the connection between action extremization and quantum amplitudes explicit (stationary phase in the classical limit). We cite it in POS-2 to link PLA to quantum mechanics within a single extremal framework. First cited in \S\ref{sec:POS2}.

\bibitem[Feynman(1949)]{Feynman1949QED}

\textbf{Feynman, R.~P.\ (1949).}
Space-time approach to quantum electrodynamics.
\textit{Physical Review}, \textit{76}(6), 769--789.
\url{https://doi.org/10.1103/PhysRev.76.769}
\par\noindent\textbf{Relevance.} Feynman's 1949 reformulation of QED in terms of space-time diagrams and propagators provided the diagrammatic technology that made all subsequent quantum-field-theory calculations practical. The renormalization procedure is constructed by absorbing divergent loop integrals into redefinitions of the bare mass and charge. We cite it in POS-5 as one of the three foundational QED-renormalization papers (Tomonaga 1946, Schwinger 1948, Feynman 1949) from which the renormalization-group concept later emerged. First cited in \S\ref{sec:POS5}.

\bibitem[Feynman \& Vernon(1963)]{FeynmanVernon1963}

\textbf{Feynman, R.\ P., \& Vernon, F.\ L.\ (1963).}
The theory of a general quantum system interacting with a linear dissipative system.
\textit{Annals of Physics, 24}, 118--173.
\par\noindent\textbf{Relevance.} Introduces the influence functional for open quantum systems, a key tool for deriving reduced dynamics
and decoherence. It is cited to connect \OGGEP{}/\GDOS{}-style open evolution to established path-integral machinery. First cited in \S\ref{sec:AX3}.

\bibitem[Frauchiger \& Renner(2018)]{FrauchigerRenner2018}

\textbf{Frauchiger, D., \& Renner, R.\ (2018).}
Quantum theory cannot consistently describe the use of itself.
\emph{Nature Communications}, \emph{9}, 3711.
\url{https://doi.org/10.1038/s41467-018-05739-8}
\par\noindent\textbf{Relevance.} The Frauchiger-Renner thought
experiment is the sharpest recent no-go statement against
interpretation-free quantum mechanics: it argues that any single
universal interpretation is inconsistent under iterated agent-based
measurement. In the constitutional reading of $\AX{3}~\OGGEP{}$,
the Frauchiger-Renner paradox is the boundary report of an attempt
to compose two $\OGGEP{}$-projections at different scales without
honoring the operational dictionary's CPTP discipline; once the
projections are required to be admissible CPTP channels and the
measurement contract is registered, the paradox dissolves into a
specification of which channel the second observer is actually
running. Cited as the canonical modern target of the
constitutional answer. First cited in \S\ref{app:more_examples_template_claim}.

\bibitem[Freed(2019)]{Freed2019}

\textbf{Freed, D.~S. (2019).}
Anomalies and invertible field theories.
\emph{Geometry and Topology of Quantum Field Theories},
Lecture Notes (Park City Mathematics Institute).
\url{https://arxiv.org/abs/1404.7224}
\par\noindent\textbf{Relevance.} Freed's modern categorical
formulation of anomalies as invertible (n+1)-dimensional topological
field theories. In the constitutional reading of $\POS{3}~\AF{}$,
this work makes the geometric-origin-of-anomaly principle precise:
every boundary anomaly is the partition function of a bulk invertible
TQFT, so anomaly cancellation is a bulk-side topological identity. First cited in \S\ref{sec:POS3}.


\bibitem[Freed, Hopkins \& Teleman(2010)]{FreedHopkinsTeleman2010}

\textbf{Freed, D.~S., Hopkins, M.~J., \& Teleman, C. (2010).}
Loop groups and twisted K-theory III.
\emph{Annals of Mathematics}, \emph{174}, 947--1007.
\url{https://arxiv.org/abs/math/0312155}
\par\noindent\textbf{Relevance.} The Freed--Hopkins--Teleman theorem
relating twisted equivariant K-theory of a compact Lie group to its
Verlinde algebra. In the constitutional reading of $\POS{3}~\AF{}$,
this work is the categorical scaffolding on which the modern
anomaly-as-TQFT picture rests, supplying the technical foundation for
the bulk-side classification of admissible gauge structures. First cited in \S\ref{sec:POS3}.

\bibitem[Freedman \& Clauser(1972)]{FreedmanClauser1972}

\textbf{Freedman, S.~J., \& Clauser, J.~F. (1972).}
Experimental test of local hidden-variable theories.
\emph{Physical Review Letters}, \emph{28}, 938--941.
\url{https://doi.org/10.1103/PhysRevLett.28.938}
\par\noindent\textbf{Relevance.} The first experimental confirmation
of Bell-inequality violation. In the constitutional reading of
$\POS{7}~\MF{}$, Freedman--Clauser is cited as the historical anchor
for the empirical-physics practice of pre-declaring a kill criterion
(``if the inequality holds, quantum mechanics is wrong'') and then
running the registered test. First cited in \S\ref{sec:POS7}.

\bibitem[Freidlin \& Wentzell(1998)]{FreidlinWentzell1998}

\textbf{Freidlin, M.~I., \& Wentzell, A.~D. (1998).}
\emph{Random Perturbations of Dynamical Systems} (2nd ed.).
New York: Springer.
\url{https://doi.org/10.1007/978-1-4612-0611-5}
\par\noindent\textbf{Relevance.} Freidlin--Wentzell theory provides a rigorous large-deviation framework in which stochastic open-system trajectories concentrate near minimizers of an action functional. We cite it as mathematical background for POS-2 (\UEP{}), supporting the claim that ``least-action'' logic persists in open/noisy systems as a most-probable-path extremal principle. First cited in \S\ref{sec:POS2}.

\bibitem[Friston(2010)]{Friston2010}

\textbf{Friston, K. (2010).}
The free-energy principle: a unified brain theory?
\textit{Nature Reviews Neuroscience}, \textit{11}(2), 127--138.
\url{https://doi.org/10.1038/nrn2787}
\par\noindent\textbf{Relevance.} Friston's free-energy principle is a prominent modern attempt to unify perception, learning, and action by variational optimization. We cite it in the bridge to \AFive{} in \citep{ArisakaA5GDOS} to illustrate that ``free-energy-like'' scalar functionals already function as organizing principles in neuroscience, motivating the \GGTheory{} claim that \GFF{} and \GDOS{} provide a systematic open-system generalization with explicit gauge/geometry and falsifiability constraints. First cited in \S\ref{sec:POS2}.

\bibitem[Fujikawa(1979)]{Fujikawa1979}

\textbf{Fujikawa, K. (1979).}
Path-integral measure for gauge-invariant fermion theories.
\textit{Physical Review Letters}, \textit{42}, 1195--1198.
\par\noindent\textbf{Relevance.} Shows how anomalies arise from the non-invariance of the fermionic path-integral measure, providing a clean and widely used derivation. This is a key technical underpinning of POS-3, which treats anomalies as an axiomatic violation unless canceled by inflow. First cited in \S\ref{sec:POS3}.

\bibitem[Galilei(1632)]{Galilei1632}

\textbf{Galilei, G.\ (1632).}
\textit{Dialogo sopra i due massimi sistemi del mondo}
(\textit{Dialogue Concerning the Two Chief World Systems}).
Florence: Landini.
\par\noindent\textbf{Relevance.} Galileo's \textit{Dialogue} of
1632 contains the first explicit statement of the principle of
relativity --- the famous ``ship's cabin'' thought experiment in
which mechanics inside a uniformly moving ship is
indistinguishable from mechanics at rest. In the AX-1 lineage,
Galileo's principle is the original recognition that the laws of
physics carry a kinematic invariance group: there is no
preferred frame, and any admissible kernel must respect that
invariance. Locality and causality, as we now understand them,
are the relativistic refinement of Galileo's insight. First cited in \S\ref{sec:AX1}.

\bibitem[Gardiner(2009)]{Gardiner2009}

\textbf{Gardiner, C.~W. (2009).}
\textit{Stochastic Methods: A Handbook for the Natural and Social Sciences} (4th ed.).
Springer.

\par\noindent\textbf{Relevance.} A comprehensive reference for stochastic differential equations and the Fokker--Planck formalism.
It supports the \GDOS{} ``kernel'' derivations in the appendices and clarifies positivity-preserving conditions (POS-\CP). First cited in \S\ref{sec:AX4}.

\bibitem[Gell-Mann \& Low(1954)]{GellMannLow1954}

\textbf{Gell-Mann, M., \& Low, F.~E. (1954).}
Quantum electrodynamics at small distances.
\emph{Physical Review}, \emph{95}, 1300--1312.
\url{https://doi.org/10.1103/PhysRev.95.1300}
\par\noindent\textbf{Relevance.} The Gell-Mann--Low paper is a foundational early statement of RG ideas in quantum field theory. It is cited as historical background for POS-5 (RG): scale dependence must be organized systematically, and physical content must be extracted in an RG-consistent way. First cited in \S\ref{sec:POS5}.

\bibitem[Georgi and Glashow(1974)]{GeorgiGlashow1974}

\textbf{Georgi, H., \& Glashow, S.~L. (1974).}
Unity of all elementary-particle forces.
\textit{Physical Review Letters}, \textit{32}(8), 438--441.
\href{https://doi.org/10.1103/PhysRevLett.32.438}{doi:10.1103/PhysRevLett.32.438}.
\par\noindent\textbf{Relevance.}
The SU(5) grand-unification proposal is a canonical example of symmetry-driven unification beyond the Standard Model.
\GGTheory{}\ references it historically, but differs axiomatically by treating gauge--gravity unification through
a composite geometric connection (\GGEP{}) and by enforcing anomaly/normalization constraints as admissibility filters. First cited in \S\ref{sec:AX4}.

\bibitem[Giustina et~al.(2015)]{Giustina2015}

\textbf{Giustina, M., et~al.\ (2015).}
Significant-loophole-free test of Bell's theorem with entangled photons.
\textit{Physical Review Letters}, \textit{115}(25), 250401.
\url{https://doi.org/10.1103/PhysRevLett.115.250401}
\par\noindent\textbf{Relevance.} The Vienna loophole-free Bell test
of 2015, performed in parallel with the Delft \citep{Hensen2015}
and NIST experiments. We cite it together with Aspect (1982) and
Hensen (2015) at every axiomatic invocation of \AX{1}~\LC{}
as the laboratory closure of the locality crisis identified by
Bell (1964). First cited in \S\ref{sec:AX1}.

\bibitem[Glashow(1961)]{Glashow1961}

\textbf{Glashow, S.~L. (1961).}
Partial-symmetries of weak interactions.
\textit{Nuclear Physics}, \textit{22}, 579--588.
\url{https://doi.org/10.1016/0029-5582(61)90469-2}
\par\noindent\textbf{Relevance.} Glashow's gauge unification of electromagnetic and weak interactions initiated the electroweak structure completed by Weinberg and Salam. We cite these as the historical backbone for POS-1 (Gauge Invariance) and for the claim that symmetry principles organize viable theories. First cited in \S\ref{sec:intro}.

\bibitem[Gorini et~al.(1976)]{Gorini1976}

\textbf{Gorini, V., Kossakowski, A., \& Sudarshan, E.~C.~G. (1976).}
Completely positive dynamical semigroups of $N$-level systems.
\textit{Journal of Mathematical Physics}, \textit{17}(5), 821--825.
\par\noindent\textbf{Relevance.} Along with Lindblad's work, this establishes the generator form required for physical open-system quantum evolution.
It supports POS-\CP\ and the \OGGEP\ demand that open evolution be part of the foundational dictionary. First cited in \S\ref{sec:intro}.

\bibitem[Green and Schwarz(1984)]{GreenSchwarz1984}

\textbf{Green, M.~B., \& Schwarz, J.~H. (1984).}
Anomaly cancellations in supersymmetric $D=10$ gauge theory and superstring theory.
\textit{Physics Letters B}, \textit{149}(1--3), 117--122.
\par\noindent\textbf{Relevance.} Green and Schwarz's 1984 construction
is the prototype of bulk-side classical absorption of a boundary anomaly
by a $B_2$-form coupling. In the constitutional reading of
$\POS{3}~\AF{}$, what Green and Schwarz wrote down in 1984 is the
bulk-side classical-topological identity --- not a quantum cancellation
of a one-loop pathology but a $B_2$-coupled rewrite of the bulk Bianchi
identity descended from Bianchi 1902. The cohort framework occupies
exactly this bulk-side classical-topological sector and inherits the
$dH_3 = X_4$ structural identity. The quantum-string content that
Green--Schwarz \emph{also} buys (graviton vertex operators, $\alpha'$
corrections, perturbative-string apparatus) is a separate physical
sector that the cohort axioms neither subsume nor contradict. Cited as
the historical anchor for the bulk-side classical-absorption framing of
$\POS{3}~\AF{}$, with the understanding that the cohort claim is
strictly the classical-topological backbone. First cited in \S\ref{sec:four_clusters}.

\bibitem[Gross \& Wilczek(1973)]{GrossWilczek1973}

\textbf{Gross, D.~J., \& Wilczek, F. (1973).}
Ultraviolet behavior of non-Abelian gauge theories.
\textit{Physical Review Letters}, \textit{30}(26), 1343--1346.
\url{https://doi.org/10.1103/PhysRevLett.30.1343}
\par\noindent\textbf{Relevance.} Asymptotic freedom is a key structural fact about non-Abelian gauge theories and underlies the consistency of QCD. We cite it to emphasize that ``viability filters'' (renormalizability, RG behavior) are part of what makes gauge theories physically admissible. First cited in \S\ref{sec:AX4}.

\bibitem[Gubser, Klebanov, \& Polyakov(1998)]{Gubser1998}

\textbf{Gubser, S.~S., Klebanov, I.~R., \& Polyakov, A.~M. (1998).}
Gauge theory correlators from non-critical string theory.
\textit{Physics Letters B}, \textit{428}(1--2), 105--114.
\url{https://arxiv.org/abs/hep-th/9802109}
\par\noindent\textbf{Relevance.} Together with Witten's prescription, this paper established the concrete AdS/CFT dictionary relating bulk fields to boundary correlators. We cite it as historical precedent for treating a bulk$\to$boundary export map as part of the theory's definition, which \OGGEP{} generalizes to include explicit measurement semantics. First cited in \S\ref{sec:AX2}.

\bibitem[Guth(1981)]{Guth1981}

\textbf{Guth, A.~H. (1981).}
Inflationary universe: A possible solution to the horizon and flatness problems.
\textit{Physical Review D}, \textit{23}(2), 347--356.
\url{https://doi.org/10.1103/PhysRevD.23.347}
\par\noindent\textbf{Relevance.} Inflation is the dominant paradigm for early-universe initial conditions. We cite it as a core cosmological motivation: the mechanism is empirically successful yet microphysically underdetermined, supporting the need for a stronger constitution that restricts ``allowed'' inflationary dynamics and connects them to observation. First cited in \S\ref{sec:motivation}.

\bibitem[Haag(1992)]{Haag1992}

\textbf{Haag, R. (1992).}
\textit{Local Quantum Physics: Fields, Particles, Algebras} (2nd ed.).
Springer.
\url{https://doi.org/10.1007/978-3-662-03081-1}
\par\noindent\textbf{Relevance.} Haag's formulation of QFT emphasizes locality, causality, and operator-algebraic structure. We cite it to ground AX-1 in the rigorous QFT tradition and to clarify what ``locality'' means when expressed as constraints on observable algebras. First cited in \S\ref{sec:intro}.

\bibitem[Haag \& Kastler(1964)]{HaagKastler1964}

\textbf{Haag, R., \& Kastler, D. (1964).}
An algebraic approach to quantum field theory.
\emph{Journal of Mathematical Physics}, \emph{5}, 848--861.
\par\noindent\textbf{Relevance.} The Haag--Kastler axioms for local
quantum physics: a net of $C^*$-algebras of local observables indexed
by bounded spacetime regions. In the constitutional reading of
$\POS{4}~\CP{}$, Haag--Kastler is the canonical algebraic statement of
positivity + closure for a relativistic local theory. First cited in \S\ref{sec:POS4}.

\bibitem[Hamilton(1834)]{Hamilton1834}

\textbf{Hamilton, W.~R. (1834).}
On a general method in dynamics.
\textit{Philosophical Transactions of the Royal Society of London}, \textit{124}, 247--308.
\url{https://doi.org/10.1098/rstl.1834.0017}
\par\noindent\textbf{Relevance.} Hamilton's formulation is the modern backbone of the principle of least action and canonical dynamics. We cite it in POS-2 to emphasize that extremal principles historically reorganized mechanics and are a natural candidate for a unification engine. First cited in \S\ref{sec:AX4}.

\bibitem[Hamilton(1835)]{Hamilton1835}

\textbf{Hamilton, W.~R. (1835).}
Second essay on a general method in dynamics.
\emph{Philosophical Transactions of the Royal Society of London},
\emph{125}, 95--144.
\par\noindent\textbf{Relevance.} Hamilton's reformulation of mechanics
on phase space, introducing what we now call the Hamiltonian and the
canonical equations. In the constitutional reading of $\POS{2}~\UEP{}$,
Hamilton 1835 is the technical bridge from Lagrangian to phase-space
formulations and the historical foundation of Hamilton--Jacobi theory
on which the WKB limit of POS-2 rests. First cited in \S\ref{sec:POS2}.

\bibitem[Hardy \& Ramanujan(1918)]{HardyRamanujan1918}

\textbf{Hardy, G.~H., \& Ramanujan, S. (1918).}
Asymptotic formulae in combinatory analysis.
\emph{Proceedings of the London Mathematical Society}, \emph{17},
75--115.
\par\noindent\textbf{Relevance.} The Hardy--Ramanujan circle method
for partition counting. In the constitutional reading of
$\POS{6}~\OGN{}$, Hardy--Ramanujan exemplify the modern arithmetic-
representation theory tradition into which the \OGNfive{} Diophantine
constraint $K = M^2 + N^2 + L^2$ naturally fits. First cited in \S\ref{sec:POS6}.

\bibitem[Harlow \& Ooguri(2019)]{HarlowOoguri2019}

\textbf{Harlow, D., \& Ooguri, H. (2019).}
Symmetries in quantum field theory and quantum gravity.
\emph{Communications in Mathematical Physics}, \emph{383}, 1669--1804 (published 2021; arXiv 2019).
\url{https://arxiv.org/abs/1810.05338}
\par\noindent\textbf{Relevance.} The modern technical statement of the
``no global symmetry in quantum gravity'' conjecture. In the
constitutional reading of $\POS{1}~\GI{}$, Harlow--Ooguri supplies the
contemporary philosophical justification for the primacy of gauge
invariance: any apparent global symmetry must, in any consistent UV
completion, either be gauged or be broken. First cited in \S\ref{sec:POS1}.

\bibitem[Hawking(1975)]{Hawking1975}

\textbf{Hawking, S.~W. (1975).}
Particle creation by black holes.
\textit{Communications in Mathematical Physics}, \textit{43}, 199--220.
\par\noindent\textbf{Relevance.} Hawking radiation is a canonical example of an open-system process in fundamental physics: a horizon mediates information/energy exchange between regions, producing an arrow of time in an otherwise unitary theory. We cite it to motivate why \GDOS{} treats openness and operational observation as foundational. First cited in \S\ref{sec:AX3}.

\bibitem[Heisenberg(1925)]{Heisenberg1925}

\textbf{Heisenberg, W. (1925).}
{\"U}ber quantentheoretische Umdeutung kinematischer und mechanischer Beziehungen.
\textit{Zeitschrift f{\"u}r Physik}, \textit{33}(1), 879--893.
\par\noindent\textbf{Relevance.} Matrix mechanics is one of the two founding formulations of quantum theory and exemplifies the radical redefinition
of dynamical variables. It sets historical context for treating quantum structure as something potentially geometric. First cited in \S\ref{sec:historical_chain}.

\bibitem[Heisenberg(1927)]{Heisenberg1927}

\textbf{Heisenberg, W.\ (1927).}
{\"U}ber den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik.
\textit{Zeitschrift f{\"u}r Physik, 43}, 172--198.
\par\noindent\textbf{Relevance.} A primary source for the uncertainty principle and early quantum-measurement ideas.
It is cited as part of the historical background for why operational definitions are central to quantum theory. First cited in \S\ref{sec:AX3}.

\bibitem[Hensen et~al.(2015)]{Hensen2015}

\textbf{Hensen, B., et~al.\ (2015).}
Loophole-free Bell inequality violation using electron spins separated by 1.3 kilometres.
\textit{Nature}, \textit{526}(7575), 682--686.
\url{https://doi.org/10.1038/nature15759}
\par\noindent\textbf{Relevance.} The Delft loophole-free Bell test
of 2015, the first to close the locality and detection loopholes
simultaneously. We cite it together with Aspect (1982) and
Giustina (2015) as the laboratory closure of the locality crisis
that motivates \AX{1}~\LC{} as a constitutional clause rather
than a metaphysical claim. First cited in \S\ref{sec:AX1}.

\bibitem[Heron(c.~1st c.)]{Heron}

\textbf{Heron of Alexandria (c.~1st century CE).}
\emph{Catoptrica} (on reflection). Recovered through Latin
translation and modern editions; see Cohen \& Drabkin,
\emph{A Source Book in Greek Science}, McGraw-Hill, 1948.
\par\noindent\textbf{Relevance.} Heron's argument that light's path of
equal angles minimizes the distance --- the first recorded extremal
principle in physics. In the constitutional reading of $\POS{2}~\UEP{}$,
Heron is the deepest ancient ancestor of every subsequent variational
principle. First cited in \S\ref{sec:POS2}.

\bibitem[Hilbert(1915)]{Hilbert1915}

\textbf{Hilbert, D. (1915).}
Die Grundlagen der Physik.
\emph{Nachrichten von der K\"oniglichen Gesellschaft der Wissenschaften
zu G\"ottingen, Mathematisch-physikalische Klasse}, 395--407.
\par\noindent\textbf{Relevance.} Hilbert's variational derivation of
general relativity, contemporaneous with Einstein's field equations
and the historical context in which Noether developed her theorems.
In the constitutional reading of $\POS{1}~\GI{}$, Hilbert 1915 is the
direct ancestor of Noether 1918 and the foundational example of a
gauge theory presented in fully variational form. First cited in \S\ref{sec:POS1}.

\bibitem[Hopkins \& Singer(2005)]{HopkinsSinger2005}

\textbf{Hopkins, M.~J., and Singer, I.~M.\ (2005).}
Quadratic functions in geometry, topology, and M-theory.
\textit{Journal of Differential Geometry}, \textit{70}(3), 329--452.
\url{https://doi.org/10.4310/jdg/1143642908}

\par\noindent\textbf{Relevance.} Hopkins and Singer built the
differential-cohomology machinery --- differential characters,
quadratic refinements, and the long exact sequences relating them ---
on which the global anomaly argument of \AThree{} rests. Where the
present paper says that the \BIsix{} anomaly cancellation is exact
rather than perturbative, this is the mathematics that carries the
word ``exact''. First cited in \S\ref{sec:POS3}.


\bibitem[Hosotani(1983)]{Hosotani1983}

\textbf{Hosotani, Y. (1983).}
Dynamical mass generation by compact extra dimensions.
\textit{Physics Letters B}, \textit{126}, 309--313.
\url{https://doi.org/10.1016/0370-2693(83)90170-3}
\par\noindent\textbf{Relevance.} Hosotani's 1983 paper introduces
the mechanism by which a higher-dimensional gauge symmetry is broken
to a lower-dimensional subgroup by a non-vanishing vacuum
expectation value of the Wilson line wrapping a compact direction.
It is the first concrete extra-dimensional realization of
$\AX{2}~\GGEP{}$: it shows that the holonomy of the gauge
connection around a compact direction is itself a dynamical field
whose vacuum value selects the low-energy gauge content. The
cohort framework's use of the $S^1_\theta$ holonomy phase as the
carrier of $\AX{3}~\OGGEP{}$'s hidden-variable content is a
direct descendant of the Hosotani construction. First cited in \S\ref{sec:AX2}.

\bibitem[Hosotani(1989)]{Hosotani1989}

\textbf{Hosotani, Y. (1989).}
Dynamics of non-integrable phases and gauge symmetry breaking.
\textit{Annals of Physics (NY)}, \textit{190}, 233--253.
\url{https://doi.org/10.1016/0003-4916(89)90015-8}
\par\noindent\textbf{Relevance.} Hosotani's 1989 follow-up
develops the full systematic treatment of dynamical gauge symmetry
breaking via compact-direction Wilson-line holonomy. Together
with the 1983 paper, it establishes the compact-holonomy-as-physical-field
reading of the gauge connection that $\AX{2}~\GGEP{}$ axiomatizes.
We cite it as the systematic technical reference for the Hosotani
mechanism in the AX-2 historical lineage. First cited in \S\ref{sec:AX2}.

\bibitem[Jarzynski(1997)]{Jarzynski1997}

\textbf{Jarzynski, C. (1997).}
Nonequilibrium equality for free energy differences.
\textit{Physical Review Letters}, \textit{78}(14), 2690--2693.
\par\noindent\textbf{Relevance.} Jarzynski's equality shows that free-energy differences can be inferred from nonequilibrium work statistics. It supports the \UEP{}/\GDOS{} viewpoint that variational/ledger principles remain meaningful and falsifiable far from equilibrium. First cited in \S\ref{sec:AX1}.

\bibitem[Jaynes(1957)]{Jaynes1957}

\textbf{Jaynes, E.\ T.\ (1957).}
Information theory and statistical mechanics.
\textit{Physical Review, 106}(4), 620--630.
DOI: \href{https://doi.org/10.1103/PhysRev.106.620}{10.1103/PhysRev.106.620}.
\par\noindent\textbf{Relevance.} Introduces the maximum-entropy (MaxEnt) principle as a constructive variational rule for statistical mechanics.
We cite it in POS-2 to connect thermodynamic extremal principles to inference and to motivate treating free-energy/entropy functionals as legitimate variational objects when paired with operational constraints. First cited in \S\ref{sec:POS2}.

\bibitem[Joos \& Zeh(1985)]{JoosZeh1985}

\textbf{Joos, E., \& Zeh, H.~D.\ (1985).}
The emergence of classical properties through interaction with the environment.
\textit{Zeitschrift f\"ur Physik B}, \textit{59}(2), 223--243.
\url{https://doi.org/10.1007/BF01725541}
\par\noindent\textbf{Relevance.} The decisive demonstration that environmental scattering decoheres a system on extraordinarily short timescales --- one of the foundational papers of the decoherence program. We cite it whenever the
historical chain Zeh (1970) $\to$ Joos \& Zeh (1985) $\to$ Zurek (1981, 2003) is invoked as the modern recognition that openness is unavoidable, in motivation of \AX{3}~\OGGEP{} and the axiomatic inversion that elevates openness from corrective layer to foundational law. First cited in \S\ref{sec:intro}.

\bibitem[Kadanoff(1966)]{Kadanoff1966}

\textbf{Kadanoff, L.\ P.\ (1966).}
Scaling laws for Ising models near $T_c$.
\textit{Physics Physique Fizika, 2}(6), 263--272.
DOI: \href{https://doi.org/10.1103/PhysicsPhysiqueFizika.2.263}{10.1103/PhysicsPhysiqueFizika.2.263}.
\par\noindent\textbf{Relevance.} An early primary source for scaling and coarse-graining ideas that evolved into modern RG.
It motivates POS-5 by showing why scale transformations are structural, not computational tricks. First cited in \S\ref{sec:POS5}.

\bibitem[Kaluza(1921)]{Kaluza1921}

\textbf{Kaluza, T. (1921).}
Zum Unit{\"a}tsproblem der Physik.
\textit{Sitzungsberichte der Preu{\ss}ischen Akademie der Wissenschaften zu Berlin}, 966--972.
\par\noindent\textbf{Relevance.} Kaluza's proposal that electromagnetism can be interpreted geometrically via an extra dimension is a foundational ancestor of the ``gauge as geometry'' program. We cite it as historical support for \GGEP{} and for the appearance of extra coordinates as geometric encodings of gauge structure. First cited in \S\ref{sec:geometrization-ladder-a1}.

\bibitem[Kass \& Raftery(1995)]{KassRaftery1995}

\textbf{Kass, R.~E., \& Raftery, A.~E. (1995).}
Bayes factors.
\emph{Journal of the American Statistical Association}, \emph{90}(430), 773--795.
\url{https://doi.org/10.1080/01621459.1995.10476572}
\par\noindent\textbf{Relevance.} Kass and Raftery provide a widely cited modern reference for Bayes factors as quantitative evidence measures. We cite it as part of the methodological background for POS-7 (MF): falsifiability can be operationalized through explicit decision thresholds, whether frequentist or Bayesian, but must be specified in advance. First cited in \S\ref{sec:POS6}.

\bibitem[Klein(1926)]{Klein1926}

\textbf{Klein, O. (1926).}
Quantentheorie und f{\"u}nfdimensionale Relativit{\"a}tstheorie.
\textit{Zeitschrift f{\"u}r Physik}, \textit{37}, 895--906.
\url{https://doi.org/10.1007/BF01397481}
\par\noindent\textbf{Relevance.} Klein introduced compactification and the resulting quantization of momentum in the extra dimension. We cite it for the standard mechanism by which compact geometry yields discrete spectra (used in AX-2 to illustrate ``quantization from topology'' rather than an imposed quantization rule). First cited in \S\ref{sec:geometrization-ladder-a1}.

\bibitem[Kobayashi \& Nomizu(1963)]{KobayashiNomizu1963}

\textbf{Kobayashi, S., \& Nomizu, K. (1963).}
\textit{Foundations of Differential Geometry} (Vol.~1).
Wiley.
\par\noindent\textbf{Relevance.} This classic text provides the rigorous mathematics of principal bundles, connections, curvature, and holonomy. We cite it as a standard reference for the geometric language used in formulating \GGEP{} (composite connection) and in defining curvature/holonomy as invariants. First cited in \S\ref{sec:AX2}.

\bibitem[Kolmogorov(1933)]{Kolmogorov1933}

\textbf{Kolmogorov, A.~N. (1933).}
\emph{Grundbegriffe der Wahrscheinlichkeitsrechnung}. Berlin:
Springer. English translation: \emph{Foundations of the Theory of
Probability}, Chelsea, 1956.
\par\noindent\textbf{Relevance.} Kolmogorov's axiomatization of
probability: probability measures on $\sigma$-algebras. In the
constitutional reading of $\POS{4}~\CP{}$, Kolmogorov 1933 is the
modern mathematical scaffolding inside which all of POS-4's positivity
and closure requirements live. First cited in \S\ref{sec:POS4}.

\bibitem[Kraus(1971)]{Kraus1971}

\textbf{Kraus, K. (1971).}
General state changes in quantum theory.
\textit{Annals of Physics}, \textit{64}(2), 311--335.
\url{https://doi.org/10.1016/0003-4916(71)90108-4}
\par\noindent\textbf{Relevance.} Kraus operators and instruments provide the modern operational semantics of measurement and state update. We cite this work as foundational for AX-3, where measurement is defined axiomatically as a CP instrument produced by the bulk$\to$boundary dictionary. First cited in \S\ref{sec:AX1}.

\bibitem[Kraus(1983)]{Kraus1983}

\textbf{Kraus, K. (1983).}
\textit{States, Effects, and Operations: Fundamental Notions of Quantum Theory}.
Springer.
\url{https://doi.org/10.1007/978-3-540-39068-5}

\par\noindent\textbf{Relevance.} Kraus' monograph systematizes the operational formulation of quantum measurement and dynamics via completely positive maps.
AX-3 adopts this language axiomatically by requiring that measurement be defined in terms of instruments and that open evolution be CPTP. First cited in \S\ref{sec:AX1}.

\bibitem[Kuhn(1962)]{Kuhn1962}

\textbf{Kuhn, T.~S. (1962).}
\emph{The Structure of Scientific Revolutions}. University of Chicago
Press. Second edition, 1970.
\par\noindent\textbf{Relevance.} Kuhn's analysis of paradigm shifts.
In the constitutional reading of $\POS{7}~\MF{}$, Kuhn is the
philosophical context in which Popper's naive demarcation criterion is
refined: the modern empirical-science culture of registered protocols and pre-registered analyses is the
working-physics response to the
Kuhnian observation that paradigm shifts can otherwise insulate
favored theories from falsification. First cited in \S\ref{sec:POS7}.

\bibitem[Kullback \& Leibler(1951)]{KullbackLeibler1951}

\textbf{Kullback, S., \& Leibler, R.~A. (1951).}
On information and sufficiency.
\textit{Annals of Mathematical Statistics}, \textit{22}(1), 79--86.
\url{https://doi.org/10.1214/aoms/1177729694}
\par\noindent\textbf{Relevance.} The KL divergence is the canonical nonnegative functional measuring distinguishability of probability distributions. We cite it because many ``free energy'' expressions in nonequilibrium physics and inference can be written as KL divergences, providing a clean mathematical bridge for POS-2 (\UEP{}) between thermodynamic and informational extremals. First cited in \S\ref{sec:POS7}.

\bibitem[Lagrange(1770)]{Lagrange1770}

\textbf{Lagrange, J.-L. (1770).}
D\'emonstration d'un th\'eor\`eme d'arithm\'etique.
\emph{Nouveaux M\'emoires de l'Acad\'emie Royale des Sciences et
Belles-Lettres de Berlin}.
\par\noindent\textbf{Relevance.} Lagrange's four-square theorem: every
positive integer is the sum of at most four squares. In the
constitutional reading of $\POS{6}~\OGN{}$, this is the direct
arithmetic ancestor of the \OGNfive{} Diophantine constraint
$K = M^2 + N^2 + L^2$ --- a number-theoretic identity from 1770
appearing in the deepest structural lock of contemporary physics. First cited in \S\ref{sec:POS6}.

\bibitem[Lagrange(1788)]{Lagrange1788}

\textbf{Lagrange, J.-L. (1788).}
\emph{M\'ecanique analytique}. Paris: Veuve Desaint.
\par\noindent\textbf{Relevance.} The full analytical recasting of
classical mechanics as a variational problem on configuration space.
In the constitutional reading of $\POS{2}~\UEP{}$, Lagrange 1788 is the
synthesis that turned Maupertuis's principle and Euler's calculus of
variations into the working mathematical apparatus of theoretical
physics. First cited in \S\ref{sec:POS2}.

\bibitem[Lakatos(1970)]{Lakatos1970}

\textbf{Lakatos, I. (1970).}
Falsification and the methodology of scientific research programmes.
In Lakatos \& Musgrave (eds.), \emph{Criticism and the Growth of
Knowledge}, Cambridge University Press.
\par\noindent\textbf{Relevance.} Lakatos's refinement of Popperian
falsifiability to the level of research programs with a ``hard core''
and protective ``auxiliary hypotheses.'' In the constitutional
reading of $\POS{7}~\MF{}$, Lakatos is cited as the principal
intermediate between Popper's naive demarcation and the modern
pre-registration discipline embedded in POS-7. First cited in \S\ref{sec:POS7}.

\bibitem[Lamb \& Retherford(1947)]{LambRetherford1947}

\textbf{Lamb, W.~E., \& Retherford, R.~C. (1947).}
Fine structure of the hydrogen atom by a microwave method.
\emph{Physical Review}, \emph{72}(3), 241--243.
\url{https://doi.org/10.1103/PhysRev.72.241}
\par\noindent\textbf{Relevance.} The Lamb shift measurement: the
empirical anomaly that triggered the development of modern
renormalized QED. In the constitutional reading of $\POS{5}~\RG{}$,
Lamb--Retherford 1947 is the experimental anchor for the entire
renormalization program that POS-5 elevates to axiomatic status. First cited in \S\ref{sec:POS5}.

\bibitem[Landauer(1961)]{Landauer1961}

\textbf{Landauer, R.\ (1961).}
Irreversibility and heat generation in the computing process.
\textit{IBM Journal of Research and Development}, \textit{5}(3), 183--191.
\url{https://doi.org/10.1147/rd.53.0183}

\par\noindent\textbf{Relevance.} Landauer established that erasing one
bit costs at least $k_B T \ln 2$ of dissipated heat --- the first
result to price information in physical currency. In the
constitutional reading it is one of the three independent nineteenth-
and twentieth-century discoveries that $\AX{3}~\OGGEP{}$ unifies:
Landauer's bound, the Bekenstein--Hawking entropy, and Zurek's
einselection are three faces of one axiom. First cited in
\S\ref{sec:AX3}.


\bibitem[Laplace(1812)]{Laplace1812}

\textbf{Laplace, P.-S. (1812).}
\emph{Th\'eorie analytique des probabilit\'es}. Paris: Veuve Courcier.
\par\noindent\textbf{Relevance.} Laplace's comprehensive treatise on
probability theory. In the constitutional reading of $\POS{4}~\CP{}$,
Laplace is the historical synthesizer between Bernoulli/Bayes and
Kolmogorov: the first complete analytical apparatus for inverse
probability, on which all subsequent quantitative inference rests. First cited in \S\ref{sec:POS4}.

\bibitem[Lindblad(1976)]{Lindblad1976}

\textbf{Lindblad, G. (1976).}
On the generators of quantum dynamical semigroups.
\textit{Communications in Mathematical Physics}, \textit{48}(2), 119--130.
\par\noindent\textbf{Relevance.} Lindblad's theorem provides the canonical generator form for Markovian \CPTP\ evolution. It is central to POS-\CP\
and to \OGGEP's claim that open-system dynamics belongs in the theory's syntax. First cited in \S\ref{sec:intro}.

\bibitem[Linde(1982)]{Linde1982}

\textbf{Linde, A.~D. (1982).}
A new inflationary universe scenario.
\textit{Physics Letters B}, \textit{108}(6), 389--393.
\url{https://doi.org/10.1016/0370-2693(82)91219-9}
\par\noindent\textbf{Relevance.} Linde's work exemplifies the proliferation of inflationary model-building. We cite it to motivate why a ``no-extra-dials'' principle (\MSMF{}) could be valuable: without axiomatic restrictions, many potentials are admissible and prediction becomes diffuse. First cited in \S\ref{sec:motivation}.

\bibitem[Lorentz(1904)]{Lorentz1904}

\textbf{Lorentz, H.~A.\ (1904).}
Electromagnetic phenomena in a system moving with any velocity
smaller than that of light.
\textit{Proceedings of the Royal Netherlands Academy of Arts and
Sciences}, \textit{6}, 809--831.
\par\noindent\textbf{Relevance.} Lorentz's 1904 paper introduced
the transformations that bear his name --- the symmetry group
under which Maxwell's electromagnetism is form-invariant and which
Einstein's 1905 special relativity would elevate to a universal
kinematic constraint. We cite it as the immediate technical
ancestor of the operational locality and causality content of
$\AX{1}~\LC{}$. First cited in \S\ref{sec:AX1}.

\bibitem[Lorenz(1867)]{Lorenz1867}

\textbf{Lorenz, L. (1867).}
\"Uber die Identit\"at der Schwingungen des Lichts mit den elektrischen
Str\"omen.
\emph{Annalen der Physik}, \emph{207}, 243--263.
\par\noindent\textbf{Relevance.} Lorenz's gauge condition (often
historically attributed to H.~A.\ Lorentz, but introduced by the Danish
physicist L.~Lorenz in 1867). In the constitutional reading of
$\POS{1}~\GI{}$, the Lorenz condition is one of the earliest explicit
demonstrations that gauge freedom can be partially fixed without
altering observables --- the first technical instance of gauge fixing. First cited in \S\ref{sec:POS1}.

\bibitem[Mach(1893)]{Mach1893}

\textbf{Mach, E.\ (1893).}
\textit{The Science of Mechanics: A Critical and Historical
Account of its Development}, transl.\ T.~J.~McCormack.
Open Court, La Salle, IL.
(German original: 1883.)
\par\noindent\textbf{Relevance.} Mach's 1893 \textit{Science of
Mechanics} formulated the relational principle: inertial frames
are defined relative to the distribution of distant matter, not
to an absolute background. In the AX-4 lineage, Mach's principle
is the philosophical ancestor of the anti-tuning ethos: a theory
is more honest when its physical content is determined by
relational structure rather than by adjustable absolute
parameters. $\AX{4}~\MSMF{}$ is the modern axiomatic refinement
of this ancestor: no continuous parameter may remain unforced. First cited in \S\ref{sec:AX4}.

\bibitem[Maldacena(1998)]{Maldacena1998}

\textbf{Maldacena, J.~M. (1998).}
The large $N$ limit of superconformal field theories and supergravity.
\textit{Advances in Theoretical and Mathematical Physics}, \textit{2}, 231--252.
\url{https://arxiv.org/abs/hep-th/9711200}
\par\noindent\textbf{Relevance.} Maldacena's AdS/CFT conjecture is the modern archetype for bulk--boundary duality and motivates the idea that geometry can encode quantum gauge dynamics. We cite it as a historical anchor for the \GGEP{}/\OGGEP{} posture: bulk geometry is not just a classical background, but a structure from which boundary quantum physics and observables can be exported. First cited in \S\ref{sec:four_clusters}.

\bibitem[Maupertuis(1744)]{Maupertuis1744}

\textbf{Maupertuis, P.-L.~M.~de (1744).}
Accord de diff\'erentes lois de la nature qui avaient jusqu'ici paru
incompatibles.
\emph{M\'emoires de l'Acad\'emie Royale des Sciences de Paris}, 417--426.
\par\noindent\textbf{Relevance.} The first stated principle of least
action in mechanics, $\delta \int p\,dq = 0$. In the constitutional
reading of $\POS{2}~\UEP{}$, Maupertuis 1744 is the foundational
articulation of action as the universal currency of dynamics, the
direct ancestor of every subsequent variational principle in physics. First cited in \S\ref{sec:POS2}.

\bibitem[Maxwell(1865)]{Maxwell1865}

\textbf{Maxwell, J.~C.\ (1865).}
A dynamical theory of the electromagnetic field.
\textit{Philosophical Transactions of the Royal Society of London}, \textit{155}, 459--512.
\url{https://doi.org/10.1098/rstl.1865.0008}
\par\noindent\textbf{Relevance.} The first complete unified-field
constitution: four equations folding electricity, magnetism, and
the optics of light into a single coherent picture. We cite
Maxwell as the historical precedent for the \AX{2}~\GGEP{} stance
that gauge structure and geometry should be treated as facets of a
single composite object rather than as independent ingredients
written together. First cited in \S\ref{sec:historical_chain}.

\bibitem[Minkowski(1908)]{Minkowski1908}

\textbf{Minkowski, H. (1908).}
Raum und Zeit.
In \textit{Jahresbericht der Deutschen Mathematiker-Vereinigung} (published 1909); English translation in \textit{The Principle of Relativity} (Dover).

\par\noindent\textbf{Relevance.} Minkowski's spacetime formalism makes causality geometric via the light cone.
AX-1 inherits this causal structure and exports it to operational open-system evolution: the admissible record must respect the Minkowski/relativistic notion of causal order. First cited in \S\ref{sec:geometrization-ladder-a1}.

\bibitem[Naimark(1940)]{Naimark1940}

\textbf{Naimark, M.~A. (1940).}
Spectral functions of a symmetric operator.
\emph{Izvestiya Akademii Nauk SSSR, Seriya Matematicheskaya},
\emph{4}, 277--318.
\par\noindent\textbf{Relevance.} Naimark's dilation theorem: every
POVM on a Hilbert space is the compression of a PVM on a larger
space. In the constitutional reading of $\POS{4}~\CP{}$, Naimark 1940
is the structural foundation of operationally-defined measurement and
the mathematical precondition for treating instruments and POVMs as
the basic objects of physics. First cited in \S\ref{sec:POS4}.

\bibitem[Nakahara(2003)]{Nakahara2003}

\textbf{Nakahara, M. (2003).}
\textit{Geometry, Topology and Physics} (2nd ed.).
Institute of Physics Publishing.

\par\noindent\textbf{Relevance.} Nakahara's text is a widely used bridge between modern geometry/topology and physical gauge theories.
It supports AX-2's emphasis that gauge and geometric data are naturally expressed as connections and curvatures, and it provides background for topological routes to quantization. First cited in \S\ref{sec:AX2}.

\bibitem[Newton(1687)]{Newton1687}

\textbf{Newton, I. (1687).}
\textit{Philosophi{\ae} Naturalis Principia Mathematica}.
Jussu Societatis Regi{\ae} ac Typis Josephi Streater.
\par\noindent\textbf{Relevance.} Newton's definitions and laws were a prototypical axiom system for mechanics, converting astronomy into dynamics and setting the model for ``principles $\to$ equations $\to$ predictions.'' We cite it as precedent for framing \GGTheory{} as a constitution that will be cited across later technical papers. First cited in \S\ref{sec:historical_chain}.

\bibitem[Neyman \& Pearson(1933)]{NeymanPearson1933}

\textbf{Neyman, J., \& Pearson, E.~S. (1933).}
On the problem of the most efficient tests of statistical hypotheses.
\emph{Philosophical Transactions of the Royal Society A}, \emph{231}, 289--337.
\url{https://doi.org/10.1098/rsta.1933.0009}
\par\noindent\textbf{Relevance.} Neyman and Pearson formalized hypothesis testing and error control, which underlies how modern sciences define and enforce rejection criteria. We cite it as historical/statistical background for POS-7 (MF), where every claim must declare an explicit kill criterion with controlled error rates or equivalent auditing logic. First cited in \S\ref{sec:POS7}.

\bibitem[Nicolis \& Prigogine(1977)]{NicolisPrigogine1977}

\textbf{Nicolis, G., \& Prigogine, I.\ (1977).}
\textit{Self-Organization in Nonequilibrium Systems}.
Wiley.
\par\noindent\textbf{Relevance.} A classic monograph on dissipative structures and nonequilibrium self-organization.
It provides historical and mathematical grounding for the idea that open systems generically develop attractors and phases under driving---a key theme of \GDOS{}. First cited in \S\ref{sec:POS2}.

\bibitem[Nielsen \& Chuang(2010)]{NielsenChuang2010}

\textbf{Nielsen, M.~A., \& Chuang, I.~L. (2010).}
\textit{Quantum Computation and Quantum Information} (10th Anniversary ed.).
Cambridge University Press.
\url{https://doi.org/10.1017/CBO9780511976667}
\par\noindent\textbf{Relevance.} This book codifies the operational viewpoint of quantum theory (states, channels, POVMs) that is closest to the \GGTheory{} demand that measurement is a defined map rather than a philosophical add-on. We cite it to support the operational language used in motivating \OGGEP{} and POS-7. First cited in \S\ref{sec:AX1}.

\bibitem[Noether(1918)]{Noether1918}

\textbf{Noether, E. (1918).}
Invariante Variationsprobleme.
\textit{Nachrichten von der Gesellschaft der Wissenschaften zu G{\"o}ttingen, Mathematisch-Physikalische Klasse}, 235--257.
\par\noindent\textbf{Relevance.} Noether's theorem is the clearest statement that symmetry is not optional: it has measurable conservation consequences.
This motivates POS-\GI\ as the first admissibility rule and supports the axiomatic emphasis on symmetry. First cited in \S\ref{sec:historical_chain}.

\bibitem[Occam(c.~14th c.)]{Occam}

\textbf{Occam, W.~of (c.~1320s).}
\emph{Summa logicae} and \emph{Quodlibetal Questions}. Modern
discussion in Boehner, \emph{Ockham: Philosophical Writings},
Nelson, 1957.
\par\noindent\textbf{Relevance.} The principle ``\emph{numquam ponenda
est pluralitas sine necessitate}'' --- multiplicity is never to be
posited without necessity. In the constitutional reading of
$\POS{6}~\OGN{}$, William of Occam is the deepest philosophical
ancestor of the Occam-gauge-normalization discipline, which POS-6
elevates from a methodological preference to an axiomatic filter. First cited in \S\ref{sec:POS6}.

\bibitem[Onsager(1931)]{Onsager1931}

\textbf{Onsager, L. (1931).}
Reciprocal relations in irreversible processes I.
\emph{Physical Review}, \emph{37}, 405--426.
\url{https://doi.org/10.1103/PhysRev.37.405}
\par\noindent\textbf{Relevance.} Onsager's reciprocity and variational viewpoint provide a foundational extremal structure for near-equilibrium nonequilibrium thermodynamics. We cite it as historical and mathematical background for POS-2 (\UEP{}), supporting the statement that open-system dissipation can be generated by a variational principle (minimum dissipation / gradient flow). First cited in \S\ref{sec:POS2}.

\bibitem[Onsager \& Machlup(1953)]{OnsagerMachlup1953}

\textbf{Onsager, L., \& Machlup, S.\ (1953).}
Fluctuations and irreversible processes.
\textit{Physical Review, 91}(6), 1505--1512.
DOI: \href{https://doi.org/10.1103/PhysRev.91.1505}{10.1103/PhysRev.91.1505}.
\par\noindent\textbf{Relevance.} Introduces an action-like functional for stochastic trajectories near equilibrium, providing a bridge
between variational principles and stochastic thermodynamics. It supports POS-2 (\UEP{}) in the open-system regime by
showing how ``least action'' persists as a most-probable-path principle for noisy dynamics. First cited in \S\ref{sec:POS2}.

\bibitem[Ozawa(1984)]{Ozawa1984}

\textbf{Ozawa, M. (1984).}
Quantum measuring processes of continuous observables.
\textit{Journal of Mathematical Physics}, \textit{25}(1), 79--87.
\url{https://doi.org/10.1063/1.526000}

\par\noindent\textbf{Relevance.} Ozawa developed the instrument-based formalism for measurements beyond ideal projective cases.
This supports AX-3's insistence that measurement be defined operationally (as an instrument), which is essential for exporting the framework to realistic open systems. First cited in \S\ref{sec:AX1}.

\bibitem[Padmanabhan(2010)]{Padmanabhan2010}

\textbf{Padmanabhan, T.\ (2010).}
Thermodynamical aspects of gravity: new insights.
\emph{Reports on Progress in Physics}, \emph{73}, 046901.
\url{https://doi.org/10.1088/0034-4885/73/4/046901}
\par\noindent\textbf{Relevance.} Padmanabhan's review of the
thermodynamic structure of gravity (Jacobson 1995 onward) provides
the systematic case that the Einstein equations can be read as an
equation of state. In the constitutional reading of
$\AX{2}~\GGEP{}$, this is the most technically developed version
of the ``gravity is emergent'' counter-stance; the cohort framework
takes the opposite view (geometry primitive, dynamics derived).
Named in Part~VI for the same reason as Verlinde 2011. First cited in \S\ref{sec:comparison_modern}.

\bibitem[Page(1993)]{Page1993}

\textbf{Page, D.~N.\ (1993).}
Information in black hole radiation.
\textit{Physical Review Letters}, \textit{71}(23), 3743--3746.
\url{https://doi.org/10.1103/PhysRevLett.71.3743}

\par\noindent\textbf{Relevance.} Page computed the entanglement-entropy
curve that an evaporating black hole must follow if the evaporation is
unitary --- the curve whose recovery from semiclassical gravity the
islands formula later achieved. The present paper places the Page
curve at Stratum~3 of the constitutional tower, beside the Born rule
and the Friston free-energy principle, as one of the three sibling
families of openness. First cited in \S\ref{sec:V-walking-the-tower}.


\bibitem[Particle Data Group(2024)]{PDG2024}

\textbf{Particle Data Group (Navas, S., et~al.). (2024).}
Review of Particle Physics.
\textit{Physical Review D}, \textit{110}(3), 030001.
\url{https://doi.org/10.1103/PhysRevD.110.030001}
(See also the PDG online database).

\par\noindent\textbf{Relevance.} The PDG is the canonical source for the Standard Model parameterization and experimental constraints.
It supports the motivation that our current fundamental theories contain a significant set of empirical inputs and motivates \MSMF/\OGN\ as
a push toward reduced dial dependence. First cited in \S\ref{sec:four_clusters}.

\bibitem[Penington et~al.(2022)]{Penington2020}

\textbf{Penington, G., Shenker, S.~H., Stanford, D., and Yang, Z.\ (2020).}
Replica wormholes and the black hole interior.
\textit{Journal of High Energy Physics}, \textit{2022}(3), 205.
\url{https://doi.org/10.1007/JHEP03(2022)205}
\par\noindent\textbf{Relevance.} The Penington--Shenker--Stanford--
Yang companion paper to Almheiri et~al.\ derives the islands
formula via replica wormholes, putting the bulk-to-boundary
information recovery on a firm calculational footing. Together
with Almheiri et~al., these two papers complete the
twenty-first-century holographic chain --- from 't~Hooft and
Susskind's holographic principle through Maldacena's AdS/CFT
correspondence and the Ryu--Takayanagi formula to the
Page-curve recovery --- that $\AX{3}~\OGGEP{}$ takes as the
operational confirmation of its constitutional reading of
measurement as bulk-to-boundary projection. First cited in \S\ref{sec:AX3}.

\bibitem[Peres(1979)]{Peres1979}

\textbf{Peres, A.\ (1979).}
Proposed test for complex versus quaternion quantum theory.
\textit{Physical Review Letters}, \textit{42}(11), 683--686.
\url{https://doi.org/10.1103/PhysRevLett.42.683}
\par\noindent\textbf{Relevance.} Peres proposed the first sharp
laboratory discrimination between complex and quaternionic quantum
mechanics, via the non-commutativity of successive scattering
phases. Cited as the class-defining reference for kill clause (3) of
F-Constitution-10 (cohort alias F-Real-1): a confirmed
quaternionic-direction deviation from the Born trace form would
falsify the minimality leg of L-Constitution-1 (candidate). First
cited in \S\ref{sec:AX4_category_axis}.

\bibitem[Perlmutter et~al.(1999)]{Perlmutter1999}

\textbf{Perlmutter, S., et~al. (1999).}
Measurements of $\Omega$ and $\Lambda$ from 42 high-redshift supernovae.
\textit{The Astrophysical Journal}, \textit{517}(2), 565--586.
\url{https://doi.org/10.1086/307221}
\url{https://arxiv.org/abs/astro-ph/9812133}
\par\noindent\textbf{Relevance.} Together with Riess et~al., this work established the observational evidence for accelerated expansion and nonzero $\Lambda$ in late-time cosmology. We cite it in the applications/discussion sections to anchor the empirical context for ``beyond-$\Lambda$'' claims in a standard observational milestone. First cited in \S\ref{sec:motivation}.

\bibitem[Peskin and Schroeder(1995)]{PeskinSchroeder1995}

\textbf{Peskin, M.~E., \& Schroeder, D.~V. (1995).}
\textit{An Introduction to Quantum Field Theory}.
Westview Press.

\par\noindent\textbf{Relevance.} A standard modern presentation of locality, causality, and renormalization in QFT.
It provides background for AX-1's microcausality/locality requirement and for later postulates concerning RG compatibility and scale covariance. First cited in \S\ref{sec:POS4}.

\bibitem[Planck(1900)]{Planck1900}

\textbf{Planck, M. (1900).}
Zur Theorie des Gesetzes der Energieverteilung im Normalspektrum.
\textit{Verhandlungen der Deutschen Physikalischen Gesellschaft}, \textit{2}, 237--245.
\par\noindent\textbf{Relevance.} Planck's work initiated the quantum revolution and exemplifies how a small modification at the axiomatic level
(quantization of energy exchange) can restructure an entire domain. First cited in \S\ref{sec:motivation}.

\bibitem[Planck Collaboration(2018)]{Planck2018}

\textbf{Planck Collaboration. (2018).}
Planck 2018 results. VI. Cosmological parameters.
\textit{Astronomy \& Astrophysics}, \textit{641}, A6.
\url{https://arxiv.org/abs/1807.06209}
\par\noindent\textbf{Relevance.} Precision cosmology strongly constrains the expansion history and composition of the Universe. We cite Planck as representative of the empirical pressure behind the inflation/dark-sector puzzles, and as motivation for a constitution that connects cosmological claims to operational observables with falsifiers. First cited in \S\ref{sec:intro}.

\bibitem[Polchinski(1984)]{Polchinski1984}

\textbf{Polchinski, J. (1984).}
Renormalization and effective Lagrangians.
\textit{Nuclear Physics B}, \textit{231}, 269--295.
\par\noindent\textbf{Relevance.} A classic effective-field-theory treatment emphasizing RG as a map between effective descriptions. POS-5 treats RG/coarse-graining as a structural operation that must commute with observation (\OGGEP{}), and Polchinski's work is a key reference for this map-based viewpoint. First cited in \S\ref{sec:POS5}.

\bibitem[Polchinski(1995)]{Polchinski1995}

\textbf{Polchinski, J. (1995).}
Dirichlet-branes and Ramond-Ramond charges.
\textit{Physical Review Letters}, \textit{75}(26), 4724--4727.
\par\noindent\textbf{Relevance.} This work helped consolidate modern string theory as a framework with extended objects and dualities.
It is a key reference for contemporary quantum gravity and unification programs that \GGTheory{}\ compares against in its discussion section. First cited in \S\ref{sec:four_clusters}.

\bibitem[Polchinski(2004)]{Polchinski2004Landscape}

\textbf{Polchinski, J. (2004).}
The cosmological constant and the string landscape.
Talk at the 23rd Solvay Conference on Physics (Brussels), Dec 2004.
Available as arXiv:hep-th/0603249 (Erice 2006 follow-up).
\url{https://arxiv.org/abs/hep-th/0603249}

\par\noindent\textbf{Relevance.} The landscape idea is an example of explanatory flexibility that can undermine predictivity. It motivates \MSMF\ and \OGN\
as axiomatic constraints aimed at reducing or eliminating continuous families of equally admissible solutions. First cited in \S\ref{sec:four_clusters}.

\bibitem[Politzer(1973)]{Politzer1973}

\textbf{Politzer, H.~D. (1973).}
Reliable perturbative results for strong interactions?
\textit{Physical Review Letters}, \textit{30}(26), 1346--1349.
\par\noindent\textbf{Relevance.} Along with Gross--Wilczek, this established asymptotic freedom. It reinforces the centrality of RG structure and motivates
scale covariance as an axiomatic requirement. First cited in \S\ref{sec:AX4}.

\bibitem[Popper(1959)]{Popper1959}

\textbf{Popper, K. (1959).}
\textit{The Logic of Scientific Discovery}.
Hutchinson. (Original work published 1934).
\url{https://doi.org/10.4324/9780203994627}
\par\noindent\textbf{Relevance.} Popper's falsifiability criterion is a classic articulation of what separates scientific statements from metaphysical ones. We cite it in POS-7 to support the demand that every theorem come with a predeclared operational kill criterion, implemented concretely via instruments and statistical tests. First cited in \S\ref{sec:POS7}.

\bibitem[Prigogine(1947)]{Prigogine1947}

\textbf{Prigogine, I. (1947).}
\emph{\'Etude thermodynamique des ph\'enom\`enes irr\'eversibles}.
Li\`ege: Editions Desoer.
\par\noindent\textbf{Relevance.} Prigogine's principle of minimum
entropy production in linear non-equilibrium steady states: a
boundary-side variational principle for irreversible processes.
In the constitutional reading of $\POS{2}~\UEP{}$, Prigogine 1947
is the canonical example of how the variational engine of physics
extends from closed Lagrangian sectors to open dissipative ones. First cited in \S\ref{sec:POS2}.

\bibitem[Renou et~al.(2021)]{Renou2021}

\textbf{Renou, M.-O., Trillo, D., Weilenmann, M., Le, T.~P.,
Tavakoli, A., Gisin, N., Ac\'in, A., \& Navascu\'es, M.\ (2021).}
Quantum theory based on real numbers can be experimentally falsified.
\textit{Nature}, \textit{600}, 625--629.
\url{https://doi.org/10.1038/s41586-021-04160-4}
\par\noindent\textbf{Relevance.} The network-Bell theorem and
experiment class excluding quantum theory over \emph{unshared} real
tensor products. In the constitutional reading the exclusion
constrains exactly the unshared case and is satisfied by
construction by the shared derived $J$ (T-P4-4 of \PaperPFour{});
kill clause (2) of F-Constitution-10 (cohort alias F-Real-1) is this
experiment class run in reverse. First cited in
\S\ref{sec:AX4_category_axis}.

\bibitem[Reuter(1998)]{Reuter1998}

\textbf{Reuter, M.\ (1998).}
Nonperturbative evolution equation for quantum gravity.
\emph{Physical Review D}, \emph{57}, 971--985.
\url{https://doi.org/10.1103/PhysRevD.57.971}
\par\noindent\textbf{Relevance.} Reuter's exact-renormalization-group
equation for the gravitational effective action launched the
\emph{asymptotic-safety} program for quantum gravity: gravity is
UV-complete via a non-Gaussian fixed point of the renormalization
flow. In the constitutional reading of $\POS{5}~\RG{}$, asymptotic
safety is a Stratum-2 alternative to the cohort's anomaly-locked 6D
bulk: it accepts gauge-covariant gravity but does not yet activate
$\AX{3}~\OGGEP{}$ or $\AX{4}~\MSMF{}$. Named in Part~VI as a
competing program. First cited in \S\ref{sec:comparison_modern}.

\bibitem[Riemann(1854)]{Riemann1854}

\textbf{Riemann, B.\ (1854).}
\"Uber die Hypothesen, welche der Geometrie zu Grunde liegen.
\textit{Habilitationsvortrag, G\"ottingen}; published posthumously in
\textit{Abhandlungen der K\"oniglichen Gesellschaft der Wissenschaften zu G\"ottingen}, \textit{13} (1868).
\par\noindent\textbf{Relevance.} The founding document of the modern
mathematics of curved manifolds and the conceptual leap that turned
geometry into a degree of freedom of physical space. We cite
Riemann's lecture as the historical precedent for the 6D bulk of \GG{} \citep{ArisakaA2GG6}: in the \GGTheory{} program,
the geometry of the bulk is not chosen for convenience but forced
by the constitution. First cited in \S\ref{sec:historical_chain}.

\bibitem[Riess et~al.(1998)]{Riess1998}

\textbf{Riess, A.~G., et~al. (1998).}
Observational evidence from supernovae for an accelerating universe and a cosmological constant.
\textit{The Astronomical Journal}, \textit{116}(3), 1009--1038.
\par\noindent\textbf{Relevance.} One of the two discovery papers for cosmic acceleration. It motivates the claim that cosmology contains deep missing pieces
(dark energy) that invite a new foundational viewpoint. First cited in \S\ref{sec:motivation}.

\bibitem[Risken(1989)]{Risken1989}

\textbf{Risken, H. (1989).}
\textit{The Fokker--Planck Equation: Methods of Solution and Applications} (2nd ed.).
Springer.

\par\noindent\textbf{Relevance.} A classic reference for diffusion processes and Fokker--Planck equations.
It provides technical background for the \GDOS{} continuity-equation form and its solution methods. First cited in \S\ref{sec:AX4}.

\bibitem[Rissanen(1978)]{Rissanen1978}

\textbf{Rissanen, J. (1978).}
Modeling by shortest data description.
\emph{Automatica}, \emph{14}(5), 465--471.
\url{https://doi.org/10.1016/0005-1098(78)90005-5}
\par\noindent\textbf{Relevance.} Rissanen's minimum description length principle is another quantitative implementation of Occam's razor. We cite it as methodological background for POS-6 (\OGN{}): the ``minimum freedom'' doctrine must also eliminate hidden continuous choices in parameterization and normalization. First cited in \S\ref{sec:AX4}.

\bibitem[Rovelli(2004)]{Rovelli2004}

\textbf{Rovelli, C. (2004).}
\textit{Quantum Gravity}.
Cambridge University Press.
\url{https://doi.org/10.1017/CBO9780511755804}
\par\noindent\textbf{Relevance.} Loop quantum gravity is a canonical example of the ``quantize geometry'' program. We cite it to emphasize that \GGTheory{} proposes an inversion: a ``geometrization of quantum physics'' in which gauge/quantum structures are encoded geometrically and read operationally at the boundary. First cited in \S\ref{sec:four_clusters}.

\bibitem[Ryu and Takayanagi(2006)]{RyuTakayanagi2006}

\textbf{Ryu, S.\ and Takayanagi, T.\ (2006).}
Holographic derivation of entanglement entropy from AdS/CFT.
\textit{Physical Review Letters}, \textit{96}, 181602.
\url{https://doi.org/10.1103/PhysRevLett.96.181602}
\par\noindent\textbf{Relevance.} The Ryu--Takayanagi formula
expresses the boundary CFT's entanglement entropy as the area of
a minimal surface in the bulk AdS geometry. In the AX-3 lineage,
Ryu--Takayanagi is the first construction in which a fully
operational boundary observable (the entanglement entropy of a
boundary region) is computed directly from bulk geometric data,
validating the $\OGGEP{}$ thesis that the boundary's operational
content is determined by the bulk's geometric content via the
projection $\Dmap$. First cited in \S\ref{sec:AX3}.

\bibitem[Salam(1968)]{Salam1968}

\textbf{Salam, A. (1968).}
Weak and electromagnetic interactions.
In \textit{Elementary Particle Theory} (Proceedings of the Nobel Symposium No.~8), 367--377.

\par\noindent\textbf{Relevance.} Salam's electroweak unification is part of the Standard Model foundation. It is cited here to emphasize that modern physics
already depends on strong ``admissibility rules'' (gauge invariance and renormalizability) even if not presented as an explicit constitution. First cited in \S\ref{sec:intro}.

\bibitem[Schlosshauer(2005)]{Schlosshauer2005}

\textbf{Schlosshauer, M.\ (2005).}
Decoherence, the measurement problem, and interpretations of quantum mechanics.
\textit{Reviews of Modern Physics, 76}(4), 1267--1305.
\par\noindent\textbf{Relevance.} A modern review clarifying what decoherence explains and what it does not.
We cite it to motivate why an operational definition of observation (\OGGEP{}) is valuable in an axiom system. First cited in \S\ref{sec:AX3}.

\bibitem[Schr{\"o}dinger(1926)]{Schrodinger1926}

\textbf{Schr{\"o}dinger, E. (1926).}
Quantisierung als Eigenwertproblem.
\textit{Annalen der Physik}, \textit{79}, 361--376.
\par\noindent\textbf{Relevance.} Wave mechanics provides the second founding formulation of quantum theory. It anchors the historical background for the
measurement issue: the wavefunction becomes a central object whose operational status requires axiomatic clarification. First cited in \S\ref{sec:historical_chain}.

\bibitem[Schwinger(1948)]{Schwinger1948QED}

\textbf{Schwinger, J.\ (1948).}
Quantum electrodynamics.\ I.\ A covariant formulation.
\textit{Physical Review}, \textit{74}(10), 1439--1461.
\url{https://doi.org/10.1103/PhysRev.74.1439}
\par\noindent\textbf{Relevance.} Schwinger's 1948 covariant formulation of QED was, with Tomonaga's 1946 paper, the first systematic treatment of mass and charge renormalization in a relativistic field theory. The procedure absorbs divergent loop integrals into redefinitions of bare parameters, leaving finite physical predictions for the anomalous magnetic moment of the electron and the Lamb shift. We cite it in POS-5 (RG) as one of the three foundational QED-renormalization papers (Tomonaga 1946, Schwinger 1948, Feynman 1949). First cited in \S\ref{sec:POS5}.

\bibitem[Seifert(2012)]{Seifert2012}

\textbf{Seifert, U. (2012).}
Stochastic thermodynamics, fluctuation theorems and molecular machines.
\textit{Reports on Progress in Physics}, \textit{75}(12), 126001.
\url{https://doi.org/10.1088/0034-4885/75/12/126001}
\par\noindent\textbf{Relevance.} Stochastic thermodynamics extends thermodynamic concepts to small fluctuating open systems and provides precise expressions for entropy production and nonequilibrium work. We cite it for the appendix toy models that connect \GDOS{}-style open dynamics to measurable entropy production and for motivating POS-2's open-system extremal forms. First cited in \S\ref{sec:AX1}.

\bibitem[Shannon(1948)]{Shannon1948}

\textbf{Shannon, C.~E. (1948).}
A mathematical theory of communication.
\textit{Bell System Technical Journal}, \textit{27}, 379--423, 623--656.
\par\noindent\textbf{Relevance.} Shannon's information theory provides a quantitative language for ``entropy'' and information flow. Because \GGTheory{}\ aims to
export a single operational grammar across sciences, information-theoretic quantities become natural candidates for inclusion in the open-system
ledger (\GDOS) implied by \OGGEP. First cited in \S\ref{sec:POS6}.

\bibitem[Sinha et~al.(2010)]{Sinha2010}

\textbf{Sinha, U., Couteau, C., Jennewein, T., Laflamme, R., \&
Weihs, G.\ (2010).}
Ruling out multi-order interference in quantum mechanics.
\textit{Science}, \textit{329}(5990), 418--421.
\url{https://doi.org/10.1126/science.1190545}
\par\noindent\textbf{Relevance.} The three-slit bound on the Sorkin
parameter: genuine three-path interference is consistent with zero,
as the Born trace form requires. Cited as the operational anchor of
kill clause (3) of F-Constitution-10 (cohort alias F-Real-1) --- the
quaternionic-excess direction of Born deviations is the live kill
against L-Constitution-1 (candidate). First cited in
\S\ref{sec:AX4_category_axis}.

\bibitem[Slavnov(1972)]{Slavnov1972}

\textbf{Slavnov, A.~A. (1972).}
Ward identities in gauge theories.
\textit{Theoretical and Mathematical Physics}, \textit{10}(2), 99--104.
\url{https://doi.org/10.1007/BF01090719}
\par\noindent\textbf{Relevance.} Slavnov's 1972 paper extends the
Ward--Takahashi identities of QED to non-Abelian gauge theory,
producing the Slavnov--Taylor identities that BRST cohomology later
cast as off-shell algebraic relations among the renormalized vertex
functions. These identities are the operational diagnostic that
the gauge bundle's geometric content registers correctly at the
boundary effective-action level; they are the bridge between
$\AX{2}~\GGEP{}$'s bulk-side principle and $\AX{3}~\OGGEP{}$'s
boundary-side operational dictionary, and they make non-Abelian
renormalizability possible. First cited in \S\ref{sec:AX2}.

\bibitem[Sober(2015)]{Sober2015}

\textbf{Sober, E. (2015).}
\textit{Ockham's Razors: A User's Manual}.
Cambridge: Cambridge University Press.
\par\noindent\textbf{Relevance.} A modern analysis of simplicity principles (Occam's razor) and how they function in scientific inference. POS-6 (\OGN{}) and AX-4 (\MSMF{}) elevate simplicity from heuristic to axiomatic restriction by forbidding hidden continuous degrees of freedom and convention drift. First cited in \S\ref{sec:POS6}.

\bibitem[Stinespring(1955)]{Stinespring1955}

\textbf{Stinespring, W.~F. (1955).}
Positive functions on $C^\ast$-algebras.
\textit{Proceedings of the American Mathematical Society}, \textit{6}(2), 211--216.
\url{https://doi.org/10.1090/S0002-9939-1955-0069403-2}
\par\noindent\textbf{Relevance.} Stinespring's dilation theorem is the mathematical backbone of quantum channels: CP maps arise from unitary evolution on an enlarged space. We cite it to justify how \OGGEP{} can treat open-system boundary dynamics as fundamental while remaining consistent with microscopic unitarity in the bulk. First cited in \S\ref{sec:historical_chain}.

\bibitem[Stora(1976)]{Stora1976}

\textbf{Stora, R. (1976).}
Continuum gauge theories.
In \emph{New Developments in Quantum Field Theory and Statistical
Mechanics}, Carg\`ese Lectures (M.~L\'evy and P.~Mitter, eds.),
NATO Advanced Study Institutes Series, Plenum Press, 201--224.
\par\noindent\textbf{Relevance.} Stora's introduction of the descent
equations relating the gauge anomaly polynomial to Chern--Simons
forms via BRST cohomology. In the constitutional reading of
$\POS{3}~\AF{}$, Stora 1976 is the algebraic-topological scaffolding
that turns anomaly classification into a question about a chain
complex. First cited in \S\ref{sec:POS3}.

\bibitem[Streater \& Wightman(1964)]{StreaterWightman1964}

\textbf{Streater, R.~F., \& Wightman, A.~S. (1964).}
\textit{PCT, Spin and Statistics, and All That}.
W.~A. Benjamin.
\par\noindent\textbf{Relevance.} This book codifies rigorous axioms for relativistic QFT, including microcausality and locality. We cite it as a foundational reference for AX-1's operator-algebraic encoding of locality and causality. First cited in \S\ref{sec:why_axioms_postulates}.

\bibitem[Stueckelberg \& Petermann(1953)]{StueckelbergPetermann1953}

\textbf{Stueckelberg, E.~C.~G., \& Petermann, A. (1953).}
La normalisation des constantes dans la th\'eorie des quanta.
\emph{Helvetica Physica Acta}, \emph{26}, 499--520.
\par\noindent\textbf{Relevance.} The first recognition that the freedom
in renormalization subtractions generates a group action on the
space of couplings. In the constitutional reading of $\POS{5}~\RG{}$,
Stueckelberg--Petermann 1953 is the founding statement of the
renormalization group as a structural feature of quantum field theory,
predating Gell-Mann and Low. First cited in \S\ref{sec:POS5}.

\bibitem[Susskind(1979)]{Susskind1979}

\textbf{Susskind, L. (1979).}
Dynamics of spontaneous symmetry breaking in the Weinberg--Salam theory.
\textit{Physical Review D}, \textit{20}(10), 2619--2625.
\par\noindent\textbf{Relevance.} Early naturalness discussions emphasize that symmetry and scale hierarchies are not independent. This supports the motivation
for a constitutional axiom (\MSMF) aimed at suppressing unexplained parameter sensitivity. First cited in \S\ref{sec:four_clusters}.

\bibitem[Susskind(1995)]{Susskind1995}

\textbf{Susskind, L. (1995).}
The world as a hologram.
\textit{Journal of Mathematical Physics}, \textit{36}(11), 6377--6396.
\url{https://arxiv.org/abs/hep-th/9409089}
\par\noindent\textbf{Relevance.} Susskind's holographic principle paper is a key historical milestone leading to modern bulk--boundary dualities. We cite it to frame \OGGEP{} as a constitution-level insistence that observation is a bulk$\to$boundary export, generalized beyond correlators to full operational semantics. First cited in \S\ref{sec:four_clusters}.

\bibitem[Susskind(2003)]{Susskind2003}

\textbf{Susskind, L. (2003).}
The anthropic landscape of string theory.
\textit{Universe or Multiverse?} (conference contribution).
\url{https://arxiv.org/abs/hep-th/0302219}
\par\noindent\textbf{Relevance.} The landscape framing represents a modern endpoint of unconstrained model freedom: many vacua, many low-energy effective theories, and selection by anthropic arguments. We cite it as a motivation for \MSMF{}, which aims to restore determinism by bounding freedom and demanding symmetry-driven minimality. First cited in \S\ref{sec:AX4}.

\bibitem[Symanzik(1970)]{Symanzik1970}

\textbf{Symanzik, K.\ (1970).}
Small distance behaviour in field theory and power counting.
\textit{Communications in Mathematical Physics, 18}(3), 227--246.
DOI: \href{https://doi.org/10.1007/BF01649434}{10.1007/BF01649434}.
\par\noindent\textbf{Relevance.} A foundational development of renormalization and scaling in quantum field theory.
Together with Callan and Wilson, it underpins the modern RG view used by POS-5 and motivates the identification of a
log-scale coordinate with flow in the space of effective theories. First cited in \S\ref{sec:POS5}.

\bibitem[Takahashi(1957)]{Takahashi1957}

\textbf{Takahashi, Y. (1957).}
On the generalized Ward identity.
\textit{Nuovo Cimento}, \textit{6}, 371--375.
\url{https://doi.org/10.1007/BF02832514}
\par\noindent\textbf{Relevance.} Takahashi's 1957 paper extends
Ward's 1950 identity to its general form, producing the
Ward--Takahashi identities that underlie every textbook proof of
QED renormalizability. In the AX-2 historical lineage, the
Ward--Takahashi identities are the quantum-loop expression of
Noether's second theorem in the Abelian gauge sector --- the
operational confirmation that the gauge bundle's geometric content
is preserved by the quantum effective action. First cited in \S\ref{sec:AX2}.

\bibitem[Taylor(1971)]{Taylor1971}

\textbf{Taylor, J.~C. (1971).}
Ward identities and charge renormalization of the Yang--Mills field.
\textit{Nuclear Physics B}, \textit{33}, 436--444.
\url{https://doi.org/10.1016/0550-3213(71)90297-5}
\par\noindent\textbf{Relevance.} Taylor's 1971 paper, together
with Slavnov's 1972 paper, produces the Slavnov--Taylor identities
that establish charge renormalization in Yang--Mills theory. In
the AX-2 historical lineage, Taylor's contribution is the
non-Abelian generalization of the Ward identity --- one of the two
papers that together demonstrated that 't~Hooft's renormalizability
proof for non-Abelian gauge theory is consistent. First cited in \S\ref{sec:AX2}.

\bibitem['t~Hooft(1971)]{tHooft1971}

\textbf{'t~Hooft, G. (1971).}
Renormalizable Lagrangians for massive Yang--Mills fields.
\textit{Nuclear Physics B}, \textit{35}, 167--188.
\par\noindent\textbf{Relevance.} Renormalizability of gauge theories is a historical turning point that made the Standard Model viable. It motivates the view
that ``admissibility filters'' (symmetry + consistency) are as important as the detailed dynamics. First cited in \S\ref{sec:motivation}.

\bibitem['t~Hooft(1993)]{tHooft1993}

\textbf{'t~Hooft, G. (1993).}
Dimensional reduction in quantum gravity.
In \textit{Salamfestschrift} (World Scientific).
\par\noindent\textbf{Relevance.} One of the early modern statements of holography/dimensional reduction. It provides historical background for the idea that a
bulk theory may be fully encoded on a lower-dimensional boundary, a central inspiration for \OGGEP. First cited in \S\ref{sec:four_clusters}.

\bibitem[THooft(1980)]{THooftNaturalness1980}

\textbf{'t~Hooft, G. (1980).}
Naturalness, chiral symmetry, and spontaneous chiral symmetry breaking.
In G. 't~Hooft et al. (Eds.), \textit{Recent Developments in Gauge Theories} (pp.~135--157).
Plenum Press.
\url{https://doi.org/10.1007/978-1-4684-7571-5_9}
\par\noindent\textbf{Relevance.} Naturalness arguments connect small parameters to enhanced symmetry. We cite this as historical background for \MSMF{}: the \GGConstitution{} pushes the symmetry$\leftrightarrow$parameter connection to a stronger law, demanding maximal symmetry and forbidding non-protected dials. First cited in \S\ref{sec:four_clusters}.

\bibitem[THooft \& Veltman(1972)]{THooftVeltman1972}

\textbf{'t~Hooft, G., \& Veltman, M. (1972).}
Regularization and renormalization of gauge fields.
\textit{Nuclear Physics B}, \textit{44}(1), 189--213.
\url{https://doi.org/10.1016/0550-3213(72)90279-9}
\par\noindent\textbf{Relevance.} The renormalizability of non-Abelian gauge theories is a cornerstone of the SM's consistency. We cite it to illustrate a key lesson: deep consistency conditions (not just phenomenological fits) decide what counts as a viable theory. This motivates postulates that function as consistency filters (e.g.\ anomaly freedom, RG compatibility). First cited in \S\ref{sec:POS1}.

\bibitem[Tomonaga(1946)]{Tomonaga1946}

\textbf{Tomonaga, S.\ (1946).}
On a relativistically invariant formulation of the quantum theory of wave fields.
\textit{Progress of Theoretical Physics}, \textit{1}(2), 27--42.
\url{https://doi.org/10.1143/PTP.1.27}
\par\noindent\textbf{Relevance.} Tomonaga's 1946 paper, written at Tokyo Bunrika University during the war years, provided the first relativistically invariant treatment of QED with the now-standard interaction picture. Together with Schwinger's 1948 covariant formulation and Feynman's 1949 diagrammatic approach (unified by Dyson 1949), it constitutes the founding of renormalized quantum field theory. The three authors shared the 1965 Nobel Prize in Physics for this work. We cite it in POS-5 (RG) as the original technical achievement from which the renormalization-group concept later emerged. First cited in \S\ref{sec:POS5}.

\bibitem[Touchette(2009)]{Touchette2009}

\textbf{Touchette, H. (2009).}
The large deviation approach to statistical mechanics.
\textit{Physics Reports} \textbf{478}, 1--69.
\href{https://doi.org/10.1016/j.physrep.2009.05.002}{doi:10.1016/j.physrep.2009.05.002}.
\par\noindent\textbf{Relevance.}
A modern review of large-deviation theory, clarifying how action-like functionals control typical and rare trajectories.
It supports POS-\UEP by providing a general mathematical framework for extremal principles in open systems. First cited in \S\ref{sec:POS2}.

\bibitem[Tsirelson(1980)]{Tsirelson1980}

\textbf{Tsirelson, B.~S.\ (1980).}
Quantum generalizations of Bell's inequality.
\textit{Letters in Mathematical Physics}, \textit{4}, 93--100.
\url{https://doi.org/10.1007/BF00417500}
\par\noindent\textbf{Relevance.} Tsirelson proved that the
Bell-CHSH inequality is bounded \emph{above} in quantum mechanics
by $|S| \leq 2\sqrt{2}$, the now-canonical Tsirelson bound. This
bound is the quantitative content of operational causality in the
quantum setting: quantum mechanics maximally violates the
classical-local-causality bound by exactly the Tsirelson factor,
no more. In the AX-1 lineage, the Tsirelson bound is the
quantitative statement that $\AX{1}~\LC{}$ admits Bell-violating
correlations only up to a specific quantum-saturated limit set by
the bulk-to-boundary projection. First cited in \S\ref{sec:AX1}.

\bibitem[Unruh(1976)]{Unruh1976}

\textbf{Unruh, W.~G.\ (1976).}
Notes on black-hole evaporation.
\textit{Physical Review D}, \textit{14}(4), 870--892.
\url{https://doi.org/10.1103/PhysRevD.14.870}
\par\noindent\textbf{Relevance.} Unruh's 1976 follow-up paper
develops the systematic theory of the accelerated-observer
vacuum and the now-canonical Unruh effect. Together with the
Davies--Fulling--Unruh paper, it establishes the operational
dependence of the boundary record on the choice of boundary frame ---
the AX-3 lineage's first quantum-field-theoretic confirmation that
the bulk-to-boundary projection is genuinely observer-dependent. First cited in \S\ref{sec:AX3}.

\bibitem[Vafa(2005)]{Vafa2005}

\textbf{Vafa, C.\ (2005).}
The string landscape and the swampland.
\textit{arXiv preprint} \texttt{hep-th/0509212}.
\url{https://arxiv.org/abs/hep-th/0509212}
\par\noindent\textbf{Relevance.} Vafa's 2005 swampland proposal
argued that not every consistent-looking low-energy effective
theory can be UV-completed to a string theory; the consistent ones
form the ``landscape,'' the inconsistent ones the ``swampland.''
In the AX-4 lineage, the swampland program is the recognition
that consistency with a deeper UV structure is itself a
constitutional admissibility filter; $\AX{4}~\MSMF{}$ extends this
logic from the string-theoretic UV to the full four-axiom
constitution, with the cohort's integer $\OGNfive{}$ ledger as the
operational substitute for the swampland constraint. First cited in \S\ref{sec:AX4}.

\bibitem[Verlinde(2011)]{Verlinde2011}

\textbf{Verlinde, E.\ (2011).}
On the origin of gravity and the laws of Newton.
\emph{Journal of High Energy Physics}, \emph{2011}, 29.
\url{https://doi.org/10.1007/JHEP04(2011)029}
\par\noindent\textbf{Relevance.} Verlinde's entropic-gravity proposal
inverts the standard relation between gravity and entropy: gravity
is treated as an emergent thermodynamic phenomenon rather than as a
fundamental geometric one. In the constitutional reading of
$\AX{2}~\GGEP{}$, Verlinde's program stands as the explicit
opposite stance to the cohort's: geometry is taken as derived
rather than as primitive. Named in Part~VI for completeness. First cited in \S\ref{sec:comparison_modern}.

\bibitem[von Neumann(1932)]{vonNeumann1932}

\textbf{von Neumann, J. (1932).}
\textit{Mathematische Grundlagen der Quantenmechanik}.
Springer.

\par\noindent\textbf{Relevance.} von Neumann formalized quantum theory axiomatically but introduced measurement as an additional process distinct from unitary
evolution. This motivates \GGTheory{}'s insistence that measurement be defined as a constitutional axiom/postulate rather than left as a dualistic rule. First cited in \S\ref{sec:historical_chain}.

\bibitem[Ward(1950)]{Ward1950}

\textbf{Ward, J.~C. (1950).}
An identity in quantum electrodynamics.
\textit{Physical Review}, \textit{78}, 182.
\url{https://doi.org/10.1103/PhysRev.78.182}
\par\noindent\textbf{Relevance.} Ward's identity is the prototype relation showing how gauge symmetry constrains quantum amplitudes. We cite it in POS-1 and POS-4 to emphasize that gauge invariance is not only a classical redundancy but an admissibility condition that has nontrivial quantum consequences. First cited in \S\ref{sec:AX2}.

\bibitem[Weinberg(1967)]{Weinberg1967}

\textbf{Weinberg, S. (1967).}
A model of leptons.
\textit{Physical Review Letters}, \textit{19}(21), 1264--1266.
\url{https://doi.org/10.1103/PhysRevLett.19.1264}
\par\noindent\textbf{Relevance.} Weinberg's paper, together with Glashow and Salam, established the electroweak gauge theory. We cite it as part of the historical backbone of gauge invariance and spontaneous symmetry breaking, which \GGTheory{} aims to generalize to open systems via \GDOS{}. First cited in \S\ref{sec:intro}.

\bibitem[Weinberg(1989)]{Weinberg1989CC}

\textbf{Weinberg, S. (1989).}
The cosmological constant problem.
\textit{Reviews of Modern Physics}, \textit{61}(1), 1--23.
\href{https://doi.org/10.1103/RevModPhys.61.1}{doi:10.1103/RevModPhys.61.1}.
\par\noindent\textbf{Relevance.}
The cosmological-constant problem is a sharp expression of the ``too many dials'' tension between quantum field theory and gravity.
It motivates the \GGTheory{}\ emphasis on \MSMF{}/\OGN{} (no hidden knobs) and on seeking geometric constraints that reduce phenomenological freedom in cosmology. First cited in \S\ref{sec:AX4}.

\bibitem[Weinberg(1995)]{Weinberg1995QFT}

\textbf{Weinberg, S. (1995).}
\textit{The Quantum Theory of Fields} (Vol.~1: Foundations).
Cambridge University Press.

\par\noindent\textbf{Relevance.} Weinberg's axiomatic and symmetry-driven approach to QFT is a key reference for how locality, causality, and gauge invariance constrain admissible theories.
AX-1 is aligned with this tradition but extends it by insisting that the same causal discipline apply to open-system operational evolution and measurement outputs. First cited in \S\ref{sec:POS3}.

\bibitem[Wess \& Zumino(1971)]{WessZumino1971}

\textbf{Wess, J., \& Zumino, B. (1971).}
Consequences of anomalous Ward identities.
\textit{Physics Letters B}, \textit{37}(1), 95--97.
\url{https://doi.org/10.1016/0370-2693(71)90582-X}
\par\noindent\textbf{Relevance.} Wess and Zumino derived integrability/consistency conditions on gauge anomalies, demonstrating that anomalies have constrained algebraic structure. We cite this work to justify treating anomaly freedom and inflow cancellation as an axiomatic filter (POS-3) rather than a model-dependent afterthought. First cited in \S\ref{sec:POS1}.

\bibitem[Witten(1983)]{WessZuminoWitten1983}

\textbf{Witten, E. (1983).}
Global aspects of current algebra.
\emph{Nuclear Physics B}, \emph{223}, 422--432.
\par\noindent\textbf{Relevance.} Witten's full development of the
Wess--Zumino term as a topological action in chiral effective field
theory, including the quantization of its coefficient. In the
constitutional reading of $\POS{3}~\AF{}$, the Wess--Zumino--Witten
action is the canonical worked example of how anomaly inflow
generates integer-quantized topological couplings on the boundary. First cited in \S\ref{sec:POS3}.

\bibitem[Wetterich(1993)]{Wetterich1993}

\textbf{Wetterich, C. (1993).}
Exact evolution equation for the effective average action.
\emph{Physics Letters B}, \emph{301}, 90--94.
\url{https://doi.org/10.1016/0370-2693(93)90726-X}
\par\noindent\textbf{Relevance.} Wetterich's exact functional flow
equation for the effective average action. In the constitutional
reading of $\POS{5}~\RG{}$, Wetterich 1993 is the contemporary
computational engine for non-perturbative RG studies, the modern
incarnation of Wilson's renormalization group as a functional flow on
theory space. First cited in \S\ref{sec:POS5}.

\bibitem[Weyl(1918)]{Weyl1918}

\textbf{Weyl, H. (1918).}
Gravitation und Elektrizit{\"a}t.
\textit{Sitzungsberichte der K{\"o}niglich Preussischen Akademie der Wissenschaften zu Berlin}, 465--480.
\par\noindent\textbf{Relevance.} Weyl's early gauge ideas represent one of the first attempts to relate geometry and electromagnetism.
While historically distinct from modern Yang--Mills theory, it anchors the long-standing theme that interactions may be geometric in origin, a theme AX-2 makes constitutional. First cited in \S\ref{sec:AX2}.

\bibitem[Wightman(1956)]{Wightman1956}

\textbf{Wightman, A.\ S.\ (1956).}
Quantum field theory in terms of vacuum expectation values.
\textit{Physical Review, 101}(2), 860--866.
\par\noindent\textbf{Relevance.} An early axiomatic formulation of relativistic quantum field theory in which locality and causality
appear as precise algebraic/analytic conditions. It supports the microcausality formulation of AX-1. First cited in \S\ref{sec:intro}.

\bibitem[Wigner(1939)]{Wigner1939}

\textbf{Wigner, E.~P. (1939).}
On unitary representations of the inhomogeneous Lorentz group.
\textit{Annals of Mathematics}, \textit{40}(1), 149--204.
\url{https://doi.org/10.2307/1968551}

\par\noindent\textbf{Relevance.} Wigner's classification of particles by symmetry representations established symmetry as the organizing principle of fundamental physics.
It motivates AX-4 (\MSMF): the search for maximal symmetry is not aesthetic alone, but historically productive in determining the space of admissible physical structures. First cited in \S\ref{sec:AX3}.

\bibitem[Wigner(1961)]{WignerFriend1961}

\textbf{Wigner, E.~P.\ (1961).}
Remarks on the mind--body question.
In \textit{The Scientist Speculates},
ed.\ I.~J.\ Good (Heinemann, London), pp.\ 284--302.
\par\noindent\textbf{Relevance.} Wigner's 1961 ``Wigner's friend''
thought experiment generalizes the von Neumann cut from a single
observer to a chain of observers, forcing the question of where
exactly the bulk-to-boundary projection takes place. In the
AX-3 lineage, Wigner's friend is the foundational scenario that
$\OGGEP{}$ resolves by definitional fiat: the projection is the
always-on bulk-to-boundary functor $\Dmap$, not a process tied to
any particular observer. First cited in \S\ref{sec:AX3}.

\bibitem[Wilson(1971)]{Wilson1971}

\textbf{Wilson, K.~G. (1971).}
Renormalization group and critical phenomena. I. Renormalization group and the Kadanoff scaling picture.
\textit{Physical Review B}, \textit{4}(9), 3174--3183.
\par\noindent\textbf{Relevance.} Wilson's RG is the prototype of scale-covariant thinking and motivates POS-\RG. Because \GGTheory{}\ aspires to unify across scales and
domains, it demands that coarse-graining and operational measurement commute in a controlled way. First cited in \S\ref{sec:historical_chain}.

\bibitem[Wilson(1974)]{Wilson1974}

\textbf{Wilson, K.~G. (1974).}
Confinement of quarks.
\textit{Physical Review D}, \textit{10}(8), 2445--2459.
\url{https://doi.org/10.1103/PhysRevD.10.2445}
\par\noindent\textbf{Relevance.} Wilson's 1974 paper introduces
the Wilson loop $W(C) = \Tr\,\mathcal{P}\exp(\oint_C A)$ in lattice
gauge theory as the non-perturbative order parameter for
confinement: an area law in the confined phase, a perimeter law in
the deconfined phase. In the AX-2 historical lineage, the Wilson
loop is the operational gauge-invariant observable that registers
the holonomy of the gauge connection around any closed curve, and
is the first construction in which the geometric content of the
connection becomes a quantitative, measurable invariant of the
non-perturbative quantum theory. Every later step in the AX-2
lineage --- the Hosotani mechanism, the Green--Schwarz construction,
the AdS/CFT correspondence, and the cohort framework's use of the
Wilson-line phase $\Theta_H$ as the hidden variable behind quantum
measurement --- descends from Wilson's reading of the holonomy as a
physical observable. First cited in \S\ref{sec:AX2}.

\bibitem[Wilson \& Kogut(1974)]{WilsonKogut1974}

\textbf{Wilson, K.~G., \& Kogut, J. (1974).}
The renormalization group and the $\epsilon$ expansion.
\textit{Physics Reports}, \textit{12}, 75--200.
\par\noindent\textbf{Relevance.} A standard reference for the modern RG viewpoint: coarse-graining as a flow on theory space. POS-5 builds on this picture but adds the axiomatic demand that RG must commute with the operational dictionary (\OGGEP{}), enabling a geometric RG coordinate in \GG{}. First cited in \S\ref{sec:POS5}.

\bibitem[Witten(1998)]{Witten1998}

\textbf{Witten, E. (1998).}
Anti-de Sitter space and holography.
\textit{Advances in Theoretical and Mathematical Physics}, \textit{2}, 253--291.
\url{https://arxiv.org/abs/hep-th/9802150}
\par\noindent\textbf{Relevance.} Witten's prescription (together with Gubser--Klebanov--Polyakov and Maldacena) provides the canonical bulk$\to$boundary correlator map. We cite it as a concrete historical anchor for \OGGEP{}'s demand that a theory includes a bulk$\to$boundary dictionary, and as motivation for extending that dictionary to operational measurement objects. First cited in \S\ref{sec:four_clusters}.

\bibitem[Wu \& Yang(1975)]{WuYang1975}

\textbf{Wu, T.~T., \& Yang, C.~N. (1975).}
Concept of nonintegrable phase factors and global formulation of gauge
fields.
\emph{Physical Review D}, \emph{12}, 3845--3857.
\url{https://doi.org/10.1103/PhysRevD.12.3845}
\par\noindent\textbf{Relevance.} The Wu--Yang ``dictionary'' between
gauge-field theory and fiber-bundle geometry. In the constitutional
reading of $\POS{1}~\GI{}$, Wu--Yang 1975 is the technical bridge
between the physicist's $A_\mu$ and the mathematician's principal
connection, the historical moment at which gauge theory became
\emph{geometric}. First cited in \S\ref{sec:AX2}.

\bibitem[Yang and Mills(1954)]{YangMills1954}

\textbf{Yang, C.~N., \& Mills, R.~L. (1954).}
Conservation of isotopic spin and isotopic gauge invariance.
\textit{Physical Review}, \textit{96}(1), 191--195.
\url{https://doi.org/10.1103/PhysRev.96.191}

\par\noindent\textbf{Relevance.} Yang--Mills theory is the modern origin of non-Abelian gauge symmetry.
AX-2 builds on the connection/curvature viewpoint of gauge forces and then strengthens it by asserting that gauge and gravity are parts of one composite geometric structure (\GGEP{}). First cited in \S\ref{sec:intro}.

\bibitem[Zeh(1970)]{Zeh1970}

\textbf{Zeh, H.~D.\ (1970).}
On the interpretation of measurement in quantum theory.
\textit{Foundations of Physics}, \textit{1}(1), 69--76.
\url{https://doi.org/10.1007/BF00708656}
\par\noindent\textbf{Relevance.} The first explicit recognition that
realistic quantum systems are inevitably open --- coupled to a much
larger environment whose accessible degrees of freedom decohere the
system on extremely short timescales. We cite Zeh as the founding
paper of the decoherence program; \AX{3}~\OGGEP{} is the
axiomatic formulation of the same intuition, with openness
elevated from corrective layer to foundational law. First cited in \S\ref{sec:intro}.

\bibitem[Zumino(1985)]{Zumino1985}

\textbf{Zumino, B. (1985).}
Cohomology of gauge groups: cocycles and Schwinger terms.
\emph{Nuclear Physics B}, \emph{253}, 477--493.
\par\noindent\textbf{Relevance.} Zumino's full cohomological treatment
of gauge anomalies and Schwinger terms. In the constitutional reading
of $\POS{3}~\AF{}$, Zumino 1985 supplies the algebraic apparatus that
turns the Stora descent equations into a complete classification of
admissible anomaly structures. First cited in \S\ref{sec:POS3}.

\bibitem[Zurek(1981)]{Zurek1981}

\textbf{Zurek, W.~H.\ (1981).}
Pointer basis of quantum apparatus: Into what mixture does the wave packet collapse?
\textit{Physical Review D}, \textit{24}(6), 1516--1525.
\url{https://doi.org/10.1103/PhysRevD.24.1516}
\par\noindent\textbf{Relevance.} The original einselection paper:
environment-induced superselection picks out a preferred ``pointer
basis'' that survives decoherence. We cite it as the mid-1981 turning
point in the decoherence program that established openness as
operationally inescapable, motivating \AX{3}~\OGGEP{} and the
\GDOS{}-as-boundary-law thesis. First cited in \S\ref{sec:intro}.

\bibitem[Zurek(2003)]{Zurek2003}

\textbf{Zurek, W.~H.\ (2003).}
Decoherence, einselection, and the quantum origins of the classical.
\textit{Reviews of Modern Physics}, \textit{75}(3), 715--775.
\url{https://doi.org/10.1103/RevModPhys.75.715}
\par\noindent\textbf{Relevance.} The mature statement of the
decoherence program: a mechanism for effective classicality in open
systems via environment-induced superselection. We cite it whenever
the chain Zeh (1970) $\to$ Joos \& Zeh (1985) $\to$ Zurek (1981, 2003)
is invoked, and as the immediate intellectual precedent for elevating
openness from corrective layer to constitutional law in \AX{3}~\OGGEP{}. First cited in \S\ref{sec:intro}.
\end{thebibliography}



\end{document}
