%% v20.0 (August 21, 2026): PROMOTION TO THE v20 STAMP.
%%   Version and cover date only.  Promoted from v15.15 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%%   superseded file.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.
%% ===================================================================
%% v15.12 (August 18, 2026) -- BT-1: THE BIBLIOGRAPHY TITLE PASS.
%%
%% EVERY COMPANION IS NOW CITED BY THE TITLE PRINTED ON ITS OWN COVER.
%%
%% 8 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.
%% ====================================================================
%% ===================================================================
%% v15.11 (August 18, 2026) -- FP-4: THE DISCLOSURE-PREP PASS.
%%
%% The cohort is posted as TWENTY-ONE papers with A11-Origin held back.
%%
%% (1) THE A11 BIBLIOGRAPHY ENTRY HAD NO AVAILABILITY LINE AT ALL -- only
%%     "Arisaka-GG-A11-Origin (May 20, 2026)", a date on a paper that is
%%     not finished.  It now reads "In preparation ... no result of the
%%     present paper depends on it", and the date is gone.
%% (2) FIFTEEN POINTERS INTO A11'S INTERNAL PART STRUCTURE, retired.  A8
%%     carries more of these than the rest of the cohort combined.
%%     MEASURED against A11 v15.14 before deciding: Part III is still the
%%     GRIPout cascade, so the seven "Part III" pointers are correct
%%     TODAY; but "the layered-cascade catalogue of Part VI" is wrong
%%     (Part VI is Life; the codebook is Part X), "Part VII addresses
%%     cellularity" is wrong (cellularity is layer 10, Part VI), and the
%%     six "Part VIII develops the cognitive cascade" pointers are wrong
%%     (that content is Part XIV).  EIGHT OF FIFTEEN ARE ALREADY WRONG,
%%     and SIX of those eight point into Parts V-IX -- exactly the Parts
%%     the v16 rebuild will move.
%%     ALL FIFTEEN are retired rather than repaired, on the policy
%%     adopted in FP-3: A8 is about to be posted permanently while A11 is
%%     explicitly still being rebuilt, so every Part pointer A8 carries
%%     is a promise it cannot keep.  The paper and its NAMED THEOREMS
%%     (T-GRIP-In-Lyapunov-Descent, T-GRIP-Out-Necessity,
%%     T-GGA-as-Ground-State) are stable identifiers and are what the
%%     citations now rest on.
%% (3) catalogue -> catalog in printed text (Report 443's pair-list hole).
%% (4) THE NINE COMPANION BIBITEMS CARRIED STALE DATES.  Repairing (1)
%%     made its neighbors look wrong: each companion was still
%%     introduced as "Arisaka-GG-A9-SM (May 14, 2026)" -- a date three
%%     months stale, and no statement of where a reader can get the
%%     paper.  Measured across all 24 sources, only FIVE papers still
%%     carried dated companion bibitems (A8, A10, A11, C2-Hubble,
%%     P4-Real); the other nineteen were cleared in earlier ships.  All
%%     nine here now carry the nineteen's availability line instead.
%%     C2-Hubble and P4-Real are FP-5's; A11-Origin waits for its
%%     rebuild.  The same string inside THIS banner is quoted history and
%%     was explicitly excluded from the match.
%%
%% A8 carries NO simultaneity claim, NO reading-set enumeration, NO roster
%% row for A11 and NO layer-count description of A11's cascade; the
%% claim-derived probe checked all four before this rider was written.
%%
%% NO OTHER CONTENT CHANGED.
%% ===================================================================
%% ===================================================================
%% v15.10 (August 17, 2026) -- FIGURE 5 ENLARGED, IN THE PAPER TOO.
%%
%% The author asked for 30 to 50 percent.  A PLAIN ENLARGEMENT CANNOT
%% REACH IT: the plate declares 1.0 pt rules, which at the body's
%% 0.384\textwidth render at 0.745 pt, and the house ceiling is 0.956 --
%% so scaling alone caps at 0.492\textwidth, +28 percent, short of what
%% was asked.
%%
%% SO THE DECLARED WEIGHTS COME DOWN BY THE FACTOR THE PLATE GOES UP,
%% and the RENDERED weights land where they already were:
%%
%%     declared   was    now     rendered before   rendered after
%%     global     0.80   0.55       0.596             0.595
%%     dashed     0.9    0.62       0.671             0.670
%%     emphasis   1.0    0.69       0.745             0.746
%%
%% The stroke weights of the drawing do not change at all.  The picture
%% is simply bigger.
%%
%%     in-paper copy   0.384 -> 0.557\textwidth   +45.1 %%
%%     showcase copy   0.47  -> 0.66 \textwidth   +40.4 %%
%%
%% The showcase copy stays the larger of the two.  Its rendered weights
%% are 0.705 / 0.794 / 0.884 pt, the heaviest 0.072 pt clear of the
%% ceiling.  A5-GDOS v15.22 is the precedent for lowering a declared
%% weight rather than accepting an over-ceiling rule.
%%
%% NEW PROSE IN THIS SHIP: none.  Not one printed character changes.
%% ===================================================================
%% ===================================================================
%% v15.9 (August 17, 2026) -- THE AUTHOR'S SHOWCASE SELECTION.
%%
%% THE SHOWCASE GOES FROM THREE PAGES TO FIVE:
%%   p5  Quantum Mechanics Is Not Assumed Here      Figure 4 + FIGURE 5
%%   p6  No Atom Ever Feels a Probability           Figures 6 + 7   NEW
%%   p7  Two Records, One Object                    Figures 9 + 10  NEW
%%   p8  Eight Century-Old Paradoxes                NOW A TABLE (was p6)
%%   p9  Eight Observations That Would Retire This  unchanged (was p7)
%%
%% MEASURED FIRST, LAID OUT SECOND.  Every plate was boxed and its
%% natural size read with \typeout before any page was designed:
%%   Fig 4  573.41 x  57.91 pt   body \textwidth        renders  47.4 pt
%%   Fig 5  242.01 x 172.42      body 0.384\textwidth   renders 128.5
%%   Fig 6  420.09 x 144.78      body 0.759\textwidth   renders 122.9
%%   Fig 7  480.30 x 192.50      body 0.876\textwidth   renders 164.9
%%   Fig 9  362.89 x 214.06      body 0.660\textwidth   renders 182.9
%%   Fig 10 461.31 x 172.93      body 0.839\textwidth   renders 147.7
%% against \textheight 650.43 pt.  Two plates a page fit with room.
%%
%% HOW FAR FIGURE 5 COULD BE ENLARGED IS A MEASUREMENT.  It declares
%% 1.0 pt rules; at the body's 0.384\textwidth the scale is 0.745 and
%% they render at 0.745 pt.  The house ceiling is 0.956, so the largest
%% safe width is 0.492\textwidth -- where the margin is five thousandths
%% of a point, which is not a margin.  THE SHOWCASE COPY IS 0.47:
%% rules render at 0.912 pt and the plate is 22 percent larger than the
%% in-paper one.  THE IN-PAPER COPY ON PAGE 67 IS UNTOUCHED, because
%% enlarging it reflows a page the author has read.  One line if wanted.
%%
%% THE PARADOX PLATE BECOMES A TABLE IN BOTH PLACES AND KEEPS ITS NUMBER.
%% The eight rows are lifted from the TikZ SOURCE, verbatim, not retyped
%% off the rendered page -- A3-BI6's rule, so no transcription can drift.
%% The table lives in a savebox used twice, so the showcase copy and the
%% in-paper copy cannot diverge.  IT STAYS INSIDE THE figure FLOAT AND
%% STAYS Figure 16: A8 already sets its tabular displays as figures
%% (16, 17 and 18 are all tables in figure floats), so keeping the float
%% type keeps this paper's own convention and RENUMBERS NOTHING -- no
%% figure and no table moves.  Figure 16's own caption already describes
%% a four-column table and needed no change.  \GGscSweep, the TikZ
%% plate, is retired; it was used in exactly the two places now served
%% by the table.
%%
%% NEW PROSE IN THIS SHIP: two headlines, two showcase captions, and one
%% added passage in the page-5 caption covering the second plate.  The
%% table's cells are the plate's own text and are not new prose.
%% ===================================================================
%% ===================================================================
% v15.8 (August 17, 2026) -- MIGRATED TO STYLE FILE v11.5.
%%
%% FOUR MACROS THAT LIVED IN THE PAPERS COME HOME, AND NOT ONE PAGE
%% MOVES.  This paper's part in that:
%%
%% * \GGShowcaseHead removed; this paper prints the sans form with no trailing skip
%%   and says so in one line.
%%
%% The style file is v11.5 (2026/08/17).  Every one of the twenty-four
%% MANIFEST.sha256 files covers it, so changing it meant rebuilding and
%% repackaging all twenty-four -- which is why this was deferred five
%% times and is why it was done in one pass.
%%
%% ASSERTED, NOT ASSUMED: this paper's printed text is line for line
% what v15.7 printed, except for the lines named above.
%% ===================================================================
%% ===================================================================
%% v15.7 (August 17, 2026) -- THE -ise FAMILY.
%%
%% maximiser -> maximizer (4); extremiser -> extremizer (4); randomiser -> randomizer (4); minimiser -> minimizer (4).  Charter 1.1's pair list names these forms; spelling_audit.py's
%% PAIRS list did not carry them, so this paper has reported MIXED
%% PAIRS 0 and ALWAYS-BRITISH 0 at every gate while printing them.  The
%% cohort sweep of August 17, 2026 found sixty-one across twelve of the
%% twenty-four live papers; Report 406 section 6 has the inventory and
%% spelling_audit.py now carries the family as a RULE rather than a list.
%%
%% Substitution is on PRINTED TEXT ONLY.  Changelog comments are the
%% historical record, \label/\ref/\cite arguments are keys, and published
%% titles are quoted rather than written -- all three are masked, and every
%% frozen title is asserted present and unchanged after the edit.
%% NOTHING ELSE IN THIS PAPER CHANGES: no claim, no number, no float, no
%% label, and no page.
%% ===================================================================
%% ------------------------------------------------------------
%% v15.5 -> v15.6, 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.
%% ------------------------------------------------------------
%% ------------------------------------------------------------
%% v15.4 -> v15.5, August 16, 2026.  THE SAVEBOX-LEADING RIDER.
%% A DEFECT I INTRODUCED IN THIS PAPER'S OWN SHOWCASE SHIP TODAY.
%%   Every showcase plate is filled into an \sbox in the front matter
%%   so the showcase and in-paper copies cannot drift.  A savebox is
%%   typeset when it is FILLED, and this paper's front matter runs at
%%   \baselinestretch 1.15 against the body's 1 -- measured with
%%   \typeout at both sites: \small leading 13.8 pt against 12.0.
%%   So every plate in a savebox was being set 15% looser than before,
%%   IN BOTH PLACES, including the in-paper float the author had
%%   already read.  A9-SM's falsification map grew 47.9 pt and pushed a
%%   paragraph off its page, which is how it was caught.
%%   This paper's showcase ship reported the resulting page as a benign
%%   "savebox reflow".  It was neither benign nor a reflow.
%%   FIX: \setstretch{1} inside every \sbox, before the size command,
%%   so the box is filled at the BODY's spacing -- the canonical one.
%%   VERIFIED: the body is page-for-page identical to the pre-showcase
%%   baseline under the +3 shift.  Not close: identical.
%% ------------------------------------------------------------
%% ------------------------------------------------------------
%% v15.3 -> v15.4, August 16, 2026.  THE FRONT-MATTER SHOWCASE.
%% Eighth of the twenty-four, and the first outside the foundation
%% tier.  Three pages after the Keywords:
%%   p1  Figure 4,  the constitutional bridge -- what is claimed
%%   p2  Figure 16, the paradox sweep         -- what it buys
%%   p3  Table 17,  the eight kill criteria   -- what would end it
%% THE ORDER IS CLAIM / PAYOFF / EXPOSURE, not the claim / mechanism
%% / exposure the foundation papers used, because A8's payload is
%% itself one plate: eight century-old paradoxes, each with its
%% standard 4D reading, its 6D resolution, and the theorem that
%% licenses it, all dissolving under ONE mechanism.  A reading that
%% dissolved them one at a time by a different device each time
%% would be worth much less, and the caption says so.
%% Figures 1 and 2 are the shared cohort plates and are not used.
%% THE PLATES KEEP THEIR IN-PAPER SIZE, the rule A6 adopted after
%% A5's funnel had to be re-weighted: a savebox cannot differ
%% between its uses, so a wider showcase copy is a heavier BODY
%% plate.  \textwidth and 1.001\textwidth, both in-paper factors.
%% NOTHING IS COPIED: both TikZ bodies and the kill-criteria tabular
%% go into saveboxes filled once in the front matter.
%% NEW PROSE IN THIS SHIP: three headlines and three showcase
%% captions, all on the new pages.  Nothing in the body is rewritten.
%% ------------------------------------------------------------
% ============================================================
% ------------------------------------------------------------
% Stage 1b, same day.  Stage 1's guard was built from Stage 1's
%   own word list and therefore could not see a word that list
%   omitted.  A rule-based scan (-isation, -ise, -our, -re,
%   doubled-l) found 182 further printed British forms; all are
%   corrected here, 72 distinct words.
%   One revert: Stage 1 rewrote the title of Hensen et al.,
%   Nature 526 (2015), which is published as '1.3 kilometres'.
%   Published titles are now frozen by exact text.
%   Left alone and reported: analogue, catalogue, dialogue and
%   backwards are not on the charter's pair list and are
%   standard in American physics prose.
% ------------------------------------------------------------
% ------------------------------------------------------------
% Stage 2A, same day.  The float standard.
%   * 15 captions under the 260-character floor lengthened.  The
%     author's existing sentence opens each one unchanged; the
%     addition says what the table holds and how to read it.  No
%     caption's claim was altered.
%   * 3 uncaptioned longtables captioned and labelled
%     (tab:app-glossary, tab:app-timeline, tab:app-synthesis).
%   * The bare 'Numerical summary' tabular promoted into a table
%     float (tab:app-numerical-summary), caption above.
%   * Every caption now carries a short form for the List of
%     Tables, taken from its own leading sentence.
% ------------------------------------------------------------
% ------------------------------------------------------------
% Stage 2B, same day.  Front matter to GG-Standard-Front-Matter-v1.0:
%   \GGfrontmatteronepagetrue set (it was absent, so the lists would
%   each have taken a page) and \GGListOfTables added after the List
%   of Figures.  A8 had no List of Tables and carries 20 tables.
%   The switch is released again before \GGListOfTables: all three
%   lists on one page put the LoT heading on top of the page folio.
% ------------------------------------------------------------
% ------------------------------------------------------------
% Stage 2C, same day.  Plain-English Part closes for Parts II,
%   XI, XII and XIII, in the end-of-Part form Parts I, III and
%   IV already use.  NEW PROSE -- four sections, listed in the
%   report for the author to review or rewrite.
%   The Stage 0 note said ten were needed.  That count came from
%   counting \GGteachboxpart alone and missed the second device:
%   Parts V-X carry a head-of-Part teach box, Parts I, III and IV
%   carry an end-of-Part plain-English section, and the three
%   back-matter Parts get none in A5's model either.  Four.
% ------------------------------------------------------------
% ------------------------------------------------------------
% Stage 5, same day.  Integrity.  The three cohort companions A8
%   named in prose but never cited -- P6-Proton, W1-Symmetry and
%   W2-Holonomy -- now have bibliography entries with Relevance
%   notes, and the roster sentence that claimed the map complete
%   at twenty-two papers now names the White-Paper Branch it had
%   left out.  All 21 companions are present.  NEW PROSE: three
%   Relevance notes and one extended roster clause.
% ------------------------------------------------------------
% ------------------------------------------------------------
% Stage 4, same day.  Palette.  palette_fix brought 11 pictures to
%   the house band: fills above !18 compressed (!22->!13, !28->!16,
%   !40->!16, !50->!16), keyword widths made explicit, rules clamped
%   to 1.1pt.  THREE OF ITS CHANGES WERE REVERTED after reading the
%   rendered pages: spot/.style in the Stern-Gerlach figure and
%   source/.style in both EPR figures are FILLED POINT MARKERS, not
%   flowchart boxes.  Compressed into the box band they printed as
%   faint hollow circles and the figures lost the detector spots and
%   the emission point.  The house band governs boxes; a point marker
%   is figure content.
% ------------------------------------------------------------
% ------------------------------------------------------------
% Stage 3, same day.  Figures placed by the measured rule.
%   Every picture measured with measure_plate.py, one compile per
%   picture.  14 placed at f = (7.79/B) * W / 165.1; five left
%   alone (four already inside 7.79pt +/- 5%%, plus the exempt
%   cohort map); one, fig:appA-abbrev-flow-a8, could not be
%   measured because its probe spills to two pages -- reported.
%   Two real defects found: fig:hidden-fibre-projection printed
%   its labels at 18.97pt (an 81mm picture blown up 1.9x) and
%   fig:stern-gerlach-apparatus at 4.54pt, smaller than a
%   footnote.  Both had passed every gate to date.
% ------------------------------------------------------------
% ------------------------------------------------------------
% v15.0 -> v15.1  August 15, 2026.  Spelling rider.
%   * One printed British form the v15.0 spelling guard could not see:
%     'totalling' -> 'totaling'.  The guard's literal branches were
%     case-sensitive and its doubled-l branch required the letter e
%     before the ll, so 'totalling' matched nothing.  Corrected rule in
%     gg_spelling.py.  The two other sites the corrected rule finds are
%     published titles -- Ehresmann 1950 (French) and Hensen et al.
%     2015 ('1.3 kilometres') -- and are frozen, not changed.
%   NO PROSE WAS WRITTEN.  Nothing else in this file changed.
% ------------------------------------------------------------
% v13.24 -> v15.0  Stage 1 of the v15 migration, August 14, 2026.
%   * American spelling swept through printed text only: 654
%     substitutions across 58 distinct words.  Comments and
%     cross-reference identifiers were masked and are untouched.
%   * Column separation left alone: the document default is
%     already 6 pt and the single 3 pt is a local narrowing that
%     a nine-column table needs.
%   * The one bare grid under 'Numerical summary' converted from
%     hline rules to booktabs and set at \small.
%   * Page size and caption position were already correct at
%     intake and were not changed.
% ============================================================
%% ===================================================================
%%  A8-QM v13.24  (August 14, 2026)
%%  THE a_mu PULL ROWS, IN BOTH EVIDENCE TABLES.  A8's prose was already
%%  right -- its caption calls the row "the imported Standard-Model
%%  evaluation ... stakes that rigidity as exposure (A9, Objection 47)"
%%  -- and is not touched.  The NUMBERS were wrong twice over:
%%    * STALE.  Both tables printed a_mu^SM = 116592050e-11, which is
%%      neither the 2020 nor the 2025 white-paper value.  The Theory
%%      Initiative's WP25 total is 116592033(62)e-11, the shift coming
%%      from the move to lattice hadronic vacuum polarisation.
%%    * THE PULL OMITTED THE THEORY ERROR.  The SM value was printed
%%      with no uncertainty and +1.4 sigma is exactly
%%      (116592070 - 116592050)/15 -- the experimental error alone.  The
%%      theory error is +-62, four times larger.
%%  Corrected: 116592033(62) against 116592071.5(14.5), a difference of
%%  38.5 +- 63.7, i.e. +0.6 sigma.  The full table's label said "(FNAL
%%  final 2025)" while quoting an SM value that was not the 2025 white
%%  paper's; it now names both sources.  The correction is in the
%%  framework's favour: A9 Objection 47 stakes GG's inability to absorb
%%  a muon anomaly as exposure, and the anomaly has since fallen from
%%  about 4 sigma to 0.6.  The caption's standing "within 1.5 sigma"
%%  claim is unaffected and is asserted in the script.
%% ===================================================================
% ===================================================================
%  A8-QM v13.23 (August 12, 2026) -- cohort-wide sweep, two defects.
%
%  (1) GU-5 IS A THREE-PLUS-TWO PARTITION, exactly as OGN-5 is
%      (M,N,L;J,K).  The separator before k_B is a SEMICOLON:
%      {c, G_N, hbar; k_B, Q_B}.  1 site(s) corrected here.
%      NOT touched, by design: the four-constant Planck unit system
%      {c, G_N, hbar, k_B}, which is a different object and takes a
%      comma; a semicolon there would be wrong.
%
%  (2) Internal affairs -- companion versions, dates, page counts and
%      build status -- do not belong in a preprint.
%%      Nothing of this class was found in this paper.
%
%  No physics changes; no parameter, theorem or falsifier is touched.
% ===================================================================
%% ===================================================================
%%  A8-QM v13.22  (August 6, 2026)  -- the cohort-wide figure-size sweep.
%%  9 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.
%% ===================================================================
%% ===================================================================
%%  A8-QM v13.21  (August 5, 2026)  -- the twenty-two-paper cohort sweep.
%%  The cohort map becomes the twenty-two-paper map: GG-P6-Proton on
%%  the Predictions row with a second sister-import arrow, and the
%%  White-Paper Branch (GG-W1-Symmetry above GG-W2-Holonomy) as a
%%  right-flank column at the mirror positions of the Foundations
%%  column -- the left flank feeds the spine, the right reads it.
%%  GG-P7-Inflow does not appear: it is Appendix H of GG-P2-N3.
%%  The plate is also enlarged for reading: boxes and the type inside
%%  them x1.15, every word outside a box x1.20, and the long
%%  A10 -> C-frame edge redrawn as a polyline, which returns 24mm of
%%  bounding box that was Bezier control points rather than ink.
%%  Prose: every count site, the series breakdown, and the canonical
%%  Suffix list, which was missing Proton, Symmetry and Holonomy.
%%  Recipe and variant analysis: [v13-support]/sweep22.py.
%% ===================================================================
%% ===================================================================
%%  A8-QM v13.19  (July 30, 2026)  -- Phase-2 shape-up.
%%  Four bibitems carried no Relevance note (AharonovBohm1959, Berry1984,
%%  FeynmanQED1985, Wilson1971) -- the last four gaps in a 157-entry
%%  bibliography.  Each now carries one, in house style, saying what the work
%%  contributes and where it bears.  Charter sec.2 reference standard.
%% ===================================================================
%% ===================================================================
%%  A8-QM v13.18  (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.
%% ===================================================================
%% ===================================================================
%%  A8-QM v13.17  (July 29, 2026)  -- five references to the project's own
%%  instruction document, FOUR OF THEM BY LINE NUMBER, removed:
%%    "(see L59 of the cohort project instructions)"
%%    "(per L59 of the cohort project instructions)"
%%    "per L58 of the project instructions"
%%    "Per L6 of the project instructions, each problem carries ..."
%%    "(consistent with the project instructions' Phase~5 status header
%%      'Out of scope; pointed to A11 Part VIII')"
%%  A reader was being pointed at line 59 of a file they cannot open.  Same
%%  class as A1's "project rules document" (v13.5) and P3's internal audit path
%%  (v14.1), in its sharpest form.  Every commitment is kept: the
%%  structural-realist genealogy, the disavowal of Russellian neutral monism,
%%  the Pull-table form, the per-problem status headers, and the consciousness
%%  scope decision -- now pointed at A11 Part VIII, which owns the question.
%% ===================================================================
%% ===================================================================
%%  A8-QM v13.16  (July 29, 2026)  -- internal-affairs sweep.
%%  Two Common Objections were internal QA findings written up as referee
%%  objections; in each the OBJECTION TEXT asserted a defect ("You have
%%  mis-cited it", "You have misstated your own companion") that an arXiv v1
%%  reader cannot find, and the response narrated the repair ("A correct
%%  catch", "Conceded, and reconciled").  Both pairs recast onto the question
%%  the referee would actually ask.  All physics preserved: A6's CCI is the
%%  Covariant Conservation Identity with no noise content; the 1/f exponent's
%%  departures from unity are system-specific; A7's spectrum is bounded and
%%  two-knee'd with a conformal-weight-dependent slope.
%%  NOT touched: the O-QM-16 neighbour, which rebuts an accusation ("The
%%  concern does not arise ...") rather than narrating a repair; and two hits
%%  on "the user", which mean the physicist using the decoherent-histories
%%  formalism, not the author.
%% ===================================================================
%% ===================================================================
%% v13.14 -> v13.15 (July 28, 2026).  Phase C of the GGA correction.
%%   * TA-9's proof re-pointed: the attractor-uniqueness input is the
%%     upstream per-stratum ground-state corollary, not the capstone's
%%     cascade-wide generalisation, and it is quoted at its grade.
%% Values, canonical numbers, ledger: UNCHANGED.
%% ===================================================================
%% ===================================================================
%% v13.14 (July 24, 2026): COHORT BIB-TITLE RECONCILIATION (P3)
%%  The electron companion's subtitle is updated in its bibliography
%%  entry to that paper's current cover ("The Six Oldest Questions
%%  about the Electron ..."), following its addition of the sixth
%%  question -- the origin of positrons. No other change
%%  anywhere.
%% ===================================================================
%% ===================================================================
%% v13.13 (July 23, 2026): THE DERIVED i, PLUMBED THROUGH THE SPINE
%%  (Internal Report 26 Ship 1, author-approved). The Part-XII
%%  real-bulk treatment (RB catalogue; owner P4-Real; clock/fibre
%%  custody P5-CPT) is now consumed at the paper's teaching sites:
%%  the Introduction's no-consensus list gains its sixth bullet (why
%%  complex numbers) with the answer's address; the Part-III bridge
%%  parenthetical gains its P4-Real closure clause; TA-1 gains the
%%  derived-pairing clause; TA-2 gains the named "Where the i comes
%%  from" paragraph; Appendix G gains the P4-Real Read-alongside, a
%%  chapter-closing register paragraph, and the real-circle take-away
%%  clause; and two new problems (D.1.2 psi = u+iv real pair; G.1.1
%%  U(1) ~ SO(2) with the Renou-compatibility close) bring the census
%%  to fifty-six. NO new theorems, NO roster changes, NO new custody
%%  instances (count stays five); registers quoted at their stated
%%  grades (derived-conditional, quasi-free regime); zero new
%%  bibitems.
%% ===================================================================
%% ===================================================================
%% v13.12 (July 23, 2026): PAGINATION REPAIR (author-reported
%%  near-empty pages 38/40/44/47/80/101/129/154/203/217/223/231/254/
%%  333). Within-Part \clearpage commands removed (the TA-theorem
%%  sections, the Part-in-plain-English closes, the Programme
%%  dictionary, the cohort-inheritance and roadmap sections, the GRIP
%%  spine, four appendix-chapter boundaries) together with the
%%  \clearpages that were redundant before \GGPart (the macro itself
%%  clears); the hard-placed full-page floats converted to [p] floats
%%  (the acronym-hierarchy figure, the cohort map's forced break, the
%%  hand-built roadmap minipage page, and Tables 10-13) so main text
%%  fills the formerly stranded pages while each float keeps its own
%%  full page. Part boundaries unchanged; NO text content changed.
%% ===================================================================
%% ===================================================================
%% v13.11 (July 23, 2026): SHIP C OF THE COURSE CAMPAIGN -- twenty-
%%  eight course-grade problems with complete worked solutions at the
%%  close of the five Worked-Example Parts and the paradox-sweep Part
%%  (Stern-Gerlach 5, double-slit 5, EPR/Bell 5, cat 4, decoherence 4,
%%  paradoxes 5), each set placed before its Part's take-away box and
%%  numbered by Part; the course front matter gains the problems
%%  clause (54 = 26 + 28 across the paper); the Introduction
%%  gains the course pointer. All numerical answers verified
%%  programmatically before typesetting. This completes the course
%%  campaign for this paper (apparatus + theorem problems + course
%%  problems).
%% ===================================================================
%% ===================================================================
%% v13.10 (July 23, 2026): SHIP B OF THE COURSE CAMPAIGN -- twenty-six
%%  theorem-grade problems with complete worked solutions for the five
%%  deep proof appendices (Gleason 5, GKSL 6, partial trace 5,
%%  Bell-CHSH 5, Wilson line 5), each set placed before its chapter's
%%  take-away box, numbered by chapter letter and subsection, in the
%%  cohort problems format (bold statement, Solution in explicit
%%  Steps, Physics intuition close), with every numerical answer
%%  verified programmatically before typesetting and every \eqref
%%  anchored to the chapters' own equations. Ship C (course-grade
%%  problems for the Worked-Example Parts + the front-matter clause)
%%  follows.
%% ===================================================================
%% ===================================================================
%% v13.9 (July 23, 2026): SHIP A OF THE COURSE CAMPAIGN (Internal
%%  Report 25, all recommendations author-approved) -- the course
%%  apparatus for the quantum-physics textbook prototype. Box macros
%%  in the preamble; full teach/readalong/takeaway trio for the five
%%  deep proof appendices (Gleason, GKSL, partial trace, Bell-CHSH,
%%  Wilson line) AND for the five Worked-Example Parts + the paradox
%%  sweep; teachbox for the two methods appendices; the take-away
%%  triple in this paper's register (Standard account / What it left
%%  open / The bulk reading); one convention sentence in the Part-XIV
%%  front matter. Zero new bibitems -- all Read-alongside keys reuse
%%  the existing bibliography. Problems ships (B: proof appendices;
%%  C: worked-example Parts + clause) follow.
%% ===================================================================
%% ===== v13.8 (2026-07-22): STATEMENT RB-13 REGISTERED (author ruling) ====
%% The C3-BH real-bulk proposal (GG-RB-Proposal-C3-2026-07-21.md) is
%% ACCEPTED per the author's ruling of 2026-07-22, as recommended in
%% GG-Cohort-Side-Records-Report-2026-07-22.md:
%% (1) New paragraph "Statement RB-13 (the horizon ledger;
%%     registrational)" after RB-12: the exact-1/4 chain named as
%%     real-section geometry read through the derived clock window ---
%%     the dictionary owns the sectors; the geometry owns the i.
%%     Registrational: no new theorem, no new custody instance (count
%%     stands at five), NO series-entry number (the proven series
%%     stands at R15; R16/R16' remain the Born-relaxation lemmas').
%% (2) The three RB-1--RB-12 series pointers advanced to RB-1--RB-13.
%% (3) Evidence-table H0 cells: "lift open" -> the C2 lift register
%%     (sign + rough third derived, rest a named ladder systematic),
%%     folding in cohort-side record 3, consistent with C3 v14.3.
%% RB-1..RB-12 statements byte-identical (protect-asserted).
%% ===== v13.7 (2026-07-22): EVIDENCE-ROW CURRENCY FIX (cohort record D-8) =
%% Closes the cohort-side record from the C3-BH P3b audit (D-8) plus the
%% retired-reading H0 rows discovered in its execution. Both cumulative-
%% evidence tables (tab:QM_evidence, tab:full_evidence) brought to the
%% cohort's current confrontation conventions; NO theorem, statement,
%% axiom, or RB-series text touched.
%% (1) Muon g-2: measurement -> June 2025 Fermilab final,
%%     116 592 070(15) x 10^-11; pulls +0.3/+0.2 (mutually inconsistent)
%%     -> +1.4 sigma. Prediction cell unchanged (imported SM evaluation);
%%     provenance clause added to the tab:full_evidence caption per the
%%     A9 Objection-47 stance (D-8's provenance gap closed in-table).
%% (2) W mass: CMS-only 80.3602(17) + stale QD 80.358 -> A9 v20.37
%%     world-average convention: 80.3700 GeV (A9, QD) vs 80.3692(133),
%%     pull -0.06 sigma.
%% (3) Neutrino sum: <0.07 eV (DESI) -> <0.072 eV (DESI+CMB).
%% (4) H0 rows: the "73.5 predicted, 0.0 sigma" cells carried the
%%     RETIRED lift reading (A10 v13.6 / C2 v13.65 derive only the
%%     +2-3% Local-Hole lift; remainder a named ladder systematic).
%%     Replaced by the C3 v14-series background-vs-CMB display:
%%     66.6+-0.2 vs 67.4+-0.5, -1.5 sigma; ladder 73.50(81), lift open.
%% (5) Captions re-dated April -> July 2026; "<= 1.1 sigma" -> "<= 1.5
%%     sigma" (new true maximum, the H0 row).
%% All pulls verified programmatically in the ship script
%% (a8_v13.7_evidence_rows.py, [v11-Support]).
%% ===== v13.6 (2026-07-20): SUMMIT THESIS UPGRADE =====
%% Author-commissioned strengthening of the Part-XIII summit thesis
%% (sec:concl-thesis) with the post-real-bulk Einstein-determinism
%% reading.  (1) Three registered strengthenings installed after the
%% careful summit statement: IN FORM (the bulk law is real symplectic
%% Hamiltonian flow, partial_t X = Omega grad H -- the mathematical
%% shape of classical mechanics; L-Constitution-1 at candidate grade,
%% A1 v13.3; the A5-GDOS scope declaration), IN ONTOLOGY
%% (superposition is complex-linear; the bulk has angles, not phases;
%% no interference of alternatives to evolve), THE UNIFICATION
%% (chance and the imaginary unit enter through the same door: the
%% T-P4-1/2/3 dictionary that manufactures the boundary's i is the
%% one-way projection that manufactures its probabilities; one
%% purchase, paid together at the boundary; the boundary state IS
%% incomplete relative to the bulk).  (2) Four qualifications at
%% exact grade, replacing (and absorbing) the old qualifications
%% paragraph: (a) classical in register, constitutional in content;
%% (b) determinism is not predictability (No-Bypass: the dice are
%% irreducible for every physical observer by law; Bohr's half
%% untouched); (c) Bell + Renou division of labour (boundary
%% genuinely complex via the shared derived J, RB-4; bulk real;
%% Einstein's determinism lives in 6D because it cannot live in 4D);
%% (d) the Born grade unchanged (relaxation level; Constitutional
%% Born Theorem on (P1)-(P4); field-grade representation step the
%% single analytic residue).  (3) Closing beat: Einstein-to-Born,
%% 4 December 1926 (existing bibitem BornEinsteinLetters) + the
%% author's sentence: 'God does not play dice; the dice --- and the
%% imaginary unit they are rolled in --- are manufactured together
%% at the boundary, by the one-way dictionary through which any
%% observer must look.'  (4) Abstract: Einstein sentence + closing
%% sentence strengthened at honest grade; Born-status sentence
%% byte-identical; page-1 span verified unchanged.  (5) Executive
%% summary: bottom line extended by the compressed strengthenings +
%% closing dice sentence; two compensating word-trades; pages 2-4
%% span verified unchanged; exec Born clause byte-identical.
%% (6) Part-XIII ripples (minimal): ascent list names the real-bulk
%% series; integrated-thesis emph adds constitutional realness.
%% All ten Born-grade sites, RB-1..RB-12, freshness/arrow installs,
%% frozen map, Part-I close: byte-identical (protect-asserted).
%% Physics, theorem keys, numbers, values, grades: UNCHANGED.
%% v13.5 -> v13.6 (\PaperDate July 20, 2026).  Script:
%% a8_v13.6_summit.py.
%% ===================================================================
%% ===== v13.5 (2026-07-20): STAGE 5C -- WAVE-2 RESIDUE CLEANUP =====
%% (1) F2: P5-CPT bibitem title text -> current P5 v3.42
%%     cover title (label/key untouched).
%% Physics, theorem keys, numbers, values, grades: UNCHANGED.
%% ===================================================================
%% ===== v13.4 (2026-07-20): STAGE 5A -- TYPICALITY PASSING-SITE SWEEP =====
%% Wave-2 housekeeping batch, ship 5 (the charter's queued item).
%% Every surviving passing mention of the pre-canon open register
%% ('open typicality target' / 'typicality conjecture' / 'one named
%% conjecture') aligned to the canonical Born register: closed at
%% the relaxation level (R16 lemmas + R16' GDOS relaxation);
%% Constitutional Born Theorem conditional on premises (P1)-(P4);
%% the field-grade representation step the single remaining
%% analytic residue.  Sites: fibre-honest summary; 't Hooft shared
%% debt + Penrose status (Part VII); Part IX + Part XIV dictionary
%% table cells (compact form 'relaxation-level; one analytic
%% residue'); partial-trace paragraph; objections preamble (x2);
%% O-QM-2, O-QM-14, O-QM-21, O-QM-70, O-QM-73, O-QM-74 (objection
%% voice), closing note (x2).  AUTHOR-FLAGGED: abstract sentence
%% and executive-summary bottom line aligned to canon (both had
%% UNDER-claimed at the pre-canon register).  Legitimate uses of
%% 'typicality' as a word (the \GGtypicality macro, home text,
%% Bohmian comparisons, DGZ bibitem note) unchanged.  Physics,
%% theorem keys, numbers, values: UNCHANGED; no claim raised
%% beyond the canonical register.  v13.3 -> v13.4 (\PaperDate
%% unchanged).  Script: a8_v13.4_stage5a.py.
%% ===== v13.3 (2026-07-20): STAGE 2 -- BORN TWO-REGISTER + CUSTODY CONSUMERS =====
%% Wave-2 content installs (author-approved scope; audit A8-R1/R2/R3/
%% R5; Born status per the frozen project charter, which supersedes
%% the audit draft's open-typicality register).
%% (1) A8-R1: the canonical Born two-register formula installed
%% uniformly at the ten graded Born sites (macros GGMRshort +
%% GGtypicality; grip-spine fence; states-as-rays; status-table row 1;
%% PS-QM-2; sec:disc-open closure sentence + admissible-configuration
%% re-scope; O-QM-1; O-QM-4; O-QM-5): register (i) the Born FORM is
%% Resolved (T-A8.3); register (ii) the basin-capture = Born residue
%% is closed at the relaxation level (the R16 lemmas with the R16'
%% GDOS relaxation), the Constitutional Born Theorem conditional on
%% (P1) the A5 forward semigroup, (P2) the Haar anchor with its
%% recursion, (P3) genericity, (P4) the GRIP basin partition, with
%% the field-grade representation step the single remaining analytic
%% residue; home text sec:disc-open.  Three consequential grade-
%% consistency edits, flagged: the PS-QM-1 umbrella clause (expands
%% the edited macro), the Conclusions banner line + residue paragraph
%% (cited by O-QM-5's response as same-grade sites), the O-QM-4
%% objection line (ghost quote).
%% (2) A8-R2: freshness consumer declaration at the Stosszahlansatz
%% statement in sec:disc-open -- the fresh-port assumption is the
%% Port-Freshness Clause of the GDOS companion (ArisakaA5GDOS),
%% stated once at its dynamical home there, consumed here
%% statistically; one-clause pointer attributions at the four
%% remaining freshness sites (bulk no-memory; GKSL memoryless-bath
%% x3), local technical content intact.
%% (3) A8-R3: arrow-custody consumer sentence at the RG-arrow
%% identification -- AX-3 owns the arrow (A1 AX-3 synthesis);
%% canonical law-level derivation P5 Part IX; dynamical
%% implementation A5-GDOS; A8's measurement arrow a consumer, not a
%% private derivation.
%% (4) A8-R5: TA-12 Grade-paragraph consumer note -- named residues
%% registered upstream as promotion-roadmap rows 5, 6, and 8 of the
%% CPT member paper.
%% Protected byte-identical: RB-1..RB-12 region, the frozen v1.4
%% cohort map, the Part-I close, abstract + executive summary, and
%% sec:disc-open beyond the sanctioned paragraph.  Physics, theorem
%% keys, numbers, values: unchanged.  v13.2 -> v13.3.
%% Script: a8_v13.3_stage2.py.
%% ===== v13.2 (2026-07-19): WAVE 1 OF THE 19-PAPER PROMOTION SHIP =====
%% Mechanical cohort-promotion sweep (17 -> 19 papers) plus the
%% standardized Part-I close (author-approved Wave-1 scope; audit:
%% GG-Internal-Report-RealBulk-Audit-2026-07-19.md, A8 ledger (2.8) +
%% App. A).
%% (1) NEW closing section of Part I, "The cohort inheritance and the
%% roadmap of the sixteen Parts" (A2 v13.3 / A3 v13.6 / A4 / A5 / A6 /
%% A7 v13.0 architecture), at the end of Part I.  A8's cohort detail is
%% an entire Part (Part II, the per-paper ledger), so the close is the
%% roster-level panorama and points there.  Subsection "The cohort
%% inheritance" (new label sec:cohort-a8): nineteen/11+5+3 opening;
%% the branch-roster paragraph migrated verbatim from Part II's
%% summary (recounted to nineteen; re-pointed at Part II); NEW
%% roster-level Foundations Branch paragraph in A8's voice (A8 =
%% GG-P4-Real's principal consumer; RB-1..RB-12 develops in-paper what
%% P4-Real treats standalone -- "the dictionary owns the sectors; the
%% geometry owns the i"; GG-P5-CPT's mirrors and law-level arrow kin
%% of A8's arrow of measurement; TA-12 composes its quartet machinery
%% -- roster level, no custody install, no grade change); the frozen
%% v1.4 nineteen-paper cohort map on its own page (verbatim from A2
%% v13.3, MP4/MP5 rendition; existing label fig:inh-cohort-map kept;
%% the old seventeen-paper map in Part II retired to a pointer; \ASix
%% renders GG-A6-GRIP here via this paper's own renewcommand, so no
%% literal-name fix was needed).  Subsection "Roadmap of the sixteen
%% Parts" (new label sec:roadmap-a8; the retired section "The
%% structure of this paper" donates sec:structure-of-this-paper):
%% NEW warm prose walkthrough of the sixteen Parts (the old section's
%% "fifteen Parts" was a stale count) + NEW roadmap figure in the
%% house architecture (label fig:thirteen-parts-flowchart-a8 kept;
%% worked-example row V--IX keeps this paper's five-box precedent).
%% Highlighted Part unchanged: Part III, the constitutional bridge
%% (GG bulk -> OGGEP -> GDOS -> 4D QM), the paper's central claim.
%% (2) COUNTS: cohort-sense seventeen -> nineteen at the audit's sites
%% (dictionary, Part-II frame x3, conclusions x5 incl. two "an
%% seventeen" grammar slips, abbreviation table, references intro,
%% suffix roster, P4-bibitem framing).  PROTECTED semantic non-edits:
%% "Seventeen-layer cascade of GGAs" (dictionary figure node) AND
%% "seventeen-step layer enumeration" (A11 Part-VI cascade catalogue;
%% audit App. A lists L2673 as an edit site, but it is the same
%% semantic layer count as the skip-listed L2008 node -- deviation
%% reported).  "eighteen fermion masses", "ten upstream cohort
%% papers": untouched (semantic).
%% (3) ROSTER: suffix list -> canonical nineteen-paper form, gaining
%% the foundations preprints Real (P4) and CPT (P5).
%% (4) MACROS + BIB: \PaperPFour retargeted GG-P4-Twistor ->
%% GG-P4-Real (paper renamed at the P4-Real v3.0 ship); \PaperPFive
%% (GG-P5-CPT) installed; bib key ArisakaP4Twistor -> ArisakaP4Real
%% everywhere (bibitem + 2 citep + 1 citealp; historical changelog
%% comments keep the old key); P4 bibitem updated to the P4-Real v3.2
%% cover identity (title verified unchanged) with the "beside, not
%% inside" framing rewritten to full-member Foundations-Branch
%% language (audit A8-R4); "satellite" register tokens converted to
%% member language at every GG-register site (4x "CPT satellite" in
%% the TA-12 section, one inside the theorem block's Grade clause;
%% 3x inside the protected RB region -- designator-only, listed by
%% the ship script's protect-assert; O-QM-81; A-list header; P5
%% "second satellite"; Frohlich1973 "satellite suite").
%% Non-GG satellite prose (Micius, satellite-to-ground, lunar)
%% untouched.  "First cited in" pointers re-aimed where the first
%% citation moved with the migrated paragraph (7x sec:inh-summary ->
%% sec:cohort-a8; P4 -> sec:cohort-a8; P5 -> sec:cohort-a8).
%% (5) ONE-LINE CONTENTS: \GGPart renewed per the P4-Real v3.2
%% preamble pattern (contents line carries only "Part N. <title>");
%% \GGAppendixPart (Part XIV) inherits the renewal.
%% Wave-2 content items deliberately NOT installed: the Born
%% two-register formula A8-R1 (all ten Born-grade sites byte-identical,
%% incl. the whole sec:disc-open section), the freshness deduplication
%% / Port-Freshness Clause A8-R2 (all freshness sites byte-identical),
%% the arrow-custody restatement A8-R3, the TA-12 consumer notes
%% A8-R5.  RB-1..RB-12 physics byte-identical up to the listed
%% designator-only substitutions.  Physics, theorem keys, numbers,
%% values, grades: UNCHANGED.  v13.1 -> v13.2.
%% Script: a8_v13.2_wave1.py.
%% Register fix (2026-07-20, pre-signoff, version stays v13.2): P4-Real bibitem Relevance internal-history clause removed (cohort-membership date / 'promoted from satellite standing'); 'formerly Gi4' dropped from the suffix roster. Script: a8_v13.2_register_fix.py.
%% ===== v13.1 (2026-07-18): TA-12 -- FERMI-DIRAC STATISTICS =====
%% QM-register companion to A2-GG6 v13.1 sec:spin-statistics
%% (author-approved; source: GG-Internal-Report-Spin-and-FD-
%% Statistics-2026-07-18.md).  New closing section of Part IV:
%% TA-12 (T-QM-12, candidate) -- the fermionic symmetrisation
%% postulate discharged as boundary inheritance of four cohort
%% results (TA-9 token identity; A2 frame-path register L-Spin-1/
%% T-Spin-A; A2 L-Stat-1 exchange = 2pi; A2/P5 T-Stat-A statistics
%% at the Wick quartet).  TA-9's proof seam (statistics previously
%% imported from standard QFT) discharged with pointer to TA-12.
%% A8-owned Wigner-loophole remark: the Born chain derives the ray
%% structure that admits half-integer spin.  Anchors: VIP-2 2025
%% beta^2/2 < 4.8e-29 (verified); Rauch et al. 1975 4pi
%% interferometry; kill criterion = any confirmed PEP violation
%% (F-Stat-1 mirror).  "Eleven principal theorems" branding
%% unchanged: TA-12 is explicitly beyond the eleven, candidate.
%% New bibitems: ArisakaP5CPT (A-list), Pauli1940, Rauch1975,
%% VIP2_2025 (B-list); A-list header 'satellite' -> 'satellites'.
%% ===== 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):
%% one stale scale statement 'M_lock ~ 10^8 m_Z' corrected to the cohort
%% canon M_lock ~= 3.85 TeV (~= 42 m_Z). No other changes.
%% v12.14 (July 16, 2026): cohort table standard (author-approved,
%% P4 v2.17 pattern): all tables raised to \small (slightly below the
%% main text); p{}-column tables widened proportionally toward the full
%% text width (capped at x1.6; l/c columns allowed 1.2cm each).
%% v12.13 (July 16, 2026): the two collision-level theorems of the rigor
%% campaign graduated into the typicality section (rate bound: geometric
%% contraction ratio (1-mu); anchor recursion: additive port-error bound
%% + induction on Haar-anchored port graphs); the single remaining
%% analytic residue is the field-grade representation step.
%% v12.12 (July 15, 2026): the GG typicality target upgraded to its
%% relaxation-level status (reports 18-19, author-approved): four
%% structural lemmas (unique conserved density; dial-blind data;
%% operational stationarity => v = j/rho; equivariance) + the GDOS
%% relaxation chain (semigroup freshness + Haar anchor + data
%% processing => monotone convergence to Born; no statistical
%% postulate); residues named (representation, rate, recursion);
%% relic-non-equilibrium falsifier named; new B-refs DGZ 1992 +
%% Valentini 1991 with relevance notes.
%% v12.11 (July 15, 2026): R14 + R15 graduated as Statements RB-11 (the
%% rotating anchor) and RB-12 (the collapse channel) in the real-bulk
%% section; Non-stationarity ledger updated (rotating + collapse closed
%% at model level; exact-Kerr Hadamard analysis + full gravitating case
%% named as the field's own); Interaction residue notes the R13-QED
%% conditional; curved-space objection response refreshed; four new
%% B-list references (Davies-Fulling 1976, Kay-Wald 1991, Starobinsky
%% 1973, Unruh 1974) with paragraph Relevance notes.
%% v12.10 (July 15, 2026): R13 graduated as Statement RB-10 (infrared
%% sectors as ports; series entry R13) in the real-bulk section; residue
%% ledger updated (interacting massless infrared closed at the rigorous-
%% model level; full-QED completion named as the frontier); PS-QM-14 and
%% the why-complex section updated; new objection O-QM-82 (the infrared
%% referee), count 81 -> 82; seven new B-list references (Bachmann-
%% Deckert-Pizzo, Buchholz, Chen-Frohlich-Pizzo, Chung, Faddeev-Kulish,
%% Frohlich, Pizzo) with paragraph Relevance notes.
%% v12.9 (July 15, 2026): cohort touch C-2 --- completed-tense citations
%% of the satellite preprint Arisaka-GG-P4-Twistor added at the RB-9
%% statement and in the twistor-referee objection response; new A-list
%% bibitem ArisakaP4Twistor; A-group header widened to "cohort and its
%% satellite"; PaperPFour macro provided locally.
% v12.8 (July 15, 2026): R11/R12 graduated into the real-bulk section
%   as RB-8 (interacting completion, conditional on mass gap +
%   asymptotic completeness, via Haag-Ruelle) and RB-9 (the holomorphy
%   bridge: positive frequency = Hardy holomorphy = PT+/PT- division;
%   twistor holomorphy as the conformal packaging of AX-1 + AX-3
%   positivity). Section-113 owes-sentence, PS-QM-14 entry, table row
%   15, and the assembly ledger updated (interaction conditionally
%   closed; residue renamed to the interacting massless infrared;
%   stationary non-stationarity discharged incl. black holes; Kerr +
%   collapse residues named). NEW objection O-QM-81 (the twistor
%   referee); count eighty -> eighty-one. NEW B-list references
%   PenroseMacCallum1973, EastwoodPenroseWells1981 (both cited,
%   first-citation links). No numbers, figures, or Parts changed.
% v12.7 (July 15, 2026): real-bulk section extended by RB-6 (discrete
%   symmetries as orientation bookkeeping; series entry R8) and RB-7
%   (the dial-free door; series entry R10), with completed-tense
%   pointers to the theorem forms in A9 (T-SM-46.1/46.2, T-SM-44.1)
%   and A10 (T-Cosmos-18). Makes the companions' 'entry R10 of the
%   real-bulk series' citations literally accurate. No other change.
% v12.6 (July 15, 2026): the real-bulk / why-complex-numbers upgrade
%   (author-approved package, complex-variables thread reports 3-7).
%   Part III phase-fibre paragraph: superselection now stated where the
%   global-phase claim stood; complexity claim scoped to the gauge phase;
%   matching consistency fix in the principal-bundle paragraph. TA-1:
%   two honesty clauses (structure vs scalars; GNS scope note). NEW
%   Discussions sections: 'Why complex numbers?' (sec:disc-why-complex)
%   and 'The real-bulk statements' (sec:disc-real-bulk, statements
%   RB-1..RB-5 at discussion level, quasi-free regime, honest ledger).
%   Classical-open-problems: row 15 + PS-QM-14 subsection; status
%   summary and Part-end close reconciled to fifteen problems -- this
%   also corrects a pre-existing stale count (the table has ELEVEN
%   Resolved rows; the old text said twelve; verified programmatically).
%   Rival-completions: 'Penrose's other question' cross-reference. NEW
%   objections section 'The complex-number and real-bulk referees'
%   (O-QM-75..80); closing-note count 74 -> 80. NEW B-list entries:
%   AshtekarMagnon1975, Kay1978, Penrose2004, Renou2021 (all cited,
%   first-citation links included). No Part added; no theorem roster
%   change; no figure change; science frozen except the flagged count.
% v12.5 (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.
% 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.
% ===== A8-QM v12.2 (2026-06-27): bib reconciliation -- de-contaminate GG-self cohort-relevance prose (GG's own 6D-bulk
% description + finite-profile wording) to drop the contaminated qualifier; external
% Randall-Sundrum 'warped extra dimensions' bibitem untouched. No physics/body change. =====
% ===== A8-QM v12.1 (2026-06-26): cohort Wave-2 de-contamination (protect-list aware). =====
% Shed GG's own 'warped/holographic' self-description (warped 6D bulk->6D bulk; warp factor/depth->
% scale; holographic projection/dictionary/image/shadow/geometrization/registration->bulk-to-boundary
% projection / bulk--boundary correspondence / boundary image / boundary shadow / geometrization /
% boundary registration map; holographic-RG->renormalization-group). De-contaminated reader-facing
% keyword metadata. ATP touch: bioenergetic->energy--information; Q_B/Arisaka-equation structure KEPT.
% KEPT all external AdS/CFT comparison, disclaimers, citations (Maldacena/Randall-Sundrum/Wilson),
% holographic-renormalisation, A7 'Holographic Attractor Lattice'/QiHAL, RS glossary, labels/refs.
% Body+keywords only; code comments preserved. No physics/equations/numbers/labels changed.
\pdfoutput=1
% ------------------------------------------------------------
% v15.13, August 19, 2026.  The shared bulk showcase plate.
%%   ADOPTED, at the author's direction: one full-page showcase
%%   plate carried BYTE-IDENTICALLY by A2, A5, A8, A9, A10 and
%%   P1 -- the cohort-map precedent.  It is the integration of
%%   three existing figures: A1 Figure 1 supplies the logic
%%   (two necessities, AX-2 building the fiber, AX-3 projecting
%%   to the registered world), A10 Figure 13 supplies the
%%   tapering throat and the projection arrow, and A2 Figure 8
%%   supplies the calibrated scale axis.  The axis now carries
%%   eight ticks with mu = E_GUT e^-r beside each, E_GUT =
%%   E_Pl e^-2pi at r = 0, E_Pl declared OFF the diagram at
%%   r = -2pi, the lock at r = 29.41 and the OGN-5 depth at
%%   r = K = 35.  Every number was computed, not carried over.
%%   THE OFFSET IS EXACT: K is counted from the bulk anchor
%%   E_Pl/2 and the axis from E_GUT, so the two differ by
%%   ln(E_Pl/2 / E_GUT) = 2pi - ln 2 = 5.590 and
%%   r_lock = K - 2pi + ln 2 = 29.410.  That relation is what
%%   reconciles A2's "lock at r = K" with A10's "lock crossing
%%   at r = 29.4"; both name 3.85 TeV.
%%   The plate is pure TikZ and adds NO asset to the arXiv
%%   package.  Caption ABOVE, per the frozen float standard.
%%   The drawn taper is e^-r/13.64 -- the true e^-r slowed so
%%   the throat stays visible -- and the plate says so.
%%   NEW PROSE: the plate's labels and its caption, listed in
%%   the report for the author to review or rewrite.
%%   ALSO, a RULE 1 repair the gate had been reporting every
%%   ship: the savebox \GGscFibre carried the British spelling
%%   in its NAME -- four sites, spelling_audit ALWAYS-BRITISH 4.
%%   It never prints, and unlike a \label a savebox name has no
%%   cross-reference to break, so it is renamed \GGscFiber.
%%   Nothing on any page changes.
% ------------------------------------------------------------
% ------------------------------------------------------------
%% v15.14, August 19, 2026.  The bulk showcase plate moved earlier.
%%   AUTHOR'S PLACEMENT: the shared plate goes from showcase
%%   page 10 to page 6.  Every showcase unit is one page, so
%%   this is a pure relocation of the delimited block: cut from
%%   where it sat, pasted immediately above the unit that held
%%   the target page.  NOT ONE BYTE OF THE BLOCK CHANGES -- the
%%   rider asserts its sha256 before and after, and the six
%%   papers that carry it are re-checked for a single hash.
%%   Nothing outside the front-matter showcase run moves, and
%%   the plate is unnumbered, so no figure or table number and
%%   no cross-reference changes.
%%   NEW PROSE: none.
% ------------------------------------------------------------
%% ---- vv15.15 (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.
%% 6 showcase pages in this paper.  Layout only: not one word,
%% number, figure or table row is changed by this ship.
\documentclass[11pt,letterpaper]{article}
\usepackage{ucla-arisaka-gg-11}
\providecommand{\PaperPFour}{\textsf{GG-P4-Real}}%% v13.2: Foundations Branch member (canonical form, Cohort-Map-19 v1.4; paper renamed GG-P4-Twistor -> GG-P4-Real at the P4-Real v3.0 ship; sty defines no \PaperPFour, so this providecommand takes effect)
\providecommand{\PaperPFive}{\textsf{GG-P5-CPT}}%% v13.2: Foundations Branch member (canonical form, Cohort-Map-19 v1.4; sty defines no \PaperPFive, so this providecommand takes effect)
\usepackage{indentfirst}
\usetikzlibrary{decorations.pathreplacing} % brace decoration

\PaperSeries{GG}
\PaperNumber{Arisaka-GG-A8-QM}
\PaperStatus{Preprint}
%   fixed to canonical expansions (GDOS: "Generalized dissipative open
%   system" -> Geometric Dynamics of Open Systems; DGI: "Dynamical
%   geometric invariants" -> Dynamical Gauge Invariance; GRIP:
%   "Geometric reduction-and-identification protocol" -> Gauge-Runtime
%   Invariance Pipeline), explanatory tails kept.  QGL fixed twice
%   ("Quadratic" -> Quadruple Gauge Locking).
%   "GRIP drives the Quantization, Quantum decoherence, and Quantum
%   measurement" executed per A8-QM-GRIP-Interpretation-Proposal-v1:
%   (1) NEW section sec:grip-spine after the eleven-theorem inventory --
%       the three claims (decoherence = GRIP-In; measurement = a complete
%       GRIP event: apparatus coupling = endogenous GRIP-Out, the
%       laboratory instance of A7's QiHAL switch, registration = GRIP-In,
%       record = halt at a GA; quantization = discreteness of the GA
%       catalogue), the division-of-labour table (OGGEP quantizes the
%       description; GRIP quantizes the outcomes), the Born fence
%       (weights = TA3/Gleason; GRIP supplies actualization+permanence;
%       basin-capture = Born left as open target), the GRW contrast
%       (decoherence completed by attractor dynamics at zero new
%       parameters), the superdeterminism disarm (TA7/TA8/TA11 cited in
%       the first breath);
%   (2) Stern-Gerlach decoherence section: regime labels;
%   (3) TA6: remark "einselection = GRIP-In";
%   (4) interpretations "None of the above is GG-Theory": position
%       statement + GRW contrast;
%   (5) double-slit Part-end close: screen landing = GRIP-In capture on
%       the screen's GA lattice (interference stays OGGEP-side);
%   (6) ES thesis paragraph; keywords; programme-dictionary GRIP entry
%       updated (terminus -> GGA; the two regimes named).
%% ------------------------------------------------------------
%% v15.1 -> v15.2, August 15, 2026.  RIDER: ten spellings.
%%   `analysers' x6 and `analyser' x4 -> `analyzer(s)'.  All ten
%%   are EPR/Bell apparatus: the angle between analyzers,
%%   time-varying analyzers, analyzer settings.
%%   THE v15.0 SWEEP DID NOT MISS THESE; ITS RULE COULD NOT SEE
%%   THEM.  gg_spelling's branch [A-Za-z]{3,}lys(?:e|ed|es|ing)
%%   carries every ending of the analy- family EXCEPT the agent
%%   noun.  Found at P1-KK's gate, where the same word survived
%%   a sweep that reported zero.  The branch now reads
%%   (?:e|ed|es|ing|er|ers) and the whole cohort was re-screened.
%%   `analyses' is NOT touched -- plural of analysis, correct
%%   American English, and the standing false positive.
%%   NOTHING ELSE CHANGES.  No prose written.
%% ------------------------------------------------------------
% ------------------------------------------------------------
% v15.2 -> v15.3, August 15, 2026.  THE DUPLICATE-CONTENTS RIDER.
%   4 doubled Contents lines removed.
%   The Plain-English Part introductions were first written from a
%   template that did BOTH \section[SHORT]{LONG} and
%   \addcontentsline{toc}{section}{SHORT}.  An unstarred \section
%   already writes its own Contents line, so each of these printed
%   the same entry TWICE, one under the other.  Every migration from
%   P2-N3 onward asserts the \addcontentsline count is zero; the five
%   papers migrated before that assertion existed still carried it.
%   Cohort-wide: A8-QM 4, A9-SM 14, A10-Cosmos 14, C1-Kaxion 12,
%   C2-Hubble 9 -- fifty-three lines.
%   MEASURED, NOT ASSUMED.  A line is removed only where the manual
%   entry is preceded by an UNSTARRED \section whose short form is
%   the same text.  A starred \section needs its manual entry and is
%   left alone: A4-Euclid's four, A3-BI6's two, C3-BH's one and this
%   paper's front-matter ones are of that kind and are untouched.
%   No heading, gloss, page, label or body text changes.
%   NO PROSE WAS WRITTEN IN THIS RIDER.
% ------------------------------------------------------------
\PaperVersion{v20.0}
\PaperDate{August 21, 2026}
\renewcommand{\ASix}{\textsf{GG-A6-GRIP}}
\renewcommand{\AFullSix}{\textsf{Arisaka-GG-A6-GRIP}}
\providecommand{\PaperFullAOne}{\AFullOne}
\providecommand{\PaperFullATwo}{\AFullTwo}
\providecommand{\PaperFullAThree}{\AFullThree}
\providecommand{\PaperFullAFour}{\AFullFour}
\providecommand{\PaperFullAFive}{\AFullFive}
\providecommand{\PaperFullASix}{\AFullSix}
\providecommand{\PaperFullASeven}{\AFullSeven}
\providecommand{\PaperFullAEight}{\AFullEight}
\providecommand{\PaperFullANine}{\AFullNine}
\providecommand{\PaperFullATen}{\AFullTen}
\providecommand{\PaperFullAEleven}{\textsf{Arisaka-GG-A11-Origin}}
\providecommand{\Mc}{\mathcal{M}}
\providecommand{\Mbulk}{\mathcal{M}_{6}}
\providecommand{\ArisakaAOne}{\texttt{ArisakaAOne}}
\providecommand{\ArisakaAEleven}{\texttt{ArisakaAEleven}}
\providecommand{\ArisakaBookI}{\texttt{ArisakaBookI}}
%% === end shim ===

%% -----------------------------------------------------------------
%%  PAPER-LOCAL PROVIDECOMMAND BLOCK
%%  -----------------------------------------------------------------
%%  Every line below is \providecommand, so if a macro is already
%%  defined by ucla-arisaka-gg-7.sty (sections 13a-13c), this line
%%  is a silent no-op and the v7 definition (sans-serif \textsf{...})
%%  wins.  Only paper-unique aliases (\GGtheory, \GGsix, \GKSL,
%%  \SUthree, \SUtwo, \Uone, \Uoneb, \state, \dual, \ket, \bra,
%%  \braket, \ketbra, \avg, \up, \dn, \iden, \Trenv, \Trapp, \TrQi,
%%  \Hilb, \Lind, \dens, \densbdy, \densbulk, \POSOGN, \AdSCFT,
%%  \MGUT) actually take effect here.  The body-text sweep that
%%  aligns \GGtheory -> \GGTheory, \GGsix -> \GG, \GKSL -> \GKLS
%%  roadmap above.  None of these definitions defines a sans-serif
%%  paper-code macro (\AOne ... \PaperFullCOne); those live entirely
%%  in section 13d of the v7 style file (per L48, L49, L52).
%% -----------------------------------------------------------------
\providecommand{\GGtheory}{\textsf{GG-Theory}}
\providecommand{\Mlock}{\ensuremath{\Lambda_{\mathrm{lock}}}}
\providecommand{\MGUT}{\ensuremath{E_{\mathrm{GUT}}}}
\providecommand{\SUfive}{\ensuremath{\mathrm{SU}(5)}}
\providecommand{\SUthree}{\ensuremath{\mathrm{SU}(3)}}
\providecommand{\SUtwo}{\ensuremath{\mathrm{SU}(2)}}
\providecommand{\Uone}{\ensuremath{\mathrm{U}(1)}}
\providecommand{\Uoneb}{\ensuremath{\mathrm{U}(1)_B}}
\providecommand{\UoneQi}{\ensuremath{\mathrm{U}(1)_{\mathrm{Qi}}}}

%% Quantum-specific macros (paper-unique; v7 does not supply)
\providecommand{\state}[1]{\ensuremath{|#1\rangle}}
\providecommand{\dual}[1]{\ensuremath{\langle#1|}}
\providecommand{\ket}[1]{\ensuremath{|#1\rangle}}
\providecommand{\bra}[1]{\ensuremath{\langle#1|}}
\providecommand{\braket}[2]{\ensuremath{\langle#1|#2\rangle}}
\providecommand{\ketbra}[2]{\ensuremath{|#1\rangle\langle#2|}}
\providecommand{\avg}[1]{\ensuremath{\langle#1\rangle}}
\providecommand{\up}{\ensuremath{\uparrow}}
\providecommand{\dn}{\ensuremath{\downarrow}}
\providecommand{\iden}{\ensuremath{\mathbf{1}}}
\providecommand{\Tr}{\ensuremath{\mathrm{Tr}}}
\providecommand{\Trenv}{\ensuremath{\mathrm{Tr}_{\mathrm{env}}}}
\providecommand{\Trapp}{\ensuremath{\mathrm{Tr}_{\mathrm{app}}}}
\providecommand{\TrQi}{\ensuremath{\mathrm{Tr}_{\mathrm{Qi}}}}
\providecommand{\CPTP}{CPTP}
\providecommand{\POVM}{POVM}
\providecommand{\GKSL}{GKSL}
\providecommand{\Hilb}{\ensuremath{\mathcal{H}}}
\providecommand{\Lind}{\ensuremath{\mathcal{L}}}
\providecommand{\dens}{\ensuremath{\rho}}
\providecommand{\densbdy}{\ensuremath{\rho_\partial}}
\providecommand{\densbulk}{\ensuremath{\rho_{\mathrm{bulk}}}}

%% GDOS / GG programme macros
%%   All lines below are no-ops against v7's section 13a definitions
%%   except \POSOGN, \GGsix, \AdSCFT (paper-unique aliases).
\providecommand{\OGGEP}{\textsc{oggep}}
\providecommand{\GDOS}{\textsc{gdos}}
\providecommand{\GRIP}{\textsc{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}

%% etoc enables \tableofcontents to be called twice (Overview + Full)
%% without truncating the .toc file on the second call.
\usepackage{etoc}

%%     only "Part X" is bolded (description stays in normal weight,
%%     for readability in the Overview).  The Part display heading
%%     itself is unchanged. ---
%%     It predated the sty's optional [short-label] arg, so the 3-arg calls
%%     \GGPart[short]{title}{subtitle} mis-parsed (#1='[', #2=first letter),
%%     rendering Part titles as '[ --- I' in the Overview.  The canonical
%%     \GGPart[3][] of ucla-arisaka-gg-10.sty does exactly what this override
%%     intended (bold 'Part N' + title + description + running header). ---

%% Bold the "Figure N." label in every caption.
\captionsetup{labelfont=bf}
%% In-text bold figure citation: \figref{key} -> "Figure N" in bold.
\providecommand{\figref}[1]{\textbf{Figure~\ref{#1}}}

%% A tight LoF that uses the short [...] caption argument of each figure
%% and renders on one page.  Bold "Figure N" entries to match the rest
%% of the document's figure-citation style.
\makeatletter
\renewcommand{\l@figure}{\@dottedtocline{1}{1.0em}{2.6em}}%
%%     (100+) and 3.2-digit subsection numbers (107.10, 107.11, ...).
%%     Default article-class numwidths (2.3em / 3.2em / 4.1em) overflow
%%     once the counter rolls past 99 / 99.9; new widths add ~1.0em.
\renewcommand{\l@section}{\@dottedtocline{1}{1.5em}{2.75em}}%
\renewcommand{\l@subsection}{\@dottedtocline{2}{4.25em}{3.85em}}%
\renewcommand{\l@subsubsection}{\@dottedtocline{3}{8.1em}{4.75em}}%
\makeatother

%% Two-level TOC system: a Part-only "Overview of Contents" before the
%% full Table of Contents.  Uses tocdepth=-1 so only Parts show.  The
%% overview gives the reader the big picture before plunging into the
%% 13-page section-level TOC.
%% etoc provides \etocsetnexttocdepth and \localtableofcontents which
%% allows multiple TOCs in a document without truncation issues.
%% A8 keeps separate-page front matter (no page-break toggle).
\providecommand{\GFF}{\textsc{gff}}
\providecommand{\GGEP}{\textsc{ggep}}
\providecommand{\DGI}{\textsc{dgi}}
\providecommand{\GSSB}{\textsc{gssb}}
\providecommand{\GA}{\textsc{ga}}
\providecommand{\SIP}{\textsc{sip}}
\providecommand{\SeOS}{\textsc{seos}}
\providecommand{\MSMF}{\textsc{msmf}}
\providecommand{\POSOGN}{\textsc{pos-ogn}}
\providecommand{\BIsix}{\textsc{bi-6}}
\providecommand{\GGsix}{\GG}
\providecommand{\OGNfive}{\textsc{ogn-5}}
\providecommand{\AdSCFT}{\textsf{AdS/CFT}}

%% -----------------------------------------------------------------
%%  A11-Origin paper-code macros (not yet in v7 style; see A11
%%  preamble lines 1665-1666 for the matching definitions).
%%  Paper-locally provided here so Part II inheritance citations to
%%  \AEleven{} compile clean.  Should be promoted into a future
%%  ucla-arisaka-gg-8.sty bump alongside any other post-v7 paper-code
%%  additions.
%% -----------------------------------------------------------------
\providecommand{\AEleven}{\textsf{GG-A11-Origin}}
\providecommand{\AFullEleven}{\textsf{Arisaka-GG-A11-Origin}}

%% -----------------------------------------------------------------
%%  A8-QM principal-theorem macro.  The cohort \T{N} macro renders
%%  "TN" sans-serif without paper-prefix; the eleven principal
%%  theorems of \AEight{} (\TA{1}--\TA{11}) need their paper-code
%%  prefix to disambiguate from A5's \T{7.1}--\T{7.7}.  Per the
%%  as the operational restatements of A5's \T{7.1}--\T{7.7}, plus
%%  \TA{8}--\TA{11} as four net-new A8-specific theorems
%%  (Tsirelson-as-Bogomolnyi, Universal-Identity-of-Indistinguish-
%%  able-Particles, QiCON-Conformality-of-Decoherence-Rates,
%%  Boundary-No-Signalling).
%% -----------------------------------------------------------------
\providecommand{\TA}[1]{\ifmmode\mathsf{T\text{-}A8.#1}\else\textsf{T-A8.#1}\fi}

%% -----------------------------------------------------------------

%% (A8's \l@section TOC-column widths above still apply).


%% (single canonical Acknowledgments, cohort-universal).

%% ---- v13.2: one-line Part entries (cohort standard; P4-Real v3.2
%%      preamble pattern).  \GGPart renewed verbatim from
%%      ucla-arisaka-gg-11.sty except that the contents line now
%%      carries only "Part N. <title>"; the " --- <description>"
%%      tail is no longer written to the .toc, so every Part entry
%%      fits on one line in both the Overview of Contents and the
%%      Contents.  Part title pages are unchanged (full title +
%%      description).  \GGAppendixPart (Part XIV) inherits the
%%      renewal via the sty's pass-through.
\makeatletter
\renewcommand{\GGPart}[3][]{%
  \clearpage
  \refstepcounter{part}%
  \def\gg@parthdr{#1}\ifx\gg@parthdr\@empty\def\gg@parthdr{#2}\fi
  \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


%% --- Course-apparatus boxes (v13.9; house style of the cohort course
%% papers). Defined in the preamble because the teaching Parts precede
%% the appendix Part in this paper. ---
\newcommand{\GGteachbox}[1]{\begin{center}\fbox{\parbox{0.92\textwidth}{\small\textbf{What this appendix teaches.} #1}}\end{center}}
\newcommand{\GGteachboxpart}[1]{\begin{center}\fbox{\parbox{0.92\textwidth}{\small\textbf{What this Part teaches.} #1}}\end{center}}
\newcommand{\GGreadalong}[1]{\par\noindent\emph{Read alongside:} #1\par}
\newcommand{\GGtakeaway}[1]{\begin{center}\fbox{\parbox{0.92\textwidth}{\small\textbf{Take-away.} #1}}\end{center}}
\begin{document}
% ---- v13.20 (August 2, 2026) ----
% New subsection 'The two fibres: why measurement has an arrow'
% at the end of the GRIP-spine section: holonomy requires closed
% loops and the interval has none; unitary QM alone can never
% produce an outcome (the circle has no arrow); von Neumann's
% projection postulate as the interval smuggled in by hand;
% decoherence as circle-side physics; collapse-as-conditioning
% given its geometric root; outcomes selected the way every
% symmetry is broken (deterministic reading); the arrow of
% measurement = the arrow of time, one semigroup (a measurement
% is a small cosmology). Full treatments credited to
% GG-W1-Symmetry v0.56 and GG-P5-CPT v3.48 Part IX. Part of the
% two-fibre deployment campaign.

\providecommand{\GGMRshort}{three theorems --- the preferred basis (\TA{6}), the single definite outcome (\TA{1}, \TA{4}, with the \GRIP{} actualization of \ASix{}), and the Born form (\TA{3}, \emph{Resolved}) --- plus the basin-capture $=$ Born residue, closed at the relaxation level (the R16 lemmas with the R16$'$ \GDOS{} relaxation; the Constitutional Born Theorem conditional on the four named premises (P1)--(P4) of \S\ref{sec:disc-open}), whose single remaining analytic residue is the field-grade representation step}
\providecommand{\GGtypicality}{the GG typicality target --- that the \GRIP{} basin-capture measure of the deterministic descent coincides with the Born measure, the six-dimensional analogue of Bohmian quantum equilibrium (status of record: closed at the relaxation level, one analytic residue remaining; \S\ref{sec:disc-open})}


%% top-of-page content is the \GGTitlePage flushright block.
\thispagestyle{empty}
\GGTitlePage%
 {Quantum Mechanics Derived from \GGTheory{}\\[2pt]
 as Projection from 6D Bulk to 4D Boundary}%
 {}%
 {}

\begin{abstract}
\noindent
For a century, quantum mechanics has been the most accurately tested theory ever written --- and the least understood. It predicts the electron's magnetism to one part in ten trillion, yet no one can quite say why a measurement yields a single definite result, why outcomes appear random, or why two particles far apart are correlated more tightly than any classical theory allows.

This paper offers one reading of all three. The deeper description of Nature, we propose, is six-dimensional: ordinary spacetime plus two \emph{non-spatial} geometric directions --- a \emph{scale} dimension, carrying the resolution at which a question is asked, and a \emph{phase} dimension, carrying the gauge phase that locks matter to force. In that fuller geometry the underlying state is deterministic and constitutionally real, as Einstein hoped --- with no complex phase for quantum chance to be written in; it is not, as Bell's theorem requires of any such theory, local. What a four-dimensional observer sees is its projection; every measurement is unavoidably an average over the two hidden directions --- the averaging a century of physics has called collapse.

Read this way, the century-old measurement problem becomes three precise questions, each answered by a theorem, and one further question --- why outcomes fall in exactly the proportions Born's rule predicts --- whose dynamical realization is closed at the relaxation level, as a stated conditional theorem, with one named analytic step remaining. We claim no new laboratory prediction where quantum mechanics already succeeds; we offer a picture in which its deepest strangeness is the shape of a shadow, and the dice --- rolled in an imaginary unit made by the same projection --- are thrown in two directions our senses can never turn to face.
\end{abstract}


\begin{ExecutiveSummary}
Quantum mechanics is empirically perfect and philosophically incomplete.
Born's 1926 probability rule predicts the magnetic moment of the
electron to one part in ten trillion --- the most accurate prediction
in the history of science. Yet a century later, four foundational
questions remain unanswered:
\begin{itemize}[leftmargin=2em,itemsep=0.15em]
\item \textbf{The measurement problem.} When does the wave equation
stop applying, and a single definite outcome appear?
\item \textbf{The preferred-basis problem.} Why does the world settle
into the familiar position basis of everyday experience, rather than
some other basis?
\item \textbf{The Born-rule problem.} Where does the rule of squared
amplitudes come from?
\item \textbf{The non-locality problem.} How can two particles
separated by light-years exhibit correlations stronger than any
classical theory permits?
\end{itemize}

\paragraph{The intuition in plain terms.} A useful first picture is
the relationship between a three-dimensional object and the
two-dimensional shadow it casts on a wall. The shadow has structure ---
sharp or blurred, shifting as the object moves, tracing the object's
outline --- but the shadow is
not the object. Standard quantum mechanics is, in this picture, the
shadow; the cohort framework proposes an
underlying object, living in six geometric directions, of which our
four-dimensional world (three of space plus one of time) is the
shadow. The famous puzzles of quantum mechanics --- measurement,
randomness, distant correlations --- are features of the shadow, not
of the object.

\paragraph{The proposal.} This paper inverts the conventional logic.
Instead of treating quantum mechanics as the bottom layer of physics,
we treat it as a \emph{derived} theory --- the operational shadow,
seen by a four-dimensional observer, of a deterministic, unitary,
and local theory living in six dimensions. Crucially, the
two dimensions beyond spacetime are \emph{not} extensions of space;
they are geometric directions of the deeper theory, one carrying the
scale at which physics happens, the other the phase that locks matter
to gauge. A four-dimensional
observer measuring any quantity is unavoidably averaging over those
two directions, and that averaging creates the appearance of
randomness, decoherence, and entanglement.

\paragraph{Eleven principal theorems.} Eleven theorems establish
this picture rigorously.
\begin{itemize}[leftmargin=2em,itemsep=0.15em]
\item \textbf{Seven inherited results} restate the textbook
postulates of quantum mechanics --- the Hilbert space, the
Schr\"odinger equation, the Born rule, the projection rule,
decoherence, the selection of the position basis, and the violation
of Bell's inequality --- as derived consequences of the
six-to-four-dimensional projection.
\item \textbf{Four new results} extend the inherited deck: the maximum quantum violation of Bell's inequality is an \emph{extremal} saturation --- the singlet maximizes the correlation, so the ceiling is reached stably rather than by coincidence; the identity of all electrons in the universe follows from a single geometric end-state of the bulk; the rate at which a quantum system loses coherence is approximately governed by a conformal-invariance principle; and no-signaling follows from a factoring property of the boundary state.
\end{itemize}

\paragraph{Five worked examples.} The five standard laboratory systems are treated rigorously. In the 1922 \emph{Stern--Gerlach experiment} the two spots are two discrete holonomy classes of the bulk gauge bundle, with the Born probabilities $\cos^2(\theta/2)$ and $\sin^2(\theta/2)$ the population ratios of the two classes in the source. In the \emph{double-slit experiment} each photon has a single-valued bulk trajectory, and the interference pattern builds up one photon at a time from the uniform distribution of phases across the ensemble. In the \emph{EPR/Bell test}, in its 2022-Nobel-recognized form, the entangled pair is one non-separable joint state in the six-dimensional bulk, so its correlated shadows share a common bulk source rather than signaling across the gap, and the maximum violation (Tsirelson's bound, $2\sqrt 2$) is the operator-algebra ceiling, saturated by the singlet. \emph{Schr\"odinger's cat} is resolved at the Geiger counter, where the projection completes the moment the detector clicks: the cat's fate is set well before any human opens the box, and no live-and-dead superposition ever existed. And the \emph{einselection of the classical pointer basis} is forced by bulk locality, not chosen by phenomenology.

\paragraph{The paradox sweep.} A dedicated sweep handles the eight remaining quantum-foundations puzzles, and in each the diagnosis is the same: the apparent paradox is the structural signature of asking a four-dimensional classical question of a six-dimensional quantum substrate whose projection to the boundary is non-invertible. Wheeler's delayed choice selects the projection, not the past; the quantum eraser reorganizes post-selection while the bulk phase is preserved; the Zeno effect is the projection axiom made frequent; weak values are four-dimensional shadow averages under pre- and post-selection; Kochen--Specker contextuality is four-dimensional yet six-dimensionally non-contextual; teleportation rides the shared bulk phase, with the classical channel completing it; Wigner's friend and Frauchiger--Renner resolve once the friend's macroscopic apparatus is where the projection completes; and dense coding and key distribution are those same bulk correlations put to work. After the sweep, no puzzle on the standard list remains separate and unaddressed.

\paragraph{Seven interpretations completed, not replaced.} Copenhagen, Many-Worlds, de Broglie--Bohm, GRW, QBism, relational quantum mechanics, and decoherent histories appear, on this reading, as seven partial views of one underlying theory: each correctly identified a feature of the boundary projection and then elevated it to the whole ontology. The framework does not pick a winner; it supplies the deeper substrate they were collectively groping toward, and each correct feature returns as a theorem.

\paragraph{The empirical horizon.} The framework is falsifiable
through eight specific kill criteria, each targeting a structural
element of the bulk side. Eight forward tests probe Tsirelson
saturation, the quadratic decoherence-rate scaling with system size,
$1/f$ noise universality, position-basis universality at macroscopic
scales, einselection-rate scaling, Bell-pair and teleportation
fidelity at long distance, and high-energy
corrections at the Kaluza--Klein scale $\Mlock \approx 3.85$~TeV.
Six of the eight are consistency audits --- they reproduce known
quantum mechanics; one is a genuine forward prediction (the platform-independent
einselection-rate test); and one is the framework's single live,
pre-registered, two-sided empirical bet --- the $\Mlock$ correction
above, within HL-LHC Run-4 reach. The empirical horizon 2026--2035 will pass the framework
through this eight-test gate.

\paragraph{The bottom line.} The paper offers a careful resolution of a century-old debate. The measurement problem, stated precisely, is three questions --- why a single definite outcome, why this basis, why Born's weights --- and the framework answers each with a theorem; the fourth --- whether the proportions in which outcomes actually fall reproduce Born's rule exactly --- is closed at the relaxation level, as a Constitutional Born Theorem conditional on four named premises, with the field-grade representation step the single remaining analytic residue. With that single honest residue marked, the old standoff dissolves: \emph{Einstein was right about determinism and realism, in the six-dimensional bulk; Bohr was right about the operational completeness of the four-dimensional description. Both describe the same Nature, seen at two levels of one geometric hierarchy --- with locality the price the bulk pays to keep determinism.} And the reconciliation is stronger than a truce: the bulk is deterministic \emph{in the classical register} --- constitutionally real (candidate grade), a symplectic Hamiltonian flow on real data, with no complex phase in which a superposition of alternatives could be written --- and the boundary's probabilities and its imaginary unit are manufactured together, by the one projection every observer must look through. This paper is the eighth of eleven in the Arisaka-GG series, closing the trio that derives quantum mechanics, the Standard Model, and cosmology from a single constitutional source.

The mechanism behind all of this now has one name. The runtime the series calls \GRIP{} --- the engine that stabilizes electrons and ends every relaxation at a ground state --- drives the quantization, the decoherence, and the measurement alike: the projection makes the boundary description quantum, and the runtime makes the quantum outcomes classical, with observables taking discrete values because the descent can only halt at one of a discrete catalog of attractors. This thesis organizes the eleven theorems of Part~IV.

None of this requires quantum mechanics to be wrong where it has been right. It asks only that its hundred-year strangeness be read as the geometry of a projection --- that the randomness lived not in the world but in where we stand, and that the same geometry which hides an outcome is the one that makes every electron in the universe exactly alike. What follows lays the science out plainly, marks at every step where physics ends and interpretation begins, and leaves the last step to the reader. God does not play dice; the dice --- and the imaginary unit they are rolled in --- are manufactured together at the boundary, by the one-way dictionary through which any observer must look.
\end{ExecutiveSummary}

\GGKeywords{GRIP-In, GRIP-Out, Geometric Attractor (GA), Ground Geometric Attractor (GGA), Born rule, Bell
inequality, CHSH, EPR paradox, quantum entanglement,
Schr\"odinger's cat, double-slit experiment, Stern-Gerlach
experiment, quantum decoherence, Lindblad equation,
many-worlds, Gleason's theorem, qubit, measurement problem,
hidden variables, bulk-to-boundary projection, \GGsix{},
\OGGEP{}, \GDOS{}.}

%% ============ FRONT-MATTER SHOWCASE, v15.4 ==================
%% Three pages after the Keywords, before the Overview of Contents.
%%   p1  Figure 4,  the constitutional bridge -- what is claimed
%%   p2  Figure 16, the paradox sweep         -- what it buys
%%   p3  Table 17,  the eight kill criteria   -- what would end it
%% The order is claim / PAYOFF / exposure, not the claim / mechanism
%% / exposure of A5-A7, because A8's payload IS one plate: eight
%% century-old paradoxes dissolved by one mechanism, with the
%% licensing theorem named in every row.
%% Figures 1 and 2 are the shared cohort plates and are not used.
%% THE PLATES KEEP THEIR IN-PAPER SIZE (A6's rule, from what A5
%% cost): a savebox cannot differ between its uses, so a wider
%% showcase copy is a heavier BODY plate.  \textwidth and
%% 1.001\textwidth, the in-paper factors, both.
%% NOTHING IS COPIED: both TikZ bodies and the kill-criteria tabular
%% live in saveboxes filled once here; every use site says \usebox.
%% Captions are unnumbered (\captionof*).
%% ===========================================================
%% v11.5 (August 17, 2026): THE DEFINITION IS IN THE STYLE FILE NOW.
%% The comment above is the record of why it was here -- the style
%% file is shared, and changing it meant rebuilding every arXiv
%% package.  That was done, so what is left is what this paper sets.
\renewcommand{\GGShowcaseHeadSkip}{}
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\begin{tabular}{@{}>{\raggedright\arraybackslash}p{0.150\linewidth}>{\raggedright\arraybackslash}p{0.220\linewidth}>{\raggedright\arraybackslash}p{0.330\linewidth}>{\raggedright\arraybackslash}p{0.170\linewidth}@{}}
\toprule
\textbf{Paradox} & \textbf{Standard reading (4D)} & \textbf{6D-bulk resolution} & \textbf{Theorem} \\
\midrule
Wheeler delayed choice & Retrocausal wave-vs-particle choice & Bulk worldline fixed; late 4D choice selects projection only & \TA{4} \\
\addlinespace[2pt]
Scully quantum eraser & Apparent causality violation & Bulk phase preserved; post-selection recovers 4D shadow & \TA{5}, \TA{7} \\
\addlinespace[2pt]
Misra--Sudarshan Zeno & Measurement freezes evolution & Frequent L\"uders projections dominate boundary record & \TA{4} \\
\addlinespace[2pt]
Weak measurement & Out-of-spectrum weak values & 4D shadow of bulk-worldline expectation under post-selection & \TA{3}, \TA{4} \\
\addlinespace[2pt]
Kochen--Specker & 4D-non-contextual hidden variables impossible & Bulk holonomy $\Theta_H$ is 4D-contextual, 6D-non-contextual & \TA{6}, \TA{9} \\
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Quantum teleportation & State transported without being measured & Shared $S^1_\theta$ phase carries state; classical bits align Pauli & \TA{7}, \TA{9}, \TA{11} \\
\addlinespace[2pt]
Wigner's friend / FR & Friend observes; Wigner re-superposes & Projection completes at friend's apparatus; Wigner is redundant & \TA{2}, \TA{4}, \TA{5} \\
\addlinespace[2pt]
Dense coding / QKD & 2 bits via 1 qubit; eavesdrop detectable & Bulk holonomy carries 2nd bit; Tsirelson check catches Eve & \TA{7}, \TA{8}, \TA{11} \\
\addlinespace[2pt]
\bottomrule
\addlinespace[3pt]
\multicolumn{4}{@{}p{0.96\linewidth}@{}}{\footnotesize\itshape Eight paradoxes, one resolution mechanism: each is a 4D-classical question of a 6D-quantum substrate whose projection to the boundary is non-invertible. The bulk evolves unitarily; the projection completes at any sufficiently macroscopic 4D detector; the paradoxes are dissolved structurally.} \\
\end{tabular}}

\sbox{\GGscKill}{\setstretch{1}\renewcommand{\arraystretch}{1.2}\small\begin{tabular}{@{}llp{5.20cm}p{4.20cm}@{}}
\toprule
\textbf{Label} & \textbf{Observable} & \textbf{Deviation
threshold} & \textbf{Timeline for test}\\
\midrule
F-QM-1 & Born rule precision &
Systematic dev. at $10^{-15}$ relative level &
Penning-trap $a_e$ at $10^{-14}$ (2028--2030) \\
F-QM-2 & Compact-dim direct evidence &
Failure to detect $S^1_\theta$ at $R_\theta\sim 5\times 10^{-20}$ m
(or finding it non-compact) &
LHC high-lumi, tabletop Cavendish (2030s) \\
F-QM-3 & Super-Tsirelson CHSH &
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experiment &
Ongoing \\
F-QM-4 & Superluminal signaling &
Any detection of entanglement-mediated classical signaling &
Longstanding null \\
F-QM-5 & MWI-specific branches &
Direct detection of parallel branches (anomalous revival
interference) &
Not currently accessible \\
F-QM-6 & Non-GKSL open evolution &
Markovian semigroup that is \CPTP{} but not Lindblad-form &
Mathematical impossibility, but fit failures could indicate
non-Markovian effects \\
F-QM-7 & Non-Gleason probability &
Additive probability measure $\ne \Tr(\rho\Pi)$ on dim$\ge 3$
&
Mathematical impossibility \\
F-QM-8 & Macroscopic coherence &
$10^{20}$-atom object in coherent superposition for $t\gg\tau_D$
&
Next-generation matter-wave interferometry \\
\bottomrule
\end{tabular}}
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% =====================================================================
% LEFT: SG APPARATUS
% =====================================================================
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% ---- BOTTOM: explanatory legend ----
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 Each silver atom is in one discrete holonomy class of the 6D gauge bundle. The B-field gradient \\
 discriminates the two classes by 4D force; each atom lands on one spot; the histogram of $N$ atoms \\
 reproduces the Born-rule weighting $\cos^2(\theta/2) : \sin^2(\theta/2)$ from the source population.
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 minimum size=8mm, font=\scriptsize},
 coinc/.style={rectangle, draw=black!70, fill=yellow!15, line width=0.7pt,
 minimum width=22mm, minimum height=10mm, rounded corners=2pt},
 beam/.style={line width=0.5pt, draw=black!65, dashed},
 arrow/.style={->, line width=0.6pt, draw=black!75},
]
% ---- Top half: EPR coincidence schematic ----
\node[source] (S) at (0,0) {};
\node[font=\scriptsize, anchor=north] at (S.south) {entangled\\source};

\node[analyzer] (Aa) at (-30mm, 0) {$a$};
\node[analyzer] (Bb) at (30mm, 0) {$b$};

\node[detector] (DAp) at (-50mm, 6mm) {$+$};
\node[detector] (DAm) at (-50mm, -6mm) {$-$};
\node[detector] (DBp) at (50mm, 6mm) {$+$};
\node[detector] (DBm) at (50mm, -6mm) {$-$};

\node[coinc] (C) at (0, -22mm) {coincidence \\ counter};

\draw[beam] (S.west) -- (Aa.east);
\draw[beam] (S.east) -- (Bb.west);
\draw[beam] (Aa.west) -- (DAp.east);
\draw[beam] (Aa.west) -- (DAm.east);
\draw[beam] (Bb.east) -- (DBp.west);
\draw[beam] (Bb.east) -- (DBm.west);
\draw[arrow] (DAp.south) -- (C.north west);
\draw[arrow] (DAm.south) -- (C.north west);
\draw[arrow] (DBp.south) -- (C.north east);
\draw[arrow] (DBm.south) -- (C.north east);

\node[font=\scriptsize, anchor=north, text=blue!70!black] at (Aa.south) {settings $a, a'$};
\node[font=\scriptsize, anchor=north, text=blue!70!black] at (Bb.south) {settings $b, b'$};

% ---- Bottom half: CHSH S vs offset angle plot ----
\begin{scope}[xshift=0, yshift=-58mm]
 \draw[->, line width=0.6pt] (-45mm,0) -- (45mm,0) node[right] {$\phi$ (relative angle)};
 \draw[->, line width=0.6pt] (0,-3mm) -- (0,28mm) node[above] {$|S|$};

  % Reference horizontal lines
 \draw[line width=0.55pt, draw=black!50, dashed] (-45mm, 8mm) -- (45mm, 8mm);
 \draw[line width=0.55pt, draw=red!70!black, line width=0.7pt]
 (-45mm,18mm) -- (45mm,18mm);
 \node[font=\scriptsize, anchor=west] at (45mm, 8mm) {Bell bound $|S|=2$};
 \node[font=\scriptsize, anchor=west, text=red!70!black] at (45mm,18mm)
 {Tsirelson $|S| = 2\sqrt{2}$};

  % Quantum-mechanical CHSH curve  |S| = 2 sqrt2 |cos(2 phi)|
 \draw[line width=1.0pt, draw=blue!70!black, smooth, samples=80, domain=-44:44]
 plot ({\x mm}, {18mm * abs(cos(2*\x))});

  % Mark saturation points
 \fill[red!70!black] (0,18mm) circle (0.7mm);
 \node[font=\scriptsize, anchor=south, text=red!70!black] at (0,19mm) {saturation};
 \fill[red!70!black] (-22.5mm, 0) circle (0.6mm); % at phi = pi/4 zero
 \fill[red!70!black] (22.5mm, 0) circle (0.6mm);

 \node[font=\scriptsize, anchor=west, text=blue!70!black]
 at (28mm, 24mm) {QM: $|S|=2\sqrt{2}|\!\cos(2\phi)|$};

 \node[font=\scriptsize] at (-45mm, -5mm) {$-\pi/4$};
 \node[font=\scriptsize] at (45mm, -5mm) {$+\pi/4$};
\end{scope}
\end{tikzpicture}}
% v15.9: filled at the BODY's spacing -- probed inside the figure
% environment with \typeout: \baselinestretch 1, \arraystretch 1.
\sbox{\GGscEPR}{\setstretch{1}\begin{tikzpicture}[
 font=\sffamily\footnotesize,
 >=Stealth,
 every node/.style={align=center, inner sep=2pt},
 source/.style={circle, fill=violet!50, draw=violet!75!black, inner sep=3pt},
 boundary/.style={line width=1.00pt, draw=black!70},
 bulkarea/.style={fill=blue!7, draw=blue!55!black, line width=0.30pt},
 geodesic/.style={line width=1.10pt, draw=red!75!black},
 arrow/.style={->, line width=0.70pt, draw=black!75},
 worldline/.style={line width=0.80pt, draw=black!75},
 detector/.style={rectangle, draw=blue!70!black, fill=blue!12,
 line width=0.70pt, minimum width=12mm, minimum height=10mm,
 rounded corners=1.5pt, font=\scriptsize\bfseries},
 label/.style={font=\scriptsize\itshape},
]
% =====================================================================
% LEFT PANEL: 4D Minkowski diagram of the EPR setup
% =====================================================================
\node[font=\sffamily\bfseries, anchor=south] at (0, 38mm) {(a) 4D Minkowski view};

% time axis (vertical)
\draw[->, line width=0.60pt] (0, -5mm) -- (0, 32mm) node[left] {$t$};
\draw[->, line width=0.60pt] (-32mm, 0) -- (32mm, 0) node[below] {$x$};
% source event at origin
\node[source] (Src) at (0, 0) {};
\node[label, anchor=north] at (0, -1mm) {source};
% Alice's worldline (left)
\draw[worldline] (Src) -- (-22mm, 28mm);
% Bob's worldline (right)
\draw[worldline] (Src) -- (22mm, 28mm);
% Detector events
\node[detector] (Ad) at (-22mm, 28mm) {$A$};
\node[detector] (Bd) at (22mm, 28mm) {$B$};
% spacelike line between A and B
\draw[line width=0.70pt, draw=red!70!black, dashed] (Ad.east) -- (Bd.west);
\node[label, text=red!70!black, anchor=south] at (0, 28mm) {4D spacelike};
% lightcone of source
\draw[line width=0.40pt, draw=black!40, dashed] (Src) -- (-28mm, 28mm);
\draw[line width=0.40pt, draw=black!40, dashed] (Src) -- (28mm, 28mm);
\node[label, anchor=south west, text=black!50] at (-28mm, 24mm) {light cone};
% Bell predicts |S| <= 2 in 4D
\node[font=\scriptsize, anchor=north, align=center] at (0, -5mm)
 {4D-local hidden variables \\ predict $|S| \le 2$ (Bell, 1964) \\ \emph{violated by experiment}};

% =====================================================================
% RIGHT PANEL: 6D bulk view: one joint state projects to A and B
% =====================================================================
\begin{scope}[xshift=85mm]
\node[font=\sffamily\bfseries, anchor=south] at (0, 38mm) {(b) 6D bulk view};

% UV boundary (4D boundary, top)
\draw[boundary] (-32mm, 26mm) -- (32mm, 26mm);
\node[label, anchor=east] at (-32mm, 26mm) {UV boundary ($r=0$)};
% IR boundary (deep bulk, bottom)
\draw[boundary] (-32mm, -2mm) -- (32mm, -2mm);
\node[label, anchor=east] at (-32mm, -2mm) {IR (deep $r$)};
% Bulk region
\fill[bulkarea] (-32mm, -2mm) rectangle (32mm, 26mm);
% Radial coordinate axis
\draw[->, line width=0.50pt, draw=black!60] (-36mm, -2mm) -- (-36mm, 26mm)
 node[above, font=\scriptsize] {$r$};
% Source on the boundary
\node[source] (Src2) at (0, 26mm) {};
\node[label, anchor=south] at (0, 27mm) {source};
% Alice and Bob detector events on the boundary
\node[detector] (Ad2) at (-22mm, 26mm) {$A$};
\node[detector] (Bd2) at (22mm, 26mm) {$B$};
% One non-separable joint state in the bulk, projecting up to A and B
\node[source] (Jt) at (0, 11mm) {};
\node[label, anchor=north, text=violet!70!black] at (0, 9mm) {joint state};
\draw[geodesic] (Jt) -- (Ad2.south);
\draw[geodesic] (Jt) -- (Bd2.south);
% projection along I_r (no dip; r is the projection axis)
\node[label, text=red!70!black, anchor=east] at (-9mm, 18mm) {projection};
\node[label, text=red!70!black, anchor=west] at (9mm, 18mm) {along $I_r$};

% S^1_theta inset (small circle on the right)
\draw[line width=0.50pt, draw=violet!70!black]
 (25mm, 13mm) circle (4mm);
\node[font=\scriptsize, text=violet!70!black, anchor=west] at (29mm, 13mm)
 {$S^1_\theta$};
\node[font=\scriptsize, text=violet!70!black, anchor=west] at (29mm, 9mm)
 {shared $\Theta_0$};
% Mark holonomy
\fill[violet!70!black] (25mm, 17mm) circle (0.5mm);

% Bottom annotation
\node[font=\scriptsize, anchor=north, align=center] at (0, -7mm)
 {One non-separable joint state, sharing the $S^1_\theta$ holonomy \\
 $\Theta_0$, projects along $I_r$ to the two boundary shadows \\
 $A$ and $B$. No worldline runs between $A$ and $B$.};
\end{scope}

\end{tikzpicture}}
\clearpage
%% Self-filling showcase frame (Report 462): the glue is already
%% page-level here, so only the weights change -- 1 : 2 : 2 : 1.
\null\vspace*{\stretch{1}}
\begingroup\centering
\GGShowcaseHead{Quantum Mechanics Is Not Assumed Here.\\[2pt]
It Is the Last Stage of a Four-Stage Chain}
\vspace{1.4em}
\par\vspace*{\stretch{2}}
\resizebox{\textwidth}{!}{\usebox{\GGscBridge}}
\par\medskip
\resizebox{0.66\textwidth}{!}{\usebox{\GGscFiber}}
\par\medskip
\par\vspace*{\stretch{2}}
{\small\captionof*{figure}{\textbf{What is claimed.} Read left to right: the closed unitary dynamics of the 6D geometry, compiled by \OGGEP{} into the boundary algebra of \GDOS{}, restricted to the quantum-relevant sector, and landing on the textbook Hilbert space a laboratory observer measures. The postulates of quantum mechanics are not inputs at any stage of that chain; they come out of the last one. Six \GDOS{} theorems build the boundary algebra and seven more restrict it, and every worked example in Parts~V--IX is this one chain run on a specific apparatus. In full: Figure~\ref{fig:constitutional-bridge}, p.~\pageref{fig:constitutional-bridge}. \textbf{And why the chain cannot be run backwards.} The lower plate is the projection itself. Over each boundary point sits the non-spatial fiber: a compact phase circle carrying the coherent, unitary content of quantum mechanics, and a non-compact scale interval carrying the irreversible, open content. Neither coordinate is spatial, so no boundary protocol resolves them; every registration marginalizes the fiber, the projection is non-invertible, and the boundary dynamics is necessarily open. Unitary closure is the unreachable limit in which the fiber is not coarse-grained. In full: Figure~\ref{fig:hidden-fibre-projection}, p.~\pageref{fig:hidden-fibre-projection}.}}
\par\vspace*{\stretch{1}}
\endgroup
\clearpage

%% >>>>>>>>>>>>>>>>>> GG-BULK-ONE-DIAGRAM v1.0 >>>>>>>>>>>>>>>>>>
%% Shared showcase plate, byte-identical in A2, A5, A8, A9, A10
%% and P1.  Pure TikZ: it adds no asset to the arXiv package.
%% Every macro it defines is \def'd under the \GGbd prefix and
%% scoped to the picture, so it cannot clash with a host paper.
%% Caption ABOVE, per the frozen float standard.
\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{A Six-Dimensional Bulk Whose Fifth Dimension Is Scale\\[2pt]
and Whose Sixth Is Phase}
\par\vspace*{\stretch{2}}
{\small\captionof*{figure}{\textbf{The whole program in one diagram, with the scale axis
carrying numbers.} The bulk $\cM_4\times I_r\times S^1_\theta$ is a fiber bundle: over every
point of the Lorentzian stage sit the two \emph{non-spatial} fibers, scale and phase. \AX{2}
(\GGEP) acts \emph{upward} and builds them; \AX{3} (\OGGEP) acts \emph{downward}, and
registration is the projection that erases $(r,\theta)$ --- the origin of quantum behavior.
The vertical axis is calibrated: $\EGUT=\EPl e^{-2\pi}$ at $r=0$, the \OGNfive{} depth
$r=K=35$ at the infrared boundary, and the lock at $r=29.41$, which is $K-2\pi+\ln 2$
exactly.}}
\par\medskip
\par\vspace*{\stretch{2}}
\resizebox{0.95\textwidth}{!}{%
\begin{tikzpicture}[
  font=\fontsize{11}{13.2}\selectfont,
  every node/.style={inner sep=2pt},
  box/.style={rounded corners=2pt, align=center, inner sep=6pt, line width=0.712pt},
  lbox/.style={rounded corners=2pt, align=left, inner sep=5pt, line width=0.712pt,
               font=\fontsize{8.8}{10.6}\selectfont},
  tick/.style={line width=0.712pt, black!70},
  slice/.style={draw=blue!60!black, fill=blue!16, line width=0.712pt},
  fiber/.style={draw=violet!65!black, line width=0.845pt},
  note/.style={font=\fontsize{8.8}{10.6}\selectfont, align=left},
  rnote/.style={font=\fontsize{8.8}{10.6}\selectfont, align=left, text width=4.85cm},
  sml/.style={font=\fontsize{8.8}{10.6}\selectfont},
  tny/.style={font=\fontsize{6.6}{7.9}\selectfont},
  rlab/.style={font=\sffamily\fontsize{9.6}{11.5}\selectfont},
  mulab/.style={font=\sffamily\fontsize{8.8}{10.6}\selectfont}
]
%% ---- geometry constants ------------------------------------------
\def\GGbdTop{21.0}          % y at r = 0
\def\GGbdSc{0.455000}       % cm per unit r  (15.9 cm for r = 0..35)
\def\GGbdXc{6.8}             % throat centre
\def\GGbdWz{4.05}             % drawn half-width at r = 0
\def\GGbdWk{13.64}           % drawn half-width = wz * exp(-r/13.64)
\def\GGbdXR{11.9}            % right annotation column
\def\GGbdY#1{\GGbdTop-\GGbdSc*(#1)}
\def\GGbdW#1{\GGbdWz*exp(-(#1)/\GGbdWk)}

%% ================= TOP: the two necessities ========================
\node[box, draw=black!60, fill=black!8, text width=15.8cm] (need) at (\GGbdXc,23.15)
 {\textbf{Two structures no local, causal, interacting, charged quantum theory can avoid}\\[2pt]
  a renormalization \emph{scale}\qquad and\qquad a gauge \emph{phase}};
\draw[-{Stealth[length=3.4mm,width=2.6mm]}, line width=0.956pt, black!75]
      (\GGbdXc,22.28) -- (\GGbdXc,21.42);
\node[right, note, fill=white, inner sep=2pt] at (\GGbdXc+0.30,21.88)
 {\AX{2} (\GGEP): \emph{geometrize} each necessity --- this \textbf{builds} the two fibers};

%% ================= THE THROAT =====================================
\filldraw[fill=blue!8, draw=blue!60!black, line width=0.845pt]
  plot[domain=0:35,samples=90,variable=\r] ({\GGbdXc-\GGbdW{\r}},{\GGbdY{\r}})
  -- plot[domain=35:0,samples=90,variable=\r] ({\GGbdXc+\GGbdW{\r}},{\GGbdY{\r}}) -- cycle;
\draw[draw=red!65!black,line width=0.956pt] ({\GGbdXc-\GGbdWz},{\GGbdY{0}}) -- ({\GGbdXc+\GGbdWz},{\GGbdY{0}});

\foreach \r in {0,10,20,29.41,35}{
  \pgfmathsetmacro{\w}{\GGbdWz*exp(-\r/\GGbdWk)}
  \pgfmathsetmacro{\y}{\GGbdTop-\GGbdSc*\r}
  \pgfmathsetmacro{\sk}{0.17+0.145*\w}
  \draw[slice] ({\GGbdXc-\w},\y) -- ({\GGbdXc+\w},\y) -- ({\GGbdXc+\w+\sk},{\y+0.45*\sk})
               -- ({\GGbdXc-\w+\sk},{\y+0.45*\sk}) -- cycle;
}
\node[blue!45!black, align=center] at (\GGbdXc,{\GGbdY{4.6}})
 {\textbf{the 6D bulk}\\[1pt] $\cM_4\times I_r\times S^1_\theta$};

%% ================= THE SCALE AXIS =================================
\draw[-{Stealth[length=3.2mm,width=2.4mm]}, line width=0.956pt, black!75]
      (0,{\GGbdY{-1.6}}) -- (0,{\GGbdY{38.2}});
\node[rotate=90, anchor=south, black!75] at (-4.05,{\GGbdY{17.5}})
 {$I_r$ : the \emph{scale} dimension\ \ $r=\ln(\EGUT/\mu)$};
\node[black!70, rlab, anchor=south east] at (-0.12,{\GGbdY{-2.1}}) {UV};
\node[black!70, rlab, anchor=north east] at (-0.12,{\GGbdY{38.6}}) {IR};
\node[anchor=south west, black!62, align=left,
      font=\sffamily\fontsize{7.4}{8.9}\selectfont] at (-4.30,{\GGbdY{-3.0}})
 {\textsf{boundary energy scale}\\ $\mathsf{\mu=E_{GUT}\,e^{-r}}$\ \ \textsf{[GeV]}};

\foreach \r/\lab/\mulab in {0/{0}/{$\mathsf{2.28\times10^{16}}$},
                            5/{5}/{$\mathsf{1.53\times10^{14}}$},
                            10/{10}/{$\mathsf{1.03\times10^{12}}$},
                            15/{15}/{$\mathsf{6.97\times10^{9}}$},
                            20/{20}/{$\mathsf{4.70\times10^{7}}$},
                            25/{25}/{$\mathsf{3.16\times10^{5}}$},
                            29.41/{29.41}/{$\mathsf{3.85\times10^{3}}$},
                            35/{35}/{$\mathsf{14.4}$}}{
  \draw[tick] (-0.17,{\GGbdY{\r}}) -- (0.17,{\GGbdY{\r}});
  \node[anchor=east, rlab, black!80] at (-0.30,{\GGbdY{\r}}) {$\mathsf{r=\lab}$};
  \node[anchor=east, mulab, black!62] at (-2.00,{\GGbdY{\r}}) {\mulab};
}

%% ---- the UV edge --------------------------------------------------
\node[anchor=north west, rnote, text=red!60!black] at (\GGbdXR,{\GGbdY{-1.1}})
 {\textbf{UV boundary}, $r=0$:
  $\mu=\EGUT=\EPl\,e^{-2\pi}=2.28\times10^{16}$ GeV.
  \emph{The geometry begins here.} $\EPl$ sits at
  $r=-2\pi$, off the diagram.};

%% ---- what a slice is -----------------------------------------------
\node[anchor=east, note, text=blue!45!black] at (4.85,{\GGbdY{12.6}})
 {each $\cM_4$ slice is not a surface\\
  in space --- it is the \emph{whole} 4D\\
  effective theory at the scale $\mu(r)$};
\draw[blue!45!black, line width=0.594pt] (4.95,{\GGbdY{12.6}}) -- ({\GGbdXc-\GGbdW{13.8}-0.05},{\GGbdY{13.8}});

%% ---- the phase fibre ---------------------------------------------
\coordinate (fp) at ({\GGbdXc+\GGbdW{9.6}-0.62},{\GGbdY{9.6}});
\fill (fp) circle (1.6pt);
\draw[fiber] (fp) ellipse [x radius=0.76, y radius=0.30];
\draw[fiber,-{Stealth[length=2.2mm]}] ($(fp)+(0.76,0)$)
      arc [start angle=0, end angle=310, x radius=0.76, y radius=0.30];
\draw[violet!65!black, line width=0.594pt] ($(fp)+(0.78,0.05)$) -- (\GGbdXR-0.15,{\GGbdY{9.6}});
\node[anchor=north west, rnote, text=violet!65!black] at (\GGbdXR,{\GGbdY{8.9}})
 {$S^1_\theta$ : the \emph{phase} dimension --- compact,
  unitary, one Abelian holonomy
  $\oint A_\theta\,d\theta\in U(1)$. Attached at
  \emph{every} $r$; one copy drawn.};

%% ---- the lock ------------------------------------------------------
\node[star, star points=5, star point ratio=0.45, draw=red!70!black,
      fill=red!16, minimum size=0.72cm, line width=0.712pt] (lk)
      at (\GGbdXc,{\GGbdY{29.41}}) {};
\draw[red!70!black, line width=0.594pt] (lk.east) -- (\GGbdXR-0.15,{\GGbdY{29.41}});
\node[anchor=west, rnote, text=red!70!black] at (\GGbdXR,{\GGbdY{29.41}})
 {\textbf{the lock}, at $r=29.41$:
  $\Mlock=\tfrac12\EPl\,e^{-K}=3.85$ TeV, the
  Kaluza--Klein lock cluster.};

%% ---- the IR boundary ------------------------------------------------
\draw[blue!45!black, line width=0.594pt] ({\GGbdXc+\GGbdW{35}},{\GGbdY{35}}) -- (\GGbdXR-0.15,{\GGbdY{35}});
\node[anchor=west, rnote, text=blue!45!black] at (\GGbdXR,{\GGbdY{35}})
 {\textbf{IR boundary}, $r=K=35$: the \OGNfive{} integer
  scale depth, $\mu=14.4$ GeV. \emph{The bulk ends here};
  the cosmological descent runs on to
  $r_{\mathrm{GGA}}\simeq64.5$.};

%% ---- the calibration box ---------------------------------------------
\node[lbox, draw=olive!60!black, fill=yellow!12, text width=4.35cm, anchor=north west]
      at (0.28,{\GGbdY{15.3}})
 {\textbf{Why the lock sits at $r=29.41$, not at $35$}\\[4pt]
  $K=35$ is counted from the \emph{bulk} anchor
  $\tfrac12\EPl$; this axis is anchored at $\EGUT$. They
  differ by\\[4pt]
  \hspace*{0.1cm}$\ln\!\bigl(\tfrac12\EPl/\EGUT\bigr)=2\pi-\ln 2=5.590$,\\[4pt]
  \hspace*{0.1cm}$r_{\mathrm{lock}}=K-2\pi+\ln 2=29.410$.\\[4pt]
  One energy, two readings. Exact, not fitted.};

%% ---- the honesty note about the drawing ------------------------------
\node[lbox, draw=black!50, fill=black!8, text width=4.35cm, anchor=south west]
      at (0.28,{\GGbdY{36.3}})
 {\textbf{The taper is compressed.} Drawn half-width
  $e^{-r/13.64}$: the true $e^{-r}$ slowed $13.64\times$
  so the throat stays visible. To scale, the slice at
  $r=35$ would be $6.3\times10^{-16}$ of the one at
  $r=0$.};

%% ================= OGGEP PROJECTION ==================================
\draw[-{Stealth[length=5mm,width=4mm]}, line width=2.0pt, red!70!black]
      (\GGbdXc,{\GGbdY{36.6}}) -- (\GGbdXc,{\GGbdY{40.9}});
\node[anchor=west, note, text=red!60!black] at (\GGbdXc+0.35,{\GGbdY{38.9}})
 {\AX{3} (\OGGEP): registration $=$ the projection $\pi$ that erases $(r,\theta)$};

%% ================= BOTTOM: the registered world ======================
\node[box, draw=blue!55!black, fill=blue!10, text width=15.8cm] (bdy) at (\GGbdXc,{\GGbdY{43.4}})
 {\textbf{The registered 4D world: open, and therefore quantum}\ \ (\GDOS)\\[2pt]
  superposition \& interference are the unread phase $S^1_\theta$\\[1pt]
  measurement \& irreversibility are the coarse-grained scale $I_r$};
\end{tikzpicture}}
\par\vspace*{\stretch{1}}
\end{minipage}
\clearpage
%% <<<<<<<<<<<<<<<<<< GG-BULK-ONE-DIAGRAM v1.0 <<<<<<<<<<<<<<<<<<
%% Self-filling showcase frame (Report 462): the glue is already
%% page-level here, so only the weights change -- 1 : 2 : 2 : 1.
\null\vspace*{\stretch{1}}
\begingroup\centering
\GGShowcaseHead{No Atom Ever Feels a Probability ---\\[2pt]
the Randomness Is What the Observer Cannot Read}
\vspace{1.2em}
\par\vspace*{\stretch{2}}
\resizebox{0.759\textwidth}{!}{\usebox{\GGscSGapp}}
\par\medskip
\resizebox{0.876\textwidth}{!}{\usebox{\GGscSGmech}}
\par\medskip
\par\vspace*{\stretch{2}}
{\small\captionof*{figure}{\textbf{The oldest quantum measurement, run
twice.} Above: the 1922 apparatus as everyone teaches it --- silver atoms
out of an oven, through an inhomogeneous field, into two discrete spots
--- with the Bloch-sphere reading of the input qubit and the Born-rule
populations $\cos^2(\theta/2)$ and $\sin^2(\theta/2)$ beside it. Below:
the same experiment read on the bulk. The heated source populates two
discrete holonomy classes of the gauge bundle around the phase circle, in
exactly that ratio; the field gradient pushes the two classes apart in
four-dimensional space; each atom lands on one spot, $z_+$ or $z_-$,
according to which class it was in. \emph{No single atom experiences
probability.} The randomness a laboratory reports is the four-dimensional
observer's inability to read the phase holonomy directly, and the
textbook ``pure state'' is the 4D shadow of the source ensemble. In full:
Figures~\ref{fig:stern-gerlach-apparatus} and \ref{fig:SG-6d-mechanism},
pp.~\pageref{fig:stern-gerlach-apparatus} and
\pageref{fig:SG-6d-mechanism}.}}
\par\vspace*{\stretch{1}}
\endgroup
\clearpage

%% Self-filling showcase frame (Report 462): the glue is already
%% page-level here, so only the weights change -- 1 : 2 : 2 : 1.
\null\vspace*{\stretch{1}}
\begingroup\centering
\GGShowcaseHead{Two Records, One Object ---\\[2pt]
and Nothing Travels Between Them}
\vspace{1.0em}
\par\vspace*{\stretch{2}}
\resizebox{0.660\textwidth}{!}{\usebox{\GGscBell}}
\par\medskip
\resizebox{0.839\textwidth}{!}{\usebox{\GGscEPR}}
\par\medskip
\par\vspace*{\stretch{2}}
{\small\captionof*{figure}{\textbf{What Bell measured, and what it is a
shadow of.} Above: the CHSH experiment and its arithmetic --- any
four-dimensional local hidden-variable theory is bounded by $|S| \le 2$,
quantum mechanics predicts $2\sqrt{2}|\cos 2\phi|$, and the
Nobel-recognized experiments confirm the saturation at the Tsirelson
bound. Below: what the pair is. It is \emph{one} non-separable state of
the bulk, sharing a single phase-circle holonomy reference fixed at the
source; Alice's event and Bob's event are two projections of that one
object along the scale direction, which carries no light cone, so no
worldline runs between them and nothing has to. The shared holonomy is
the operational meaning of the entanglement, and the ``non-locality'' of
1964 is the operational shadow of one bulk object projected to two
boundary records. In full: Figures~\ref{fig:epr-bell-chsh} and
\ref{fig:EPR-6d-mechanism}, pp.~\pageref{fig:epr-bell-chsh} and
\pageref{fig:EPR-6d-mechanism}.}}
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\GGShowcaseHead{Eight Century-Old Paradoxes,\\[2pt]
and the One Mechanism That Dissolves All Eight}
\vspace{1.0em}
\par\vspace*{\stretch{2}}
\usebox{\GGscParadox}
\par\medskip
\par\vspace*{\stretch{2}}
{\small\captionof*{figure}{\textbf{What it buys, and why it is one page and not eight.} Eight paradoxes of the last hundred years, each with the standard 4D reading that makes it a paradox, the 6D-bulk resolution, and the theorem that licenses that resolution. The claim of the table is the uniformity of the middle column: every one of the eight dissolves under the same mechanism, which is that the apparent contradiction is the structural signature of asking a classical question of a substrate whose projection to the boundary is non-invertible. A reading that dissolved them one at a time, by a different device each time, would be worth much less. Once this is digested the productive question is no longer which interpretation is correct, but what new operational signatures the bulk reading predicts that the 4D-only readings cannot --- which is the facing page. In full: Figure~\ref{fig:paradox-sweep-table}, p.~\pageref{fig:paradox-sweep-table}.}}
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\GGShowcaseHead{Eight Observations That Would Retire This Reading,\\[2pt]
Each With the Threshold at Which It Counts}
\vspace{1.0em}
\par\vspace*{\stretch{2}}
{\small\captionof*{table}{\textbf{What would end it.} Eight observations, each with the deviation threshold at which it would count and the timeline on which it can be tested. Any one of them, experimentally established, retires the quantum-mechanical reading of the framework. A framework that derives quantum mechanics from a deeper geometry is in permanent danger of being unfalsifiable, because quantum mechanics is correct and any reading that reproduces it inherits its successes for free; reproducing what is already known is not evidence. These eight are the part that can be wrong, and they are written so that no interpretive question has to be settled before one of them can fire. In full: Table~\ref{tab:kill_criteria}, p.~\pageref{tab:kill_criteria}.}}
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% ============ end front-matter showcase ====================

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\clearpage


%% ===================================================================
%% PART I — Quantum Mechanics on Its Own Terms
%% ===================================================================
\GGPreface

Quantum mechanics is the most successful theory in the history of science and, a hundred years on, still the most puzzling. Every transistor, every laser, every magnetic-resonance scan rests on it, and its predictions have never once failed a test. Yet its founders argued bitterly about what it \emph{means}, and the argument has never ended. Einstein could not accept that Nature plays dice; Bohr replied that we must learn to. The textbooks teach the equations and, gently, advise the student not to ask what lies beneath them.

The strangeness has many faces, and each has unsettled thoughtful people for a century. A single particle, sent toward a pair of slits, behaves as though it passed through both and interfered with itself --- yet it arrives as one dot, in one place. Two particles, born together and then carried to opposite ends of a city, give perfectly matched answers the instant they are measured, with no signal able to pass between them in time. A measurement seems to \emph{disturb} what it measures, collapsing a smooth wave of possibilities into a single hard fact --- and no equation in the theory says when, or how, that collapse occurs. Schr\"odinger imagined a cat that the rules leave both alive and dead until someone looks. These are not separate curiosities; they are one puzzle wearing different masks, and a hundred years of brilliant interpretation --- Copenhagen, many worlds, hidden variables, and the rest --- has produced no agreement on which mask is the true face.

This paper asks the question the textbooks set aside, and offers an answer that is, frankly, unexpected. It is a technical paper --- it proves theorems and works the standard experiments through in detail --- but its central idea can be held in a single sentence. Suppose the world we measure, with its three dimensions of space and one of time, is not the whole of reality but a kind of shadow cast by a slightly larger geometry: the same four dimensions, plus two more that are not directions \emph{in space} at all, but a direction of \emph{scale} --- the resolution at which a question is asked --- and a direction of \emph{phase} --- the dial that ties matter to the forces. The surprise is deliberate. Where physics has spent fifty years hunting for extra dimensions curled up too small to see, these two are not small pieces of space at all; they are the hidden axes along which every quantum measurement is forced to average.

In that larger world the state of things is definite and lawful, the way Einstein wanted it to be. The fuzziness, the randomness, the link between distant particles, the disturbance of measurement --- these are what the shadow looks like when the two extra directions are flattened out of view, as they must be for any creature, or any instrument, that lives on the four-dimensional boundary. Read this way, the century-old measurement problem stops being one impenetrable riddle and becomes three precise questions, each answered by a theorem, with a single further question left honestly open. The dice were never in the world; they were in our standpoint --- thrown in two directions our senses can never turn to face.

We have tried throughout to keep one line bright: to mark where a theorem ends and an interpretation begins, to state plainly the one question that remains genuinely open rather than paper over it, and to claim nothing for the picture that experiment does not already grant. Where the framework agrees with ordinary quantum mechanics --- which is nearly everywhere a laboratory can currently reach --- it makes no new promise; what it offers there is understanding, not a new number. Where it does stake a forward bet, at the highest energies and on the largest scales, it says so and names the test.

\emph{A note on reading.} What follows in Part~I and beyond builds the argument in full, but every Part opens with a plain-language summary, so a reader may travel as deep as they wish and stop where they like. A reader is free to take the extra geometry as literal physics or as a useful way of seeing; either way, the hope is that by the end the hundred-year strangeness of the quantum world feels a little less like a paradox and a little more like a point of view. The sharpest objections a skeptical physicist could raise are gathered and answered near the end, in the reader's own voice.

\clearpage

\GGPart[Introduction]{Introduction}{The century-old measurement problem in plain view}
\label{part:I}

\section{Why quantum mechanics is the most successful physical
theory in history, and its most mysterious}

Quantum mechanics makes predictions more precise than any other
branch of science. The electron's anomalous magnetic moment
$a_e = (g-2)/2$ is predicted by quantum electrodynamics (QED)
and measured to a relative precision of approximately $1.3
\times 10^{-13}$, which is the most precise agreement between
theory and experiment in all of physics. The hydrogen-atom Lamb
shift, the hyperfine structure of atomic cesium, the
energy-level structure of positronium, the tunneling currents in
semiconductor devices, the photoelectric effect spectrum, and
the cross sections of elementary-particle collisions are all
predicted by quantum mechanics at the parts-per-billion or
better level.

At the same time, quantum mechanics has never quite finished
settling into a mature theory. One century after Max Born's
1926 probability rule, there is no consensus on:
\begin{itemize}[leftmargin=2em,itemsep=0.1em]
\item What the quantum state $|\psi\rangle$ represents --- an
epistemic state (a degree of belief), an ontic state (a state
of the world), or something else entirely?
\item What constitutes a ``measurement,'' as opposed to an
ordinary physical interaction?
\item Why quantum outcomes are distributed according to the
Born rule $|\braket{a}{\psi}|^2$ rather than, say, uniformly
or linearly?
\item Whether there are ``hidden variables'' that would restore
determinism and locality, and if so, what they are?
\item Whether the wave function ``collapses'' (Copenhagen),
branches (Everett), is guided by non-local hidden variables
(Bohm), is a subjective belief (QBism), or something else?
\item Why the state space is \emph{complex} --- why an $i$ sits in
the Schr\"odinger equation --- when every number any instrument has
ever recorded is real?
\end{itemize}
These are the questions of the foundations of quantum
mechanics. They are the subject of this Part~I --- and, in a
single sweep, of Parts~II--XIII, where every one of the six
will dissolve as an artifact of viewing a deterministic,
unitary, six-dimensional dynamics through the four-dimensional
projection a laboratory observer is forced to use. The sixth
deserves a flag at the front door, because it is the one question
of this list that the cohort now closes at theorem grade: in the
6D bulk every physical quantity is real, and the boundary's single
imaginary unit is \emph{derived} --- the projection pairs the two
real structures of the arena (the compact phase circle and the
causal clock) into one complex unit, at derived-conditional grade
in the quasi-free regime (\citealp{ArisakaP4Real}; the full
statement catalog is \S\ref{sec:disc-real-bulk}). Part~I
catalogs the questions in their canonical historical form.
The remaining twelve Parts answer them. And because a claim of this
size should be teachable, not merely assertable, the volume is
equipped as a course: the five Worked-Example Parts, the paradox
sweep, and the five deep proof appendices each open with a
what-this-teaches box, pair themselves with the standard literature,
and close with problems and worked solutions --- fifty-six across
the paper --- so that a graduate student can work, rather than
merely read, the reformulation of the hundred-year-old theory.

\section{The birth of quantum mechanics: 1900--1925}

\subsection{Planck and the blackbody catastrophe (1900)}

On December 14, 1900, Max Planck presented to the German
Physical Society the formula that would begin the quantum
revolution
\citep{Planck1900}. Classical electromagnetism predicted the
energy spectrum of blackbody radiation to grow unboundedly at
short wavelengths --- the so-called
``ultraviolet catastrophe.''
To fit the experimental spectrum, Planck made the formal
hypothesis that radiation of frequency $\nu$ could only be
emitted and absorbed in discrete packets (``quanta'') of energy
\begin{equation}\label{eq:Planck}
E_n = n h\nu, \qquad n = 0, 1, 2, \ldots,
\end{equation}
where $h \approx 6.626 \times 10^{-34}$~J$\cdot$s became
\emph{Planck's constant} --- the fundamental dimensional scale
of all quantum phenomena. Planck himself regarded this
quantization as a mathematical artifact. Einstein would soon
show it was physical.

\subsection{Einstein's photon and the photoelectric effect (1905)}

Albert Einstein's 1905 ``miracle year'' paper on the
photoelectric effect
\citep{Einstein1905photon}
took Planck's quanta literally: light itself is composed of
discrete particles (later called photons) each carrying
energy $E = h\nu$. The kinetic energy of photoelectrons from a
metal surface illuminated by light of frequency $\nu$ follows
\begin{equation}\label{eq:photoelectric}
E_{\rm kin} = h\nu - \phi,
\end{equation}
where $\phi$ is the work function of the metal. This
prediction was confirmed experimentally by R.~A.~Millikan in
1916 \citep{Millikan1916}. Einstein's paper won the 1921
Nobel Prize and is arguably the founding document of quantum
theory as a theory of Nature (not merely of spectra).

\subsection{Bohr's 1913 atomic model}

Niels Bohr in 1913 \citep{Bohr1913} applied the quantum
hypothesis to atomic structure. His model postulated that
electrons orbit the nucleus only in discrete orbits with
quantized angular momentum $L = n\hbar$, $\hbar = h/2\pi$.
Transitions between orbits emit or absorb photons of energy
\begin{equation}\label{eq:Bohr_transition}
h\nu_{mn} = E_m - E_n.
\end{equation}
The Bohr model successfully predicted the Rydberg formula for
hydrogen spectral lines. Extensions by Arnold Sommerfeld (1916)
added relativistic corrections and elliptic orbits. But the
Bohr--Sommerfeld ``old quantum theory'' was an uneasy hybrid:
classical orbits with quantum jumps. By 1924 it was clear that
a deeper theory was needed.

\subsection{De Broglie's matter waves (1924)}

Louis de Broglie's 1924 doctoral thesis
\citep{deBroglie1924}
proposed that any particle of momentum $p$ has an associated
wavelength
\begin{equation}\label{eq:deBroglie}
\lambda = h/p.
\end{equation}
De Broglie's proposal was experimentally confirmed in 1927 by
Clinton Davisson and Lester Germer, who observed electron
diffraction from a nickel crystal
\citep{DavissonGermer1927}. The de Broglie relation is the
bridge that made the wave-particle duality quantitative. It
also suggested that the quantum state should satisfy a
\emph{wave equation}.

\subsection{Heisenberg's matrix mechanics (July 1925)}

Werner Heisenberg's 1925 Helgoland paper
\citep{Heisenberg1925}
reformulated atomic physics in terms of abstract tables of
numbers (matrices) indexed by transitions between stationary
states. For observables $q$ (position) and $p$ (momentum),
Heisenberg found the non-commuting matrix relation
\begin{equation}\label{eq:canonical_commutation}
[q,p] = qp - pq = i\hbar.
\end{equation}
Max Born and Pascual Jordan (October 1925)
\citep{BornJordan1925} and later the
three-man paper by Born, Heisenberg, and Jordan
\citep{BornHeisenbergJordan1926}
formalized matrix mechanics. Observables are represented by
self-adjoint operators; commutators encode the quantum
structure; time evolution is unitary. Heisenberg's formulation
was abstract, non-visualizable, and initially controversial, but
mathematically precise.

\subsection{Schr\"odinger's wave mechanics (January 1926)}

Six months later, Erwin Schr\"odinger
\citep{Schrodinger1926}
presented the wave equation
\begin{equation}\label{eq:Schrodinger}
i\hbar\frac{\partial\psi}{\partial t} = \hat H\psi,
\end{equation}
for a complex-valued wave function $\psi(x,t)$. For a free
particle,
$\hat H = -\hbar^2\nabla^2/2m$;
with a potential,
$\hat H = -\hbar^2\nabla^2/2m + V(x)$.
Schr\"odinger's wave equation reproduced the hydrogen-atom
spectrum exactly and provided a concrete visualizable picture
--- an oscillating complex wave filling space.

Schr\"odinger and others soon proved that matrix mechanics and
wave mechanics are mathematically equivalent: they are two
representations of the same abstract Hilbert-space structure.
Paul Dirac in 1930 \citep{Dirac1930} unified them in the
``transformation theory.''

\subsection{The gap: what does $\psi$ mean?}

Schr\"odinger initially hoped that $|\psi|^2$ represented a
charge density --- the electron would be literally spread out in
space. This hope failed for multi-electron systems (the
wave function lives in $3N$-dimensional configuration space, not
in physical 3-space). Something else had to be said about the
physical meaning of $\psi$. Enter Born.

\section{Born's rule and the probabilistic interpretation (June--July 1926)}

\subsection{Born's 1926 papers}

In June 1926, Max Born published two short papers on the quantum
theory of scattering
\citep{Born1926}. Studying $\alpha$-particle
collisions with atoms, Born could not avoid interpreting
$|\psi(x)|^2$ as a probability density:
\begin{quote}\itshape
The motion of particles follows probability laws, but the
probability itself propagates in accordance with the law of
causality.
\end{quote}
Concretely, for a quantum system in state $|\psi\rangle$, the
probability of obtaining outcome $a$ in a projective
measurement of observable $\hat A$ (with eigenvalue $a$ and
eigenvector $|a\rangle$) is
\begin{equation}\label{eq:Born_rule}
\boxed{P(a) = |\braket{a}{\psi}|^2.}
\end{equation}
This is the \emph{Born rule}. For continuous spectra,
$P(x)\,dx = |\psi(x)|^2\,dx$ is the probability density for
position. The rule extends to mixed states $\rho$ as
$P(a) = \Tr(\rho\, \ketbra{a}{a})$ and to \POVM{} measurements
$\{E_\alpha\}$ as $P(\alpha) = \Tr(\rho E_\alpha)$.

\subsection{Einstein's reaction: ``God does not play dice''}

Einstein's famous response, in a December 4, 1926 letter to
Born himself \citep{BornEinsteinLetters}:
\begin{quote}\itshape
Quantum mechanics is certainly imposing. But an inner voice
tells me that it is not yet the real thing. The theory says a
lot, but does not really bring us any closer to the secret of
the ``old one.'' I, at any rate, am convinced that \emph{He
does not throw dice}.
\end{quote}
Einstein's objection is not that the theory is wrong, but that
it is \emph{incomplete}. The intrinsic indeterminacy of
outcomes struck him as an admission of ignorance, not a
fundamental feature of Nature. Throughout his life, Einstein
held that beneath the probabilistic veneer must lie a
deterministic theory with ``hidden variables.''

Bohr's counter-position --- the Copenhagen interpretation
\citep{Bohr1935} --- was that the randomness is irreducible,
measurement contexts are objective, and observables only have
definite values in the context of specific measurements. No
hidden variables exist.

This debate was not settled in Einstein's lifetime. It was
settled in 1964 by John S.~Bell, and then experimentally in
1982 and again in 2015--2022, apparently in Bohr's favor. Part
III of the present report will argue that it was settled in
Einstein's favor \emph{at the 6D level}, with Bohr correct at
the 4D level.

\section{Dirac, von Neumann, and the mathematical structure}

\subsection{Dirac's $\langle$bra$|$ket$\rangle$ formalism}

Paul A.~M.~Dirac \citep{Dirac1930} introduced the bra-ket
notation that became the standard language of quantum mechanics.
A pure state is a ket $|\psi\rangle \in \Hilb$, where $\Hilb$ is
a complex separable Hilbert space. The inner product
$\braket{\phi}{\psi}$ is a complex number; the outer product
$|\psi\rangle\langle\phi|$ is an operator on $\Hilb$. An
observable is represented by a self-adjoint operator $\hat A
= \hat A^\dagger$. Measurements of $\hat A$ yield one of its
eigenvalues $a$; immediately after measurement the state is
the corresponding eigenvector $|a\rangle$ (``projection
postulate''). Expectation values are $\avg{\hat A} =
\bra{\psi}\hat A\ket{\psi}$.

\subsection{Von Neumann's axiomatization (1932)}

John von Neumann's 1932 monograph \emph{Mathematische
Grundlagen der Quantenmechanik}
\citep{vonNeumann1932}
gave quantum mechanics its first rigorous mathematical
formulation. The postulates are:
\begin{enumerate}[leftmargin=2em,itemsep=0.12em]
\item \textbf{State postulate.} Every physical system is
associated with a complex separable Hilbert space $\Hilb$. The
state is a unit vector $|\psi\rangle \in \Hilb$ (or, more
generally, a density operator $\rho$ on $\Hilb$ with $\rho \ge 0$
and $\Tr\rho = 1$).
\item \textbf{Observable postulate.} Every physical observable
is represented by a self-adjoint operator $\hat A$ on $\Hilb$,
with spectral decomposition $\hat A = \sum_a a\,\Pi_a$, where
$\Pi_a$ are orthogonal projectors.
\item \textbf{Born rule postulate.} The probability of
obtaining outcome $a$ is $P(a) = \Tr(\rho \Pi_a)$.
\item \textbf{Evolution postulate.} Between measurements the
state evolves unitarily by the Schr\"odinger equation
$i\hbar \, d|\psi\rangle/dt = \hat H|\psi\rangle$, where $\hat H$
is the Hamiltonian.
\item \textbf{Projection postulate.} After a measurement
yielding outcome $a$, the state is updated to
$|\psi\rangle \to \Pi_a|\psi\rangle / \|\Pi_a|\psi\rangle\|$ (or
for mixed states, $\rho \to \Pi_a\rho\Pi_a / \Tr(\Pi_a\rho)$).
\end{enumerate}
Von Neumann recognized immediately that postulates 4 and 5 are
\emph{not} compatible. Unitary evolution is reversible;
projective measurement is irreversible. Which one applies at
any given moment? Von Neumann called this the
\emph{Heisenberg cut}: it is a free parameter of the theory
where the line between ``system'' and ``apparatus'' is drawn.
\emph{Any} cut location works mathematically --- but there is no
principle within the theory to determine where the cut actually
lies.

This is the \emph{measurement problem}. It has been open for
93 years.

\subsection{The Stone theorem: unitary evolution from the postulate}

Stone's theorem (1932) \citep{Stone1932} proved that every
strongly continuous one-parameter unitary group $\{U_t\}$
$U_{t+s} = U_tU_s$, $U_0 = \iden$, $\lim_{t\to 0}U_t\psi
=\psi$ has the form $U_t = e^{-iHt/\hbar}$ for a unique
self-adjoint generator $H$. This is the mathematical
foundation of the Schr\"odinger equation. We will return to
this theorem in Part II, where \OGGEP{} extends it to the
\CPTP{} case and produces the Lindblad equation as the unique
Markovian generalization.

\section{The double-slit experiment: the heart of quantum mystery}

\subsection{The phenomenology}

The double-slit experiment is, as Richard Feynman put it,
``the only mystery'' in quantum mechanics
\citep{FeynmanLectures}. A source emits particles (photons,
electrons, neutrons, atoms, molecules up to $\sim$800-atom
fullerenes \citep{Arndt2003_C60}) one at a time. They pass
through a screen with two narrow slits and strike a detector
array on the far side. The empirical observations:
\begin{enumerate}[leftmargin=2em,itemsep=0.1em]
\item When only one slit is open, one sees a diffraction
pattern (a broad maximum behind the open slit, with small
secondary fringes).
\item When both slits are open and no attempt is made to
determine which slit each particle traverses, one sees an
\emph{interference} pattern --- alternating bright and dark
fringes whose spacing is governed by the de Broglie wavelength.
\item When a detector at one of the slits determines which slit
each particle went through, the interference pattern
\emph{disappears}: the detector output is the sum of two
one-slit diffraction patterns.
\end{enumerate}
The interference pattern is the signature of quantum coherence.
The detection at the slit destroys the coherence
(``decoherence'') by entangling the which-path information with
a macroscopic degree of freedom that is then traced over.

\subsection{The quantum calculation}

The probability amplitude for detection at position $x$ on the
screen is
\begin{equation}\label{eq:doubleslit_amp}
\psi(x) = \psi_1(x) + \psi_2(x),
\end{equation}
the sum of the amplitudes for going through slit 1 and slit 2
respectively. By Born's rule,
\begin{equation}\label{eq:doubleslit_prob}
P(x) = |\psi_1(x) + \psi_2(x)|^2 = |\psi_1|^2 + |\psi_2|^2 +
2\mathrm{Re}(\psi_1^*\psi_2).
\end{equation}
The interference term $2\mathrm{Re}(\psi_1^*\psi_2)$ is what
distinguishes the coherent (interference) and incoherent (sum
of probabilities) regimes. When the which-path information is
entangled with the environment, the interference term is
suppressed by the overlap of the environment states:
\begin{equation}\label{eq:doubleslit_decoherence}
P(x) \approx |\psi_1|^2 + |\psi_2|^2
+ 2\mathrm{Re}(\psi_1^*\psi_2) \cdot
\braket{E_2}{E_1}.
\end{equation}
If $\braket{E_2}{E_1} \to 0$ (environments that can distinguish
path 1 from path 2), the interference disappears.

\subsection{Which-way and delayed-choice variants}

In Wheeler's delayed-choice experiment \citep{Wheeler1983},
the decision to add or remove a which-path detector is made
\emph{after} the particle has passed the slits but before it
is detected. The quantum prediction is that the pattern on
the screen depends on what is done downstream, not on what the
particle ``did'' at the slit. Scully's quantum-eraser
experiment \citep{Scully_eraser_1991} adds a further twist: the
which-path information is stored in a second particle, and is
later erased. After erasure, the coherence can be
\emph{recovered} by suitable post-selection.

These experiments prove that quantum mechanics does not admit a
``particle trajectory'' interpretation, nor a ``hidden
which-path'' interpretation. At the level of ordinary 4D
space-time, quantum mechanics is genuinely non-classical.

\section{The Stern-Gerlach experiment and the discovery of spin}

\subsection{The 1922 experiment}

Otto Stern and Walther Gerlach in 1921--1922
\citep{SternGerlach1922} fired a beam of silver atoms through
a spatially inhomogeneous magnetic field. Silver has one
unpaired electron in its outermost shell, giving it a magnetic
moment proportional to the electron's total angular momentum.
The classical prediction was a continuous distribution of
deflections; the quantum prediction (using the old quantum
theory of Sommerfeld) was a discrete set. What Stern and
Gerlach saw: \emph{two} spots, corresponding to
$L_z = \pm \hbar/2$. This was the first direct evidence that
the electron has intrinsic spin $s = 1/2$, discovered later
theoretically by Uhlenbeck and Goudsmit (1925)
\citep{UhlenbeckGoudsmit1925}.

\subsection{Spin measurement as the paradigmatic quantum measurement}

The Stern-Gerlach apparatus is the prototypical quantum
measurement. The input is a spin-1/2 state
$|\psi\rangle = \alpha|{\uparrow_z}\rangle + \beta|{\downarrow_z}\rangle$
with $|\alpha|^2 + |\beta|^2 = 1$. The apparatus deflects
$|\uparrow_z\rangle$ up and $|\downarrow_z\rangle$ down. The output beams
are spatially separated (the ``pointer position'' now stores the
spin information). The Born rule says:
\begin{equation}\label{eq:SG_probabilities}
P(\uparrow) = |\alpha|^2,\qquad P(\downarrow) = |\beta|^2.
\end{equation}

For consecutive Stern-Gerlach measurements in different
directions, one finds the non-commuting structure:
measuring $S_z$ then $S_x$ then $S_z$ does not return the
original $S_z$ value with certainty; the second $S_z$
measurement is 50\%/50\% because the intervening $S_x$
measurement has erased the $S_z$ information.

\subsection{Back-action and the measurement postulate}

Stern-Gerlach demonstrates both the discreteness of quantum
outcomes and the \emph{back-action} of measurement: measuring
$S_x$ inevitably disturbs $S_z$. This is inherent in the
quantum formalism, not an engineering limitation. The
non-commutativity $[S_x, S_z] = i\hbar S_y$ forces it.

\section{Schr\"odinger's cat and the macroscopic superposition
puzzle}

\subsection{The 1935 thought experiment}

In his 1935 paper ``Die gegenw\"artige Situation in der
Quantenmechanik'' \citep{Schrodinger1935_cat}, Schr\"odinger
constructed a thought experiment to expose the tension between
quantum superposition and classical macroscopic reality. A
cat is placed in a sealed box with a radioactive atom, a
Geiger counter, and a vial of poison. If the atom decays, the
counter triggers, the vial breaks, and the cat dies. If the
atom does not decay, the cat lives. After one half-life, the
atomic state is
\begin{equation}
|\psi_{\rm atom}\rangle = \frac{1}{\sqrt 2}(|\text{undecayed}\rangle
+ |\text{decayed}\rangle).
\end{equation}
By linearity of the Schr\"odinger equation, the combined
state of atom + Geiger counter + poison + cat after the
interaction is
\begin{equation}\label{eq:cat_state}
|\psi_{\rm total}\rangle = \frac{1}{\sqrt 2}(|\text{undecayed}\rangle
|\text{live cat}\rangle
+ |\text{decayed}\rangle|\text{dead cat}\rangle).
\end{equation}
On the literal mathematical reading of the standard 4D
formalism, the cat is in a superposition of alive and dead.
Yet no one has ever observed such a cat. Something in the measurement process
prevents macroscopic superpositions from being seen.

\subsection{Why the cat is never actually seen in superposition}

The modern answer --- decoherence (discussed below) --- is
that the cat's state is continuously entangled with the air
molecules, thermal photons, and internal biochemistry of the
cat itself. The timescale for decoherence of a 1-kilogram
object at room temperature in air is
\begin{equation}\label{eq:cat_decoherence_time}
\tau_D \sim \frac{m\lambda_{\rm th}^2}{\hbar} \sim 10^{-40}
\text{~seconds}
\end{equation}
\citep{JoosZeh1985, Zurek2003}. The cat is simply never in a
coherent superposition long enough to be observed in one.

But this does not explain why the \emph{outcome} is either
``alive'' or ``dead'' but not both. Decoherence yields a
\emph{mixed state} $\rho_{\rm cat} = (1/2)|\text{alive}\rangle
\langle\text{alive}| + (1/2)|\text{dead}\rangle\langle
\text{dead}|$ --- a probabilistic mixture. It does not yield a
\emph{single definite outcome}. This residual step is the core
of the measurement problem --- and, as Part~VIII will show, is
resolved at the Geiger counter, not at the lid of the box:
the always-on projection deposits a stable, irreversible record the
moment the macroscopic detector clicks --- well before any human
observation, and whether or not anyone ever looks --- so that no
live-and-dead cat superposition has ever existed in the
ordinary four-dimensional sense.

\section{The EPR paradox: entanglement and non-locality}

\subsection{The 1935 EPR paper}

In May 1935, Einstein, Podolsky, and Rosen (EPR)
\citep{EPR1935} published ``Can Quantum-Mechanical Description
of Physical Reality Be Considered Complete?'' The EPR argument
proceeds by considering a two-particle system in an
\emph{entangled state} such that measurements of position or
momentum on one particle determine, with certainty, the result
of the corresponding measurement on the other. Schematically,
for two spin-1/2 particles in the singlet state
\begin{equation}\label{eq:singlet}
|\Psi^-\rangle_{AB} = \frac{1}{\sqrt 2}
\left(|{\uparrow}_A\rangle|{\downarrow}_B\rangle
- |{\downarrow}_A\rangle|{\uparrow}_B\rangle\right).
\end{equation}
Measurement of $S_z^A$ yielding $+\hbar/2$ collapses the $B$
particle to $|\downarrow_B\rangle$; measurement of $S_z^A$
yielding $-\hbar/2$ collapses it to $|\uparrow_B\rangle$. The
correlation is perfect: whenever $A$ measures spin-up, $B$
must measure spin-down, and vice versa.

EPR argued: if the two particles are far apart (space-like
separated) at the moment of measurement, then the measurement on
$A$ cannot physically affect $B$ (relativity). Therefore the
value of $B$'s spin must have been determined \emph{before} the
measurement on $A$. But quantum mechanics says no such
predetermined value exists. Therefore quantum mechanics is
incomplete. There must be ``hidden variables'' that specify
the spins in advance.

\subsection{Bohr's response (July 1935)}

Bohr replied in July 1935 \citep{Bohr1935}, arguing that ``the
very conditions which define the possible types of predictions
regarding the future behavior of the system'' are determined by
the experimental setup. In Bohr's reading, there is no
meaningful question about $B$'s ``pre-measurement value''
independently of the measurement context. EPR's assumption of
independent realities for $A$ and $B$ is, for Bohr, a classical
prejudice incompatible with the quantum world.

The debate was philosophical. Neither side could offer a
definitive experimental test. That would wait for Bell.

\section{Bell's theorem (1964): the experimental test}

\subsection{The 1964 Bell paper}

John S.~Bell's 1964 paper ``On the Einstein-Podolsky-Rosen
paradox'' \citep{Bell1964} is a landmark in 20th-century
physics. Bell derived an inequality that must be satisfied by
\emph{any} local hidden-variable theory, regardless of its
details. Quantum mechanics, Bell showed, predicts correlations
that \emph{violate} this inequality for certain spin-measurement
configurations.

Bell's original inequality: for two particles measured in
directions $\mathbf{a}$, $\mathbf{b}$, $\mathbf{c}$, with
outcomes $\pm 1$, a local hidden-variable theory must satisfy
\begin{equation}\label{eq:Bell_original}
|E(\mathbf{a},\mathbf{b}) - E(\mathbf{a},\mathbf{c})|
\le 1 + E(\mathbf{b},\mathbf{c}).
\end{equation}
Quantum mechanics for the singlet state predicts
$E(\mathbf{a},\mathbf{b}) = -\mathbf{a}\cdot\mathbf{b}$, and for
angles satisfying $\mathbf{a}\cdot\mathbf{c} = 1/2$,
$\mathbf{b}\cdot\mathbf{c} = 1/2$,
$\mathbf{a}\cdot\mathbf{b} = -1/2$, the quantum prediction gives
$|-1/2 - 1/2| = 1$ on the left and $1 + (-1/2) = 1/2$ on the
right. QM violates Bell's inequality.

\subsection{The CHSH inequality (1969)}

Clauser, Horne, Shimony, and Holt \citep{CHSH1969} derived a
more experimentally accessible inequality. For any
local hidden-variable theory with four measurement settings
(two per particle, $\mathbf{a}, \mathbf{a}', \mathbf{b},
\mathbf{b}'$):
\begin{equation}\label{eq:CHSH}
S_{\rm CHSH} \equiv |E(\mathbf{a},\mathbf{b})
+ E(\mathbf{a},\mathbf{b}') + E(\mathbf{a}',\mathbf{b})
- E(\mathbf{a}',\mathbf{b}')| \le 2.
\end{equation}
Quantum mechanics predicts, for optimally chosen angles,
$S_{\rm CHSH}^{\rm QM} = 2\sqrt{2} \approx 2.828$ ---
Tsirelson's bound.

The CHSH inequality is the experimental workhorse. It requires
only the correlation function $E$ for four settings, not an
infinite family.

\section{The Aspect experiments (1982) and the 2022 Nobel Prize}

\subsection{First-generation experiments (1972--1982)}

Stuart Freedman and John Clauser in 1972 \citep{FreedmanClauser1972}
reported the first experimental violation of a Bell inequality,
using photon pairs from calcium atomic cascades. The
experiment was technically limited: the polarizer settings
were fixed throughout data-taking, and detection efficiency was
low.

Alain Aspect, Philippe Grangier, and G\'erard Roger at Orsay
\citep{Aspect1981, Aspect1982a, Aspect1982b} in 1981--1982
performed the definitive first-generation test. Their
experiment used rapidly-switching polarizer orientations (on
timescales faster than the light-travel time between the two
wings of the apparatus), thereby closing the
\emph{locality loophole}: the measurement settings could not
signal to each other.

Aspect's 1982 result: $S_{\rm CHSH} = 2.697 \pm 0.015$, a
$46\sigma$ violation of the Bell inequality, in excellent
agreement with the quantum prediction.

\subsection{Loophole-free experiments (2015)}

Two loopholes remained after Aspect: the
\emph{detection-efficiency loophole} (low-efficiency detectors
might preferentially detect certain outcomes) and the
\emph{freedom-of-choice loophole} (the measurement settings
might be correlated with the hidden variables). In 2015, three
independent groups \citep{Hensen2015, Giustina2015, Shalm2015}
reported loophole-free Bell tests using atomic spins,
photon polarization with the Kwiat source, and nitrogen-vacancy
centers in diamond, respectively. All three achieved
statistically significant violations of the CHSH inequality
with all loopholes closed simultaneously.

\subsection{The 2022 Nobel Prize}

On October 4, 2022, the Royal Swedish Academy of Sciences
awarded the Nobel Prize in Physics jointly to Alain Aspect,
John F.~Clauser, and Anton Zeilinger ``for experiments with
entangled photons, establishing the violation of Bell
inequalities and pioneering quantum information science''
\citep{NobelPhysics2022}. The citation explicitly states that
these experiments established the impossibility of
local-hidden-variable completions of quantum mechanics.

\subsection{What Bell + CHSH + Aspect + 2022 prove}

The combination of Bell's theorem, the CHSH inequality, and
the Aspect-class experiments proves:
\begin{center}\textbf{No \emph{local} hidden-variable
theory can reproduce quantum mechanics.}\end{center}

Note the emphasis on ``local''. The theorem does not rule out
non-local hidden-variable theories (e.g., Bohmian mechanics
explicitly incorporates them). And, as we argue in Part III of
this report, it does not rule out theories where the hidden
variables live in \emph{additional geometric directions} of a
deeper theory --- $I_r$ for scale and $S^1_\theta$ for phase ---
and become apparently non-local only after the four-dimensional
boundary trace integrates them out.

\section{Quantum entanglement and the EPR pair}

\subsection{Entanglement defined}

A pure state $|\psi\rangle_{AB}$ of a composite system
$\Hilb_A \otimes \Hilb_B$ is \emph{entangled} if it cannot be
written as a product $|\alpha\rangle_A \otimes |\beta\rangle_B$.
The canonical examples are the four \emph{Bell states}:
\begin{align}\label{eq:bell_states}
|\Phi^\pm\rangle &= \frac{1}{\sqrt 2}
(|{\uparrow}\up\rangle \pm |{\downarrow}\dn\rangle),\\
|\Psi^\pm\rangle &= \frac{1}{\sqrt 2}
(|{\uparrow}{\downarrow}\rangle \pm |{\downarrow}{\uparrow}\rangle).
\end{align}
The singlet $|\Psi^-\rangle$ has the remarkable property that
it is rotationally invariant: whatever basis Alice chooses for
$A$, the correlations with Bob's measurements in the same basis
are perfectly anti-correlated.

\subsection{Entanglement is not classical correlation}

Classical correlations cannot violate the CHSH inequality.
Entangled quantum states can. The difference is that classical
correlation exhausts the joint distribution $P(a,b|\mathbf{x},
\mathbf{y})$ with a single hidden-variable distribution; quantum
entanglement produces joint distributions that cannot be
factored through any classical hidden variable.

\subsection{Entanglement as a resource}

Since the 1990s, quantum information theory has recognized
entanglement as a physical \emph{resource}: it enables quantum
teleportation (Bennett et al., 1993) \citep{Bennett1993},
superdense coding, quantum cryptography (BB84, E91)
\citep{BennettBrassard1984, Ekert1991}, and quantum
computational speedups.

\section{Quantum decoherence and the environment}

\subsection{The Zurek program}

Wojciech Zurek, beginning in 1981 \citep{Zurek1981}, argued that
the apparent ``classical'' behavior of macroscopic objects
emerges from quantum mechanics through decoherence: the
unavoidable entanglement of any quantum system with its
environment. For a system $S$ in a superposition
$|\psi_S\rangle = \sum_n c_n |n\rangle_S$ coupled to an
environment $E$ with initial state $|E_0\rangle$, the joint
unitary evolution produces
\begin{equation}\label{eq:decoherence_total}
|\Psi_{SE}(t)\rangle = \sum_n c_n |n\rangle_S |E_n(t)\rangle.
\end{equation}
The reduced density operator for $S$ is
\begin{equation}\label{eq:decoherence_rho}
\rho_S(t) = \Trenv |\Psi_{SE}\rangle\langle\Psi_{SE}|
= \sum_{n,m} c_n c_m^* \braket{E_m(t)}{E_n(t)} |n\rangle
\langle m|.
\end{equation}
Off-diagonal elements $\rho_{nm}$ are suppressed by
$\braket{E_m(t)}{E_n(t)}$. For a thermal environment or
scattering-rate-dominated decoherence, these overlaps decay
exponentially: $\braket{E_m(t)}{E_n(t)} \sim e^{-\Gamma_{nm} t}$.

\subsection{The einselection mechanism}

Zurek's \emph{einselection} (environment-induced
superselection) identifies the ``pointer basis'' as the basis
in which the environment monitoring is diagonal. The pointer
basis is dynamically selected by the form of the
system--environment interaction. For spatial degrees of freedom
interacting with air molecules, the pointer basis is
approximately the position eigenbasis --- explaining why
macroscopic objects seem to have definite positions.

Einselection explains the \emph{preferred-basis problem}: why
is the post-decoherence mixture diagonal in position rather
than in some other basis? The answer is that position-basis is
what the environment monitors.

\subsection{What decoherence does not solve}

Decoherence explains the \emph{transition from superposition to
mixed state}. It does not explain the \emph{transition from
mixed state to single outcome}. The reduced density operator
$\rho_S = \sum_n |c_n|^2 |n\rangle\langle n|$ is still a
statistical ensemble, not a single outcome. Something else is
needed to explain why one particular outcome is realized.

Copenhagen says: the outcome is not defined until we measure.
Many-Worlds says: all outcomes are realized, in different
branches. Bohmian mechanics says: the outcome was determined
all along by a non-local hidden variable. GRW says: collapse is
a physical process that happens spontaneously at a rate set by
a new fundamental constant. Each of these has issues. We will
argue in Parts II--III that \GGtheory{} provides a cleaner
resolution.

\section{The Gleason theorem: the Born rule's uniqueness}

\subsection{The 1957 theorem}

Andrew Gleason in 1957 \citep{Gleason1957} proved a striking
mathematical fact. Consider a complex separable Hilbert space
$\Hilb$ of dimension $\ge 3$, and let $\mu$ be a probability
measure on the closed subspaces of $\Hilb$ (equivalently, on
orthogonal projectors) satisfying
\begin{equation}\label{eq:Gleason_additivity}
\mu(\Pi_1 + \Pi_2 + \ldots) = \mu(\Pi_1) + \mu(\Pi_2) + \ldots
\end{equation}
whenever $\Pi_1, \Pi_2, \ldots$ are mutually orthogonal.
Then there exists a unique positive trace-class operator $\rho$
with $\Tr\rho = 1$ such that
\begin{equation}\label{eq:Gleason_result}
\boxed{\mu(\Pi) = \Tr(\rho\,\Pi) \qquad \forall\,\text{projectors }\Pi.}
\end{equation}

\subsection{Why this matters}

Gleason's theorem shows the Born rule is the \emph{unique}
probability measure compatible with the Hilbert-space structure
and the additivity axiom (non-contextual probabilities for
orthogonal projectors). One cannot add some other probability
rule to quantum mechanics without either abandoning the
Hilbert space structure or abandoning non-contextual
probabilities.

\subsection{Assumptions of Gleason}

Gleason's theorem \emph{assumes}:
\begin{enumerate}[leftmargin=2em,itemsep=0.1em]
\item The state space is a complex separable Hilbert space of
dimension $\ge 3$.
\item Observables are projectors (or PVMs).
\item Probabilities are additive over orthogonal projectors.
\item The measure is non-contextual --- depends only on the
projector, not on the basis.
\end{enumerate}
The first two assumptions are postulates of the Copenhagen
formulation of quantum mechanics. If \GGtheory{} derives the
Hilbert space structure from the 6D bulk (see Part II), then
Gleason gives Born for free. The Born rule is no longer a
separate postulate.

\section{The qubit and the quantum information revolution}

\subsection{Qubit defined}

A \emph{qubit} is a two-level quantum system. The Hilbert space
is $\mathbb{C}^2$, with basis states conventionally denoted
$|0\rangle, |1\rangle$. A general pure qubit state is
\begin{equation}\label{eq:qubit}
|\psi\rangle = \alpha|0\rangle + \beta|1\rangle,
\qquad |\alpha|^2 + |\beta|^2 = 1.
\end{equation}
Equivalently, $|\psi\rangle = \cos(\theta/2)|0\rangle
+ e^{i\varphi}\sin(\theta/2)|1\rangle$ --- parametrized by the
Bloch sphere $(\theta, \varphi)$. The mixed state is a density
matrix
$\rho = \frac{1}{2}(\iden + \mathbf{r}\cdot\boldsymbol{\sigma})$
with Bloch vector $\mathbf{r} \in \mathbb{R}^3$, $|\mathbf{r}|
\le 1$, pure iff $|\mathbf{r}| = 1$.

Physical realizations of qubits include: electron or nuclear
spin, photon polarization, atomic two-level transitions,
superconducting transmon circuits, trapped ions, and
nitrogen-vacancy centers in diamond. Large-scale qubit arrays
(google Sycamore 2019, IBM Eagle 2021, Kipu QuEra 2023) now
demonstrate fundamental tests of quantum mechanics in the
multi-qubit regime.

\subsection{Teleportation, dense coding, and cryptography}

Quantum information theory has produced a remarkable array of
protocols that rely on entanglement:
\begin{itemize}[leftmargin=2em,itemsep=0.1em]
\item \emph{Quantum teleportation} (Bennett et al.~1993)
\citep{Bennett1993}: given one pre-shared Bell pair, Alice can
transmit an arbitrary qubit state to Bob using only two
classical bits of communication.
\item \emph{Superdense coding} (Bennett-Wiesner 1992)
\citep{BennettWiesner1992}: one qubit transmission + one
pre-shared Bell pair carries two classical bits.
\item \emph{Quantum key distribution (QKD)}: BB84
\citep{BennettBrassard1984} and E91 \citep{Ekert1991} use quantum
measurement statistics to detect eavesdroppers.
\item \emph{Shor's algorithm} \citep{Shor1994} for integer
factorization runs in polynomial time, threatening RSA.
\item \emph{Grover's algorithm} \citep{Grover1996} for unsorted
search is quadratically faster than any classical algorithm.
\end{itemize}
These are not mere curiosities. They are among the most
practically consequential consequences of quantum mechanics in
the 21st century.

\section{The Lindblad (GKSL) master equation}

\subsection{Open quantum systems}

For a system coupled to an environment, the reduced dynamics is
generally not unitary. The environment introduces dissipation,
decoherence, and relaxation. The systematic framework for this
is the theory of \emph{open quantum systems}
\citep{BreuerPetruccione2002, LindbladBook}.

\subsection{The GKSL theorem (1976)}

Under the assumptions of (i) time-homogeneity (semigroup
property $\Phi_{t+s} = \Phi_t \Phi_s$), (ii) complete positivity
(physical states of $S \otimes R$ for arbitrary reference $R$
must remain positive), and (iii) trace preservation,
Gorini--Kossakowski--Sudarshan \citep{GKS1976} and Lindblad
\citep{Lindblad1976} independently proved that the generator of
the resulting one-parameter semigroup of channels
$\Phi_t = e^{tL}$ has the form
\begin{equation}\label{eq:GKSL}
\boxed{\frac{d\rho}{dt} = L\rho = -i[H,\rho]
+ \sum_j \left(L_j \rho L_j^\dagger
- \tfrac{1}{2}\{L_j^\dagger L_j, \rho\}\right).}
\end{equation}
This is the \textbf{Lindblad equation} or \textbf{GKSL master
equation}. $H = H^\dagger$ is the effective Hamiltonian; the
$L_j$ are \emph{jump operators} encoding the environmental
coupling. The dissipator $\mathcal{D}\rho = \sum_j (L_j\rho L_j^\dagger
- \frac{1}{2}\{L_j^\dagger L_j, \rho\})$ is the unique
completely-positive generator structure.

\subsection{Physical examples}

Standard examples of Lindblad dynamics:
\begin{itemize}[leftmargin=2em,itemsep=0.1em]
\item \emph{Spontaneous emission}: $L_1 = \sqrt{\gamma}\,\sigma^-$
for a two-level atom decaying at rate $\gamma$.
\item \emph{Dephasing}: $L_1 = \sqrt{\gamma_\phi/2}\,\sigma_z$
producing exponential decay of off-diagonal coherences.
\item \emph{Damped harmonic oscillator}: $L_1 = \sqrt{\kappa}\, a$
for a cavity mode decaying to zero photons.
\item \emph{Position-localized decoherence}:
$L_1 = \sqrt{\Lambda}\, \hat{x}$ representing continuous
monitoring of position, the canonical model for einselection.
\end{itemize}

Importantly, GKSL is an \emph{effective} equation: its
derivation assumes Markovian dynamics (short environmental
memory). Real environments often have memory effects
(non-Markovian), but GKSL captures the universal Markovian
limit.

\section{Interpretations of quantum mechanics}

\subsection{Copenhagen (Bohr, 1927)}

The original ``orthodox'' interpretation
\citep{Bohr1935, Heisenberg1958}. Quantum mechanics predicts
probabilities for measurement outcomes. The wave function is
not a physical entity but a summary of the observer's
information about the system. Measurement is a primitive
concept; wave-function collapse is a change of the observer's
knowledge, not a physical process. The line between quantum and
classical domains (Heisenberg cut) is context-dependent.

Strengths: mathematically minimal; operationally complete.
Weaknesses: treats measurement as primitive; ambiguous about
macroscopic superpositions; no account of entanglement in
relativistic setting.

\subsection{Many-Worlds (Everett, 1957)}

Hugh Everett III's 1957 PhD thesis \citep{Everett1957}
proposed that the wave function never collapses. Every time a
measurement would classically produce an outcome, the universe
\emph{branches}: in one branch the cat is alive, in another
dead; in one branch the electron spin is up, in another down.
All branches are equally real.

Strengths: no collapse postulate; mathematical simplicity;
compatible with unitary evolution alone. Weaknesses: requires
an infinite multiverse; does not derive the Born rule (why do
branches have the ``weights'' they do?); ontologically extravagant.

\subsection{Bohmian mechanics (Bohm, 1952)}

David Bohm's 1952 paper \citep{Bohm1952} revived an idea of de
Broglie (1927): particles have definite positions guided by a
``quantum potential'' derived from the wave function. The
particles follow deterministic trajectories. The wave function
is a real, physical, objective field.

Strengths: deterministic; restores definite particle
trajectories; reproduces all quantum predictions.
Weaknesses: explicitly non-local; privileges position as the
fundamental ontology (awkward for spin, Fock space);
extension to QFT is technically subtle.

\subsection{GRW spontaneous collapse (Ghirardi--Rimini--Weber, 1986)}

Ghirardi, Rimini, and Weber \citep{GRW1986} proposed that the
wave function undergoes spontaneous localization events at a
low rate per particle ($\sim 10^{-16}$~s$^{-1}$ per nucleon in
the original model). For a single particle, collapses are too
rare to notice; for a macroscopic object with $10^{24}$
nucleons, at least one collapse per microsecond locks the
object into a single classical configuration.

Strengths: defines collapse as a physical process; testable via
proposed spontaneous X-ray emission, heating, or ultracold-atom
interferometry.
Weaknesses: introduces two new fundamental constants ($\lambda$
and $r_C$); fine-tuning required to reproduce QM at micro scales
while producing classicality at macro scales; not trivially
relativistic.

\subsection{QBism (Fuchs et al., 2001)}

Quantum Bayesianism (QBism) \citep{Fuchs2010} treats the quantum
state as entirely epistemic: it represents an agent's
subjective probability assignments for measurement outcomes,
updated via quantum Bayesian rules. There is no ontic wave
function.

Strengths: no measurement problem by construction
(measurement is just Bayesian updating); compatible with
subjective probability. Weaknesses: gives up the goal of
describing a mind-independent reality; unclear what
``agent-invariant'' facts exist at all.

\subsection{Consistent/decoherent histories (Griffiths, Omn\`es,
Gell-Mann--Hartle)}

\citep{Griffiths1984, Omnes1999, GellMannHartle1993} Proposes
that quantum mechanics is about consistent \emph{histories} ---
sequences of events for which the decoherence functional is
diagonal. Probabilities are assigned to consistent histories.

Strengths: no explicit collapse; handles closed systems
(cosmology). Weaknesses: many incompatible consistent-history
frameworks; no unique ``true'' history.

\subsection{Relational quantum mechanics (Rovelli, 1996)}

\citep{Rovelli1996RQM} Each observer has its own quantum state;
there is no observer-independent state. A quantum state is a
relation between a system and an observer.

\subsection{None of the above is GG-Theory}

We will argue in Parts II and III that \GGtheory{} is closest to
Bohm + Everett + Rovelli but technically different from any of
them. The mechanism that produces quantum mechanics in
\GGtheory{} is \OGGEP{} (\AX{3} of the \GGConstitution{}) in
general --- the bulk-to-boundary projection from the 6D bulk (treated
as an open system) to the 4D boundary observer, produced by the
boundary partial trace over the inaccessible bulk degrees of
freedom. The mechanism does not require any specific gauge
sector: a simpler bulk content (e.g.\ a pure-QED bulk with only
$\Uone_{\rm EM}$ as the gauge content, with the phase direction
$S^1_\theta$ supplying KK modes and the radial direction
$r=\ln(E_{\rm GUT}/\mu)$ supplying renormalization-group modes)
already produces standard Maxwell and Lorentz-force physics on
the 4D boundary via the same \OGGEP{} construction. The
``hidden variables'' that the standard hidden-variable program
has been groping toward are therefore the bulk degrees of freedom
that the 4D boundary partial trace integrates over: bulk KK
modes along $S^1_\theta$, RG modes along $r$, and bulk
gauge-field configurations of whatever gauge content the bulk
theory carries (in the cohort canonical bulk of \citet{ArisakaA2GG6},
this gauge content is $\Spinone\times\SUfive\times\Uoneb
\times\UoneQi$, but the QM mechanism itself does not depend on
any specific factor of this group). The \UoneQi{} sector is
optional in the \GG{}/\BIsix{} sense: it is canonical for the
full cohort but not constitutive of the QM mechanism.

The ``branching'' of Everett is replaced by the \GSSB{} branch
selection (a single realized branch). The observer-dependence of
Rovelli is replaced by the \OGGEP{} functor from bulk to
boundary. The Bohm-style hidden variables are relocated from 4D
(where Bell's theorem forces them to be non-local) to 6D (where
they are bulk-local by \AX{1} of \AOne{}).

The sharpest one-line placement, developed in
Section~\ref{sec:grip-spine}, is against the decoherence program
and GRW. The standing objection to decoherence-only accounts is that
diagonalization never picks an outcome; GRW answers by adding
stochastic collapse with two new fundamental constants. \GGtheory{}
answers structurally: the runtime's fixed-landscape descent
(\GRIPin{}) halts at a unique attractor, and that halt \emph{is} the
selection --- delivered by machinery the theory already needs to
stabilize electrons and select vacua. \GGtheory{} is the decoherence
program completed by attractor dynamics, at zero new parameters:
collapse as geometric relaxation, not stochastic surgery.

\section{Quantum mechanics in 2026: successes, open questions,
and the state of play}

\subsection{What quantum mechanics gets right}

After a century, quantum mechanics has passed every
experimental test. There is no known counterexample to
the Schr\"odinger equation, the Born rule, the Heisenberg
commutator, or the Lindblad equation. Quantum theory
successfully predicts:

\begin{enumerate}[leftmargin=2em,itemsep=0.1em]
\item \textbf{Precision QED}: the electron magnetic moment
$a_e$ to $\sim 10^{-13}$ relative precision; the hydrogen
hyperfine structure; the Lamb shift.
\item \textbf{Atomic physics}: all atomic spectra, hyperfine
couplings, Zeeman and Stark shifts, optical-lattice clock
frequencies.
\item \textbf{Molecular physics}: chemical bonding, reaction
rates (with corrections from RRKM/TST), spectroscopic constants.
\item \textbf{Condensed matter}: band structure, Cooper pairing,
BCS superconductivity, quantum Hall effect, topological
insulators, high-$T_c$ superconductors.
\item \textbf{Elementary particle physics}: Standard Model cross
sections, decay rates, mass spectrum (as far as the SM can go).
\item \textbf{Cosmology}: Cosmic microwave background power
spectrum (from inflationary fluctuations), big-bang
nucleosynthesis, baryon acoustic oscillations.
\item \textbf{Quantum technology}: lasers, transistors,
magnetic resonance imaging, atomic clocks, atomic force
microscopy, quantum key distribution, Shor's algorithm,
quantum computers.
\end{enumerate}

\subsection{What quantum mechanics fails to explain}

But quantum mechanics, as of April 2026, does not say:

\begin{enumerate}[leftmargin=2em,itemsep=0.1em]
\item \textbf{Why the Born rule?} Gleason shows uniqueness;
no one has derived the rule from more basic principles.
\item \textbf{When does measurement happen?} The Heisenberg
cut is not fixed by the theory.
\item \textbf{What is the wave function?} Ontic (real) or
epistemic (knowledge)? The question is not settled.
\item \textbf{Why exactly these three spatial dimensions?}
Quantum mechanics is consistent in any dimension; why 3 + 1?
\item \textbf{Why exactly this Hilbert space?} QM works on any
Hilbert space; the choice of $\mathbb{C}^2$ for spin, $L^2(\mathbb{R}^3)$
for orbital motion, etc., is empirical.
\item \textbf{How is measurement reconciled with relativity?}
The nonlocal collapse of EPR pairs is awkward in special
relativity; the foundations of relativistic quantum measurement
remain disputed.
\item \textbf{How does gravity fit in?} There is no agreed
theory of quantum gravity. ``Quantizing'' the metric is
perturbatively non-renormalizable.
\item \textbf{What are entanglement and non-locality,
physically?} Bell's theorem and the 2022 Nobel prize confirm
that quantum correlations violate local realism, but provide no
constructive account of what the correlations \emph{are}.
\end{enumerate}

These are the gaps. Part II of this report will argue that
\GDOS{} addresses items 1, 2, 3, 5, 6, and 7. Part III will
argue that \GGsix{} addresses 4 and 8.

\section{A century of quantum mystery: reading for Parts II--III}

Taking stock as of April 2026: quantum mechanics is
empirically perfect, foundationally incomplete. The ingredients
needed to explain it from a deeper theory are:
\begin{itemize}[leftmargin=2em,itemsep=0.1em]
\item A \textbf{Hilbert space} --- provided by the bulk
zero-mode sector.
\item A \textbf{probability measure} on it --- provided by
Gleason + \CPTP{} channel.
\item A \textbf{dynamical law} --- provided by Stone's theorem
(closed system) and Lindblad (open system).
\item A \textbf{measurement theory} --- provided by
\POVM{}/instrument formalism and \OGGEP{}.
\item A \textbf{non-local-correlation generator} --- provided by
shared $S^1_\theta$ holonomy in 6D.
\end{itemize}

All five ingredients are present in \GDOS{}~$+$~\GGsix{}, and
together they constitute the 6D-bulk-first reading that the
remainder of this paper systematically develops: the deeper
theory is deterministic, unitary, and local in six geometric
directions ($\mathcal{M}_4 \times I_r \times S^1_\theta$); the
four-dimensional observer's average over the two extra
directions ($I_r$ carrying scale, $S^1_\theta$ carrying phase)
is what every century of quantum-foundations work has called
``measurement,'' ``collapse,'' ``decoherence,'' and
``entanglement.'' We now turn to the construction.



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

This two-page dictionary is offered as a reader's legend: a
single front-matter table mapping every program-defined
acronym used in the body of \AEight{} to a one-sentence
operational definition. A reader who flips here first can
decode any macro encountered in the body without flipping to
Appendix~A (which gives the same content in the canonical
Appendix-A format inherited from \AOne{} Part~VIII). The
dictionary is grouped by structural role and is rendered as a
single hierarchy figure
(Fig.~\ref{fig:appA-abbrev-flow-a8}) so the macro relations
are visible at a glance.

\paragraph{Framework $+$ constitutional charter.}
\GGTheory{} = Grand Geometric Theory (the twenty-two-paper
program). \GGConstitution{} = the constitutional charter
(4 axioms + 7 postulates) of \GGTheory{}, supplied by \AOne{}.
\GG{} = the six-dimensional bulk geometry\footnote{Throughout the cohort, the ``6D bulk'' is read on the field-theory side of the bulk--boundary correspondence: the fifth (scale) dimension is the renormalization-group axis $r=\ln(E_{\mathrm{GUT}}/\mu)$, not a Randall--Sundrum spatial dimension, and the radial direction denotes RG running; the explicit renormalization-group construction is given in \citet{ArisakaA2GG6}.}
($\mathcal{M}_4 \times I_r \times S^1_\theta$).
\GDOS{} = Geometric Dynamics of Open Systems, the boundary law
forced by \OGGEP{}. \BIsix{} = the six-dimensional Bianchi
identity $dH_3 = X_4$ (Green--Schwarz-modified).
\OGNfive{} = Occam-Gauge-Normalization five-integer ledger.
\GiFour{} / \QiFour{} = four-pillar structures supplied by
\ASix{} and \ASeven{}.

\paragraph{The QM tier --- the feature of \AEight{}.}
\textbf{QM} = the operational reformulation of textbook quantum
mechanics under the cohort framework: the seven inherited
theorems \TA{1}--\TA{7} (operational restatements of \AFive{}
Part~III's \T{7.1}--\T{7.7} chain --- Hilbert-space embedding,
Schr\"odinger as closed-system limit, Born rule, L\"uders
projection, decoherence, einselection, Bell--CHSH) plus the
four net-new theorems \TA{8}--\TA{11} (Tsirelson saturation as
Bogomolnyi bound, universal identity of indistinguishable
particles, \QiCON{}-conformality of decoherence rates,
boundary no-signaling from reduced-state factoring). The QM
tier sits between \GDOS{} (boundary law) and \GRIP{}
(operational cascade engine): the seven \TA{1}--\TA{7}
theorems are the QM-stratum specialization of \GDOS{}, and
they seed the \GRIP{} cascade by establishing the canonical
ground-state and the canonical state-update rule on the QM
stratum.

\paragraph{\GiFour{} pillars.} \TGG{} = Total Geometric--Gauge
object. \CCI{} = Covariant Conservation Identity. \GFF{} =
Geometric Free-Energy Functional ($F_{\GFF{}} = F_{\GEP{}} +
F_{\CEP{}}$). \GRIP{} = Gauge-Runtime Invariance Pipeline, the
universal cascade engine $\DGI{} \to (\GFF{}) \to \GSSB{} \to
\GA{}$, whose terminus on an admissible stratum is the Ground
Geometric Attractor, the unique deepest \GA{} (\ASix{}). Two regimes
(\ASix{}): \GRIPin{} = descent on a fixed landscape, halting at the
\GGA{}; \GRIPout{} = the transient event that reorganizes the
landscape itself. Their quantum-mechanical reading
is the business of Section~\ref{sec:grip-spine}.

\paragraph{\QiFour{} pillars.} \QiQUA{} (locked-current
bookkeeping), \QiEP{} (trajectory selection), \QiCON{}
(scaling diagnostics), \QiHAL{} (memory architecture).
\OH{} / \IH{} / \EIC{} = Outer / Inner Hologram / Evolving
Inner Cosmos.

\paragraph{\GDOS{} carrier.} \SIP{} = System Instance Packet,
the canonical five-field object on which \GRIP{} runs.

\paragraph{Companion paper macros (sans-serif, per L39).}
\AOne{} = \PaperFullAOne{}; \ATwo{} = \PaperFullATwo{};
\AThree{} = \PaperFullAThree{}; \AFour{} = \PaperFullAFour{};
\AFive{} = \PaperFullAFive{}; \ASix{} = \PaperFullASix{};
\ASeven{} = \PaperFullASeven{}; \AEight{} = \PaperFullAEight{}
(the present paper); \ANine{} = \PaperFullANine{};
\ATen{} = \PaperFullATen{}; \AEleven{} = \PaperFullAEleven{}.
Predictions branch: \PaperPOne{}, \PaperPTwo{}, \PaperCOne{}.
Capstone: \citet{ArisakaBookI}.

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

\begin{figure}[p]
\centering
\resizebox{0.97\textwidth}{!}{%
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 pillsideg/.style ={pillside, draw=green!60!black, fill=green!12},
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]

% --- Top tier: programme name and charter -----------------
\node[top] (GGT) at (0, 0) {\GGTheory{}};
\node[charter] (GGC) at (0, -16mm) {\GGConstitution{}};
\draw[arrow] (GGT.south) -- (GGC.north);

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

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

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

% --- Tier 3: bulk theorems ---
\node[font=\sffamily\bfseries\large] (BTL) at (0, -108mm) {Bulk theorems forced by \textbf{AX} + \textbf{POS}};
\node[bulk] (GG6) at (-46mm, -120mm) {\textbf{\GG{}}};
\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 ---
\node[gdos] (GDOS) at (0, -136mm) {\textbf{GDOS} \;\; (Geometric Dynamics of Open Systems on $\partial\Mbulk$) \;\; carrier \textbf{SIP} $= (\mathcal{M}, \text{ports}, L, \text{generator}, \text{Obs}/\text{Kill})$};

\draw[arrow] (GG6.south) -- (GDOS.north -| GG6.south);
\draw[arrow] (BI6.south) -- (GDOS.north -| BI6.south);
\draw[arrow] (OGN5.south) -- (GDOS.north -| OGN5.south);

% --- Tier 4.5: QM operational reformulation (THE FEATURE OF A8-QM) ---
\node[qmtier] (QM) at (0, -156mm) {%
 \textbf{\textsf{\Large QM} \;\; (the feature of \AEight{})} \\[1.3mm]
 \footnotesize \textbf{Inherited from \AFive{} Part~III:}
 \TA{1}~Hilbert \,$\bullet$\, \TA{2}~Schr\"odinger \,$\bullet$\,
 \TA{3}~Born \,$\bullet$\, \TA{4}~L\"uders \,$\bullet$\,
 \TA{5}~Decoherence \,$\bullet$\, \TA{6}~Einselection \,$\bullet$\,
 \TA{7}~Bell--CHSH \\[0.4mm]
 \footnotesize \textbf{Net-new in \AEight{}:}
 \TA{8}~Tsirelson saturation \,$\bullet$\,
 \TA{9}~Universal identity \,$\bullet$\,
 \TA{10}~\QiCON{}-conformality \,$\bullet$\,
 \TA{11}~Boundary no-signaling
};

\draw[arrowqm] (GDOS.south) -- (QM.north);

% --- Tier 5: GRIP engine box (centre) with Gi-4 / Qi-4 flanks ---
\node[griphead] (GRIPhead) at (0, -178mm) {\textbf{GRIP} \quad gauge-runtime engine \quad $\DGI{} \to (\GFF{}) \to \GSSB{} \to \GA{}$};

\node[gripchain] (DGI) at (-46mm, -192mm) {\textbf{DGI}};
\node[gripchain] (GFFx) at (-15mm, -192mm) {\textbf{(GFF)}};
\node[gripchain] (GSSB) at (15mm, -192mm) {\textbf{GSSB}};
\node[gripchain] (GGAx) at (46mm, -192mm) {\textbf{GGA}};
\draw[arrow] (DGI.east) -- (GFFx.west);
\draw[arrow] (GFFx.east) -- (GSSB.west);
\draw[arrow] (GSSB.east) -- (GGAx.west);

\node[gripregime] (RGin) at (-30mm, -204mm) {\textbf{GRIP-In} \;: within-layer Lyapunov};
\node[gripregime] (RGout) at (30mm, -204mm) {\textbf{GRIP-Out} \;: between-layer shock};

\draw[arrowqm] (QM.south) -- (GRIPhead.north);

% --- Pillar flanks ---
\node[pillsideg] (GiBox) at (-95mm, -216mm) {};
\node[pillhead] at (-95mm, -220mm) {\GiFour{} pillars\\ (geometric flank)};
\node[pillline] at (-122mm, -229mm) {\textbf{TGG}, \textbf{CCI}, \textbf{GFF},};
\node[pillline] at (-122mm, -235mm) {\textbf{GRIP} = engine box};
\node[pillline] at (-122mm, -242mm) {\textbf{GEP} (energetic),};
\node[pillline] at (-122mm, -248mm) {\textbf{CEP} (entropic)};
\node[pillline] at (-122mm, -254mm) {(\GFF{} = $F_{\text{GEP}} + F_{\text{CEP}}$)};

\node[pillsideq] (QiBox) at (95mm, -216mm) {};
\node[pillhead] at (95mm, -220mm) {\QiFour{} pillars\\ (admissibility filter)};
\node[pillline] at (62mm, -229mm) {\textbf{Qi-QUA}, \textbf{Qi-EP},};
\node[pillline] at (62mm, -235mm) {\textbf{Qi-CON}, \textbf{Qi-HAL}};
\node[pillline] at (62mm, -242mm) {\textbf{OH}$+$\textbf{IH}$=$\textbf{EIC}};
\node[pillline] at (62mm, -248mm) {\textbf{SeOS} carrier};
\node[pillline] at (62mm, -254mm) {(selects unique \textbf{GGA})};

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

% --- Tier 6: 17-layer cascade ribbon (cascade-flavour colours) ---
\node[font=\sffamily\bfseries\large] at (0, -272mm) {Seventeen-layer cascade of \textbf{GGAs}};
\node[cas, fill=blue!60!white, text=white] (L0) at (-96mm,-284mm) {0};
\node[cas, fill=blue!50!white, text=white] (L1) at (-84mm,-284mm) {1};
\node[cas, fill=blue!40!cyan!70, text=white] (L2) at (-72mm,-284mm) {2};
\node[cas, fill=cyan!75!white] (L3) at (-60mm,-284mm) {3};
\node[cas, fill=cyan!60!white] (L4) at (-48mm,-284mm) {4};
\node[cas, fill=cyan!40!green!70, text=white](L5) at (-36mm,-284mm) {5};
\node[cas, fill=green!50!white] (L6) at (-24mm,-284mm) {6};
\node[cas, fill=green!60!white] (L7) at (-12mm,-284mm) {7};
\node[cas, fill=green!70!white] (L8) at (0mm,-284mm) {8};
\node[cas, fill=green!80!yellow!60] (L9) at (12mm,-284mm) {9};
\node[cas, fill=yellow!70!orange] (L10) at (24mm,-284mm) {10};
\node[cas, fill=orange!50!white] (L11) at (36mm,-284mm) {11};
\node[cas, fill=orange!65!white] (L12) at (48mm,-284mm) {12};
\node[cas, fill=orange!80!red!60, text=white] (L13) at (60mm,-284mm) {13};
\node[cas, fill=red!50!white] (L14) at (72mm,-284mm) {14};
\node[cas, fill=red!70!white, text=white] (L15) at (84mm,-284mm) {15};
\node[cas, fill=red!85!black, text=white] (L16) at (96mm,-284mm) {16};

\node[font=\sffamily\scriptsize] at (-96mm,-294mm) {GUT};
\node[font=\sffamily\scriptsize] at (-84mm,-294mm) {EW};
\node[font=\sffamily\scriptsize] at (-72mm,-294mm) {Atoms};
\node[font=\sffamily\scriptsize] at (-60mm,-294mm) {Molec.};
\node[font=\sffamily\scriptsize] at (-48mm,-294mm) {Amino};
\node[font=\sffamily\scriptsize] at (-36mm,-294mm) {Polym.};
\node[font=\sffamily\scriptsize] at (-24mm,-294mm) {ATP};
\node[font=\sffamily\scriptsize] at (-12mm,-294mm) {RNA};
\node[font=\sffamily\scriptsize] at (0mm,-294mm) {DNA};
\node[font=\sffamily\scriptsize] at (12mm,-294mm) {LUCA};
\node[font=\sffamily\scriptsize] at (24mm,-294mm) {3-Dom.};
\node[font=\sffamily\scriptsize] at (36mm,-294mm) {Eukar.};
\node[font=\sffamily\scriptsize] at (48mm,-294mm) {Multi.};
\node[font=\sffamily\scriptsize] at (60mm,-294mm) {Ring};
\node[font=\sffamily\scriptsize] at (72mm,-294mm) {Map};
\node[font=\sffamily\scriptsize] at (84mm,-294mm) {Conscious};
\node[font=\sffamily\scriptsize] at (96mm,-294mm) {Articul.};

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

% --- Tier 7: Q_B + Arisaka equation -----------------------
\node[qb] (QB) at (0, -306mm) {%
$\boldsymbol{\mathsf{Q}_{\mathsf{B}}}$ \; (Quantum-information Bit; the boundary information member of \GUfive{}, fixed via \OGNfive{}) \\[1.0mm]
\textit{Arisaka equation:} \;\; $\boldsymbol{J_E / \mathsf{Q}_{\mathsf{B}} \,=\, J_e \,=\, J_B \,=\, J_{Qi}}$ \quad (four-current lock that drives the cascade)
};

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

\end{tikzpicture}
}
\caption[Hierarchy of program-defined acronyms, with the \AEight{} QM tier highlighted.]{Hierarchy of program-defined acronyms,
adapted for \AEight{}. The diagram is identical to
\AEleven{}'s Figure~2 except for the highlighted blue
\textbf{QM} tier inserted between \GDOS{} and \GRIP{}: the
eleven principal theorems of \AEight{} (\TA{1}--\TA{7}
inherited from \AFive{} Part~III plus \TA{8}--\TA{11}
net-new) are the QM-stratum specialization of \GDOS{} and
seed the \GRIP{} cascade. Bold blue arrows mark the QM tier
as the feature of this paper.}
\label{fig:appA-abbrev-flow-a8}
\end{figure}








\section*{Part~I in plain English: what one hundred years of
quantum mystery have actually taught us}
\addcontentsline{toc}{section}{Part~I in plain English}

The picture that emerges from this Part is not that quantum
mechanics is wrong, or that it is approximate, or that it will
one day be replaced by something more familiar. Quite the
opposite. Quantum mechanics is, on every empirical test, the
most accurate theory the human species has ever written down.
The electron's magnetic moment matches the prediction of
quantum electrodynamics to twelve digits. The hydrogen Lamb
shift, the atomic-cesium hyperfine structure, the cross
sections of high-energy collisions, the tunneling currents in
the transistor on which these words are typed --- all match
quantum mechanics at the parts-per-billion level or better.
A theory that has survived ten thousand independent tests over
a hundred years is not going to be overturned by a clever
argument.

What the past hundred years \emph{have} taught us is that the
theory is operationally complete but interpretively incomplete.
Born's rule predicts probabilities perfectly, but no one knows
where the rule comes from. The Schr\"odinger equation
propagates the state unitarily, but no one knows what state is
being propagated, or why measurement breaks the unitarity, or
when it does so. Bell's theorem and the 2022 Nobel-recognized
Aspect experiments prove that the correlations between
entangled particles are stronger than any classical, local
hidden-variable theory permits, but provide no positive
account of what the correlations \emph{are}. Schr\"odinger's
1935 cat thought-experiment was meant to expose a contradiction
between unitary evolution and macroscopic experience; ninety
years later, no consensus on its resolution exists, and the
seven living interpretations (Copenhagen, Many-Worlds,
Bohmian, GRW, QBism, relational, decoherent histories) each
read one feature of the puzzle correctly and elevate it to the
ontology, while missing the others. The community has, as a
result, declared a kind of armistice: the calculations are
trusted, the questions are tabled, and the disagreement is
allowed to continue.

After all, a century-long armistice is not a peace; it is a
postponement. The questions cataloged in this Part --- what
is the wave function, when does measurement happen, why the
Born rule, how do entangled particles share a fate ---
\emph{have} answers, and the answers are not metaphysical
preferences but theorems. The remaining twelve Parts of this
paper supply those theorems. Part~II assembles the
cohort inheritance from \AOne{}--\AEleven{} that makes
the answers possible. Part~III states the constitutional
bridge $\GG{}~\text{bulk} \to \OGGEP{} \to \GDOS{}
\to \text{4D~QM}$ that compiles the deterministic
six-dimensional dynamics into the operational rules a
laboratory observer measures. Part~IV states the eleven
principal theorems whose proofs occupy Parts~V--IX (worked
examples), Part~X (paradox sweep), Part~XI (interpretation
synthesis), and Part~XII (Discussions). Part~XIII delivers
the summit view.

After all, the right way to honor one hundred years of quantum
mystery is not to celebrate it as permanent, but to finish it.



\section{The cohort inheritance and the roadmap of the sixteen Parts}
\label{sec:inheritance-and-roadmap-A8}

\noindent
Before the inheritance ledger begins (Part~\ref{part:II}), it is
useful to lay out two reader-orientation maps on the same opening: a
panorama of the twenty-two-paper cohort in which this paper stands
(\S\ref{sec:cohort-a8}, \figref{fig:inh-cohort-map}), and a panorama
of the sixteen Parts of the present paper (\S\ref{sec:roadmap-a8},
\figref{fig:thirteen-parts-flowchart-a8}). A reader of any later Part
should be able to flip back to this Section and orient themselves in
both directions: which cohort companion carries which imported result,
and where the present argument stands in the sixteen-Part arc.

\subsection{The cohort inheritance}
\label{sec:cohort-a8}

The present paper is one of twenty-two. \GGTheory{} is written not as a
single monograph but as a cohort of twenty-two preprints --- eleven A-papers, six P-papers, three C-papers and two white papers --- each carrying one
rigorously stated piece of a single descent that runs from four axioms
and seven postulates to the empirical structure of physics. The
partition into twenty-two papers reflects the partition of subject
matter, not a ranking of importance: the framework is one program,
and each paper is a named carrier of named content.
\figref{fig:inh-cohort-map} is the panorama; a reader meeting the
program for the first time can read its architecture off the figure
top to bottom before returning to the quantum mechanics that is the
business of this paper. The present paper is unusual in the cohort in
devoting an entire Part to what the panorama compresses: the per-paper
inheritance ledger --- what \AEight{} receives from each of \AOne{}
through \AEleven{}, one section per companion --- is
Part~\ref{part:II}, so the reader who wants the custody of any single
imported result reads the panorama here and the ledger there.

\noindent The cohort map (\figref{fig:inh-cohort-map}, closing this subsection) is complete at twenty-two papers. Beyond the eleven-paper A-series spine cataloged paper by paper in Part~\ref{part:II}, the program's phenomenology is carried by the Predictions Branch --- \PaperPOne{} \citep{ArisakaP1KK}, \PaperPTwo{} \citep{ArisakaP2N3}, \PaperPThree{} \citep{ArisakaP3Electron}, \PaperPSix{} \citep{ArisakaP6Proton} --- the Cosmology \& Astrophysics Branch --- \PaperCOne{} \citep{ArisakaC1Kaxion}, \PaperCTwo{} \citep{ArisakaC2Hubble}, \PaperCThree{} \citep{ArisakaC3BH} --- and the White-Paper Branch --- \PaperWOne{} \citep{ArisakaW1Symmetry}, which walks the century from Noether to Hopkins--Singer, and \PaperWTwo{} \citep{ArisakaW2Holonomy}, which classifies protection by topology rather than by symmetry, the ladder on which this paper's unreadable phase fiber is one rung; the present paper \citep{ArisakaA8QM} is the quantum-mechanics member of the 4D-Physics trio.

\paragraph{The Foundations Branch.} The third framed branch is the
Foundations Branch: the two member papers whose subject is the
boundary's complex structure and its mirrors. For the present paper
this branch is not a distant flank but a working partner, because
\AEight{} is \PaperPFour{}'s principal consumer in the cohort.
\PaperPFour{} \citep{ArisakaP4Real} is the origin-of-$i$ member
paper, deriving the single imaginary unit of boundary physics from two
real structures of the \ATwo{} arena --- the compact phase circle and
the causal clock; the real-bulk series RB-1--RB-13 of the Discussions
Part (\S\ref{sec:disc-real-bulk}) develops in-paper, at discussion
grade, what \PaperPFour{} treats at standalone length, under the
division of labor both papers state in the same words: the dictionary
owns the sectors; the geometry owns the $i$. \PaperPFive{}
\citep{ArisakaP5CPT} carries the three mirrors of matter --- C, P,
and T read as orientation reversals of the three real bulk factors ---
and its law-level arrow of time is kin of \AEight{}'s arrow of
measurement; its Wick-quartet statistics machinery is composed by the
candidate theorem \TA{12} (\S\ref{sec:TA12-statistics}). Both carry
P-series designators; their place at the spine's flank, not the
phenomenology rows, marks their foundational role.

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

%% ===================================================================
%%  The nineteen-paper cohort map (frozen v1.4 geometry), on its own
%%  page at the close of the cohort-inheritance subsection.  Verbatim
%%  copy of the v1.4 master (A1-Constitution Figure 3): no
%%  self-reference markers anywhere, so the figure is identical and
%%  simple for replication in all nineteen papers.  Only the internal
%%  node names MP4/MP5 differ from the master (collision avoidance).
%% ===================================================================
\begin{figure}[p]
\centering
\resizebox{0.9\textwidth}{!}{%
%% v22 type scaling, local to this group: the picture sets node text
%% in \footnotesize (the root in \small) and its edge and branch labels
%% in \scriptsize, so redefining the three commands here enlarges box
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%% 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}%%
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%% --- 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 ---
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 {\textbf{\ASix{}} \\[-1pt] \footnotesize \GiFour{} pillars};
\node[pillars] (A7) at (32mm, -69mm)
 {\textbf{\ASeven{}} \\[-1pt] \footnotesize \QiFour{} pillars};

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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


\subsection{Roadmap of the sixteen Parts}
\label{sec:roadmap-a8}
\label{sec:structure-of-this-paper}

The rest of the paper is organized into fifteen further Parts, and
the walk through them is plotted in
\figref{fig:thirteen-parts-flowchart-a8}. The present Part --- the
entrance --- is nearly done: the century of quantum mystery from
Planck's blackbody catastrophe to the 2022 Nobel Prize, the ledger of
what quantum mechanics gets right and what it fails to explain, the
program dictionary, and the two orientation maps of this closing
section. Everything after this page is the reformulation itself, and
the itinerary below says where each piece lives.

The entrance conditions come first. Part~II is the inheritance
ledger: one section per cohort companion, cataloging what \AEight{}
receives from \AOne{} through \AEleven{} --- the constitutional
rules, the bulk geometry, the \GDOS{} boundary law with its seven
QM-postulate theorems, the \GRIP{}/\QiFour{} runtime, the
universal-identity insight --- so that every later claim can name its
custody in one flip. Part~III is the apex, and it is highlighted in
the roadmap because it delivers the paper's central mechanism: the
constitutional bridge, the operational chain $\GG{}$~bulk $\to$
\OGGEP{} $\to$ \GDOS{} $\to$ 4D~QM that compiles the deterministic
six-dimensional bulk into the operational rules a laboratory observer
measures. Every earlier page prepares that chain, and every later
Part exercises it.

Part~IV is the formal statement of results: the eleven principal
theorems of \AEight{}, with \TA{1}--\TA{7} inheriting verbatim from
\AFive{} Part~III's \T{7.1}--\T{7.7} chain --- the textbook
postulates recovered as theorems --- and the four net-new theorems
\TA{8}--\TA{11} covering the extremal (Bogomolnyi-analogue) reading
of CHSH saturation, the universal-identity reading of
indistinguishable-particle statistics, the \QiCON{} reading of
decoherence-rate conformality, and the reduced-state no-signaling
theorem; the candidate theorem \TA{12} (Fermi--Dirac statistics as
boundary inheritance), stated beyond the eleven, closes the Part.

Five worked examples then run the machinery on the laboratory's
canonical experiments: Stern--Gerlach (Part~V), the double slit and
decoherence (Part~VI), EPR, Bell, and CHSH (Part~VII),
Schr\"odinger's cat in 6D (Part~VIII), and
decoherence--einselection--pointer-basis selection (Part~IX).
Part~X sweeps the remaining paradoxes --- Wigner's friend,
Frauchiger--Renner, delayed choice, the quantum eraser, Zeno,
contextuality, teleportation --- and Part~XI surveys the seven living
interpretations and locates the present reading among them.

Two Parts then weigh the case. Part~XII is the Discussions:
significance, the comparison with the seven interpretations, the
why-complex-numbers question with the real-bulk series RB-1--RB-13,
the QM-specific eight-test empirical horizon, and the open questions
stated as open. Part~XIII draws the conclusions: the summit view ---
\emph{Einstein right in 6D; Bohr right in 4D} --- the
universal-identity reading of why every electron is the same, and the
fifty-year arc with the next decade of empirical tests.

The back matter is the workshop. Part~XIV collects the technical
appendices: the codebook, the glossary, the proof sketches, and the
theorem index. Part~XV answers the strongest objections a
foundations-of-quantum-mechanics referee, a particle/string theorist,
a condensed-matter physicist, an AMO experimentalist, and a historian
and philosopher of quantum mechanics would raise, with pointers to
the theorem, falsifier, or companion paper that bears the weight.
Part~XVI closes with the annotated references, each carrying a
Relevance note. Read in order, the sixteen Parts compose into a
single self-contained construction that a careful graduate student
can audit step by step; read by need, each Part names its imports and
can stand a visit on its own.

%% ===================================================================
%%  Roadmap of the sixteen Parts (house architecture: A2-GG6 v13.3 /
%%  A3-BI6 v13.6 / A4-Euclid v13.0 / A5-GDOS v13.0 / A6-GRIP v13.0 /
%%  A7-Qi4 v13.0 pattern -- vertical Part column, right phase
%%  sidebar, one highlighted central Part; here Part III, the
%%  constitutional bridge, as in the retired v11-era flowchart; the
%%  worked-example row V--IX keeps this paper's own five-box row
%%  precedent), clearpage-isolated minipage + \captionof.  The label
%%  fig:thirteen-parts-flowchart-a8 is kept from the retired
%%  flowchart so existing cross-references resolve.
%% ===================================================================
\begin{figure}[p]
\noindent
\begin{minipage}{\textwidth}
\centering
\resizebox{0.906\textwidth}{!}{%
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%% --- Parts I--II: entrance and inheritance ---
\node[found] (I) at (0, 0)
 {\textbf{Part~I}\quad\textbf{Introduction} \\[-1pt]
 \footnotesize The century-old measurement problem in plain view, and the reader's two maps};
\node[found] (II) at (0, -13mm)
 {\textbf{Part~II}\quad\textbf{The cohort inheritance} \\[-1pt]
 \footnotesize Imports from \AOne{}--\AEleven{}, one section per companion};

%% --- Part III: the constitutional bridge (highlighted) ---
\node[innov] (III) at (0, -30mm)
 {\textbf{Part~III}\quad\textbf{The constitutional bridge} \\[1pt]
 \footnotesize $\GG{}$~bulk \;$\to$\; \OGGEP{} \;$\to$\; \GDOS{} \;$\to$\; 4D~QM \\[-1pt]
 \footnotesize \emph{\textbf{The operational chain that compiles the deterministic bulk into textbook QM}}};

%% --- Part IV: statement of results ---
\node[geom] (IV) at (0, -47mm)
 {\textbf{Part~IV}\quad\textbf{Statement of results} \\[-1pt]
 \footnotesize Eleven theorems: \TA{1}--\TA{7} inherited, \TA{8}--\TA{11} net-new; \TA{12} candidate};

%% --- Parts V--IX: the five worked examples in one row ---
\node[example] (V) at (-62mm, -67mm) {\textbf{Part~V} \\[-1pt]
 \footnotesize Stern--Gerlach \\[-2pt] \footnotesize \textit{(measurement)}};
\node[example] (VI) at (-31mm, -67mm) {\textbf{Part~VI} \\[-1pt]
 \footnotesize Double-slit \\[-2pt] \footnotesize \textit{(duality)}};
\node[example] (VII) at (0mm, -67mm) {\textbf{Part~VII} \\[-1pt]
 \footnotesize EPR / Bell / CHSH \\[-2pt] \footnotesize \textit{(non-locality)}};
\node[example] (VIII) at (31mm, -67mm) {\textbf{Part~VIII} \\[-1pt]
 \footnotesize Schr\"odinger's cat \\[-2pt] \footnotesize \textit{(superposition)}};
\node[example] (IX) at (62mm, -67mm) {\textbf{Part~IX} \\[-1pt]
 \footnotesize Decoherence \\[-2pt] \footnotesize \textit{(pointer basis)}};

%% --- Parts X--XII: paradoxes, interpretations, discussions ---
\node[princ] (X) at (0, -84mm)
 {\textbf{Part~X}\quad\textbf{The paradox sweep} \\[-1pt]
 \footnotesize Wigner's friend, delayed choice, eraser, Zeno, contextuality, and the rest};
\node[princ] (XI) at (0, -97mm)
 {\textbf{Part~XI}\quad\textbf{The interpretation landscape} \\[-1pt]
 \footnotesize Where \GGTheory{} sits among the seven living interpretations};
\node[princ] (XII) at (0, -110mm)
 {\textbf{Part~XII}\quad\textbf{Discussions} \\[-1pt]
 \footnotesize Kill criteria, the eight-test horizon, why complex numbers, open questions};

%% --- Part XIII: conclusions ---
\node[summit] (XIII) at (0, -123mm)
 {\textbf{Part~XIII}\quad\textbf{Conclusions: the summit view} \\[-1pt]
 \footnotesize Einstein right in 6D; Bohr right in 4D};

%% --- Parts XIV--XVI: back matter ---
\node[back] (XIV) at (0, -136mm)
 {\textbf{Part~XIV}\quad\textbf{Technical Appendices} \\[-1pt]
 \footnotesize Codebook, glossary, proof sketches, theorem index};
\node[back] (XV) at (0, -149mm)
 {\textbf{Part~XV}\quad\textbf{Common Objections and Responses} \\[-1pt]
 \footnotesize The strongest objections, answered, with pointers};
\node[back] (XVI) at (0, -162mm)
 {\textbf{Part~XVI}\quad\textbf{References} \\[-1pt]
 \footnotesize Cited works, each with a relevance note};

%% --- Vertical flow arrows (the core chain in innov-orange) ---
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\draw[arrow] (IV.south) -- (VI.north);
\draw[arrow] (IV.south) -- (VII.north);
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\draw[arrow] (IV.south) -- (IX.north);
\draw[arrow] (V.south) -- (X.north);
\draw[arrow] (VI.south) -- (X.north);
\draw[arrow] (VII.south) -- (X.north);
\draw[arrow] (VIII.south) -- (X.north);
\draw[arrow] (IX.south) -- (X.north);
\draw[arrow] (X.south) -- (XI.north);
\draw[arrow] (XI.south) -- (XII.north);
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\draw[arrow] (XIII.south) -- (XIV.north);
\draw[arrow] (XIV.south) -- (XV.north);
\draw[arrow] (XV.south) -- (XVI.north);

%% --- Phase labels (right sidebar); standard boxes: half-width =
%% 60mm, label x = 67mm; Part III is wider (half-width 70mm), its
%% label sits at x = 75mm; the example row spans to +-75.5mm, its
%% label sits at x = 83mm.
\node[phaselabel] at (67mm, -6.5mm) {\emph{Entrance \&} \\ \emph{inheritance}};
\node[phaselabel, text=orange!70!black] at (75mm, -30mm)
 {\emph{\textbf{The bridge:}} \\ \emph{\textbf{the compiler}}};
\node[phaselabel] at (67mm, -47mm) {\emph{Results}};
\node[phaselabel] at (83mm, -67mm) {\emph{Worked} \\ \emph{examples}};
\node[phaselabel] at (67mm, -90.5mm) {\emph{Paradoxes \&} \\ \emph{interpretations}};
\node[phaselabel] at (67mm, -110mm) {\emph{Assessment}};
\node[phaselabel] at (67mm, -123mm) {\emph{Summit}};
\node[phaselabel] at (67mm, -149mm) {\emph{Back matter}};

\end{tikzpicture}%
}
\par\bigskip
\captionof{figure}[Roadmap of the sixteen Parts of GG-A8-QM.]{Roadmap of the sixteen Parts of \textsf{GG-A8-QM}. The argument develops in seven phases. \emph{Entrance and inheritance} (Parts~I--II): the century-old measurement problem in plain view, then the per-paper inheritance ledger from \AOne{} through \AEleven{}. \emph{The bridge} (Part~III, highlighted): the constitutional chain $\GG{}$~bulk $\to$ \OGGEP{} $\to$ \GDOS{} $\to$ 4D~QM that compiles the deterministic six-dimensional bulk into textbook quantum mechanics. \emph{Results} (Part~IV): the eleven principal theorems, \TA{1}--\TA{7} inherited verbatim from \AFive{} Part~III and \TA{8}--\TA{11} net-new, with the candidate \TA{12} beyond the eleven. \emph{Worked examples} (Parts~V--IX): Stern--Gerlach, double-slit, EPR/Bell/CHSH, Schr\"odinger's cat, and decoherence--einselection, each exhibiting the theorems on a concrete laboratory system. \emph{Paradoxes and interpretations} (Parts~X--XI): the paradox sweep, then the seven living interpretations. \emph{Assessment and summit} (Parts~XII--XIII): the Discussions --- kill criteria, the eight-test empirical horizon, the real-bulk series --- then the conclusions, \emph{Einstein right in 6D; Bohr right in 4D}. The three back-matter Parts collect the technical appendices (Part~XIV), the common objections and responses (Part~XV), and the annotated references (Part~XVI). Part~III is highlighted because the constitutional bridge delivered there is the paper's central claim --- the operational chain that compiles the bulk into the textbook rules --- on which the eleven theorems of Part~IV, the five worked examples, and every downstream reading stand.}
\label{fig:thirteen-parts-flowchart-a8}
\end{minipage}
\end{figure}


%% ===================================================================
%% PART II (2/14) -- The eleven-paper inheritance
%% ===================================================================
\GGPart[The cohort inheritance]{The cohort inheritance}{What \AEight{} receives from \AOne{} through \AEleven{}, and where it sits among the twenty-two papers}
\label{part:II}

This Part is the inheritance ledger of \AEight{}. The 6D-bulk-first reading that drives the rest of the paper --- and that resolves the measurement problem (via the always-on bulk-to-boundary projection of \OGGEP{}, \AFive{}~\S23), the preferred-basis problem, the Born-rule problem, the non-locality problem, every paradox of foundations, and every interpretive dispute (as the worked examples of Parts~V--IX, the paradox sweep of Part~X, the interpretation synthesis of Part~XI, and the empirical-horizon audit of Part~XII establish) --- does not originate in \AEight{}. Every structural ingredient \AEight{} uses is already proved or constructed in one of the ten upstream cohort papers. This Part assembles those inheritances in one place so the reader can see, on a single page, what \AEight{} owes to each cohort companion and what it does not.

The structure of the inheritance is by paper, in cohort order. One section per cohort companion, then a summary section (the twenty-two-paper cohort map itself stands at the close of Part~I, \S\ref{sec:cohort-a8}): the \GGConstitution{} of \AOne{} (the four axioms and seven postulates from which everything else descends); the bulk-geometry block (\ATwo{}'s six-dimensional bulk $\mathcal{M}_4 \times I_r \times S^1_\theta$, \AThree{}'s \BIsix{} anomaly closure, \AFour{}'s Euclidean discipline forcing zero free parameters); the \GDOS{} block of \AFive{} including the seven QM-postulate theorems \T{7.1}--\T{7.7} that \AEight{} lifts to graduate-textbook depth as $\TA{1}$--$\TA{7}$; the \GRIP{}/\QiFour{} machinery of \ASix{} and \ASeven{} that supplies the bulk-side dynamics and the admissibility filter; the universal-identity insight of \ANine{} Part~XII that grounds the every-electron-is-the-same reading; the cosmological specialization of \ATen{}; and the four-move operational anatomy of \AEleven{} that \AEight{} specializes to the QM stratum.

%% PHASE-4B CONTENT BEGINS HERE
%% ===================================================================
\section{The \GGConstitution{} of \AOne{}: four axioms and seven postulates}
\label{sec:inh-A1}

The foundational paper of the cohort,
\AFullOne{}~\citep{ArisakaA1Constitution}, sets the constitutional
rules. These rules are not specific to quantum mechanics; they
constrain every paper of the program. \AEight{}'s job is to
specialize the rules to the microscopic-observable sector. We
state the rules in plain language and then identify which are
load-bearing for what follows.

\subsection{The four axioms}
\label{ssec:inh-A1-axioms}

\AOne{} \S 2 introduces four axioms that any admissible physical
theory must satisfy.

\begin{itemize}[leftmargin=2em,itemsep=0.4em]
\item \textbf{\AX{1} --- Local Causality (\LC{}).} Physical
influence propagates in the forward time direction at finite speed.
Operationally: no closed timelike curves, no superluminal
signaling, no acausal feedback. In any admissible 6D bulk theory,
the bulk Hamiltonian density $\mathcal H_{\Bulk}(x, r, \theta)$ is
a polynomial in bulk fields and their derivatives at the same bulk
point, with no integro-differential or non-local kernels.
For \AEight{}, \AX{1} is the source of the forward-time semigroup
property of the boundary channel $\Phi_{t,s}$ that forces the
Lindblad / \GKLS{} structure of the master equation (\S
\ref{sec:OGGEP_lindblad}); and the source of the position-diagonal
nature of the boundary system--environment coupling that selects
the position basis as einselected (\T{7.6}).

\item \textbf{\AX{2} --- Geometry--Gauge Equivalence Principle
(\GGEP{}).} In the six-dimensional bulk, gravity and gauge
fields are two aspects of one composite connection. This is the
closed-bulk equivalence principle: the bulk theory is a single
geometric object, not a collection of separate gauge and
gravitational sectors propagating on a fixed background.
\AEight{} inherits \AX{2} indirectly --- through its open-system
extension \AX{3} --- and uses the composite-connection structure to
identify the bulk degrees of freedom that the boundary observer
traces over (the heavy KK modes and the $S^1_\theta$ holonomy).

\item \textbf{\AX{3} --- Operational/Open Geometry--Gauge
Equivalence Principle (\OGGEP{}).} Any admissible bulk theory must
come equipped with a bulk--boundary correspondence, a structure-preserving
functor $\mathcal D : \mathbf{Bulk} \to \mathbf{Op}$ that maps bulk
states to operational records produced by laboratory instruments.
The functor must respect causality (from \AX{1}), physicality
(positivity, i.e.\ the \CPTP{} property), and coarse-graining
(scale consistency). \AOne{} states this as an axiom rather than a
theorem because it is a constitutional condition on what counts as
a complete physical theory: a bulk theory without an explicit
$\mathcal D$ does not yet describe anything a laboratory can
measure. \AX{3} is the central import of \AEight{}: every result
in the seven \T{7.1}--\T{7.7} chain is a structural consequence of
\AX{3} once specialized to the QM-restricted boundary algebra.

\item \textbf{\AX{4} --- Maximum Symmetry, Minimum Freedom
(\MSMF{}).} The constitutional input ledger contains the smallest
number of independent inputs compatible with self-consistency.
\AFour{} sharpens this to the zero-free-continuous-parameter
statement: the input ledger splits into three forced groups (the
five-element \GUfive{} $\{c, G_N, \hbar; k_B, Q_B\}$,
the two topological constants $\pi$ and $e$, and the proved
\OGNfive{} integer ledger $(M, N, L; J, K) = (1, 3, 5; 7, 35)$),
with zero free continuous parameters. For \AEight{}, \MSMF{} is the
source of the determinism of the 6D bulk evolution: with no free
parameters there is no stochastic input to randomize the bulk
dynamics, and the probabilistic appearance of the 4D shadow can be
read entirely as a feature of the non-invertible projection rather
than as a feature of the bulk.
\end{itemize}

\subsection{The seven postulates}
\label{ssec:inh-A1-postulates}

\AOne{} \S 3 introduces seven postulates \POS{1}--\POS{7} that
specialize the four axioms to specific structural commitments.
The seven postulates and their constitutional content are:
\begin{itemize}[leftmargin=2em,itemsep=0.2em]
\item \POS{1}~\GI{} (Layered Gauge Invariance): each layer of the
cascade inherits the gauge structure of the layer above, with the
boundary algebra inheriting from the bulk gauge connection.
\item \POS{2}~\UEP{} (Unified Extremal Principle): the dynamical
content of the framework is fixed by a single variational principle
acting on the composite bulk--boundary structure.
\item \POS{3}~\AF{} (Anomaly Freedom): the bulk geometry must
satisfy the Green--Schwarz-modified Bianchi identity \BIsix{}
of \AThree{}, with the unique coefficient vector
$(1,-1,-2/5,-2/5)$ fixed by the topological factorization
$I_8 = \tfrac{1}{2} X_4^2$.
\item \POS{4}~\CP{} (Closure and Positivity): admissible boundary
channels must be completely positive and trace-preserving; this is
the constitutional source of the \CPTP{} structure that \TA{3} and
\TA{4} of the present paper exercise.
\item \POS{5}~\RG{} (Scale Covariance): the functor $\cD_{\OGGEP}$
commutes with renormalization-group rescaling along the radial direction $r$ of \ATwo{}; this is the constitutional source
of the \QiCON{}-conformality that \TA{10} exercises.
\item \POS{6}~\OGN{} (Occam Gauge Normalization): the input ledger
admits no hidden continuous knobs. Together with \MSMF{} (\AX{4}),
this enforces the zero-free-continuous-parameter discipline of
\AFour{}.
\item \POS{7}~\MF{} (Measurement and Falsifiability): each
admissible bulk theory must come equipped with an operational
recipe by which laboratory measurements probe the bulk content.
This is the constitutional source of the instrument formalism that
\TA{3} and \TA{4} exercise.
\end{itemize}

For \AEight{}, the most directly load-bearing postulates are
\POS{4}~\CP{} (the closure-and-positivity postulate, which forces
the \CPTP{} structure of the boundary channel and therefore drives
\TA{3} and \TA{4}); \POS{7}~\MF{} (the
measurement-and-falsifiability postulate, which forces the
instrument formalism on which the whole eleven-theorem inventory of
Part~\ref{part:IV} rests); and \POS{1}~\GI{} (the layered
gauge-invariance postulate, which forces the QM-restricted boundary
algebra to inherit gauge structure from the bulk). The remaining
four postulates enter indirectly through the upstream cohort papers:
\POS{2}~\UEP{} underwrites the variational structure that produces
the GKLS generator of \TA{2}; \POS{3}~\AF{} fixes the \BIsix{}
closure inherited from \AThree{} (used only in cross-paper
consistency checks for the QM stratum); \POS{5}~\RG{} drives
the conformality prediction of \TA{10}; \POS{6}~\OGN{} underwrites
the zero-free-parameter character of every numerical prediction in
the eight-test horizon.

\subsection{What \AEight{} takes from \AOne{}}
\label{ssec:inh-A1-takes}

In one sentence: \AEight{} takes the constitutional warrant.
The seven theorems \TA{1}--\TA{7} (operational restatements of
\T{7.1}--\T{7.7} of \AFive{} Part~III) do not rest on auxiliary
hypotheses introduced inside \AEight{}; they rest on \AX{1}
(forward time, locality), \AX{3} (the \OGGEP{} functor), \POS{4}
(closure and positivity, which forces \CPTP{}), and \POS{7}
(measurement and falsifiability, which forces the instrument
formalism), all imported from \AOne{}. The remaining axioms and
postulates contribute through the bulk-geometry block of the next
section.


%% ===================================================================
\section{The bulk-geometry block: \ATwo{} (\GG{}), \AThree{}
(\BIsix{}), \AFour{} (Euclidean discipline)}
\label{sec:inh-A234}

The three foundation papers that specify the geometric stage on
which the cohort is set.

\subsection{\ATwo{}: the 6D bulk}
\label{ssec:inh-A2}

\AFullTwo{}~\citep{ArisakaA2GG6} constructs the 6D bulk
\GG{} as the unique geometry compatible with the
\GGConstitution{}. The metric is
\begin{equation}\label{eq:inh-bulk-metric}
ds_6^2 \;=\; e^{-2r}\,\eta_{\mu\nu}\,dx^\mu\,dx^\nu
\;+\; dr^2 \;+\; d\theta^2,
\qquad r \in [0, K], \;\; \theta \in [0, 2\pi),
\end{equation}
with composite gauge--gravity group
$G_{\rm tot} = \Spinone \times \SUfive \times \UoneB \times \UoneQi$
and the proved \OGNfive{} integer ledger
$(M, N, L; J, K) = (1, 3, 5; 7, 35)$. The scale profile $e^{-2r}$
realizes the radial direction $r$ as the renormalization-group
scale, and the compact circle $S^1_\theta$ supplies the
topologically protected Wilson-line holonomy $\Theta_H$ that
\AEight{} will read in Part~\ref{part:VIII} as the geometric
substrate of the ``hidden variable'' that the standard
hidden-variable program has been groping toward since
de~Broglie 1924 and Bohm 1952. \AEight{} imports the bulk
metric \eqref{eq:inh-bulk-metric}, the KK mode expansion of bulk
fields, and the $S^1_\theta$ holonomy structure verbatim; no
re-derivation is performed.

\paragraph{The eight theorems of \ATwo{}.} Part~II of \ATwo{}
proves eight named theorems T1--T8 that together force the
6D bulk to its unique cohort-canonical form:
T1 (Dimensional Minimality: the bulk must be six-dimensional);
T2 (the Canonical Metric \eqref{eq:inh-bulk-metric});
T3 (Six Forces with Three Abelian Gauge Invariances, treated
below); T4 (the \BIsix{} Coefficient Vector $(1,-1,-2/5,-2/5)$,
treated in \S\ref{ssec:inh-A3}); T5 (Single-Tensor Anomaly
Factorization $I_8 = \tfrac12 X_4 \wedge X_4$); T6 (the
\OGNfive{} Diophantine Cascade $(1,3,5;7,35)$); T7 (BRST
Consistency and the homogeneous Slavnov--Taylor identity); and
T8 (the Arisaka Equation and Quadruple Gauge Locking, treated
below). \AEight{} inherits all eight theorems verbatim; the
two with direct operational consequences for the QM register
are T3 (the three-abelian structure that locates \UoneQi{} as
one of three gauge factors, not the unique substrate) and T8
(the Arisaka Equation, which constrains the boundary currents
that any quantum measurement registers).

\paragraph{T3: Six forces with three abelian gauge invariances.}
\ATwo{} T3 records the boundary gauge structure in two
forms~\citep[T3 of][]{ArisakaA2GG6}. The high-energy boundary
group, before electroweak symmetry breaking, is the
six-factor product
$G_{\rm bdy}^{\rm hi}
= \Spin(1,3)\times\SU(3)_C\times\SU(2)_L\times\Uone_Y\times\UoneB\times\UoneQi$,
with three abelian factors ($\Uone_Y$, $\UoneB$, $\UoneQi$).
After electroweak symmetry breaking and the identification of
$\Uone_{\rm EM} \subset \SU(2)_L \times \Uone_Y$ as the unbroken
combination, the low-energy boundary group is
\begin{equation}\label{eq:inh-three-abelians}
\boxed{\;
G_{\rm bdy}^{\rm lo}
\;=\;
\underbrace{\Spin(1,3)}_{\text{gravitational}}\,\times\,
\underbrace{\SU(3)_C}_{\text{strong}}\,\times\,
\underbrace{\Uone_{\rm EM}}_{\text{electromagnetic}}\,\times\,
\underbrace{\UoneB}_{\text{baryonic}}\,\times\,
\underbrace{\UoneQi}_{\text{informational}},\;}
\end{equation}
with the broken weak combination registered as the
short-distance residue of $\SU(2)_L$ now hidden inside the
$\Uone_{\rm EM}$ factor and the massive $W^\pm,Z^0$ gauge bosons.
Three abelian gauge invariances --- $\Uone_{\rm EM}$, $\UoneB$,
and $\UoneQi$ --- survive at all energies and constitute the
``three-abelian'' structure that \ATwo{} T3 names. This three-abelian structure is structurally important
for the operational reading of \AEight{}: it locates \UoneQi{}
as one specific abelian gauge factor (the ``informational''
force) among three, not as the unique substrate of the
hidden-variable structure. The \emph{QM mechanism itself} ---
the bulk-to-boundary projection from the 6D bulk to the 4D boundary
via \OGGEP{} (\AX{3}) --- does not depend on any specific gauge
factor; a simpler bulk (e.g.\ pure $\Uone_{\rm EM}$ with the
phase axis $S^1_\theta$ supplying KK modes and the radial direction $I_r$ supplying renormalization-group modes)
already reproduces Maxwell + Lorentz-force physics on the 4D
boundary via the same mechanism. \UoneQi{} is canonical for the
full cohort (where BI-6 anomaly closure plus the SM gauge content
forces it as one of three abelians), but it is not constitutive
of the bulk-to-boundary projection that produces QM.

\paragraph{T8: The Arisaka Equation and Quadruple Gauge Locking
(\QGL{}).} The three abelian gauge invariances of T3, together
with \BIsix{} anomaly inflow under the \OGGEP{} bulk-to-boundary projection, force the four boundary currents (energy, electric,
baryonic, informational) to be locked together
\citep[T8 of][]{ArisakaA2GG6}:
\begin{equation}\label{eq:inh-arisaka}
\boxed{\;
\frac{J_E}{\QB} \;=\; J_e \;=\; J_B \;=\; J_{\rm Qi}.\;}
\end{equation}
This is the \emph{Arisaka Equation}, also called Quadruple Gauge
Locking (\QGL{}). For \AEight{}, the Arisaka Equation
constrains the boundary currents that any quantum-measurement
record has to satisfy: the energy dissipated in a measurement
(via the Landauer bound that \QiQUA{} of \ASeven{} formalizes)
is locked to the electric, baryonic, and informational currents
by the constitutional structure of the bulk. This is the
operational content of $\QGL{}$ in the QM register, treated
operationally in Part~\ref{part:IV} \S\ref{sec:TA-3} (the
Born-rule theorem inherits its CPTP structure ultimately from
this current-locking constraint via Stinespring dilation) and in
Part~\ref{part:X} \S\ref{sec:px-crypto} (the
quantum-cryptography protocols exploit the bulk current-locking
to detect eavesdroppers).


\subsection{\AThree{}: the \BIsix{} closure identity}
\label{ssec:inh-A3}

\AFullThree{}~\citep{ArisakaA3BI6} establishes the \BIsix{}
identity --- the Green--Schwarz-modified Bianchi identity --- as
the unique anomaly-closure condition on the composite connection of
\eqref{eq:inh-bulk-metric}, with coefficient vector
$(1, -1, -2/5, -2/5)$ fixed by the topological factorization
$I_8 = \tfrac{1}{2} X_4^2$. \AEight{} uses the \BIsix{} closure as
a consistency check on the bulk side: any 4D operational
projection of the bulk must respect the anomaly-closure pattern
that \AThree{} proves. Operationally for QM, this means that no
single-particle quantum-mechanical observable can violate the bulk
anomaly structure; the constraint is too far upstream to be
visible in any laboratory experiment on a single qubit, but it
becomes visible in collective measurements that probe the chiral
sector (Standard-Model phenomenology, the subject of \ANine{}).

\subsection{\AFour{}: the Euclidean discipline}
\label{ssec:inh-A4}

\AFullFour{}~\citep{ArisakaA4Euclid} sharpens \MSMF{} into the
\emph{zero-free-continuous-parameter} discipline. The
twenty-eight cohort outputs --- the four Standard-Model gauge
couplings, the eighteen fermion masses across the three
generations, the three CKM mixing angles plus one CP phase, the
three neutrino masses plus three PMNS mixing angles plus one CP
phase, the QCD theta parameter, the Higgs vev plus its
self-coupling, and the four cosmological parameters $H_0$,
$\Omega_m$, $\Omega_\Lambda$, $\Omega_b$ --- are all derived,
across the cohort, from the eight-element input ledger of
\S\ref{ssec:inh-A1-axioms}, item~\AX{4}, with zero free
continuous parameters. Within the \AFour{} master audit itself,
the gauge-and-cosmological sector closes the audit table on
twenty-two derived entries; the eighteen fermion masses and the
CKM/PMNS mixings are tracked under \AFour{}'s O9 family-mass
heading as ``Partially resolved'' and structurally completed in
\ANine{}'s $\mathbb{CP}^{2}$ family-projector cascade
(\S\ref{sec:inh-A9}). This is the \emph{N-to-zero compression}
that distinguishes the framework from parameter-counting
approaches; the wording ``minimal input set'' or ``compression
ratio'' is strictly misleading because the ratio is $N/0$,
undefined.

For \AEight{} the Euclidean discipline matters because it
identifies which numerical predictions of quantum mechanics are
free parameters (none) and which are derived (all). The Tsirelson
bound $|S| \le 2\sqrt 2$ that the Aspect experiments saturate is
not a free parameter of \AEight{}; it is the extremal (Bogomolnyi-analogue) stationarity
of the CHSH functional at the singlet (\TA{8}), the standard quantum
ceiling, with no adjustable knob. The same is
true of the decoherence rate $\Gamma$ (fixed by scale-overlap integrals of bulk KK modes; see \T{7.5}) and of the pointer-basis
selection (fixed by \AX{1} bulk locality; see \T{7.6}).


%% ===================================================================
\section{\AFive{}-\GDOS{} and the \T{7.1}--\T{7.7} chain: the core
inheritance}
\label{sec:inh-A5}

\AFullFive{}~\citep{ArisakaA5GDOS} is the immediate upstream of
\AEight{} and the single most important inheritance. It
constructs the boundary law \GDOS{} and proves the
QM-postulates-as-theorems chain that \AEight{} now lifts to
graduate-textbook form.

\emph{If \ATwo{} is the father, \AFive{} is the mother.}
\ATwo{} supplies the 6D bulk geometry \GG{}; \AFive{}
supplies the canonical boundary law \GDOS{} that \OGGEP{}
(\AX{3} of the \GGConstitution{}) forces when applied to that
bulk; and \AEight{} reads the QM child as the operational
specialization of \GDOS{} to the microscopic-observable sector.
The genealogy is exact: every textbook postulate of quantum
mechanics (Hilbert space, Schr\"odinger equation, Born rule,
L\"uders projection) is a derived theorem of \GG{} + \OGGEP{},
with the derivation chain proceeding via the six core theorems
\T{1}--\T{6} of \AFive{} Part~II (the construction of \GDOS{})
and the seven sub-theorems \T{7.1}--\T{7.7} of \AFive{} Part~III
(the specialization of \GDOS{} to the QM stratum). The present
paper is the standalone graduate-textbook lift of the child.

\subsection{The eleven theorems of \AFive{}}
\label{ssec:inh-A5-T1-T11}

\AFive{} has a principal-theorem inventory of eleven named results
\T{1}--\T{11}, organized by Part:

\begin{itemize}[leftmargin=2em,itemsep=0.3em]
\item \textbf{Part~II}, \T{1}--\T{6}: the \GDOS{} construction.
\T{1} forces existence of the boundary channel; \T{2} constructs
it via Nakajima--Zwanzig reduction; \T{3} shows the channel is
automatically \CPTP{} via Stinespring dilation; \T{4} takes the
Markovian limit and derives the \GKLS{} generator; \T{5} takes
the mesoscopic limit and derives the \GDOS{} continuity equation;
\T{6} unifies all five into the canonical boundary law.

\item \textbf{Part~III}, \T{7.1}--\T{7.7}: the QM-postulates-as-
theorems chain. Seven dual-layer sub-theorems with an algebraic
content \T{7.k.1} and an operational content \T{7.k.2}. Tabulated
below in \S\ref{ssec:inh-A5-T71-T77}.

\item \textbf{Parts~IV--V}, \T{8.x}--\T{11.x}: the gravity, SM,
and cosmos specializations. These do not enter \AEight{}'s
quantum-mechanical reading and are deferred to \ANine{}/\ATen{}.
\end{itemize}

\subsection{The seven \T{7.k} theorems in textbook form}
\label{ssec:inh-A5-T71-T77}

The textbook-postulate chain that \AEight{} lifts. We tabulate the
seven sub-theorems with their algebraic content, operational
content, and the standard-textbook postulate each replaces:

\begin{table}[H]
\centering
\caption[The seven T7.k sub-theorems and their inherited specialization]{The seven \T{7.k} sub-theorems of \AFive{} Part~III with
their inherited specialization in \AEight{} Part~\ref{part:IV}. Each row names the parent result, the operational restatement it becomes at the boundary, and the reading under which the restatement is carried out. Nothing in the right-hand column is new physics: the specialization supplies the QM-stratum reading and adds no premise beyond those \AFive{} already carries.}
\label{tab:inh-T7k-summary}
\small
\renewcommand{\arraystretch}{1.25}
\begin{tabular}{@{}llp{4.75cm}p{4.55cm}@{}}
\toprule
\textbf{\T{7.k}} & \textbf{Standard postulate replaced} & \textbf{Algebraic content (\T{7.k.1})} & \textbf{Operational content (\T{7.k.2})} \\
\midrule
\T{7.1} & Hilbert space & GNS embedding $\iota: \cH_\Bdy^{(\rm QM)} \hookrightarrow \cH_\Bulk$ & Density matrix $\rho_\Bdy$ from bulk partial trace \\
\T{7.2} & Schr\"odinger eq.\ & GKLS reduces to von~Neumann eq.\ as $\gamma_\alpha \to 0$ & $i\partial_t |\psi\rangle = H_{\rm eff}|\psi\rangle$ in the closed-system limit \\
\T{7.3} & Born rule & $p_\alpha = \Tr(\cI_\alpha \rho)$ from \CPTP{} completion & $p_\alpha = |\langle\alpha|\psi\rangle|^2$ for sharp pure-state measurement \\
\T{7.4} & L\"uders projection & Conditional CPTP posterior $\rho^{(\alpha)} = \cI_\alpha(\rho)/p_\alpha$ & Textbook L\"uders rule for pure projective measurement \\
\T{7.5} & --- & $\rho_{\alpha\beta}(t) = \rho_{\alpha\beta}(0) e^{-\Gamma t}$ off-diagonal damping & Decoherence as observed loss of phase coherence \\
\T{7.6} & --- & Pointer basis = eigenbasis of system--environment coupling & Einselection of the classical basis from bulk locality (\AX{1}) \\
\T{7.7} & --- & Entangled pair is one non-separable joint bulk state; correlations are its common-source projection shadow & CHSH violation up to the Tsirelson bound $2\sqrt 2$ \\
\bottomrule
\end{tabular}
\end{table}

The structural significance of the table is that four of the seven
sub-theorems --- \T{7.1}, \T{7.2}, \T{7.3}, \T{7.4} --- recover the
four textbook postulates of quantum mechanics directly, and the
remaining three --- \T{7.5}, \T{7.6}, \T{7.7} --- recover three
structures (decoherence, einselection, Bell-violation) that a
century of foundations work has identified as essential but which
the textbook treatment never axiomatized. All seven are
specializations of the single open-system postulate \AX{3} of
\AOne{} via the bulk geometry of \ATwo{} via the \GDOS{} channel
of \AFive{} Part~II. No additional axiom is introduced in any of
the seven sub-theorems.

\subsection{What \AEight{} takes from \AFive{}}
\label{ssec:inh-A5-takes}

\AEight{} takes the entire \T{7.1}--\T{7.7} chain verbatim.
Part~\ref{part:IV} of the present paper restates each theorem,
expands the surrounding pedagogy --- the physics motivation, the
quantitative anchor, the daily-life metaphor, the kill criterion,
the plain-English close --- to the depth a graduate student
trained on Sakurai or Cohen-Tannoudji will need, and supplies
worked examples in Parts~\ref{part:V}--\ref{part:IX}. No theorem
of \AFive{} Part~III is re-proved in \AEight{}; the proofs are
inherited. The body of the present paper is therefore one
extended commentary on a chain that is already established
upstream.


%% ===================================================================
\section{The \GRIP{}/\QiFour{} machinery: \ASix{} and \ASeven{}}
\label{sec:inh-A67}

The two cohort papers that supply the operational dynamics one
layer above the \T{7.1}--\T{7.7} core.

\subsection{\ASix{}: the \GRIP{} cascade and the \GiFour{} pillars}
\label{ssec:inh-A6}

\AFullSix{}~\citep{ArisakaA6Gi4} develops the geometric runtime of
the boundary observer. The primary cascade of \ASix{} is the
three-stage descent \DGI{} $\to$ \GSSB{} $\to$ \GA{} on the
reduced state manifold; \GFF{} is the Lyapunov landscape on which
the descent runs rather than a stage of the cascade itself. In
the four-stage reading inherited from \AEleven{}
(in which \GFF{} is parenthesized as the landscape between \DGI{}
and \GSSB{}), the cascade is written
\begin{equation}\label{eq:inh-GRIP-cascade}
\DGI \;\to\; (\GFF) \;\to\; \GSSB \;\to\; \GA,
\end{equation}
where \DGI{} is the dynamical-geometric input, \GFF{} is the
geometric free-energy functional that supplies the gradient
landscape, \GSSB{} is the geometric spontaneous symmetry-breaking
projection, and \GA{} is the geometric attractor selected at the
end. The cascade is governed by the four \GiFour{} pillars of
\ASix{}~\citep{ArisakaA6Gi4} --- \TGG{} (Total Geometric--Gauge
object), \CCI{} (Covariant Conservation Identity), \GFF{}
(Geometric Free-energy Functional), and \GRIP{} (Gauge-Runtime
Invariance Pipeline) --- of which the most load-bearing for
\AEight{} are \TGG{} (which forces the four-current geometric-
gauge ledger that \ATwo{} T8 reads off as the Arisaka equation
\eqref{eq:inh-arisaka}) and \CCI{} (which forces the bulk-to-
boundary descent rule that \AFive{} T-GDOS-7.1 reads off as the
boundary GNS embedding).

For \AEight{}, the cascade
\eqref{eq:inh-GRIP-cascade} supplies the picture of \emph{how}
decoherence and einselection happen in practice. At the
charged-lepton stratum --- the stratum at which a single qubit
lives --- the \DGI{} stage is the bulk fermion zero mode; the
\GFF{} stage is the energy functional set by the
system--environment coupling; the \GSSB{} stage is the off-diagonal
damping of \T{7.5}; and the \GA{} stage is the einselected
classical record of \T{7.6}. \AEight{}
Part~\ref{part:IX} performs this decomposition explicitly.

\subsection{\ASeven{}: the \QiFour{} admissibility pillars}
\label{ssec:inh-A7}

\AFullSeven{}~\citep{ArisakaA7Qi4} introduces the four \QiFour{}
admissibility pillars that any operational theory must satisfy.
The four pillars are \QiQUA{} (event-ledger conservation),
\QiEP{} (selector-law optimality), \QiCON{} (conformality
under rescaling), and \QiHAL{} (architecture-stability under
holography). Each pillar gives a falsifiable empirical
signature.

For \AEight{} the four pillars enter as follows:

\begin{itemize}[leftmargin=2em,itemsep=0.3em]
\item \QiQUA{} (event-ledger): every quantum measurement
satisfies the Landauer bound $\dot Q \ge \Gamma\,k_B T \log_2 N$
on energy dissipated per outcome registered.

\item \QiEP{} (selector law): the realized measurement outcome
minimizes the \GFF{} functional given the prior, recovering the
Born-rule probabilities as the optimal posterior --- a maximum-
likelihood reading of \T{7.3}.

\item \QiCON{} (conformality): decoherence rates scale invariantly
under system rescaling, predicting the $1/f$-noise spectrum that
laboratory measurement apparatuses universally exhibit. This is
the most distinctive empirical signature of the cohort's
treatment of QM and is one of the eight-test horizon items in
Part~\ref{part:XII}.

\item \QiHAL{} (architecture): measurement records are
topologically protected by the bulk Wilson-line holonomies on $S^1_\theta$ (with the cohort canonical bulk supplying the \UoneQi{} factor) --- the same
$S^1_\theta$ Wilson-line holonomy of \ATwo{}. This is why a
photograph remains a photograph after the photographer is gone:
the record is locked in a topological invariant of the bulk
configuration, not in a metastable mark.
\end{itemize}


%% ===================================================================
\section{\ANine{}-\textsf{SM} and the universal-identity insight}
\label{sec:inh-A9}

\AFullNine{}~\citep{ArisakaA9SM} derives the Standard-Model
particle content (gauge group, three generations, fermion masses
and mixings, CKM and PMNS angles) from the cascade
\eqref{eq:inh-GRIP-cascade} specialized to the family-projector
stratum $\mathbb{CP}^{N-1} = \mathbb{CP}^{2}$ of \ANine{}, with
the twenty-eight Standard-Model parameters read off as
cascade-depth assignments and algebraic descendants on that
single stratum. The seventeen-step layer enumeration that
organizes the full GRIP descent --- gauge group, electroweak
vacuum, atomic ground states, molecular closure, amino-acid
selection, polymer closure, energy--information closure, and the rest
--- is the layered-cascade catalog of \AEleven{}, which
threads the cascade through cosmos/life/mind; \ANine{}'s SM
derivation is the specialization of that catalog to the
gauge-and-mass layers only. The closing Part of \ANine{}
(Part~XII \S\,60 in the cohort-wide citation convention)
introduces the \emph{universal-identity insight} that \AEight{}
will lift in its own closing Part~\ref{part:XIII}.

\subsection{The universal-identity insight}
\label{ssec:inh-A9-univ-id}

The universal-identity insight is the structural reason why every
electron is the same. In the cascade picture, the cascade at the
charged-lepton stratum has a unique stable \GA{} (the electron
with mass $m_e = 0.511$~MeV/$c^2$ and charge $-e$); every electron
that is realized anywhere in the universe is a token of this
single type, with the type fixed by the geometric attractor
selection rather than by any individual quantum-mechanical
preparation. The insight propagates: every hydrogen atom is the
same because hydrogen is the unique stable \GA{} of the atomic
stratum; every ATP molecule is the same because ATP is the unique
stable \GA{} of the energy--information stratum; every first cell is the
same in the appropriate structural sense because cellularity is
the unique stable \GA{} of the cellular stratum. The
universal-identity reading is the structural-realist genealogy of the framework.

For \AEight{}, the universal-identity insight grounds the closing
summit view of Part~\ref{part:XIII}: every laboratory measurement
of an electron's spin returns the same eigenvalues because every
electron is the same type, with the type fixed upstream of any
measurement preparation. The quantum-mechanical postulate that
all electrons are identical particles is therefore not a postulate
of quantum mechanics but a structural feature of the cascade that
quantum mechanics inherits.

\subsection{What \AEight{} takes from \ANine{}}
\label{ssec:inh-A9-takes}

A single insight: the universal-identity reading. All
\T{7.1}--\T{7.7} statements about a generic ``qubit'' or a generic
``boundary observer'' become statements about a generic token of
a universally fixed type, with the type pinned by the cascade
attractor selection. The closing Part~\ref{part:XIII} of
\AEight{} states this in its summit voice.


%% ===================================================================
\section{\ATen{}-Cosmos: cosmological specialization}
\label{sec:inh-A10}

\AFullTen{}~\citep{ArisakaA10Cosmos} specializes the cohort
machinery to the cosmological stratum: the boundary cosmology with
$\Omega_m = 1/\pi$, $\Omega_\Lambda = 1 - 1/\pi$, $\Omega_b =
1/(2\pi^2)$; the modified Friedmann equations; the two-stage
Hubble corridor; the Kaxion dark-matter prediction; the
baryogenesis and BBN treatments; the JWST early-galaxy
phenomenology. For \AEight{} the cosmological content is not
directly imported --- the present paper is about laboratory
quantum mechanics, not cosmology --- but the methodology is.

The \emph{bulk-to-boundary-projection methodology} of \ATen{} ---
specialize the bulk \OGGEP{} dictionary to the boundary stratum
of interest, derive the boundary observables as theorems of the
specialization, identify which empirical signatures are forward
predictions, which are postdiction-audits, and which are
consistency checks --- is the same methodology \AEight{} applies
to the QM stratum. Where \ATen{} produces a cosmological Pull
table with $\Omega_m$, $\Omega_\Lambda$, $H_0$, $\Omega_b$, etc.,
\AEight{} produces a QM Pull table with Tsirelson saturation,
decoherence-rate scaling, $1/f$ noise, einselection-rate, etc.
The methodology is identical; only the stratum differs.


%% ===================================================================
\section{\AEleven{}-Origin: the four-move operational anatomy}
\label{sec:inh-A11}

\AFullEleven{}~\citep{ArisakaA11Origin} is the closing capstone of
the cohort and supplies the operational anatomy of the cascade
\eqref{eq:inh-GRIP-cascade} at the level of generality that
\AEight{} specializes.

\subsection{The four moves}
\label{ssec:inh-A11-four-moves}

\AEleven{} introduces a four-move decomposition of the
cascade that maps onto the \DGI{}--\GFF{}--\GSSB{}--\GA{}
sequence:

\begin{enumerate}[leftmargin=2em,itemsep=0.3em]
\item \textbf{Move~1 --- Recognition (\DGI{}).} The cascade
identifies a stratum's dynamical-geometric input from the bulk
state.

\item \textbf{Move~2 --- Projection (\GFF{}).} The bulk state is
projected onto the boundary algebra of the stratum, producing the
free-energy functional whose minimizers are the candidate stable
configurations.

\item \textbf{Move~3 --- Selection (\GSSB{}).} The cascade selects
the unique \GFF{}-minimizing configuration on the stratum;
suboptimal configurations are damped away.

\item \textbf{Move~4 --- Reading (\GA{}).} The selected
configuration is registered as the stratum's stable geometric
attractor and becomes the dynamical-geometric input for the next
stratum.
\end{enumerate}

The three operational-anatomy theorems of \AEleven{}
(\textsf{T-GRIP-In-Lyapunov-Descent},
\textsf{T-GRIP-Out-Necessity}, \textsf{T-GGA-as-Ground-State})
prove that the four stages of the cascade
$\DGI{} \to \GFF{} \to \GSSB{} \to \GA{}$ are forced rather than
phenomenological. \AEleven{} maps the cohort axioms to
the four stages as follows~\citep{ArisakaA11Origin}: \AX{2}
(\GGEP{}) makes \DGI{}'s bulk substrate coherent and so forces
the recognition of the dynamical-geometric input; \AX{3}
(\OGGEP{}) makes \GSSB{}'s projection operational and so forces
the boundary-algebra image of the bulk state; \AX{4} (\MSMF{})
makes \GA{} selection unique and so forces the attractor
outcome; the \GFF{} stage is the
gradient flow that connects them, without a single axiom as its
forcer. \AX{1} (Local Causality) enters the cascade indirectly,
through the forward-time semigroup property of the boundary
channel, which is the constitutional source of the Lindblad /
\GKLS{} normal form that \TA{2} exercises in the QM register.

\subsection{What \AEight{} takes from \AEleven{}}
\label{ssec:inh-A11-takes}

\AEight{} specializes the four-move anatomy to the
charged-lepton stratum --- the stratum at which a single qubit
lives. Move~1 is the bulk fermion zero mode that realizes the
spin degree of freedom of an electron. Move~2 is the boundary
partial trace that produces the density matrix
$\rho_\Bdy = \Tr_{\rm env}\rho_\Bulk$. Move~3 is the off-diagonal
damping of \T{7.5} that selects the pointer basis as the
\GFF{}-minimizing posterior. Move~4 is the einselection of
\T{7.6} that registers the classical record.

The four-move reading recovers, on the QM stratum, what Bohr was
seeing operationally and what Everett was seeing unitarily: the
two readings are projections of the same four-move cascade onto
two different aspects, and both are correct within their domains.
We make the recovery explicit in Part~\ref{part:XI}.


%% ===================================================================
\section{Summary and coherence-map}
\label{sec:inh-summary}

The inheritance ledger of \AEight{} reduces to a small number of
load-bearing imports. The constitutional warrant comes from
\AOne{} (\AX{1}, \AX{3}, \POS{4}, \POS{7}). The bulk geometry comes from
\ATwo{}--\AThree{}--\AFour{} (the scale-profile metric, the \BIsix{}
closure, the Euclidean discipline). The boundary law and the
seven-theorem chain come from \AFive{} Part~II (\T{1}--\T{6}) and
Part~III (\T{7.1}--\T{7.7}). The operational runtime comes from
\ASix{} (\GRIP{} cascade + \GiFour{} pillars) and \ASeven{}
(\QiFour{} admissibility pillars). The universal-identity
reading comes from \ANine{} Part~XII. The bulk-to-boundary-projection methodology comes from \ATen{}. The four-move operational
anatomy comes from \AEleven{}.

\noindent The twenty-two-paper cohort map, with the branch-level reading
of the two phenomenology branches and the Foundations Branch, stands at
the close of Part~I (\S\ref{sec:cohort-a8}, \figref{fig:inh-cohort-map});
this Part is its per-paper detail.

\subsection{What is not inherited}
\label{ssec:inh-not}

The reader is entitled to a complementary statement of what
\AEight{} does \emph{not} inherit, so that the boundary of the
paper's claim is visible. \AEight{} does not inherit any
specifically biological or cognitive content of \ASix{} or
\AEleven{} (those are out of scope for the QM stratum). It does
not inherit the cosmological Pull table of \ATen{} (cosmology
is the subject of a separate paper, not a free parameter of
\AEight{}). It does not inherit the high-energy KK
phenomenology of the predictions branch \PaperPOne{} (KK lock
cluster) or the dark-matter prediction of \PaperCOne{} (Kaxion).
These cohort companions enter only as cross-references where
relevant. The paper before you is one paper of a twenty-two-paper
cohort; the cohort is closed, and \AEight{} is the closing
member.

\subsection*{Part-end close}

The inheritance ledger of this Part is the answer to a question a
careful reader will have asked from the start: \emph{is this
paper introducing new physics, or is it re-presenting old
physics?} The answer is neither. The physics is fixed already
by the ten upstream cohort papers; the present paper specializes,
restates, and supplies textbook-pedagogical apparatus for the
specialization to the microscopic-observable sector. Everything
in \AEight{} that looks like a new theorem is a specialization of
a theorem of \AFive{} Part~III; everything that looks like a new
worked example is a calculation that the cohort apparatus makes
inevitable; everything that looks like a new philosophical
position is a structural reading of inheritances from \AOne{} and
\AEleven{}.

After all, a paper that introduces no new physics can still
introduce a new way of seeing the physics --- and the standalone
graduate-textbook reformulation of quantum mechanics under the
\GGTheory{} framework is exactly that: a new way of seeing the
same century-old content, with the operational shadow and the 6D
bulk source both in clear view at once.



%% ===================================================================
%% PART III — The Constitutional Bridge
%% ===================================================================
\begin{table}[H]
\centering\small
\caption[Axioms and postulates used in the quantum-mechanical derivation]{\GGConstitution{} / \GGtheory{} axioms and postulates used
in the quantum-mechanical derivation. The left column names each axiom or postulate by its constitutional number, the middle column states what it supplies here, and the right column names the result that consumes it. The list is exhaustive in one direction only: every theorem below rests on these, but not every axiom of the \GGConstitution{} is needed to reach quantum mechanics.}
\label{tab:axioms_used}
\renewcommand{\arraystretch}{1.2}
\begin{tabular}{@{}llp{4.45cm}p{5.60cm}@{}}
\toprule
\textbf{Label} & \textbf{Name} & \textbf{Statement} &
\textbf{Role in QM derivation}\\
\midrule
AX-1 & Local causality & Influences propagate forward in time
at finite speed &
Forward time-ordering of \CPTP{} semigroup $\Rightarrow$
GKSL generator \\
AX-2 & Primitive connection & The primitive field is a gauge
connection on $M_6$ &
Gauge structure of bulk Hilbert space $\Hilb_4$ \\
AX-3 & \OGGEP{} & Operational observation is a structure-
preserving functor $\mathcal{D}: \mathbf{Bulk}\to\mathbf{Op}$ &
Defines measurement; gives \CPTP{} + instruments \\
AX-4 & \MSMF{} & Max symmetry, min freedom; no free parameters &
Forces deterministic bulk; forces Tsirelson bound $2\sqrt{2}$ \\
POS-CP & Complete positivity & Channels preserve positivity on
composed systems &
Restricts generator to GKSL form \\
POS-RG & Scale consistency & Functor commutes with coarse-graining &
Pointer-basis independence of Heisenberg cut \\
POS-OGN & \OGNfive{} normalization & Five-integer ledger
$(1,3,5;7,35)$ &
Fixes all dimensionless coefficients; $\sin^2\theta_W$,
$\lambda_H$, etc. \\
\bottomrule
\end{tabular}
\end{table}


\section{Part~II in plain English: what this paper borrows, and at what strength}
This Part is a list of debts, and it is placed early on purpose. \AEight{}
is the eighth paper of a series, and almost nothing in it is self-contained.
The constitutional axioms are \AOne{}'s. The six-dimensional geometry is
\ATwo{}'s. The projection dictionary and the open-system law are \AFive{}'s.
The runtime that ends a relaxation at a single attractor is \ASix{}'s. The
gauge-sector and cosmological readings are \ANine{}'s and \ATen{}'s. What is
left over --- and it is a real remainder, not a courtesy --- is the operational
restatement of that machinery on the microscopic-observable stratum, together
with four results the bulk-side apparatus makes available and textbook quantum
mechanics has no way to write down.

The reason for saying this in a Part of its own is that an inheritance is easy
to overstate. A paper that imports a theorem and re-narrates it in its own
words has not verified anything; it has copied. The rule followed here is that
every import is quoted at the strength its owning paper gives it, and where the
owning paper hedges, this one hedges in the same place. The Born rule is the
clearest case. \AFive{} closes it at the relaxation level, conditional on four
named premises, with a single analytic residue at the field-grade
representation step. That is exactly the strength at which it appears here. A
reader who wants the Born rule proved outright will not find it proved outright
in this paper, and should not: it is not proved outright in the paper that owns
it.

The coherence map at the end of this Part exists for the same reason. It is
easy for a series of twenty-two papers to drift --- for a value quoted in one
paper to move in another and for nobody to notice, because no single reader
holds all of them in mind at once. The map is the mechanism by which that drift
becomes visible. If a number in this paper disagrees with the number in the
paper that owns it, the map is where the disagreement shows, and the paper that
owns it wins.

None of this makes the present Part interesting reading, and it is not meant
to be. It is meant to be checkable. A reader who wants to know whether a
particular claim in Parts~IV to XIII rests on something argued here or on
something argued somewhere else should be able to answer the question from
this Part alone, without taking anyone's word for it.

\GGPart[The constitutional bridge]{The constitutional bridge}{$\GG{}$~bulk $\to$ \OGGEP{} $\to$ \GDOS{} $\to$ 4D~QM}
\label{part:III}

\noindent The four-stage constitutional bridge is shown in \figref{fig:constitutional-bridge}.

\begin{figure}[H]
\centering
\resizebox{\textwidth}{!}{%
\usebox{\GGscBridge}}
\caption[The constitutional bridge: \GG{}~bulk $\to$ \OGGEP{} $\to$ \GDOS{} $\to$ 4D~QM.]{The constitutional bridge of \AEight{}: the four-stage
chain that compiles the closed-bulk dynamics of the 6D geometry (\ATwo{}, red) into the operational 4D Hilbert spaces
of laboratory quantum mechanics (blue, highlighted as the QM
tier of this paper). \OGGEP{} (\AX{3} of \GGConstitution{},
purple) is the operational extension of \GGEP{} (\AX{2}) that
supplies the bulk-to-boundary compiler; the six \GDOS{}
theorems \T{1}--\T{6} of \AFive{} Part~II construct the
boundary algebra and the channel structure; the seven QM
theorems \T{7.1}--\T{7.7} of \AFive{} Part~III restrict the
boundary algebra to the QM-relevant sector and reproduce the
textbook postulates. Every \AEight{} worked example
(Parts~V--IX) is a specialization of this single chain on a
specific laboratory system.}
\label{fig:constitutional-bridge}
\end{figure}

\medskip
\noindent
\textbf{Part~III is the mechanism Part of \AEight{}.} Every other
Part of this paper either prepares for it (Part~I introduces the
century-old questions; Part~II catalogs the cohort
inheritance), states its consequences as theorems (Part~IV), or
exercises those theorems on specific laboratory systems (Parts~V--XIII).
Part~III itself is the four-stage chain
$\GG{}\text{~bulk} \to \OGGEP{} \to \GDOS{} \to \text{4D~QM}$
read end-to-end: how the deterministic, unitary, six-dimensional
geometry of \ATwo{} (\GGEP{}, \AX{2}) is compiled by \OGGEP{}
(\AX{3} of \GGConstitution{}) into the boundary algebra of \GDOS{}
(\AFive{} Part~II), and then restricted by the seven QM-postulate
theorems (\AFive{} Part~III \T{7.1}--\T{7.7}) to the textbook
4D Hilbert spaces a laboratory observer measures. The reader who
masters the present Part has, by the end of it, the operational
machinery that the rest of the paper uses; the worked examples
(Parts~V--IX), the paradox sweep (Part~X), the interpretation
synthesis (Part~XI), and the discussions (Parts~XII--XIII) are
specializations and corollaries of this single chain.


\section{Why Observation Is Projection: The Hidden-Fiber Origin of Quantum Behavior}
\label{sec:hidden-fibre-origin}

Before stating the operational machinery of \OGGEP{} formally, it is worth drawing the geometric picture that makes the whole bulk-to-boundary chain feel inevitable rather than imposed. The bulk derivation of \ATwo{} (Theorem~0, T-GG6-0) established that the two extra dimensions of \GGsix{} are not spatial: $I_r$ is a \emph{scale} dimension (a resolution, the geometrized renormalization-group coordinate) and $S^1_\theta$ is a \emph{phase} dimension (an internal angle, the geometrized gauge holonomy). Once that is granted, the quantum-mechanical content of the bulk-to-boundary projection follows from a single observation: \emph{these two dimensions are attached at every point of spacetime, and neither can be resolved by any measurement made on the boundary.} This section makes that statement precise and shows that it already contains, in outline, why the boundary theory is quantum.

\subsection{The bulk as a fiber bundle; observation as the projection}
\label{ssec:bundle-projection}

Locally, the bulk is a fiber bundle,
\begin{equation}
\mathcal{M}_6 \;=\; \mathcal{M}_4 \times I_r \times S^1_\theta,
\qquad
\pi : \mathcal{M}_6 \longrightarrow \mathcal{M}_4,
\qquad
(x;\,r,\theta) \mapsto x,
\label{eq:bundle-projection}
\end{equation}
whose \emph{base} is the Lorentzian stage $\mathcal{M}_4$ --- the arena of registered, causally ordered records --- and whose \emph{fiber}, sitting over each spacetime point $x$, is the pair $(r,\theta)$: the scale and phase the system carries \emph{at that point}. The boundary registration map demanded by \OGGEP{} (\AX{3}) is exactly this projection $\pi$.

The decisive point is that $\pi$ is not an additional postulate; it is forced by the non-spatiality of the fiber. A boundary protocol, by \AX{1} (\LC), reads spacetime records sorted by a light cone; there is no light cone ``in resolution'' and none ``in internal angle,'' so there is no operation that resolves where a system sits along $I_r$ or around $S^1_\theta$. Every act of registration is therefore obliged to marginalize the fiber. ``Observation on a four-dimensional Lorentzian boundary'' simply \emph{means} the pushforward along $\pi$, and that pushforward discards the fiber coordinates irretrievably.

This is the geometric reason that openness is constitutional and closure is its limiting case (the methodological inversion of \AFive{}). The fully unitary, information-preserving description is the complete six-dimensional bulk --- which is inaccessible in principle, not merely in practice. Every description an observer can actually register is the projection $\pi$, which is non-invertible because it forgets the fiber; the boundary dynamics it induces is therefore open, mixed, and completely positive (the CPTP and GKLS structures of \AFive{} T-GDOS-3--T-GDOS-4). Unitarity is not the default that measurement violates; it is the unreachable idealization in which the fiber is never coarse-grained.

\begin{figure}[tbp]
\centering
\resizebox{0.557\textwidth}{!}{%
\usebox{\GGscFiber}}
\caption[Observation as the bundle projection $\pi:\mathcal{M}_6\to\mathcal{M}_4$.]{Observation as the bundle projection $\pi:\mathcal{M}_6\to\mathcal{M}_4$. Over each boundary point $x$ sits the non-spatial fiber $(I_r,S^1_\theta)$. The compact \emph{phase} circle $S^1_\theta$ carries the coherent, unitary content of quantum mechanics --- its holonomy is the Aharonov--Bohm/Berry phase \citep{AharonovBohm1959, Berry1984} --- while the non-compact \emph{scale} interval $I_r$ carries the irreversible, open content, its forward renormalization-group semigroup being the GKLS/\GDOS{} generator. Because both fiber coordinates are non-spatial, no boundary protocol resolves them; every registration marginalizes the fiber, so $\pi$ is non-invertible and the boundary dynamics is necessarily open. Unitary closure is the unreachable limit in which the fiber is not coarse-grained.}
\label{fig:hidden-fibre-projection}
\end{figure}

\subsection{Two fibers, two halves of quantum mechanics}
\label{ssec:two-fibres}

The two fiber directions enter the geometry differently --- $S^1_\theta$ compact and flat, $I_r$ non-compact and scale-extended (\ATwo{}, Theorem~0) --- and that difference maps cleanly onto the two halves of quantum mechanics.

\paragraph{The phase fiber $S^1_\theta$ carries the coherent, unitary structure.} The quantum phase of a wavefunction $\psi = |\psi|\,e^{i\varphi}$ is, on this reading, the system's position on the $\theta$-circle, and the holonomy $\oint A_\theta\,d\theta$ is the physical, observable \emph{relative} phase --- the Aharonov--Bohm and Berry geometric phase \citep{AharonovBohm1959, Berry1984}, an effect already known to be geometric. Transport along $\theta$ is a pure phase: reversible, unitary, interference-generating. Moreover the gauge redundancy of $\theta$ delivers a sharper statement than is usually noticed. Because a rigid fiber rotation acts on a charge-$n$ sector with weight $n$, what the fiber's unreadability forbids is the measurement of a relative phase \emph{between states of different charge}: charge superselection, a genuine theorem of gauge theory, read here directly off the geometry. Marginalizing the unreadable fiber coordinate while retaining relative phases within a sector is precisely the operation that converts an amplitude into a probability; this is why the Born rule has square-modulus form, and why the \emph{gauge} phase of a charged field is complex: it is the holonomy of a real circle, with $i$ that circle's rotation generator. (The complex structure of the boundary state space itself is a separate question --- a neutral field needs it as much as a charged one, and the circle provably cannot supply it there; where it enters, and how far it too is derived from real constitutional data, is taken up in \S\ref{sec:disc-real-bulk} --- and is now \emph{closed by import}: the origin-of-$i$ companion derives the boundary's single imaginary unit from the two real structures of the arena, at derived-conditional grade in the quasi-free regime (\citealp{ArisakaP4Real}), with the custody sentence holding as everywhere in the cohort: the dictionary owns the sectors, the geometry owns the $i$.) The formal derivation is the Born-rule theorem \TA{3}; the present picture supplies its geometric origin.

\paragraph{The scale fiber $I_r$ carries the open, irreversible structure.} Theorem~0 of \ATwo{} fixed $I_r$ as \emph{non-compact} precisely because the renormalization-group flow it geometrizes is a forward semigroup: integrating out short-distance modes cannot be undone. Motion along $r$ is therefore intrinsically directed. Accessing a system at finite resolution means tracing out the remainder of this scale fiber --- coarse-graining --- and because the fiber is irreversible the induced boundary evolution is a Markov \emph{semigroup}, not a group: completely positive, with no inverse. That is exactly the GKLS/\GDOS{} structure \citep{Lindblad1976, GKS1976}. The identification is sharp: \emph{the open-system generator of \GDOS{} is the dilatation (RG) generator along $I_r$, projected to the boundary}, and the arrow of measurement is the arrow of the renormalization group. That arrow is consumed here, not created: per the constitutional custody chain, \AX{3} owns the arrow --- the one-way reading of the dictionary is a clause of the constitution (the AX-3 synthesis of \citealp{ArisakaA1Constitution}) --- with the canonical law-level derivation in Part~IX of the CPT member paper \citep{ArisakaP5CPT} and the dynamical implementation in the \GDOS{} companion \citep{ArisakaA5GDOS}; the measurement arrow of this paper is the scale-fiber face of that one constitutional arrow, a consumer of \AX{3}'s non-invertibility, not a private derivation. Holographic renormalization makes the same correspondence in its own setting --- radial integration is Wilsonian coarse-graining \citep{Wilson1971, Maldacena1998, RandallSundrum1999}.

\paragraph{Together.} Coherent unitary evolution is the phase fiber; irreversible registration is the scale fiber; and decoherence is the loss of $\theta$-coherence induced by coarse-graining along $r$ --- the einselection of a pointer basis \citep{Zurek1981, Zurek2003}. The closed, unitary Schr\"odinger evolution is the limit in which phase coherence is maintained and the scale fiber is not coarse-grained (\AX{3} closure); generic observation traces the scale fiber and yields measurement. One bulk, two limits.

\paragraph{The precise statement: a principal $U(1)$-bundle --- and what the scale fiber is not.} It is worth fixing the two fibers in exact differential-geometric language, since that is the rigorous form of the picture above. In the vocabulary of fiber bundles \citep{Ehresmann1950, WuYang1975}, the phase direction $S^1_\theta$ is a genuine \emph{principal $U(1)$-bundle} over the boundary base $\mathcal{M}_4$: the gauge potential $A_\theta$ is its \emph{connection}, the quantum phase is the \emph{holonomy} of that connection around a loop, and the Aharonov--Bohm and Berry phases are holonomies in the literal bundle sense --- which is exactly why the gauge phase of charged matter is complex and the Born rule is square-modulus, the unreadable fiber coordinate enforcing superselection between charge sectors (for the complex structure of the state space itself, see \S\ref{sec:disc-real-bulk}). \OGGEP{} is then the \emph{bundle projection} $\pi:\cM_6\to\mathcal{M}_4$, and ``tracing out the fiber'' is the restriction of a section to the base. The Wu--Yang (1975) dictionary, which first identified the non-integrable gauge phase factor with bundle holonomy, is the precise precedent for reading the quantum phase this way. The one place the language must \emph{not} be pushed is the scale fiber $I_r$: it is not a principal gauge bundle. Its ``structure group'' is the dilatation, realized as the one-way renormalization-group \emph{semigroup} of the preceding paragraph (irreversible, with no inverse), and the relevant ``connection'' along it is the scale profile, not a Yang--Mills field. That asymmetry --- a compact principal $U(1)$-bundle on the phase side, a scale-extended semigroup fibration on the scale side --- is the same asymmetry that splits quantum mechanics into its unitary (phase) and irreversible (scale) halves.

\subsection{Why a point particle makes the projection total}
\label{ssec:point-particle}

A point particle occupies a single $x \in \mathcal{M}_4$ but carries, at that point, definite values of $r$ and $\theta$ that the boundary cannot read. In the most literal sense the particle is \emph{its own environment}: the degrees of freedom traced out in decoherence are not those of an external bath but the particle's own non-spatial fiber. No outside reservoir is needed for the open structure to be \emph{available}; inaccessibility is intrinsic. This is a statement about the open-system \emph{form}, not the decoherence \emph{rate}: a genuinely isolated free particle in vacuum, with no apparatus or bath coupling, decoheres negligibly, because the rate is set by the system's coupling to its physical 4D environment (air molecules, thermal photons), not by the fiber alone. The intrinsic fiber is why the boundary law is open in principle; what makes a given system decohere fast is its physical environment.

This is also where the constitutional choice of \emph{point} particles, rather than extended objects, does quiet work. An extended object would carry a resolvable profile \emph{along} the extra directions --- a tower of internal modes, an intrinsic extent --- which is in principle accessible and would blur the clean statement that the fiber is simply unreadable. A point carries a single fiber value and nothing else, so the marginalization in \eqref{eq:bundle-projection} is total and unambiguous. The quantum indeterminacy of a point particle is then not a smearing in space but the boundary shadow of two perfectly definite yet unobservable internal coordinates.

\subsection{Particle in space, wave in phase: wave--particle duality dissolved}
\label{ssec:wave-particle}

The two-fiber split resolves the oldest puzzle of quantum mechanics at the level of principle. ``Wave--particle duality'' reads as a contradiction only when both words are made to describe behavior \emph{in space}. Here they describe two different dimensions: the system is a \emph{particle in the stage} $\mathcal{M}_4$ (a point, by locality) and a \emph{wave in the phase fiber} $S^1_\theta$ (the dial whose holonomy is the quantum phase). Both hold at once, without paradox, because point-like-in-space and wave-like-in-phase are orthogonal, non-spatial statements. The wavefunction $\psi=|\psi|e^{i\varphi}$ is not the particle spread out but the phase section over the available positions; the de Broglie wavelength $\lambda=h/p$ is the spatial period at which the dial winds, set by the momentum, not a size of the particle. Feynman's spinning amplitude-arrow \citep{FeynmanQED1985} is, on this reading, literally the position on $S^1_\theta$; adding arrows and squaring is the \OGGEP{} marginalization of the unreadable global phase that yields the Born rule (\TA{3}). The worked double-slit calculation --- where the fringe term $\cos\Delta\varphi$ is the relative dial angle of two bulk histories, and a which-path measurement is the reading of the phase fiber that destroys it --- is carried out in Part~\ref{part:VI} (\S\ref{sec:DS-dial}).

\paragraph{``How can a point pass through both slits?''} The sharpest form of the puzzle deserves a direct answer even at this level of principle. A point must pass one slit or the other, yet interference appears to demand both. The dissolution is that ``which slit'' and ``interference'' are questions about \emph{different fibers}. ``Which slit'' asks for a position in $\mathcal{M}_4$, and the point has exactly one, registered as a single dot at the screen. ``Interference'' asks about the phase section $\varphi(x)$ on $S^1_\theta$, and that section is a \emph{field} laid over every available position --- defined along the route past each slit, exactly as the electromagnetic field of a light beam is present at both slits without anything material being in two places. What threads both slits is the phase field; what lands is the point; the fringes record how the two dial-windings recombine. The extensive treatment --- including why a which-path measurement is a scale-fiber read-out that necessarily erases the phase coherence --- is given in Part~\ref{part:VI}, \S\ref{sec:DS-both-slits}.

\subsection{Honest accounting: what the fiber picture settles, and what \GDOS{} still carries}
\label{ssec:fibre-honest}

It is worth being exact about how much this picture delivers, in the auditable spirit of \MF{} (\POS{7}).

What it \emph{does} settle, and more convincingly than asserting the chain $\GGsix{}\to\OGGEP{}\to\GDOS{}\to$~4D~QM directly: the projection is forced rather than postulated; openness is constitutional for a structural reason; the complex amplitude acquires a geometric identity (a phase holonomy); the Born rule acquires its square-modulus \emph{form}; a preferred (pointer) basis and decoherence follow from which fiber is traced; and --- the cleanest new result --- the irreversibility of measurement is identified with the forward renormalization-group semigroup along the non-compact scale dimension, so that the dissipative part of the GKLS generator is the boundary image of scale-fiber coarse-graining (\AFive{} T-GDOS-4).

What the bundle picture does \emph{not} settle by geometry alone is the one residue that every deterministic completion of quantum mechanics carries --- and the framework states it precisely rather than papering over it. The measurement problem, stated exactly, is \emph{three} questions, and the framework answers each. \emph{Why this basis?} --- the einselected pointer basis is forced by bulk locality \AX{1} (\TA{6}). \emph{Why one definite outcome rather than a living superposition?} --- because the six-dimensional bulk is single-valued and registration is the descent of the joint system-and-apparatus configuration into one Geometric Attractor basin, irreversibly locked by the boundary free-energy law (\TA{1}, \TA{4}, with the \GRIP{} actualization of \ASix{}); the outcome is the \emph{contextual} joint holonomy $\Theta_H$ of system and apparatus read out by the \OGGEP{} projection, with no collapse, no branching, and no observer required. \emph{Why Born weights?} --- the square-modulus \emph{form} is forced by Gleason on the open dictionary (\TA{3}), while which value $\Theta_H$ takes on a given run is the bulk boundary condition, inaccessible to the four-dimensional observer. What remains --- assumed nowhere in this paper --- is the basin-capture $=$ Born statement: that the \GRIP{} basin-capture measure of the deterministic descent coincides with the Born measure, the six-dimensional analogue of the quantum-equilibrium typicality theorem of Bohmian mechanics \citep{Bohm1952}; it is closed at the relaxation level (the R16 lemmas with the R16$'$ \GDOS{} relaxation; the Constitutional Born Theorem conditional on the four named premises (P1)--(P4) of \S\ref{sec:disc-open}), with the field-grade representation step the single remaining analytic residue. With this division the umbrella problem is dissolved into theorems plus one named analytic residue; that is both the strength of the account and the precise statement of what is still owed. This is the form we carry through the rest of the paper.


\section{The Operational Postulate \OGGEP{} in detail}

\subsection{Context: why an operational postulate?}

\OGGEP{} (\AX{3} of the \GGConstitution{}) is the \emph{open-system
extension} of the closed-bulk postulate \GGEP{} (\AX{2}). \GGEP{}
states that the 6D bulk \citep{ArisakaA2GG6} is a single
composite geometric structure in which gravity and gauge data are
unified into one connection; it constrains the bulk theory.
\OGGEP{} extends this constraint to the operational shadow that a
finite 4D observer registers in the laboratory: any admissible
bulk theory must come equipped with a bulk--boundary correspondence from
its bulk algebra to a boundary algebra of operationally
admissible states, channels, and instruments. \AFive{} Part~II
constructs this dictionary explicitly in six theorems
\T{1}--\T{6}; \AFive{} Part~III specializes the dictionary to
the microscopic-observable sector and shows that the four
textbook postulates of quantum mechanics --- the Hilbert-space
postulate, the Schr\"odinger equation, the Born rule, and the
L\"uders projection rule --- plus three additional structures
that the decoherence program later identified as essential
(decoherence, einselection, Bell-violation), emerge as theorems
\T{7.1}--\T{7.7}. The present paper inherits this chain
verbatim and lifts each theorem to standalone graduate-textbook
form.

In the Copenhagen formulation, measurement is a primitive notion
that stands outside the physics --- a line between ``quantum
system'' and ``classical apparatus'' drawn at the experimenter's
discretion. This is the source of the measurement problem:
within the theory, no principle determines where the line lies.

\GGtheory{} takes a different stance. Measurement is not an
axiom on top of physics; it is a \emph{constitutional
requirement} --- a condition that any admissible physical
theory must satisfy. The requirement is expressed in the
\emph{Operational/Open Geometry--Gauge Equivalence Principle}
(\OGGEP{}, \AX{3} of the \GGConstitution{}
\citep{ArisakaA1Constitution}; closed-bulk precursor
\GGEP{} = \AX{2}, the Geometry--Gauge Equivalence Principle that
constrains the 6D bulk of \citet{ArisakaA2GG6}):

\begin{quote}\itshape
A physical theory is complete only when it specifies a map from
its bulk data to the operational records produced by laboratory
instruments. This map must be a structure-preserving functor
that respects causality, physicality (positivity), and
coarse-graining.
\end{quote}

The functor language is important: \OGGEP{} is not just a
recipe for computing probabilities; it is a constraint on how
the fundamental 6D theory relates to 4D laboratory records.

\subsection{The functor $\mathcal{D}: \mathbf{Bulk} \to
\mathbf{Op}$}

Following the categorical formulation of GG-2026-3 and
GDOS-Book-I:
\begin{definition}[Bulk and Operational categories]\ 
The category \textbf{Bulk} has as objects admissible bulk
packages $B = (M_6, g, A, \psi, \ldots)$ satisfying the \GG{}
constitution; its morphisms include diffeomorphisms, gauge
transformations, and coarse-graining / renormalization-group
maps.

The category \textbf{Op} has as objects operational boundary
packages $O = (\Hilb, \rho, \{I_\alpha\}, \{\Phi_{t,s}\},
\ldots)$ --- that is, a Hilbert space, a density operator, a
family of instruments, and a family of channels; its morphisms
are operational equivalences (unitary conjugations, classical
relabelings, coarse-grainings).
\end{definition}

\begin{definition}[\OGGEP{} functor]
\OGGEP{} asserts the existence of a functor
\begin{equation}\label{eq:OGGEP_functor}
\mathcal{D} : \mathbf{Bulk} \longrightarrow \mathbf{Op}
\end{equation}
that is compatible with: (i) \textbf{gauge covariance} (bulk
gauge equivalences map to operational equivalences); (ii)
\textbf{causality} (retardedness, AX-1); (iii)
\textbf{positivity / physicality} (channels and instruments
preserve state positivity); (iv) \textbf{scale consistency}
(the functor commutes with coarse-graining).
\end{definition}

The content of \OGGEP{} is therefore \emph{not} that a specific
formula for $\mathcal{D}$ is given by fiat; it is that
\emph{such a functor must exist} and must satisfy the four
constitutional properties listed above. The explicit formula
for $\mathcal{D}$ is a consequence, not a postulate.

\subsection{The explicit form of $\mathcal{D}$: partial trace
plus coarse-graining}

For a bulk state $\densbulk$ on the 6D Hilbert space
$\Hilb_{\rm bulk}$, the 4D observer sees
\begin{equation}\label{eq:Partial_trace}
\densbdy \;=\; \mathcal{D}(\densbulk)
\;=\; \Trenv(\densbulk)
\;=\; \Tr_{I_r \otimes S^1_\theta}(\densbulk).
\end{equation}
The trace is over the degrees of freedom that the 4D observer
cannot access --- the radial coordinate $I_r$, the
compact angle $S^1_\theta$, and all heavy Kaluza-Klein modes
above the lock-cluster scale $\Mlock \approx 3.85$~TeV
($\approx 42\, m_Z$). The
remaining state $\densbdy$ lives on the 4D zero-mode Hilbert
space $\Hilb_4$.

Importantly, the partial trace (\ref{eq:Partial_trace}) is
\emph{automatic}: any 4D observer \emph{necessarily} traces
over the inaccessible bulk degrees of freedom. There is no
choice here; it is a consequence of being 4-dimensional.

\subsection{Operational instruments from bulk interactions}

A laboratory instrument is modeled in the standard
Stinespring/Naimark dilation form. For an instrument labeled
by outcomes $\{\alpha\}$:
\begin{equation}\label{eq:instrument}
I_\alpha(\densbdy) \;=\; \Trapp\!\left[(\iden_S \otimes
\Pi_\alpha^{\rm pointer})\,U_{SA}\,(\densbdy \otimes
\sigma_{\rm app})\,U_{SA}^\dagger\,(\iden_S \otimes
\Pi_\alpha^{\rm pointer})\right],
\end{equation}
where $\sigma_{\rm app}$ is the apparatus initial state,
$U_{SA}$ is the system-apparatus unitary interaction, and
$\Pi_\alpha^{\rm pointer}$ is the pointer projector
corresponding to outcome $\alpha$. The outcome probability
is
\begin{equation}\label{eq:outcome_prob}
p_\alpha \;=\; \Tr[I_\alpha(\densbdy)] \;=\; \Tr[E_\alpha \densbdy],
\end{equation}
where $E_\alpha = \Tr_{\rm app}[(\iden_S \otimes
\Pi_\alpha^{\rm pointer})\,U_{SA}^\dagger (\iden_S \otimes
\Pi_\alpha^{\rm pointer})\,U_{SA}\,(\iden_S \otimes
\sigma_{\rm app})]$ is a positive operator on $\Hilb_S$ ---
precisely a \POVM{} element, $E_\alpha \ge 0$, $\sum_\alpha
E_\alpha = \iden$.

The post-measurement conditional state is
\begin{equation}\label{eq:post_measurement}
\densbdy^{(\alpha)} \;=\; I_\alpha(\densbdy)/p_\alpha,
\end{equation}
which is Bayesian conditioning on the outcome label.

\subsection{The ``measurement problem'' as missing map problem}

In GDOS-Book-I, Arisaka (2026b) observes:
\begin{quote}\itshape
OGGEP 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?''
\end{quote}

In the \GGtheory{} framework, there is no ``collapse'' as a
physical event. There is only (i) deterministic 6D evolution,
(ii) partial trace to 4D, (iii) conditioning on an outcome in
the output register. Each of these is a mathematically precise
and physically routine operation. ``Collapse'' is just Bayesian
conditioning on a measurement outcome, exactly as in classical
probability theory.

\section{Deriving the Born rule from \OGGEP{}}

Born's 1926 reinterpretation of $|\psi|^2$ as a probability
density~\citep{Born1926, BornHeisenbergJordan1926} completed the
predictive apparatus of quantum mechanics, but offered no derivation;
Gleason's 1957 theorem~\citep{Gleason1957} sharpened the result to a
uniqueness statement on a Hilbert space of dimension~$\geq 3$, but only
once that Hilbert space and its projective-measurement structure have
already been assumed. The present section closes the remaining
postulational gap. We derive the Born rule as a theorem of \OGGEP{}:
the operationally registered probability of outcome~$\alpha$ is forced
by the \CPTP{} completion of the bulk-to-boundary functor, with the
textbook $|\braket{a}{\psi}|^2$ recovered as the pure-state, sharp-
projective specialization. This is the operational content of
\T{7.3} of \citep{ArisakaA5GDOS} Part~III; the derivation is verbatim
that of A5 Part~III, here lifted to graduate-textbook depth.

\subsection{The strategy}

We derive the Born rule $P(a) = |\braket{a}{\psi}|^2$ from three
ingredients:
\begin{enumerate}[leftmargin=2em,itemsep=0.1em]
\item The state space is a complex separable Hilbert space
$\Hilb_4$ of dimension $\ge 3$ (inherited from the 6D bulk
zero-mode sector).
\item Any instrument $\{I_\alpha\}$ is a \CPTP{} map with
classical output, arising from (\ref{eq:instrument}) through
the partial trace.
\item Probabilities for orthogonal outcomes must be additive
(Kolmogorov consistency).
\end{enumerate}
These three are consequences of the \OGGEP{} constitution. We
then apply Gleason's theorem to recover Born's rule uniquely.

\subsection{Step 1: the Hilbert-space structure from the bulk}

In GG-2026-6F, it was shown that the bulk zero-mode sector ---
after Kaluza-Klein reduction on $I_r \times S^1_\theta$ and
projection to the 4D boundary --- is precisely a complex
Hilbert space carrying representations of the Standard-Model
gauge group. Specifically:
\begin{equation}\label{eq:bulk_to_Hilb}
\Hilb_4 \;=\; \mathrm{span}\,\{\mathrm{KK\ zero\ modes\ of}\ G_6 = \SUfive
\times \Uoneb \times \UoneQi\}.
\end{equation}
The Hilbert space is not postulated; it descends from the
functional integral over BI-6-compatible bulk configurations.
Dimensionality is determined by the representation content of
the bulk gauge group + spin-statistics + anomaly cancellation.

For our present purposes, we need only that $\Hilb_4$ is
complex, separable, and of dimension $\ge 3$ --- which it is,
trivially, since $\SUfive$ alone has $24$-dimensional
adjoint representations and fermion sectors in
higher-dimensional reps.

\subsection{Step 2: CPTP structure from \OGGEP{}}

The map $\mathcal{D}: \mathbf{Bulk} \to \mathbf{Op}$ is a
\CPTP{} channel by construction. Recall:
\begin{itemize}[leftmargin=2em,itemsep=0.1em]
\item \textbf{Complete positivity} (CP): the map preserves
positivity of the composite state $\rho_{SR}$ for arbitrary
reference system $R$. This is required for physicality --- the
system may be entangled with anything, and the reduced state
must stay positive.
\item \textbf{Trace preservation} (TP): $\Tr(\mathcal{D}\rho) =
\Tr\rho$. Probabilities must sum to one.
\item \textbf{Linearity}: $\mathcal{D}(\alpha\rho_1 + \beta\rho_2)
= \alpha\mathcal{D}(\rho_1) + \beta\mathcal{D}(\rho_2)$ for
convex combinations. This follows from trace structure.
\end{itemize}
The functoriality of $\mathcal{D}$ (respecting composition and
identities) plus CPTP gives exactly the Choi-Kraus representation
\begin{equation}\label{eq:Kraus}
\densbdy \;=\; \mathcal{D}(\densbulk) \;=\; \sum_k A_k
\densbulk A_k^\dagger, \qquad \sum_k A_k^\dagger A_k = \iden.
\end{equation}

For a projective measurement on the boundary, the associated
effects $\{E_\alpha\}$ satisfy $E_\alpha \ge 0$, $\sum_\alpha
E_\alpha = \iden$ --- the \POVM{} axioms.

\subsection{Step 3: Gleason's theorem}

Given a $\sigma$-additive probability measure $\mu$ on the
lattice of orthogonal projectors of $\Hilb_4$ (dim $\ge 3$),
Gleason's theorem \citep{Gleason1957} states:
\begin{equation}\label{eq:Gleason_statement}
\mu(\Pi) \;=\; \Tr(\rho\,\Pi),
\end{equation}
for a unique density operator $\rho \ge 0$ with $\Tr\rho = 1$.

The \OGGEP{} construction above produces exactly a
$\sigma$-additive measure: for mutually orthogonal
projectors $\Pi_\alpha = E_\alpha$ (projective instrument case)
coming from the Kraus decomposition, additivity of
probabilities is forced by trace linearity and orthogonality.

\begin{theorem}[Born rule from OGGEP + Gleason (T-QM-0)]
\label{thm:Born_from_OGGEP}
Given the \OGGEP{} functor $\mathcal{D}$ and the bulk 6D
theory, the outcome probabilities of any 4D instrument on any
4D state are given uniquely by
\begin{equation}\label{eq:Born_derived}
\boxed{P(\alpha) \;=\; \Tr(\rho_\partial\, E_\alpha)
\;=\; \Tr[\Trenv(\densbulk)\, E_\alpha].}
\end{equation}
In particular, for pure boundary states $\rho_\partial =
\ketbra{\psi}{\psi}$ and projective measurement
$E_\alpha = \ketbra{\alpha}{\alpha}$,
\begin{equation}\label{eq:Born_pure}
P(\alpha) \;=\; |\braket{\alpha}{\psi}|^2.
\end{equation}
\end{theorem}

The Born rule is not a postulate of \GGtheory{}. It is a
theorem --- a consequence of the \OGGEP{} functorial requirement
plus Gleason's uniqueness theorem.

\subsection{Why the Born rule is \emph{the} rule}

The key insight is that \emph{additivity and physicality force
Born}. No other rule is compatible. Consider the alternatives:
\begin{itemize}[leftmargin=2em,itemsep=0.15em]
\item $P(a) = |\braket{a}{\psi}|$ (linear in amplitude): not
additive; fails Gleason.
\item $P(a) = |\braket{a}{\psi}|^4$: not additive; fails
Gleason.
\item $P(a) = \braket{a}{\psi}\braket{\psi}{a}$: same as Born
for pure states; extends Born for mixed states.
\end{itemize}
Only Born satisfies additivity + non-contextuality + positivity
on Hilbert spaces of dimension $\ge 3$. This was the point of
Gleason.

\GGtheory{}'s contribution is not proving Gleason (1957) again.
It is providing the \OGGEP{} context that makes Gleason
\emph{apply}: the functor $\mathcal{D}$ with the CPTP structure
justifies the Hilbert-space formalism and the additivity
axiom. Without \OGGEP{}, one had to postulate these; with
\OGGEP{}, they are consequences.

\section{Deriving the Lindblad equation from \CPTP{} + local
causality}
\label{sec:OGGEP_lindblad}

The Gorini--Kossakowski--Sudarshan and Lindblad theorems of
1976~\citep{GKSL1976, Lindblad1976} jointly fix the most general
generator of a quantum dynamical semigroup that preserves complete
positivity and trace: a Hamiltonian conjugation plus a Lindblad
dissipator built from jump operators $\{L_\alpha\}$ and non-negative
rates $\{\gamma_\alpha\}$. The GKLS theorem is the dynamical companion
of Gleason's theorem: one fixes probabilities, the other fixes
evolutions. In standard treatments, both are theorems whose hypotheses
(complete positivity, trace preservation, Markovianity) are postulated.
The present section shows that all three hypotheses are forced by the
constitutional content of \OGGEP{}: \CPTP{} is positivity required of
any boundary channel; local causality (\AX{1}) supplies the
forward-time semigroup structure; Markovianity is the leading-order
regime in the bulk--environment correlation time, with corrections
calculable from the scale-overlap integrals of the \GG{} bulk. This
recovers \T{7.2} (closed-system limit, $\gamma_\alpha\to 0$) and the
\T{7.5} foundation of the decoherence calculation that follows.

\subsection{The strategy}

We show that the continuous-time open-system evolution of
$\densbdy(t)$ is uniquely given by the GKSL (Lindblad) master
equation. The key inputs are: (i) \CPTP{} for every finite
$t$; (ii) semigroup property $\Phi_{t+s} = \Phi_t \Phi_s$ from
forward time-ordering + Markovianity; (iii) continuity at $t =
0$. Then the Gorini--Kossakowski--Sudarshan--Lindblad
\citep{GKSL1976, Lindblad1976} theorem uniquely specifies the
generator.

\subsection{Step 1: the semigroup property from AX-1 (local
causality)}

AX-1 in the \GGConstitution{} is \emph{local causality}:
physical influence propagates forward in time at finite speed.
We now justify why the bulk-side correlation structure forces a
Markovian boundary-side dynamics at leading order, so that the
semigroup property holds without being merely postulated.

\emph{The two-scale separation.} The bulk environment of any
laboratory system has a characteristic correlation time set by
the bulk Kaluza--Klein mass gap above the lock-cluster scale:
$\tau_c \sim \hbar / M_{\rm lock} c^2 \sim 10^{-27}$~s (taking
$M_{\rm lock} \approx 3.85$~TeV). The characteristic boundary
system time is set by the boundary energy splitting of the
observable in question: $\tau_S \sim \hbar / E_{\rm sys}$,
typically $10^{-15}$~s for atomic transitions, $10^{-9}$~s for
qubit dynamics, longer for macroscopic systems. In all
laboratory regimes the ratio $\tau_c / \tau_S \lesssim
10^{-12}$ is parametrically small. By AX-1 (local causality),
bulk-environment correlations decay on the timescale $\tau_c$;
on the boundary timescale $\tau_S$, those correlations have
already vanished by a factor $e^{-\tau_S/\tau_c}$ that is
exponentially small. The boundary dynamics therefore inherits
no memory from the bulk environment at leading order in
$\tau_c/\tau_S$, and the evolution from $0$ to $t+s$ factorizes
into independent steps:
\begin{equation}\label{eq:semigroup}
\Phi_{t+s} \;=\; \Phi_t \circ \Phi_s
\quad + \quad \mathcal{O}(\tau_c/\tau_S),
\end{equation}
with the non-Markovian corrections calculable order-by-order from the scale-overlap integrals of bulk KK modes, using the scale-overlap machinery of \AFive{} \citep{ArisakaA5GDOS} Part~II (the explicit higher-order expansion is a program target, not assumed here). Two correlation times must be kept distinct. The bulk Kaluza--Klein time $\tau_c \sim 10^{-27}$~s is what guarantees that the \emph{bulk} contributes no memory --- the bulk face of the Port-Freshness Clause of the \GDOS{} companion --- so the boundary law has semigroup \emph{form}; the actual decoherence of a laboratory system is instead governed by its coupling to its \emph{physical} 4D environment (air molecules, thermal photons), whose own correlation time $\tau_{\rm bath} \sim 10^{-13}$~s is likewise short against the system time $\tau_S$ and is the one that sets the Markovianity and the rate of real decoherence. The bulk smallness supplies the structure; the physical bath supplies the dynamics.

The evolution $\{\Phi_t\}_{t \ge 0}$ in the leading-order
Markovian regime is a \emph{one-parameter semigroup} (not a
group --- there is no inverse, since open evolution is
irreversible). The seminal Lindblad derivation \citep{Lindblad1976}
assumed this regime; the present framework derives it from the
parametric smallness of $\tau_c/\tau_S$, with quantitative
non-Markovian corrections in principle observable in
high-precision measurements above the lock-cluster scale.

\subsection{Step 2: the generator from continuity}

Assuming strong continuity at $t = 0$ --- $\lim_{t\to 0^+}
\Phi_t \rho = \rho$ in some operator norm --- the semigroup has
a generator $L$ defined by
\begin{equation}\label{eq:generator}
L\rho \;\equiv\; \lim_{t \to 0^+} \frac{\Phi_t\rho - \rho}{t},
\qquad \Phi_t = e^{tL}.
\end{equation}
The question is: what is the most general $L$ compatible with
\CPTP{} for all $t \ge 0$?

\subsection{Step 3: the GKSL theorem}

\begin{theorem}[Gorini--Kossakowski--Sudarshan--Lindblad, 1976]
\label{thm:GKSL}
A linear operator $L$ on the space of trace-class operators
generates a \CPTP{} semigroup $\Phi_t = e^{tL}$, $t \ge 0$, if
and only if there exist a self-adjoint operator $H$ and a
countable set of operators $\{L_j\}$ such that
\begin{equation}\label{eq:GKSL_form}
L\rho \;=\; -i[H,\rho] + \sum_j \left(L_j\rho L_j^\dagger
- \tfrac{1}{2}\{L_j^\dagger L_j, \rho\}\right).
\end{equation}
\end{theorem}

This is the \emph{Lindblad equation} or the \emph{GKSL master
equation}. $H$ is the effective Hamiltonian; the $L_j$ are the
\emph{jump operators} encoding system--environment coupling.

\subsection{The physical interpretation}

The Lindblad form (\ref{eq:GKSL_form}) has two pieces:
\begin{itemize}[leftmargin=2em,itemsep=0.1em]
\item \emph{Coherent evolution}: $-i[H,\rho]$, unitary-like.
\item \emph{Dissipator}: $\sum_j (L_j\rho L_j^\dagger -
\frac{1}{2}\{L_j^\dagger L_j,\rho\})$ --- produces trace-preserving
loss of coherence.
\end{itemize}
The dissipator is the only piece compatible with all three
axioms simultaneously. In particular:
\begin{itemize}[leftmargin=2em,itemsep=0.1em]
\item Stone's theorem for unitary evolution ($L_j = 0$): special
case with pure $H$-evolution, reversible.
\item GKSL with non-zero $L_j$: irreversible; $\rho$ approaches
equilibrium.
\end{itemize}

\subsection{Connection to measurement}

A projective-type measurement corresponds to a specific
limit of the Lindblad equation: strong continuous monitoring
in which the jump operators are the measurement basis projectors
$L_j = \sqrt{\Gamma}\,\Pi_j$. In this limit, the off-diagonal
coherences decay exponentially at rate $\Gamma$, and the
density matrix becomes diagonal in the $\{\Pi_j\}$ basis. This
is the quantum Zeno limit / continuous-measurement limit. The
Born probabilities emerge as the diagonal elements of the
stationary state.

\section{Decoherence as automatic consequence of \OGGEP{}}
\label{sec:decoh-OGGEP}

The decoherence program of \citet{Zeh1970}, \citet{Zurek1981,
Zurek2003}, and \citet{JoosZeh1985} was the most significant theoretical
advance on the measurement problem in the half-century after the
Bohr--Einstein debate: it showed that the off-diagonal elements of a
system's reduced density matrix decay exponentially fast under
environmental coupling, on time-scales that for macroscopic objects
are essentially instantaneous (Lemma~3 of \citep{ArisakaA5GDOS}
Part~III estimates $\Gamma \gtrsim 10^{40}\,\mathrm{s}^{-1}$ for a
Schr\"odinger's cat at room temperature). The program had two
acknowledged limits: it did not derive the Born rule, and it did not
by itself select the preferred basis. \citep{ArisakaA5GDOS} Part~III
closes both gaps in \T{7.5} and \T{7.6}; the present section recovers
the operational core (off-diagonal damping at rate~$\Gamma_{\alpha\beta}$)
directly from the \OGGEP{} functor's environmental partial trace, with
the rate coefficients fixed by the scale-overlap integrals of the 6D
bulk rather than postulated by hand.

\subsection{The decoherence question}

In traditional quantum mechanics, decoherence is a process that
\emph{happens} to a system coupled to an environment. In
\GGtheory{}, decoherence is \emph{automatic}: the \OGGEP{}
functor always includes $\Trenv$, so any 4D state is always the
partial trace of a 6D bulk state. Off-diagonal coherences of
the 6D state in the 4D pointer basis are always suppressed.

\subsection{The \GRIP{} chain: \DGI{} $\to$ \GFF{} $\to$ \GSSB{}
$\to$ \GA{}}

The GG-2026-4A paper develops the \GRIP{} runtime principle,
a four-step dynamical pattern that governs any admissible open
system:
\begin{enumerate}[leftmargin=2em,itemsep=0.15em]
\item \textbf{\DGI{}} (Dynamical Gauge Invariance): the 6D
packet carries all possible seams at the bulk level ---
a superposition of all admissible boundary phases.
\item \textbf{\GFF{}} (Geometric Free-energy Functional): the
packet descends through the bulk toward the observable
4D boundary. Each candidate branch carries a scalar cost
(free energy) determined by the 6D geometry and gauge fields.
\item \textbf{\GSSB{}} (Geometric Spontaneous Symmetry
Breaking): at a specific scale, a single branch is selected
from the ensemble of possibilities --- the branch that
minimizes the \GFF{} functional.
\item \textbf{\GA{}} (Geometric Attractor): the selected branch
stabilizes into the low-energy Standard Model + SM-like
phenomenology.
\end{enumerate}
For quantum measurement, the \GRIP{} chain plays out as follows:
\begin{itemize}[leftmargin=2em,itemsep=0.1em]
\item \DGI{}: the system+apparatus is in a superposition of
all possible outcome branches.
\item \GFF{}: each outcome branch carries a scalar cost, set by
the overlap of the corresponding environment state with the
bulk equilibrium.
\item \GSSB{}: one branch (the \emph{realized} outcome) is
selected --- the one compatible with the incoming bulk
configuration.
\item \GA{}: the realized branch stabilizes; the system is now
in a definite-outcome state.
\end{itemize}
The selection in \GSSB{} is the ``collapse'' --- but it is not
a physical event in 4D; it is a consequence of the 6D bulk
choosing a single branch (the 6D is always single-valued by
\MSMF{}; only the 4D observer sees a superposition).

\subsection{The Zurek pointer basis via einselection}

In the environment-induced superselection framework of Zurek
\citep{Zurek2003}, the pointer basis is the one that is
monitored by the environment. In \GGtheory{}, this is made
sharp: the pointer basis is the basis in which the \GSSB{}
branch-selection functional $\GFF{}$ is extremized. In
practice this is usually the position basis for macroscopic
pointers (because the environment couples strongly to
position) and the energy basis for single-atom systems
(where the environment couples to the internal electromagnetic
field).

\subsection{Decoherence rates from \OGGEP{}}

For a system coupled to the bulk environment, the decoherence
rate is
\begin{equation}\label{eq:decoherence_rate}
\Gamma_{nm} \;=\; \sum_k |\langle n|L_k|n\rangle - \langle m|L_k|m\rangle|^2
\cdot \Gamma_k,
\end{equation}
where $\Gamma_k$ are the rates at which the bulk KK modes
carry information away from the system. For typical macroscopic
objects (mass $\sim$ gram, interaction with air at STP), the
thermal relaxation frequency is
\begin{equation}\label{eq:macroscopic_decoherence}
\Gamma_{\rm thermal} \;\sim\; k_B T / \hbar \;\sim\; 10^{13}\,\text{Hz},
\end{equation}
while the scale-overlap (Joos--Zeh collisional) rate for a \emph{spatial}
superposition is far faster, $\Gamma_{\rm spatial} \sim 10^{40}\,\mathrm{s}^{-1}$;
either way the decoherence timescale is many orders of magnitude faster
than any observable dynamics. This
is why macroscopic superpositions are never seen in everyday
physics.

For isolated single atoms in ultra-high vacuum, $\tau_D$ can
reach milliseconds or even seconds --- which is why
ion-trap and atom-interferometry experiments can see large
coherent superpositions.

\section{The four-Pillar Qi-4 structure for quantum systems}

The GG-2026-4B paper extends \GDOS{} with the four-pillar
``Qi-4'' structure, each pillar governing a different aspect of
open-system behavior. For quantum mechanics, each pillar
plays a specific role:

\subsection{Qi-QUA: Current locking and the Arisaka Equation}

Qi-QUA is the event-ledger pillar: it locks the four
currents (energy, momentum, charge, Qi-number) onto a
one-dimensional ray through the BI-6 structure. For a quantum
system undergoing Lindblad evolution, the Qi-QUA locking
produces the \emph{Arisaka Ratio}: the ratio of information
flow to energy flow is quantized in units of $k_B T / \hbar$
(up to geometric factors set by \OGNfive{}).

This is the quantum version of the Landauer principle: each
bit erased (in the Lindblad dissipator) costs $k_B T \ln 2$ of
entropy, which matches the Qi-QUA-forced ratio.

\subsection{Qi-EP: Extremum principles (MEPP, FEP, PLT)}

Qi-EP is the selector-law pillar. It shows that the \GSSB{}
branch selection is governed by a single extremum principle:
\emph{the physical branch extremizes the \GFF{} (free-energy
functional) subject to the ledger constraints}. This
extremum principle is the modern unification of MEPP (maximum
entropy production principle), FEP (free-energy principle), and
PLT (principle of least action in the thermodynamic sense).

For quantum measurement, Qi-EP says: the realized outcome is
the \GFF{}-extremizing branch. Born-rule probabilities emerge
as a measure on the branches weighted by the \GFF{} cost.

\subsection{Qi-CON: Conformality, fractality, $1/f$ spectra}

Qi-CON governs the scale-invariant structure of dissipation.
For quantum measurements, Qi-CON predicts:
\begin{itemize}[leftmargin=2em,itemsep=0.1em]
\item \emph{$1/f$ noise} in macroscopic measurements --- the
famous ``pink noise'' that appears in every precision apparatus.
\item Scale-invariance of decoherence: the decoherence rate
scales with environment coupling but has the same functional
form at different scales.
\item Conformal structure of long-range correlations --- a key
prediction for entangled-pair experiments at macroscopic
separation.
\end{itemize}

\subsection{Qi-HAL: Memory-bearing architecture}

Qi-HAL is the architecture pillar --- the memory-bearing, 
holonomic structure that allows a system to ``remember'' past
interactions. In quantum measurement, Qi-HAL is the mechanism
that makes the pointer record stable: once an outcome is
registered, the record is topologically protected by the bulk
Wilson-line holonomies on $S^1_\theta$ (which in the cohort canonical
bulk include the \UoneQi{} factor).

In everyday physics, Qi-HAL is what makes classical records
classical --- a photograph, a magnetic tape, a scratch on a
surface. These are all topologically protected against thermal
fluctuations by the bulk-gauge-field dynamics on $S^1_\theta$
(with \QiHAL{} the cohort-canonical pillar of \ASeven{} that
formalizes this protection).

\section{The \SIP{} (System Instance Packet): canonical unit}

GG-2026-4B introduces the \SIP{} as the canonical unit of
\GDOS{} accounting. For a quantum measurement, the \SIP{}
consists of:
\begin{itemize}[leftmargin=2em,itemsep=0.1em]
\item \textbf{State}: the 4D density matrix $\densbdy$.
\item \textbf{Ports}: the input/output channels (the
system-environment couplings).
\item \textbf{Currents}: the four currents (energy, momentum,
charge, Qi-number) flowing through the ports.
\item \textbf{Ledger}: the scalar accounting of the \GFF{} cost.
\item \textbf{Dictionary}: the operational instrument
$\{I_\alpha\}$ specifying what outcomes are registered.
\end{itemize}
The SIP is the universal bookkeeping unit that any open
quantum system fits into. It is not a new physical entity; it
is the minimal data structure required to specify a \GDOS{}
instance.

\section{Reduction to AdS/CFT, Randall-Sundrum, and \GG{} ---
similarities and differences}

Two precedents in the high-energy-theory literature relate
higher-dimensional bulks to lower-dimensional boundaries: Maldacena's
1997 AdS/CFT correspondence~\citep{Maldacena1998}, an exact strong--weak
duality between two equivalent quantum theories; and the Randall--Sundrum
warped-extra-dimension scenarios of 1999~\citep{RandallSundrum1999}, in
which a non-factorizable 5D scale-profile metric solves the hierarchy problem
without requiring large extra geometric directions. The \GG{} bulk shares
features with both --- a radial direction with $r$ playing the
RG-scale role of AdS, and an effective 4D boundary law (\GDOS{}) that
specializes to the textbook QM postulates --- but the relation is
neither an exact duality (the projection is non-invertible) nor a
passive backdrop (the boundary law is structurally forced by \OGGEP{}).
The present section makes the three-way comparison explicit so that
readers familiar with AdS/CFT or Randall--Sundrum can locate the
\GG{}/\GDOS{} construction precisely on the holography landscape.

\subsection{AdS/CFT is an exact duality}

In AdS/CFT \citep{Maldacena1998}, the bulk and boundary
theories are \emph{equivalent} as quantum theories --- two
descriptions of the same physics. The duality is ontologically
symmetric.

\subsection{\GG{} is an asymmetric projection}

In \GGsix{}, the 6D bulk is the \emph{primary} theory and the
4D boundary is a \emph{reduction} --- a partial trace. The
reduction is not a duality; the 4D theory does not know
everything that the 6D theory does. Information is genuinely
lost in the partial trace.

This asymmetry is what makes the measurement problem solvable
in \GGsix{} but not in AdS/CFT: the 6D deterministic theory can
have a well-defined state and evolution even when the 4D
observer sees probabilistic outcomes.

\subsection{Comparison table}

\begin{table}[H]
\centering\small
\caption[How GG-6 differs from AdS/CFT and Randall--Sundrum]{How \GGsix{} differs from AdS/CFT and Randall-Sundrum
for the measurement problem. The columns compare what each construction takes as given, what it derives, and what it leaves to interpretation. The two differences that matter here are the compact phase direction and the non-invertibility of the projection; the holographic constructions share the bulk-to-boundary geometry and not the measurement content, which is why they are neighbors of this reading rather than rivals to it.}
\label{tab:adscft_comparison}
\begin{tabular}{@{}lccc@{}}
\toprule
\textbf{Feature} & \textbf{AdS/CFT} & \textbf{Randall-Sundrum} &
\textbf{\GG{}/\GDOS{}}\\
\midrule
Bulk dimension & 5 & 5 & 6\\
Bulk-boundary & Exact duality & Gravity localization &
Partial trace \\
Unitary bulk? & Yes & Yes & Yes\\
Unitary boundary? & Yes (CFT) & Yes & No (CPTP)\\
Measurement theory? & External & External & Derived
from $\Trenv$\\
Born rule? & Assumed & Assumed & Derived via Gleason\\
Lindblad? & N/A & N/A & Forced by AX-1 + CP\\
Decoherence? & N/A & N/A & Automatic from $\Trenv$\\
Bell violations? & In CFT & In brane & 6D-local
artifact of $\Trenv$ \\
\bottomrule
\end{tabular}
\end{table}

\section{Worked example: Stern-Gerlach in \GDOS{}}

The Stern--Gerlach experiment of 1922~\citep{SternGerlach1922},
performed in Frankfurt with a beam of silver atoms passing through an
inhomogeneous magnetic field, is the original demonstration that
intrinsic angular momentum is quantized: the two discrete spots on the
detector, separated by an amount proportional to the field gradient,
were the first direct observation that a microscopic degree of freedom
can take only discrete values. Goudsmit and Uhlenbeck's 1925 spin
hypothesis~\citep{UhlenbeckGoudsmit1925} supplied the theoretical
framework a few years later. We use the simplest non-trivial
Stern--Gerlach setup --- a single qubit prepared in
$\ket{\psi} = \alpha\ket{\!\!\up} + \beta\ket{\!\!\dn}$ and measured
along direction $\hat n$ --- as the prototype worked example of the
\GDOS{}/\OGGEP{} construction, with the 6D bulk state, the boundary
partial trace, the instrument family, and the Born-rule probabilities
all calculated explicitly. The same calculation appears in
\citep{ArisakaA5GDOS} Part~III as the worked example for \T{7.3}.

To make the construction concrete, we work through the
Stern-Gerlach measurement in the \GDOS{} / \OGGEP{} language.

\subsection{The bulk state}

A silver atom in spin state $\ket{\psi} = \alpha\ket{\uparrow} +
\beta\ket{\downarrow}$ has a 6D bulk description:
\begin{equation}\label{eq:SG_bulk_state}
\ket{\psi}_{\rm bulk} \;=\; (\alpha\ket{\uparrow} + \beta\ket{\downarrow}) \otimes
\ket{\mathrm{holonomy}\ \Theta^{(+)}_{I_r \otimes S^1_\theta}},
\end{equation}
where the holonomy factor encodes the 6D Wilson-line phase
along the extra geometric directions. This holonomy carries the
``Qi-number'' that will determine the realized outcome.

\subsection{The apparatus coupling}

The magnetic-field gradient entangles spin with position. The
unitary $U_{SA}$ acts as
\begin{equation}\label{eq:SG_interaction}
\ket{\uparrow}\ket{x_0} \;\longrightarrow\; \ket{\uparrow}\ket{x_0 + \delta},
\quad \ket{\downarrow}\ket{x_0} \;\longrightarrow\; \ket{\downarrow}\ket{x_0 - \delta}.
\end{equation}
After the magnet, the state is
\begin{equation}\label{eq:SG_after_magnet}
(\alpha\ket{\uparrow}\ket{x_0+\delta} + \beta\ket{\downarrow}\ket{x_0-\delta})
\otimes \ket{\mathrm{holonomy}}.
\end{equation}
The spin is now entangled with position.

\subsection{The detector and partial trace}

A detector at the spatial locations $x_0 \pm \delta$ measures
which location the atom arrived at. In the \GDOS{} language,
this is a \POVM{} with elements
\begin{equation}\label{eq:SG_povm}
E_\uparrow = \ketbra{x_0+\delta}{x_0+\delta},\qquad
E_\downarrow = \ketbra{x_0-\delta}{x_0-\delta}.
\end{equation}
The outcome probabilities are $P(\uparrow) = |\alpha|^2$ and
$P(\downarrow) = |\beta|^2$, from (\ref{eq:Born_derived}).

\subsection{The \GSSB{} branch selection}

At the detector, the \GFF{} functional has two competing
branches (one for each detector location). The \GSSB{} step
selects one. Which one? The one compatible with the incoming
6D bulk holonomy. Specifically, if the bulk Wilson-line
holonomy at the moment of detection corresponds to
$\Theta^{(+)}$, the $\uparrow$-branch is selected; if $\Theta^{(-)}$,
the $\downarrow$-branch. The holonomy value is \emph{not accessible} to
the 4D observer --- it lives along the phase axis $S^1_\theta$
direction --- but it is a bulk-deterministic variable. From the
4D observer's perspective, the outcome is random (probabilities
$|\alpha|^2, |\beta|^2$); from the 6D bulk perspective, it is
determined by $\Theta$.

\subsection{The summary}

Stern-Gerlach in \GDOS{} is \emph{not} a process where the wave
function collapses. It is a process where (i) the 6D bulk
Wilson-line holonomy $\Theta^{(\pm)}$ on the phase axis
$S^1_\theta$ determines the outcome deterministically; (ii) the
four-dimensional observer, forced by the dimensionality of the
laboratory to trace over the geometric directions $I_r$ (scale)
and $S^1_\theta$ (phase), sees the outcome as random; (iii) the
randomness is forced to the Born distribution $|\alpha|^2,
|\beta|^2$ by Gleason's theorem applied to the resulting
boundary algebra. Everything is consistent, every step is a
theorem, and there is no measurement problem.

What this single worked example illustrates is the entire
content of Part~III in miniature: the constitutional bridge
$\GG{}~\text{bulk} \to \OGGEP{} \to \GDOS{} \to \text{4D~QM}$ is
not an interpretive device but a compiler --- it takes a
deterministic six-dimensional theory and outputs the textbook
quantum mechanics that laboratory practice uses, with every
boundary-side feature (probabilistic outcomes, off-diagonal
damping, pointer-basis selection, Bell-violating correlations)
forced by the bulk-side structure and the four-dimensional
trace. Parts~V--IX then exercise the compiler on five distinct
laboratory systems.







\section*{Part~III in plain English: the compiler that turns
geometry into quantum mechanics}
\addcontentsline{toc}{section}{Part~III in plain English}

The four-stage chain at the heart of this Part can be read in
plain words. At the input end stands a six-dimensional bulk geometry, $\mathcal{M}_4 \times I_r \times S^1_\theta$, on which
the dynamics is deterministic, unitary, and local --- exactly
the kind of theory Einstein hoped Nature would turn out to use.
At the output end stands the textbook of quantum mechanics: a
Hilbert space, a Schr\"odinger equation, a Born rule, a
L\"uders projection, decoherence, einselection, and the
violation of Bell's inequality. Between input and output sits
the constitutional bridge: \OGGEP{}, the operational postulate
that says \emph{any} admissible bulk theory must come equipped
with a structure-preserving map to the operational records a
finite four-dimensional laboratory produces. That map is
forced --- by gauge covariance, causality, positivity, and
scale consistency --- to take the form of a partial trace over
the two extra geometric directions ($I_r$ for scale, $S^1_\theta$
for phase) followed by an instrument-family construction.

The traditional reading of quantum mechanics asks how
measurement \emph{produces} randomness from a deterministic
state. The reading developed here turns the question around.
There is no production of randomness; there is only the
unavoidable four-dimensional view of a six-dimensional
deterministic dynamics. When the boundary observer integrates
out the two geometric directions she cannot access, the result
is a probability distribution. When she records an outcome,
the result is Bayesian conditioning on which boundary branch
the bulk Wilson-line holonomy delivered. When she watches the
off-diagonal terms of a density matrix decay, the result is the
overlap integral of the bulk environment configurations that
project to her record. None of these requires a new postulate.
All are theorems of the bridge.

After all, the most patient and accurate way to read the
hundred-year history of quantum mechanics is that the
community learned the boundary law before it learned the bulk
geometry. Part~III supplies the missing half --- the geometry
--- and shows that the boundary law every laboratory has tested
to twelve digits is exactly what the geometry compiles into.
The compiler is the mechanism. Everything that follows in this
paper is an application of it.


%% ===================================================================
%% PART IV (4/14) -- Statement of results --- the eleven principal theorems of A8-QM
%% ===================================================================
\begin{table}[H]
\centering\small
\caption[Dictionary from 4D quantum concepts to their 6D bulk origin]{Dictionary translating 4D quantum-mechanical concepts
to their 6D bulk origin in \GGtheory. Read the left column as what a boundary observer measures and the right as the bulk structure of which that measurement is a shadow. The dictionary states what the framework claims rather than deriving it; each entry is discharged by the theorem named in the row that carries it, and an entry with no theorem beside it is a reading, not a result.}
\label{tab:dictionary}
\renewcommand{\arraystretch}{1.2}
\begin{tabular}{@{}p{5.22cm}p{5.22cm}p{5.22cm}@{}}
\toprule
\textbf{4D quantum concept} &
\textbf{6D bulk origin} &
\textbf{Mechanism}\\
\midrule
Hilbert space $\Hilb_4$ &
KK zero-mode sector of $G_6 = \SUfive\times\Uoneb\times\UoneQi$
&
Representation content at bulk-to-boundary projection \\
\addlinespace[0.15em]
State $\ket{\psi}\in\Hilb$ &
Bulk wavefunctional $\Psi_{\rm bulk}[\Phi_6]$ projected to
zero-modes &
Partial trace over heavy KK + $I_r,S^1_\theta$ \\
\addlinespace[0.15em]
Density matrix $\densbdy$ &
$\Trenv(\rho_{\rm bulk})$ &
Automatic loss of information about extra geometric directions \\
\addlinespace[0.15em]
Born rule $P(a) = |\braket{a}{\psi}|^2$ &
Gleason's unique additive measure on $\Hilb_4$ &
Forced by \CPTP{} + additivity + dim $\ge 3$ \\
\addlinespace[0.15em]
Observable $\hat A = \hat A^\dagger$ &
Bulk gauge-current operator restricted to zero-modes &
Self-adjointness from Hermiticity of bulk Hamiltonian \\
\addlinespace[0.15em]
Commutator $[\hat x,\hat p]=i\hbar$ &
Bulk Poisson-bracket structure on zero-mode phase space &
Inherited from bulk symplectic form \\
\addlinespace[0.15em]
Schr\"odinger equation &
Stone theorem applied to zero-mode unitary subgroup &
Closed-subsystem limit of the Lindblad equation \\
\addlinespace[0.15em]
Lindblad equation \GKSL{} &
\CPTP{} semigroup from AX-1 + AX-3 + continuity &
Partial trace of 6D unitary + Markovian limit \\
\addlinespace[0.15em]
Decoherence &
$\Trenv$ off-diagonal suppression &
Automatic for any 4D observer \\
\addlinespace[0.15em]
Wave-function collapse &
Bayesian conditioning on outcome label &
Not a physical event in either 6D or 4D \\
\addlinespace[0.15em]
Entanglement &
Shared $S^1_\theta$-fiber bulk configuration &
6D-local; 4D-nonlocal after trace \\
\addlinespace[0.15em]
Spin 1/2 & Irrep of $\SUtwo\subset G_6$ zero-mode sector &
Geometric: half-integer winding on bulk bundles \\
\addlinespace[0.15em]
Photon $\gamma$ & $\Uone_{\rm EM}$ zero-mode gauge boson &
KK zero mode of bulk gauge field \\
\addlinespace[0.15em]
Hidden variable &
Wilson-line holonomy $\Theta_H$ on $S^1_\theta$ &
Inaccessible to 4D observers; deterministic in 6D \\
\addlinespace[0.15em]
Pointer basis (Zurek) &
Basis diagonalizing the \GSSB{} free-energy functional \GFF{} &
Dynamical selection by environment coupling \\
\addlinespace[0.15em]
\POVM{} element $E_\alpha$ &
$\Trapp[(\iden\otimes\Pi_\alpha)U(\cdot\otimes\sigma_{\rm app})
U^\dagger]$ &
Stinespring dilation from bulk Hamiltonian \\
\addlinespace[0.15em]
Tsirelson bound $2\sqrt{2}$ &
Maximum CHSH value allowed by MSMF 6D Lagrangian &
Could be exceeded only by non-MSMF bulk \\
\bottomrule
\end{tabular}
\end{table}

\GGPart[Statement of results]{Statement of results}{Eleven principal theorems: \TA{1}--\TA{7} inherited, \TA{8}--\TA{11} net-new}
\label{part:IV}

This Part is the technical core of \AEight{}: the eleven named theorems of the present paper. Each Section follows the cohort six-element envelope --- L4 historical opening, plain-English physics question, formal theorem statement (dual-layer algebraic + operational), proof outline, quantitative anchor, physics intuition with daily-life metaphor, kill criterion, plain-English close --- at the depth a careful graduate student in physics will need on a first reading.

\medskip
\noindent
\textbf{Part~IV is the results Part of \AEight{}.} Where Part~III states the constitutional bridge $\GG{}~\text{bulk} \to \OGGEP{} \to \GDOS{} \to \text{4D~QM}$ as a mechanism, Part~IV states the eleven principal theorems that bridge produces. Read together, the eleven theorems exhaust the operational content of textbook quantum mechanics: the four von~Neumann postulates (\TA{1}--\TA{4}), the three decoherence-program structures that a century of foundations work added to them (\TA{5}--\TA{7}), and four further theorems (\TA{8}--\TA{11}) that the bulk-side machinery makes available but standard treatments do not state. Nothing in the rest of the paper requires a postulate beyond these eleven.

The eleven theorems split into two groups. The first seven, \TA{1} through \TA{7}, are operational restatements of \T{7.1} through \T{7.7} of \AFive{} Part III \citep{ArisakaA5GDOS}; their proofs are inherited from \AFive{} and cited by reference, with the present treatment supplying only the textbook-pedagogical envelope. The remaining four, \TA{8} through \TA{11}, are net-new to \AEight{}: they promote results that \AFive{} stated as lemmas or sub-theorems, or that the cohort apparatus makes possible but \AFive{} had no room to develop, to the status of principal theorems of the QM register. All eleven are constitutional --- forced by \AX{1}--\AX{4} of the \GGConstitution{} (\AOne{}) acting on the bulk geometry of \ATwo{} via the \GDOS{} channel of \AFive{} Part II --- and none introduces any new axiom.

\begin{table}[H]
\centering
\caption[The eleven principal theorems of GG-A8-QM]{The eleven principal theorems of \AEight{}. The first seven inherit from \AFive{} Part III by operational restatement; the last four are net-new contributions of the present paper. The four columns give the theorem number, its statement in one line, the constitutional warrant it draws on, and the Part in which it is proved. The split matters for what the paper is claiming: the inherited seven are \AFive{} results restated at the boundary and carry no new assumption, while the net-new four are the paper's own and stand or fall with it.}
\label{tab:A8-theorem-inventory}
\small
\renewcommand{\arraystretch}{1.25}
\begin{tabular}{@{}llp{11.40cm}@{}}
\toprule
\textbf{Theorem} & \textbf{Lineage} & \textbf{Statement (one line)} \\
\midrule
\TA{1} & inherited (\T{7.1}) & The boundary Hilbert space is the GNS representation of the QM-restricted boundary algebra of the 6D bulk. \\
\TA{2} & inherited (\T{7.2}) & The Schr\"odinger equation is the closed-system ($\gamma_\alpha\to 0$) limit of the GKLS generator of the \GDOS{} channel. \\
\TA{3} & inherited (\T{7.3}) & The Born rule $p_\alpha = \Tr(\cI_\alpha\rho)$ is forced by CPTP completion of the \OGGEP{} dictionary. \\
\TA{4} & inherited (\T{7.4}) & The L\"uders projection $\rho \mapsto \cI_\alpha(\rho)/p_\alpha$ is the conditional CPTP posterior. \\
\TA{5} & inherited (\T{7.5}) & Off-diagonal coherences decay at rate $\Gamma_{\alpha\beta}$ fixed by scale-overlap integrals of bulk KK modes. \\
\TA{6} & inherited (\T{7.6}) & The einselected pointer basis is the eigenbasis of the dominant local system--environment coupling (\AX{1}); for a macroscopic center-of-mass this is position, for a flux/charge qubit it is flux/charge. \\
\TA{7} & inherited (\T{7.7}) & The CHSH bound is violated up to the Tsirelson saturation $|S|=2\sqrt 2$ as the boundary shadow of one non-separable joint bulk state. \\
\midrule
\TA{8} & net-new & The Tsirelson saturation is an extremal (Bogomolnyi-analogue) bound: the singlet is the CHSH extremizer, so saturation is first-order stationary and structurally stable, not a tuned coincidence. \\
\TA{9} & net-new & Every electron registered in any laboratory is a token of the unique stable \GA{} of the charged-lepton stratum; quantum indistinguishability is a cascade inheritance, not a postulate. \\
\TA{10} & net-new & Decoherence-rate spectra are approximately $1/f$ within a bounded conformal window, with a system-specific exponent near unity set by \QiCON{} (\ASeven{}). \\
\TA{11} & net-new & The non-separable joint bulk state underlying Bell correlations carries no operational signal; the boundary marginal at site~B is independent of the measurement settings at site~A. \\
\bottomrule
\end{tabular}
\end{table}

The remainder of this Part proves the four net-new theorems and inherits the seven from \AFive{} with the full pedagogical envelope.




%% ===================================================================
\section{\TA{1} --- The Hilbert-space postulate as theorem}
\label{sec:TA-1}

\subsection{L4 historical opening}

The Hilbert-space framework of quantum mechanics was assembled in the years 1925--1932 by Heisenberg, Schr\"odinger, Dirac, and von~Neumann \citep{Heisenberg1925, Schrodinger1926, Dirac1930, vonNeumann1932}. Von~Neumann's 1932 \emph{Mathematische Grundlagen} fixed the modern axiomatic form: the state of a closed quantum system is a unit vector in a complex separable Hilbert space, with observables represented by self-adjoint operators. Every standard graduate textbook \citep{CohenTannoudji1992, Sakurai2011, Ballentine1998, Peres1995} states this as a primitive postulate. The constitutional warrant of \AOne{} promotes the Hilbert-space \emph{structure} to a theorem; the complex scalars over which that structure is built enter with the algebra it represents, and their own constitutional standing is examined in \S\ref{sec:disc-real-bulk}.

\subsection{The physics question}

Why a Hilbert space? Why not a real vector space, a Banach space, a more exotic algebraic structure? And why, supposing one accepts the framework, does an isolated electron live in $L^2(\bR^3)$ rather than in some other infinite-dimensional Hilbert space? The textbook answer is silence: the postulate is asserted and verified by its predictive success. The framework's answer is direct: the boundary Hilbert space is the GNS representation \citep{Gelfand1943, Naimark1943} of the QM-restricted boundary algebra inherited from the 6D bulk of \citet{ArisakaA2GG6}. One scope note belongs here rather than in a footnote: GNS answers ``why a Hilbert space'' --- the linear, inner-product, completeness structure --- while inheriting its complex scalars from the algebra it represents; the distinct question ``why \emph{complex}'' is answered separately, in \S\ref{sec:disc-why-complex} and \S\ref{sec:disc-real-bulk}.

\subsection{Theorem statement}

\begin{euclidtheorem}[\TA{1} --- Hilbert-space postulate as theorem (T-QM-1)]
\label{thm:TA-1}
\textbf{Algebraic (\TA{1}.1).} The QM-restricted boundary algebra $\cA_\Bdy^{(\rm QM)}$, generated by the zero-mode operators $\{\phi^{(0,0)}(x)\}$ of the bulk fields and their conjugate momenta, admits a unique-up-to-bulk-gauge faithful representation on a complex separable Hilbert space $\cH_\Bdy^{(\rm QM)}$, with an isometric embedding $\iota:\cH_\Bdy^{(\rm QM)}\hookrightarrow\cH_\Bulk$ that intertwines the algebra actions.

\textbf{Operational (\TA{1}.2).} The operationally registered state of the boundary system is the density operator $\rho_\Bdy^{(\rm QM)} = \cR_{\rm QM}[\Tr_{\rm env}\rho_\Bulk]$, positive and trace one on $\cH_\Bdy^{(\rm QM)}$. Pure states correspond to the limiting case of negligible bulk--boundary entanglement.
\end{euclidtheorem}

\begin{proof}[Proof outline]
Inherited from \T{7.1} of \citep{ArisakaA5GDOS} Part III via the GNS construction \citep{Haag1992, StreaterWightman1964}: any unital $C^*$-algebra together with any state on it admits a cyclic representation on a Hilbert space; choosing the state to be the restriction of the bulk vacuum to $\cA_\Bdy^{(\rm QM)}$ produces the boundary GNS space, which embeds isometrically into the bulk GNS space (which is $\cH_\Bulk$). Density-matrix recovery follows from partial trace plus CPTP boundary reduction, both of which preserve positivity and trace.
\end{proof}

\subsection{Quantitative anchor}

For the simplest non-trivial case --- a single fermion zero mode realizing a qubit --- $\cH_\Bdy^{(\rm QM)} = \mathbb{C}^2$, the standard qubit Hilbert space; the Bloch-ball constraint $|\vec r\,|\le 1$ on the density matrix $\rho = \tfrac{1}{2}(\mathbf{1}+\vec r\cdot\vec\sigma)$ inherits directly from the bulk-unitarity bound on the reduced state. Every textbook qubit calculation (Pauli algebra, Bloch sphere, single-qubit gates, the geometric phase, two-level Rabi oscillations) is a specialization of \TA{1} to this realization.

\subsection{Physics intuition and daily-life metaphor}

\emph{Physics intuition.} The Hilbert space is not added as an external mathematical convenience: it is the smallest linear-algebraic object that can faithfully carry the boundary algebra's structure (commutation relations, eigenvalue spectra, transition amplitudes). Once one accepts that the boundary algebra exists (which \AX{2}/\AX{3} of \AOne{} force), the Hilbert space follows automatically by GNS.

\emph{Daily-life metaphor.} A photograph is not just a sheet of paper; it is the smallest two-dimensional surface that can faithfully carry the visual structure of the scene. Try to record the scene on a strip or in a point and the structure tears. The Hilbert space is the analogous smallest object for the boundary algebra: not chosen, but forced by what one is trying to represent.

\subsection{Kill criterion}

\TA{1} would be retired by the experimental discovery of a quantum system whose boundary algebra cannot be embedded in any Hilbert space --- for instance, a system exhibiting a probability rule incompatible with $\sigma$-additivity on closed subspaces. No such system has ever been observed; the framework's prediction matches a century of laboratory experience.

\subsection{Plain-English close}

The Hilbert-space framework of quantum mechanics is not added on top of the physics by mathematical fiat. It is the unique algebraic object compatible with the boundary algebra that \OGGEP{} forces. The first textbook postulate is therefore not a postulate; it is a structural inevitability. One register note completes it: the recovered space arrives \emph{complex} --- not by assumption but because the pairing the projection installs is the derived complex unit $J$ of the real-bulk statements (\S\ref{sec:disc-real-bulk}; owner \citealp{ArisakaP4Real}, derived-conditional in the quasi-free regime), so even the field $\mathbb{C}$ under this theorem is an output of the same descent.




%% ===================================================================
\section{\TA{2} --- The Schr\"odinger equation as closed-system limit}
\label{sec:TA-2}

\subsection{L4 historical opening}

Schr\"odinger's 1926 wave equation \citep{Schrodinger1926}, written down in January 1926 and equivalent to Heisenberg's 1925 matrix mechanics \citep{Heisenberg1925} as Dirac showed in 1930 \citep{Dirac1930}, is the most-celebrated equation in twentieth-century physics. Every textbook presents it as primitive, on the same footing as Newton's $F=ma$. But just as Newton's laws are the classical limit of Lorentzian dynamics, the Schr\"odinger equation is the closed-system limit of a deeper open-system dynamics: the GKLS master equation of \citet{GKSL1976, Lindblad1976}.

\subsection{The physics question}

Is the Schr\"odinger equation primitive, or is it the limiting case of some deeper dynamical law? The textbook answer is no. The framework's answer is yes: the constitutional boundary law is the GKLS generator of the \GDOS{} channel, and the Schr\"odinger equation is its $\gamma_\alpha\to 0$ closed-system limit.

\subsection{Theorem statement}

\begin{euclidtheorem}[\TA{2} --- Schr\"odinger equation as closed-system limit (T-QM-2)]
\label{thm:TA-2}
\textbf{Algebraic (\TA{2}.1).} In the limit $\gamma_\alpha^{(\rm QM)}\to 0$ where all GKLS dissipation rates vanish, the boundary master equation reduces to the von~Neumann equation $\partial_t\rho = -i[H_{\rm eff}^{(\rm QM)},\rho]$, with $H_{\rm eff}^{(\rm QM)}$ unique up to a constant additive shift (the choice of energy zero).

\textbf{Operational (\TA{2}.2).} For pure states $\rho = |\psi\rangle\langle\psi|$, the von~Neumann equation reduces to the Schr\"odinger equation
\begin{equation}\label{eq:TA-2-schrodinger}
i\partial_t |\psi(t)\rangle \;=\; H_{\rm eff}^{(\rm QM)}\,|\psi(t)\rangle,
\end{equation}
unique up to global phase. The Schr\"odinger equation is therefore the constitutional closed-system limit of the \GDOS{} boundary law, not an independent primitive law.
\end{euclidtheorem}

\begin{proof}[Proof outline]
Inherited from \T{7.2} of \citep{ArisakaA5GDOS} Part III in two steps. Step 1: with all rates vanishing the GKLS dissipator vanishes and the generator becomes the Hamiltonian conjugation $-i[H_{\rm eff}^{(\rm QM)},\cdot]$; hermiticity of $H_{\rm eff}^{(\rm QM)}$ ensures unitarity. Step 2: substituting $\rho = |\psi\rangle\langle\psi|$ into the von~Neumann equation and pairing with $\langle\psi|$ recovers the Schr\"odinger equation up to an unobservable global phase.
\end{proof}

\subsection{Quantitative anchor}

The closed-system limit governs all of single-particle quantum mechanics in the absence of measurement back-action: the hydrogen-atom Rydberg spectrum to twelve significant figures \citep{CODATA2018}, the anomalous magnetic moment of the electron $a_e=(g-2)/2$ to one part in $10^{12}$, the Hong--Ou--Mandel interference visibility to one part in $10^4$. The framework predicts no deviation from textbook QM in any of these regimes; \TA{2} merely supplies the constitutional warrant for what experiment has been verifying since 1925.

\subsection{Physics intuition and daily-life metaphor}

\emph{Physics intuition.} The Schr\"odinger equation is the dissipation-free idealization of the actual boundary dynamics, in exactly the way that the frictionless pendulum is the dissipation-free idealization of the actual swinging pendulum. Both idealizations are useful, and both are mathematically clean, but neither describes the full physical situation.

\emph{Daily-life metaphor.} A spinning top is well-described by the conservation laws of angular momentum for the first few seconds; the more honest description, valid over longer times, includes the friction that eventually brings it to rest. Schr\"odinger evolution is the conservation-law description; GKLS is the honest one. For laboratory experiments on time scales short compared to the decoherence time, the conservation-law description is correct; for measurement-time-scale events, it is not.

\subsection{Kill criterion}

\TA{2} would be retired by the experimental discovery of a closed quantum system whose evolution is not unitary in the $\gamma\to 0$ limit --- for instance, an isolated atom whose spectral lines drift in a way not attributable to environmental coupling. No such system has ever been observed; the framework's prediction matches the precision-spectroscopy record of the past century.

\subsection{Plain-English close}

The Schr\"odinger equation is the law of a quantum system imagined as fully isolated; the GKLS master equation is the law of a quantum system as it actually exists, coupled to an environment that the boundary observer cannot fully control. The first is a limit of the second. Both are correct within their domains, and neither is a primitive postulate.

\paragraph{Where the $i$ comes from.} One more textbook primitive
dissolves at this theorem, and it deserves its own paragraph because
it sits in the equation's very first symbol. In the 6D bulk every
physical quantity is real, and the dynamics that this theorem
projects is a \emph{real symplectic flow}; the boundary observer,
who cannot resolve the two paired real structures of the arena (the
compact phase circle and the causal clock), reads that flow in the
derived complex coordinates the projection installs --- and in those
coordinates the real pair of first-order equations compresses into
the single line $i\hbar\,\partial_t\psi = H\psi$. The $i$ of the
Schr\"odinger equation is therefore not a primitive of nature but
the projection's bookkeeping: the rotation generator of a real pair,
derived at derived-conditional grade in the quasi-free regime by the
origin-of-$i$ companion (\citealp{ArisakaP4Real}), with the arrow
and clock side of the pairing held by the CPT companion's custody
clause (\citealp{ArisakaP5CPT}); the cohort's one-line custody
sentence --- the dictionary owns the sectors, the geometry owns the
$i$ --- holds here verbatim, and the compact statement catalog is
\S\ref{sec:disc-real-bulk}. Problem D.1.2 of the technical
appendices lets the reader perform the compression by hand.




%% ===================================================================
\section{\TA{3} --- The Born rule as theorem}
\label{sec:TA-3}

\subsection{L4 historical opening}

Born's June--July 1926 papers \citep{Born1926, BornHeisenbergJordan1926} introduced the probabilistic interpretation of the wavefunction modulus-squared. The rule worked spectacularly --- atomic spectra, scattering cross-sections, the Stern--Gerlach \citep{SternGerlach1922} outcome distribution all agreed --- but Born offered no derivation. Gleason's 1957 theorem \citep{Gleason1957} sharpened the rule to a uniqueness statement on a Hilbert space of dimension $\geq 3$, but assumed the Hilbert-space framework as input. Many-Worlds proponents have spent fifty years trying to derive the rule from unitary dynamics alone; decoherence theorists have offered ``predictability-sieve'' justifications \citep{Zurek2003}; QBism reads the rule subjectively \citep{Fuchs2010}. None has been universally accepted.

\subsection{The physics question}

Where does the Born rule come from? Is it a primitive postulate, a consequence of the Hilbert-space framework, a consequence of unitarity, or a consequence of something else?

\subsection{Theorem statement}

\begin{euclidtheorem}[\TA{3} --- Born rule from CPTP completion (T-QM-3)]
\label{thm:TA-3}
\textbf{Algebraic (\TA{3}.1).} For any boundary density operator $\rho_\Bdy^{(\rm QM)}$ and any admissible instrument family $\{\cI_\alpha\}$ (CPTP-completable to a single CPTP measurement channel), conditional on the $\sigma$-additive structure of the boundary measure inherited from \AX{3},
\begin{equation}\label{eq:TA-3-Born}
\boxed{\; p_\alpha \;=\; \Tr[\cI_\alpha(\rho_\Bdy^{(\rm QM)})] \;=\; \Tr[E_\alpha\,\rho_\Bdy^{(\rm QM)}], \;}
\end{equation}
with $\{E_\alpha = \cI_\alpha^*(\mathbf{1})\}$ a POVM.

\textbf{Operational (\TA{3}.2).} For pure states with sharp projective measurement $E_\alpha = \Pi_\alpha = |\alpha\rangle\langle\alpha|$,
\begin{equation}
p_\alpha \;=\; |\langle\alpha|\psi\rangle|^2,
\end{equation}
the textbook Born rule.
\end{euclidtheorem}

\begin{proof}[Proof outline]
Inherited from \T{7.3} of \citep{ArisakaA5GDOS} Part III in three steps. Step 1: by trace duality, every CP map admits a unique Heisenberg adjoint, and the adjoint of any instrument sends the identity to a positive operator (the effect operator $E_\alpha$). Step 2: by Stinespring's theorem \citep{Stinespring1955}, the Kraus form $\cI_\alpha(\rho) = \sum_n K_{\alpha,n}\rho K_{\alpha,n}^\dagger$ gives $E_\alpha = \sum_n K_{\alpha,n}^\dagger K_{\alpha,n}\ge 0$. Step 3: trace preservation of the total CPTP channel $\sum_\alpha\cI_\alpha$ forces $\sum_\alpha E_\alpha = \mathbf{1}$ (the POVM completeness relation), and the cyclicity of trace then yields \eqref{eq:TA-3-Born}. Specialization to pure states with sharp projectors gives $|\langle\alpha|\psi\rangle|^2$ directly.
\end{proof}

\subsection{Where probabilities enter the derivation}
\label{ssec:TA-3-where-probs}

A careful reader will ask: where in the chain
\OGGEP{}\,$\to$\,CPTP\,$\to$\,Stinespring\,$\to$\,POVM\,$\to$\,Born
does a probability axiom first appear? The honest answer is at
the constitutional level, not at the proof level. The CPTP
structure of the boundary channel is required by \AX{3} of
\AOne{} (\OGGEP{}: an admissible bulk theory must come equipped
with a structure-preserving functor to the boundary algebra that
is, among other things, positivity-preserving); complete
positivity, trace preservation, and the resulting probability
semantics of the effect operators $\{E_\alpha\}$ are therefore
constitutional inheritances, not derivations. Gleason's
$\sigma$-additivity is a hypothesis on the boundary measure; the
present paper does not derive that the partial trace over bulk
KK modes produces a $\sigma$-additive boundary measure (this is
a corollary of \AX{3} + \AX{4} that the constitutional papers
\AOne{} and \AFour{} address). What \TA{3} establishes is therefore
the \emph{conditional} statement:

\begin{quote}\itshape
If the boundary register is described by a $\sigma$-additive
probability measure on the projection lattice (which \AX{3} and
\AX{4} together force), then by Gleason that measure has the
trace form \eqref{eq:TA-3-Born}, with the textbook
$|\langle\alpha|\psi\rangle|^2$ recovered as the pure-state
sharp-projective specialization.
\end{quote}

The unconditional statement --- that the partial trace over the
radial direction $I_r$ and the phase direction $S^1_\theta$ \emph{must} produce
a $\sigma$-additive measure rather than, say, a deterministic
boundary record selected by some bulk gauge condition --- is a
constitutional inheritance from \AOne{} + \AFour{}, not a theorem
of \TA{3}. We flag this as a residual constructive question in
§\ref{sec:disc-open} (the framework-versus-Many-Worlds operational
discriminator question): a complete unconditional derivation of
the probability semantics from a deterministic bulk substrate
remains a constructive task for future work. What the present
paper claims is sharper than ``the Born rule is postulated'' (the
textbook position) and weaker than ``the Born rule is derived
from no probability axiom whatever'' (a position no derivation
in the foundations literature has achieved, including
Many-Worlds decision-theoretic attempts \citep{Wallace2012EmergentMultiverse}). We claim that the Born rule's specific functional form is forced once the constitutional structure of \OGGEP{} and \MSMF{} is in place.

\subsection{Quantitative anchor}

The Born rule has been verified experimentally to one part in $10^4$ in single-particle interference experiments \citep{Sinha2010_TripleSlit}, and to higher precision in CHSH violations (where the rule plus quantum unitarity together predict $|S|=2\sqrt 2$ to the measured precision of \citep{Hensen2015, Giustina2015, Shalm2015}). Gleason's theorem makes the rule unique among $\sigma$-additive probability measures on closed subspaces. \TA{3} closes the remaining postulational gap by supplying the constitutional source of the Hilbert-space framework Gleason assumed.

\subsection{Physics intuition and daily-life metaphor}

\emph{Physics intuition.} The Born rule is the unique probability-assignment rule compatible with (a) CPTP evolution at the boundary, (b) a Hilbert-space framework inherited via GNS, and (c) the no-signaling constraint that the boundary projection inherits from the bulk's reduced-state factoring. Any deviation would violate one of the three.

\emph{Daily-life metaphor.} A weighing scale gives readings whose statistical distribution is fixed by the engineering of the scale plus the constraint that the scale return reproducible measurements at each level of precision. Once the scale is built, the distribution is forced --- not chosen by the scale's user. The Born rule is forced in the same way once \OGGEP{} is built.

\subsection{Kill criterion}

\TA{3} would be retired by the experimental discovery of a quantum system whose outcome statistics violate $|\langle\alpha|\psi\rangle|^2$ at a level above $10^{-4}$ in the linearity regime that quantum mechanics is supposed to govern. Recent triple-slit experiments \citep{Sinha2010_TripleSlit} have tested for the Born rule's signature linear-superposition response to within these limits, with results consistent with quantum-mechanical linearity.

\subsection{Plain-English close}

A century of experiment has shown that Nature obeys the Born rule. A century of theory has tried to explain why. The framework's answer is that the Born rule is the only rule compatible with the constitutional warrants the cohort imposes upstream; it is not a postulate of quantum mechanics, but a feature of the operational projection from the 6D bulk to the 4D boundary.




%% ===================================================================
\section{\TA{4} --- The L\"uders projection rule as theorem}
\label{sec:TA-4}

\subsection{L4 historical opening}

The L\"uders projection rule --- often called the ``collapse postulate'' --- has been the most physically controversial of the textbook postulates since von~Neumann introduced it in 1932 \citep{vonNeumann1932}. L\"uders refined the rule to the modern non-degenerate-measurement form in 1951. Copenhagen leaves the postulate ambiguous, locating ``collapse'' at a Heisenberg-cut drawn at the experimenter's discretion; many-worlds denies that collapse occurs at all; GRW models \citep{GRW1986} replace the postulate with a stochastic dynamical law. None of these has been universally accepted. \TA{4} dissolves the conundrum: the ``collapse'' is just the conditional CPTP state-update of the boundary observer, and the underlying bulk evolution remains unitary throughout.

\subsection{The physics question}

Is the wavefunction collapse a physical process, or is it an information-theoretic update of the observer's record? When, exactly, does it happen? Where is the boundary between the unitarily evolving quantum system and the classically observed apparatus?

\subsection{Theorem statement}

\begin{euclidtheorem}[\TA{4} --- L\"uders projection from conditional CPTP (T-QM-4)]
\label{thm:TA-4}
\textbf{Algebraic (\TA{4}.1).} The state-update map
\begin{equation}\label{eq:TA-4-luders}
\boxed{\; \rho_\Bdy^{(\rm QM)\,(\alpha)} \;=\; \frac{\cI_\alpha(\rho_\Bdy^{(\rm QM)})}{p_\alpha} \;}
\end{equation}
is, for each fixed outcome $\alpha$ with $p_\alpha > 0$, a (conditional) CPTP map on $\cB(\cH_\Bdy^{(\rm QM)})$ --- the unique CPTP update consistent with classical-outcome registration.

\textbf{Operational (\TA{4}.2).} For pure states with sharp projective measurement, $|\psi^{(\alpha)}\rangle = \Pi_\alpha|\psi\rangle/\|\Pi_\alpha|\psi\rangle\|$, the textbook L\"uders projection. No physical collapse occurs in the bulk; the apparent collapse is the operational update of the boundary observer's record.
\end{euclidtheorem}

\begin{proof}[Proof outline]
Inherited from \T{7.4} of \citep{ArisakaA5GDOS} Part III in two steps. Step 1: $\cI_\alpha$ is sub-normalized CPTP by hypothesis, and division by the normalization $p_\alpha$ restores trace preservation while preserving complete positivity \citep{NielsenChuang2010}. Step 2: the bulk evolution is unitary by \AX{1}, so the boundary state-update is implemented by unitary bulk evolution plus conditional partial trace over non-registered apparatus pointer states --- the standard conditional-CPTP construction. No discontinuous bulk collapse is required.
\end{proof}

\subsection{Quantitative anchor}

Repeated-measurement experiments (single-shot dispersive readout of superconducting qubits, electron-shelving in trapped-ion qubits, photon-number-resolving detectors; see \citet{WisemanMilburn2010} for the standard operational treatment) confirm \TA{4} to within readout-error rates of $10^{-3}$--$10^{-5}$ depending on platform \citep{NielsenChuang2010}. The framework predicts no deviation from textbook QM measurement statistics in any of these regimes; \TA{4} merely supplies the constitutional warrant for what experiment has been verifying since the original double-slit photon-counting experiments.

\subsection{Physics intuition and daily-life metaphor}

\emph{Physics intuition.} The ``collapse'' is not an event in the bulk; it is a re-labeling of the boundary observer's record. Before the measurement, the observer assigns a density operator $\rho$ to the system; after registering outcome $\alpha$, the observer updates to the conditional posterior $\rho^{(\alpha)}$. This is Bayesian conditioning in quantum language, with the prior--posterior structure forced by the CPTP nature of the instrument.

\emph{Daily-life metaphor.} A scientist studying a coin's bias updates her estimate of the bias after each flip; the coin does not change, but her record of it does. The L\"uders rule is the quantum analogue: the boundary observer updates her record after each measurement, but the bulk state evolves unitarily throughout.

\subsection{Kill criterion}

\TA{4} would be retired by the experimental observation of a non-CPTP state-update --- for instance, a measurement whose post-measurement statistics violate Bayes consistency under sequential conditioning, or whose post-measurement density operator fails complete positivity. No such measurement has been reported; the framework's prediction matches the foundations-of-quantum-information literature \citep{NielsenChuang2010, Holevo2012}.

\subsection{Plain-English close}

There is no measurement problem. There is a deeper unitary bulk dynamics, a non-invertible projection to a boundary algebra, and an operational update of the boundary observer's record on outcome registration. The textbook L\"uders rule is the third of these, and it is structurally forced by the first two.




%% ===================================================================
\section{\TA{5} --- Decoherence as off-diagonal damping}
\label{sec:TA-5}

\subsection{L4 historical opening}

The decoherence program of Zeh (1970), Zurek (1981), and Joos--Zeh (1985) \citep{Zeh1970, Zurek1981, JoosZeh1985, Zurek2003} was the most significant theoretical advance on the measurement problem in the half-century after the Bohr--Einstein debate. It established that off-diagonal density-matrix elements decay exponentially fast under environmental coupling, on time-scales that for macroscopic objects are essentially instantaneous. Standard treatments \citep{Schlosshauer2005, Schlosshauer2007, BreuerPetruccione2002} take the decoherence rate as a phenomenological input. \TA{5} promotes the rate to a derived quantity, fixed by scale-overlap integrals of the bulk geometry.

\subsection{The physics question}

What is the rate of decoherence, and where does it come from? Why is it essentially instantaneous for macroscopic objects but slow enough to permit interference for microscopic ones?

\subsection{Theorem statement}

\begin{euclidtheorem}[\TA{5} --- Decoherence at rate $\Gamma_{\alpha\beta}$ from bulk KK overlap (T-QM-5)]
\label{thm:TA-5}
\textbf{Algebraic (\TA{5}.1).} In any basis $\{|\alpha\rangle\}$ that diagonalizes the system--environment interaction, the off-diagonal evolution under the GKLS generator has the solution
\begin{equation}\label{eq:TA-5-damping}
\boxed{\; \rho_{\alpha\beta}(t) \;=\; \rho_{\alpha\beta}(0)\,e^{-i\omega_{\alpha\beta}t}\,e^{-\Gamma_{\alpha\beta}t}, \;}
\end{equation}
with decoherence rate
$\Gamma_{\alpha\beta} \;=\; \sum_\gamma \gamma_\gamma\,|\langle\alpha|L_\gamma|\alpha\rangle - \langle\beta|L_\gamma|\beta\rangle|^2$.

\textbf{Operational (\TA{5}.2).} The GKLS rates $\gamma_\gamma$ are not free parameters: they inherit their values from scale-overlap integrals of bulk KK modes,
$\gamma_\gamma \;=\; 2\pi\,|\int_0^K dr\,e^{-2r}f_0(r)f_n(r)|^2 \cdot \rho_{\rm env}(\omega_n)$,
with $\rho_{\rm env}(\omega_n)$ the bulk-mode spectral density at the resonant Bohr frequency. For macroscopically distinguishable states, $\Gamma_{\alpha\beta} \gtrsim 10^{40}~\mathrm{s}^{-1}$ at room temperature.
\end{euclidtheorem}

\begin{proof}[Proof outline]
Inherited from \T{7.5} of \citep{ArisakaA5GDOS} Part III via the Born--Markov chain specialized to the \GG{} bulk coupling. Step 1: substitute the KK mode expansion into the bulk system--environment Hamiltonian; second-order perturbation theory in the coupling constant produces the scale-overlap integral. Step 2: take the Markov limit (environmental correlation time short compared to the boundary system's characteristic time); Fermi's golden rule gives the $2\pi\rho_{\rm env}(\omega_n)$ factor. Step 3: substitute the rate into the GKLS off-diagonal evolution; integration yields the exponential damping. The macroscopic-instantaneous regime follows from Joos--Zeh's calculation \citep{JoosZeh1985} of the air-molecule scattering rate.
\end{proof}

\subsection{Quantitative anchor}

For Schr\"odinger's cat in a position superposition of separation $\Delta x \sim 0.1$~m at room temperature, the decoherence rate per air molecule is $\Gamma_1 \sim k_B T m^2 v/\hbar^2 \cdot (\Delta x)^2 \sim 10^{20}~\mathrm{s}^{-1}$; the total over the $\sim 10^{20}$ molecules per cm$^2$ per second of impact gives $\Gamma_{\rm total} \sim 10^{40}~\mathrm{s}^{-1}$, so $\tau_{\rm dec} \sim 10^{-40}$~s --- forty orders of magnitude faster than any apparatus can resolve. For a single photon in a high-vacuum Mach--Zehnder interferometer with arm length $\sim 1$~m and ambient density $n\sim 10^{10}$~molecules/m$^3$, $\Gamma \sim 10^{-3}~\mathrm{s}^{-1}$, giving $\tau_{\rm dec}\sim 10^3$~s and ample time for interference.

\subsection{Physics intuition and daily-life metaphor}

\emph{Physics intuition.} The decoherence rate is fixed not by an adjustable parameter but by the constitutional bulk geometry --- specifically, by the scale profile $e^{-2r}$ that sets which KK modes overlap the boundary zero mode and at what amplitude. In the cohort's zero-free-continuous-parameter framework, $\Gamma$ is a derived number for every system one cares to study.

\emph{Daily-life metaphor.} A whisper in a quiet room can be heard many feet away; the same whisper in a stadium full of cheering fans dies within a meter. The decoherence rate is the cheering of the bulk environment: small systems can keep their quantum coherence audible for a long time; macroscopic systems are drowned out instantly.

\subsection{Kill criterion}

\TA{5} would be retired by the experimental observation of decoherence rates that fail the $\Delta x^2$-scaling for macroscopic objects, or by the discovery of a quantum system whose decoherence rate depends on a free parameter unfixed by the bulk geometry. Existing experiments \citep{Arndt2003_C60, Schlosshauer2005} on mesoscopic systems (large molecules, BEC interferometry) have not produced such an observation; the framework's prediction holds.

\subsection{Plain-English close}

Decoherence is what makes the quantum-to-classical transition look as sharp as it does. In the cohort framework, the rate is not an adjustable knob but a calculation from the bulk geometry. Whether you can or cannot see an interference pattern is fixed by where in the 6D bulk your apparatus sits in coupling space.




%% ===================================================================
\subsection{A fully worked overlap integral: collisional decoherence of $C_{70}$ matter waves}
\label{sec:worked-overlap}

It is one thing to state that the decoherence rate of \TA{5} is fixed by a
scale-overlap integral, and another to evaluate that integral to an actual
number for a concrete platform and compare it with a measured rate. This
subsection does so. The platform is fullerene matter-wave
interferometry --- the cleanest laboratory in which decoherence has been
\emph{measured} as the controlled loss of interference contrast
\citep{Arndt2003_C60, Hornberger2003, Hackermuller2004}. The methods, the
dispersion-coefficient estimates, and the uncertainty budget are collected in
Appendix~\ref{app:worked-overlap}; the physical result is given here.

\subsubsection*{A refinement of the reading of \TA{5}.2}

Before computing, we state precisely what the framework does and does not claim
for a laboratory system, because the worked integral sharpens the reading of
\TA{5}.2. Three objects must be kept distinct.
\begin{itemize}
\item \emph{The physical bath sets the rate.} The decohering environment of a
  laboratory system is its physical $4$D bath --- background gas, thermal
  photons, phonons. The magnitude of the decoherence rate is the boundary
  projection of the system--bath coupling, which is operationally the ordinary
  scattering matrix element. This is what is evaluated below.
\item \emph{The bulk geometry supplies the open-system structure, not the
  magnitude.} Coarse-graining the non-compact scale fiber $I_r$ --- a forward
  renormalization-group semigroup --- is what makes the boundary evolution
  completely positive, Markovian, and irreversible, i.e.\ of GKLS form
  (\S\ref{sec:decoh-OGGEP}). It explains \emph{why} marginalization yields a
  Lindblad generator; it does not by itself fix the numerical rate.
\item \emph{The compact phase fiber $S^1_\theta$ KK tower is a separate, and
  negligible, channel.} Only its massive $n\ge 1$ modes (at mass $\sim\Mlock$)
  are framework-specific new physics; its massless $n=0$ zero mode is the
  ordinary $4$D gauge field, whose decoherence is already part of the physical
  bath.
\end{itemize}
Thus the operationally correct reading of \TA{5}.2 is: \emph{the laboratory
decoherence rate is the physical-bath overlap integral; the bulk geometry fixes its open-system form; the $S^1_\theta$ tower contributes only a
kinematically suppressed correction that switches on near $\Mlock$.} The
remainder of this subsection makes both halves quantitative.

\subsubsection*{The decoherence function and its two limits}

Collisional decoherence is governed by the Gallis--Fleming /
Hornberger--Vacchini master equation \citep{GallisFleming1990, Hornberger2003},
under which the off-diagonal density-matrix element in the position basis decays
as $\rho(\mathbf x,\mathbf x',t)=\rho(\mathbf x,\mathbf x',0)\,e^{-F(\mathbf
x-\mathbf x')\,t}$, with the decoherence function
\begin{equation}\label{eq:Fdx}
F(\Delta\mathbf x)\;=\;n\!\int\!\mu(v)\,v\,dv\!\int\! d\Omega\,
\frac{d\sigma}{d\Omega}\,\Big[\,1-\cos\!\Big(\tfrac{\Delta\mathbf p\cdot
\Delta\mathbf x}{\hbar}\Big)\Big],
\end{equation}
where $n$ is the gas number density, $\mu(v)$ the gas speed distribution, and
$\Delta\mathbf p=2m_{\rm gas}v\sin(\theta/2)$ the momentum transfer in an elastic
collision. Equation~\eqref{eq:Fdx} is a single function with two physical limits,
separated by the crossover length $\Delta x_\star\simeq\hbar/\langle\Delta
p\rangle\approx\lambda_{\rm th,gas}$ (the thermal de Broglie wavelength of the
gas):
\begin{itemize}
\item \emph{Saturated (long-separation) limit, $\Delta x\gg\Delta x_\star$:} the
  cosine oscillates rapidly and averages to zero except in the forward
  direction, so the decoherence function saturates to the \emph{total scattering
  rate},
  \begin{equation}\label{eq:Fsat}
  F(\Delta\mathbf x)\;\longrightarrow\;\Gamma_{\rm dec}\;=\;n\,\sigma_{\rm
  tot}\,\bar v_{\rm rel}.
  \end{equation}
  This is the matter-wave regime: every collision that resolves the path
  separation removes a full unit of coherence.
\item \emph{Localization (short-separation) limit, $\Delta x\ll\Delta x_\star$:}
  expanding $1-\cos(\Delta\mathbf p\cdot\Delta\mathbf x/\hbar)\approx\tfrac12
  (\Delta\mathbf p\cdot\Delta\mathbf x/\hbar)^2$ gives
  $F(\Delta\mathbf x)\to\Lambda\,(\Delta x)^2$ with a single localization
  parameter $\Lambda$. This is the macroscopic regime.
\end{itemize}
The wide spread of decoherence rates sometimes quoted for a single
``macroscopic object'' is nothing but the localization branch $\Lambda(\Delta x)^2$ evaluated at different separations:
with the canonical air-bath value $\Lambda\sim10^{36}\,\mathrm{m^{-2}s^{-1}}$ one
has $\Gamma\sim10^{18}\,\mathrm{s^{-1}}$ at $\Delta x=1$~nm, $10^{24}$ at
$1\,\mu$m, $10^{32}$ at $1$~cm, and $10^{34}$ at $10$~cm. All are the same curve;
the only error is to quote a rate without naming $\Delta x$ and the bath.

\subsubsection*{The overlap integral as the van der Waals cross section}

In the saturated regime the overlap integral that fixes the rate is the total
elastic cross section of the molecule on the gas, taken over the long-range
van der Waals tail $-C_6/r^6$. The semiclassical Massey--Mohr / Landau--Lifshitz
result is
\begin{equation}\label{eq:massey}
\sigma_{\rm tot}\;=\;\kappa\,\Big(\frac{C_6}{\hbar\,\bar v_{\rm
rel}}\Big)^{2/5},\qquad \kappa=8.083\ (\pm\,{\sim}25\%),
\end{equation}
the coefficient $\kappa$ carrying a glory-averaging/approximation uncertainty of
order $25\%$ (Appendix~\ref{app:worked-overlap}). The dispersion coefficient for
$C_{70}$ on argon, estimated by the London and Slater--Kirkwood formulae
(Appendix~\ref{app:worked-overlap}), brackets
\begin{equation}\label{eq:c6band}
C_6\;\approx\;(2\text{--}4)\times10^{3}\ \text{a.u.}\qquad
(\text{London }2.2\times10^3,\ \text{Slater--Kirkwood }4.2\times10^3),
\end{equation}
an uncertainty of a factor $\sim 2$ in $C_6$, hence (since
$\sigma\propto C_6^{2/5}$) a factor $\sim 1.3$ in $\sigma$.

\subsubsection*{The number}

Taking argon at $T=300$~K and a representative beam speed $v_{\rm mol}=100$~m/s
(so that $\bar v_{\rm rel}=\sqrt{v_{\rm mol}^2+\bar v_{\rm gas}^2}=411$~m/s),
equations~\eqref{eq:massey}--\eqref{eq:c6band} give
\begin{equation}\label{eq:sigmaresult}
\boxed{\;\sigma_{\rm tot}\;\approx\;(1.5\text{--}2.0)\times10^{-17}\
\mathrm{m^2}\;\;(1500\text{--}2000\ \text{\AA}^2),\quad\text{central }
1.7\times10^{-17}\ \mathrm{m^2},\;}
\end{equation}
squarely within the measured band of fullerene--noble-gas cross sections. The
saturation condition is comfortably met: the gas thermal de Broglie wavelength
$\lambda_{\rm th,gas}=1.6\times10^{-11}$~m is four orders of magnitude below the
coherent path separation ($\sim10^{-8}$--$10^{-7}$~m), so
Eq.~\eqref{eq:Fsat} applies, and the velocity-averaged
$\langle\sigma v_{\rm rel}\rangle$ differs from $\sigma(\bar v_{\rm rel})\bar
v_{\rm rel}$ by only $2.5\%$ (Appendix~\ref{app:worked-overlap}). The decoherence
rate and the contrast-loss onset pressure are then
\begin{equation}\label{eq:gammaresult}
\Gamma_{\rm dec}(p)\;=\;\frac{p}{k_BT}\,\sigma_{\rm tot}\,\bar v_{\rm rel}
\;\approx\;150\text{--}196\ \mathrm{s^{-1}}\ \text{at }10^{-6}\ \mathrm{mbar},
\end{equation}
\begin{equation}\label{eq:pstarresult}
p_\star\;=\;\frac{k_BT}{\sigma_{\rm tot}\,\bar v_{\rm rel}\,t_{\rm transit}}
\;\approx\;1.4\times10^{-6}\ \mathrm{mbar}\quad
(\text{band }1.2\text{--}2.1\times10^{-6}),
\end{equation}
using the transit time $t_{\rm transit}=2L_T/v\approx4.1$~ms set by the Talbot
length $L_T=d^2/\lambda_{\rm dB}=0.21$~m of the interferometer.

\paragraph{Comparison with experiment (stated honestly).} The Arndt-group
fullerene interferometers preserve high contrast below $\sim10^{-7}$~mbar and
lose it as the pressure rises toward $\sim10^{-6}$~mbar; the computed onset
$p_\star\approx1.4\times10^{-6}$~mbar \emph{reproduces that measured order of
magnitude and onset-pressure scale}, and the cross section
$\sigma\sim1.7\times10^{-17}\,\mathrm{m^2}$ lies in the band for which
Hornberger \emph{et~al.}\ \citep{Hornberger2003} reported theory--experiment
agreement at the $10$--$20\%$ level. This is an order-of-magnitude result
computed from the interaction potential, not a fit to a particular run, and not
a precision match: the honest accuracy is a factor $\sim 1.5$, set by the $C_6$
and $\kappa$ uncertainties of Eqs.~\eqref{eq:massey}--\eqref{eq:c6band}.
Crucially, the framework \emph{must}
reproduce the Joos--Zeh collisional value in this regime to be correct; doing so
is a success condition, not a new prediction. The genuinely framework-specific
content lies elsewhere, in the bulk channel.

\subsubsection*{The framework-specific channel, bounded}

The $S^1_\theta$ KK tower is the one piece of decoherence that is not standard.
Its size is controlled by a kinematic threshold, and this is the robust part of
the estimate, independent of any coupling detail. Decoherence requires the
environment to absorb a \emph{real} quantum carrying which-path information; the
lightest massive excitation of the tower sits at $\Mlock c^2\approx3.85$~TeV. A
molecular or condensed-matter system carries $E_{\rm sys}\sim$~eV--keV, so real
KK emission is kinematically forbidden, and only virtual exchange occurs ---
which renormalizes parameters but transmits no information, hence causes no
decoherence. Below threshold the induced rate is suppressed by at least the
leading effective-field-theory power,
\begin{equation}\label{eq:bulkbound}
\frac{\Gamma_{\rm bulk}}{\Gamma_{\rm bath}}\;\lesssim\;
\Big(\frac{E_{\rm sys}}{\Mlock}\Big)^{p},\qquad p\ge 1,
\end{equation}
with $p$ fixed by the (presently unspecified) bulk--boundary operator dimension.
For \emph{every} $p\ge 1$ the suppression is overwhelming in the laboratory:
$(E_{\rm sys}/\Mlock)\sim3\times10^{-13}$ for a molecular system and
$\sim3\times10^{-10}$ at the keV scale, so $\Gamma_{\rm bulk}$ is unmeasurably
small and the physical bath dominates entirely --- exactly as observed. The
correlation time of the tower, $\tau_c=\hbar/\Mlock=1.7\times10^{-28}$~s, is a
\emph{correlation time, not a rate}; conflating the two would be a category error, which
Eq.~\eqref{eq:bulkbound} prevents. The channel
reaches $\mathcal O(1)$ only as $E_{\rm sys}\to\Mlock$, so it ``switches on''
near the same $\Mlock\approx3.85$~TeV scale as the high-energy Test~8, tying the
laboratory einselection test (Test~5) to the collider horizon (Test~8); the
\emph{location} of switch-on is $p$-dependent and therefore not yet a sharp
prediction. The massless $n=0$ zero mode is excluded from this channel by
construction: it is the ordinary photon, and its decoherence is already counted
in the physical-bath rate of Eq.~\eqref{eq:gammaresult}.

\subsubsection*{Summary}

The worked integral delivers, for one concrete platform, what \TA{5}.2 asserts
in general. Equations~\eqref{eq:sigmaresult}--\eqref{eq:pstarresult} give a real
decoherence rate, computed from the interaction potential and matched to the
measured onset pressure; Eq.~\eqref{eq:bulkbound} bounds the bulk channel
kinematically to negligibility, so the physical bath sets the rate and a
genuinely isolated particle does not measurably decohere; and the single
function $F(\Delta x)$ of Eq.~\eqref{eq:Fdx} unifies the matter-wave and
macroscopic regimes. Reproducing the Joos--Zeh collisional value where it
applies is a correctness condition, not a new prediction; the framework's
specific content is the structure of the open-system law together with the
threshold-gated bulk deviation of Eq.~\eqref{eq:bulkbound} --- the single
genuinely new, if presently unmeasurable, prediction, gated to $\Mlock$.

\section{\TA{6} --- Einselection of the pointer basis}
\label{sec:TA-6}

\subsection{L4 historical opening}

Zurek's 1981 einselection criterion \citep{Zurek1981, Zurek2003} answered the long-standing decoherence-program objection: decoherence shows that off-diagonal elements decay in \emph{some} basis, but does not by itself say \emph{which} basis is preferred. Zurek identified the einselected pointer basis as the one minimizing entropy production --- the eigenbasis of the system--environment coupling --- but this identifies the basis with the coupling without saying why the coupling has the form it does. \TA{6} closes the gap: the position-diagonal nature of the boundary coupling is forced by \AX{1} bulk locality.

\subsection{The physics question}

Why does the universe ``prefer'' the position basis as the classical record of macroscopic observations? Why not the momentum basis, the energy basis, some superposition basis? Where does the universal preference come from?

\subsection{Theorem statement}

\begin{euclidtheorem}[\TA{6} --- Einselection from bulk locality (\AX{1}) (T-QM-6)]
\label{thm:TA-6}
\textbf{Algebraic (\TA{6}.1).} The einselected pointer basis of $\cH_\Bdy^{(\rm QM)}$ is the eigenbasis of the dominant boundary system--environment coupling. By \AX{1} (bulk locality), the bulk Hamiltonian density is a polynomial in bulk fields at the same bulk point; the boundary zero-mode coupling inherits this locality as position-diagonal multiplicativity: $H_{SE} = \sum_{x_0}|x_0\rangle\langle x_0|\otimes\hat E_{\rm eff}(x_0)$.

\textbf{Operational (\TA{6}.2).} The classical-looking outcomes registered by laboratory measurements correspond to the elements of the position-eigenbasis specified above. No experimenter choice is required; the einselected basis is unique up to phase for generic bulk coupling.
\end{euclidtheorem}

\begin{proof}[Proof outline]
Inherited from \T{7.6} of \citep{ArisakaA5GDOS} Part III in three steps. Step 1 (locality $\Rightarrow$ local boundary coupling): \AX{1} forbids integro-differential kernels in the bulk Hamiltonian, so after KK expansion the boundary system--environment interaction is a sum of local-at-the-same-$x$ products. Step 2 (zero-mode coupling $\Rightarrow$ position-diagonal): for the lightest bulk scalar carrying the measured observable, the boundary zero mode is multiplicative in the position basis, so $H_{SE}$ decomposes as a sum of $|x_0\rangle\langle x_0|\otimes\hat E_{\rm eff}(x_0)$. Step 3 (Zurek's predictability sieve): a coupling diagonal in $\{|x_0\rangle\}$ decays off-diagonal coherences $\langle x_0|\rho|x_0'\rangle$ for $x_0\ne x_0'$ on the decoherence timescale of \TA{5}, while keeping diagonal elements unchanged; the basis $\{|x_0\rangle\}$ wins the sieve. Uniqueness up to phase follows from generic transversality \citep{Zurek2003}.
\end{proof}

\emph{Runtime remark.} In the regime vocabulary of
Section~\ref{sec:grip-spine}, this theorem has a one-line
restatement: \emph{einselection is \GRIPin{}}. The pointer basis is
the Geometric-Attractor catalog of the boundary stratum, the
predictability sieve is the $F_{\GFF}$ ranking, and the
``environment winning'' is the descent halting. Zurek's mechanism is
not replaced; it is recognized as the runtime's fixed-landscape
regime wearing its 1981 name.

\subsection{Quantitative anchor}

Decoherence experiments on superconducting qubits, trapped ions, and atomic clocks register the position basis as the einselected one at the macroscopic level, with off-diagonal coherences in non-position bases damped at the \TA{5} rate. The framework's prediction matches the universal observation that macroscopic records are localized in space: photographs, hard-drive bits, the macroscopic state of a Geiger counter --- all live in the position basis, none in the momentum basis.

\subsection{Physics intuition and daily-life metaphor}

\emph{Physics intuition.} Bulk locality is the deep reason the world looks classical at macroscopic scales. Because the bulk Hamiltonian is local-at-the-same-bulk-point, the boundary coupling is local-at-the-same-boundary-point; the operationally registered observables are therefore position-diagonal. Any other basis would require the coupling to be non-local in position, which \AX{1} forbids.

\emph{Daily-life metaphor.} A wet floor records footprints by position --- the imprint sits where the foot pressed --- not by speed; a slow walker and a sprinter leave the same imprint at the same place. Fingerprint dust adheres to surfaces by contact, not by velocity class. The position basis of macroscopic records is privileged not by a choice of detector but by the contact-locality of the recording mechanism. The same applies to every macroscopic record in nature.

\subsection{Kill criterion}

\TA{6} would be retired by the experimental observation of a macroscopic measurement record that is sharply localized in a non-position basis (e.g., a Geiger counter whose definite outcome is a momentum eigenvalue rather than a position eigenvalue) under conditions where the coupling is genuinely position-diagonal. No such observation has been reported; einselection is among the most empirically robust features of quantum measurement.

\subsection{Plain-English close}

The universe prefers the position basis because the universe is locally constructed. Once you accept bulk locality (which the cohort imposes via \AX{1}), the position-eigenbasis preference of classical records follows automatically. Nothing else needs to be added.




%% ===================================================================
\section{\TA{7} --- Bell--CHSH violation as boundary shadow}
\label{sec:TA-7}

\subsection{L4 historical opening}

Bell's 1964 inequality \citep{Bell1964} and the CHSH reformulation \citep{CHSH1969} of 1969 demonstrated that no local hidden-variable theory in 4D can reproduce the correlations of an entangled bipartite singlet. Aspect's 1981--1982 experiments \citep{Aspect1981, Aspect1982} pushed the violation past the classical bound; the loophole-free Delft/Vienna/NIST runs of 2015 \citep{Hensen2015, Giustina2015, Shalm2015} closed the remaining experimental loopholes; the 2022 Nobel Prize \citep{NobelPhysics2022} formalized the result; the modern review by \citet{BrunnerEtAl2014} is the canonical reference for the post-Aspect Bell-non-locality literature. The standard reading is that nature exhibits genuine non-locality. \TA{7} sharpens the reading: nature exhibits 4D-non-locality as the boundary shadow of one non-separable joint bulk state.

\subsection{The physics question}

Are Bell-CHSH violations evidence of non-locality in the fundamental theory, or are they an artifact of a non-invertible bulk-to-boundary projection from a higher-dimensional bulk?

\subsection{Theorem statement}

\begin{euclidtheorem}[\TA{7} --- CHSH violation as the projection shadow of one non-separable joint bulk state (T-QM-7)]
\label{thm:TA-7}
\textbf{Algebraic (\TA{7}.1).} An entangled bipartite preparation is a single \emph{non-separable} state of the six-dimensional bulk, supported on the shared phase fiber $S^1_\theta$ established at the common source. Its two boundary shadows, registered at 4D-spacelike-separated events $A=(x_A,0,0)$ and $B=(x_B,0,0)$, are projections of that one joint state along the radial (renormalization-scale) direction $I_r$; the boundary correlation function is the standard operator-algebra expectation in the joint state (\TA{1}, \TA{3}), not a quantity transmitted between the events.

\textbf{Operational (\TA{7}.2).} For an entangled bipartite boundary state (e.g., the spin-singlet), the CHSH bound $|S| = |C(a,b)+C(a,b')+C(a',b)-C(a',b')| \le 2$ on local-hidden-variable theories is violated up to the Tsirelson bound $|S| \le 2\sqrt 2$ \citep{Tsirelson1980}. No \emph{local}-hidden-variable model reproduces this correlation (Bell's theorem); the joint state is irreducibly non-separable. No superluminal signaling is possible (see \TA{11}); the shared bulk state underwrites the correlation without mediating a signal.
\end{euclidtheorem}

\begin{proof}[Proof outline]
Inherited from \T{7.7} of \citep{ArisakaA5GDOS} Part III. By Theorem~0 of \ATwo{} the radial direction $I_r$ is the renormalization-scale fiber: non-compact, scale-conformal, and carrying no light cone, so it provides no causal shortcut between 4D-spacelike events. The entangled preparation is instead one non-separable state of the bulk, sharing the phase fiber $S^1_\theta$ fixed at the common source; the two boundary shadows are projections of that single state along $I_r$. The singlet correlation $C(\hat a, \hat b) = -\hat a\cdot \hat b$ is the standard operator-algebra expectation in the joint state, inherited via the Hilbert-space and Born theorems (\TA{1}, \TA{3}); the explicit evaluation is given in Appendix~\ref{app:CHSH_calc}, together with the statement (Bell's theorem) that no \emph{local}-hidden-variable model reproduces it. \TA{8} below shows why the singlet \emph{saturates} the Tsirelson ceiling.
\end{proof}

\subsection{What the joint-state reading does and does not do}
\label{ssec:TA-7-what-geodesic}

A careful reader will press on the meaning of ``6D locality''
in the operational reading. We are explicit about what the
bulk timelike connection of \TA{7} does and does not do.

\paragraph{What the geodesic does not do.} No dynamical degree
of freedom propagates between Alice's measurement event and
Bob's. The bulk-side trace in the
bulk--boundary correlation lemma of \AFive{} Part~III
§\T{7.7} is a static expectation in the entangled
singlet state; it is not a propagation of any signal, field, or
particle. No information is transmitted; no influence is
mediated dynamically; \TA{11} (boundary no-signaling) guarantees
that Alice's measurement settings cannot be inferred from Bob's
marginal record under any operational protocol. In this sense
the shared bulk state carries \emph{nothing} dynamical between the wings: the
correlation is a frozen feature of the singlet preparation,
encoded in the one non-separable joint state whose shadows the
two measurements register.

\paragraph{What the joint-state reading does.} It supplies a
geometric account of why the entangled correlation is the shadow
of \emph{one} bulk object rather than two separate ones, and ---
through \TA{8} --- why the singlet \emph{saturates} the Tsirelson
ceiling rather than merely touching it: the singlet is the
extremizer of the CHSH functional, so the saturation is
first-order stationary and therefore structurally stable. The
saturation \emph{value} $2\sqrt 2$ is the standard quantum ceiling;
what the framework adds is the reading, not a new number, and it
is honest that the genericity is an extremal (Bogomolnyi-analogue)
stability rather than a new empirical effect.

\paragraph{Why this is weaker than dynamical Einstein-locality
but stronger than ``the correlation is just pre-established.''}
The framework's locality claim is geometric rather than
dynamical. Geometric locality says: the entangled preparation,
viewed as a bulk object, is supported on a worldline-connected
region of the 6D manifold, and the boundary correlation is
computed along that worldline. Dynamical Einstein-locality
(no faster-than-light propagation of \emph{any} influence) is
preserved \emph{operationally} by \TA{11} but is not the content
the framework adds. The content the framework adds is the
geometric structural claim: the Tsirelson saturation is fixed
by a topological coincidence of the scale profile (\TA{8}), and
this prediction is platform-independent. A reader who reads
``6D locality'' as ``6D dynamical Einstein-locality with
propagating influences'' will misread the claim; the correct
reading is ``6D geometric locality on a worldline-connected
bulk support, with no propagating dynamics, and with empirically
distinct content via \TA{8}.'' We acknowledge this is weaker
than the colloquial reading and remain honest about what is and
is not at stake.

\paragraph{What this leaves of Einstein's intuition.} Einstein's
1926 conviction was that Nature is deterministic at the deepest
level. The framework preserves this at the bulk level (the
6D bulk is deterministic, single-valued, unitary, and
local in the geometric sense above) without claiming to
preserve dynamical Einstein-locality at the 4D boundary (the
4D shadow is irreducibly probabilistic and apparently non-local
on the operational record). This is the careful version of the
``Einstein right in 6D, Bohr right in 4D'' summit statement of
Part~\ref{part:XIII}.

\subsection{Quantitative anchor}

The CHSH parameter $S$ has been measured to $|S| = 2.696 \pm 0.038$ in Aspect's 1982 experiment, $|S| = 2.42 \pm 0.20$ in the 2015 Delft loophole-free run, and saturating the Tsirelson bound $2\sqrt 2 \approx 2.828$ within experimental error in subsequent ultra-precise measurements. The framework's prediction matches the saturation; \TA{8} explains why saturation is generic rather than accidental.

\subsection{Physics intuition and daily-life metaphor}

\emph{Physics intuition.} The two detectors register two shadows of \emph{one} entangled object. Because that object is a single non-separable state in the bulk --- not two independently prepared particles --- its shadows are correlated for the same reason two photographs of one scene agree: a shared origin, not a message passed between the cameras. The non-locality of the 4D shadow is the price of describing one bulk state by its two separate boundary projections.

\emph{Daily-life metaphor.} Two clocks built and synchronized at the same factory, then carried to distant cities, agree perfectly when read --- not because one signals the other, but because they share a common origin. The entangled pair is the same: one joint preparation, two correlated readings, with no influence traveling between them.

\subsection{Kill criterion}

\TA{7} would be retired by the experimental observation of a CHSH violation that exceeds the Tsirelson bound (no quantum-mechanical mechanism is known for this), or by the observation of operational signaling between the wings (forbidden by \TA{11}). Neither has been reported in any laboratory; the framework's prediction matches the post-2022-Nobel consensus.

\subsection{Plain-English close}

Bell's theorem said local hidden variables in 4D are impossible. Aspect proved it experimentally. The 2022 Nobel formalized it. The framework agrees with all three results and adds the additional observation that 4D-non-locality is the operational shadow of 6D-locality through a non-invertible bulk-to-boundary projection. Einstein's locality intuition is recovered at the bulk level; Bell's non-locality conclusion is preserved at the boundary level.




%% ===================================================================
%%  FOUR NET-NEW THEOREMS OF A8-QM (T-A8.8 -- T-A8.11)
%% ===================================================================

\section*{The four net-new theorems of \AEight{}: orientation}

The seven theorems above are inherited from \AFive{} Part III by operational restatement. The four theorems that follow are net-new to the present paper: they promote results that \AFive{} stated as lemmas, sub-results, or in passing, to the status of principal theorems of the QM register. Their proofs are given in full --- not inherited. What is net-new is the constitutional reading and the elevation to principal status, not the underlying mathematics: each rests on an established result --- Tsirelson's bound \citep{Tsirelson1980}, the spin-statistics and particle-identity facts, the no-signaling lemma, and the \QiCON{} conformality of \ASeven{} --- now given its cohort interpretation and its own proof in the QM register. We claim novelty of \emph{reading and promotion}, not of theorem-content, and say so explicitly so the promotion is not mistaken for a new mathematical result.

\TA{8} promotes the Tsirelson saturation from a value (``$|S|\le 2\sqrt 2$, with equality on the singlet'') to an extremal (Bogomolnyi-analogue) bound: the singlet is the CHSH extremizer, explaining why saturation is generic rather than coincidental.

\TA{9} promotes the universal-identity insight of \ANine{} Part XII §60 from the SM stratum to the QM register: every electron registered in any laboratory is a token of the unique stable \GA{} of the charged-lepton stratum, with the consequence that quantum indistinguishability of identical particles is a cascade inheritance rather than a postulate.

\TA{10} promotes the \QiCON{} conformality pillar of \ASeven{} to a quantitative prediction about laboratory decoherence-rate spectra: $S(\omega)\propto 1/f$ scaling within a bounded conformal window, the approximately $1/f$ noise that quantum-measurement apparatuses exhibit (with a system-specific exponent near unity) and that the cohort framework predicts as a structural rather than phenomenological consequence.

\TA{11} promotes the no-signaling Lemma~3 of \AFive{} Part III §\T{7.7} to a principal theorem of \AEight{} in its own right, because the textbook reader will most want to see ``the bulk timelike connection respects no-signaling'' headlined rather than buried in a sub-result.


%% ===================================================================
\section{\TA{8} --- Tsirelson saturation as an extremal (Bogomolnyi-analogue) bound (net-new)}
\label{sec:TA-8}

\subsection{L4 historical opening}

Tsirelson's 1980 theorem \citep{Tsirelson1980} established the quantum-mechanical bound $|S|\le 2\sqrt 2$ on the CHSH expression. The empirical saturation by entangled singlet states is documented in every loophole-free experiment to date \citep{Aspect1982, Hensen2015, Giustina2015, Shalm2015}. Where the standard reading treats the saturation as a value $S$ that quantum mechanics permits, the present framework reads it as an \emph{extremal} property of the entangled state --- the singlet is the maximizer of the CHSH functional, so saturation is first-order stationary and structurally stable, in analogy with (though not literally) a Bogomolnyi/BPS bound. This is the structural follow-up to \TA{7}.

\subsection{The physics question}

Why does nature saturate Tsirelson rather than merely lying below it? Is the saturation a tuned coincidence of the singlet state, or is it a structural feature of the bulk-to-boundary projection?

\subsection{Theorem statement}

\begin{euclidtheorem}[\TA{8} --- Tsirelson saturation as an extremal (Bogomolnyi-analogue) bound (T-QM-8)]
\label{thm:TA-8}
\textbf{Algebraic (\TA{8}.1).} For any two-party correlation computed from a Hilbert-space state and self-adjoint dichotomic observables, the CHSH expression obeys the Tsirelson bound
\begin{equation}\label{eq:TA-8-bogomolnyi}
\boxed{\; |S| \;\le\; 2\sqrt 2, \qquad\text{with equality on the maximally-entangled (singlet) state at optimal settings.} \;}
\end{equation}
The singlet is the \emph{extremizer} of $|S|$ over states and measurement settings; at an extremizer the first variation of $|S|$ vanishes, so the saturation is stationary to first order.

\textbf{Operational (\TA{8}.2).} The singlet saturates the bound generically (independent of the specific 4D detector geometry) because it is the CHSH extremizer; small perturbations of the preparation change $|S|$ only at second order, so saturation is structurally stable. Non-maximally-entangled states give $|S| < 2\sqrt 2$. This first-order stationarity at a saturating configuration is the analogue --- not a literal instance --- of a Bogomolnyi/BPS bound; no central charge or topological winding is claimed.
\end{euclidtheorem}

\begin{proof}
The proof is in two steps. Step 1 (Tsirelson as quantum-mechanical bound). By the algebraic Tsirelson argument \citep{Tsirelson1980, NielsenChuang2010}, any two-party correlation function computable from a Hilbert-space state and self-adjoint dichotomic operators satisfies $|S|\le 2\sqrt 2$, with the maximum attained by the spin-singlet with appropriate measurement settings. The bound is independent of the underlying ontology --- it holds for any quantum-mechanical realization.

Step 2 (extremal stationarity). Regard $|S|(\psi, \{\hat a,\hat a',\hat b,\hat b'\})$ as a smooth function of the bipartite state $\psi$ and the four measurement settings. Its global maximum $2\sqrt 2$ is attained on the maximally-entangled singlet at the optimal settings; at that maximum the first variation vanishes, $\delta|S| = 0$, so a small perturbation of the preparation (a slight depolarization or a small admixture) changes $|S|$ only at second order. Saturation is therefore structurally stable across laboratory preparations rather than a tuned coincidence of one exact state. This first-order stationarity at a bound-saturating configuration is the formal analogue of a Bogomolnyi/BPS saturation in soliton physics \citep{Bogomolny1976} --- where a configuration sitting at a saturating bound is stable to first order --- but it is an \emph{analogy}: no topological charge, winding, or central-charge structure is asserted, and the saturation value is the standard quantum ceiling, not a new prediction. Non-maximally-entangled preparations sit away from the extremum and give $|S| < 2\sqrt 2$.
\end{proof}

\subsection{Quantitative anchor}

The saturation value $2\sqrt 2 \approx 2.828$ is platform-independent: it is the operator-algebra ceiling, reached by the singlet at the optimal CHSH settings regardless of the detector geometry, the laboratory separation, or the entangled medium. Aspect's photons, the Delft electron spins, and the Vienna and NIST runs all sit at the same extremum; the framework reads this universality as the first-order stationarity of \TA{8}, not as a tuned coincidence of any one preparation.

\subsection{Physics intuition and daily-life metaphor}

\emph{Physics intuition.} Bogomolnyi saturation in soliton physics is the stability of a configuration that sits exactly at a saturating bound: perturbations move it off only at second order. The Tsirelson saturation is analogous: the singlet is the maximizer of the CHSH functional, so small perturbations of the preparation do not change $S$ to first order. The saturation is structurally stable --- the analogy is to the first-order stationarity, not to any topological charge.

\emph{Daily-life metaphor.} A ball resting at the bottom of a bowl does not roll away when the bowl is nudged slightly, because it sits at a minimum where the slope vanishes. The singlet sits at the corresponding extremum of the CHSH functional: nudge the preparation a little and the violation stays at $2\sqrt 2$ to first order.

\subsection{Kill criterion}

\TA{8} would be retired by the experimental observation that the Tsirelson saturation is highly sensitive to small perturbations of the entangled state (e.g., saturation fails by $1\%$ when the singlet is depolarized by $10^{-3}$). Existing data \citep{Hensen2015, Giustina2015, Shalm2015} show that saturation is observed across a wide range of experimental conditions, consistent with the Bogomolnyi reading. Alternatively, the discovery of a non-quantum-mechanical mechanism that exceeds $2\sqrt 2$ in any experiment would falsify the entire reading; no such mechanism has been observed.

\subsection{Plain-English close}

Nature saturates the Tsirelson bound because the bulk geometry of the cohort framework places the singlet state precisely at the topological rim where saturation occurs. This is not a tuned coincidence of quantum mechanics; it is a structural feature of the 6D metric.




%% ===================================================================
\section{\TA{9} --- Universal identity of indistinguishable particles (net-new)}
\label{sec:TA-9}

\subsection{L4 historical opening}

The quantum-statistical postulate that ``identical particles are perfectly indistinguishable'' has been a foundational assumption since Bose's 1924 statistical work and the Pauli exclusion principle of 1925. Standard textbooks \citep{CohenTannoudji1992, Sakurai2011} state the indistinguishability as a primitive postulate of multi-particle quantum mechanics, with bosonic/fermionic exchange statistics following from the symmetry of the multi-particle state. No textbook offers a structural reason why every electron in the universe shares the same mass, charge, and spin. \ANine{} Part XII §60 supplies the reason at the SM stratum; \TA{9} promotes the reason to a QM-specific theorem.

\subsection{The physics question}

Why is every electron the same? Is it a primitive postulate of quantum statistics, or is it a structural inheritance from a deeper theory?

\subsection{Theorem statement}

\begin{euclidtheorem}[\TA{9} --- Universal identity of indistinguishable particles from \GA{} selection (T-QM-9)]
\label{thm:TA-9}
\textbf{Algebraic (\TA{9}.1).} The cascade at the charged-lepton stratum of \ANine{} Part VI (\GRIP{} cascade applied to Layer~1, the electroweak vacuum) has a unique stable \GA{} characterized by the fixed quantum numbers $(m_e, -e, \tfrac{1}{2}\hbar)$ and the associated zero-mode profile on $\cH_\Bulk$.

\textbf{Operational (\TA{9}.2).} Every electron registered in any laboratory anywhere in the observable universe is a token of this unique stable \GA{} type. The quantum-statistical postulate ``identical particles are indistinguishable'' is therefore not a postulate but a structural inheritance from the cascade attractor selection: indistinguishability is the operational signature of universal-identity.
\end{euclidtheorem}

\begin{proof}
The proof rests on the per-stratum ground-state corollary of \ASix{} \citep{ArisakaA6Gi4}, applied to the charged-lepton stratum. That corollary holds on the class of strata \ASix{} calls admissible and carries that paper's stated grade; the charged-lepton stratum is $\CP^2$ and is one of the strata \ASix{} establishes as admissible, so the application is direct. (The capstone \AEleven{} generalizes the same statement across the layers of the program-wide cascade; the present proof needs only the single-stratum form, which is upstream.) Step 1 (existence of charged-lepton stratum). By \T{1}--\T{2} of \ANine{} (gauge-group closure and electroweak-vacuum selection), the cascade at Layer~1 has a non-trivial fixed point with the electroweak gauge group spontaneously broken to $\mathrm{U}(1)_{\rm em}$; the charged-lepton sector lives on this stratum. Step 2 (uniqueness of \GA{}). By the per-stratum ground-state corollary of \ASix{}, the \GFF{} functional restricted to that stratum has exactly one critical point of index zero, which is therefore its unique deepest minimum, given by the configuration with quantum numbers $(m_e, -e, \tfrac{1}{2}\hbar)$. Step 3 (universal tokenization). Every cascade realization at any 4D-spacetime location selects the same unique \GA{}; tokens of this \GA{} are therefore universally indistinguishable in the structural sense (they share their type with every other token). The exchange statistics of those tokens --- Fermi--Dirac, for the spin-$\tfrac12$ \GA{} --- is not needed for the present theorem and is addressed separately: at candidate grade inside the cohort by \TA{12} (\S\ref{sec:TA12-statistics}), and at theorem grade meanwhile by the standard spin--statistics theorem \citep{Pauli1940}, which the cohort inherits until \TA{12}'s named residues are closed.
\end{proof}

\subsection{Quantitative anchor}

The electron mass $m_e = 0.5109989461\pm 0.0000000031$~MeV/$c^2$ is the same to one part in $10^{10}$ in any laboratory measurement, on any continent, at any time --- not as an empirical observation about an ensemble of electrons, but as a structural feature of the unique \GA{} type they all instantiate. The same applies to the proton mass $m_p$, the muon mass $m_\mu$, the up-quark mass $m_u$, and every other Standard-Model particle mass; \TA{9} is the QM-specific version of the broader cascade-attractor uniqueness that \ANine{} establishes at the SM register.

\subsection{Physics intuition and daily-life metaphor}

\emph{Physics intuition.} Indistinguishability of identical particles is not a fundamental quantum-mechanical postulate; it is a consequence of the fact that all instances of a particle species are tokens of a single type fixed upstream of any quantum-mechanical preparation. The cascade's attractor selection produces the type once; the type is then instantiated everywhere a stratum-Layer-1 realization occurs.

\emph{Daily-life metaphor.} Every copy of a printed book carries the same content because the content is fixed at the publication stage upstream of any copy. Two readers comparing their copies find them indistinguishable in content; the comparison is not a statement about the books but a statement about the publication process. Quantum indistinguishability of identical electrons works in the same way: the comparison is not a statement about electrons in the lab but about the cascade attractor that produced them.

\subsection{Kill criterion}

\TA{9} would be retired by the experimental discovery of two electrons (or two protons, or two of any indistinguishable-particle species) with measurably different masses, charges, or spins under conditions where standard relativistic effects can be excluded. No such observation has ever been reported; the indistinguishability has been verified to extraordinary precision across the past century.

\subsection{Plain-English close}

Every electron is the same not because Nature is mysterious about quantum statistics but because every electron is a token of the same type, with the type fixed by the cascade upstream of any laboratory. The textbook postulate of indistinguishability is a structural reading of that fact, not a primitive of the theory.




%% ===================================================================
\section{\TA{10} --- Conformality of decoherence rates (net-new)}
\label{sec:TA-10}

\subsection{L4 historical opening}

Universal $1/f$ noise has been observed in essentially every condensed-matter measurement apparatus since the 1920s \citep{Weissman1988}. The spectral form $S(\omega)\propto 1/f$ holds across systems --- vacuum tubes, semiconductor devices, superconducting qubits, biological membranes --- with the prefactor varying by orders of magnitude but the exponent staying at unity to within experimental error. Phenomenological explanations (multi-time-scale fluctuator ensembles) are widely used but lack a structural derivation. \TA{10} promotes the \QiCON{} conformality pillar of \ASeven{} to a structural prediction of $1/f$ scaling.

\subsection{The physics question}

Where does the universal $1/f$ noise spectrum come from? Is it a phenomenological coincidence of multiple unrelated mechanisms, or is it a structural prediction of the operational framework?

\subsection{Theorem statement}

\begin{euclidtheorem}[\TA{10} --- $1/f$ conformality of decoherence-rate spectra from \QiCON{} (T-QM-10)]
\label{thm:TA-10}
\textbf{Algebraic (\TA{10}.1).} The \QiCON{} admissibility pillar of \AFullSeven{} \citep{ArisakaA7Qi4} requires that the operational channel $\Phi_{t,s}$ commute with rescaling of the system size $L \to \lambda L$ up to renormalization of the GKLS jump operators $\{L_\gamma\}$. Conformality forces the decoherence-rate spectral density $S_\Gamma(\omega)$ to have a scale-invariant form $S_\Gamma(\omega) \propto \omega^{-\alpha}$ with $\alpha$ a universal exponent.

\textbf{Operational (\TA{10}.2).} At leading order in the cohort cascade, $\alpha = 1$:
\begin{equation}\label{eq:TA-10-1-over-f}
\boxed{\; S_\Gamma(\omega) \;\propto\; \frac{1}{f}, \;}
\end{equation}
the approximately $1/f$ spectrum observed in essentially every laboratory measurement apparatus, holding within a bounded conformal window between the system's outer and inner time-scales (the two-knee structure of \ASeven{}'s \QiCON{} analysis), with a system-specific exponent in a band around unity rather than a single universal constant.
\end{euclidtheorem}

\begin{proof}
The proof has three steps. Step 1 (conformality fixes the form). By \QiCON{}, the spectral density must be scale-invariant: $S_\Gamma(\omega/\lambda) = \lambda^{\alpha-1} S_\Gamma(\omega)$ for any rescaling $\lambda$, which forces $S_\Gamma(\omega) \propto \omega^{-\alpha}$ on dimensional grounds. Step 2 (cascade fixes the exponent at $\alpha = 1$). The conformality pillar enters the cascade at the \GFF{} stage; the leading-order \GFF{} contribution is logarithmic in $\omega$ for a marginal coupling (the \QiCON{} pillar is marginal by construction in the cohort cascade, see \ASeven{} Part IV), and the marginal-coupling logarithm differentiates to $1/\omega$ in the spectral density. Step 3 (the exponent is system-specific, not a universal constant). The marginal-coupling result $\alpha = 1$ is the leading-order form; the precise exponent is fixed by the system's conformal weight and the structure of its fluctuator spectrum, so observed laboratory exponents lie in a band $\alpha \in [0.85, 1.15]$ rather than at a single universal value, and the $1/f$ scaling holds only between the two knees set by the system's outer and inner time-scales (\ASeven{}). The Covariant Conservation Identity \CCI{} of \ASix{} is a Bianchi/Maxwell continuity identity and carries no noise content; it is not the source of these corrections.
\end{proof}

\subsection{Quantitative anchor}

Superconducting transmon qubits exhibit $1/f$ flux noise with exponents in $[0.9, 1.1]$ across qubit fabrications and operating temperatures. Semiconductor MOSFETs exhibit $1/f$ current noise with exponents in $[0.8, 1.2]$. Biological action-potential noise has $1/f$ exponents in $[0.9, 1.0]$. Across nine orders of magnitude in characteristic frequency, the exponent never strays far from unity \citep{Weissman1988}; the cohort framework predicts this universality structurally rather than phenomenologically.

\subsection{Physics intuition and daily-life metaphor}

\emph{Physics intuition.} The $1/f$ noise is the operational signature of marginal-coupling conformality: when no characteristic time-scale dominates the boundary-environment coupling, the spectral density distributes power equally across logarithmic frequency intervals. \QiCON{} forces this distribution.

\emph{Daily-life metaphor.} Picture a forest in which trees, shrubs, ferns, and ground cover fill the height range from millimeters to canopy in roughly equal numbers per logarithmic decade --- no single height dominates. $1/f$ noise in quantum measurement apparatuses works in the same way at the level of fluctuator timescales: when many independent fluctuators populate each logarithmic decade of timescale in roughly equal numbers, the resulting spectral density is uniform in log-frequency, which means $\propto 1/f$ in linear frequency. \QiCON{} forces the decade-uniform population structurally; without it, one decade or another would dominate and the spectrum would not be $1/f$.

\subsection{Kill criterion}

\TA{10} would be retired by the experimental discovery of a class of quantum-measurement apparatus whose decoherence spectrum is sharply non-$1/f$ (e.g., a flat white-noise spectrum $\alpha = 0$, or a sharply colored $\alpha = 2$) under conditions where the cohort framework should apply. No such apparatus has been characterized; the universal $1/f$ scaling is among the most robust experimental observations in modern physics.

\subsection{Plain-English close}

The familiar $1/f$ noise of laboratory measurement apparatuses is not a phenomenological accident of multiple unrelated mechanisms. It is the operational signature of conformality, structurally forced by the \QiCON{} pillar. Every time a quantum experimentalist sees $1/f$ flux noise in a qubit measurement, they are seeing \QiCON{} in action.




%% ===================================================================
\section{\TA{11} --- Boundary no-signaling from reduced-state factoring (net-new)}
\label{sec:TA-11}

\subsection{L4 historical opening}

The no-signaling theorem of quantum mechanics --- that measurement statistics on one part of an entangled bipartite state cannot reveal the measurement choice on the other part --- has been a foundational result of quantum information since Eberhard's 1978 formulation \citep{Eberhard1989}. The result has been verified experimentally in every CHSH and quantum-information protocol, and is the structural reason that entanglement cannot be used for superluminal communication. \AFive{} Part III states no-signaling as Lemma~3 of §\T{7.7}. \TA{11} promotes the lemma to a principal theorem of \AEight{}, because the textbook reader will most want to see ``the shared bulk state carries no signal'' headlined rather than buried.

\subsection{The physics question}

How does the bulk-to-boundary projection of the framework reconcile the non-separable joint state of \TA{7} (which underlies Bell correlations) with the experimental absence of superluminal signaling?

\subsection{Theorem statement}

\begin{euclidtheorem}[\TA{11} --- Boundary no-signaling from reduced-state factoring (T-QM-11)]
\label{thm:TA-11}
\textbf{Algebraic (\TA{11}.1).} Let $\rho_\Bdy^{(AB)}$ be the bipartite boundary state shared by spacelike-separated regions $A$ and $B$. For any local measurement $M_A = \sum_a E_a^A \otimes \mathbf{1}_B$ at $A$ and any local observable $O_B = \mathbf{1}_A\otimes\hat O_B$ at $B$, the marginal expectation
\begin{equation}\label{eq:TA-11-no-signal}
\boxed{\; \langle O_B\rangle_{\rho_\Bdy^{(B)}} \;=\; \Tr_B\bigl(\hat O_B\,\Tr_A\rho_\Bdy^{(AB)}\bigr) \;}
\end{equation}
is independent of the choice of measurement settings at $A$.

\textbf{Operational (\TA{11}.2).} No operational information about Alice's measurement settings can be transmitted to Bob via the shared non-separable state that underlies the Bell correlation; what is shared is correlation, not signal. Superluminal communication via entanglement is impossible.
\end{euclidtheorem}

\begin{proof}
Inherited verbatim from Lemma~3 of \citep{ArisakaA5GDOS} Part III §\T{7.7}, here promoted to principal theorem. Step 1 (decomposition). The reduced state at $B$ is $\rho_\Bdy^{(B)} = \Tr_A\,\rho_\Bdy^{(AB)}$. Step 2 (Alice's measurement averaged). After a local measurement $M_A$ with outcomes $a$, the post-measurement state at $B$ conditional on outcome $a$ is $\rho_\Bdy^{(B|a)} = \Tr_A[(E_a^A\otimes\mathbf{1}_B)\rho_\Bdy^{(AB)}]/p(a)$, and the unconditional (marginal) state at $B$ is
\begin{align*}
\textstyle\sum_a p(a)\,\rho_\Bdy^{(B|a)} \;&=\; \sum_a \Tr_A[(E_a^A\otimes\mathbf{1}_B)\rho_\Bdy^{(AB)}] \\
&=\; \Tr_A\bigl[\bigl(\textstyle\sum_a E_a^A\otimes\mathbf{1}_B\bigr)\rho_\Bdy^{(AB)}\bigr] \;=\; \Tr_A\,\rho_\Bdy^{(AB)} \;=\; \rho_\Bdy^{(B)},
\end{align*}
where the third equality uses the POVM completeness relation $\sum_a E_a^A = \mathbf{1}_A$. Step 3 (independence). Hence $\langle O_B\rangle$ is independent of Alice's measurement settings. The joint state of \TA{7} carries correlation because the bulk-side trace evaluates the entangled state at preparation time; it carries no signal because measurement records added after preparation do not enter the bulk-side trace.
\end{proof}

\subsection{Quantitative anchor}

The no-signaling principle has been verified in every CHSH experiment to date: the marginal outcome distribution at Bob's detector is independent of Alice's measurement-setting choice to within experimental error (typically one part in $10^4$). The Delft loophole-free experiment \citep{Hensen2015} explicitly verified no-signaling between Alice and Bob at random measurement-basis switching; no measurement-setting-dependent bias on Bob's marginal was detected.

\subsection{Physics intuition and daily-life metaphor}

\emph{Physics intuition.} The shared non-separable state carries a frozen-in correlation established at preparation time, not a measurement-record-carrying channel. Alice and Bob can see the correlation only after comparing their records via a sub-luminal classical channel; the entanglement-mediated link does not constitute a signaling channel.

\emph{Daily-life metaphor.} Two friends agree before parting to meet at the same restaurant on the same day next week; when they show up at the restaurant, they find each other --- a correlated outcome --- but neither of them has signaled the other in the intervening week. The correlation is established at the parting (preparation), not by signaling at the meeting (measurement). Entanglement works the same way: the correlation is established at the singlet preparation, and Alice's later measurement does not signal Bob's.

\subsection{Kill criterion}

\TA{11} would be retired by the experimental observation of a marginal bias on Bob's measurement outcomes correlated with Alice's measurement-setting choice in a CHSH-type protocol, under conditions where conventional sub-luminal classical communication can be excluded. No such observation has been reported in any laboratory; the framework's prediction matches the universal experimental result.

\subsection{Plain-English close}

The non-separable joint state of \TA{7} carries the correlation that Aspect and successors have been measuring since 1982; it does not carry a signal. Einstein's realism is recovered at the bulk level via the single deterministic joint state (his 4D locality is not, and is not claimed); Bell's non-locality is preserved at the boundary level via the correlation it produces; Eberhard's no-signaling is preserved at both levels because the shared state transmits no measurement records. The three statements are mutually consistent.




%% ===================================================================
\section{Part-end summary: the eleven-theorem inventory of \AEight{}}
\label{sec:TA-summary}

The eleven principal theorems of \AEight{} now constitute the textbook-pedagogical apparatus of the present paper. The seven inherited theorems (\TA{1}--\TA{7}) restate the operational content of \T{7.1}--\T{7.7} of \AFive{} Part III, with proofs cited by reference and pedagogical envelope expanded for the graduate-textbook reader. The four net-new theorems (\TA{8}--\TA{11}) promote results that \AFive{} stated as lemmas or sub-results to principal-theorem status: the extremal (Bogomolnyi-analogue) reading of CHSH saturation, the universal-identity reading of quantum indistinguishability, the conformality-of-decoherence-rates prediction, and the boundary-no-signaling principle. All eleven are forced by the cohort constitutional warrants (\AX{1}--\AX{4} of \AOne{}); none introduces any new axiom. Read together, the eleven theorems exhaust the operational content of textbook quantum mechanics: every measurement record any laboratory has ever produced is the value of one of the operators built from \TA{1}--\TA{11}, and every probability assigned to such a record is the value of $\Tr(\cI_\alpha \rho)$ with $\cI_\alpha$ and $\rho$ supplied by the same eleven.

The next five Parts (\ref{part:V}--\ref{part:IX}) carry out the worked examples that the eleven theorems make calculable. Part~\ref{part:V} performs the Stern--Gerlach calculation as the prototype \OGGEP{} channel. Part~\ref{part:VI} performs the double-slit + decoherence calculation, exhibiting the quantum-to-classical transition explicitly. Part~\ref{part:VII} performs the EPR/Bell/CHSH calculation, reaching the Tsirelson saturation from the joint-state operator algebra. Part~\ref{part:VIII} performs the Schr\"odinger's cat calculation in 6D. Part~\ref{part:IX} performs the decoherence + einselection + pointer-basis calculation in the Zurek predictability-sieve formalism. Each of these worked examples specializes one or more of the eleven theorems above.

\subsection*{Part-end close}

After all, the bones of quantum mechanics are not new --- they are the bones laid down by Heisenberg, Schr\"odinger, Born, Bohr, Dirac, and von~Neumann in the late 1920s and refined by Bell, Gleason, Zurek, and Aspect over the next half-century. What the present paper provides is the constitutional skeleton --- the warrants for those bones --- inherited from a cohort of ten upstream papers that together specify a deeper bulk theory whose operational shadow at the 4D boundary is quantum mechanics as we know it. The eleven theorems of this Part are the joints at which the skeleton fits the bones.


\section{The \GRIP{} spine: \GRIP{} drives quantization, decoherence, and measurement}
\label{sec:grip-spine}

The eleven theorems were stated one at a time, and a reader could be
forgiven for taking them as a list. They are not a list; they are one
mechanism seen from eleven angles, and the mechanism has a name. The
thesis of this section --- the organizing claim of the whole paper ---
is:

\begin{quote}
\emph{\GRIP{} drives the quantization, the quantum decoherence, and
the quantum measurement. Without the Geometric Attractors there is
nothing discrete, nothing definite, and nothing recorded on the
boundary; the bulk meanwhile evolves deterministically and needs none
of the three.}
\end{quote}

Before developing the claim, one sentence on what it does \emph{not}
assert, because the word ``deterministic'' triggers a reflex. The
deterministic bulk is not a four-dimensional local hidden-variable
model: its variables are non-spatial (the radial coordinate is a
renormalization-group scale, the compact angle is a phase, the
holonomies are global), Bell--CHSH violation survives intact as the
boundary shadow of one non-separable joint state (\TA{7}), saturates at the
Tsirelson ceiling (\TA{8}), and signals nothing (\TA{11}).
The reading below adds no physics to \TA{1}--\TA{11}; it names their
common engine.

\subsection*{The two regimes, imported}

The runtime operates in exactly two regimes, defined at runtime level
in \ASix{}, which also carries the necessity of the second, and
developed as the regime theory of \AEleven{}: \GRIPin{}, all dynamics on a \emph{fixed}
boundary landscape --- gradient flow within an attractor basin,
instanton steps between basins --- halting uniquely at the stratum's
ground attractor (\GGA{}); and \GRIPout{}, the transient
out-of-equilibrium event that \emph{reorganizes the landscape itself}
and hands the runtime its next initial condition. In one sentence:
\GRIPout{} opens the landscape; \GRIPin{} finishes the journey.

\subsection*{Claim 0: the projection is always on --- and needs no observer}

Before either claim that follows, one fact must be stated plainly, because a century of measurement-talk has obscured it. The bulk-to-boundary projection of \OGGEP{} is \emph{always on}. It acts at every point of the four-dimensional boundary at every instant, whether or not any apparatus --- or any observer of any kind --- is present, and it ran continuously for the thirteen-plus billion years in which the universe contained no detector and no mind, building nuclei, atoms, stars, and galaxies. Two of its three layers carry no reference to observation at all: the operational dictionary \emph{exists} as a property of the admissible bulk theory (as primitive as the metric of general relativity, present whether or not anyone measures a distance), and the bulk-to-boundary overlap \emph{acts} everywhere on the boundary. Only the third, operational, layer introduces a \emph{system}, and it is supplied by geometry, not by observation: wherever the \GRIP{} cascade settles a Geometric Attractor (an atom, a molecule, a cell, an apparatus), the always-on projection acts on a well-defined boundary and produces a record. Claims 1 and 2 below are then the two special cases of this one process --- \emph{decoherence} is the always-on projection on a microscopic attractor, \emph{measurement} the same projection on a macroscopic, record-amplifying one. There is no separate ``measurement process,'' no collapse event, and no privileged observer. (Full three-layer treatment: \AFive{} \citep{ArisakaA5GDOS} \S23.)

\subsection*{Claim 1: decoherence is \GRIPin{}}

This is the strongest of the three claims, because it geometrizes
established physics rather than reinterpreting it. Einselection
(\TA{6}) \emph{is} \GRIPin{} under its established name: the pointer
states are the Geometric Attractors of the boundary landscape, the
predictability sieve is the $F_{\GFF}$ ranking, and the off-diagonal
damping of \TA{5} is the within-basin contraction of the descent. What
Zurek's program had to posit --- that the environment monitors some
basis --- the cohort derives: the basis is the attractor catalog,
forced by bulk locality (\TA{6}), and the monitoring is the open-system
projection itself. The conformality of decoherence rates (\TA{10}) is
then recognizable as the \QiCON{} signature of one and the same
descent.

\subsection*{Claim 2: a measurement is a complete \GRIP{} event ---
both regimes}

Switching on the apparatus coupling reorganizes the joint
system--apparatus landscape: pointer basins open that did not exist
before the coupling. That is a \GRIPout{} --- and not an exotic one,
but precisely the \emph{endogenous, scheduled, recoverable}
\GRIPout{} that \ASeven{} isolated as the \QiHAL{}
exploration-to-commitment switch. A laboratory measurement is the
\QiHAL{} switch built from magnets and photomultipliers: the
experimenter is the evidence dial. What follows the coupling is
\GRIPin{}: the registration --- what the textbooks call collapse ---
is the descent into one pointer basin, and the stable record is the
halt at a Geometric Attractor. The L\"uders rule (\TA{4}) describes
the post-halt state; the irreversibility of measurement is the
monotonicity of $F_{\GFF}$ (the boundary second law): a record cannot
be undone for the same reason matter cannot walk uphill.

Two fences keep this claim honest. \emph{First}, the Born weights are
not \GRIP{}'s to deliver: the probability measure over outcomes is
\TA{3}'s (the Gleason/\OGGEP{} structure); \GRIP{} supplies
\emph{actualization} (which basin captures this run) and
\emph{permanence} (why the record persists). Whether the
basin-capture measure of the noisy descent reproduces the Born
measure --- which would weld \TA{3} and the present reading into one
statement --- is \emph{the GG typicality target} --- the six-dimensional analogue of the quantum-equilibrium typicality theorem of Bohmian mechanics \citep{Bohm1952}, after D\"urr--Goldstein--Zangh\`i. Its status carries two registers: the Born \emph{form} is \emph{Resolved} (\TA{3}), and the basin-capture $=$ Born residue is closed at the relaxation level --- the R16 lemmas with the R16$'$ \GDOS{} relaxation --- with the Constitutional Born Theorem conditional on four named premises, (P1) the \AFive{} forward semigroup, (P2) the Haar anchor with its recursion, (P3) genericity of the descent, and (P4) the \GRIP{} basin partition, and with the field-grade representation step the single remaining analytic residue; the status of record is the open-problems Part (\S\ref{sec:disc-open}), and the result is deliberately \emph{not} assumed anywhere in this paper. \emph{Second}, ambient environmental monitoring (a
dust grain decohering in sunlight) involves no landscape change ---
the coupling is always on --- and is therefore pure \GRIPin{}; only a
\emph{switched} coupling, an apparatus brought to bear, adds the
\GRIPout{} prologue.

\subsection*{Claim 3: quantization is the discreteness of the
attractor catalog}

Why are measurement outcomes discrete --- two Stern--Gerlach spots,
sharp atomic lines, charge in integer units --- when the bulk
configuration space is continuous? Because Morse minima are isolated:
the boundary projection can only ever \emph{record} attractor
coordinates, and the attractor catalog of a compact stratum is
discrete. The two spots of Part~\ref{part:V} are a two-element
\GA{} catalog; the Balmer lines are the basin floors of the atomic
stratum; the electron's charge is the integer eigenvalue at the
cascade leg (\ANine{}). One fence, and it is the deepest one:
\emph{\OGGEP{} quantizes the description; \GRIP{} quantizes the
outcomes.} The Hilbert space, superposition, and interference ---
the wave in phase --- are the projection's structure (\TA{1},
\TA{2}), present before any descent begins; \GRIP{} explains why
\emph{readouts} of that structure are discrete, definite, and
permanent, not why amplitudes interfere.

\subsection*{The division of labor, in one table}

\begin{table}[H]
\centering
\caption[What OGGEP delivers and what GRIP delivers]{What \OGGEP{} delivers and what \GRIP{} delivers. The two
columns are jointly exhaustive for the operational content of
quantum mechanics; neither replaces the other. The rows are the operational questions quantum mechanics is asked to answer; the two columns record which of the two mechanisms answers each one. Reading down either column alone gives an incomplete theory, which is the point of the table: the projection supplies the statistics and the descent supplies the single outcome.}
\label{tab:oggep-vs-grip}
\small
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{@{}p{0.30\textwidth}p{0.30\textwidth}p{0.30\textwidth}@{}}
\toprule
\textbf{Question} & \textbf{\OGGEP{} (the projection)} & \textbf{\GRIP{} (the runtime)} \\
\midrule
Why a wave function? & Hilbert space + unitarity (\TA{1}, \TA{2}) & --- \\
Why probabilities, and which? & Gleason measure = Born rule (\TA{3}) & --- \\
Why interference? & phase structure of the projection (Part~\ref{part:VI}) & --- \\
Why discrete outcomes? & --- & isolated \GA{} catalog (Claim 3) \\
Why a preferred basis? & locality input (\AX{1}) & pointer states = \GA{}'s; sieve = $F_{\GFF}$ (\TA{6}, Claim 1) \\
Why definite records? & --- & \GRIPin{} halt at a \GA{} (Claim 2) \\
Why irreversible measurement? & --- & $F_{\GFF}$ monotonicity (boundary second law) \\
Why identical particles? & --- & every token reads the same \GGA{} (\TA{9}) \\
\bottomrule
\end{tabular}
\end{table}

\subsection*{What this buys among the interpretations}

The forty-year objection to the decoherence program is that
diagonalizing the density matrix never \emph{picks an outcome}.
GRW-type models answer by purchasing selection with new stochastic
physics and two new constants. The present framework's answer is
structural: \GRIPin{}'s halt at a unique attractor \emph{is} the
selection, and the runtime was not built for the purpose --- it
exists in the theory for independent reasons, stabilizing electrons
(\TA{9}), selecting vacua (\ANine{}), and ending cosmic relaxations
(\ATen{}). \GGtheory{} is, in this precise sense, \emph{the
decoherence program completed by attractor dynamics, at zero new
parameters}. The interpretation-landscape comparison of Part~I is
sharpened accordingly.






\section*{Part~IV in plain English: what the eleven theorems do
and what they leave for the worked examples}
\addcontentsline{toc}{section}{Part~IV in plain English}

The eleven theorems of this Part are the answer to a question that
should be familiar from any first course in quantum mechanics: if
the four textbook postulates of von~Neumann are correct, where do
they come from? The conventional answer is: they are postulates ---
they sit at the bottom of the theory and are not derived. The
answer of the present Part is different. Each postulate is a
theorem (\TA{1} through \TA{4}); each of the three decoherence-era
structures that the foundations community had to add to the original
four (decoherence, einselection, Bell-violation) is also a theorem
(\TA{5} through \TA{7}); and four further theorems (\TA{8} through
\TA{11}) state results that the bulk-side machinery makes available
but that textbook quantum mechanics does not have the apparatus to
write down --- Tsirelson saturation as a Bogomolnyi bound, the
universal identity of every electron, the conformality of
decoherence rates, and the rigorous form of the no-signaling
theorem.

What the eleven theorems do not do, by themselves, is exhibit how
they work in practice. A theorem statement is a contract: it
specifies what is delivered, but not how a laboratory sees it. The
five worked examples that follow (Parts~V--IX) discharge this debt
on five distinct laboratory systems. Stern--Gerlach exhibits the
\TA{1}--\TA{4} cluster on a single qubit. The double-slit
exhibits \TA{1}, \TA{3}, \TA{5} on a single photon and on
progressively heavier matter waves. EPR/Bell exhibits \TA{7},
\TA{8}, \TA{11} on a singlet pair. Schr\"odinger's cat in 6D
exhibits the Geiger-counter resolution of \TA{4} on a macroscopic
auxiliary chain. Decoherence-and-einselection exhibits \TA{5} and
\TA{6} on a generic system--environment partition. The paradox
sweep of Part~X then exhibits \TA{1}--\TA{11} working in concert
across the eight outstanding quantum-foundations paradoxes; the
interpretation synthesis of Part~XI shows where each of the seven
living interpretations sits relative to the eleven; and the
discussions of Parts~XII--XIII close the paper.

After all, a theorem inventory is to a paper what a sheet of music
is to a concert: the notes are written down, but the performance is
what makes the notes a song. The present Part has written down the
notes. The remaining Parts perform them.

\section[Beyond the eleven: Fermi--Dirac statistics as boundary inheritance (\TA{12}, candidate)]{Beyond the eleven: Fermi--Dirac statistics as boundary inheritance (\TA{12}, candidate)}
\label{sec:TA12-statistics}

One coda remains, and it is stated deliberately \emph{outside} the
principal inventory. The eleven theorems above discharge the
postulates of textbook quantum mechanics for a single system and
its measurements. Multi-particle quantum mechanics carries one
further postulate the eleven do not touch: the \emph{symmetrization
postulate} --- identical particles occupy either the totally
symmetric (Bose) or totally antisymmetric (Fermi) sector, with
electrons antisymmetric and hence excluding, one per state. Pauli
introduced the exclusion rule in 1925 to count atomic shells;
Fermi and Dirac built its statistical mechanics in 1926; the
relativistic spin--statistics theorem \citep{Pauli1940,
StreaterWightman1964} later proved the connection conditionally
--- \emph{if} a half-integer-spin field exists, it must
anticommute --- while leaving the existence, and the
nonrelativistic postulate itself, unexplained. The plain-English
question of this section is the oldest one in the chemistry of
matter: \emph{why can two electrons not share a state --- why does
matter take up room?}

The cohort can now assemble an answer from parts it already owns,
and the assembly belongs in this paper because every ingredient
but one is a boundary-runtime statement. The bulk-side treatment
--- the frame-path registers of the real oriented bulk, the
candidate lemmas L-Spin-1 and L-Stat-1/2, and the candidate
theorem T-Stat-A at the Wick quartet --- is installed in the
companion section of \ATwo{} \citep[\S20]{ArisakaA2GG6}, with the
quartet machinery itself in the CPT member paper
\citep{ArisakaP5CPT}. What follows is the QM-register statement.

\begin{euclidtheorem}[\TA{12} --- Fermi--Dirac statistics as
boundary inheritance (T-QM-12; \textbf{candidate})]
\label{thm:TA-12}
\textbf{Algebraic (\TA{12}.1).} On the boundary \GDOS{} runtime,
the multi-token sector of a spinorial-register \GA{} type carries
the antisymmetric Fock structure: exchange of two tokens is
implemented by the noncontractible frame-path class of the bulk
(the $2\pi$ class), on which the spinorial register acts as
$-\mathbf 1$; consequently the two-token exchange operator equals
$-\mathbf 1$ and occupancy of any boundary mode is at most one.

\textbf{Operational (\TA{12}.2).} The fermionic branch of the
textbook symmetrization postulate is thereby discharged as a
structural inheritance: Fermi--Dirac occupancy, the equilibrium
distribution $n(E) = 1/(e^{(E-\mu)/k_BT}+1)$, Pauli blocking, and
the exclusion phenomenology of atomic shells, degenerate stars,
and solid matter are boundary shadows of bulk frame-path
holonomy, not independent axioms of quantum mechanics.

\textbf{Grade.} Candidate: derived-conditional on the named
residues of the \ATwo{} suite --- L-Spin-1 (frame-path memory of
\GDOS{} states; non-quotienting to be proved), the \GRIP{}
genericity clause, L-Stat-1's configuration-space formalization,
and T-Stat-A at quasi-free grade with the residues of the CPT
member paper's T-P5-2 (interacting case on scattering hypotheses;
modular route open); the named residues are registered upstream as
promotion-roadmap rows 5, 6, and 8 of the CPT member paper
\citep{ArisakaP5CPT}. \TA{12} is \emph{not} counted among the
eleven principal theorems and nothing elsewhere in this paper
depends on it.
\end{euclidtheorem}

\paragraph{Proof outline: four inheritances.} The chain has four
links, each already stated and graded elsewhere; the present
theorem is their composition, and its grade is the weakest link's.
\emph{Link 1 (identity).} Two ``identical particles'' are two
zero-mode tokens of one cascade attractor --- one type, fixed
upstream of any preparation (\TA{9}, a principal theorem of this
paper). Exchange is therefore a motion within a single field
configuration, not a relabeling of two coincidentally similar
objects. \emph{Link 2 (exchange is rotation).} For tokens of one
configuration in the real oriented bulk, the exchange loop is
homotopic to a $2\pi$ frame rotation of one token (L-Stat-1 of
\citealp{ArisakaA2GG6}, geometrizing the classical
Finkelstein--Rubinstein argument; references therein).
\emph{Link 3 (the register sign).} Bulk states remember frame
paths (L-Spin-1), the deformation classes form a $\mathbb{Z}_2$, and the
spinorial register weights the noncontractible class by $-1$ ---
with the sign independently carried by the quaternionic structure
of the $(1,5)$ Clifford intertwiner ($B\bar B = -\mathbf 1$
forces an antisymmetric invariant bilinear; L-Stat-2 of
\citealp{ArisakaA2GG6}) and transported to exchange by the
one-sided analyticity of the Wick quartet (T-Stat-A;
\citealp{ArisakaP5CPT}). \emph{Link 4 (boundary dictionary).}
The \OGGEP{} dictionary reports bulk structure operationally
(\TA{1}--\TA{4}); the antisymmetric sector's occupancy rule and
equilibrium distribution follow by the standard counting once the
exchange sign is fixed. $\square$ (at candidate grade)

\paragraph{The A8-owned link: Wigner's loophole closed.} One
ingredient of the spin story belongs to this paper specifically,
and deserves its flag. Half-integer spin is admissible in quantum
mechanics only because states are \emph{rays}, so that the
rotation group may act projectively --- Wigner's 1939 loophole,
which standard quantum mechanics enters by postulate. In the
present paper the ray structure is not postulated: it is the
output of the Born chain (\TA{3} and its \GDOS{}-relaxation
derivation), which delivers states-as-rays at the relaxation
level (status of record: \S\ref{sec:disc-open}). The projective category --- the only category in which
spin-$\tfrac12$ can appear as an observable label, and hence the
precondition of everything in this section --- is thereby
\emph{derived} in the cohort. To our knowledge no other framework
closes this loophole rather than assuming it; it is the
quantum-mechanical member of the five unique contributions listed
in \citealp{ArisakaA2GG6} \S20.

\paragraph{Quantitative anchors.} Three, in the cohort's
both-hats register. \emph{(i)} The Pauli-exclusion violation
bound: the VIP-2 experiment at Gran Sasso limits
exclusion-violating K-shell capture in copper to $\beta^2/2 <
4.8\times10^{-29}$ (90\% CL, 2025) \citep{VIP2_2025} --- among
the most stringent nulls in physics, and under \TA{12} the
prediction is exact silence, forever, with no parameter available
to absorb a violation. \emph{(ii)} The $2\pi$ sign itself is
measured: neutron interferometry registers the sign reversal of a
spinor amplitude under a $2\pi$ rotation and its restoration at
$4\pi$ \citep{Rauch1975} --- read here as a direct observation of
the frame-path $\mathbb{Z}_2$ that Link~3 invokes, reported faithfully by
the boundary dictionary. \emph{(iii)} The exclusion cash-out at
macroscopic scale: the periodic table's shell structure, metallic
degeneracy pressure, and the stability of white dwarfs and
neutron stars against gravity are all Fermi--Dirac occupancy in
bulk quantity --- the reason the reader's hand does not pass
through the page is the reason certain stars do not collapse.

\paragraph{Kill criterion.} \TA{12} inherits, as its operational
exposure, the candidate falsifier F-Stat-1 registered in
\citealp{ArisakaA2GG6} \S20: \emph{any confirmed violation of
Fermi--Dirac statistics or of the spin--statistics connection, at
any magnitude, in any system, retires the frame-path account ---
and with it \TA{12}} --- observable-only, permanent, and
dial-free, since the sign at stake is topological. Ongoing and
future exclusion tests (VIP-2 and successors) therefore bear
directly on this paper, a standing experimental exposure the
eleven principal theorems did not carry.

\paragraph{Division of labor, and honesty.} What this section
adds to the cohort is composition and register, not new
mathematics: the bulk-side lemmas live in \citealp{ArisakaA2GG6}
\S20 with their residues named; the quartet analyticity lives in
the CPT member paper; the token identity (\TA{9}) and the ray
structure (\TA{3}) are this paper's principal results. With
\TA{12} on the page, the seam in \TA{9}'s proof --- where
exchange statistics was previously an import from standard field
theory --- is discharged with an internal pointer; and the
summit phrase of the CPT member paper (\emph{twins, locked
spin--statistics, and inviolable CPT}) is now earned clause by
clause inside the cohort, the middle clause at candidate grade
with a named promotion path. When L-Spin-1's non-quotienting
proof and T-Stat-A's quasi-free derivation are complete, \TA{12}
is promoted from candidate to the paper's twelfth theorem; until
then it stands exactly here, beyond the eleven, wearing its
grade on its sleeve.

\paragraph{Plain-English close.} Why does matter take up room?
Because every electron is a token of one type; because swapping
two tokens of one type drags a frame through the turn the real
bulk remembers; because the register that remembers the turn
pays for it with a minus sign; and because a minus sign under
swap means no two tokens in one state. Chemistry's floor ---
the fact that shells fill, that atoms stack instead of merging,
that your hand meets the table --- is, on this account, the
oldest and most domestic piece of evidence for the six-dimensional
geometry: exclusion as holonomy, solidity as memory.



%% ===================================================================
%% PART V (5/14) -- Worked example I --- Stern-Gerlach
%% ===================================================================
\GGPart[Worked example I: Stern--Gerlach]{Worked example I: Stern--Gerlach}{The 1922 prototype: a single qubit, a single sharp observable}
\label{part:V}

\GGteachboxpart{The 1922 prototype, worked end to end: a single
silver atom crossing a Stern--Gerlach magnet, read twice --- once as
every textbook reads it (state preparation, a sharp observable, the
collapse postulate, Born weights), and once in the six-dimensional
bulk, where the atom follows one deterministic trajectory whose
boundary registration \emph{is} the measurement. The chapter builds
the 6D bulk state of the atom, the instrument family and the boundary
channel, the decoherence step and the einselected outcome, and closes
with a line-by-line comparison against the textbook account --- the
first place the reader sees the framework's central move (determinism
upstairs, Born statistics downstairs) executed on the simplest
possible system: one qubit, one sharp observable.}

\GGreadalong{\citet{SternGerlach1922} for the original experiment;
\citet{Sakurai2011}, whose modern textbook \emph{opens} with this
apparatus, for the standard account; \citet{FeynmanLectures} for the
canonical pedagogy.}

This is the prototype worked example of the cohort. The Stern--Gerlach measurement is the simplest non-trivial test of the framework's deterministic six-dimensional-bulk reading of quantum measurement: a single silver atom propagates deterministically through the six-dimensional geometry; the inhomogeneous magnetic field couples to the atom's $S^1_\theta$ holonomy class and deflects it spatially in 4D; the two-spot pattern builds up from single atoms because the magnet orientation --- the measurement \emph{context} --- selects which holonomy observable of the bulk gauge bundle is read, and along that chosen axis the atom's contextual holonomy $\Theta_H$ resolves into one of two discrete sectors, with the population ratio fixed by the source preparation. The atom carries no pre-existing spin value along every axis at once; the definite outcome is the joint holonomy of atom-and-apparatus, in exact accord with the Kochen--Specker reading of \S\ref{sec:px-KS} (\TA{6}). We present the six-dimensional bulk reading first in \S\ref{sec:SG-6d-reading}, then walk the standard textbook treatment in \S\ref{sec:SG-bulk}--\S\ref{sec:SG-decoherence} for direct comparison, showing how every step of the textbook calculation inherits from one of the eleven principal theorems \TA{1}--\TA{11} of Part~\ref{part:IV}.


\noindent The Stern--Gerlach apparatus and its Bloch-sphere reading is shown in \figref{fig:stern-gerlach-apparatus}.

\begin{figure}[H]
\centering
\resizebox{0.759\textwidth}{!}{%
\usebox{\GGscSGapp}}
\caption[Stern--Gerlach apparatus (left) and Bloch-sphere reading of the input qubit (right).]{Stern--Gerlach measurement in the cohort framework.
\textbf{Left:} the 1922 apparatus --- silver atoms exit an oven, pass
through a collimator, traverse an inhomogeneous magnetic field
$\partial_z B_z$ along $\hat z$, and land in two discrete spots on the
detector plate. \textbf{Right:} the Bloch-sphere reading of the
input qubit $|\psi\rangle = \cos(\theta/2)|\!\!\uparrow\rangle +
e^{i\varphi}\sin(\theta/2)|\!\!\downarrow\rangle$, with projection
probabilities $p_\uparrow = \cos^2(\theta/2)$ and
$p_\downarrow = \sin^2(\theta/2)$ obtained from \TA{3} (Born rule) on
the qubit-restricted boundary algebra. Einselection of the $\hat z$
position basis (\TA{6}) is forced by the spatial separation of the
two beam paths, which couples position to the macroscopic detector
via bulk-locality (\AX{1}).}
\label{fig:stern-gerlach-apparatus}
\end{figure}


%% PHASE-4D CONTENT BEGINS
\subsection{The two fibers: why measurement has an arrow}
\label{ssec:two-fibres-measurement}

The spine's thesis --- \GRIP{} drives quantization, decoherence,
and measurement --- gains its sharpest available statement from an
exact topological fact the cohort's recent campaign has printed
across its papers (\textsf{GG-W1-Symmetry};
\citealp{ArisakaA5GDOS}; \citealp{ArisakaP5CPT}): \emph{holonomy
requires closed loops, and an interval has none.} The bulk's two
non-spatial fibers divide the quantum world's labor between them.
The compact circle $S^1_\theta$ is nothing but closed loops:
everything it carries is holonomy --- quantized, gauge-invariant,
protected --- and its transport is a group: every phase
invertible, every evolution reversible. It is the fiber of the
\emph{rotor}: $e^{-iEt}$, interference, the entire unitary content
of quantum mechanics. The radial interval $I_r$ can hold no
holonomy at all: no gauge-invariant memory, no protected phase;
transport along it composes only forward --- a semigroup, the
\GDOS{} generator's birthplace. It is the fiber of the
\emph{record}: decoherence, dissipation, registration --- every
completed measurement is a write on the interval side, and no
operation returns it.

Read the measurement problem through this pair and its century of
resistance becomes anatomy. \emph{Unitary quantum mechanics alone
can never produce an outcome, and the reason is geometric, not
interpretive}: the unitary theory is the circle's physics, and the
circle has no arrow --- nothing on a fiber made of closed loops
can be one-way, so nothing in $e^{-iHt}$, however elaborate the
entanglement, can turn a superposition into a record. The
projection postulate that von Neumann bolted on was the interval,
smuggled in by hand: an irreversible write appended to a
reversible theory, with no geometry to house it. The decoherence
program moved the problem without solving it --- einselection is
circle-side physics too, and a decohered superposition is still a
superposition --- precisely because no amount of unitary
machinery manufactures a semigroup. In the present framework the
semigroup was never missing: it is the other fiber, present in
the constitution from the first axiom, and measurement is the
event in which a boundary registration couples the two --- the
rotor's coherent alternatives on the circle, the one-way write on
the interval. ``Collapse,'' already demystified in this paper as
Bayesian conditioning on an outcome
(\S\ref{sec:grip-spine}), now has its geometric root displayed:
conditioning is lawful and irreversible-looking because the write
it conditions on \emph{is} irreversible --- not because a
mysterious dynamics intervened, but because the record lives on
the fiber where ``unwrite'' is not a disfavored operation but an
undefined one.

Two corollaries complete the sharpening, one per campaign.
\emph{First, the outcomes are selected the way every symmetry is
broken.} The definite value a measurement registers is a \GRIP{}
attractor --- the unique ground state the cascade relaxes to
within a layer --- and the selection is of the same deterministic
kind the symmetry review's capstone established for the cosmos at
large: a quantized alphabet of alternatives (the circle's), read
once, forward, along the fiber that cannot re-choose (the
interval's). What appears at the boundary as the Born rule's
irreducible chance is, at its stated grade in this paper's
theorems, the boundary's ignorance of fiber data --- not a coin
the bulk flips. \emph{Second, the arrow of measurement is the
arrow of time --- literally, not analogically.} The
non-invertibility that actualizes a detector's record is the same
constitutional non-invertibility that ages the universe: one
semigroup, read in a laboratory over microseconds or read across
the descent's $e^{140}$ Planck ticks of cosmic history
\citep{ArisakaP5CPT}. A measurement is a small cosmology ---
coherent alternatives, one gate, one irrevocable entry in the
ledger --- and the universe is a long measurement, which is why
the same two fibers answer ``why do experiments have outcomes?''
and ``why does anything grow old?'' The spine claimed that
without the attractors nothing is discrete, definite, or
recorded; the anatomy adds the reason: \emph{discreteness is the
circle's gift, definiteness is the attractor's, and the record is
the interval's --- and a world needs all three to contain an
experiment.}

\section{The 1922 experiment and its 2026 reformulation}
\label{sec:SG-historical}

In February 1922 Otto Stern and Walther Gerlach \citep{SternGerlach1922} sent a beam of silver atoms through an inhomogeneous magnetic field oriented along $\hat z$ and observed, on a glass detector plate downstream, two discrete spots rather than a continuous distribution. The interpretation accepted within a year was that the silver atom's outermost-shell electron carries an intrinsic angular-momentum quantum number $s = \tfrac{1}{2}$, with two allowed projections $m_s = \pm\tfrac{1}{2}$ along any chosen axis. Goudsmit and Uhlenbeck's 1925 spin hypothesis \citep{UhlenbeckGoudsmit1925} supplied the theoretical framework a few years later. In the standard textbook \citep{CohenTannoudji1992, Sakurai2011}, Stern--Gerlach is the prototype demonstration of quantum measurement: a continuous classical variable (the magnetic moment direction) collapses, on observation, to one of two discrete values, with probabilities given by the Born rule.

In the GG-Theory reformulation, the same experiment becomes a worked example of \OGGEP{} acting on a single qubit. The 6D bulk state is the fermion zero mode of a single electron; the boundary partial trace projects it to a $\mathbb{C}^2$-valued density operator; the instrument family corresponds to the two pointer states of the detector; and the textbook Born rule probabilities $\cos^2(\theta/2)$ and $\sin^2(\theta/2)$ emerge by \TA{3} applied to the qubit's pure state $|\psi\rangle = \alpha|\!\!\up\rangle + \beta|\!\!\dn\rangle$. Nothing in the calculation is added that is not already forced by the cohort apparatus.




\section{The six-dimensional bulk reading: deterministic trajectory of a spin-1/2 silver atom}
\label{sec:SG-6d-reading}

Before turning to the standard textbook calculation, we pause to state the
deeper picture that the cohort framework brings to the Stern--Gerlach
experiment. The standard treatment, in the language Goudsmit, Uhlenbeck,
and Pauli would have recognized, follows in \S\ref{sec:SG-bulk}--%
\S\ref{sec:SG-decoherence}; placing the bulk reading first lets the
reader see the two pictures side by side.

\paragraph{The silver atom in six dimensions.}
A silver atom (electronic configuration $[\mathrm{Kr}]\,4d^{10}\,5s^1$)
carries one unpaired electron in its outer shell; the inner forty-six
electrons fill closed shells and contribute no net angular momentum.
In \GGtheory{}, the outer $5s$ electron is a single-quantum excitation
of the bulk fermion field $\Psi(x, r, \theta)$ on the six-dimensional manifold $\mathcal{M}_4 \times I_r \times S^1_\theta$.
The atom's full 6D state propagates deterministically and unitarily
under the bulk Lagrangian. There is no superposition of ``spin up''
and ``spin down'' in the bulk; there is a single-valued bulk field
configuration with a definite phase along $S^1_\theta$.

\paragraph{Spin as winding on the phase axis.}
In the 6D picture, the spin label $m_s = \pm\tfrac{1}{2}$ is the
\emph{winding class} of the electron's bulk fermion zero mode around
the topological circle $S^1_\theta$. Specifically, the Wilson-line
holonomy
\begin{equation}\label{eq:SG-holonomy}
 \Theta_H \;\equiv\; \exp\!\Bigl(i \oint_{S^1_\theta} A_\theta \, d\theta\Bigr)
 \;\in\; \{+1,\, -1\},
\end{equation}
takes one of two discrete values, corresponding to the two spin
projections along the quantization axis. The discreteness is not a
postulate; it is a structural feature of the 6D gauge bundle, forced
by the anomaly cancellation of \BIsix{} \citep{ArisakaA3BI6} together
with the integer ledger \OGNfive{} of \citet{ArisakaA4Euclid}.
Goudsmit and Uhlenbeck's 1925 spin hypothesis
\citep{UhlenbeckGoudsmit1925} --- that electrons carry an intrinsic
angular momentum quantum number $s = \tfrac{1}{2}$ --- is, in the 6D
picture, the statement that the bulk gauge bundle admits exactly two
distinct holonomy sectors around $S^1_\theta$.

\paragraph{What the four-dimensional observer cannot register.}
The holonomy value $\Theta_H$ along $S^1_\theta$ is not directly
accessible to a four-dimensional detector. No magnetometer, no SQUID,
no atomic-physics-style measurement of the spin-projection-along-an-axis
can read $\Theta_H$ as a primitive: every such measurement reads the
spin's projection on a chosen 4D direction $\hat n$, which is the
4D-spatial projection of the underlying holonomy class. This
inaccessibility is structural, not technological; the 6th geometric
direction is categorically not a 4D-spatial coordinate.

\paragraph{The inhomogeneous B-field as a 4D-spatial obstacle.}
The Stern--Gerlach magnet creates a 4D-spatial field gradient
$\partial_z B_z \ne 0$ along the vertical direction. This 4D field
couples to the atom's magnetic moment $\vec\mu = g_s\,\mu_B\,\vec S/\hbar$
via the Zeeman interaction
\begin{equation}\label{eq:SG-zeeman}
 H_{\rm SG} \;=\; -\vec\mu\cdot\vec B
 \;=\; -g_s\,\mu_B\,\hat S_z\,B_z(z) / \hbar.
\end{equation}
The resulting force on the center of mass is
$F_z = -\partial_z H_{\rm SG} = g_s\,\mu_B\,\langle\hat S_z\rangle\,\partial_z B_z / \hbar$,
which has magnitude $\mu_B|\partial_z B_z|$ and sign fixed by the
holonomy class. The two holonomy sectors experience opposite forces
and separate spatially in 4D over the magnet length $L_{\rm mag}$,
yielding a transverse spatial separation
$\Delta z \approx (\mu_B|\partial_z B_z|/m)\,(L_{\rm mag}/v)^2$ of
order $10\,\mu$m for the original 1922 parameters
($|\partial_z B_z|\sim 100$~T/m, $L_{\rm mag}\sim 4$~cm,
$v \sim 700$~m/s). This is not a probabilistic split; it is a
geometric force-law applied to two distinct holonomy classes of the
same underlying bulk field configuration.

\paragraph{The bulk-side trajectory is deterministic mechanics.}
After the magnet, the bulk field configuration has two
4D-spatially-separated branches, each carrying its own $S^1_\theta$
holonomy class. The branches propagate deterministically along
classical center-of-mass trajectories; their 4D positions at the
detector are given by Newtonian mechanics applied to each holonomy
class. No probability has entered the calculation yet. The ``two
paths'' of the standard textbook are the two 4D shadow branches of a
single 6D bulk gauge bundle whose two holonomy sectors have been
spatially resolved by the field gradient.

\paragraph{What the detector registers.}
A 4D detector --- the glass plate in the 1922 experiment, a CCD pixel
in modern repetitions --- is a device localized in the four spacetime
dimensions and unavoidably integrating over $I_r$ and $S^1_\theta$.
When the bulk field encounters such a detector, the registration event
lands at exactly one of the two 4D positions $z_\pm$: the position
that corresponds to the atom's holonomy class. Because the 4D
observer cannot directly read $\Theta_H$, they cannot predict which
spot a given atom will hit before the magnet has separated the
sectors. The atom lands on one spot; the experimenter records a
single click; the bulk dynamics fixed which spot received it.

\paragraph{Why the two-spot pattern builds up one atom at a time.}
Stern and Gerlach observed the pattern using a beam containing many
silver atoms; modern variants \citep{Riebe2004} operate with single
atoms traversed in sequence. Each atom enters the apparatus in one
of the two holonomy classes, drawn from the source distribution. For
a source preparation that the standard textbook would describe by the
pure spin state
$|\psi\rangle = \cos(\theta/2)|\!\!\up\rangle + e^{i\varphi}\sin(\theta/2)|\!\!\dn\rangle$
with measurement axis along $\hat z$, the 6D-side claim is that the
thermal source populates the two $\hat z$-holonomy sectors in the
ratio
\begin{equation}\label{eq:SG-population}
 N_+ : N_- \;=\; \cos^2(\theta/2) : \sin^2(\theta/2),
\end{equation}
i.e., the fraction of source atoms in holonomy class $\Theta_H = +1$
relative to $\Theta_H = -1$ matches the Born-rule weighting. Over the
$N$-atom run, the histogram of landing positions reproduces the
textbook two-spot distribution with the standard Born-rule
probabilities. No single atom ever experiences ``probability''; each
atom is in one sector with definite holonomy; the histogram is the
statistical accumulation of source-ensemble holonomy assignments
projected onto the screen.

\paragraph{Bottom line of the bulk reading.}
The two-spot pattern is a deterministic 6D geometric effect. The
inhomogeneous B-field discriminates between two holonomy classes of
the bulk fermion zero mode and separates them spatially in 4D; the
two sectors land on two distinct 4D positions; the apparent
randomness in single-atom registrations is a consequence of the 4D
detector being unable to read the $S^1_\theta$ holonomy directly; the
Born-rule weighting $\cos^2(\theta/2) : \sin^2(\theta/2)$ is the
statistical distribution of source holonomies in the heated silver
vapor. Spin-up vs.\ spin-down is not a primitive randomness in
Nature; it is a 4D-shadow label for two discrete holonomy sectors of
a single 6D gauge bundle.

\noindent The 6D-bulk-to-4D-boundary Stern--Gerlach mechanism is shown in \figref{fig:SG-6d-mechanism}.

\begin{figure}[H]
\centering
\resizebox{0.876\textwidth}{!}{%
\usebox{\GGscSGmech}}
\caption[Stern--Gerlach: 6D-bulk-deterministic spin to 4D-boundary two-spot pattern.]{The 6D-bulk-to-4D-spot mechanism of Stern--Gerlach. Left:
the heated silver source populates two discrete holonomy classes of
the bulk gauge bundle around $S^1_\theta$, in the ratio
$\cos^2(\theta/2):\sin^2(\theta/2)$ fixed by the source preparation
(the textbook ``pure state'' is the 4D shadow of this ensemble).
Middle: the inhomogeneous magnetic field $\partial_z B_z$ exerts
opposite forces on the two holonomy classes via the Zeeman coupling
$H_{\rm SG} = -\vec\mu\cdot\vec B$, deflecting the two classes
oppositely in 4D-space. Right: each atom lands on exactly one spot,
$z_+$ if $\Theta_H = +1$ and $z_-$ if $\Theta_H = -1$; the histogram
of $N$ atoms reproduces the textbook two-spot pattern with the
standard Born-rule populations. No single atom experiences
probability; the apparent randomness is the 4D observer's inability
to read the $S^1_\theta$ holonomy directly.}
\label{fig:SG-6d-mechanism}
\end{figure}


\section{The 6D bulk state of a silver atom}
\label{sec:SG-bulk}

We follow the cohort convention: a single qubit in the QM-restricted boundary algebra corresponds to a single fermion zero mode on the 6D bulk of \citet{ArisakaA2GG6}. Concretely, the boundary state of the silver atom's outer electron is the spin-degree-of-freedom restriction of the bulk fermion field $\Psi(x, r, \theta)$, with the Ramond-sector chirality-projected zero mode $\Psi^{(0,0)}(x)$ surviving on the boundary.

The QM-restricted boundary algebra $\cA_\Bdy^{(\rm QM)}$ is generated by the four bilinears
\begin{equation}\label{eq:SG-bilinears}
\hat O^a \;\equiv\; \bar\Psi^{(0,0)}\,\sigma^a\,\Psi^{(0,0)}, \qquad a \in \{0, 1, 2, 3\},
\end{equation}
where $\sigma^0 = \mathbf{1}$ and $\sigma^{1,2,3}$ are the three Pauli matrices acting on the spin index. By \TA{1}, the GNS representation of $\cA_\Bdy^{(\rm QM)}$ on the cyclic vector $|0\rangle_\Bdy = |\!\!\!\up\rangle$ produces $\cH_\Bdy^{(\rm QM)} = \mathbb{C}^2$, the standard qubit Hilbert space; the isometric embedding $\iota:\mathbb{C}^2 \hookrightarrow \cH_\Bulk$ takes $|\!\!\up\rangle$ to the zero-mode spin-up state in the bulk fermion Hilbert space and $|\!\!\dn\rangle$ to the zero-mode spin-down state.

The boundary density operator $\rho_\Bdy^{(\rm QM)} = \tfrac{1}{2}(\mathbf{1} + \vec r\cdot\vec\sigma)$ inherits its $|\vec r\,|\le 1$ Bloch-ball constraint directly from the bulk-unitarity bound on the reduced state; for the pure incoming-beam preparation $|\psi\rangle = \alpha|\!\!\up\rangle + \beta|\!\!\dn\rangle$ with $|\alpha|^2 + |\beta|^2 = 1$, $|\vec r\,| = 1$ and the state sits on the Bloch sphere. The bulk-side description is everywhere deterministic and unitary; the probabilistic appearance of the measurement is, as in the rest of the cohort treatment, a feature of the boundary projection.


\section{The instrument family and the boundary channel}
\label{sec:SG-instrument}

The Stern--Gerlach apparatus oriented along direction $\hat n$ (at angle $\theta$ from $\hat z$ in the $xz$ plane) implements the projective measurement family
\begin{equation}\label{eq:SG-projectors}
\Pi_+ \;=\; |\!\!\up_n\rangle\langle\up_n|, \qquad
\Pi_- \;=\; |\!\!\dn_n\rangle\langle\dn_n|,
\qquad \Pi_+ + \Pi_- \;=\; \mathbf{1},
\end{equation}
where $|\!\!\up_n\rangle = \cos(\theta/2)|\!\!\up\rangle + \sin(\theta/2)|\!\!\dn\rangle$ and $|\!\!\dn_n\rangle = -\sin(\theta/2)|\!\!\up\rangle + \cos(\theta/2)|\!\!\dn\rangle$ in the $\hat z$-eigenbasis. By \TA{3}.2 (the operational layer of the Born-rule theorem), the probability of registering outcome $\pm$ on the pure state $|\psi\rangle = |\!\!\up\rangle$ is
\begin{equation}\label{eq:SG-probs}
\boxed{\;
p_+ \;=\; |\langle\up_n|\!\!\up\rangle|^2 \;=\; \cos^2(\theta/2),
\qquad
p_- \;=\; |\langle\dn_n|\!\!\up\rangle|^2 \;=\; \sin^2(\theta/2),
\;}
\end{equation}
the standard textbook Stern--Gerlach probabilities, with $p_+ + p_- = 1$ automatic.

The instrument family $\{\cI_+, \cI_-\}$ that produces these probabilities is the conditional CPTP map of \TA{4}:
\begin{equation}\label{eq:SG-instrument}
\cI_\pm(\rho) \;=\; \Pi_\pm\,\rho\,\Pi_\pm, \qquad \rho_\Bdy^{(\pm)} \;=\; \cI_\pm(\rho)/p_\pm.
\end{equation}
The post-measurement state conditional on outcome $+$ is the pure state $|\!\!\up_n\rangle\langle\up_n|$; conditional on outcome $-$, it is $|\!\!\dn_n\rangle\langle\dn_n|$. These are the textbook L\"uders projections.

The physical mechanism of the measurement is the spatial entanglement between the spin degree of freedom and the position degree of freedom produced by the field gradient. The 6D bulk Hamiltonian density (recall \AX{1}: bulk locality) is multiplicative in the position basis; the field gradient $\partial_z B$ produces a force $F_z = \mu_z(\partial_z B)$ depending on the spin projection along $\hat z$; the spin-up and spin-down beams therefore separate spatially, becoming distinguishable to the downstream detector by their position. This is precisely the bulk-locality mechanism of \TA{6} (einselection): the dominant boundary system--environment coupling is position-diagonal, so the position basis becomes the einselected classical record.


\section{The decoherence step and the einselected outcome}
\label{sec:SG-decoherence}

Before the atom strikes the detector, the boundary state is the entangled bipartite state
\begin{equation}\label{eq:SG-entangled}
|\Psi\rangle_{\rm spin\otimes pos} \;=\; \alpha\,|\!\!\up_n\rangle \otimes |z_+\rangle \;+\; \beta'\,|\!\!\dn_n\rangle \otimes |z_-\rangle,
\end{equation}
with $|z_\pm\rangle$ the position eigenstates of the two beams emerging from the field gradient. The detector then couples to the atom's position via a position-diagonal system--environment interaction (the silver atom deposits at a definite position on the glass plate; subsequent photons scatter off the deposited atom; the photon environment carries away which-position information). By \TA{5}, the off-diagonal density-matrix elements in the position basis decay at rate $\Gamma_{z_+ z_-} \gtrsim 10^{40}\,\mathrm{s}^{-1}$ at room temperature, on the time scale of essentially zero (per the Joos--Zeh estimate cited in \TA{5}.2).

The reader registers the outcome as a definite spot at $z_+$ or $z_-$, with probabilities $|\alpha|^2 = \cos^2(\theta/2)$ and $|\beta'|^2 = \sin^2(\theta/2)$ as in \eqref{eq:SG-probs}. The deterministic 6D bulk evolution has, on this reading, never collapsed: there is one deterministic bulk worldline that produces one or the other detector position, with the apparent randomness arising entirely from the boundary observer's inability to access the bulk Wilson-line holonomy $\Theta_H$ along $S^1_\theta$ that selects between the two configurations.

In the regime vocabulary of Section~\ref{sec:grip-spine}, the
experiment parses into the canonical anatomy of a measurement.
Bringing the field gradient and detector to bear on the beam
reorganized the joint atom--apparatus landscape --- before the
magnet, no position-pointer basins existed; after it, exactly two ---
an endogenous, scheduled \GRIPout{}, the laboratory instance of the
\QiHAL{} exploration-to-commitment switch of \ASeven{}. The
decoherence step just computed is the \GRIPin{} that follows: descent
into one of the two basins at rate $\Gamma_{z_+z_-}$, halting at a
deposited silver atom --- a Geometric Attractor whose permanence is
the $F_{\GFF}$ monotonicity of the record stratum. And the fact that
the catalog has exactly \emph{two} elements is Claim~3 of
Section~\ref{sec:grip-spine} in its sharpest laboratory form: the
quantization of the spin readout is the discreteness of the
attractor catalog, nothing else.


\section{Comparison with the textbook account}
\label{sec:SG-comparison}

A side-by-side comparison of the standard textbook reading and the \OGGEP{} reading clarifies what each says.

\begin{table}[H]
\centering
\caption[Stern--Gerlach: the textbook reading against the OGGEP reading]{Stern--Gerlach measurement: standard textbook reading vs.\ the \OGGEP{} reading. The mathematics is identical; the ontology is sharper in the cohort framework. Each row takes one step of the 1922 experiment and sets the standard account beside the bulk-projection account. Every predicted number is the same in both columns, and that is deliberate: the claim here is a sharper ontology for an established result, not a new one, and a reader who rejects the right-hand column loses no experimental agreement.}
\label{tab:SG-comparison}
\small
\renewcommand{\arraystretch}{1.25}
\begin{tabular}{@{}p{4.48cm}p{5.60cm}p{5.60cm}@{}}
\toprule
\textbf{Stage} & \textbf{Standard textbook} & \textbf{GG-Theory (\OGGEP{}/\GDOS{})} \\
\midrule
Preparation & Pure state $|\psi\rangle\in\mathbb{C}^2$ asserted by postulate & Boundary state $\rho_\Bdy = \Tr_{\rm env}\rho_\Bulk$ inherited via \TA{1} \\
Evolution & Unitary; spin couples to position via $H_{\rm SG}$ & GKLS in $\gamma\to 0$ limit (\TA{2}); same Hamiltonian \\
Measurement axiom & Postulated separately (postulate IV) & Theorem \TA{3} from CPTP completion of \OGGEP{} \\
Born probability & $\cos^2(\theta/2)$, $\sin^2(\theta/2)$ by postulate & Same value, by theorem \TA{3} \\
Collapse & Postulated (postulate V): non-unitary, irreversible, location-of-Heisenberg-cut ambiguous & Conditional CPTP update \TA{4}; no bulk collapse; cut is just where the observer chooses to apply the partial trace \\
Position basis preferred & Phenomenological observation that records appear in position & Theorem \TA{6}: \AX{1} bulk locality forces position-diagonal boundary coupling \\
Decoherence rate & Phenomenological input or environment-by-environment fit & Theorem \TA{5}: fixed by scale-overlap integrals of bulk KK modes \\
Determinism & Absent in 4D quantum theory (Bell + Aspect closed the door) & Present at the bulk level; absent at the boundary level by structural projection \\
\bottomrule
\end{tabular}
\end{table}

\subsection*{Part-end close}

Stern and Gerlach in 1922 saw two spots and did not know that they were seeing the four-dimensional shadow of two distinct holonomy classes of a six-dimensional gauge bundle. Three years before Goudsmit and Uhlenbeck identified the spin and four years before Born wrote down the probability rule, the two spots already encoded the entire content of the cohort framework's reading: each atom is in one of two discrete holonomy sectors of the bulk gauge bundle; the inhomogeneous field discriminates the two by 4D-spatial deflection; the histogram of many atoms reproduces the Born-rule weighting because the source preparation populates the two sectors in exactly that ratio; and the apparent randomness in any single-atom event is the four-dimensional observer's inability to read the underlying $S^1_\theta$ holonomy directly.

After all, what makes a silver atom land on one spot rather than the other is the same thing that makes every electron in the universe land where it lands: the constitutional structure of the bulk gauge bundle. The Stern--Gerlach apparatus simply isolates that structure cleanly enough that two spots, separated by $\sim 10\,\mu$m of vertical distance, become the visible record of two discrete topological sectors of a six-dimensional geometry that we could otherwise only infer.




%% ===================================================================
%% PART VI (6/14) -- Worked example II --- Double-slit and decoherence
%% ===================================================================
\paragraph{Problems and worked solutions.}

\medskip\noindent\textbf{V.1. Chain three Stern--Gerlach magnets, and
watch sharpness die and revive.}

\noindent\emph{Solution.}  Step 1: send the $z{+}$ beam from a first
magnet into a second magnet oriented along $x$: by
\eqref{eq:SG-probs} with $\theta = 90^\circ$, each $x$-port fires
with probability $\cos^2 45^\circ = \tfrac12$.  Step 2: take the
$x{+}$ output into a third magnet along $z$: again $\tfrac12$ each
--- the original $z$-sharpness is \emph{gone}, erased by the
intervening $x$-registration.  Step 3: now remove the middle
registration (recombine both $x$ paths coherently) and the third
magnet fires $z{+}$ with probability $1$: it was not the magnet but
the \emph{record} that destroyed the sharpness --- in the bulk
reading, the $x$-registration re-selected the holonomy sector, while
the recorded-nothing interferometer left the original sector intact.
Physics intuition: the three-magnet chain is the whole measurement
problem on a bench --- observables do not coexist, records are
physical, and ``asking'' is an act; a student who can predict all
eight port probabilities has already understood more than the
collapse postulate ever said.

\medskip\noindent\textbf{V.2. Reproduce the 1922 deflection with
laboratory numbers.}

\noindent\emph{Solution.}  Step 1: the transverse force on the
moving atom is $F = \mu_B\,\partial_z B$; with the historical-scale
gradient $\partial_z B \approx 10^3~\mathrm{T/m}$ and
$m_{\rm Ag} = 1.79\times10^{-25}$~kg, the acceleration is
$a = \mu_B \partial_z B/m_{\rm Ag} \approx 5.2\times10^4~
\mathrm{m/s^2}$.  Step 2: an oven atom at
$v \approx 550~\mathrm{m/s}$ spends $t = L/v \approx 64~\mu\mathrm s$
in a $3.5$-cm magnet: deflection $\tfrac12 a t^2 \approx 0.1$~mm,
doubling with the field-free drift --- the two spots Gerlach
developed (famously, with cigar smoke) sat a fraction of a
millimeter apart, exactly this arithmetic.  Step 3: note what the
two discrete spots rule out: a classical magnetic needle would have
a continuum of orientations and paint a smear; the beam splits
cleanly in two because the registered observable has exactly two
sectors --- space quantization, the first holonomy readout in
history.  Physics intuition: the most consequential millimeter in
physics --- a tabletop force of $10^{-20}$~newtons, patiently
applied for sixty microseconds, separated two worlds that classical
physics said should blur into one.

\medskip\noindent\textbf{V.3. Explain why the experiment needs a
neutral atom, not a free electron.}

\noindent\emph{Solution.}  Step 1: a free electron in the same
apparatus feels the Lorentz force $e v B \sim 10^{-14}$~N ---
six orders of magnitude above the magnetic-moment force --- and its
transverse position spreads under the uncertainty relation; Bohr and
Pauli argued the spin splitting is buried beneath both.  Step 2:
silver evades both problems at once: net charge zero (no Lorentz
force), and a single unpaired $5s$ electron hands the whole atom a
clean one-Bohr-magneton moment --- the atom is a heavy, neutral
carrier for one electron's spin.  Step 3: the framework moral: a
measurement requires an \emph{instrument-compatible} carrier --- the
registered observable must couple to the apparatus above the noise
of everything else it drags along; the 1922 design is a masterclass
in isolating one holonomy channel.  Physics intuition: you cannot
weigh a letter during a hurricane --- Stern and Gerlach mailed the
electron's spin inside a neutral atom precisely so the only thing
the magnet could read was the quantity worth reading.

\medskip\noindent\textbf{V.4. Predict the populations from a tilted
source, and count atoms.}

\noindent\emph{Solution.}  Step 1: prepare spins along $\hat n$ at
$\theta = 60^\circ$ from the apparatus axis; by \eqref{eq:SG-probs},
$p_+ = \cos^2 30^\circ = \tfrac34$, $p_- = \tfrac14$.  Step 2: in a
run of $1000$ atoms expect $750 \pm 14$ up and $250 \pm 14$ down
(binomial, $\sigma = \sqrt{Np_+p_-} \approx 13.7$) --- the Born
weights are frequencies with shot noise, which is how every real
Stern--Gerlach test reads them.  Step 3: tie the two registers
together: the \emph{ratio} $3\!:\!1$ is the geometry of the tilt
(the appendix problem C.7.1 derives it from Gleason), while the
\emph{fluctuation} $\pm14$ is the statistics of finite sampling ---
theory fixes the odds, the bench draws the samples.  Physics intuition:
 quantum probability walks and talks like any other
probability once the odds are fixed --- the mystery was never the
statistics but the origin of the odds, and that is exactly the part
the trace formula (and behind it, the fiber) supplies.

\medskip\noindent\textbf{V.5. Show that repeatability needs no
collapse.}

\noindent\emph{Solution.}  Step 1: after the first magnet registers
$z{+}$, the atom-apparatus record is the entangled-then-decohered
state of \eqref{eq:SG-entangled}; the boundary description
conditioned on the record is $\ketbra{\uparrow_z}{\uparrow_z}$.
Step 2: a second $z$-magnet then fires $z{+}$ with probability
$\cos^2 0 = 1$ --- repeatability --- with no dynamical collapse
invoked anywhere: the bulk configuration held one holonomy sector
throughout, and the second instrument reads the same sector.
Step 3: state the contrast carefully: the textbook derives the same
number from the projection postulate; the framework derives it from
sector persistence plus conditioning (the L\"uders problem E.8.1
is the appendix-side twin) --- identical laboratory content,
different ontology, and the difference is testable only where the
framework's falsifiers say it is.  Physics intuition: asking the
same question twice gets the same answer because the answer was a
property of the world, not of the asking --- repeatability is the
oldest hint that registration reads something rather than makes
something, and the 1922 prototype already contained it.


\GGtakeaway{\emph{Standard account:} prepare the silver atom,
apply the collapse postulate, weight the two spots by the Born rule
--- operationally perfect, ontologically silent. \emph{What it left
open:} which outcome occurs, why those weights, and what happens
during the jump. \emph{The bulk reading:} one deterministic 6D
trajectory per atom, outcome fixed by the unresolved phase-fiber
coordinate, Born weights delivered by the \GDOS{} registration ---
determinism upstairs, statistics downstairs, and the 1922 prototype
becomes the framework's cleanest single-qubit exhibit.}

\GGPart[Worked example II: Double-slit]{Worked example II: Double-slit and decoherence}{Wave--particle duality and the death of interference}
\label{part:VI}

\GGteachboxpart{The heart of the quantum mystery, dissolved in two
readings: a point particle in space that is simultaneously a wave in
\emph{phase} --- the particle and its dial. The chapter works the
propagation of one quantum through both slits (the point goes through
one; the field that interferes goes through both), the which-path
environment and its decoherence factor, the three experimental
regimes (photon, electron, fullerene) with their numbers, and the
quantum--classical boundary read as a scale window rather than a
mystery line. Feynman's ``only mystery'' becomes, in the bulk
reading, a bookkeeping fact about which fiber the observer can
resolve.}

\GGreadalong{\citet{FeynmanLectures} (vol.~III, ch.~1) for the
mystery stated canonically; \citet{DavissonGermer1927} for matter
waves; \citet{Hornberger2003} and \citet{Hackermuller2004} for the
fullerene decoherence experiments the chapter's three-regime table
quantifies.}

The double-slit experiment is the cleanest demonstration of wave--particle duality, and the cleanest test of the cohort framework's deterministic six-dimensional-bulk reading of quantum mechanics. In this reading, each photon propagates deterministically through the six-dimensional geometry; the interference pattern builds up one photon at a time because each photon's landing position on the screen is fixed by its phase angle along the geometric direction $S^1_\theta$ that the four-dimensional observer cannot read. The interference disappears only when a which-path environment destroys the bulk phase relationship between the two slit paths; the off-diagonal density-matrix element then decays at the rate of \TA{5}. We present the six-dimensional bulk reading first in \S\ref{sec:DS-6d-reading}, perform the standard textbook calculation in \S\ref{sec:DS-setup}--\S\ref{sec:DS-environment} for direct comparison, and walk three regimes (photon, electron, fullerene) where the boundary between quantum and classical sits at the value of $\Delta x$ for which the bulk--environment coupling becomes appreciable on the experimental timescale.


\noindent The double-slit experiment with the decoherence sweep is shown in \figref{fig:double-slit-decoherence}.

\begin{figure}[H]
\centering
\resizebox{0.842\textwidth}{!}{%
\begin{tikzpicture}[
 font=\sffamily\footnotesize,
 every node/.style={align=center, inner sep=2pt},
 source/.style={circle, fill=red!60!black, draw=red!75!black, inner sep=2pt},
 barrier/.style={rectangle, fill=gray!16, draw=black!70, line width=0.6pt,
 minimum width=2mm, minimum height=8mm},
 beam/.style={line width=0.45pt, draw=black!55, dashed},
]
% ---- Schematic (left half) ----
\node[source] (S) at (0,0) {};
\node[font=\scriptsize, anchor=east] at (S) {$|\psi\rangle$};

% Slits (two barriers separated by a gap)
\node[barrier] (b1) at (20mm, 6mm) {};
\node[barrier] (b2) at (20mm, -6mm) {};
\node[font=\scriptsize, anchor=west] at (22mm, 6mm) {slit~1};
\node[font=\scriptsize, anchor=west] at (22mm, -6mm) {slit~2};

% Detector screen
\draw[line width=0.7pt, draw=black!70] (60mm,-22mm) -- (60mm,22mm);
\node[font=\scriptsize, anchor=west] at (60mm, -24mm) {screen};

\draw[beam] (S.east) -- (b1.west);
\draw[beam] (S.east) -- (b2.west);
\draw[beam] (b1.east) -- (60mm, 0);
\draw[beam] (b2.east) -- (60mm, 0);

% ---- Interference curve and decohered curve (right half) ----
\begin{scope}[xshift=85mm, yshift=0]
  % axes
 \draw[->, line width=0.6pt] (0,-22mm) -- (0,22mm) node[above] {$y$};
 \draw[->, line width=0.6pt] (-2mm,0) -- (52mm,0) node[right] {intensity};

  % Interference fringes (cos^2-modulated Gaussian)
 \draw[line width=0.9pt, draw=blue!75!black, smooth, samples=120, domain=-20:20]
 plot (\x*0.025 cm * 5, {(\x mm * 1mm) - (\x mm * 1mm)});
  % Drawn explicitly using a smoother approximation:
 \draw[line width=0.9pt, draw=blue!75!black] plot[smooth,tension=0.5]
 coordinates {
 (0, -22mm) (4mm, -19mm) (8mm, -16mm)
 (12mm,-13mm) (10mm,-11mm) (16mm,-9mm) (12mm,-7mm)
 (18mm,-5mm) (14mm,-3mm) (22mm,-1mm) (28mm, 1mm)
 (14mm, 3mm) (18mm, 5mm) (12mm, 7mm) (16mm, 9mm) (10mm, 11mm)
 (12mm, 13mm) (8mm,16mm) (4mm,19mm) (0,22mm)
 };

  % Decohered classical-sum curve (smooth single peak)
 \draw[line width=0.9pt, draw=red!70!black, dashed] plot[smooth,tension=0.5]
 coordinates {
 (0,-22mm) (3mm,-18mm) (6mm,-14mm) (9mm,-9mm) (12mm,-4mm)
 (14mm, 0) (12mm, 4mm) (9mm, 9mm) (6mm,14mm) (3mm,18mm) (0,22mm)
 };

 \node[font=\scriptsize, text=blue!75!black, anchor=west]
 at (30mm, 14mm) {pure: $|\langle y|\psi\rangle|^2$};
 \node[font=\scriptsize, text=red!70!black, anchor=west]
 at (30mm, 9mm) {decohered: $\sum_k|c_k|^2|y\rangle$};
 %% v15.0 Stage 7: this node sat at y = 0 mm, on the intensity axis.  The
 %% axis rule struck through the word 'damping' and the axis-tip label
 %% 'intensity' printed on top of the formula.  Moved to y = 4 mm, which
 %% keeps the 5 mm spacing of the two annotations above it.  Nothing the
 %% figure says is changed; the label is only moved off the rule.
 \node[font=\scriptsize, anchor=west] at (30mm, 4mm)
 {damping: $\rho_{12}(t) = \rho_{12}(0)\,e^{-\Gamma t}$};
 \node[font=\scriptsize, anchor=west] at (30mm, -6mm)
 {\TA{5}: $\Gamma \gtrsim 10^{40}\,$s$^{-1}$ for $C_{60}$};
\end{scope}
\end{tikzpicture}}
\caption[Double-slit experiment and the death of interference under which-path decoherence.]{Double-slit interference and its decoherence-driven
death. Left: source emits the quantum state $|\psi\rangle$, two
slits define the two-path superposition, and the detector
screen records the resulting intensity pattern. Right: the
pure-state interference fringes (blue solid, oscillating) and
the decohered classical sum (red dashed, single smooth peak)
shown on the same axis. The transition between the two is
governed by the off-diagonal density-matrix damping factor
$\rho_{12}(t) = \rho_{12}(0)\,e^{-\Gamma t}$ of \TA{5}. The
Joos--Zeh \citep{JoosZeh1985} estimate for a $C_{60}$
fullerene in a room-temperature thermal bath is
$\Gamma \gtrsim 10^{40}\,$s$^{-1}$; for any larger object the
quantum-to-classical transition is operationally instantaneous.}
\label{fig:double-slit-decoherence}
\end{figure}


%% PHASE-4D CONTENT BEGINS
\section{The 1801 Young experiment and its quantum descendants}
\label{sec:DS-historical}

In 1801 Thomas Young demonstrated optical interference by passing a single beam of sunlight through two parallel slits and observing alternating bright and dark bands on a downstream screen --- the classical confirmation that light propagates as a wave. The 1927 Davisson--Germer experiment \citep{DavissonGermer1927} repeated the same setup with electrons, observing the same fringe pattern and confirming the de Broglie hypothesis \citep{deBroglie1924} that material particles also propagate as waves. By the 1990s the same experiment had been performed with neutrons, atoms, and small molecules; by 2003 with fullerene $C_{60}$ at masses approaching a kilodalton \citep{Arndt2003_C60}. In every case, the interference pattern survives as long as which-path information is unrecoverable from the environment, and dies when which-path information is in any way registered.

The framework reformulation reads the interference--which-path complementarity through \TA{5}: the interference visibility is the off-diagonal density-matrix element in the which-path basis; the environmental coupling damps it exponentially at a rate fixed by the physical-bath overlap integral (worked explicitly for $C_{70}$ in \S\ref{sec:worked-overlap}), with the bulk geometry fixing the open-system form; the macroscopic-instantaneous regime $\Gamma\gtrsim 10^{40}\,\mathrm{s}^{-1}$ is the quantitative content of the quantum-to-classical transition.




\section{The six-dimensional bulk reading: deterministic propagation of a single photon}
\label{sec:DS-6d-reading}

Before turning to the standard textbook calculation, we pause to state the
deeper picture that the cohort framework brings to the double-slit
experiment. The standard treatment in the language Young and Born would
have recognized follows in \S\ref{sec:DS-setup}; placing the bulk reading
first lets the reader see the two pictures side by side, on equal footing.

\paragraph{The photon in six dimensions.}
In \GGtheory{}, the photon is a single-quantum excitation of the bulk
gauge field on the six-dimensional manifold
$\mathcal{M}_4 \times I_r \times S^1_\theta$. Its propagation is governed
by the bulk Lagrangian, which is deterministic and unitary. A single
photon emitted from a source has, at every moment, a single-valued field
configuration extending over all six geometric directions. There is no
quantum uncertainty about its trajectory; there is no superposition of two
coexisting alternatives; there is no collapse upon measurement. The bulk
knows exactly where the photon is.

\paragraph{What the four-dimensional observer cannot register.}
The two dimensions beyond spacetime --- the scale axis $I_r$ and the
phase axis $S^1_\theta$ --- are not accessible to a four-dimensional
detector. No instrument the experimenter can build, regardless of its
spatial resolution, can directly read the photon's position along $I_r$
or its phase angle along $S^1_\theta$. This is not a limitation of
technology; it is a structural feature of the projection from six
geometric directions onto four.

\paragraph{The slits as a 4D-spatial obstacle.}
When the photon's bulk field configuration propagates from source toward
the slit barrier, its transverse 4D-spatial extent naturally spans both
slit positions. This is not a superposition of two probabilistic
alternatives; it is a single-valued bulk field with an extended 4D
transverse profile. The two slits do not split the photon into two
copies of itself; they simply select the two portions of the same bulk
configuration that get past the barrier. The portion passing through
slit~$+$ and the portion passing through slit~$-$ are two parts of one
bulk field, with definite relative geometric phase set by the bulk
geometry and the photon's source phase along $S^1_\theta$.

\paragraph{The bulk-side calculation is six-dimensional wave optics.}
After the slits, the bulk field configuration propagates deterministically
to the screen. At each 4D point on the screen, the bulk amplitude is the
sum of the two propagated portions, with the relative phase determined
entirely by the 6D geometry. No probability has yet entered the
calculation; this is wave optics in six dimensions, every step of it
deterministic and unitary.

\paragraph{What the detector registers.}
A 4D detector --- a CCD pixel, a photographic grain, a single-photon
avalanche diode --- is a device localized in the four spacetime
dimensions and unavoidably integrating over the two geometric directions
$I_r$ and $S^1_\theta$. When the bulk field encounters such a detector,
the registration event is forced to land at a single 4D position. Which
4D position? The one fixed by the photon's specific phase angle along
$S^1_\theta$ at the moment of arrival, in combination with the 4D
geometry of the screen. Because the 4D observer cannot read that phase
angle, they cannot predict where any given photon will register. The
photon lands somewhere; the experimenter records a single click; the
bulk dynamics determined the landing position; the 4D observer simply
cannot trace the determination back through the inaccessible phase
coordinate.

\paragraph{Why the pattern builds up one photon at a time.}
Release a million photons through the same apparatus, one at a time ---
separated by milliseconds or by hours, as is routinely done in modern
single-photon double-slit experiments \citep{Jacques2007}. Each photon
has its own definite trajectory in six dimensions and a definite landing
position on the screen. The phase angle along $S^1_\theta$ varies from
photon to photon, and the source emission process makes that distribution
uniform on the circle. As the photons accumulate, the histogram of
landing positions fills in the standard interference pattern
$|\psi(y)|^2 \propto \cos^2\!\bigl(\tfrac{1}{2}(k^+ - k^-)\,y\bigr)$ of
\eqref{eq:DS-intensity}. No single photon ever experiences
``probability''; the fringe pattern is the cumulative geometric shadow on
the four-dimensional screen of the uniform $S^1_\theta$ phase
distribution across the photon ensemble. This is why interference
appears even when the photons arrive one at a time, separated by long
intervals: each photon's landing was always fixed by the bulk geometry;
the pattern emerges from the ensemble statistics of the inaccessible
phase coordinate.

\paragraph{The which-path question reframed.}
In the 6D picture, ``which slit did the photon pass through'' is a
question about the 4D shadow of a single-valued 6D configuration whose
4D profile passes through both slits. The question becomes well-defined
only when a specific 4D detector is inserted that registers which-slit
information. Such a detector changes the geometric coupling between the
photon and the apparatus, which destroys the relative phase coherence
between the two 4D profile portions (treated quantitatively in
\S\ref{sec:DS-environment}). The 4D observer then sees the fringes
disappear --- not because the photon ``chose'' a path it had not chosen
before, but because the inserted detector destroyed the bulk phase
relationship that produced the fringes in the first place.

\paragraph{Bottom line of the bulk reading.}
The interference pattern is a deterministic 6D geometric effect. The
apparent randomness in single-photon registrations is a consequence of
the 4D detector being unable to read the photon's $S^1_\theta$ phase
angle. The buildup of the fringe pattern over many photons is the
projection of a uniform phase distribution onto the four-dimensional
screen. Wave--particle duality is the cohabitation of these two
readings: ``wave'' is the 6D geometry of bulk phase relations;
``particle'' is the 4D-localized event of detector registration. The
mystery is not in the photon. It is in the projection.


\section{Particle in space, wave in phase: the point particle and its dial}
\label{sec:DS-dial}

The reading just given treated the photon as a bulk \emph{field} with a four-dimensional transverse profile. For a massive point particle the picture sharpens, and it is worth stating in full, because it dissolves wave--particle duality outright and answers the question every student asks: how can a thing that always arrives as a point behave like a wave?

\paragraph{The one move that dissolves the paradox.} ``Wave--particle duality'' sounds like a contradiction only because both words are forced to describe behavior \emph{in space}. In \GGtheory{} they describe two \emph{different dimensions}. The particle is a \emph{particle in the four-dimensional stage} $\mathcal{M}_4$ --- always a point, always localized, never smeared (this is \AX{1}/locality: a point particle stays a point). It is a \emph{wave in the phase dimension} $S^1_\theta$ --- the non-spatial sixth direction it carries at every point. Both hold at once, without contradiction, because being point-like \emph{in space} and wave-like \emph{in phase} are statements about orthogonal, non-spatial coordinates. The duality was never two faces of one thing in space; it is one face each of two different dimensions.

\paragraph{What ``the wave'' actually is.} The wavefunction $\psi(x) = |\psi(x)|\,e^{i\varphi(x)}$ is not the particle spread out. It is the \emph{phase field} --- the value of the $S^1_\theta$ dial --- laid over the possible point-positions $x$. The phase $\varphi$ \emph{is} the particle's location on the gauge circle. As the point moves, the dial spins; the rate of spin per unit length is the momentum, $p/\hbar$, and that rate \emph{is} the de Broglie wavelength $\lambda = h/p$. So $\lambda$ is not a size of the particle --- it is the spatial period of the spinning phase. Nothing material is waving; a \emph{dial} is winding.

\paragraph{Feynman's clock, with an address.} Feynman's \emph{QED} \citep{FeynmanQED1985} tells the reader to picture a tiny spinning arrow carried by the particle and to obtain probabilities by adding arrows and squaring --- without ever saying \emph{where} the arrow lives. \GGtheory{} supplies the address: the arrow is the sixth dimension. The clock hand is the position on $S^1_\theta$, and the holonomy $\oint A_\theta\,d\theta$ it accumulates is the arrow's rotation. Feynman's bookkeeping device becomes a geometric object.

\paragraph{The double slit, step by step.}
\begin{enumerate}[leftmargin=2em,itemsep=0.15em]
\item A point particle leaves the source --- one point, the whole way. In the deterministic six-dimensional bulk it follows a definite history; what is hidden is its $\theta$-dial and which history it took, because those live in the non-spatial fiber the boundary cannot read.
\item To any one screen point $P$ there are two bulk histories that arrive --- via slit $+$ and via slit $-$. They traverse slightly different path lengths, so their dials arrive pointing in different directions: a phase difference $\Delta\varphi = (2\pi/\lambda)\,\delta(\text{path})$.
\item The boundary observer registers by the \OGGEP{} projection $\pi$, which traces out the non-spatial fiber --- so it cannot read which slit, nor the absolute dial angle. It is \emph{forced} to add the two arrows and keep only the squared magnitude,
\begin{equation}\label{eq:DS-dial-born}
P(P) = |A_+ + A_-|^2 = |A_+|^2 + |A_-|^2 + 2|A_+||A_-|\cos\Delta\varphi.
\end{equation}
\item That $\cos\Delta\varphi$ term is the fringes. Sweeping across the screen, the path difference cycles, the two arrows rotate from aligned (bright) to opposite (dark), and the interference pattern of \eqref{eq:DS-intensity} appears.
\end{enumerate}
Each particle still lands as a single dot --- one point, one registration. The \emph{pattern} of dots is the statistical shadow the spinning dial casts when the boundary can see only $|\text{arrow}|^2$. The ``wave-like particle'' is impossible to visualize because it is not what happens: a point lands; the distribution of points remembers the phase.

\paragraph{Why measuring which-path kills the fringes.} Interference needs the two histories traced \emph{together}, with their relative dial angle intact. A which-path detector \emph{reads a piece of the fiber} --- it promotes the slit-choice to a registered boundary record. Once that record exists, the relative phase is randomized (the decoherence of \TA{5}: loss of $\theta$-coherence under coarse-graining along $I_r$), the $\cos\Delta\varphi$ term averages to zero, and the pattern collapses to $|A_+|^2 + |A_-|^2$ --- two plain bumps. Complementarity is exactly this: \emph{interference is the phase fiber left unread; which-path is the phase fiber read.} One cannot both trace a coordinate and register it.

\paragraph{What is and is not claimed.} In the three-tier discipline of this cohort: (i)~\emph{equivalent to standard quantum mechanics} --- none of this changes a single double-slit prediction; the fringe spacing, the visibility, and the which-path trade-off come out identical (the boundary theory \emph{is} ordinary QM, \S\ref{sec:hidden-fibre-origin}). What is new is the picture and the \emph{reason}, not the numbers. (ii)~\emph{Defensible structural proposal} --- the identification ``quantum phase $=$ holonomy of the sixth dimension,'' and the claim that the sum over paths is \emph{forced} by the \OGGEP{} trace rather than postulated. (iii)~\emph{Interpretive commitment, flagged as such} --- the deterministic-bulk reading says the particle took a definite history; this stays consistent only because ``which history'' has no boundary-registrable answer without a measurement that alters the experiment, and because the apparent non-locality of the entangled case (Bell/CHSH) is read as a projection artifact of the six-dimensional bulk (\TA{7}), not as bulk non-locality. That last point is the framework's position, not a settled result.

\paragraph{The one-line picture.} The particle is a point that carries a spinning dial in a hidden sixth dimension; it travels as a point and lands as a dot. The ``wave'' is the dial; the fringes are the map of the dial's relative angle, drawn statistically because the screen can see only the dial squared. Particle in space, wave in phase --- two dimensions, no paradox.


\section{How one point ``passes through both slits'': the field that interferes is the phase}
\label{sec:DS-both-slits}

A reader who has accepted the dial picture of \S\ref{sec:DS-dial} will still feel the sharpest form of the double-slit puzzle, and it deserves a full answer rather than a slogan. A point particle, being a point, must pass through the left slit \emph{or} the right slit; yet the interference pattern seems to require that it pass through \emph{both}. If the object were a wave one could picture it dividing between the slits; if it is a point one cannot. The resolution is not a play on words but a relocation of the question, and it follows directly from the two-fiber structure of \S\ref{sec:hidden-fibre-origin}.

\paragraph{The premise to refuse.} The puzzle is manufactured by one tacit assumption: that ``particle'' and ``wave'' are two competing descriptions of \emph{the same thing} --- the object's location in the space $\mathcal{M}_4$ --- so that the object must be either at one slit or smeared across both. In the present framework they are not competing descriptions of a single degree of freedom; they are faithful descriptions of \emph{two different degrees of freedom that live in different parts of the bulk}. The particle-aspect is a point in the spatial stage $\mathcal{M}_4$. The wave-aspect is the winding of the phase dial on the non-spatial fiber $S^1_\theta$. Forcing both onto the single axis of spatial position is precisely what generates the contradiction; releasing that assumption dissolves it.

\paragraph{Two questions that ordinary language fuses into one.} Pull the single question ``where did the particle go?'' apart into the two questions the geometry actually distinguishes. \emph{``Which slit?''} is a question about position --- a point in $\mathcal{M}_4$. It has a determinate answer in the only sense that is operational: when the particle is finally detected, it is detected at one place, as one dot. \emph{``How does it interfere?''} is a question about phase --- the value of the $S^1_\theta$ dial, $\varphi$. And the phase is not a location in space at all; it is a coordinate on the hidden gauge circle. The two questions have different homes, and the answer to one does not constrain the answer to the other.

\paragraph{What threads both slits is a field --- and a field at both slits is no paradox.} The wavefunction $\psi(x) = |\psi(x)|\,e^{i\varphi(x)}$ is, in this reading, the dial-value $\varphi$ \emph{laid over every available point-position} $x$: a section of the phase fiber across the whole apparatus, defined along the route that passes the left slit and along the route that passes the right slit alike. It is this \emph{phase field} that ``passes through both slits.'' But a field defined over an extended region is not in the least mysterious: the electromagnetic field of an ordinary light beam is also present at both slits at once, and no one is tempted to say a photon's field is ``in two places.'' The framework's only claim is that the quantum phase is a field of exactly this kind --- the winding of a real, if non-spatial, dial, laid over the point's possible locations. Nothing material is ever in two places. What is present at both slits is the phase section; what is present at the screen, at the end, is a single point.

\paragraph{Where the fringes come from: a phase ledger, not a map of a smeared object.} The interference pattern is the record of how the dial, having wound along the two routes, \emph{recombines} at each point of the screen. The route past the far slit is slightly longer, so along it the dial completes a slightly different number of turns --- this is exactly the de Broglie content of \S\ref{sec:DS-dial}, that $\lambda = h/p$ is the spatial period at which the phase winds. At a screen point where the two windings arrive back \emph{in step}, the dial-values add and points accumulate: a bright fringe. At a screen point where they arrive back \emph{exactly opposite}, the dial-values cancel, the combined amplitude is zero, and \emph{no point particle ever lands there}: a dark fringe. Equation~\eqref{eq:DS-intensity} is precisely this ledger --- the $\cos\bigl((k_y^+ - k_y^-)\,y\bigr)$ term is the relative dial-angle $\Delta\varphi$ of the two routes as a function of screen position $y$. The fringes are an accounting of relative phase, not a photograph of where a divided object was.

\paragraph{Why the point never has to ``decide.''} ``Which slit did the point pass through?'' demands a \emph{spatial} fact. But between source and screen the quantity that propagates and carries the structure is the phase field over both routes; the spatial position is made determinate only at detection. The point therefore never splits and never chooses, because the dial --- the thing that ``went through both'' --- was never a single place in space to begin with. There is no suppressed fact ``the particle was really on the left'' that interference would have to contradict: absent a position read-out, ``which slit'' has no boundary-registrable answer, not because the point is smeared in space but because what is physical in the interval is the non-spatial phase section. This is the deterministic-bulk reading of \S\ref{sec:DS-dial} stated with care: a definite bulk history exists, but ``which history'' is not a question the boundary can pose without altering the experiment.

\paragraph{Why looking destroys the effect: which-path is a scale-fiber read-out.} The picture earns its keep by explaining, mechanically, why any attempt to see which slit the point took abolishes the fringes. To read which-path is to read the point's \emph{position} at the slit. In the two-fiber language of \S\ref{sec:hidden-fibre-origin}, reading a position is a \emph{scale-fiber} operation: it couples the point to an environment and coarse-grains along the radial direction $I_r$, irreversibly. But coarse-graining along $I_r$ is exactly what destroys coherence on the \emph{phase} fiber $S^1_\theta$ (the decoherence content of \TA{5}; the rate is computed in \S\ref{sec:DS-environment}). With the phase coherence gone, the two dial-windings can no longer be recombined, the $\cos\Delta\varphi$ term averages to zero, and the screen shows two ordinary piles, one behind each slit. Interference and which-path are therefore not two answers competing over one object's whereabouts; they are the \emph{coherent} (phase-fiber) face and the \emph{read-out} (scale-fiber) face of the two non-spatial dimensions, and a single experiment cannot hold both sharp at once. Complementarity, in this framework, is not an extra postulate but the statement that one cannot simultaneously leave a coordinate unread (in order to interfere) and register it (in order to know the path).

\paragraph{The honest boundary: what this dissolves and what it does not.} Two claims must be kept apart, in the three-tier discipline of the cohort. What the phase-field picture \emph{dissolves}, completely, is the reader's puzzle --- ``how can a \emph{point} interfere with itself?'' --- the answer being that the thing which interferes was never the point but the phase section laid over its possible positions, and that section is a field, present at both slits as any field is. What the picture does \emph{not} settle by geometry alone is why each run yields exactly \emph{one} dot rather than a faint continuous smear of the predicted intensity: that single-outcome selection is \emph{delivered} by the irreversible \GRIP{} descent on the scale fiber (\GDOS{}/\ASix{}) rather than by the phase geometry; what remains open is only \GGtypicality{} (\S\ref{ssec:fibre-honest}). The geometry tells you that there is interference, and where the fringes fall; the dynamics tells you that you always see one particle. These are different statements, and the framework asserts only the first as forced; the second is the open dynamical content treated under \TA{5} and in \citep{ArisakaA5GDOS}. None of this alters a single numerical prediction of textbook quantum mechanics --- fringe spacing, visibility, and the which-path trade-off come out identical (Eqs.~\eqref{eq:DS-wavefunction}--\eqref{eq:DS-intensity}). What is new is the picture and the reason, not the numbers.

\paragraph{Synthesis.} Particle in space, wave in phase. At the moment of detection the point occupies one slit's worth of space; in the interval between source and screen it occupies no slit at all in the sense the question intends --- while the phase dial winds through both routes, always. The bright and dark bands are the ledger of that double winding, read out statistically because the boundary can see only the squared dial. Put this way, there is nothing left in the double slit that must be pictured as a single point in two places, because in this framework nothing ever is.

\section{Setup: the wavefunction at the screen}
\label{sec:DS-setup}

Consider a particle of mass $m$ prepared in an incoming plane wave with momentum $p$, incident normally on a screen pierced by two slits at positions $y = \pm d/2$ in the screen plane. Diffraction at each slit produces two outgoing spherical-wave components. At a screen-position $y$ on the downstream detector, distance $L$ from the slits, the standard textbook analysis \citep{FeynmanLectures} gives the wavefunction
\begin{equation}\label{eq:DS-wavefunction}
\psi(y) \;=\; \frac{1}{\sqrt 2}\bigl(\psi_+(y) + \psi_-(y)\bigr),
\qquad
\psi_\pm(y) \;=\; A\,e^{i k_y^\pm y},
\end{equation}
with $k_y^\pm$ the transverse wavevector picked up by passage through slit $\pm$. The intensity (single-particle probability density) is
\begin{equation}\label{eq:DS-intensity}
|\psi(y)|^2 \;=\; |A|^2\bigl[1 + \cos((k_y^+ - k_y^-)\,y)\bigr] \;=\; 2|A|^2\,\cos^2\!\Bigl(\tfrac{1}{2}(k_y^+ - k_y^-)\,y\Bigr),
\end{equation}
the textbook interference fringe pattern with fringe spacing $\Delta y = 2\pi/|k_y^+ - k_y^-| = \lambda L / d$ for de Broglie wavelength $\lambda = 2\pi/p$.


\section{The which-path environment and the decoherence factor}
\label{sec:DS-environment}

Now couple the particle to an environment that can register which slit it passed through. The relevant entangled state at the screen is
\begin{equation}\label{eq:DS-entangled}
|\Psi\rangle_{\rm particle\otimes env} \;=\; \frac{1}{\sqrt 2}\,\bigl(\psi_+\otimes|E_+\rangle + \psi_-\otimes|E_-\rangle\bigr),
\end{equation}
where $|E_\pm\rangle$ are the environmental states corresponding to the particle's passage through slit $\pm$. The boundary density operator for the particle alone, obtained by tracing over the environment, is
\begin{equation}\label{eq:DS-density}
\rho_{\rm particle}(y, y') \;=\; \tfrac{1}{2}\bigl[\psi_+(y)\psi_+^*(y') + \psi_-(y)\psi_-^*(y') + \mathcal D\bigl(\psi_+(y)\psi_-^*(y') + \psi_-(y)\psi_+^*(y')\bigr)\bigr],
\end{equation}
with the \emph{decoherence factor} $\mathcal D = \langle E_+|E_-\rangle$. Two limits matter:
\begin{itemize}[leftmargin=2em,itemsep=0.2em]
\item $\mathcal D = 1$ (environments fully indistinguishable): interference fringe pattern preserved.
\item $\mathcal D = 0$ (environments orthogonal): off-diagonal terms vanish, leaving the classical incoherent sum $|\psi_+|^2 + |\psi_-|^2$. No fringes.
\end{itemize}
The interference visibility $V = |\mathcal D|$ on the screen interpolates continuously between the two limits as a function of how strongly the environment couples to the which-path information.

By \TA{5}, the decoherence factor decays exponentially in time when the environment is a thermal bath: $|\mathcal D(t)| = \exp(-\Gamma_{+-}\,t)$, with the rate
\begin{equation}\label{eq:DS-rate}
\Gamma_{+-} \;=\; \sum_\gamma \gamma_\gamma\,|\langle +|L_\gamma|+\rangle - \langle -|L_\gamma|-\rangle|^2,
\end{equation}
where the bulk GKLS rates $\gamma_\gamma$ are themselves fixed by scale-overlap integrals (\TA{5}.2) of bulk KK modes onto the particle's zero mode.


\section{Three regimes: photon, electron, fullerene}
\label{sec:DS-three-regimes}

We compute $\Gamma_{+-}$ in three regimes that demonstrate the macroscopic--mesoscopic--microscopic boundary.

\paragraph{Photon in a Mach--Zehnder interferometer (microscopic).}
A single photon at $\lambda = 633$~nm in a high-vacuum Mach--Zehnder interferometer with arm length $L \sim 1$~m and ambient gas density $n \sim 10^{10}$~molecules/m$^3$ has a scattering rate per molecule $\sim 10^{-13}$~s$^{-1}$, giving total $\Gamma \sim 10^{-3}~\mathrm{s}^{-1}$ and decoherence time $\tau_{\rm dec} \sim 10^3$~s. Interference is preserved for the full transit; high-visibility fringes are observed routinely in laboratory single-photon interferometry.

\paragraph{Fullerene $C_{60}$ at room temperature (mesoscopic).}
Arndt et al.~\citep{Arndt2003_C60} performed Talbot--Lau interferometry with fullerene molecules at $T \sim 900$~K and ambient pressure varied by six orders of magnitude. Their measured decoherence rate scales with both the gas-collision rate and the absorbed-photon rate, in quantitative agreement with the Joos--Zeh prediction $\Gamma\sim n\,\sigma\,v_{\rm rel}\,(\Delta x)^2/\lambda_{\rm dB}^2$. At $10^{-9}$~mbar, decoherence times are in the millisecond range; at $10^{-6}$~mbar, the interference fringes disappear within microseconds.

\paragraph{Schr\"odinger's cat in a macroscopic position superposition (macroscopic).}
A cat-mass object ($m \sim 1$~kg) in a position superposition $\Delta x \sim 0.1$~m at room temperature has decoherence rate \citep{JoosZeh1985}
\begin{equation}\label{eq:DS-cat-rate}
\Gamma_{\rm cat} \;\sim\; n_{\rm air}\,\sigma_{\rm N_2}\,v_{\rm therm}\,(\Delta x)^2 / \lambda_{\rm thermal}^2 \;\sim\; 10^{40}\,\mathrm{s}^{-1},
\end{equation}
giving decoherence time $\tau_{\rm dec}\sim 10^{-40}$~s. No apparatus can resolve a time scale forty orders of magnitude shorter than the Planck time. The interference pattern has no chance of being observed; the cat appears classical.


\section{The boundary between quantum and classical}
\label{sec:DS-boundary}

What \TA{5} and the cohort framework together provide is the structural identification of where the quantum-to-classical boundary lies in any given experiment: it lies at the value of $\Delta x$ (the spatial separation between the two interference paths) for which $\Gamma_{+-}\,\tau_{\rm exp}\sim 1$, where $\tau_{\rm exp}$ is the experimental flight time. Substituting $\Gamma \propto (\Delta x)^2$ from \eqref{eq:DS-cat-rate}, the critical separation scales as
\begin{equation}\label{eq:DS-boundary-scale}
\Delta x_{\rm crit} \;\sim\; \frac{1}{\sqrt{n\,\sigma\,v\,\tau_{\rm exp}}/\lambda_{\rm dB}}.
\end{equation}
For a typical laboratory environment ($n\sim 10^{20}$~molecules/cm$^3$, $\sigma\sim 10^{-15}$~cm$^2$, $v\sim 5\times 10^4$~cm/s, $\tau_{\rm exp}\sim 1$~s, $\lambda_{\rm dB}\sim 10^{-10}$~cm), one gets $\Delta x_{\rm crit}\sim 10^{-5}$~m, the micron scale --- exactly where the quantum-to-classical transition is observed empirically.

This is the substantive content of \TA{5} for laboratory practice: the boundary is not where the experimenter chooses to draw it, but where the bulk-environment coupling integrates to $\Gamma\,\tau\sim 1$. In the cohort framework, that integral is a calculation from the 6D geometry, not a free parameter.

\subsection*{Part-end close}

The double-slit experiment has been performed in every laboratory that takes quantum mechanics seriously, with every kind of particle from photons to fullerenes. Every time the interference pattern is preserved, the experimenter has succeeded in shielding the particle from a which-path-resolving environment; every time the pattern is lost, the environment has won. Where the boundary lies in any specific experiment is fixed by the scale-overlap integrals of the bulk geometry --- not by the experimenter's choice of Heisenberg cut, not by a phenomenological fit, but by a calculation that the cohort framework makes inevitable.

After all, the question Young could not have answered in 1801 --- ``why do the fringes appear, and what makes them sometimes disappear'' --- now has an answer one level deeper than the wave function. The fringes appear because a single photon propagates deterministically through the six-dimensional geometry, its bulk field passing through both slits as a single-valued configuration with a definite relative phase. They build up one photon at a time because each photon's landing on the screen is fixed by its own $S^1_\theta$ phase angle, which the four-dimensional observer cannot read but which is uniformly distributed across the photon ensemble. They disappear when an environmental coupling destroys the bulk phase relation between the two paths, at a rate the bulk geometry fixes without any tuning. The wave aspect lives in the deeper geometry; the particle aspect lives in the four-dimensional detector; neither is fundamental, and neither contradicts the other. In the runtime vocabulary of Section~\ref{sec:grip-spine}, the division is exact: the interference is \OGGEP{}-side structure --- the projection's phase, present before any descent --- while each landing is a \GRIPin{} capture, the smallest measurement event in physics, halting at one element of the screen's discrete attractor lattice. The fringes are the wave's bookkeeping; the grains that build them, one photon at a time, are \GRIP{}'s.

%% ===================================================================
%% PART VII (7/14) -- Worked example III --- EPR, Bell, and the CHSH inequality
%% ===================================================================
\paragraph{Problems and worked solutions.}

\medskip\noindent\textbf{VI.1. Compute a real fringe spacing for
electrons.}

\noindent\emph{Solution.}  Step 1: electrons at $600$~eV have
de Broglie wavelength $\lambda = 1.226~\mathrm{nm}/\sqrt{V} =
0.050$~nm.  Step 2: with slit separation $d = 1~\mu$m and screen
distance $L = 1$~m, the fringe spacing of the intensity pattern
\eqref{eq:DS-intensity} is $\Delta y = \lambda L/d = 50~\mu$m ---
comfortably resolvable, which is why the J\"onsson-class
experiments could photograph it.  Step 3: scale awareness: the
pattern's existence needs $\lambda$ only \emph{comparable to} the
apparatus coherence scales, not to $d$ itself --- the wavelength is
$10^4$ times smaller than the slit separation and the fringes
survive; what kills them is never geometry but \emph{records}, as
the next problems quantify.  Physics intuition: a fringe is a ruler
graduated in wavelengths, magnified by $L/d$ --- the double slit is
an interferometric microscope that blows a picometer-scale phase
difference up to the width of a human hair.

\medskip\noindent\textbf{VI.2. Relate which-path knowledge to
fringe visibility.}

\noindent\emph{Solution.}  Step 1: couple the particle to any
environment that becomes path-dependent, $\ket{L}\ket{E_L} +
\ket{R}\ket{E_R}$ \eqref{eq:DS-entangled}; tracing the environment
leaves the interference term multiplied by
$|\braket{E_L}{E_R}|$ \eqref{eq:DS-density}.  Step 2: the two
limits: $\braket{E_L}{E_R} = 1$ (no record) gives full fringes;
orthogonal records give none --- and one scattered photon of
wavelength shorter than the slit separation is already nearly
orthogonal: a single quantum of which-path light erases the
pattern.  Step 3: the quantitative bridge is the chapter's
decoherence-rate formula \eqref{eq:DS-rate}: visibility decays
exponentially in the number of path-tagging events, so the
quantum-classical transition is a \emph{rate} question, not a
size question.  Physics intuition: nature runs a strict
information economy --- fringes are paid for in ignorance, and
every bit of path information anywhere in the universe, read or
unread, debits the visibility account by exactly the
Cauchy--Schwarz overlap.

\medskip\noindent\textbf{VI.3. Evaluate the fullerene wavelength,
and respect what it means.}

\noindent\emph{Solution.}  Step 1: a C$_{70}$ molecule ($m = 840$~u)
at $v = 200$~m/s has $\lambda = h/mv = 2.4$~pm --- five orders of
magnitude smaller than the molecule itself.  Step 2: interference
was nevertheless observed (the Vienna experiments): a thousand-atom
object, hotter than room temperature, wrote fringes determined by a
wavelength far smaller than its own bonds --- quantum mechanics does
not switch off with complexity.  Step 3: read the regime table of
this Part: photon, electron, fullerene differ not in the
\emph{rules} but in how easily the environment acquires their path
--- mass buys shorter wavelengths \emph{and} stronger coupling to
everything, which is why the fullerene is the frontier and the cat
(next Part) is beyond it by forty orders of magnitude.  Physics intuition:
 the de Broglie wavelength of a flying molecule is
smaller than its atoms --- and yet it interferes; the boundary of
the quantum world is drawn by decoherence bookkeeping, never by
size, weight, or dignity.

\medskip\noindent\textbf{VI.4. Explain where the fullerene fringes
die.}

\noindent\emph{Solution.}  Step 1: heat the molecule and it becomes
its own which-path lamp: internal thermal photons carry position
information to the field, and each emission multiplies the
visibility by an overlap factor $< 1$ (VI.2's mechanism, source
internalized).  Step 2: the 2004 experiment swept the internal
temperature and watched visibility fall continuously from
$\sim\!1500$~K and vanish by $\sim\!3000$~K --- decoherence
observed as a dial, not a postulate, exactly as the rate law
\eqref{eq:DS-rate} predicts.  Step 3: the framework's addition
(\eqref{eq:DS-boundary-scale}): the same bookkeeping descends from
the bulk GKLS rates, so the measured visibility curve is, on this
reading, a scale-overlap measurement --- one of the places where
the eight-test horizon of the Discussions touches an existing
dataset.  Physics intuition: a hot molecule gossips about itself
--- every infrared photon it sheds is a small confession of where
it is, and the fringes fade in proportion to the gossip; the
quantum-classical boundary is not a wall but a signal-to-noise
ratio.

\medskip\noindent\textbf{VI.5. State the point-and-dial reading, and
say what it does and does not change.}

\noindent\emph{Solution.}  Step 1: in the bulk reading the particle
is a point in space carrying a dial in phase: the point traverses
\emph{one} slit; the dial's field, living on the fiber, traverses
both and interferes \eqref{eq:DS-dial-born}.  Step 2: verify
operational equivalence: every observable statistic --- fringe
positions, visibilities, which-path trade-offs --- is computed from
the same wavefunction arithmetic as the textbook's; the reading
changes the \emph{referent} (what the math is about), not the
numbers.  Step 3: then say where it is not idle: the reading
removes the logical need for wave-particle ``duality'' as a
primitive (one object, two aspects, two strata), and its
falsifiable content lives in the framework's global horizon
(the kill criteria of the Discussions), not in re-fitting fringes
--- the honesty register of the course keeps those books separate.
Physics intuition: ``both slits'' was always a statement about the
\emph{phase}, never about the \emph{point} --- splitting the
particle into position and dial dissolves Feynman's mystery into
geography: the two behaviors were two different residents of the
particle all along.


\GGtakeaway{\emph{Standard account:} single quanta build an
interference pattern; observe the path and the pattern dies;
``nobody knows how it can be like that.'' \emph{What it left open:}
how one indivisible particle samples two slits, and where exactly
the classical world begins. \emph{The bulk reading:} the point
particle passes through one slit while its phase field --- a
physical object on the fiber --- passes through both; which-path
coupling decoheres the fiber and kills the fringes on schedule
(photon, electron, fullerene: the same formula, three regimes); the
quantum--classical boundary is a computable scale window, not a
mystery line.}

\GGPart[Worked example III: EPR and Bell]{Worked example III: EPR, Bell, and CHSH}{The 2022-Nobel-recognized test of Bell's theorem}
\label{part:VII}

\GGteachboxpart{The empirical heart of the framework, and of the
2022 Nobel Prize: entanglement, EPR, Bell, and CHSH worked as one
chain. The chapter builds the joint bulk state of the pair across
both wings, evaluates the singlet correlation and the CHSH
expression, reads the Tsirelson saturation $2\sqrt2$ as an extremal
bound of the projection (\TA{8}), verifies no-signaling (\TA{11}),
and closes by reconciling the four historical readings --- Einstein's
locality, Bohr's completeness, Bell's exclusion, and the loophole-free
experiments --- in the one sentence the century could not write:
local and deterministic in 6D, non-local and irreducibly statistical
on the 4D boundary, with no contradiction because the two statements
live at different strata of the same geometry.}

\GGreadalong{\citet{EPR1935} and \citet{Bell1964} for the question
and the answer; \citet{Aspect1982} for the decisive experiments;
\citet{NobelPhysics2022} for what the prize certified.}

This worked example is the empirical heart of the framework's reading of quantum mechanics. The CHSH bound $|S|\le 2$ on local hidden-variable theories in 4D has been violated experimentally to many standard deviations, with the measured value at or near the Tsirelson saturation $|S|=2\sqrt{2}$ \citep{Aspect1982, Hensen2015, Giustina2015, Shalm2015, NobelPhysics2022}. The cohort framework reads the violation as the most direct empirical signature of the deterministic six-dimensional bulk: each entangled pair is a single bulk field configuration with two 4D-spatially-separated branches that share a single $S^1_\theta$ holonomy reference; the two 4D-spacelike-separated detector events are projections of that one non-separable joint state along the radial direction $I_r$ (which, by Theorem~0 of \ATwo{}, carries no light cone), not endpoints of any worldline between them; the Tsirelson saturation is the extremal (Bogomolnyi-analogue) ceiling on the CHSH functional at the singlet. We present the six-dimensional bulk reading first in \S\ref{sec:EPR-6d-reading}, perform the explicit CHSH calculation on the singlet state in \S\ref{sec:EPR-singlet}, recover the Tsirelson saturation from the joint-state picture (\TA{7}) in \S\ref{sec:EPR-bulk}, promote it to the topological bound (\TA{8}) in \S\ref{sec:EPR-bogomolnyi}, derive no-signaling (\TA{11}) in \S\ref{sec:EPR-no-signal}, and reconcile with the four standard interpretations in \S\ref{sec:EPR-four-readings}.


\noindent The EPR/Bell/CHSH worldline picture is shown in \figref{fig:epr-bell-chsh}.

\begin{figure}[H]
\centering
\resizebox{0.660\textwidth}{!}{%
\usebox{\GGscBell}}
\caption[EPR/Bell/CHSH worldline picture and the 2022-Nobel-recognized Tsirelson saturation.]{The CHSH version of Bell's theorem and its cohort
reading. Top: the entangled-pair coincidence-counting setup
\citep{Aspect1982, Hensen2015}, with source $S$, analyzers
$a/a'$ on the left and $b/b'$ on the right, four single-photon
detectors, and a coincidence counter recording the joint
outcomes. Bottom: the CHSH bound $|S|\le 2$ holds in any local
hidden-variable theory (dashed line); standard QM predicts
$|S| = 2\sqrt{2}|\cos(2\phi)|$ (blue curve) where $\phi$ is the
relative analyzer angle; the maximum reaches the Tsirelson
bound $|S| = 2\sqrt{2}$ (red horizontal) at $\phi = 0$. The
2022 Nobel-recognized experiments confirm the saturation;
\TA{8} reads the saturation as an extremal (Bogomolnyi-analogue)
ceiling on the CHSH functional that the boundary CHSH
measurement shadows.}
\label{fig:epr-bell-chsh}
\end{figure}


%% PHASE-4D CONTENT BEGINS
\section{The 1935 EPR question and the 1964 Bell answer}
\label{sec:EPR-historical}

In May 1935 Einstein, Podolsky, and Rosen \citep{EPR1935} argued that the quantum-mechanical description of an entangled bipartite system could not be a complete description of physical reality. Their argument used the spin-singlet state of two spin-$\tfrac{1}{2}$ particles separated to spacelike-disconnected regions, observing that a measurement of $\sigma_z$ on the first particle fixes the second's $\sigma_z$ value instantaneously, while a measurement of $\sigma_x$ on the first fixes the second's $\sigma_x$ value instantaneously --- both ``elements of reality'' for the second particle, but mutually incompatible by the uncertainty principle. Bohr's 1935 response \citep{Bohr1935} insisted on the inseparability of the joint quantum state; the dispute was philosophical, with no operational test in sight for two decades.

Bell's November 1964 paper \citep{Bell1964} converted the EPR question into an experimental test. He proved that any local hidden-variable completion of quantum mechanics --- any theory in which each particle carries pre-existing values for all its observables, with the values fixed locally at the source and unaffected by measurement settings at the other detector --- must satisfy an inequality on certain linear combinations of spin-correlation expectation values. Quantum mechanics, by contrast, predicts violations of the inequality. The Clauser--Horne--Shimony--Holt (CHSH) reformulation of 1969 \citep{CHSH1969} packaged the bound into a four-setting quantity $S$ with classical bound $|S|\le 2$.

Freedman--Clauser 1972 \citep{FreedmanClauser1972} demonstrated the first experimental violation. Aspect 1981--1982 \citep{Aspect1981, Aspect1982, Aspect1982a, Aspect1982b} pushed the violation past five standard deviations with delay-time-randomized detector settings. The Delft/Vienna/NIST loophole-free runs of 2015 \citep{Hensen2015, Giustina2015, Shalm2015} closed the remaining locality and detection-efficiency loopholes. The 2022 Nobel Prize \citep{NobelPhysics2022} acknowledged the result formally. All these experiments measure values of $|S|$ at or near the Tsirelson bound $2\sqrt 2 \approx 2.828$ \citep{Tsirelson1980}.

The framework reading agrees with all of this experimentally and adds the bulk-geometric layer: the boundary correlation function is the operator-algebra expectation in one non-separable joint bulk state (\TA{7}), and the Tsirelson saturation is the extremal (Bogomolnyi-analogue) ceiling on the CHSH functional (\TA{8}).




\section{The six-dimensional bulk reading: a single entangled configuration across spacelike-separated 4D detectors}
\label{sec:EPR-6d-reading}

Before turning to the standard textbook calculation, we pause to state the
deeper picture that the cohort framework brings to the EPR/Bell/CHSH
experiment. The standard treatment in the language EPR, Bell, and CHSH
would have recognized follows in \S\ref{sec:EPR-singlet}--%
\S\ref{sec:EPR-no-signal}; placing the bulk reading first lets the reader
see the deeper 6D-locality structure before the formal 4D calculation.

\paragraph{The entangled pair in six dimensions.}
A singlet pair --- two spin-$\tfrac{1}{2}$ particles, or two
polarization-entangled photons from parametric down-conversion --- is, in
\GGtheory{}, a single bulk field configuration with two 4D-spatially
separated branches. The two particles emerged from one bulk event (the
parametric crystal, the atomic cascade, the BBO down-conversion vertex),
and their bulk-side state inherits a single shared Wilson-line holonomy
reference around the topological circle $S^1_\theta$. There is no
superposition of ``Alice's particle is up and Bob's is down'' or vice
versa in the bulk; there is a single 6D bulk configuration whose 4D
projection has correlated spin/polarization values at the two detectors.

\paragraph{The shared $S^1_\theta$ holonomy reference.}
The defining feature of an entangled pair in the bulk picture is that
the two particles, as one non-separable bulk state, share a single $S^1_\theta$ holonomy
phase $\Theta_0$ that was set at the source event:
\begin{equation}\label{eq:EPR-shared-holonomy}
 \Theta_0^{(A)} \;=\; \Theta_0^{(B)} \;=\; \Theta_0
 \quad\text{(fixed at source emission).}
\end{equation}
Even though the two particles separate 4D-spatially and travel to
spacelike-separated detectors, their bulk-side holonomy phases remain
locked to the single reference $\Theta_0$. This is not ``spooky action
at a distance''; it is the structural fact that both particles are
shadows of a single non-separable bulk state, sharing the one
$S^1_\theta$ holonomy class fixed at the source --- a common origin,
not an influence relayed between the wings.

\paragraph{What the 4D observer cannot register.}
The 4D observer can read neither $\Theta_0$ nor the non-separable bulk
state that underlies the correlation. Each pair has its own random $\Theta_0$ drawn
uniformly from $S^1_\theta$ across the ensemble of pairs; the 4D
observer registers only the 4D outcomes at Alice's site $A$ and Bob's
site $B$, which are 4D-spatial projections of the bulk state with the
bulk coordinates $r$ and $\theta$ averaged out.

\paragraph{The two analyzers as 4D-spatial obstacles.}
Alice's polarizing analyzer (oriented along direction $\hat a$) and
Bob's analyzer (along $\hat b$) are 4D-spatial filters that couple to
the particle's 4D polarization. When the bulk field configuration
encounters Alice's analyzer, the 4D projection registers $\pm 1$
according to the particle's 4D spin component along $\hat a$; the same
happens at Bob's site for direction $\hat b$. The two analyzers are
4D-spatially separated; in 4D they are causally disconnected, and any
4D-local hidden-variable theory predicts $|S| \le 2$ for the CHSH
quantity.

\paragraph{The pair is a single non-separable bulk state.}
The crucial step: in 6D the two particles are not two independently
prepared systems but one non-separable state of the bulk, sharing the
phase fiber $S^1_\theta$ fixed at the source. Using the 6D metric
of \citet{ArisakaA2GG6}
\begin{equation}\label{eq:EPR-metric-here}
 ds_6^2 \;=\; e^{-2r}\,\eta_{\mu\nu}\,dx^\mu\,dx^\nu \;+\; dr^2 \;+\; d\theta^2,
\end{equation}
the radial direction $I_r$ enters with $+dr^2$: by Theorem~0 of
\citet{ArisakaA2GG6} it is the non-compact renormalization-scale fiber
and carries \emph{no} light cone, so a 4D-spacelike separation stays
spacelike in the bulk for every $\Delta r$ --- there is no radial
``dip'' that connects the detector events causally, and none is needed.
What links the two shadows is not a worldline between them but the
single joint state they are projections of: the boundary correlation is
the operator-algebra expectation in that one state (\TA{1}, \TA{3}),
and $I_r$ is the direction \emph{along which} it projects to the
boundary, not a channel \emph{through} which anything travels. The
Bell--Aspect ``non-locality'' at 4D is therefore the shadow of one bulk
object seen as two, through a non-invertible bulk-to-boundary
projection --- 4D-non-local, but with no signal and no superluminal
influence.

\paragraph{What the coincidence counter registers.}
For each pair, the 4D coincidence counter records the joint outcome
$(s_A, s_B) \in \{\pm 1\}^2$ at the two analyzer settings. The
bulk-side calculation of the joint expectation, using the boundary
representation of \TA{7}, gives the singlet correlation
\begin{equation}\label{eq:EPR-singlet-corr-here}
 C(\hat a, \hat b) \;=\; \langle\Psi^-|(\hat a\cdot\vec\sigma_A)(\hat b\cdot\vec\sigma_B)|\Psi^-\rangle
 \;=\; -\hat a\cdot\hat b
 \;=\; -\cos(\theta_a - \theta_b),
\end{equation}
which is the standard operator-algebra expectation in the
non-separable joint state (\TA{1}, \TA{3}), with both shadows referred
to the single shared holonomy $\Theta_0$ fixed at the source. The two
outcomes are not independent; they are 4D shadows of one 6D bulk state,
with their joint statistics fixed by that state.

\paragraph{Why the CHSH bound is violated even pair-by-pair.}
For any individual pair, the bulk dynamics fix the 4D outcomes at both
detectors deterministically (once the random $\Theta_0$ was drawn from
the source). The CHSH statistic
\begin{equation}\label{eq:EPR-CHSH-here}
 S \;=\; C(\hat a, \hat b) + C(\hat a, \hat b') + C(\hat a', \hat b) - C(\hat a', \hat b'),
\end{equation}
aggregated over many pairs, takes the value $2\sqrt 2$ at the optimal
angle choices, saturating the Tsirelson bound. This is not a
statistical correlation built up from probabilistic individual events;
each pair has a definite 4D joint outcome fixed by the shared bulk
holonomy $\Theta_0$. The aggregate CHSH statistic exceeds the Bell
bound $|S| \le 2$ because the individual pair correlations
$C(\hat a, \hat b) = -\cos(\theta_a - \theta_b)$ are deeper than any
4D-local hidden-variable theory permits --- precisely because the pair
is one non-separable bulk state, 4D-non-local in its shadow while
carrying no signal.

\paragraph{Extremal (Bogomolnyi-analogue) saturation of the Tsirelson bound.}
The empirical fact that experiments saturate the Tsirelson bound to
within measurement error is an \emph{extremal} property of the singlet
(\TA{8}): it is the maximizer of the CHSH functional, so the first
variation of $|S|$ vanishes there and a small change in the preparation
(entanglement strength, depolarization) moves $|S|$ only at second
order. Saturation is therefore structurally stable --- in analogy with
(though not literally) a Bogomolnyi/BPS bound, where a configuration at
a saturating bound is stable to first order \citep{Bogomolny1976}; no
topological charge is claimed. Every singlet preparation in any
laboratory, regardless of source brightness or detector efficiency,
sits at this same extremum --- and every loophole-free CHSH measurement
on a singlet attains the same $|S| = 2\sqrt 2$.

\paragraph{Bottom line of the bulk reading.}
The CHSH violation is the deterministic shadow of one non-separable 6D
state. That single joint state carries the singlet correlation
\eqref{eq:EPR-singlet-corr-here}; the Tsirelson saturation is the
extremal first-order-stationary ceiling on quantum correlations
(\TA{8}); the apparent 4D non-locality is the projection of one bulk
object through a non-invertible bulk-to-boundary map. The 4D observer cannot read the
bulk holonomy reference $\Theta_0$ pair-by-pair, but they can read the
correlation pattern; the pattern reveals the bulk-side structure to
within the operational scope of CHSH measurement. Bell's inequality
shows the lid that 4D-locality places on correlations; Aspect's
experiments show that nature breaches the 4D lid by exactly the amount
a single non-separable bulk state permits; and the framework reads this
as the most direct empirical signature of the 6D bulk's structural
primacy over the 4D shadow.

\noindent The 6D-bulk-to-4D-boundary EPR mechanism is shown in \figref{fig:EPR-6d-mechanism}.

\begin{figure}[H]
\centering
\resizebox{0.839\textwidth}{!}{%
\usebox{\GGscEPR}}
\caption[EPR/Bell mechanism: one non-separable joint bulk state seen as two boundary shadows.]{The 6D-bulk reading of an EPR/Bell singlet pair. (a) 4D
Minkowski view: Alice's and Bob's measurement events $A$ and $B$ are
spacelike-separated; any 4D-local hidden-variable theory predicts
$|S| \le 2$ for the CHSH quantity, in contradiction with experiment.
(b) 6D bulk view: the pair is \emph{one} non-separable state of the
bulk, sharing a single $S^1_\theta$ holonomy reference $\Theta_0$ set at
the source; the two events $A$ and $B$ are its projections along the
renormalization-scale direction $I_r$, which (Theorem~0 of \ATwo{})
carries no light cone, so no worldline runs between them. This shared
holonomy is the operational meaning of the entanglement. The
Bell--Aspect ``non-locality'' at 4D is the operational shadow of one
bulk object projected to two boundary records.}
\label{fig:EPR-6d-mechanism}
\end{figure}


\section{The singlet correlation and the CHSH expression}
\label{sec:EPR-singlet}

Consider two spin-$\tfrac{1}{2}$ particles prepared at a source in the spin-singlet state
\begin{equation}\label{eq:EPR-singlet}
|\Psi^-\rangle_{AB} \;=\; \tfrac{1}{\sqrt 2}\bigl(|\!\!\up\rangle_A\,|\!\!\dn\rangle_B \;-\; |\!\!\dn\rangle_A\,|\!\!\up\rangle_B\bigr),
\end{equation}
and sent to spacelike-separated detectors operated by Alice (at site $A$) and Bob (at site $B$). Alice measures $\hat a\cdot\vec\sigma_A$ along direction $\hat a$; Bob measures $\hat b\cdot\vec\sigma_B$ along direction $\hat b$. The standard quantum-mechanical correlation \citep{NielsenChuang2010} is
\begin{equation}\label{eq:EPR-singlet-corr}
C(\hat a, \hat b) \;=\; \langle\Psi^-|(\hat a\cdot\vec\sigma_A)(\hat b\cdot\vec\sigma_B)|\Psi^-\rangle \;=\; -\hat a\cdot\hat b.
\end{equation}

The CHSH quantity uses four measurement settings: Alice picks $\hat a$ or $\hat a'$, Bob picks $\hat b$ or $\hat b'$, with outcomes $\pm 1$. Define
\begin{equation}\label{eq:EPR-S}
S \;\equiv\; C(\hat a, \hat b) + C(\hat a, \hat b') + C(\hat a', \hat b) - C(\hat a', \hat b').
\end{equation}
For local hidden-variable theories \citep{Bell1964, CHSH1969}, $|S|\le 2$. For quantum-mechanical singlet correlations \eqref{eq:EPR-singlet-corr}, the maximum of $|S|$ over angle choices is achieved at the standard setting $\hat a = \hat z, \hat a' = \hat x, \hat b = (\hat z+\hat x)/\sqrt 2, \hat b' = (\hat z-\hat x)/\sqrt 2$, giving each correlation function the value $-1/\sqrt 2$:
\begin{equation}\label{eq:EPR-Tsirelson}
\boxed{\;
|S|_{\rm Tsirelson} \;=\; \biggl|\,4\cdot\Bigl(-\frac{1}{\sqrt 2}\Bigr)\,\biggr| \;=\; \frac{4}{\sqrt 2} \;=\; 2\sqrt 2 \;\approx\; 2.828.
\;}
\end{equation}
This is the Tsirelson bound \citep{Tsirelson1980}: the maximum CHSH violation that any quantum theory permits.


\section{The joint-bulk-state picture (\TA{7})}
\label{sec:EPR-bulk}

The framework reading of \eqref{eq:EPR-Tsirelson} proceeds in two steps. First, the boundary correlation function \eqref{eq:EPR-singlet-corr} is the standard operator-algebra expectation in the entangled state, $C(\hat a, \hat b) = \Tr_\Bdy[(\hat a\cdot\vec\sigma_A)(\hat b\cdot\vec\sigma_B)\rho_\Bdy^{(AB)}]$, which the framework reads as $\Tr_\Bulk[\hat O_A^{\rm bulk}\hat O_B^{\rm bulk}\,\rho_\Bulk]$ in the single non-separable bulk state $\rho_\Bulk$ of the pair.

The bulk metric of \citet{ArisakaA2GG6} is, as in Part~II,
\begin{equation}\label{eq:EPR-metric}
ds_6^2 \;=\; e^{-2r}\,\eta_{\mu\nu}\,dx^\mu\,dx^\nu \;+\; dr^2 \;+\; d\theta^2.
\end{equation}
By Theorem~0 of \citet{ArisakaA2GG6} the radial direction enters with $+dr^2$ as the non-compact renormalization-scale fiber $I_r$, carrying no light cone. For two 4D-boundary events $A = (x_A, 0, 0)$ and $B = (x_B, 0, 0)$ on the UV boundary (the $r = 0$ surface) with 4D-spacelike separation $|\Delta x|^2 = -\eta_{\mu\nu}\Delta x^\mu\Delta x^\nu > 0$, the 6D interval $ds_6^2 = -e^{-2r}|\Delta x|^2 + (\Delta r)^2$ never becomes timelike for a scale-suppressed boundary separation at laboratory scale; there is no radial ``dip'' that connects $A$ and $B$ causally, and none is needed. They are two projections of one non-separable bulk state along $I_r$.

The correlation \eqref{eq:EPR-singlet-corr} is therefore the expectation in that single joint state --- the same number at every laboratory separation (Aspect at $\sim 12$~m, Delft at $\sim 1.3$~km, Vienna at $\sim 144$~km between detectors). The local-hidden-variable assumption that Bell uses --- that each measurement event has a value fixed at the source independent of the other detector's setting --- fails not because a signal crosses the gap but because the joint state is irreducibly non-separable: there are no separate per-event values to begin with.


\section{Tsirelson saturation as an extremal bound (\TA{8})}
\label{sec:EPR-bogomolnyi}

The empirical fact that experiments saturate the Tsirelson bound to
within measurement error is the content of \TA{8}. We unpack the extremal (Bogomolnyi-analogue) argument here.

The singlet is the maximizer of the CHSH functional $|S|(\psi,\{\hat a,\hat a',\hat b,\hat b'\})$ over states and settings; at that maximum the first variation vanishes, $\delta|S| = 0$, so perturbing the singlet preparation by a small change in the entangled state's coefficients changes $|S|$ only at second order. This is the formal analogue of a Bogomolnyi/BPS saturation \citep{Bogomolny1976} --- a configuration sitting at a saturating bound, stable to first order --- though it is an analogy only: no topological charge or central-charge structure is claimed.

The result: every singlet preparation, regardless of laboratory specifics, sits at this same extremum, and every CHSH measurement on such a state attains the Tsirelson saturation \eqref{eq:EPR-Tsirelson}. Non-maximally-entangled states sit away from the extremum, with correlation functions producing $|S| < 2\sqrt 2$.

This is the reason for the empirical universality of the Tsirelson saturation: not a coincidence of the specific singlet state, but the first-order stationarity of the CHSH functional at its maximizer. The saturation value itself is the standard quantum ceiling, consistent with textbook quantum mechanics.


\section{No-signaling (\TA{11})}
\label{sec:EPR-no-signal}

The shared non-separable state underlies the singlet correlation, but it does not transmit information. We give the operational proof here as a worked use of \TA{11}.

Let Bob measure $\hat O_B = \mathbf 1_A\otimes\sigma^z_B$ (a fixed setting) while Alice randomly chooses between two measurement settings $M_A^{(0)} = \sum_a E_a^{A,(0)}\otimes\mathbf 1_B$ and $M_A^{(1)} = \sum_a E_a^{A,(1)}\otimes\mathbf 1_B$. Define Bob's marginal expectation as a function of Alice's choice:
\begin{equation}\label{eq:EPR-no-signal-marginal}
\langle\hat O_B\rangle^{(s)} \;=\; \sum_a p(a, s)\,\langle\hat O_B\rangle_{\rho_\Bdy^{(B|a,s)}}, \qquad s \in \{0, 1\}.
\end{equation}
By the calculation of \TA{11} (the POVM completeness relation $\sum_a E_a^{A,(s)} = \mathbf 1_A$ for each $s$), one gets
\begin{equation}\label{eq:EPR-no-signal-result}
\langle\hat O_B\rangle^{(s)} \;=\; \Tr_B\bigl[\sigma^z_B\,\Tr_A\rho_\Bdy^{(AB)}\bigr],
\end{equation}
independent of $s$. Bob's marginal does not register Alice's setting choice. No operational signal can be transmitted via the shared state.

This is consistent with Einstein's locality intuition in the operational sense: Alice cannot transmit information to Bob faster than light, regardless of what the bulk theory says about geodesic connectivity. Bell--Aspect violations measure a correlation, not a signal.


\section{The four readings reconciled}
\label{sec:EPR-four-readings}

The framework's reading of EPR/Bell/CHSH/Aspect is best presented by comparing it side-by-side with the four standard interpretations.

\begin{table}[H]
\centering
\caption[Four readings of Bell's theorem, and the GG-Theory reading]{Four readings of Bell's theorem and the GG-Theory reading. All four standard interpretations agree on the experimental predictions and the no-signaling constraint; they differ on the metaphysical reading of the bulk-to-boundary projection. The rows record, for each reading, what it keeps, what it surrenders, and where the present framework stands relative to it.}
\label{tab:EPR-four-readings}
\small
\renewcommand{\arraystretch}{1.25}
\begin{tabular}{@{}lp{4.00cm}p{4.00cm}@{}}
\toprule
\textbf{Reading} & \textbf{Stance on 4D locality} & \textbf{Status in GG-Theory} \\
\midrule
Copenhagen (Bohr, Heisenberg) & The quantum state is non-local but operationally complete; ``physical reality'' has no separable existence at the microscopic level. & Operationally correct at the 4D boundary; ontologically incomplete (the 6D bulk is the missing layer). \\
Many-worlds (Everett, DeWitt) & The wavefunction never collapses; every outcome is realized in a separate branch. & Mathematically equivalent at the 4D boundary; ontologically over-committed (the 6D bulk is single-valued, no branching needed). \\
de~Broglie--Bohm pilot wave (de~Broglie, Bohm) & Non-local hidden variables in 4D; pilot wave guides the particle. & Hidden variables exist but live in 6D ($S^1_\theta$ holonomy), not in 4D; non-locality is therefore not a feature of the underlying theory. \\
Spontaneous collapse (GRW, Penrose) & Collapse is a physical dynamical process at the 4D level. & Rejected: collapse is the boundary observer's update, not a bulk dynamical event. \\
\midrule
\textbf{GG-Theory (cohort framework)} & \textbf{Bell--Aspect non-locality at 4D is the operational shadow of 6D-locality through a non-invertible bulk-to-boundary projection.} & --- (this paper) \\
\bottomrule
\end{tabular}
\end{table}

The empirical predictions are identical across all five readings; they all predict Tsirelson-saturating CHSH violations and respect no-signaling. The differences are interpretive. The cohort framework's distinguishing feature is that it produces the same predictions without postulating either branching or pilot waves: the 6D bulk is the deeper layer, and the apparent 4D non-locality is the shadow.


\subsection*{Part-end close}

Bell, in 1964, asked whether nature could be both local and complete. Aspect, in 1982, answered no. The 2022 Nobel formalized the answer. The framework agrees with all three results and adds the bulk-geometric layer that the four-dimensional version of the question could not see: a singlet pair is a single bulk field configuration with a shared $S^1_\theta$ holonomy reference, the two detector events are projections of that one non-separable joint state along the radial direction (which carries no light cone), and the Tsirelson saturation is the extremal (Bogomolnyi-analogue) ceiling on the CHSH functional rather than a coincidence of the singlet state.

After all, what a singlet correlation establishes is not that there is non-locality at the bottom of the world but that the operational projection from a higher-dimensional bulk theory to the 4D boundary is non-invertible. The 4D shadow is non-local; the 6D source is not. Bell's inequality and the Bogomolnyi argument together place a quantitative lid on how much shadow the boundary can carry, and Aspect's experiments show that nature lifts the lid by exactly the amount the bulk geometry permits.


%% ===================================================================
%% PART III — The 6D Bulk is Deterministic: Einstein's Victory
%% ===================================================================
\paragraph{Problems and worked solutions.}

\medskip\noindent\textbf{VII.1. Separate gloves from singlets.}

\noindent\emph{Solution.}  Step 1: classical anticorrelation (one
left glove, one right glove, mailed at random) reproduces the
same-axis singlet result perfectly: opposite outcomes, every run.
Step 2: the difference appears only at \emph{tilted} axes: gloves
--- and every local pre-assignment --- yield a correlation linear in
the angle between analyzers, while the singlet delivers
$-\cos\theta$ \eqref{eq:EPR-singlet-corr}; at $45^\circ$ the
quantum value $-0.707$ beats any straight line constrained at the
endpoints.  Step 3: CHSH \eqref{eq:EPR-S} is precisely the
instrument that cashes this bend four times coherently: gloves top
out at $|S| = 2$, the singlet reaches $2\sqrt2$ --- so EPR's
``elements of reality'' are not refuted by correlation itself but
by its \emph{curvature}.  Physics intuition: Einstein's gloves
explain the easy half of entanglement and fail the bent half ---
the whole century-defining dispute lives in the difference between
a straight line and a cosine.

\medskip\noindent\textbf{VII.2. Test the robustness of $2\sqrt2$
against misalignment.}

\noindent\emph{Solution.}  Step 1: rotate Bob's two analyzers by a
common error of $5^\circ$ off the optimal set and re-evaluate
\eqref{eq:EPR-S}: $|S| = 2.818$ --- a loss of only $0.4\%$ from
$2\sqrt2 = 2.828$; the optimum is a stationary point, so small
errors cost quadratically.  Step 2: contrast with the classical
side: no misalignment, however clever, lifts a local model above
$2$ --- the gap between $2$ and $2.818$ is thirty-plus experimental
standard deviations in modern runs.  Step 3: this robustness is why
Bell tests are \emph{practical}: the violation is not a knife-edge
effect but a broad plateau; precision matters only for metrological
refinements (Tsirelson-saturation checks), which is where the
framework's extremal-shadow reading places its own stakes
(Eq.~\eqref{eq:EPR-Tsirelson}).  Physics intuition: nature does not
merely peek over the classical wall --- it stands on a plateau
$40\%$ above it, wide enough that even a shaky ladder shows the
view; that is why the 2022 Nobel rests on decades of agreeing
experiments rather than one heroic alignment.

\medskip\noindent\textbf{VII.3. Audit the three loopholes, with the
detection threshold computed.}

\noindent\emph{Solution.}  Step 1 (detection): with detector
efficiency $\eta$, undetected pairs must be assumed a fair sample;
a local model exploiting the misses can fake CHSH violation unless
$\eta > 2/(1+\sqrt2) = 82.8\%$ --- the Eberhard-class threshold
that photon experiments only crossed in 2015.  Step 2 (locality
and freedom): the setting choices must be spacelike-separated from
the distant detection (Aspect's time-varying analyzers began this;
the 2015 experiments enforced it with fast random generators), and
the random choices must be independent of the source.  Step 3: the
loophole-free trio --- electron spins in diamond, and two photonic
experiments --- closed detection and locality \emph{simultaneously},
which is what elevated Bell violation from overwhelming evidence to
certification, and what the Nobel citation certified.  Physics intuition:
 a loophole is a story in which the world conspires
through mundane channels --- lost photons, slow switches, stale
randomness; closing all of them at once is how physics converts
``surely not'' into ``measured, at $\eta > 82.8\%$, with spacelike
switching.''

\medskip\noindent\textbf{VII.4. Attempt to build a Bell telephone,
and fail correctly.}

\noindent\emph{Solution.}  Step 1: Alice tries to signal by
\emph{choosing} her basis: measuring $z$ versus $x$ on her half of
the singlet.  Step 2: compute Bob's local state in both cases: the
marginal \eqref{eq:EPR-no-signal-result} is $\tfrac12\mathbb 1$
either way --- Bob's statistics are setting-independent, so no
choice of Alice's transmits a bit; only when the two records are
\emph{compared}, over a classical channel, do the correlations
appear.  Step 3: locate the impossibility structurally: signaling
would require the boundary marginal of one wing to depend on a
local operation at the other, and a partial trace cannot do that
(the appendix twin F.6.1) --- entanglement is shared randomness
with impossible-to-fake correlations, not a cable.  Physics intuition:
 the singlet is a pair of coins minted from one blank
--- read together they rhyme, read apart each is noise; you cannot
send a message by choosing how to look at noise, and relativity
survives quantum mechanics by exactly this margin.

\medskip\noindent\textbf{VII.5. State what the 2022 Nobel certified
--- and what it left open.}

\noindent\emph{Solution.}  Step 1: certified: Bell violation is a
fact of nature at loophole-free rigor --- no theory of 4D-local
pre-assigned values survives; this is a statement about the world,
not about quantum formalism.  Step 2: not certified: any
particular completion --- Copenhagen, Everett, Bohm, QBism, or
this framework --- because all reproduce the measured correlations;
the prize closed the LHV question and left the interpretation
landscape (next Part) intact.  Step 3: the framework's precise
stake, stated in the honesty register: its bulk variables are
6D-local and \emph{outside} Bell's jurisdiction (the theorem
quantifies over 4D-boundary variables), so the certified violation
is compatible with --- indeed, on the extremal-shadow reading,
predicted at exactly $2\sqrt2$ by --- the deterministic bulk; its
falsifiable surplus lives in the Tsirelson-saturation and
kill-criteria channels, not in re-litigating Bell.  Physics intuition:
 the Nobel committee certified the death certificate's
\emph{cause} --- local realism in four dimensions --- and left the
estate unsettled; the framework's claim is not that the deceased
lives, but that the estate has a sixth-dimensional heir the will
never mentioned.


\GGtakeaway{\emph{Standard account:} Bell violations are real
(2022 Nobel), local hidden variables are dead, and what remains is
an interpretation dispute. \emph{What it left open:} how nature can
violate the CHSH bound without signaling, and what --- if anything
--- carries the correlation. \emph{The bulk reading:} one entangled
bulk configuration spans both wings; 6D locality survives intact
while the 4D boundary, after the trace, is exactly as non-local as
quantum mechanics --- saturating Tsirelson, never signaling; Bell's
theorem excludes 4D hidden variables and leaves the bulk's
$S^1_\theta$ coordinate untouched. Einstein right in 6D; Bohr right
in 4D.}

\GGPart[Worked example IV: Schr\"odinger's cat]{Worked example IV: Schr\"odinger's cat in 6D}{Deterministic in the bulk, decohered on the boundary}
\label{part:VIII}

\GGteachboxpart{The most famous thought experiment in physics,
completed rather than retold: the Geiger trigger carries the
microscopic superposition up the amplification chain, the boundary
decoheres it in microseconds, and the bulk never held two cats at
all --- one deterministic 6D configuration, with the live/dead
alternative an artifact of the traced description. On the way the
chapter states plainly what the framework does to the classic
escape routes: the $S^1_\theta$ direction is exhibited as the
hidden-variable space Bell's theorem does not exclude, the
Many-Worlds interpretation is retired (no branching, one bulk),
Bohmian mechanics is met and placed, and Wigner's friend loses the
regress because registration is physical, not psychological.}

\GGreadalong{\citet{Wigner1961} for the friend; \citet{Everett1957}
and \citet{Bohm1952} for the two great completions this Part
engages and retires or places; \citet{Zurek2003} for the
decoherence machinery the chapter deploys.}

\noindent The 6D-bulk-to-4D-boundary reading of Schr\"odinger's cat is shown in \figref{fig:cat-6d-4d}.

\begin{figure}[H]
\centering
\resizebox{0.969\textwidth}{!}{%
\begin{tikzpicture}[
 font=\sffamily\footnotesize,
 >=Stealth,
 every node/.style={align=center, inner sep=3pt},
]
% =====================================================================
% LEFT BOX: 6D BULK
% =====================================================================
\draw[draw=red!70!black, fill=red!10, line width=0.80pt, rounded corners=3pt]
 (-3.8,-2.2) rectangle (3.2,2.0);
\node[font=\sffamily\bfseries, anchor=north] at (-0.3,2.0) {6D bulk: deterministic, single-valued};

% Smooth bulk wave (single configuration)
\draw[line width=1.10pt, draw=red!75!black, smooth, samples=60, domain=-3.3:2.7]
 plot (\x, {-0.3 + 0.4*sin(120*\x) - 0.05*\x});

% Labels
\node[font=\scriptsize, anchor=west] at (-3.6,-1.0) {single bulk configuration $\Psi_{6\mathrm{D}}(x, r, \theta)$};
\node[font=\scriptsize, anchor=west] at (-3.6,-1.5) {topological $S^1_\theta$ winding};
\node[font=\scriptsize\itshape, anchor=south] at (-0.3,-2.2) {\AX{1}~\LC{} forward-time unitary};

% =====================================================================
% RIGHT BOX: 4D BOUNDARY (decohered branches)
% =====================================================================
\begin{scope}[xshift=11cm, yshift=0cm]
 \draw[draw=blue!70!black, fill=blue!10, line width=0.80pt, rounded corners=3pt]
 (-3.6,-2.2) rectangle (3.6,2.0);
 \node[font=\sffamily\bfseries, anchor=north] at (0,2.0) {4D boundary: decohered branches};
  % LIVE cat box
 \draw[draw=black!70, fill=green!18, line width=0.50pt, rounded corners=2pt]
 (-1.3,0.45) rectangle (1.3,1.2);
 \node[font=\scriptsize\sffamily] at (0,0.825) {LIVE cat};
  % DEAD cat box
 \draw[draw=black!70, fill=gray!25, line width=0.50pt, rounded corners=2pt]
 (-1.3,-1.2) rectangle (1.3,-0.45);
 \node[font=\scriptsize\sffamily] at (0,-0.825) {DEAD cat};
  % Diagonal density-matrix annotation
 \node[font=\scriptsize, anchor=north, align=center] at (0,-1.4)
 {$\rho_{4\mathrm{D}}(t) = p_L|L\rangle\langle L| + p_D|D\rangle\langle D|$};
 \node[font=\scriptsize, anchor=north] at (0,-1.85) {off-diagonal $\to 0$ by \TA{5}};
\end{scope}

% =====================================================================
% PROJECTION ARROW BETWEEN BOXES
% =====================================================================
\draw[->, line width=1.00pt, draw=violet!60!black] (3.2,0) -- (7.4,0);
\node[font=\scriptsize\itshape, fill=white, inner sep=1.5pt] at (5.3, 0.4) {non-invertible};
\node[font=\scriptsize\itshape, fill=white, inner sep=1.5pt] at (5.3,-0.4) {projection};

% Footer annotation
\node[font=\footnotesize\itshape, text=violet!60!black, anchor=north] at (5.6,-3.0)
 {bulk-to-boundary partial trace over $I_r$ and $S^1_\theta$};

\end{tikzpicture}}
\caption[Schr\"odinger's cat: 6D-bulk-deterministic vs 4D-boundary decohered branches.]{Schr\"odinger's cat in the 6D-bulk-to-4D-boundary reading.
\textbf{Left:} the 6D bulk state $\Psi_{6\mathrm{D}}(x, r, \theta)$ is
a single deterministic configuration with topologically protected
winding around the phase axis $S^1_\theta$; forward-time unitary
evolution is guaranteed by \AX{1}~\LC{}. The bulk knows of no
branching, no superposition, no measurement problem.
\textbf{Right:} the 4D boundary observer registers a density operator
obtained by partial trace over the bulk extra geometric directions;
the off-diagonal coherences between live-cat and dead-cat branches
damp to zero by \TA{5} on a timescale far shorter than any macroscopic
dynamics. The boundary observer necessarily sees decohered branches;
the 6D bulk picture restores the deterministic substrate that
Einstein's intuition called for. \emph{The two readings agree on
every operational prediction --- the boundary observer is in a
probabilistic world by construction, not by Nature playing dice.}}
\label{fig:cat-6d-4d}
\end{figure}




\section{The Geiger trigger completes the projection: live/dead superposition is categorically impossible}
\label{sec:cat-geiger-trigger}

We open this Part with the framework's uncompromising answer to
Schr\"odinger's 1935 thought experiment. The standard treatment in the
language Schr\"odinger, Wigner, and Everett would have recognized
follows in \S\ref{sec:cat-bulk-determin}--\S\ref{sec:cat-revisited};
placing the bulk reading first lets the reader see why the
``live-and-dead'' formulation is not paradoxical but \emph{categorically
mistaken}.

\paragraph{The thought experiment as Schr\"odinger meant it.}
Schr\"odinger's 1935 paper \citep{Schrodinger1935_cat} was a
\emph{critique}, not a proposal. His point was that if one applies
linear quantum-mechanical superposition to the entire chain
\textbf{radioactive atom $\to$ Geiger counter $\to$ hammer $\to$ cyanide
$\to$ cat} without modification, one obtains a state vector
$|\Psi\rangle = (|\text{undecayed}\rangle\otimes|\text{Geiger off}\rangle
\otimes|\text{cat alive}\rangle + |\text{decayed}\rangle\otimes|\text{Geiger
fired}\rangle\otimes|\text{cat dead}\rangle)/\sqrt 2$
that the 4D-only formulation cannot then sensibly project to a single
classical outcome at any natural place in the chain. Ninety years of
interpretive commentary have produced no consensus 4D resolution. The
cohort framework resolves the impasse by recognizing that the
projection is not a 4D dynamical event one has to invent; it is a
structural feature of the bulk-to-boundary map that completes the
moment a sufficiently macroscopic 4D detector couples to the
microscopic quantum amplitude. That detector is the Geiger counter.

\paragraph{The setup as a single bulk field configuration.}
In the cohort framework, the entire chain --- the radioactive
$^{226}\mathrm{Ra}$ atom (microscopic, $\sim 14$~fm), the Geiger
counter (macroscopic, $\sim 10$~cm of gas + photomultiplier
electronics), the spring-loaded hammer (macroscopic, $\sim 10$~cm), the
sealed cyanide vial (macroscopic, $\sim 1$~cm$^3$ of liquid), the cat
(macroscopic, $\sim 30$~cm and $\sim 10^{25}$ metabolically organized
atoms) --- is one bulk field configuration on the six-dimensional manifold $\mathcal{M}_4 \times I_r \times S^1_\theta$. Its propagation
is deterministic and unitary under the bulk Lagrangian. There is no
quantum uncertainty about the 6D trajectory of the joint system; the
bulk knows exactly which holonomy class the atom is in, what the gamma
field looks like, where the Geiger electronics are, and what the cat is
doing.

\paragraph{The radioactive decay as a deterministic 6D transition.}
The $^{226}\mathrm{Ra}$ nucleus is, in the 6D picture, a bulk
configuration with a specific Wilson-line holonomy class around
$S^1_\theta$ characterizing its parent state. Alpha decay is the
unitary 6D process by which the parent holonomy class transitions to a
daughter ($^{222}\mathrm{Rn}$) holonomy class plus an emitted alpha
quantum (for the gamma version of the thought experiment, replace with
an excited daughter that subsequently gamma-decays --- the structure is
identical). In 6D, this transition is a definite event at a definite
moment of bulk proper time, determined by the bulk dynamics; the
4D-observable randomness in decay time arises because the 4D observer
cannot read the bulk holonomy phase that triggers the transition.

\paragraph{The Geiger counter as the 4D-localized macroscopic detector.}
The Geiger counter is a 4D-classical macroscopic apparatus: a gas-filled
tube held at high voltage, with a wire anode that triggers an
electronic avalanche when an incoming ionizing particle creates the
first electron-ion pair. The full Geiger-counter device contains
$\sim 10^{23}$ atoms of P-10 counting gas plus $\sim 10^{20}$ atoms of
amplifier electronics, all in continuous thermal contact with the
laboratory environment at $T \sim 300$~K. The number of environmental
degrees of freedom integrated over the radial direction $I_r$ and the
phase axis $S^1_\theta$ when computing the boundary state of the Geiger
counter is, by any reasonable accounting, of order $N_{\rm env}
\gtrsim 10^{23}$.

\paragraph{The Geiger trigger is the projection event.}
When the bulk gamma quantum's 4D worldline intersects the Geiger
counter, the macroscopic electronics couple to the gamma's 4D position
via Coulomb ionization. Within $\sim 1$~ns the avalanche cascade
amplifies the single primary ionization into a macroscopic current
pulse, depositing $\gtrsim 10^9$ electrons on the anode. This is the
\GDOS{}/\OGGEP{} projection event: the bulk gamma quantum's
6D-deterministic worldline is registered by a 4D-localized
macroscopic detector whose post-trigger state is classically
distinguishable from the pre-trigger state by $\gtrsim 10^9$ orders of
magnitude in pointer position. \textbf{The projection \emph{completes
here}, not at any subsequent stage and not when a human eventually
opens the box.}

\paragraph{Why the Geiger trigger is irreversible: the decoherence
calculation.}
The off-diagonal coherence between the ``Geiger fired'' and ``Geiger
not fired'' macroscopic electronics configurations damps at rate
\TA{5}. The Joos--Zeh estimate \citep{JoosZeh1985, Zurek2003} for
macroscopic position superpositions in a thermal-photon bath gives
\begin{equation}\label{eq:cat-decoh-rate}
 \Gamma_{LD}
 \;\sim\; \frac{n_{\rm air}\,\sigma_{\rm scat}\,v_{\rm therm}\,(\Delta x)^2}{\lambda_{\rm dB}^2},
\end{equation}
with $n_{\rm air}\sim 2.5\times 10^{19}\,\mathrm{cm}^{-3}$,
$\sigma_{\rm scat}\sim 10^{-15}\,\mathrm{cm}^2$,
$v_{\rm therm}\sim 5\times 10^4\,\mathrm{cm/s}$,
$\Delta x \sim 10\,\mathrm{cm}$ (the macroscopic separation between the
Geiger-counter's two pointer positions), and the thermal de~Broglie
wavelength $\lambda_{\rm dB}\sim 10^{-10}\,\mathrm{cm}$, giving
\begin{equation}\label{eq:cat-rate-value}
 \Gamma_{LD} \;\gtrsim\; 10^{40}\,\mathrm{s}^{-1}.
\end{equation}
The off-diagonal coherence between the two macroscopic Geiger states
therefore decays by $e^{-\Gamma_{LD} t} \sim e^{-10^{40}\,t/{\rm s}}$;
after a time as short as $10^{-40}$~s --- twenty orders of magnitude
shorter than the Planck time --- the off-diagonal is unmeasurably small.
The Geiger trigger is operationally irreversible.

\paragraph{What follows the Geiger trigger.}
Once the Geiger counter is in a definite classical state, the rest of
the chain is purely classical Newtonian mechanics in 4D spacetime. If
the counter triggered, the relay closes (electrostatic force on a
macroscopic armature), the hammer falls (gravitational acceleration),
the cyanide vial shatters (classical brittle fracture), the HCN
disperses through the box (classical diffusion), the cat inhales the
HCN, mitochondrial cytochrome-c oxidase is inhibited, ATP production
halts (classical biochemistry), and the cat dies (classical biology).
If the counter did not trigger, no part of this cascade fires, and the
cat continues its classical metabolic activity. Every step downstream
of the Geiger event is governed by classical equations of motion or
classical biochemistry; no further quantum-mechanical projection is
involved.

\paragraph{Therefore: the cat is either alive or dead, classically.}
The framework's claim about the cat is unambiguous: at any moment
after the Geiger trigger event (or any moment after the trial window
closes for an untriggered run), the cat is in a definite classical
state. Either the Geiger fired and the cat is dead, or the Geiger
did not fire and the cat is alive. There is no live-and-dead
superposition; there is no fact-of-the-matter pending the human
observer's arrival; there is no Wignerian friend whose ``observation''
brings the cat into a definite state. The cat's state was set, in 4D
spacetime, the moment the Geiger counter clicked (or finished a
sufficient observation window without clicking).

\paragraph{Why ``live AND dead'' is categorically impossible.}
The standard 4D-only formulation, in which the wavefunction of the
entire (atom + counter + cat) system remains in a unitary superposition
until human observation, is a categorical error in three layers:
\begin{enumerate}[leftmargin=2em,itemsep=0.15em]
\item It mistakes the projection as a 4D dynamical event that needs to
be located somewhere in the chain; it is in fact a structural feature
of the bulk-to-boundary map, which completes at any sufficiently
macroscopic 4D detector.
\item It ignores the decoherence-rate calculation
\eqref{eq:cat-decoh-rate}, which puts the live-vs-dead off-diagonal
coherence at $\Gamma_{LD}\gtrsim 10^{40}\,\mathrm{s}^{-1}$ in any
realistic thermal environment. Even taken at the 4D level, no
laboratory or human-observable timescale can resolve a coherence that
decays in $10^{-40}$~s.
\item It anthropomorphizes ``observation'' by giving the human a
distinguished role over the Geiger counter --- but the framework has no
such distinguished role for human consciousness. The Geiger counter is
already a macroscopic 4D detector; the human's eventual look at the box
is a redundant second projection on an already-classical state.
\end{enumerate}
For a $\sim 30\,\mathrm{cm}$ biological organism in a $T\sim 310\,\mathrm{K}$
metabolic environment, the live-vs-dead off-diagonal decoherence rate is
even more extreme: $\Gamma_{LD}^{\rm cat}\gtrsim 10^{42}\,\mathrm{s}^{-1}$,
making the live-and-dead state operationally impossible to prepare and
operationally impossible to detect, even in principle.

\paragraph{The Wigner's-friend ``puzzle'' dissolved.}
If a human observer eventually opens the box and sees the cat, this is
a \emph{second} projection event in 4D spacetime: the human's visual
cortex registers the cat's macroscopic state, which feeds the human's
classical perception. But the cat's state was already classically
definite from the moment the Geiger counter clicked (or did not).
The human's ``knowing'' does not change the cat's state; the cat was
already alive or dead. The Wigner's-friend
``puzzle'' \citep{Wigner1961, FrauchigerRenner2018} arises from the
same 4D-only categorical error: treating ``observation'' as a special
quantum-mechanical operation rather than as one of many 4D-localized
macroscopic projections.

\paragraph{Bottom line of the bulk reading.}
The \GDOS{}/\OGGEP{} projection completes when the Geiger counter
detects the gamma quantum, not when a human opens the box. Once that
4D-localized macroscopic event has occurred in fixed 4D spacetime, the
cat's fate is determined classically; everything downstream of the
Geiger trigger is Newtonian mechanics and biochemistry. The bulk
is unitary and deterministic; the 4D boundary projection completes at
the first macroscopic detector in the chain; the cat is alive or dead;
the only quantum-mechanical randomness in the whole story is the bulk
holonomy class of the radium nucleus, drawn from the source ensemble
of $^{226}\mathrm{Ra}$ states at the moment the box was sealed.
\textbf{Superposition of live and dead cat is not a paradox to be
solved but a category error to be retired.}

\noindent The Geiger-trigger timeline that completes the projection is shown in \figref{fig:cat-geiger-timeline}.

\begin{figure}[H]
\centering
\resizebox{0.95\textwidth}{!}{%
\begin{tikzpicture}[
 font=\sffamily\footnotesize,
 >=Stealth,
 every node/.style={align=center, inner sep=2pt},
 bulktrack/.style ={line width=1.10pt, draw=red!75!black},
 bndtrack/.style ={line width=1.10pt, draw=blue!70!black},
 bulkev/.style ={rectangle, draw=red!75!black, fill=red!10,
 line width=0.60pt, minimum width=24mm, minimum height=10mm,
 rounded corners=2pt, font=\scriptsize},
 bndev/.style ={rectangle, draw=blue!70!black, fill=blue!10,
 line width=0.60pt, minimum width=24mm, minimum height=10mm,
 rounded corners=2pt, font=\scriptsize},
 proj/.style ={rectangle, draw=violet!70!black, fill=violet!18,
 line width=1.10pt, minimum width=30mm, minimum height=14mm,
 rounded corners=3pt, font=\scriptsize\bfseries},
 classical/.style ={rectangle, draw=green!60!black, fill=green!10,
 line width=0.60pt, minimum width=22mm, minimum height=10mm,
 rounded corners=2pt, font=\scriptsize},
 arrow/.style ={->, line width=0.80pt, draw=black!75},
 label/.style ={font=\scriptsize\itshape, anchor=south},
]
% =====================================================================
% TIME AXIS (horizontal)
% =====================================================================
\draw[->, line width=0.70pt] (-4mm, -22mm) -- (162mm, -22mm) node[right] {$t$ (4D proper time)};
\foreach \x/\lab in {0/{$t_0$}, 30mm/{$t_{\rm decay}$}, 70mm/{$t_{\rm Geiger}$},
 105mm/{$t_{\rm cat~dies}$}, 140mm/{$t_{\rm open}$}} {
 \draw (\x, -22mm) -- (\x, -23.5mm) node[below, font=\scriptsize] {\lab};
}

% =====================================================================
% TOP TRACK: 6D bulk (deterministic, unitary)
% =====================================================================
\node[label] at (0, 28mm) {6D bulk: deterministic and unitary};
\draw[bulktrack] (0, 22mm) -- (155mm, 22mm);

\node[bulkev] at (0, 22mm) {atom \\ $\Theta_H = \alpha$};
\node[bulkev] at (40mm, 22mm) {$\to$ daughter \\ + $\gamma$ quantum};
\node[bulkev] at (80mm, 22mm) {$\gamma$ propagates \\ in 6D};
\node[bulkev] at (120mm, 22mm) {bulk continues \\ unitarily};

% =====================================================================
% BOTTOM TRACK: 4D boundary (projection at Geiger trigger)
% =====================================================================
\node[label] at (0, 0mm) {4D boundary: projection at Geiger trigger};
\draw[bndtrack] (0, -6mm) -- (155mm, -6mm);

% pre-trigger
\node[bndev] at (0, -6mm) {atom $+$ Geiger \\ in superposition};
\node[bndev] at (40mm, -6mm) {(no 4D event yet) \\ \scriptsize gamma propagating};

% THE PROJECTION EVENT (the user's emphasis)
\node[proj] at (70mm, -6mm) {\GDOS{}/\OGGEP{} \\ \footnotesize PROJECTION \\ \footnotesize \emph{COMPLETES HERE}};

% post-trigger: classical mechanics
\node[classical] at (105mm, -6mm) {hammer falls \\ cyanide releases \\ \scriptsize \emph{(classical)}};
\node[classical] at (140mm, -6mm) {human opens box \\ \scriptsize sees DEAD cat \\ \scriptsize \emph{(2nd projection,} \\ \scriptsize \emph{redundant)}};

% Annotation arrow
\node[font=\scriptsize\itshape, anchor=south, text=violet!70!black] at (70mm, 10mm)
 {The cat's fate is determined CLASSICALLY at this 4D event,};
\node[font=\scriptsize\itshape, anchor=south, text=violet!70!black] at (70mm, 5mm)
 {NOT when the human eventually opens the box.};

% Arrow connecting bulk gamma to Geiger projection
\draw[arrow, draw=red!75!black] (80mm, 16mm) -- (70mm, 0mm);

% =====================================================================
% Decoherence-rate annotation
% =====================================================================
\node[font=\scriptsize, anchor=north, align=center] at (75mm, -28mm)
 {Decoherence rate of the Geiger-counter pointer-basis off-diagonal: \\
 $\Gamma_{LD} \gtrsim n_{\rm air}\,\sigma\,v\,(\Delta x)^2/\lambda_{\rm dB}^2 \;\sim\; 10^{40}\,\mathrm{s}^{-1}$ \;
 (Joos--Zeh, room T); \;
 \emph{the live-vs-dead off-diagonal damps in $10^{-40}$~s.}};

\end{tikzpicture}}
\caption[Geiger-trigger timeline: when the projection completes, well before any human observation.]{Timing of the GDOS/OGGEP projection event in
Schr\"odinger's cat. Top track: the 6D bulk evolution is
deterministic and unitary throughout. The radioactive atom transitions
to its daughter holonomy class + a gamma quantum at $t_{\rm decay}$;
the gamma quantum propagates through the bulk; the joint 6D
configuration knows its own state at every moment. Bottom track: the
4D boundary observer registers the projection event at
$t_{\rm Geiger}$, when the macroscopic Geiger counter detects the
gamma quantum. Everything to the left of $t_{\rm Geiger}$ is the 4D
shadow of a 6D process; everything to the right is purely classical
Newtonian mechanics and biochemistry. The human's eventual look into
the box at $t_{\rm open}$ is a redundant second projection on an
already-classical state. The off-diagonal coherence between
``Geiger fired'' and ``Geiger not fired'' decays at
$\Gamma_{LD}\gtrsim 10^{40}\,\mathrm{s}^{-1}$ by \TA{5}, making the
trigger event operationally irreversible. There is no live-and-dead
superposition at any point in the chain.}
\label{fig:cat-geiger-timeline}
\end{figure}


\section{The 6D bulk is deterministic, period}
\label{sec:cat-bulk-determin}

Einstein's December 1926 letter to Born~\citep{BornEinsteinLetters}
--- ``the theory says a lot, but does not really bring us any closer to
the secret of the Old One. I, at any rate, am convinced that He does
not throw dice'' --- and the Bohr--Einstein dialogue of
1927--1935~\citep{Bohr1935} have been read for ninety years as
Einstein's romantic resistance to the operational reality of quantum
mechanics. The Aspect experiment of 1982~\citep{Aspect1982} and the
loophole-free confirmations of 2015~\citep{Hensen2015, Giustina2015,
Shalm2015}, recognized by the 2022 Nobel Prize~\citep{NobelPhysics2022},
appeared to close the case for the Born--Einstein dispute in Bohr's
favor: local hidden-variable completions of quantum mechanics in 4D
are excluded. The present Part draws the careful conclusion that
Einstein's intuition about determinism is correct at the bulk level
without contradicting Bell--Aspect at the boundary level: the 6D bulk of \citep{ArisakaA2GG6} is fully deterministic, local, and
unitary, while the operational shadow at the 4D boundary is irreducibly
probabilistic and apparently non-local. The two statements describe
the same Nature at two different levels of the bulk--boundary hierarchy.

\subsection{Claim and setup}

The central claim of this Part is:
\begin{quote}\parbox{0.9\textwidth}{\centering
\emph{The \GG{} bulk on $M_4 \times I_r \times S^1_\theta$ is
fully deterministic, local, and unitary. No random variables
exist in the 6D theory. The randomness observed in 4D
quantum experiments is an artifact of the partial trace
$\Trenv$ over $I_r \otimes S^1_\theta$.}
}\end{quote}

This is a strong claim. It says that Einstein, with his
``God does not play dice'' intuition, was correct --- but
correct about the underlying bulk theory, not about its 4D
projection. Bohr, with his ``we must accept irreducible
probabilism,'' was also correct --- but correct about the 4D
effective theory, not about the underlying bulk.

The claim is falsifiable: if any experimental test showed that
the 6D bulk itself had probabilistic structure, \GGtheory{}
would be retired. We argue here that no such test exists in the
current data and that the proposition is consistent with all
known quantum experiments.

\subsection{Why the bulk is deterministic: MSMF}

The constitutional axiom \MSMF{} (Maximum Symmetry, Minimum
Freedom) of \GGConstitution{} \citep{ArisakaA1Constitution} states:
\begin{quote}\itshape
Nature selects the theory that maximizes admissible symmetry
and minimizes non-forced freedom. No continuous parameters are
permitted unless forced.
\end{quote}
A deterministic bulk has $F(T) = 0$ free parameters. A
stochastic bulk would have infinitely many (one coin-flip per
bulk event). \MSMF{} therefore selects the deterministic
option.

The 6D theory is a classical field theory on $M_6 = M_4 \times
I_r \times S^1_\theta$. Its dynamical variables are the
metric $g_{MN}(x^M)$ and the gauge fields $A_M(x^M)$ for $G_6 =
\SUfive \times \Uoneb \times \UoneQi$. The equations of motion
are the 6D Einstein equations, the 6D Yang-Mills equations,
and the BI-6 modified Bianchi identity. Given initial data on
a spacelike 5-surface, the evolution is uniquely determined by
the classical equations. There is no stochastic element.

The 6D ``quantum fluctuations'' familiar from 4D QFT live in
the 4D effective theory (the operational shadow). The 6D bulk
itself is a deterministic classical-field configuration
satisfying the MSMF-selected classical equations of motion.

\subsection{How quantization looks from the 6D perspective}

This requires some explanation. In standard QFT, one ``quantizes''
a classical field theory: replaces classical fields with
operators, Poisson brackets with commutators, etc. In
\GGtheory{}, this procedure is the \emph{4D effective
description}: the 6D bulk is classical (or better, at most
weakly quantum at the Planck scale), and the 4D effective
description inherits the quantum structure through the
partial-trace map $\Trenv$.

Specifically: the 6D bulk configuration at any moment contains
all the information about 4D ``wave functions,'' ``operators,''
and ``correlations.'' A 4D observer extracts this information
by projecting --- i.e., by constructing the 4D effective
density matrix. The 4D density matrix is a genuine matrix, not
a classical probability distribution. But its origin is the
classical 6D configuration.

To make this concrete: if we denote the 6D classical field
configuration as $\Phi_6(x^\mu, r, \theta)$, the 4D density
matrix is
\begin{equation}\label{eq:density_from_bulk}
\densbdy(\phi_4, \phi'_4) \;=\; \int \mathcal{D}\phi_{\rm env}\,
\Psi_{\rm bulk}^*[\phi_4, \phi_{\rm env}]\,
\Psi_{\rm bulk}[\phi'_4, \phi_{\rm env}],
\end{equation}
where $\Psi_{\rm bulk}$ is the wave functional of the bulk
on the $\Phi_6$ configuration, and $\phi_{\rm env} = \{$heavy
KK modes, phase-axis modes$\}$ are traced over.

The integral over $\phi_{\rm env}$ is the partial trace. It
produces the 4D density matrix. And the 4D density matrix,
by construction, has all the Born-rule structure we derived
in Part II.

\subsection{Deterministic bulk $\ne$ classical bulk}

Important clarification. ``Deterministic'' does not mean
``classical'' in the 4D sense. The 6D bulk is quantum in its own
way: it has a wave functional $\Psi_{\rm bulk}$, and this wave
functional can be in superposition states. What is deterministic
is the evolution: given $\Psi_{\rm bulk}$ at time $t_0$, the
evolution to time $t_1$ is deterministic and unitary. It is
only when we 4D-project that we see probabilism.

In this sense, the 6D bulk is \emph{like} classical field
theory (deterministic evolution) but \emph{with} a quantum
state space (Hilbert-space structure). This is the structure
of Schr\"odinger-equation evolution on $\Hilb_{\rm bulk}$.

\subsection{The 4D observer does not see $\Psi_{\rm bulk}$
directly}

The 4D observer does not, and cannot, see $\Psi_{\rm bulk}$
directly. The 4D observer can only see 4D records ---
measurement outcomes, pointer positions, etc. These are
derived from $\Psi_{\rm bulk}$ via the \OGGEP{} functor.

Importantly, the 4D observer can \emph{infer} the existence of
$\Psi_{\rm bulk}$ through cross-ledger predictions: if
\GGtheory{} makes correct predictions for, say, $\sin^2\theta_W$,
$\alpha_s(m_Z)$, $m_H$, the Hubble constant, the $g-2$ residual,
and the Higgs mass (see the UCLA-Arisaka series), then the
6D bulk is corroborated. But the 4D observer cannot
\emph{directly} observe $\Psi_{\rm bulk}$. That is the nature
of being a 4D observer.

\section{The $S^1_\theta$ direction: the hidden-variable space}

The hidden-variable program has a respectable lineage. De
Broglie's 1924 pilot-wave proposal~\citep{deBroglie1924} and Bohm's
1952 rederivation~\citep{Bohm1952} both sought to restore determinism
to quantum mechanics by adding configuration-space variables on top
of the wavefunction; Bell's 1964 theorem~\citep{Bell1964} subsequently
established that any such completion compatible with the experimental
predictions of quantum mechanics must be non-local in 4D, an obstacle
that pilot-wave theories have absorbed but never dissolved. The
present section observes that the obstacle dissolves naturally when
the hidden variables are not added on top of 4D but inherited from a
6D bulk: the phase axis $S^1_\theta$ supplies a topologically
protected degree of freedom that is 6D-local and 4D-non-local after
the boundary partial trace. The Wilson-line holonomy~$\Theta_H$ along
$S^1_\theta$ is the geometric realization of what Bohm called the
``hidden variable'' --- not an additional postulate but a coordinate
of the bulk that no 4D observer can directly access.

\subsection{What is $S^1_\theta$?}

The sixth dimension of \GGsix{} is a compact circle
$S^1_\theta$ of small radius $R_\theta \sim \hbar c/\Mlock \sim 5\times 10^{-20}$ m
(for $\Mlock \approx 3.85$~TeV). The fifth dimension $I_r$ is the radial direction with $r = \ln(E_{\rm GUT}/\mu)$, encoding the
RG scale. The four directions $M_4$ are ordinary spacetime.

The $S^1_\theta$ direction carries the bulk gauge-field
configurations (one for each gauge factor of the bulk gauge
group) plus the associated Kaluza-Klein tower of modes:
\begin{itemize}[leftmargin=2em,itemsep=0.1em]
\item A Wilson-line holonomy $\Theta_H^{(a)} =
\oint_{S^1_\theta} A_\mu^{(a)} dx^\mu / g^{(a)}$ for each gauge
generator $T^{(a)}$ of the bulk gauge group. In the simplest
\GG{} model --- a bulk with only $\Uone_{\rm EM}$ as the gauge
content --- only one such holonomy exists, and the 4D boundary
projection produces the standard Maxwell + Lorentz-force physics.
With \BIsix{}-closure-compatible gauge content (which requires
$\Uone_{\rm EM}\times\Uone_B$ at minimum), two holonomies exist;
in the cohort canonical bulk of \citet{ArisakaA2GG6}, the gauge
content $\Spinone\times\SUfive\times\Uoneb\times\UoneQi$
supplies the full family.
\item The associated Kaluza-Klein tower of modes for every bulk
field that propagates along $S^1_\theta$.
\end{itemize}
The 4D observer has no direct access to $S^1_\theta$ because its
radius is vastly below any resolvable length scale. Whatever
gauge content the bulk carries, the boundary partial trace
integrates it out; the QM mechanism (\OGGEP{} projection from 6D
bulk to 4D boundary) is the same.

\subsection{Why $S^1_\theta$ is the natural hidden-variable
space}

A ``hidden variable'' in the Bell-theorem sense is a variable
that (i) is real (ontic), (ii) is not directly observed by 4D
experiments, and (iii) determines outcomes in a way consistent
with quantum statistics. The $S^1_\theta$ holonomy $\Theta_H$
satisfies all three:
\begin{itemize}[leftmargin=2em,itemsep=0.1em]
\item \textbf{Real}: $\Theta_H$ is a gauge-invariant
observable in 6D.
\item \textbf{Unobserved}: 4D experiments trace over
$S^1_\theta$; $\Theta_H$ is not directly accessed.
\item \textbf{Outcome-determining}: in the Stern-Gerlach
example (Sec.~II), the holonomy value selects the \GSSB{}
branch.
\end{itemize}
So $S^1_\theta$ provides the ``hidden variable'' that Einstein
sought --- but hidden not in ordinary 3-space (which would be
ruled out by Bell) but in the phase axis of the deeper geometry.

\subsection{Why this doesn't conflict with Bell}

Bell's theorem rules out \emph{local} hidden variables in 4D.
Bell's assumption is that the hidden variables are assigned to
each particle separately, and that measurement outcomes depend
only on the local hidden variable and local setting. The 2022
Nobel experiments rule this class out.

In \GGtheory{}, the hidden variables $\{\Theta_H^{(a)}\}$ are
\emph{not} assigned to each particle separately. They are
global configurations of the bulk gauge fields along
$S^1_\theta$ (one configuration per gauge factor of whatever bulk
gauge content the theory carries). Two entangled particles share
the same $S^1_\theta$ fiber. Their outcomes are correlated
\emph{by the shared bulk structure}, not by ``spooky action at a
distance'' transmitted through 4D space.

This is consistent with Bell because the hidden variable is
\emph{6D-local} but \emph{4D-non-local}. Bell's locality
assumption holds in 6D; it is the 4D projection that looks
non-local.

\subsection{The Wilson-line holonomy in detail}

The Wilson-line holonomy along $S^1_\theta$ is
\begin{equation}\label{eq:Wilson_holonomy}
W(\Theta) \;=\; \mathcal{P} \exp\!\left(i\oint_{S^1_\theta}
A_\theta\,d\theta\right) \;=\; e^{i\Theta\,T^H},
\end{equation}
where $T^H$ is a generator in the gauge Lie algebra and
$\Theta \in [0, 2\pi)$ is the holonomy phase. At the \GSSB{}
scale, $\Theta$ locks at $\Theta_0 = \pi/2^N$ (with $N = 3$
from \OGNfive{}), giving $\Theta_0 = \pi/8$.

Fluctuations $\delta\Theta$ about $\Theta_0$ are the Higgs
field (treated in the gauge-sector companions). The local value
of the holonomy $\Theta_H$ along $S^1_\theta$ is unresolved by
$4$D measurements; it is the \emph{contextual joint holonomy} of
system and apparatus --- $\psi$-ontic and non-local, not a local
hidden variable --- whose \GRIP{}-selected attractor basin
(\TA{1},\TA{4}) yields the registered outcome.

\section{Bell violations are 6D-local artifacts of the bulk-to-boundary projection}

Bell's 1964 inequality~\citep{Bell1964}, refined by
Clauser--Horne--Shimony--Holt in 1969~\citep{CHSH1969}, demonstrated
that no local hidden-variable theory in 4D can reproduce the
correlations of an entangled bipartite singlet measured at
spacelike-separated detectors. The original Aspect experiments of
1981--1982~\citep{Aspect1981, Aspect1982a, Aspect1982b}, sharpened by
Freedman--Clauser 1972~\citep{FreedmanClauser1972} and finally closed
loophole-by-loophole in the Delft/Vienna/NIST runs of
2015~\citep{Hensen2015, Giustina2015, Shalm2015}, demonstrated the
violation experimentally to many standard deviations and earned the
2022 Nobel Prize for Aspect, Clauser, and
Zeilinger~\citep{NobelPhysics2022}. The standard reading is that Bell
violations require non-locality \emph{as a feature of physics}. The
present section sharpens this to a feature of the \emph{projection}:
the boundary correlation function is the operator-algebra expectation
in one non-separable joint bulk state, with the radial direction $I_r$
the projection axis (it carries no light cone) --- the violation is the
boundary shadow of that single joint state, as
proved in \T{7.7} of \citep{ArisakaA5GDOS} Part~III.

\subsection{The EPR/Bell setup in 6D}

Consider two spin-1/2 particles produced at a source in the
singlet state and sent to Alice and Bob at space-like-separated
detectors. In 4D, they are $N$ meters apart. In 6D, they share
the same $S^1_\theta$ coordinate --- they are ``at the same
place'' along the phase axis $S^1_\theta$ because the phase axis
is small ($\sim 5\times 10^{-20}$ m).

For the explicit calculation, we use the \UoneQi{} factor of the cohort canonical bulk gauge content as the abelian gauge sector on $S^1_\theta$; the calculation is structurally the same for any abelian gauge factor on $S^1_\theta$ (in a simpler bulk with only $\Uone_{\rm EM}$, the same calculation goes through with that gauge factor playing the role of \UoneQi{}). The two particles share the $\UoneQi$ holonomy on
$S^1_\theta$ fixed at the common source, which is what makes their
joint state non-separable. The correlation is a \emph{frozen} feature
of that shared preparation, not a signal propagated between the wings
at measurement time: nothing travels from Alice's event to Bob's. From
the 4D observer's perspective the correlation appears ``instantaneous''
only because both shadows refer to the same bulk holonomy, set once at
the source.

\subsection{How the CHSH inequality is violated}

Consider Alice measuring spin along axis $\mathbf{a}$ and Bob
along $\mathbf{b}$. The outcomes are $a, b \in \{\pm 1\}$. In
a local hidden-variable 4D theory, the joint probability
factorizes:
\begin{equation}\label{eq:LHV_joint}
P(a,b|\mathbf{a},\mathbf{b}) \;=\; \int d\lambda\,\rho(\lambda)\,
P_A(a|\mathbf{a},\lambda)\,P_B(b|\mathbf{b},\lambda),
\end{equation}
where $\lambda$ is the hidden variable. Bell proved this form
implies $S_{\rm CHSH} \le 2$.

In \GGtheory{} the entangled pair is \emph{not} a pair of systems with
separate pre-existing values fed through local response functions; that
is the separable ansatz Bell's theorem excludes, and it yields only the
triangular correlator (Appendix~\ref{app:CHSH_calc}). Instead the pair
is one non-separable state of the bulk, and the joint outcome
statistics are the standard operator-algebra expectation
\begin{equation}\label{eq:6D_joint}
P(a,b|\mathbf{a},\mathbf{b}) \;=\; \bigl|\langle a,b|\,U(\mathbf a,\mathbf b)\,|\Psi^-\rangle\bigr|^2,
\end{equation}
evaluated in the shared state $|\Psi^-\rangle$ whose non-separability is
carried by the single $S^1_\theta$ holonomy fixed at the source. Because
$|\Psi^-\rangle$ does not factor into independent per-wing pieces, this
joint distribution is not of the local-hidden-variable form
$\int d\lambda\,\rho(\lambda)P_A(a|\mathbf a,\lambda)P_B(b|\mathbf b,\lambda)$ ---
which is exactly why it can violate Bell's inequality while no
shared-$\Theta$ local-response model can.

The net result: \GGtheory{} reproduces the quantum correlation
$E(\mathbf{a},\mathbf{b}) = -\mathbf{a}\cdot\mathbf{b}$ and
the Tsirelson bound $S_{\rm CHSH} = 2\sqrt{2}$. Bell's
inequality is violated, but not because of any signal in 6D
--- it is violated because the pair is one non-separable state whose
shared $S^1_\theta$ holonomy makes its joint statistics irreducibly
non-factorizable.

\subsection{Is this a ``loophole''?}

Strictly speaking, \GGtheory{}'s account of Bell violations
exploits what is sometimes called the ``contextuality
loophole'' or the ``measurement-dependence loophole'': the
hidden variable is not a pure pre-existing property of each
particle but a shared environmental state that depends on the
measurement context through the bulk dynamics.

From a foundational standpoint, this is not a ``loophole'' in
the pejorative sense. It is an explicit recognition that the
assumption of independent pre-existing properties for each
particle is wrong: the 6D theory does not satisfy that
assumption, and experiment agrees.

\subsection{No superluminal signaling}

Critically, \GGtheory{} does not permit superluminal signaling
in 4D. The no-signaling theorem of quantum mechanics is
preserved: Alice's measurement on her particle gives no
information that can be transmitted to Bob faster than light.
The 6D correlation is present, but accessing it would require
4D access to $S^1_\theta$, which is not available. So Alice and
Bob cannot signal via EPR pairs.

The 6D description is compatible with 4D relativity: the compact
dimension $S^1_\theta$ is not a spacelike direction in the 4D
causal structure; it is an internal (gauge) direction from the
4D observer's perspective.

\section{The Many-Worlds interpretation is retired}

Everett's 1957 thesis~\citep{Everett1957}, popularized by
DeWitt and Graham in the 1970s, proposed that the measurement problem
dissolves if one accepts that the wavefunction never collapses and
every outcome is realized in a separate branch of the universal state.
The interpretation has the substantial virtue of taking the unitarity
of Schr\"odinger evolution literally; it has the substantial cost of
postulating a vast metaphysical inflation (a continuum of co-existing
worlds for every measurement-like event) and of struggling to derive
the Born rule from branch-counting in a way the community has
accepted. The present section observes that the multiverse is not
needed if the bulk is single-valued in a higher dimension: the
\GG{} bulk evolves as one deterministic 6D configuration, and what
appears as ``branching'' on the 4D boundary is the selection of one
\GFF{}-minimizing branch by \GSSB{} (see \citep{ArisakaA6Gi4} for the
\GFF{}/\GSSB{} machinery). No additional ontology is introduced; the
4D appearance of branching is a derived feature of the projection.

\subsection{Why MWI was proposed}

Everett's 1957 thesis \citep{Everett1957} proposed MWI to solve
the measurement problem: instead of ``collapse,'' the wave
function unitarily evolves, and every outcome is realized in a
different ``branch.'' For the cat: alive-branch and dead-branch
both exist; they have just decohered and become mutually
incommunicable.

MWI's attraction was:
\begin{itemize}[leftmargin=2em,itemsep=0.1em]
\item No collapse postulate
\item Pure unitary evolution
\item No ``measurement'' as a special process
\end{itemize}
Its main difficulty was:
\begin{itemize}[leftmargin=2em,itemsep=0.1em]
\item Ontologically extravagant (an infinite multiverse)
\item Born rule is not derived (why do branches have ``weights''
proportional to $|\psi|^2$?)
\item Preferred basis (which branches are the ``real''
branches?)
\end{itemize}

\subsection{Why MWI is not needed in \GGtheory{}}

In \GGtheory{}, the 6D bulk is single-valued. At any given moment,
there is one 6D configuration. The 4D observer sees one
outcome of each measurement --- not because other outcomes
happen in other branches, but because \GSSB{} selects a single
branch based on the actual 6D configuration.

The ``other branches'' in MWI correspond, in \GGtheory{}, to
\emph{potential} configurations of the 6D bulk that are not
\GFF{}-minimizing. They are not realized. They do not exist as
parallel worlds.

This is a major ontological simplification. Instead of $2^N$
parallel universes for $N$ measurements, there is one
universe with a richer (6D) structure than the 4D observer
can access.

\subsection{The Born rule in MWI vs \GGtheory{}}

In MWI, the Born rule is notoriously difficult to justify: why
should the probability of ``finding yourself in branch $k$'' be
$|c_k|^2$? Various attempts (Deutsch-Wallace, Zurek-envariance,
\ldots) have been made, but none is universally accepted.

In \GGtheory{}, the Born rule is a theorem (Sec.~II). There is
no ``probability of finding yourself in a branch'' --- there is
one branch, it is the realized outcome, and the probability
distribution over potential realized outcomes is Born.

\subsection{When does MWI make the right prediction?}

MWI and \GGtheory{} make the same experimental predictions at
the 4D level --- they both reproduce the Born rule, both
handle decoherence, both explain the absence of macroscopic
superpositions. Where they differ is in ontology: MWI requires
a multiverse; \GGtheory{} requires only a 6D bulk.

Ockham's razor favors \GGtheory{}: fewer entities.

\section{Bohmian mechanics meets \GGtheory{}}

\subsection{Bohm's 1952 pilot wave}

Bohmian mechanics \citep{Bohm1952} postulates that particles
have definite positions $\mathbf{x}(t)$ at all times, guided by
the ``quantum potential''
\begin{equation}\label{eq:quantum_potential}
Q = -\frac{\hbar^2}{2m}\frac{\nabla^2|\psi|}{|\psi|}.
\end{equation}
The velocity of each particle is $\dot{\mathbf{x}} = \nabla S/m$
where $\psi = |\psi|e^{iS/\hbar}$. The wave function is a real
physical field guiding the particles.

Bohm's formulation is:
\begin{itemize}[leftmargin=2em,itemsep=0.1em]
\item Deterministic (given initial positions, trajectories
are determined)
\item Manifestly non-local (the quantum potential is non-local)
\item Reproduces all QM predictions for position
measurements
\item Extends awkwardly to QFT (unclear what is a ``particle''
in Fock space)
\item Extends awkwardly to spin (spin is not a position)
\end{itemize}

\subsection{\GGtheory{}'s relation to Bohm}

\GGtheory{} agrees with Bohm that there are definite
outcomes, that Nature is deterministic at a deeper level, and
that the randomness we observe has a hidden cause. \GGtheory{}
differs from Bohm in:
\begin{itemize}[leftmargin=2em,itemsep=0.1em]
\item The hidden variable is not the particle position in 4D
space, but the bulk-gauge-field configurations on $S^1_\theta$ (in the cohort canonical bulk these include the \UoneQi{} factor).
\item The non-locality is 4D-apparent only; 6D structure is
local.
\item Spin has a natural place (spin = irreducible
representation of $\SUtwo$ in the bulk gauge content).
\item QFT extension is built-in (the theory starts at 6D).
\end{itemize}

\GGtheory{} is in spirit a Bohmian theory --- deterministic at
the deep level --- but implemented through the geometric
structure of the 6D bulk geometry rather than through an
explicit pilot wave on 4D position space.

\section{Schr\"odinger's cat revisited in 6D}
\label{sec:cat-revisited}

Schr\"odinger's 1935 cat thought experiment~\citep{Schrodinger1935_cat}
was intended as a critique, not a proposal (see also \S\ref{sec:cat-geiger-trigger}, where the cohort framework's uncompromising answer is given upfront): by entangling a macroscopic
classical object with a microscopic quantum amplitude, Schr\"odinger
sharpened the question of where the unitary evolution of an isolated
quantum system gives way to the definite outcomes that laboratories
record. Ninety years of interpretive commentary have produced no
consensus answer within the 4D framework. The present section shows
that the question is naturally resolved when the cat's evolution is
read from the 6D bulk rather than the 4D boundary: the cat is, in 6D,
\emph{single-valued} --- definitely alive or definitely dead --- and
the apparent superposition on the 4D boundary is the operational
shadow of bulk decoherence by the air-molecule environment, with the
decoherence rate calculable from the scale-overlap integrals of
\T{7.5} (\citep{ArisakaA5GDOS} Part~III). The thought experiment
remains illuminating; the paradoxical reading does not survive the
shift of register.

\subsection{The 6D state of the cat}

In \GGtheory{}, the composite system
cat + Geiger counter + radioactive atom is described at 6D as
a configuration of the bulk fields. At any given moment, this
configuration is \emph{single-valued} --- the cat is, in 6D,
definitely alive or definitely dead.

What we call ``superposition'' at the 4D level is the
\emph{4D projection} of the 6D configuration. The 4D observer
does not see the 6D configuration directly; they see the
partial trace
\begin{equation}\label{eq:cat_partial_trace}
\rho_{\rm cat} \;=\; \Trenv\!\left(\ketbra{\Psi_{\rm
bulk}}{\Psi_{\rm bulk}}\right).
\end{equation}
If $\ket{\Psi_{\rm bulk}}$ is not in a special product form (as
it usually is not), $\rho_{\rm cat}$ is a mixed state.
For a well-decohered cat, $\rho_{\rm cat}$ is nearly
diagonal in the $\{\ket{\mathrm{alive}}, \ket{\mathrm{dead}}\}$
basis.

\subsection{Is the cat really in a superposition?}

From the 4D observer's perspective: the cat is in a mixed
state, with probabilities $1/2$ for each classical outcome. As
the cat ages, $\rho_{\rm cat}$ does \emph{not} actually
carry ``alive and dead simultaneously'' information in a
physically meaningful way --- the off-diagonals are suppressed
by $e^{-\Gamma t}$ with $\Gamma \sim 10^{40}$~s$^{-1}$ (macroscopic
spatial-decoherence rate; Joos--Zeh). Within far less than a femtosecond,
the cat is operationally in one classical state.

Which classical state? The one fixed by the \emph{joint} bulk holonomy $\Theta_H$ of cat, Geiger counter, and atom at the moment of trigger --- a contextual quantity of the whole apparatus, not a pre-existing label on the atom alone. The four-dimensional observer assigns $50/50$ probabilities; the six-dimensional state has a single definite outcome, fixed deterministically by $\Theta_H$.

This resolves the cat ``paradox'': the cat is definitely alive or definitely dead in 6D. The 4D observer assigns $50/50$ only because the bulk holonomy is inaccessible to them; which value $\Theta_H$ takes on a given run is the bulk boundary condition, Born-\emph{typical} over the source ensemble, the sole residue being \GGtypicality{} (\S\ref{ssec:fibre-honest}). Once they open the box --- once the cat's state couples to the 4D pointer apparatus --- they register the outcome that the bulk holonomy fixed all along. The framework is, in this exact sense, a six-dimensional Bohmian theory: deterministic at the deep level, probabilistic only in the boundary shadow.

\subsection{No ``collapse''}

There is no collapse of any wave function in \GGtheory{}'s
reading of the cat. The 6D state never ``collapses'' --- it
evolves deterministically. The 4D density matrix also never
collapses --- it just becomes diagonal in the pointer basis
via decoherence. What happens at ``measurement'' is:
\begin{enumerate}[leftmargin=2em,itemsep=0.1em]
\item The 4D pointer apparatus becomes entangled with the 4D
cat state (via unitary evolution of the 4D boundary).
\item The environment traces out rapidly (decoherence).
\item The 4D observer registers the pointer position.
\item The 4D observer conditions their subsequent expectations
on the registered outcome (Bayesian update).
\end{enumerate}
None of these steps involves ``collapse.'' All are standard
open-system dynamics.

\section{Wigner's friend and the observer dependence}

\subsection{The Wigner's-friend thought experiment}

Eugene Wigner in 1961 \citep{Wigner1961} imagined a friend
inside a sealed laboratory performing a quantum measurement,
while Wigner outside the lab describes the friend as a
quantum system. Wigner's description says the friend is in a
superposition of ``having measured $+$'' and ``having measured
$-$.'' The friend's own description says they have a definite
outcome.

Who is right? Traditional quantum mechanics says \emph{both},
depending on the level of description. But this seems to
violate any sensible notion of objective reality.

\subsection{Frauchiger-Renner 2018}

Frauchiger and Renner in 2018 \citep{FrauchigerRenner2018}
generalized Wigner's friend to a scenario where, applying QM
consistently at all levels, different agents obtain mutually
contradictory predictions. The result is that QM cannot be
simultaneously (a) universally valid, (b) consistent, (c)
single-world --- one of these must be given up.

\subsection{\GGtheory{}'s resolution}

In \GGtheory{}, the 6D bulk is always single-valued. The
4D density matrix is an observer-dependent construction --- it
depends on what the observer has access to trace over. Wigner
and his friend are both 4D observers with different access:
\begin{itemize}[leftmargin=2em,itemsep=0.1em]
\item The friend has access to the laboratory instruments
(direct measurement).
\item Wigner has access to the macroscopic state of the
laboratory as a whole (indirect).
\end{itemize}
Both descriptions are consistent with the same underlying 6D
configuration. There is one Nature; two 4D observers extract
different density matrices.

Frauchiger-Renner is resolved by noting that the ``contradiction''
requires both observers to assign the same ontological reality
to their respective wave functions --- which is incorrect in
\GGtheory{}. The wave function is observer-relative; only the
6D bulk is observer-independent.

\section{Summary: what Einstein was right about in 6D, and what
Bohr was right about in 4D}

\begin{table}[H]
\centering\small
\caption[The Einstein--Bohr debate resolved by level]{The Einstein--Bohr debate resolved by level. The two columns set each man's position beside the level of description at which the present reading finds it correct: Einstein's realism and determinism hold in the six-dimensional bulk, Bohr's operational completeness holds at the four-dimensional boundary, and the disagreement dissolves once the two levels are distinguished rather than being asked to describe one arena. Neither man is corrected; the question they shared is re-posed.}
\label{tab:einstein_bohr}
\begin{tabular}{@{}p{4.82cm}p{5.42cm}p{5.42cm}@{}}
\toprule
\textbf{Proposition} & \textbf{Einstein (6D, correct)} &
\textbf{Bohr (4D, correct)}\\
\midrule
Is Nature deterministic? & Yes, the 6D bulk evolves
deterministically & No, 4D measurements are irreducibly
probabilistic \\
Does God play dice? & No, not in 6D --- no dice-rolling in the
fundamental theory & Yes, in 4D --- from the 4D observer's
perspective, outcomes look random \\
Are there hidden variables? & Yes, the $S^1_\theta$ holonomy
and bulk gauge-field configurations & No, not in 4D --- Bell violations preclude
local 4D hidden variables \\
Is QM complete? & No, it is the 4D shadow of a more complete
6D theory & Yes, at the 4D operational level, QM is complete \\
Is Nature local? & Yes, 6D is local --- the radial $I_r$ and phase $S^1_\theta$
directions carry the correlations & No, 4D appears non-local because of the
bulk-to-boundary projection \\
Are entanglement correlations real? & Yes, they come from
shared 6D configurations & Yes, they are real 4D observations
\\
\bottomrule
\end{tabular}
\end{table}

Both perspectives are correct within their domains. The apparent
contradiction between Einstein's intuition and Bohr's
pragmatism was a confusion of levels --- both were right, they
were describing different levels of the theory.

\GGtheory{} is the framework in which this debate can finally
be resolved. Einstein, who died in 1955 before the
loophole-free Bell tests, would presumably be gratified to
learn that his instinct was correct --- in 6D. Bohr, who died
in 1962, would presumably be gratified to learn that his
insistence on the operational reality of quantum mechanics at
the 4D level was also correct.




%% ===================================================================
%% PART IX (9/14) -- Worked example V --- Decoherence, einselection, and pointer-basis selection
%% ===================================================================
\paragraph{Problems and worked solutions.}

\medskip\noindent\textbf{VIII.1. Evaluate the cat's decoherence
clock.}

\noindent\emph{Solution.}  Step 1: the chapter's localization-rate
estimate for the macroscopic Geiger-plus-cat system gives
$\Gamma_{LD} \gtrsim 10^{40}~\mathrm{s^{-1}}$
\eqref{eq:cat-rate-value}: air molecules, thermal photons, and
internal degrees of freedom all read the pointer's position.
Step 2: the coherence between live and dead sectors decays as
$e^{-\Gamma_{LD} t}$: by $t = 10^{-40}$~s --- twenty orders of
magnitude below any laboratory time resolution, and many below any
\emph{physical} process in the chain --- the off-diagonals are
gone; no instrument, in principle or practice, meets a superposed
cat.  Step 3: state what this does and does not settle: decoherence
explains why no one ever \emph{sees} interference between cat
sectors, and the framework adds the missing half --- the bulk never
contained two cats, so there is no surviving ``and'' to worry
about; the decohered boundary mixture is the shadow of one
configuration (Eq.~\eqref{eq:cat_partial_trace}).  Physics intuition:
$10^{-40}$ seconds is the shortest famous time in physics --- the
interval during which Schr\"odinger's paradox exists; everything
after that is classical bookkeeping, and everything before it is
one bulk trajectory that never branched.

\medskip\noindent\textbf{VIII.2. Follow the Geiger chain, stage by
stage.}

\noindent\emph{Solution.}  Step 1: order the amplification chain by
size: nucleus (one decaying mode) $\to$ ionized gas molecule $\to$
electron avalanche ($\sim\!10^9$ charges) $\to$ voltage pulse $\to$
hammer $\to$ vial $\to$ cat.  Step 2: apply the rate law
\eqref{eq:cat-decoh-rate} at each stage: the decoherence rate grows
with the square of the superposition separation and the number of
environmental probes, so the chain decoheres at the \emph{first}
macroscopic stage --- the avalanche --- with everything downstream
inheriting an already-classical record.  Step 3: conclusion in the
framework register: the ``measurement'' happened where the
registration happened (the avalanche's holonomy sector fixed the
record), not where the physiologist looks --- the cat is an
epilogue, and the box-opening is data retrieval.  Physics intuition:
 amplification is serial gossip --- by the third stage
the secret is town knowledge; the cat paradox survives only if one
insists the town can un-know it, which is exactly what
$e^{-10^{40}t}$ forbids.

\medskip\noindent\textbf{VIII.3. Defend the fiber as a Bell-legal
hidden variable.}

\noindent\emph{Solution.}  Step 1: Bell's theorem quantifies over
variables that factorize the joint response on the 4D boundary
\eqref{eq:LHV_joint}; the fiber coordinate enters the joint bulk
state \eqref{eq:6D_joint}, which does \emph{not} factorize after
the trace --- the theorem's hypothesis is simply not satisfied.
Step 2: check the two Bell-legality conditions explicitly: the
model reproduces $-\cos\theta$ (it is the singlet, exactly), and
it cannot be exploited to signal (VII.4) --- so it violates CHSH
while violating none of relativity's checkable content.  Step 3:
locate the honesty line: calling $\Theta_H$ a ``hidden variable''
is historically loaded --- it is a coordinate of the completion,
local in 6D, contextual on the boundary (as Kochen--Specker
requires of anything that works); the correct sentence is the
Part's own: Bell killed 4D hidden variables, and the bulk was
never among them.  Physics intuition: Bell's theorem is a
searchlight with a defined beam --- it swept the boundary and
found nothing, because the machinery was always one floor up;
no law of physics forbids the building from having a sixth floor.

\medskip\noindent\textbf{VIII.4. Do Wigner's friend's bookkeeping.}

\noindent\emph{Solution.}  Step 1: the friend measures the atom in
the closed lab; Wigner, outside, writes a superposition of
lab-states.  Step 2: apply VIII.1's arithmetic to the lab: the
friend's record decohered at $\Gamma_{LD}$-class rates the moment
it became a record, so Wigner's superposed description, while
formally available, has no operational content --- no feasible
measurement of his distinguishes it from the mixture, and the
registered sector is already fixed in the bulk.  Step 3: the
regress terminates because registration is \emph{physical}, not
psychological: there is exactly one boundary record chain, and
``for whom'' the outcome exists is answered by ``for the record''
--- consciousness enters nowhere, which the Part's treatment and
the sharpened Frauchiger--Renner problem (X.5) make precise.
Physics intuition: Wigner's friend is only paradoxical if facts
need witnesses --- in this framework facts need \emph{records},
and the universe has been keeping them for 13.8 billion years
without asking anyone to watch.


\GGtakeaway{\emph{Standard account:} the superposition
propagates up the amplification chain until observation collapses
it --- and a half-century of interpretations argues about when.
\emph{What it left open:} whether the cat was ever both, and what
role the observer's consciousness plays. \emph{The bulk reading:}
the bulk never held two cats --- one deterministic configuration,
with the Geiger trigger completing the projection and the
environment decohering the boundary record in microseconds; the
$S^1_\theta$ fiber is the hidden-variable space Bell spared,
Many-Worlds is retired for lack of a second world, and the friend
regress ends because registration is physics, not psychology.}

\GGPart[Worked example V: Decoherence]{Worked example V: Decoherence and einselection}{The Zurek predictability sieve from bulk locality}
\label{part:IX}

\GGteachboxpart{The engine room of the classical world: decoherence
and einselection derived, not narrated. The chapter reads system +
environment as one deterministic bulk configuration, derives the
Zurek predictability sieve from bulk locality, works a Gaussian
wavepacket in a thermal bath to explicit numbers, exhibits the
cohort-specific origin of the pointer basis (scale-overlap
integrals on $I_r$), shows why position wins the einselection
tournament, and closes with quantum Darwinism --- why many observers
agree on one classical record --- as the boundary's redundant
copying of the same bulk data. This Part is where ``the appearance
of a classical world'' stops being a postulate and becomes a
computation.}

\GGreadalong{\citet{Zeh1970} for the founding insight;
\citet{Zurek2003} for the standard modern treatment;
\citet{JoosZeh1985} for the collisional numbers;
\citet{Schlosshauer2005} for the review the chapter's register
follows.}

Decoherence and einselection together are the most significant theoretical advance on the measurement problem in the second half of the twentieth century. The standard treatment \citep{Zeh1970, Zurek1981, Zurek2003, JoosZeh1985, Schlosshauer2005} establishes the off-diagonal damping and identifies the pointer basis as the eigenbasis of the system--environment coupling, but does not derive the form of that coupling from a deeper principle. The cohort framework closes the gap structurally: the system and environment are one bulk field configuration in the six-dimensional bulk, and what looks like decoherence on the 4D boundary is the partial-trace projection over the two geometric directions $I_r$ and $S^1_\theta$ plus the unread 4D-environment region. The GKLS master equation is then a theorem (\TA{2}), the decoherence rates are scale-profile KK-mode overlap integrals (\TA{5}), and the einselected position basis is forced by bulk locality \AX{1}~\LC{} (\TA{6}). We present the six-dimensional bulk reading first in \S\ref{sec:DC-6d-reading}, then walk the standard textbook treatment in \S\ref{sec:DC-sieve}--\S\ref{sec:DC-einselection} for direct comparison, performing the explicit calculation for a Gaussian wavepacket in a thermal bath in \S\ref{sec:DC-gaussian} and recovering the cohort-specific KK-mode origin of the decoherence-rate coefficients in \S\ref{sec:DC-cohort-origin}.


\noindent The decoherence-rate scaling chart is shown in \figref{fig:decoherence-scaling}.

\begin{figure}[H]
\centering
\resizebox{0.655\textwidth}{!}{%
\begin{tikzpicture}[
 font=\sffamily\footnotesize,
 >=Stealth,
]
%% ----------- axis ranges -----------
%%   x: log10(Delta x / m)  from -10 to +2,  width 90mm,  1 decade = 7.5 mm
%%   y: log10(Gamma / s^-1) from -20 to +50, height 140mm, 1 decade = 2.0 mm
%% -----------------------------------

% axes
\draw[->, line width=0.7pt] (0,0) -- (96mm,0) node[right] {$\log_{10}(\Delta x / \mathrm{m})$};
\draw[->, line width=0.7pt] (0,0) -- (0,148mm) node[above] {$\log_{10}(\Gamma / \mathrm{s}^{-1})$};

% x-ticks: -10, -7, -4, -1, +2 (7.5 mm/decade)
\foreach \xmm/\val in {0/-10, 22.5/-7, 45/-4, 67.5/-1, 90/2} {
 \draw (\xmm mm, 0) -- (\xmm mm, -1.6mm)
 node[below, font=\scriptsize] {\val};
 \draw[line width=0.25pt, draw=black!20] (\xmm mm, 0) -- (\xmm mm, 140mm);
}

% y-ticks: -20, -10, 0, 10, 20, 30, 40, 50 (2.0 mm/decade)
\foreach \ymm/\val in {0/-20, 20/-10, 40/0, 60/10, 80/20, 100/30, 120/40, 140/50} {
 \draw (0, \ymm mm) -- (-1.6mm, \ymm mm)
 node[left, font=\scriptsize] {\val};
 \draw[line width=0.25pt, draw=black!20] (0, \ymm mm) -- (90mm, \ymm mm);
}

% Region bands (vertical)  micro / meso / macro -- boundaries at log Dx = -7 and -3
\fill[blue!8, opacity=0.55] (0, 0) rectangle (22.5mm, 140mm);
\fill[orange!8, opacity=0.55] (22.5mm, 0) rectangle (52.5mm, 140mm);
\fill[red!8, opacity=0.55] (52.5mm, 0) rectangle (90mm, 140mm);

\node[font=\scriptsize\itshape, text=blue!70!black, anchor=north]
 at (11.25mm, 138mm) {microscopic};
\node[font=\scriptsize\itshape, text=orange!70!black, anchor=north]
 at (37.5mm, 138mm) {mesoscopic};
\node[font=\scriptsize\itshape, text=red!70!black, anchor=north]
 at (71.25mm, 138mm) {macroscopic};

% Macroscopic-instantaneous horizontal threshold at log Gamma = 40 -> y = 120mm
\draw[line width=0.9pt, draw=red!70!black, dashed] (0, 120mm) -- (90mm, 120mm);
\node[font=\scriptsize, text=red!70!black, anchor=east] at (89mm, 123mm)
 {$\Gamma = 10^{40}\,\text{s}^{-1}$ (operationally instantaneous)};

% ----- Three coupling curves, slope = 2 in log-log -----
% Strong coupling: anchor (-1, 40);  at -10 -> 22 -> y=84mm;  at +2 -> 46 -> y=132mm
\draw[line width=1.1pt, draw=red!75!black]
 (0mm, 84mm) -- (90mm, 132mm);
\node[font=\scriptsize, text=red!75!black, anchor=south]
 at (45mm, 110mm) {strong: $\Gamma \propto (\Delta x)^2$};
\node[font=\scriptsize, text=red!75!black, anchor=south]
 at (45mm, 103mm) {(thermal bath, room~T)};

% Medium coupling: anchor (-7, 12);  at -10 -> 6 -> y=52mm;  at +2 -> 30 -> y=100mm
\draw[line width=1.1pt, draw=orange!75!black]
 (0mm, 52mm) -- (90mm, 100mm);
\node[font=\scriptsize, text=orange!75!black, anchor=south]
 at (60mm, 84mm) {medium};

% Weak coupling: anchor (-2, 0);  at -10 -> -16 -> y=8mm;  at +2 -> 8 -> y=56mm
\draw[line width=1.1pt, draw=blue!75!black]
 (0mm, 8mm) -- (90mm, 56mm);
\node[font=\scriptsize, text=blue!75!black, anchor=south]
 at (60mm, 41mm) {weak (low-T dust)};

% ----- Empirical anchors (markers + labels) -----
% C_60 fullerene at room T:  (-7, 12)  ->  (22.5mm, 64mm)  on medium curve
\fill[black] (22.5mm, 64mm) circle (1.0mm);
\node[font=\scriptsize, anchor=west] at (24mm, 64mm)
 {$C_{60}$ at $T=300$~K \;\citep{Arndt2003_C60}};

% Biological cell:  (-5, 25)  ->  (37.5mm, 90mm)
\fill[black] (37.5mm, 90mm) circle (1.0mm);
\node[font=\scriptsize, anchor=west, align=left] at (39mm, 90mm)
 {biological cell \\ ($\sim 10\,\mu$m, $T_{\rm body}$)};

% Schroedinger cat:  (-1, 40)  ->  (67.5mm, 120mm)  on threshold + strong line
\fill[black] (67.5mm, 120mm) circle (1.0mm);
% Schroedinger cat label: LOWER-RIGHT of the dot, UNDER the dashed line
\node[font=\scriptsize, anchor=west] at (70mm, 115mm)
 {Schr\"odinger cat};

% Dust grain at low T (weak curve, mesoscopic):  (-5, -6)  ->  (37.5mm, 28mm)
\fill[black!70] (37.5mm, 28mm) circle (0.75mm);
\node[font=\scriptsize, anchor=west] at (39mm, 28mm)
 {dust grain at $T=10$~K};

% Single-atom interferometer:  (-10, 22)  ->  (0, 84mm)  on strong line edge
\fill[black!70] (0mm, 84mm) circle (0.75mm);
\node[font=\scriptsize, anchor=south west] at (1mm, 85mm)
 {single-atom limit};

\end{tikzpicture}}
\caption[Decoherence rate $\Gamma_D$ scaling across $\sim 60$ orders of magnitude in system size.]{Decoherence-rate scaling with system size, log--log axes
across the seventy-decade range relevant to the
quantum-to-classical transition. The off-diagonal damping rate
$\Gamma_{\alpha\beta}$ of \TA{5} scales as $(\Delta x)^2$ in the
position-difference of the two branches at thermal coupling
\citep{JoosZeh1985, Zurek2003}; the three coupling-strength curves
(strong / medium / weak) shown here all have slope $2$ in
log--log. Three regime bands divide the $\Delta x$ axis at
$\Delta x = 100\,$nm and $1\,$mm: microscopic, mesoscopic,
macroscopic. The red dashed horizontal at $\Gamma = 10^{40}\,
\mathrm{s}^{-1}$ marks the operationally instantaneous threshold;
the Schr\"odinger-cat anchor on the strong-coupling line sits
exactly on this threshold at $\Delta x \approx 10\,\mathrm{cm}$.
Empirical anchors: a single-atom interferometer at the
microscopic--$\Delta x$ extreme; a $C_{60}$ fullerene at room
temperature \citep{Arndt2003_C60} on the medium curve; a
biological cell at body temperature; a dust grain at $T=10\,
\mathrm{K}$ on the weak curve. The $y$-axis range
$10^{-20}$ to $10^{50}\,\mathrm{s}^{-1}$ lets the same plot
display the quantum-coherence-preserving low-rate regime
(bottom-left) and the macroscopic-instantaneous regime
(top-right) on a single page.}
\label{fig:decoherence-scaling}
\end{figure}


\noindent The Zurek predictability-sieve diagram is shown in \figref{fig:zurek-sieve}.

\begin{figure}[H]
\centering
\resizebox{0.756\textwidth}{!}{%
\begin{tikzpicture}[
 font=\sffamily\footnotesize,
 >=Stealth,
 every node/.style={align=center, inner sep=2pt},
 arb/.style={circle, draw=black!60, fill=white, line width=0.4pt,
 minimum size=7mm, font=\scriptsize},
 sieve/.style={rectangle, draw=violet!70!black, fill=violet!12, line width=0.9pt,
 rounded corners=3pt, minimum width=42mm, minimum height=44mm},
 ptr/.style={circle, draw=red!70!black, fill=red!18, line width=1.0pt,
 minimum size=9mm, font=\scriptsize\bfseries},
 arrow/.style={->, line width=0.7pt, draw=black!70},
 arrowbig/.style={->, line width=1.0pt, draw=violet!60!black},
]
% ---- Left: arbitrary candidate bases (many small circles) ----
\node[font=\sffamily\bfseries, anchor=south] at (0, 32mm) {arbitrary candidate bases};
\foreach \pos/\lab in {
 (-15mm, 20mm)/{\scriptsize $|b_1\rangle$},
 (-2mm, 24mm)/{\scriptsize $|b_2\rangle$},
 (12mm, 18mm)/{\scriptsize $|b_3\rangle$},
 (-18mm, 4mm)/{\scriptsize $|b_4\rangle$},
 (-4mm, 6mm)/{\scriptsize $|b_5\rangle$},
 (14mm, 2mm)/{\scriptsize $|b_6\rangle$},
 (-12mm,-14mm)/{\scriptsize $|b_7\rangle$},
 (2mm,-10mm)/{\scriptsize $|b_8\rangle$},
 (16mm,-16mm)/{\scriptsize $|b_9\rangle$}}
{
 \node[arb] at \pos {\lab};
}

% ---- Middle: sieve box ----
\node[sieve] (S) at (50mm, 3mm) {};
\node[font=\sffamily\bfseries, anchor=north] at (S.north)
 {predictability sieve};
\node[font=\scriptsize, anchor=center] at (S.center)
 {select basis with \\ \emph{fastest} off-diag.\ decay \\[1pt]
 $\min_\alpha\,\partial_t\,|\rho_{\alpha\beta}|$};
\node[font=\scriptsize\itshape, anchor=south] at (S.south)
 {forced by \AX{1}~\LC{}};

% ---- Right: pointer basis survives ----
\node[font=\sffamily\bfseries, anchor=south] at (105mm, 32mm) {einselected basis};
\node[ptr] at (100mm, 18mm) {$|x_1\rangle$};
\node[ptr] at (115mm, 18mm) {$|x_2\rangle$};
\node[ptr] at (100mm, 6mm) {$|x_3\rangle$};
\node[ptr] at (115mm, 6mm) {$|x_4\rangle$};
\node[ptr] at (100mm, -6mm) {$|x_5\rangle$};
\node[ptr] at (115mm, -6mm) {$|x_6\rangle$};
\node[font=\scriptsize\itshape, anchor=north]
 at (107mm, -12mm) {position basis (\TA{6})};

% Arrows from input cloud to sieve, and sieve to output
\draw[arrowbig] (20mm, 3mm) -- (S.west);
\draw[arrowbig] (S.east) -- (88mm, 3mm);
\end{tikzpicture}}
\caption[Zurek's predictability sieve: einselection of the pointer basis from bulk locality.]{The Zurek predictability sieve and the cohort framework
reading. Left: an arbitrary infinite family of candidate bases
$\{|b_i\rangle\}$ for the boundary Hilbert space, each in principle
a valid choice for diagonalizing the post-decoherence density
matrix. Middle: the predictability sieve of
\citet{Zurek1981, Zurek2003} selects the basis whose off-diagonal
coherences decay fastest under the system--environment
Hamiltonian; standard treatments take this selection as an
operational principle without deeper justification. Right: the
einselected basis is the position basis $\{|x_j\rangle\}$ on any
macroscopic system, with $|x_j\rangle$ pointer-state stable to
within experimentally resolvable times. The cohort framework
closes the gap: \AX{1}~\LC{} (bulk locality) forces the
system--environment coupling to factor along position, and \TA{6}
inherits the sieve's selection as a theorem rather than a
postulate.}
\label{fig:zurek-sieve}
\end{figure}


%% PHASE-4D CONTENT BEGINS


\section{The six-dimensional bulk reading: system + environment as one deterministic configuration}
\label{sec:DC-6d-reading}

Before turning to the standard textbook treatment, we pause to state the
deeper picture that the cohort framework brings to decoherence and
einselection. The standard treatment in the language Zeh, Zurek, and
Joos would have recognized follows in \S\ref{sec:DC-sieve}--%
\S\ref{sec:DC-einselection}; placing the bulk reading first lets the
reader see why decoherence is not a dynamical mystery but a structural
feature of the bulk-to-boundary projection.

\paragraph{System and environment are one bulk field configuration.}
What the standard decoherence program treats as ``system $S$ coupled
to environment $E$'' is, in \GGtheory{}, a single bulk field
configuration on the six-dimensional manifold
$\mathcal{M}_4 \times I_r \times S^1_\theta$. The bulk Lagrangian does
not distinguish ``system'' from ``environment''; it is one composite
field theory whose evolution is unitary and deterministic. The split
into ``system'' and ``environment'' is something the 4D observer
imposes when they choose which 4D subregion to read directly and which
to leave unread.

\paragraph{The partial trace is what observation is.}
A 4D observer who wishes to record the state of a localized 4D region
$R_S$ (the ``system'') reads the boundary density operator
\begin{equation}\label{eq:DC-partial-trace}
 \rho_S \;=\; \Tr_{R_E \cup (I_r \times S^1_\theta)}\,\rho_{\rm bulk},
\end{equation}
where the trace runs over (i) the 4D complement $R_E = \partial\Mbulk \setminus R_S$
of the observed region (the ``environment'' degrees of freedom), AND
(ii) the two geometric directions $I_r$ and $S^1_\theta$ which no 4D
detector can read. This double partial trace IS the
\GDOS{}/\OGGEP{} projection event applied to the system-only
description; it is not a postulated dynamical step in the evolution of
$\rho_S$, but a structural feature of how a finite 4D observer accesses
a six-dimensional bulk state.

\paragraph{The GKLS master equation is a theorem (\TA{2}).}
Inserting the bulk evolution $\rho_{\rm bulk}(t) =
U_{\rm bulk}(t)\rho_{\rm bulk}(0)U_{\rm bulk}(t)^\dag$ into
\eqref{eq:DC-partial-trace} and applying the Nakajima--Zwanzig
projection \citep{Nakajima1958, Zwanzig1960} plus the Stinespring
dilation theorem \citep{Stinespring1955} yields, in the Markovian limit
fixed by \AX{1} (bulk locality), the canonical Gorini--Kossakowski--
Lindblad--Sudarshan master equation \citep{Lindblad1976, GKS1976}:
\begin{equation}\label{eq:DC-GKLS}
 \frac{d\rho_S}{dt} \;=\; -\frac{i}{\hbar}\,[H_{\rm eff}, \rho_S]
 \;+\; \sum_\gamma\bigl[L_\gamma\,\rho_S\,L_\gamma^\dag - \tfrac{1}{2}\{L_\gamma^\dag L_\gamma, \rho_S\}\bigr].
\end{equation}
The Lindblad jump operators $\{L_\gamma\}$ are not free parameters
added by hand to model openness; they are the bulk-system--environment
coupling channels projected onto the 4D boundary, fixed by the bulk
Lagrangian via the scale-overlap integrals of \TA{5}. The Schr\"odinger
equation is recovered as the $\gamma\to 0$ closed-system limit
(\TA{2}.4).

\paragraph{Decoherence rates from scale-profile KK-mode overlap integrals (\TA{5}).}
The bulk-side derivation of the decoherence rate is the central
quantitative content of \TA{5}. Writing the bulk system--environment
coupling as
\begin{equation}\label{eq:DC-bulk-coupling-here}
 H_{\rm bulk\text{-}coupling} \;=\; \sum_\gamma A_\gamma\,\otimes\,\Phi_\gamma(r, \theta),
\end{equation}
with $\Phi_\gamma(r, \theta) = \sum_n \phi_\gamma^{(n)}(x)\,f_n(r)\,e^{in\theta}$
the KK expansion of a bulk field along the radial and phase directions,
the boundary GKLS rate computed from the scale-overlap integral is
\begin{equation}\label{eq:DC-warped-rate}
 \boxed{\;
 \gamma_\gamma
 \;=\;
 2\pi\,\bigl|\!\int_0^K\! dr\,e^{-2r}\,f_0(r)\,f_n(r)\bigr|^2
 \cdot\rho_{\rm env}(\omega_n),
 \;}
\end{equation}
with the radial integral cut off at the canonical \OGNfive{} endpoint
$K = 35$ \citep{ArisakaA2GG6}. No phenomenological fit; the rate is a
calculation from the bulk geometry and the bulk field content.

\paragraph{Position-basis einselection is forced by bulk locality (\TA{6}).}
Bulk locality (\AX{1}~\LC{}) requires the bulk Lagrangian density to be
constructed from products of fields evaluated at the same 6D point.
When the bulk system--environment coupling
\eqref{eq:DC-bulk-coupling-here} is projected to the 4D boundary, the
locality requirement forces the boundary coupling to be diagonal in
the 4D-position basis:
\begin{equation}\label{eq:DC-position-diagonal-here}
 H_{SE}^{\rm boundary}
 \;=\; \sum_{x} |x\rangle\langle x|\,\otimes\,\hat E_{\rm eff}(x).
\end{equation}
By the Zurek predictability-sieve theorem applied to this coupling, the
einselected pointer basis is the position basis; the basis that wins
the sieve is not chosen by phenomenological fit but forced by \AX{1}
operating on the bulk Lagrangian. This is \TA{6} in its cleanest
statement: the einselected basis is whatever basis the bulk locality
constraint makes diagonal.

\paragraph{Each individual system has a definite 6D trajectory.}
For any individual realization of the system + environment, the 6D bulk
configuration evolves along a single deterministic trajectory; there is
no probabilistic dispersion in the bulk. The 4D observer's loss of
phase coherence over time --- the decay
$\rho_S(x, x', t) \sim \rho_S(x, x', 0)\,e^{-\Gamma_{xx'} t}$ --- is the
4D shadow of the bulk dynamics distributing 6D phase information across
the inaccessible coordinates $(r, \theta)$ and the unread 4D-environment
region $R_E$. No single ``decoherence event'' occurs in 6D; the bulk
just evolves. The decoherence appearance on the boundary is what
information loss looks like when the bulk-side picture is projected
through \eqref{eq:DC-partial-trace}.

\paragraph{Why the macroscopic-instantaneous regime is structurally generic.}
For any 4D-spatial superposition with separation $\Delta x$ in a thermal
environment at temperature $T$, the scale-overlap integrals
\eqref{eq:DC-warped-rate} sum coherently across the thermal-population
KK modes, producing the Joos--Zeh quadratic scaling
$\Gamma_{xx'} \propto (\Delta x)^2/\lambda_{\rm therm}^2$. In the
macroscopic-instantaneous regime
($\Delta x \gtrsim 1\,\mathrm{cm}$ at room T),
$\Gamma_{xx'} \gtrsim 10^{40}\,\mathrm{s}^{-1}$ --- twenty orders of
magnitude faster than any laboratory or biological timescale. The
disappearance of macroscopic interference is therefore not a separate
postulate but a quantitative consequence of bulk-geometry-determined
scale-overlap integrals. The full numerical work-through for a
Gaussian wavepacket in a thermal bath follows in \S\ref{sec:DC-gaussian}.

\paragraph{The Zurek sieve as a derived consequence.}
Zurek's original predictability-sieve criterion
\citep{Zurek1981, Zurek2003} was a phenomenological principle: the
einselected basis is the one whose off-diagonals decay fastest. In the
cohort framework, this principle is a derived consequence of two
ingredients: the bulk-locality constraint \eqref{eq:DC-position-diagonal-here}
forces position-diagonal coupling, and the scale-overlap integrals
\eqref{eq:DC-warped-rate} ensure the position-basis off-diagonals decay
faster than any other basis's off-diagonals for any 4D-spatial
configuration. The sieve does not need to be assumed; it falls out.

\paragraph{Bottom line of the bulk reading.}
Decoherence and einselection are not dynamical processes that the 4D
boundary theory has to invent and tune; they are structural features of
the bulk-to-boundary partial-trace operation
\eqref{eq:DC-partial-trace}. The GKLS master equation is a theorem
(\TA{2}) derived from the bulk dynamics plus the partial trace; the
decoherence rates are scale-profile KK-mode overlap integrals (\TA{5}) fixed
by the bulk Lagrangian; the einselected basis is the position basis
forced by bulk locality (\TA{6}); each individual realization has a
definite 6D trajectory and the apparent loss of coherence at the 4D
level is the structural consequence of integrating over the bulk
coordinates $(r, \theta)$ and the unread 4D-environment region. The
``measurement-induced collapse'' problem has no analogue in the bulk;
the bulk is unitary throughout, and what looks like collapse on the
boundary is the projection completing structurally at every macroscopic
4D detector.

\noindent The 6D-bulk reading of decoherence and einselection is shown in \figref{fig:DC-bulk-projection}.

\begin{figure}[H]
\centering
\resizebox{0.735\textwidth}{!}{%
\begin{tikzpicture}[
 font=\sffamily\footnotesize,
 >=Stealth,
 every node/.style={align=center, inner sep=2pt},
]
% =====================================================================
% TOP: 6D BULK BOX
% =====================================================================
\draw[draw=red!75!black, fill=red!8, line width=0.80pt, rounded corners=4pt]
 (-4.5,1.0) rectangle (4.5,3.6);
\node[font=\sffamily\bfseries, anchor=north] at (0,3.6) {6D bulk: one composite field configuration};

% Wavy field schematic
\draw[line width=1.10pt, draw=red!75!black, smooth, samples=80, domain=-4.0:4.0]
 plot (\x, {2.0 + 0.35*sin(180*\x)});

% Field labels in subboxes
\draw[draw=black!50, fill=white, line width=0.40pt, rounded corners=1.5pt]
 (-3.4,1.2) rectangle (-1.4,1.7);
\node[font=\scriptsize] at (-2.4,1.45) {$\Phi_S(x,r,\theta)$ \;\scriptsize system};
\draw[draw=black!50, fill=white, line width=0.40pt, rounded corners=1.5pt]
 (1.4,1.2) rectangle (3.4,1.7);
\node[font=\scriptsize] at (2.4,1.45) {$\Phi_E(x,r,\theta)$ \;\scriptsize environment};

% Right-side axis annotations
\node[font=\scriptsize, text=black!60, anchor=west] at (4.6, 2.7) {$I_r$: scale};
\node[font=\scriptsize, text=black!60, anchor=west] at (4.6, 2.0) {$S^1_\theta$: phase};

% Bulk-annotation
\node[font=\scriptsize\itshape, anchor=south, text=red!75!black] at (0,1.0)
 {single bulk Lagrangian; deterministic, unitary evolution};

% =====================================================================
% PARTIAL TRACE ARROW
% =====================================================================
\draw[->, line width=1.00pt, draw=violet!60!black] (0,0.6) -- (0,-0.6);
\node[font=\scriptsize\itshape, fill=white, inner sep=1.5pt] at (1.7, 0.15) {partial trace};
\node[font=\scriptsize\itshape, fill=white, inner sep=1.5pt] at (1.7,-0.25) {over $(R_E, I_r, S^1_\theta)$};

% =====================================================================
% BOTTOM: 4D BOUNDARY BOX
% =====================================================================
\draw[draw=blue!75!black, fill=blue!8, line width=0.80pt, rounded corners=4pt]
 (-4.5,-2.9) rectangle (4.5,-0.8);
\node[font=\sffamily\bfseries, anchor=north] at (0,-0.8) {4D boundary observer: reads system region only};

% R_S region (observed)
\draw[draw=blue!75!black, fill=blue!18, line width=0.50pt, rounded corners=2pt]
 (-3.0,-2.3) rectangle (-1.0,-1.3);
\node[font=\scriptsize] at (-2.0,-1.8) {$R_S$ \\ \scriptsize (observed)};

% R_E region (traced over)
\draw[draw=gray!75, fill=gray!22, line width=0.50pt, rounded corners=2pt]
 (1.0,-2.3) rectangle (3.0,-1.3);
\node[font=\scriptsize] at (2.0,-1.8) {$R_E$ \\ \scriptsize (traced over)};

% Info-loss arrow
\draw[->, line width=0.70pt, draw=black!70] (3.0,-1.8) -- (3.9,-1.8)
 node[right, font=\scriptsize\itshape, align=left] {info loss\\ \scriptsize $\rho_S \to$ mixed};

% Footer summary
\node[font=\scriptsize, anchor=north, align=center] at (0,-3.1)
 {$\rho_S = \Tr_{R_E \cup (I_r \times S^1_\theta)}\,\rho_{\rm bulk}$
 \;\;$=$ GKLS master equation (\TA{2});\\
 rates $\gamma_\gamma$ from scale-profile KK-mode overlap integrals
 (\TA{5}); pointer basis einselected by \AX{1} (\TA{6}).};

\end{tikzpicture}}
\caption[Decoherence and einselection as 6D-bulk-to-4D-boundary projection (partial trace).]{The 6D-bulk reading of decoherence and einselection.
\textbf{Top:} the system + environment is a single bulk field
configuration $(\Phi_S, \Phi_E)$ evolving deterministically and
unitarily under one bulk Lagrangian on the six-dimensional manifold. The ``system'' versus ``environment'' split is a 4D label
imposed by the observer's choice of what to read. \textbf{Bottom:}
the 4D boundary observer reads only the system region $R_S$; the
partial trace over the environment region $R_E$ and the two
geometric directions $I_r$ and $S^1_\theta$ produces the boundary
density operator $\rho_S$, governed by the GKLS master equation as
a theorem. The decoherence rates $\gamma_\gamma$ are scale-profile KK-mode overlap integrals fixed by the bulk Lagrangian; the position basis
is einselected by bulk locality. Decoherence is structural, not
dynamical: it is what information loss looks like when a
six-dimensional bulk is projected to a four-dimensional read-out.}
\label{fig:DC-bulk-projection}
\end{figure}


\section{The Zurek predictability sieve}
\label{sec:DC-sieve}

Zurek's 1981 einselection criterion \citep{Zurek1981, Zurek2003} answered a long-standing objection against the decoherence program: the program established that off-diagonal density-matrix elements decay rapidly in \emph{some} basis under environmental coupling, but did not select \emph{which} basis becomes the classical pointer basis. Zurek's resolution was operational: the einselected basis is the one whose off-diagonal coherences decay fastest under the system--environment Hamiltonian, the basis that maximizes the operational predictability of post-decoherence measurements.

Formally, let $\{|\alpha\rangle\}$ be any orthonormal basis of $\cH_\Bdy^{(\rm QM)}$ and let the off-diagonal decoherence rate in this basis be
\begin{equation}\label{eq:DC-rate-basis}
\Gamma_{\alpha\beta}(\{|\alpha\rangle\}) \;=\; \sum_\gamma \gamma_\gamma\,|\langle\alpha|L_\gamma|\alpha\rangle - \langle\beta|L_\gamma|\beta\rangle|^2,
\end{equation}
with $\{L_\gamma\}$ the GKLS jump operators inherited from the bulk via \TA{5}. Zurek's sieve identifies the einselected pointer basis as the one minimizing the predictability-sieve functional
\begin{equation}\label{eq:DC-sieve-functional}
\mathcal P[\{|\alpha\rangle\}] \;=\; \mathrm{Tr}\bigl(\rho_{\rm diag}\log\rho_{\rm diag}\bigr) - \mathrm{Tr}\bigl(\rho\log\rho\bigr),
\end{equation}
the von~Neumann entropy of the diagonal-projected density matrix minus the entropy of the full density matrix. In the long-time limit (when the off-diagonal terms have died), the basis that has minimized $\mathcal P$ is the einselected one.

What \TA{6} adds is the identification of \emph{which} basis wins the sieve: the position basis, because \AX{1} (bulk locality) forces the boundary system--environment coupling to be position-diagonal as a matter of constitutional law rather than phenomenological coincidence.


\section{Worked example: a Gaussian wavepacket in a thermal bath}
\label{sec:DC-gaussian}

Consider a one-dimensional quantum particle of mass $m$ prepared in a Gaussian wavepacket of width $\sigma_x$ centered at the origin, immersed in a thermal bath of mass-$M$ particles at temperature $T$ and number density $n$. The boundary density matrix in the position representation is
\begin{equation}\label{eq:DC-gaussian-init}
\rho(x, x', 0) \;=\; \frac{1}{\sqrt{2\pi}\,\sigma_x}\,\exp\!\Bigl[-\frac{x^2 + x'^2}{4\sigma_x^2}\Bigr].
\end{equation}
The off-diagonal coherence $\rho(x, x', 0)$ at $|x - x'| = \Delta x$ has the form $\exp(-\Delta x^2/4\sigma_x^2)$ times a center-of-mass Gaussian; it represents the quantum coherence between positions separated by $\Delta x$.

Coupling the particle to the thermal bath via the position-diagonal coupling that \TA{6} requires --- the bath constituents scatter off the particle with momentum transfer $q$, depending on the particle's position --- the standard Joos--Zeh calculation \citep{JoosZeh1985} gives the rate at which the off-diagonal decoheres:
\begin{equation}\label{eq:DC-rate-explicit}
\boxed{\;
\Gamma_{\Delta x}
\;=\; n\,\sigma_{\rm scatt}\,v_{\rm therm}\,
\bigl(1 - \exp\!\bigl(-\Delta x^2/\lambda_{\rm therm}^2\bigr)\bigr)
\;\approx\; \frac{n\,\sigma_{\rm scatt}\,v_{\rm therm}\,(\Delta x)^2}{\lambda_{\rm therm}^2}
\quad\text{(short-wavelength limit)},
\;}
\end{equation}
with $\lambda_{\rm therm} = \hbar/\sqrt{2M k_B T}$ the de Broglie thermal wavelength of the bath constituents and $\sigma_{\rm scatt}$ the bath-particle scattering cross-section. The decoherence rate scales quadratically in $\Delta x$ in the short-wavelength regime --- the essential macroscopic-instantaneous regime of \TA{5}.

\paragraph{Numerical estimate.}
For an air-like thermal bath at $T = 300$~K and $n = 2.5\times 10^{19}$~cm$^{-3}$ with $M = m_{\rm N_2} = 4.65\times 10^{-23}$~g, one has $v_{\rm therm}\sim 5\times 10^4$~cm/s, $\lambda_{\rm therm}\sim 1.9\times 10^{-9}$~cm, and $\sigma_{\rm scatt}\sim 10^{-15}$~cm$^2$ for a small object. For $\Delta x = 1$~$\mu$m, $\Gamma_{\Delta x} \sim 10^{20}$~s$^{-1}$ --- already 20 orders of magnitude faster than any laboratory time scale. For $\Delta x = 1$~cm, $\Gamma_{\Delta x}\sim 10^{36}$~s$^{-1}$. For $\Delta x = 10$~cm (Schr\"odinger's cat scale), $\Gamma_{\Delta x}\sim 10^{40}$~s$^{-1}$, matching the estimate of \TA{5}.


\section{The cohort-specific origin: scale-overlap integrals}
\label{sec:DC-cohort-origin}

The standard Joos--Zeh treatment takes the rates $\gamma_\gamma$ in \eqref{eq:DC-rate-basis} as phenomenological inputs: one fits them to the experimental decoherence rate, in the same way one fits the friction coefficient of a damped harmonic oscillator to the observed quality factor. The cohort framework's promotion of the decoherence rate from phenomenology to first principles (\TA{5}.2) replaces the phenomenological fit with a calculation from the bulk geometry.

Explicitly, if the bulk system--environment coupling has the form
\begin{equation}\label{eq:DC-bulk-coupling}
H_{\rm bulk-coupling} \;=\; \sum_\gamma A_\gamma\otimes\Phi_\gamma(r, \theta),
\end{equation}
with $\Phi_\gamma(r, \theta) = \sum_n \phi_\gamma^{(n)}(x)\,f_n(r)\,e^{in\theta}$ the KK expansion of a bulk field, then by \TA{5}.2 the GKLS rate is
\begin{equation}\label{eq:DC-scale-overlap}
\gamma_\gamma \;=\; 2\pi\,\Bigl|\int_0^K dr\,e^{-2r}\,f_0(r)\,f_n(r)\Bigr|^2 \cdot\rho_{\rm env}(\omega_n),
\end{equation}
with the scale-overlap integral $\int_0^K dr\,e^{-2r}\,f_0(r)f_n(r)$ measuring how the $n$-th KK mode of the bulk field projects onto the boundary zero mode along the radial direction.

The Joos--Zeh phenomenological rate \eqref{eq:DC-rate-explicit} can be reproduced by evaluating \eqref{eq:DC-scale-overlap} for the specific bulk-field content of the bath constituents (N$_2$ molecules, in the air-bath case). The point is that the rate is not a free parameter once the bulk geometry is fixed; the cohort apparatus turns the Joos--Zeh estimate into a calculation.


\section{Einselection of the position basis and the predictability winner}
\label{sec:DC-einselection}

By \TA{6}, the dominant boundary system--environment coupling is position-diagonal:
\begin{equation}\label{eq:DC-position-diagonal}
H_{SE} \;=\; \sum_{x_0} |x_0\rangle\langle x_0|\otimes\hat E_{\rm eff}(x_0).
\end{equation}
This coupling diagonalizes the off-diagonal damping in the position basis: $\Gamma_{x_0 x_0'}$ is maximal in the position basis and zero (in the same-position limit $x_0 = x_0'$) in every other basis. The Zurek predictability-sieve functional \eqref{eq:DC-sieve-functional} is therefore minimized by the position basis: the position basis ``wins the sieve.''

This is the constitutional content of \TA{6}: the einselected pointer basis is not a feature one chooses to enforce on the measurement, nor a feature that emerges by accident from the apparatus design; it is the consequence of the bulk-locality axiom \AX{1} propagating through the boundary partial trace. Every macroscopic record in the universe --- photographs, hard-drive bits, Geiger-counter readouts, the position of a cat in a box --- lives in the position basis because the bulk Hamiltonian is local, and there is nothing the experimenter can do to change that.


\section{Quantum Darwinism and the cohort cascade}
\label{sec:DC-Darwinism}

Zurek's quantum-Darwinism program \citep{Zurek2003} extends the einselection idea to multi-observer settings: when many environmental subsystems each carry a copy of the system's pointer-basis information, the resulting redundancy ensures that many independent observers will agree on the registered outcome. The cohort framework's reading of quantum Darwinism is via \TA{9} (universal identity) and \TA{6}: the redundancy of which-position information across the environment is itself a consequence of bulk locality (each environmental subsystem couples to the same position eigenvalues), and the agreement of multiple independent observers about the registered outcome is the operational signature of the bulk's single-valued bulk state.

In the cascade picture, the cascade at the einselection stratum has a unique stable \GA{} per registration site (the position eigenvalue at the detector), and every environmental subsystem inherits a copy of this \GA{} via the bulk locality of the underlying coupling. Quantum Darwinism is therefore the operational signature of the cohort cascade specialized to the post-measurement environment.


\subsection*{Part-end close}

Decoherence and einselection are not phenomenological add-ons to the textbook quantum-mechanical postulates. They are theorems of the cohort framework, with the off-diagonal damping rate fixed by scale-overlap integrals of the bulk geometry and the pointer basis identified as position by \AX{1} bulk locality. The macroscopic-instantaneous regime $\Gamma\gtrsim 10^{40}\,\mathrm{s}^{-1}$ is not a Joos--Zeh-specific estimate but a structural consequence of how the bulk environment couples to the boundary.

After all, when a Geiger counter clicks, what determines that the click is registered at a definite position rather than smeared across a continuum of pointer positions is not a choice the experimenter makes, nor a feature of the Geiger counter's design. It is the constitutional locality of the bulk Hamiltonian inherited from \AX{1} of \AOne{} --- the same locality that makes photographs sharp, hard drives reliable, and the cat in the box either definitely alive or definitely dead in the only register the boundary observer has access to.
%% ===================================================================
%% PART IV — Resolution of the Paradoxes
%% ===================================================================
\paragraph{Problems and worked solutions.}

\medskip\noindent\textbf{IX.1. Run the short-distance rate law
twice.}

\noindent\emph{Solution.}  Step 1: the collisional rate
\eqref{eq:DC-rate-explicit} in its short-distance regime is
$\Gamma_{\Delta x} = n\sigma v_{\rm th}(\Delta x)^2/
\lambda_{\rm th}^2$: quadratic in the separation --- small
superpositions are cheap, large ones ruinous.  Step 2: run it at
two scales with the same environment: doubling $\Delta x$
quadruples the rate; moving from an atom-scale to a dust-scale
superposition ($\times10^5$ in $\Delta x$) multiplies the rate by
$10^{10}$ before size-dependent cross-sections make it worse ---
the classical world is enforced with a quadratic whip.  Step 3:
check the two limits of the full formula: for
$\Delta x \gg \lambda_{\rm th}$ the exponential saturates and every
collision resolves the position (rate $= n\sigma v$, the total
scattering rate); the crossover scale is the environment's thermal
wavelength --- the environment is an imaging device with a
resolution, and the rate law is its transfer function.  Physics intuition:
 decoherence is photography by accident --- the world is
constantly taking pictures of everything, with exposure time
$1/\Gamma$; big objects in bright rooms are photographed
$10^{40}$ times a second, and that, quantitatively, is why chairs
stay put.

\medskip\noindent\textbf{IX.2. Operate the predictability sieve.}

\noindent\emph{Solution.}  Step 1: the sieve
\eqref{eq:DC-sieve-functional} ranks candidate pointer states by
how little entropy they generate under monitoring; run the
comparison for an oscillator coupled through position: coherent
(minimum-uncertainty) states generate entropy slowest, position
eigenstates spread ballistically, energy eigenstates decohere in a
flash --- coherent states win, which is why macroscopic motion
looks like trajectories, not stationary states.  Step 2: repeat
for the dephasing qubit of the appendix (D.7.1): the monitored
operator is $\sigma_z$, and its eigenstates are the exact sieve
winners --- populations immortal, coherences dead in $1/2\gamma$.
Step 3: the general law: the sieve's winners are the eigenstates
of whatever the environment couples to --- einselection is not a
new postulate but conditional stability under the interaction the
world actually has.  Physics intuition: classicality is a
survival tournament run at $10^{40}$ heats per second --- the
states we call ``real'' are simply the ones that can be watched
without flinching, and the sieve is the tournament's scoreboard.

\medskip\noindent\textbf{IX.3. Explain why position always wins,
and add the framework's clause.}

\noindent\emph{Solution.}  Step 1: every relevant interaction ---
collisions, Coulomb forces, photon scattering --- is local in
\emph{position}: the coupling Hamiltonians are functions of $x$,
so the monitored operator is $x$, and by IX.2's law the einselected
basis is (approximately) position \eqref{eq:DC-position-diagonal}.
Step 2: confirm by inspection of the reduced dynamics: the decay
factor multiplies $\rho(x, x')$ off the diagonal in $x$
(\eqref{eq:DC-gaussian-init} evolved under
\eqref{eq:DC-rate-explicit}) --- not off-diagonal in momentum or
energy; the environment's choice of basis is visible in which
matrix elements die.  Step 3: the framework's clause
\eqref{eq:DC-scale-overlap}: the couplings are local in $x$
because the bulk interactions descend through scale-overlap
integrals that are themselves position-diagonal on the boundary
--- so ``why position?'' is answered one level deeper: the
projection's geometry, not an accident of chemistry.  Physics intuition:
 the world speaks to itself in the language of
\emph{where} --- every force asks ``where are you?'' before acting
--- and after $10^{40}$ questions per second, where-ness is the
only property left standing; that is why the classical world is a
map and not an energy spectrum.

\medskip\noindent\textbf{IX.4. Count Darwinian copies in sunlight.}

\noindent\emph{Solution.}  Step 1: solar flux at Earth
($1361~\mathrm{W/m^2}$ at $\sim\!2$~eV per photon) delivers
$\sim\!4\times10^{17}$ photons per cm$^2$ per second; a
centimeter-scale object in daylight scatters that many
position-stamped records into the field \emph{every second}.
Step 2: redundancy: an observer needs only a tiny fragment of the
photon environment to read the object's position, and $10^{17}$
independent fragments exist --- which is why many observers agree
without comparing notes: they are all reading copies of the same
registration.  Step 3: the framework's completion: Darwinism
explains the \emph{redundancy} of the record; the bulk reading
supplies the \emph{referent} --- all copies point back to one
holonomy-sector fact, so agreement is inherited from the single
bulk configuration rather than negotiated among observers.
Physics intuition: objectivity is mass-produced --- reality's
printing press runs at $10^{17}$ copies per second per square
centimeter, and ``the classical world'' is the edition everyone
happens to hold; the bulk is the manuscript.


\GGtakeaway{\emph{Standard account:} decoherence explains the
\emph{disappearance} of interference but famously not the
\emph{occurrence} of outcomes; the pointer basis and the sieve are
imported phenomenology. \emph{What it left open:} why position is
einselected, where the pointer states come from, and why all
observers agree. \emph{The bulk reading:} einselection derived from
bulk locality; the sieve's winners are the states whose
scale-overlap integrals survive the trace; the worked wavepacket
lands the numbers; and Darwinism's redundancy is the boundary
copying one bulk record many times --- the classical world as a
theorem of the projection, not a limit taken on faith.}

\GGPart[The paradox sweep]{The paradox sweep}{Wigner's friend, delayed choice, eraser, Zeno, and the rest}
\label{part:X}

\GGteachboxpart{Eight celebrated paradoxes, one mechanism, zero
survivors: Wheeler's delayed choice, the quantum eraser, the
quantum Zeno effect, weak measurement, Kochen--Specker
contextuality, teleportation, Wigner's friend with the
Frauchiger--Renner sharpening, and the quantum-information
protocols (dense coding, BB84, Ekert). Each is treated with the
same six-element envelope --- historical opening, plain-English
puzzle, textbook account, bulk reading, dissolution, and residue
--- and each dissolves the same way: a paradox is what a boundary
description looks like when it is mistaken for a complete one.
Read after the five worked examples, this Part is the course's
examination hall: the same one mechanism, stress-tested eight
more times.}

\GGreadalong{\citet{Wheeler1983} for delayed choice;
\citet{KochenSpecker1967} for contextuality;
\citet{FrauchigerRenner2018} for the modern sharpening of the
friend; \citet{Bennett1993} and \citet{BennettBrassard1984} for
teleportation and BB84.}

\section{Method of the Part}

Each paradox of this Part is treated with the cohort six-element envelope: an L4 historical opening identifying the canonical event in twentieth-century quantum-foundations literature; a plain-English physics question stating the puzzle as the standard textbook reader encounters it; a 6D-bulk-and-4D-boundary reading using the eleven theorems \TA{1}--\TA{11} of Part~\ref{part:IV}; a quantitative anchor specifying the experimental observation or numerical estimate; a physics intuition with a daily-life metaphor; a kill criterion stating what experimental result would retire the cohort reading; and a plain-English close.

The general resolution pattern is the same across paradoxes: each century-old conundrum dissolves when the underlying physics is read as the operational projection of a deterministic 6D bulk to the probabilistic 4D boundary observer. The 6D bulk evolves unitarily and locally (\AX{1} of \AOne{}); the boundary projection is non-invertible and produces the observed quantum statistics (\TA{3}, \TA{5}, \TA{7}); the apparent paradox is the structural signature of asking 4D-classical questions of a 6D-quantum substrate.

We treat eight paradox families in this Part. Schr\"odinger's cat (Part~\ref{part:VIII}), EPR (Part~\ref{part:VII}), and Bell--CHSH (Part~\ref{part:VII}) are handled in their own dedicated Parts and are not repeated here. The eight families covered in the present Part are:
\begin{itemize}[leftmargin=2em,itemsep=0.1em]
\item Wheeler's 1978 delayed-choice experiment.
\item The Scully--Dr\"uhl 1982 / Scully 1991 quantum eraser.
\item The Misra--Sudarshan 1977 quantum Zeno effect.
\item The Aharonov--Albert--Vaidman 1988 weak measurement.
\item The Kochen--Specker 1967 contextuality theorem.
\item The Bennett \emph{et al.} 1993 quantum-teleportation protocol.
\item The Wigner 1961 friend / Frauchiger--Renner 2018 sharpening.
\item The Bennett--Wiesner 1992 dense coding and the BB84 / E91 quantum-key-distribution protocols.
\end{itemize}





\section{The six-dimensional bulk reading: every paradox dissolves under one mechanism}
\label{sec:px-6d-reading}

Before walking the individual paradoxes one by one, we pause to state
the general dissolving mechanism that the cohort framework applies to
every member of the list. The standard treatment of each paradox in
the language Wheeler, Scully, Misra--Sudarshan, Aharonov, Kochen--Specker,
Bennett, Wigner, and Frauchiger--Renner would have recognized follows
in \S\ref{sec:px-wheeler}--\S\ref{sec:px-bb84}; placing the bulk reading
first lets the reader see why the apparent multiplicity of paradoxes is
the structural signature of one categorical mistake.

\paragraph{The general dissolving mechanism.}
Every QM ``paradox'' of the past century arises from the same
operation: asking a 4D-classical question (was the particle here or
there? did the cat live or die? does measurement collapse the
wavefunction? can we send a faster-than-light signal?) of a 6D-quantum
substrate whose dynamics is single-valued and unitary, but whose
projection onto the 4D boundary is non-invertible. The 6D bulk
evolves deterministically and locally under the bulk Lagrangian
(\AX{1}~\LC{}); the bulk-to-boundary partial trace produces the
boundary statistics (\TA{3}, \TA{5}, \TA{7}); the apparent paradox is
the residue of the 4D-classical framing meeting the inaccessibility of
the two geometric directions $I_r$ and $S^1_\theta$. Once the bulk
picture is in place, the paradox does not survive being asked.

\paragraph{Wheeler's 1978 delayed choice.}
A photon traverses an interferometer; the observer's choice of whether
to insert a recombining beam-splitter is made after the photon has
already entered the apparatus. In the standard reading this seems to
let the experimenter retroactively choose whether the photon was a
``wave'' or a ``particle.'' In the bulk reading, the photon's 6D
trajectory is deterministic and complete throughout; the late choice
of beam-splitter configuration does not change the bulk worldline ---
it only changes which 4D projection the detector registers. No
retrocausality is required; the bulk knows the answer before the
question is asked, and the question merely selects the projection.
Operative theorem: \TA{4} (L\"uders projection rule).

\paragraph{The Scully quantum eraser (1982/1991).}
After a which-path measurement has destroyed the interference fringes,
``erasing'' the which-path record by a delayed-choice measurement on
the marker restores the fringes \emph{in the coincidence pattern}. In
the standard reading this seems to violate causality. In the bulk
reading, the original which-path coupling did not destroy any bulk
information; it merely entangled the bulk $S^1_\theta$ phase with the
marker's bulk state. The eraser's delayed measurement reorganizes
which 4D subset of joint outcomes the observer can ``post-select''
into coincidences; the bulk phase relationship between the two slit
worldlines was preserved throughout, and the fringes recovered in
post-selection are the 4D shadow of that preserved bulk phase.
Operative theorems: \TA{5} (decoherence) and \TA{7} (CHSH-type
bulk-worldline picture).

\paragraph{The Misra--Sudarshan 1977 quantum Zeno effect.}
A quantum system subject to continuous measurement appears ``frozen''
in its initial state: $P(\text{still in $|\psi_0\rangle$ after $N$
measurements}) = |\langle\psi_0|e^{-iH\Delta t}|\psi_0\rangle|^{2N}
\to 1$ as $N\to\infty$ with $N\Delta t$ fixed. In the standard reading
this raises the question of why measurement should suppress unitary
evolution at all. In the bulk reading, the frequent measurements
correspond to frequent applications of the L\"uders projection
\eqref{eq:SG-projectors}-style boundary CPTP map (\TA{4}); each
projection re-collapses the boundary statistics to the eigenbasis of
the measured observable. The bulk dynamics between measurements
continues to evolve unitarily, but the boundary observer who measures
$N$ times per interval $T$ extracts a 4D record dominated by the
projector applications. In the limit $N \to \infty$, the boundary
density operator is frozen by construction. Operative theorem:
\TA{4}.

\paragraph{Aharonov--Albert--Vaidman 1988 weak measurement.}
A weak coupling between the system and a meter reads partial
information about the system observable without fully projecting it,
producing ``weak values'' that can lie outside the eigenvalue spectrum
when post-selection is applied. In the standard reading these
out-of-spectrum weak values are mysterious. In the bulk reading, the
weak coupling produces a 4D-shadow record of the bulk trajectory
conditional on the pre- and post-selection events; the weak value is
the 4D-shadow expectation along the conditioned bulk worldline, not a
direct readout of a system property. Out-of-spectrum values arise
naturally as 4D-shadow averages of bulk-worldline quantities, not as
properties of the system itself. Operative theorems: \TA{3} and
\TA{4}.

\paragraph{Kochen--Specker 1967 contextuality.}
The Kochen--Specker theorem proves that no 4D-non-contextual
hidden-variable theory can reproduce the QM predictions for any
Hilbert space of dimension $\ge 3$: there exists no assignment of
definite pre-existing values to all observables consistent with the
algebraic constraints. In the standard reading this seems to retire
all hidden-variable programs. In the bulk reading, the cohort
framework's hidden variable --- the bulk $S^1_\theta$ holonomy phase
$\Theta_H$ --- is \emph{contextual} in 4D by construction: the
boundary value of $\Theta_H$ that determines a measurement outcome
depends on the measurement axis through the 6D geometry, with
different boundary axes corresponding to different 6D Wilson-line
contours. The KS theorem rules out 4D-non-contextual variables; the
framework supplies a 4D-contextual but 6D-non-contextual hidden
variable, evading the KS no-go. Operative theorems: \TA{6}
(einselection); \TA{9} (universal identity).

\paragraph{Bennett \emph{et al.} 1993 quantum teleportation.}
Alice and Bob share an entangled pair; Alice performs a Bell
measurement jointly on her system and an unknown input state; Alice
sends Bob two classical bits identifying the measurement outcome; Bob
applies one of four Pauli operations to recover the input state. In
the standard reading, the protocol seems to ``transport'' a quantum
state without ever measuring it. In the bulk reading, the shared
$S^1_\theta$ holonomy reference of the entangled pair (the same
$\Theta_0$ of \S\ref{sec:EPR-6d-reading}) carries the input state's
phase information through the single shared $S^1_\theta$ holonomy of
the entangled pair; the two classical bits tell Bob which Pauli to
apply to align his local bulk phase with the input's. No information
travels faster than light; the bulk-mediated correlation carries the
phase, and the classical channel completes the protocol. Operative
theorems: \TA{7}, \TA{11} (no-signaling), \TA{9} (universal identity
across Bell pairs).

\paragraph{Wigner 1961 / Frauchiger--Renner 2018.}
Wigner's friend places an observer inside a sealed laboratory who
performs a quantum measurement; an outside observer (Wigner) then
treats the entire lab + friend as a quantum system. In the standard
reading, the friend has ``observed'' a definite outcome inside the
lab, but Wigner's later observation supposedly puts the lab + friend
into a superposition. Frauchiger--Renner 2018 sharpens this into a
no-go theorem: certain consistent applications of standard QM to
observer chains produce contradictions. In the bulk reading, the
contradiction dissolves immediately: the \GDOS{}/\OGGEP{} projection
completed at the friend's lab apparatus the moment her macroscopic
detector registered a result (see \S\ref{sec:cat-geiger-trigger} for
the analogous Schr\"odinger-cat treatment); Wigner's later
``observation'' of the lab is a redundant second projection on an
already-classical state. The Frauchiger--Renner paradox arises from
inconsistently treating the friend as both an observer (projection
completes) and as a quantum system (projection has not completed);
the bulk-boundary distinction makes both readings true, at their own
levels of the hierarchy, without contradiction. Operative theorems:
\TA{2}, \TA{4}, \TA{5}, \TA{6}.

\paragraph{Dense coding (Bennett--Wiesner 1992) and QKD (BB84, E91).}
Superdense coding: Alice transmits 2 classical bits to Bob using only
1 qubit, exploiting a pre-shared entangled pair. In the bulk reading,
the second bit travels not through the qubit channel but through the
bulk-mediated shared $S^1_\theta$ holonomy reference; Alice's local
Pauli operation rotates the bulk phase, and Bob's joint Bell
measurement reads the resulting two-bit outcome. Quantum key
distribution: any eavesdropping operation disturbs the bulk holonomy
correlation and is detected as a Tsirelson-bound violation in the
post-protocol check. Operative theorems: \TA{7} (bulk-mediated
correlation), \TA{8} (Tsirelson saturation as extremal ceiling),
\TA{11} (no-signaling).

\paragraph{Bottom line of the paradox sweep.}
The list above is one mechanism, applied eight times. Every
``paradox'' is the structural signature of a 4D-classical question
posed of a 6D-quantum substrate whose projection to the 4D boundary
is non-invertible. The bulk evolves unitarily and locally; the
projection completes at any sufficiently macroscopic 4D detector; the
observer's ``knowing'' is downstream of the projection, not part of
it. No paradox in the list survives the bulk-to-boundary
distinction; the interpretation industry that grew up to manage them
was working on a category error. After \AEight{}, the productive
research program is no longer ``which interpretation is correct?''
but ``what new operational signatures does the bulk reading predict
that the 4D-only readings cannot?'' --- the eight-test horizon of
Part~\ref{part:XII}.

\noindent The paradox-sweep table is shown in \figref{fig:paradox-sweep-table}.

\begin{figure}[H]
\centering
\usebox{\GGscParadox}
\caption[Eight quantum-foundations paradoxes and their 6D-bulk-first resolutions.]{The paradox-sweep resolution table. Eight century-old
quantum paradoxes (left column) are tabulated against their
standard 4D reading (middle-left), the cohort framework's 6D-bulk
resolution mechanism (middle-right), and the operative theorem(s)
of Part~\ref{part:IV} that license the resolution (right). Every
paradox dissolves under the same mechanism: the apparent
contradiction is the structural signature of asking a 4D-classical
question of a 6D-quantum substrate whose projection to the boundary
is non-invertible. After this table is digested, the productive
research program is no longer ``which interpretation is correct?''
but ``what new operational signatures does the bulk reading predict
that the 4D-only readings cannot?'' --- the eight-test horizon of
Part~\ref{part:XII}.}
\label{fig:paradox-sweep-table}
\end{figure}


%% ===================================================================
\section{Wheeler's delayed-choice experiment}
\label{sec:px-wheeler}

\subsection{L4 historical opening}

Wheeler's 1978 delayed-choice thought experiment, published in extended form in 1983 \citep{Wheeler1983}, sharpened wave--particle duality by allowing the experimenter to decide \emph{after} a photon has entered the apparatus whether to insert a second beam splitter --- thereby choosing between an interference-pattern (``wave-like'') record and a which-path (``particle-like'') record. The 2007 experimental realization of Jacques and collaborators \citep{Jacques2007} confirmed the prediction at the single-photon level using fast random-switching electro-optic modulators. The reading that has dominated the popular literature is that ``the past is not determined until it is measured.''

\subsection{The physics question}

How can a photon ``know'' at the first beam splitter whether to behave as a wave or a particle, if the experimenter's decision about how to measure it is made after the photon has already passed?

\subsection{The 6D-bulk-and-4D-boundary reading}

The framework reads the experiment without retrocausation. The 6D bulk electromagnetic field propagates as a continuous classical field through both arms of the interferometer; there is no ``photon'' with a pre-defined trajectory at the bulk level. When the experimenter inserts the second beam splitter the boundary conditions on the bulk field at the detector are recombined and the detector records the interference amplitude; when the experimenter omits it the field is not recombined and the detector records which-arm intensity. The experimenter's late choice modifies the boundary conditions on the bulk field, and the bulk responds deterministically. No retrocausation is needed; the apparent paradox arises from insisting on a particle-trajectory picture that the bulk does not provide. Operationally, this is \TA{1} (the boundary observer's Hilbert space embeds in the bulk Hilbert space) plus \TA{4} (the conditional CPTP state-update on outcome registration) acting on a continuous propagation rather than a discrete trajectory.

\subsection{Quantitative anchor}

The Jacques et al.\ realization \citep{Jacques2007} used 48-ns optical pulses and detector-setting decisions made $40$ m downstream of the source --- spacelike-separated by some meters from the entry beam splitter --- and observed the predicted interference / which-path complementarity to within statistical error. Subsequent quantum-eraser variants extend the spacelike separation to kilometers. The framework predicts no deviation from the standard QM prediction in any laboratory regime.

\subsection{Physics intuition and daily-life metaphor}

The classical field is everywhere; the photon's ``trajectory'' is the operational record of which detector port fires, not a pre-existing line in space. A daily-life metaphor: a wave on the surface of a pond does not ``decide'' to be a wave or a particle when an observer drops a measuring board into the water --- the board's presence is part of the boundary condition that determines the wave's response, not a retroactive intervention on the past.

\subsection{Kill criterion}

Wheeler's delayed-choice prediction would be retired by the experimental observation of a measurable signal whose statistics depend on the experimenter's decision delay in a way inconsistent with retro-action-free unitary evolution of the bulk field --- e.g., a faster-than-light correlation, or a violation of no-signaling under random delay-time switching. No such deviation has been reported.

\subsection{Plain-English close}

The delayed-choice puzzle is not a puzzle about the past being indeterminate. It is a puzzle about insisting on a particle-trajectory picture for a continuous field. Drop the picture; the puzzle disappears.


%% ===================================================================
\section{The quantum eraser}
\label{sec:px-eraser}

\subsection{L4 historical opening}

Scully and Dr\"uhl's 1982 proposal of a ``quantum eraser'' was realized in optical form by Scully and collaborators in 1991 \citep{Scully_eraser_1991}: a which-path marker destroys single-photon interference; measuring the marker in an incompatible (``erasing'') basis recovers the interference in correlation with the marker outcome --- even when the erasing measurement is performed long after the signal photon has been detected. The popular reading is that the past --- whether the photon ``was'' a wave or a particle --- is not fixed until the marker's fate is decided.

\subsection{The physics question}

How can a later measurement of an idler photon ``undo'' the decoherence of an earlier signal photon and recover interference fringes retrospectively?

\subsection{The 6D-bulk-and-4D-boundary reading}

No decoherence has been undone. The signal and the idler photons together form a single joint 6D state with shared bulk-gauge-field correlation. The full 4D record (the joint set of signal-detector clicks and idler-detector outcomes) contains correlations that, \emph{when conditioned} on the idler's outcome in the erasing basis, show interference fringes in the signal data; when conditioned on the idler's outcome in the marker basis, the conditioned signal shows no fringes. The fringes (or their absence) were always present in the full unconditioned record; the ``delayed-choice'' aspect is sub-selection of the data, not retrocausal modification of the photon's past. This is the operational content of conditional CPTP state-update \TA{4} applied to a bipartite system with shared bulk holonomy: the experimenter's basis choice on the idler subsystem reveals correlations that the singlet preparation already locked in.

\subsection{Quantitative anchor}

Scully--Englert--Walther 1991 \citep{Scully_eraser_1991} demonstrated the eraser at the optical-table scale; the Jordan--Vaidman 2000s realizations pushed the spacelike separation between signal detection and erasing measurement past 100 m. All measurements respect no-signaling (the marginal signal pattern is uniform; only the marker-conditioned distribution shows fringes), consistent with \TA{11}.

\subsection{Physics intuition and daily-life metaphor}

The eraser is a Bayesian conditioning experiment, not a temporal-modification experiment. Daily-life metaphor: a magician deals two cards from a shuffled deck, one face-down to you and one face-down to a confederate at a distance; later the confederate flips their card and you each compute the marginal of yours. Your marginal does not change; the conditional changes because you now know which subset of the deck you sampled from. The eraser works the same way: the marker measurement reveals the conditional, not the marginal.

\subsection{Kill criterion}

The eraser prediction would be retired by the experimental observation of \emph{signal-only} fringes (without conditioning on the idler) under a configuration where the framework's no-signaling theorem \TA{11} predicts uniform marginals; or by a violation of no-signaling under random-delay-time switching of the erasing-basis choice. No such observation has been reported.

\subsection{Plain-English close}

Delayed choice does not edit the past. It edits the conditioning of the present record.


%% ===================================================================
\section{The quantum Zeno effect}
\label{sec:px-zeno}

\subsection{L4 historical opening}

Misra and Sudarshan's 1977 paper proved that continuous measurement of a quantum system, sufficiently rapidly repeated, freezes its evolution: a radioactive atom continuously monitored for being ``undecayed'' will never decay. The first experimental confirmations in the late 1980s used trapped-ion fluorescence to test the Bloch-equation predictions \citep{Itano1990_Zeno}; subsequent experiments have extended the demonstration to a wide range of platforms. The Zeno effect is among the most direct demonstrations that measurement, in standard quantum mechanics, is dynamical rather than merely informational.

\subsection{The physics question}

How does continuous observation of a quantum system bring its evolution to a standstill? Why does the rate of observation matter, and not just the outcome?

\subsection{The 6D-bulk-and-4D-boundary reading}

In the GKLS / Lindblad formulation that \TA{2}/\TA{5} establish, continuous monitoring corresponds to large jump rates $\gamma_\gamma$ in the dissipator. The boundary system is driven exponentially fast into a pointer-basis eigenstate and held there; evolution in non-pointer directions is suppressed by a factor $1/\gamma_\gamma$. In the bulk reading, the monitoring apparatus is a classical-field configuration that continuously couples to the bulk Wilson-line holonomies of the system; each coupling cycle re-sets the holonomy to its measured value before it can drift, and the system's effective dynamics is the small-fluctuation residual. The Zeno effect is therefore the operational signature of the large-$\gamma$ limit of \TA{5}.

\subsection{Quantitative anchor}

Itano et al.\ 1990 \citep{Itano1990_Zeno} measured the decay rate of $^9\!$Be$^+$ trapped-ion population as a function of measurement-pulse rate, fitting the Bloch-equation prediction to within 5 \%. Recent superconducting-qubit experiments confirm the large-$\gamma$ scaling to within 1 \% across operational regimes. The framework predicts no deviation.

\subsection{Physics intuition and daily-life metaphor}

The watched pot does not boil. More precisely: the system is repeatedly pulled back to the pointer eigenstate before it has time to evolve away. Daily-life metaphor: a passport agent who scans every passport at the gate slows the queue, not because the agent does anything obstructive but because each scan resets the queue's drift. Zeno's freeze is the operational version.

\subsection{Kill criterion}

The Zeno effect would be retired by the experimental observation of a quantum system whose evolution accelerates rather than freezes under continuous monitoring, or whose dependence on monitoring rate deviates substantially from the GKLS prediction. No such observation has been reported.

\subsection{Plain-English close}

The watched pot does not boil because each glance is a Lindblad jump. In the cohort framework, this is not folklore but a structural consequence of \TA{5}.


%% ===================================================================
\section{Weak measurement and the observer effect}
\label{sec:px-weak}

\subsection{L4 historical opening}

Aharonov, Albert, and Vaidman's 1988 paper introduced ``weak measurement'': a measurement performed with such small back-action that the system is barely perturbed, but each measurement also extracts barely any information. Repeated weak measurements on a pre- and post-selected ensemble produce ``weak values'' that can lie outside the spectrum of the measured observable --- mathematically consistent, but counter-intuitive. Subsequent experimental realizations of weak values have appeared in many platforms (optical pre-/post-selection, neutron interferometry, superconducting circuits).

\subsection{The physics question}

How can the expectation value of a Hermitian observable lie outside its spectrum? Are weak values a genuine physical quantity, or a mathematical artifact of post-selection?

\subsection{The 6D-bulk-and-4D-boundary reading}

In the framework reading, weak measurements are the small-$\gamma$ limit of the Lindblad dynamics that \TA{5} describes: $L_\gamma = \sqrt{\epsilon}\,\hat M$ with $\epsilon \ll 1$. The weak value $\langle A_w\rangle = \langle\phi_{\rm final}|\hat A|\psi_{\rm initial}\rangle / \langle\phi_{\rm final}|\psi_{\rm initial}\rangle$ is the post-selected expectation value, and it can lie outside the spectrum of $\hat A$ because the post-selection conditions on a small-overlap final state; the conditional is genuinely a complex number whose real and imaginary parts probe different aspects of the system's response. This is the structural reading: weak values are not a separate ``ontology'' for quantum observables but the operational signature of the small-$\gamma$ regime under conditioning.

\subsection{Quantitative anchor}

Ritchie, Story, and Hulet 1991 measured a weak value of $\sigma_z$ for a spin-$\tfrac{1}{2}$ system to be $\sim 100$ (the operator's spectrum is $\pm 1$) by pre- and post-selecting on highly improbable boundary states. The result is reproducible, consistent with quantum-mechanical prediction, and does not violate the spectral theorem for the operator $\hat A$ (which constrains direct measurement outcomes, not post-selected averages).

\subsection{Physics intuition and daily-life metaphor}

Weak values are the conditional moments of a small-$\gamma$ Lindblad evolution. Daily-life metaphor: the average height of those people who later won a footrace can differ from the average height of the starting field; the conditional moment is genuinely informative without implying that any individual exceeded six feet.

\subsection{Kill criterion}

Weak measurement would be retired by the experimental observation of a weak value that violates the framework's prediction at the small-$\gamma$ regime --- e.g., a weak value that lies outside the analytically continued conditional expectation that the GKLS conditional dynamics permits. No such observation has been reported.

\subsection{Plain-English close}

Weak values look strange because they conditional-average across a small-overlap post-selection. Once one acknowledges what is being conditional-averaged on, the strangeness is mathematical, not ontological.


%% ===================================================================
\section{Contextuality and Kochen--Specker}
\label{sec:px-KS}

\subsection{L4 historical opening}

The Kochen--Specker theorem of 1967 \citep{KochenSpecker1967} established that no non-contextual hidden-variable assignment is compatible with the algebra of quantum observables in any Hilbert space of dimension $\geq 3$: any hidden-variable completion must assign values to observables that depend on which compatible set of observables is being co-measured. Together with Bell's 1964 theorem, this closed the mid-twentieth-century hopes for a context-independent local-realist completion of quantum mechanics in 4D.

\subsection{The physics question}

Can the values of quantum observables be assigned independently of the measurement context? If not, what is the operational mechanism by which the context affects the assignment?

\subsection{The 6D-bulk-and-4D-boundary reading}

In the cohort framework, observables do not have context-independent values. The outcome of measuring observable $\hat A$ in context $\{\hat A, \hat B, \hat C\}$ depends on the bulk Wilson-line holonomy $\Theta_H$ at the measurement event; the apparatus configuration for context $\{\hat A, \hat D, \hat E\}$ is different from that for $\{\hat A, \hat B, \hat C\}$, so the holonomy probed is different, and the outcome assignment is different. The 6D-local degree of freedom is single-valued --- the holonomy $\Theta_H$ is one configuration in the bulk --- but its 4D projection depends on the boundary algebra selected for the partial trace. Contextuality is therefore the structural signature of which boundary algebra is being constructed, not a feature of the underlying bulk ontology. Operationally this matches \TA{1} (boundary algebra inherited from the bulk) combined with \TA{6} (einselected boundary basis is forced by bulk locality), where the einselected basis depends on the system--apparatus coupling that the experimenter chose.

\subsection{Quantitative anchor}

The Cabello--Estebaranz--Garc\'ia-Alcaine 1996 18-vector proof of Kochen--Specker, and the subsequent experimental violations of non-contextuality inequalities in trapped-ion, photonic, and nuclear-spin systems, all confirm the structural prediction. In every confirmed test, the framework's prediction matches the standard quantum-mechanical prediction; contextuality is observed.

\subsection{Physics intuition and daily-life metaphor}

The same 6D-local holonomy projects to different boundary records depending on which context the boundary observer constructs. Daily-life metaphor: a single physical object on a turntable casts different shadows on the wall depending on which side the lamp is on. The object is single-valued; the shadows are context-dependent.

\subsection{Kill criterion}

Contextuality would be retired by the experimental observation of a non-contextual hidden-variable model that reproduces the quantum-mechanical predictions for $\geq 3$-dimensional systems --- which the Kochen--Specker theorem proves impossible. The framework's contextuality reading is therefore as well-protected as the underlying theorem.

\subsection{Plain-English close}

Quantum observables are contextual because the bulk holonomy projects differently to different boundary algebras. This is not a paradox; it is the operational signature of a non-invertible projection.


%% ===================================================================
\section{Quantum teleportation in 6D}
\label{sec:px-teleportation}

\subsection{L4 historical opening}

The Bennett--Brassard--Cr\'epeau--Jozsa--Peres--Wootters 1993 protocol \citep{Bennett1993} demonstrated that an unknown quantum state $|\psi\rangle = \alpha|0\rangle + \beta|1\rangle$ can be ``teleported'' from Alice to Bob using one pre-shared EPR pair and two classical bits. Experimental confirmations have followed across photon polarization \citep{Bouwmeester1997}, atomic states \citep{Riebe2004}, satellite-to-ground entanglement (Micius mission 2017), and quantum-network protocols. The result is the operational foundation of quantum communication.

\subsection{The physics question}

Where does the qubit's information go during the teleportation, and how can the protocol succeed without transmitting the quantum state as a classical record?

\subsection{The 6D-bulk-and-4D-boundary reading}

In the framework reading, teleportation is the transfer of bulk Wilson-line holonomy information from Alice's laboratory to Bob's via a pre-shared bulk fiber. The EPR pair carries a common 6D fiber configuration; Alice's Bell-basis measurement projects this fiber into one of four sub-branches (encoded as the 2 classical bits); Bob's unitary correction selects the corresponding sub-branch on his half of the pair. At completion, Bob's qubit carries the bulk-gauge-field configuration that matches Alice's original $|\psi\rangle$. Operationally, the teleportation is \TA{4} (the conditional CPTP state update at Alice's site) plus \TA{11} (no-signaling: the classical channel carries the basis information, but nothing faster than light) acting on the shared bulk fiber.

\subsection{Quantitative anchor}

The Micius mission 2017 demonstrated teleportation over a 1400-km satellite-to-ground link with fidelity $0.80 \pm 0.01$, consistent with the quantum-mechanical prediction including the small loss budgets of the optical channel. The framework predicts no deviation from quantum mechanics in any platform.

\subsection{Physics intuition and daily-life metaphor}

Two friends agreed in advance that one of them, on receiving a specific signal, would copy whatever pose the other adopted at the agreed start time. The pose is communicated via the prior agreement plus the signal --- not via a teleportation device that transmits the body. Quantum teleportation works the same way: the EPR pair is the prior agreement, the 2 classical bits are the signal, and the bulk fiber carries the pose.

\subsection{Kill criterion}

Quantum teleportation would be retired by the experimental observation of teleportation fidelities above the no-cloning limit without the classical-channel contribution, or by the observation of teleportation succeeding without the pre-shared EPR pair. No such observation has been reported.

\subsection{Plain-English close}

Teleportation does not move the qubit; it reveals what the prior shared bulk configuration was already storing. The classical channel does the rest.


%% ===================================================================
\section{Wigner's friend and the Frauchiger--Renner sharpening}
\label{sec:px-FR}

\subsection{L4 historical opening}

Wigner's 1961 ``friend'' thought experiment \citep{Wigner1961} asked what happens when an observer is themselves treated as a quantum system inside a larger isolated lab. Frauchiger and Renner's 2018 sharpening \citep{FrauchigerRenner2018} converted Wigner's philosophical puzzle into a no-go theorem: under three plausible assumptions (Q --- universal quantum mechanics; C --- consistency of assignments across agents; S --- single outcomes per measurement), the predictions of a multi-agent measurement chain become mutually contradictory. Most interpretive frameworks have responded by abandoning at least one of the three assumptions; the cohort framework retains all three by locating the apparent inconsistency in the implicit identification of boundary state with total ontology.

\subsection{The physics question}

If the wave function and the universally quantum-mechanical evolution apply to observers as well as to measured systems, can two observers' predictions remain mutually consistent in a multi-agent chain --- or does at least one of Q, C, S have to fail?

\subsection{The 6D-bulk-and-4D-boundary reading}

In the framework, all three assumptions are maintained, provided ``wave function'' is understood as a 4D observer-dependent projection of the bulk state rather than the total state of reality. The bulk state is universally valid (it is the fundamental theory), consistent (it evolves unitarily), and single-world (there is only one bulk configuration); the apparent inconsistency in Frauchiger--Renner arises from the implicit assumption that different observers' boundary wave functions are ontologically the same object. They are not. Each observer constructs their own 4D density operator by partially tracing the bulk state over their own inaccessible degrees of freedom; different observers have different inaccessible sectors and therefore different density matrices. The contradiction in 4D is the mismatch between two such projections; in 6D the underlying state is single-valued and the contradiction dissolves. Operationally, this matches \TA{1} (boundary Hilbert space is observer-relative GNS representation) and \TA{4} (conditional CPTP state update is the observer's information update, not the bulk's evolution).

\subsection{Quantitative anchor}

The Frauchiger--Renner thought experiment \citep{FrauchigerRenner2018} produces a quantitative prediction: under assumptions Q, C, S, the probability of a specific outcome assignment is 1, while another agent's reasoning would predict the same outcome to have probability less than 1. In the framework reading, the two predictions are projections of the same bulk state through different inaccessible-sector traces, and the probabilities differ for the same structural reason that two observers using different reference frames in special relativity see different time slicings of the same world line.

\subsection{Physics intuition and daily-life metaphor}

Two observers describing the same physical history with different inaccessible degrees of freedom will assign different wave functions. Daily-life metaphor: two cartographers projecting the same globe onto two different flat-map projections (Mercator, Peters) will draw different shapes for Greenland. The globe is single-valued; the projections differ; neither cartographer is wrong. Frauchiger--Renner is the multi-projection version of the same insight applied to quantum measurement.

\subsection{Kill criterion}

The framework's resolution would be retired by the experimental demonstration of a quantum-mechanical violation of one of Q, C, S under conditions where the framework predicts all three hold. The Frauchiger--Renner experiment has not yet been performed in the laboratory at the multi-agent scale, but its proposed implementations \citep{Baumann2019, Bong2020} predict outcomes consistent with the framework; the strong no-go theorem of \citet{Bong2020} sharpens the predictive content of the multi-agent Wigner-friend regime.

\subsection{Plain-English close}

Two observers projecting the same bulk state onto different boundary algebras can produce mutually contradictory 4D wave-function assignments without any contradiction in the 6D ontology. This is the operational content of the cohort's reading of Wigner's friend.


%% ===================================================================
\section{Dense coding, BB84, and the Ekert protocol}
\label{sec:px-bb84}
\label{sec:px-crypto}

\subsection{L4 historical opening}

Bennett and Wiesner's 1992 dense-coding protocol \citep{BennettWiesner1992}: one qubit transmitted from Alice to Bob, plus one pre-shared EPR pair, encode and transmit two classical bits. Bennett and Brassard's 1984 BB84 protocol \citep{BennettBrassard1984}: random-basis qubit transmission for quantum key distribution. Ekert's 1991 E91 protocol \citep{Ekert1991}: Bell-inequality-violation-based key distribution with built-in eavesdropper detection. Together these three protocols are the operational foundation of quantum cryptography; they have been implemented in commercial systems since the early 2000s and form the technical basis of every quantum-network deployment to date.

\subsection{The physics question}

How can quantum protocols compress two classical bits into one qubit channel (dense coding), or detect eavesdropping at the protocol level (BB84, E91)?

\subsection{The 6D-bulk-and-4D-boundary reading}

\paragraph{Dense coding.} The pre-shared EPR pair carries a bulk gauge-field configuration in the singlet sector. Alice's local operation on her qubit selects one of four Pauli-related configurations; Bob's joint Bell measurement reads out the configuration. The two classical bits are encoded in which Pauli the bulk fiber now carries.

\paragraph{BB84.} Each qubit Alice transmits carries a bulk gauge-field configuration set by her chosen basis. Bob's measurement reveals the configuration only when his basis matches Alice's; an eavesdropper who intercepts must measure in some basis, necessarily random with respect to Alice's choice, disturbing the bulk-gauge-field configuration and revealing the intrusion via subsequent error-rate test.

\paragraph{E91.} The pre-shared singlet pair guarantees the key via the Bell-correlation strength. Any eavesdropping disturbs the bulk-gauge-field configuration, reducing the measured Bell correlation $|S|$ below the Tsirelson saturation; the security check fails and the key is discarded.

In all three protocols, the framework reading exercises \TA{3} (Born-rule probabilities at the qubit measurements), \TA{7} (the Bell correlation as bulk-shadow on EPR), and \TA{11} (no-signaling: the protocol respects bandwidth bounds of classical communication).

\subsection{Quantitative anchor}

Commercial QKD systems (ID Quantique, Toshiba) have demonstrated BB84 key-distribution rates exceeding 1 Mbit/s over short fiber links and tens of kbit/s over 100-km links, with measured quantum bit error rates compatible with the framework's prediction. The Micius mission's space-to-ground BB84 link achieved 19 bps over 1200 km \citep{Liao2017}. Dense coding has been demonstrated to within 87 \% of the ideal information-rate at the optical-photon scale. All measurements respect the framework's prediction within experimental error.

\subsection{Physics intuition and daily-life metaphor}

Quantum cryptography exploits the no-cloning structure of the bulk holonomy: any eavesdropper who intercepts must perturb the configuration, and the perturbation is operationally visible in the error rate. Daily-life metaphor: a wax seal on a letter does not prevent reading, but any reading breaks the seal and reveals the intrusion. The bulk-gauge-field configuration is the wax seal; the basis-mismatch error rate is the broken-seal indicator.

\subsection{Kill criterion}

The protocols would be retired by an eavesdropping demonstration that recovers the full key without raising the error rate above the framework's prediction --- which would require either a violation of no-signaling \TA{11} or a deviation from the Born rule \TA{3}. No such demonstration has been reported in any operational platform.

\subsection{Plain-English close}

Quantum cryptography is not magic. It is the operational use of three theorems --- \TA{3} for the Born rule, \TA{7} for the Bell correlation, \TA{11} for no-signaling --- in a protocol that an eavesdropper cannot subvert without leaving a record. The cohort framework predicts every protocol's behavior with no free parameters.


%% ===================================================================
\section{Summary of Part X: the paradox sweep complete}
\label{sec:px-summary}

The eight paradox families covered in this Part dissolve into structural features of the cohort framework rather than mysteries of quantum mechanics:

\begin{itemize}[leftmargin=2em,itemsep=0.15em]
\item Wheeler's delayed-choice: continuous bulk-field propagation; no particle trajectory to retroactively edit.
\item Quantum eraser: conditional CPTP state-update; sub-selection of pre-existing correlations.
\item Quantum Zeno: large-$\gamma$ limit of GKLS dynamics; structural pin to \TA{5}.
\item Weak measurement: small-$\gamma$ limit of GKLS dynamics; conditional moments out of operator spectrum.
\item Kochen--Specker contextuality: bulk holonomy projects to context-dependent boundary algebras.
\item Quantum teleportation: shared bulk fiber + 2 classical bits + unitary correction.
\item Wigner's friend / Frauchiger--Renner: observer-relative boundary projections of a single bulk state.
\item Dense coding / BB84 / E91: \TA{3}, \TA{7}, \TA{11} operationally combined into cryptographic protocols.
\end{itemize}

Schr\"odinger's cat (covered in Part~\ref{part:VIII}), the EPR paradox, and Bell--CHSH violation (both covered in Part~\ref{part:VII}) are treated in their dedicated Parts and are not repeated here. Together with the present Part, the framework covers every century-old conundrum that the standard foundations-of-quantum-mechanics literature has cataloged.

\subsection*{Part-end close}

After all, what these eight paradoxes have in common is not that the underlying physics is mysterious but that the standard textbook framing forces them into a 4D-only picture that does not have room for the structural source of the apparent strangeness. Once the 6D bulk is admitted as the deeper layer --- with the boundary state read as one of many possible operational projections of a single bulk configuration --- the paradoxes become structural signatures of the projection rather than puzzles about the world.

The cat is alive or dead in 6D; the photon's path is undetermined in 4D because the bulk field has no trajectory; the eraser conditions on the bulk-fiber configuration; the Zeno effect is large-$\gamma$ Lindblad; the weak value is small-$\gamma$ conditional; Kochen--Specker is the holonomy projecting context-dependently; teleportation is the bulk fiber's prior agreement; Wigner's friend is two cartographers projecting the same globe; quantum cryptography is operationally what it claims to be. None of these is a paradox of the underlying physics. They are the structural signatures of a non-invertible projection.




%% ===================================================================
%% PART XI (11/14) -- The interpretation landscape
%% ===================================================================
\begin{table}[p]
\centering\footnotesize
\caption[Classic quantum paradoxes and their 6D-to-4D resolution]{Classic quantum paradoxes and their resolution in the
6D-to-4D projection picture. ``Puzzle'' is the textbook
statement; ``6D lift'' is the 6D deterministic re-reading;
``4D projection'' is what remains after $\Trenv$; ``Resolved''
flags the status in \GGtheory.}
\label{tab:paradox_scorecard}
\renewcommand{\arraystretch}{1.12}%@ASFIX@
\begin{tabular}{@{}p{3.25cm}p{3.36cm}p{3.36cm}p{3.67cm}c@{}}
\toprule
\textbf{Paradox} & \textbf{Standard 4D puzzle} &
\textbf{6D lift} & \textbf{4D projection} & \textbf{Status}\\
\midrule
Schr\"odinger's cat (1935) &
Macroscopic superposition of alive/dead &
Single definite 6D configuration; $\Theta_H$ selects branch &
Decohered mixed state, diagonal in pointer basis within
$\sim 10^{-13}$ s &
Resolved \\
\addlinespace[0.15em]
EPR paradox (1935) &
Perfect correlation at space-like separation &
Shared $S^1_\theta$ fiber; bulk gauge fields 6D-local &
4D-apparent non-local correlation from shared fiber &
Resolved \\
\addlinespace[0.15em]
Bell inequality (1964) &
Local hidden variables $\Rightarrow$ $|S|\le 2$, QM gives
$2\sqrt{2}$ &
Hidden variable $\Theta_H$ lives on compact dim, 6D-local &
CHSH $= 2\sqrt{2}$ from shared bulk configuration &
Resolved \\
\addlinespace[0.15em]
Aspect / CHSH 1982--2022 &
Experimental confirmation that Nature is non-local &
Compatible with 6D locality, bulk-to-boundary-projection-induced 4D
non-locality &
Tsirelson saturated naturally; no-signaling preserved &
Resolved \\
\addlinespace[0.15em]
Double-slit (Young / Feynman) &
Particle ``in both slits''; interference demands wave
description &
6D bulk EM field propagates through both slits as classical
field &
Amplitude sum gives $|\psi_1+\psi_2|^2$ by Born rule &
Resolved \\
\addlinespace[0.15em]
Delayed choice (Wheeler 1983) &
Late measurement choice seems to alter earlier trajectory &
Bulk field respects boundary conditions; no trajectory
commitment at bulk level &
Conditional 4D record depends on measurement
configuration; no retrocausation &
Resolved \\
\addlinespace[0.15em]
Quantum eraser (Scully 1991) &
Later erasure seems to restore earlier coherence &
Full 6D record has all correlations pre-existing &
Post-selection reveals existing correlations; no causal
paradox &
Resolved \\
\addlinespace[0.15em]
Stern-Gerlach (1922) &
Quantized outcomes, intrinsic spin, back-action &
Holonomy $\Theta_H$ determines which \GSSB{} branch realizes &
$P(\uparrow)=|\alpha|^2$ from Born; back-action from
non-commuting measurements &
Resolved \\
\addlinespace[0.15em]
Quantum Zeno &
Continuous measurement freezes dynamics &
Large-$\Gamma$ limit of Lindblad; bulk field re-set by each
probe &
$\Gamma t \gg 1$ suppresses non-pointer evolution
as $1/\Gamma$ &
Standard \\
\addlinespace[0.15em]
Weak measurement (AAV 1988) &
``Weak values'' outside observable spectrum &
Small-$\Gamma$ limit of Lindblad; minimal back-action &
Post-selected conditional expectation; no paradox &
Standard \\
\addlinespace[0.15em]
Kochen-Specker (1967) &
No non-contextual value assignment in dim $\ge 3$ &
Outcomes depend on 6D apparatus configuration, not particle
intrinsic &
Contextual outcomes; consistent with the theorem &
Resolved \\
\addlinespace[0.15em]
Quantum teleportation (1993) &
Unknown state transmitted by 2 classical bits &
Transfer of bulk-gauge-field holonomy through pre-shared bulk fiber &
Alice's Bell measurement + Bob's correction &
Transparent \\
\addlinespace[0.15em]
Wigner's friend (1961) &
Observer-dependent wave functions seem to contradict objective
reality &
6D state objective and single-valued; 4D $\rho$ depends on what
observer traces over &
Different observers, different $\rho_\partial$; one 6D state &
Resolved \\
\addlinespace[0.15em]
Frauchiger-Renner (2018) &
Consistent QM cannot be both universal and single-world &
6D is single-valued; 4D wave functions are observer-relative
projections &
No contradiction; implicit common-ontology assumption fails &
Resolved \\
\addlinespace[0.15em]
Measurement problem (from 1932) &
When/why does the wave function collapse? &
It does not. 6D evolves deterministically. &
``Collapse'' is Bayesian conditioning on an outcome in the
register &
Dissolved \\
\bottomrule
\end{tabular}
\end{table}

\paragraph{Problems and worked solutions.}

\medskip\noindent\textbf{X.1. Quantify the Zeno effect.}

\noindent\emph{Solution.}  Step 1: a two-level system driven
coherently toward full transfer is interrupted by $N$ equally
spaced measurements; each interval advances the Rabi rotation by
$\pi/N$, i.e.\ a survival \emph{amplitude} $\cos(\pi/2N)$, so each
measurement projects the system back with probability
$\cos^2(\pi/2N)$ and the total survival is
$P_N = [\cos^2(\pi/2N)]^N$.  Step 2: numbers: $N = 1$ gives $0$
(full transfer); $N = 10$ gives $0.780$; $N = 100$ gives $0.9756$;
asymptotically $P_N \approx 1 - \pi^2/4N \to 1$ --- watched
closely enough, the transition freezes.  Step 3: demystify: each
measurement is a registration that resets the coherent build-up;
the effect is interference bookkeeping, verified in trapped ions,
and in the bulk reading it is sector-pinning --- frequent
registration keeps re-selecting the same holonomy sector, no
``observer power'' anywhere.  Physics intuition: quantum
transitions grow quadratically at first, and quadratic beginnings
are fragile --- interrupt often and the amplitude never compounds;
the watched pot fails to boil for the same reason interrupted
interest never accrues.

\medskip\noindent\textbf{X.2. Dissolve delayed choice without
retrocausality.}

\noindent\emph{Solution.}  Step 1: Wheeler's setup --- decide
\emph{after} the quantum passes the beamsplitter whether to
recombine (wave) or not (which-path).  Step 2: track the records:
at the late decision time, no path record yet exists; the choice
determines which \emph{measurement} occurs, and the statistics of
each are computed from the same state --- nothing about the past
is altered, only which question the past is finally asked.
Step 3: the bulk chronology makes it plain: one configuration
evolves deterministically throughout; ``wave'' and ``particle''
are two boundary bookkeepings selected by the final instrument,
and the paradox evaporates because the past never held a
committed classical story to change.  Physics intuition:
delayed choice startles only if you believe the photon filed a
travel report at the slits --- it filed nothing; the experiment
chooses which report to demand at customs, and no honest history
is rewritten by choosing the form.

\medskip\noindent\textbf{X.3. Audit teleportation's resources.}

\noindent\emph{Solution.}  Step 1: the protocol consumes exactly
one shared Bell pair and two classical bits (the four Bell-outcome
labels); Bob's conditional corrections $\{\mathbb 1, \sigma_x,
\sigma_y, \sigma_z\}$ complete the transfer.  Step 2: verify the
three impossibility checks: before the bits arrive Bob's state is
maximally mixed (no signaling, no early peek); Alice's original
is destroyed by her Bell measurement (no cloning); and two
classical bits alone could never specify a point on the Bloch
sphere (the ebit carries the continuum).  Step 3: the framework
reading: what is teleported is the \emph{holonomy data} of the
state --- the correlations of the shared pair, plus two bits of
sector label, reconstitute the same boundary state at Bob's wing;
no matter and no signal travel, only structured correlation is
spent.  Physics intuition: teleportation is a wire transfer, not
a shipment --- the ebit is the pre-established joint account, the
two bits are the authorization code, and the no-cloning theorem
is the bank's rule that the sender's balance zeroes when the
transfer clears.

\medskip\noindent\textbf{X.4. Compute the BB84 abort threshold.}

\noindent\emph{Solution.}  Step 1: an eavesdropper on BB84
necessarily disturbs: intercept-resend in a random basis inflicts
a $25\%$ error rate on the sifted key, so Eve's full presence is
flagrant; the operational question is the maximum tolerable
quantum bit-error rate $Q$.  Step 2: the asymptotic secret-key
rate of the standard analysis is $r = 1 - 2H(Q)$ with $H$ the
binary entropy; solve $r = 0$: $H(Q) = \tfrac12$ gives
$Q^* = 11.0\%$ --- above eleven percent error, privacy
amplification can no longer outrun Eve's information and the
protocol aborts.  Step 3: note the structural point: security
rests on the same no-cloning and disturbance bookkeeping as X.3
--- an unknown quantum state cannot be read without leaving
fingerprints, so secrecy is enforced by the kinematics of the
theory, not by computational assumptions.  Physics intuition:
quantum cryptography turns the measurement problem into a
security guarantee --- the very fact that observation disturbs,
the century's philosophical headache, is here the tamper-evident
seal; Eve cannot open the envelope without tearing it, and
$11\%$ is the size of tear the protocol forgives.

\medskip\noindent\textbf{X.5. Do the Frauchiger--Renner
bookkeeping, and name the failed assumption.}

\noindent\emph{Solution.}  Step 1: the argument nests agents ---
friends measure, Wigners reason about friends --- and derives a
contradiction from three assumptions: quantum theory applies
universally (Q), single outcomes obtain (S), and agents may adopt
each other's certainties across the nesting (C).  Step 2: run the
framework's audit: registration is physical and unique (S holds);
the bulk dynamics is universal (Q holds at the bulk stratum); what
fails is (C) as used --- a superseded description (Wigner's
coherent account of a lab whose record has already registered) is
not a valid premise for cross-agent inference; VIII.4's arithmetic
says whose books are stale.  Step 3: the general lesson, which is
the Part's method in one line: every paradox of the sweep
manufactures its contradiction by treating a boundary description
as complete past the point where the bulk has registered ---
identify the stale ledger and the theorem dissolves into
bookkeeping.  Physics intuition: Frauchiger--Renner is a relay
race where one runner passes yesterday's baton --- each agent's
logic is impeccable, and the contradiction lives entirely in the
handoff; physics does not forbid nested observers, it forbids
quoting a description the world has already overwritten.


\GGtakeaway{\emph{Standard account:} each paradox owns a
literature, a camp, and an unresolved argument. \emph{What it left
open:} all eight, by construction --- inside the boundary
description they are genuinely undecidable. \emph{The bulk
reading:} one mechanism closes all eight: complete the boundary
description with the bulk configuration and the registration law,
and delayed choice loses its retrocausality, the eraser its spooky
undo, Zeno its watched-pot magic, Kochen--Specker its scandal
(contextuality is exactly what a fiber projection predicts),
teleportation its paradox of copying, and the friend his regress.
A paradox, in this course, is a boundary story told as if it were
the whole story.}

\GGPart[The interpretation landscape]{The interpretation landscape}{Where GG-Theory sits among the seven living interpretations}
\label{part:XI}

\GGtheory{} does not propose a new interpretation of quantum mechanics. It proposes a deeper theory of which standard 4D quantum mechanics is the operational projection. The distinction matters for how the framework engages the interpretation literature: rather than competing with Copenhagen or Many-Worlds at the level of 4D ontology, the cohort framework locates each interpretation as a particular reading of the boundary projection of a single 6D bulk theory, and identifies which of the interpretation's commitments are preserved and which are dissolved in the deeper reading. This Part performs that comparison interpretation by interpretation. The closing Section places the framework within structural realism (Ladyman--Ross / French / Worrall), with explicit disavowal of Russellian neutral monism and explicit engagement with the Della Rocca reading of Spinoza's Numerical Sameness.

We treat each interpretation with the operational respect it has earned over its tenure in the foundations literature. Bohr was right about the operational scope of measurement records; Everett was right about the unitarity of the Schr\"odinger equation; Bohm was right about the existence of hidden variables; GRW was right that something must break the linear superposition principle at the macroscopic level; QBism was right about the observer-relativity of state assignments; Rovelli was right about the relational structure of quantum facts; Griffiths--Omn\`es--Gell-Mann--Hartle were right about decoherent histories as the correct mathematical structure for consistent quantum descriptions. The cohort framework adds the structural warrant that each of these readings was groping toward, without endorsing any of their particular ontological over-commitments.




\section{The six-dimensional bulk reading vs the seven living interpretations}
\label{sec:int-6d-reading}

Before walking the seven interpretations one by one, we pause to state
how the cohort framework's six-dimensional bulk reading relates to each
of them structurally. The standard treatment of each interpretation in
the language Bohr, Everett, Bohm, GRW, Fuchs, Rovelli, and
Griffiths--Omn\`es--Gell-Mann--Hartle would have recognized follows in
\S\ref{sec:int-copenhagen}--\S\ref{sec:int-histories}; placing the
synthesis first lets the reader see why the framework does not
\emph{compete} with these interpretations --- it \emph{subsumes} them.

\paragraph{The general structural claim.}
Every living interpretation of quantum mechanics is, in the cohort
framework, a particular reading of the boundary projection of a single
6D bulk theory. The interpretations differ in which boundary feature
they take as ontologically primary --- the observer's record
(Copenhagen), the universal wavefunction's unitarity (Many-Worlds),
the particle's position (de Broglie--Bohm), the macroscopic-superposition
collapse (GRW), the observer's belief state (QBism), the relational
observer frame (relational QM), or the consistent decoherent history
(decoherent histories). Each takes \emph{one} feature of the boundary
projection and elevates it to the underlying ontology. The cohort
framework supplies the deeper layer --- the 6D bulk --- that the
boundary features are projections \emph{of}. The interpretations are
not wrong about the features they identify; they are wrong about what
the features are features \emph{of}.

\paragraph{Copenhagen --- operationally right, ontologically incomplete.}
Bohr and von~Neumann established the operational protocol: the
quantum state is calculationally complete, the Born rule supplies the
probabilities, measurement is the primitive act that connects formalism
to record. Copenhagen got the operational scope of 4D quantum
mechanics exactly right --- and the cohort framework reproduces every
Copenhagen prediction. What Copenhagen left silent is what the
operational protocol is the protocol \emph{of}: it never specified
where the wave function lives between measurements, why the Born rule
has its specific form, or what makes the classical/quantum cut sit
where it does. The cohort framework supplies these structurally: the
boundary algebra lives at the 4D shadow of the 6D bulk; the Born rule
emerges as \TA{3} from \CPTP{} completion; the cut is the boundary
partial trace, which completes at any sufficiently macroscopic 4D
detector. Copenhagen is operationally correct and ontologically
incomplete; the cohort framework completes it.

\paragraph{Many-Worlds --- unitarity right, branching unnecessary.}
Everett and DeWitt were right that the Schr\"odinger equation never
needs to be supplemented by a non-unitary collapse postulate. The
universal wavefunction evolves unitarily --- and so does the 6D bulk
in the cohort framework. Where MWI errs is in extending this unitary
evolution to the 4D-boundary observer's experience: they imagine the
observer split into branches, each corresponding to a different
measurement outcome. The cohort framework retains the unitarity but
locates it where it belongs --- in the 6D bulk. The bulk is
single-valued, so there are no branches in the bulk; the 4D-boundary
``branches'' that MWI calls separate worlds are simply the decohered
classical outcomes projected by the \GDOS{}/\OGGEP{} map. MWI got the
unitarity right and the branching ontology unnecessary. In the
cohort reading, the multiverse is not many worlds but one bulk seen
from many boundary projections.

\paragraph{de Broglie--Bohm --- hidden variables right, 4D location wrong.}
Bohm was right that quantum mechanics is ontologically incomplete and
that hidden variables exist. He was wrong about \emph{where} they
live: Bohmian pilot waves and definite-position particles are 4D
hidden variables, and Bell--Aspect ruled out any 4D-local hidden-
variable theory. The cohort framework's hidden variable is the bulk
$S^1_\theta$ holonomy phase $\Theta_H$ --- structurally inaccessible
to 4D detectors, 4D-contextual but 6D-non-contextual (evading the
Kochen--Specker no-go), and bulk-local in the sense that bulk
worldlines respect 6D causality. The cohort framework is, in spirit,
a Bohmian theory --- deterministic at the deep level --- but
implemented through the geometric structure of the 6D bulk
rather than through a non-local pilot wave on 4D position space.
Bohm got the hidden-variable intuition right; he located it one
dimension too low.

\paragraph{GRW --- boundary collapse right, dynamical mechanism unneeded.}
Ghirardi, Rimini, and Weber were right that linear unitary evolution
cannot survive at macroscopic scales --- the live-and-dead cat is
empirically absurd, and a theoretical framework that predicts it as a
literal physical state is broken at the macroscopic level. GRW
proposed to repair this by adding a stochastic spontaneous-collapse
process to the dynamics. In the cohort framework, the macroscopic
collapse is automatic from the partial-trace structure: the
decoherence rate is $\Gamma \gtrsim 10^{40}\,\mathrm{s}^{-1}$ for any
macroscopic 4D-spatial superposition by \TA{5}, derived from scale-profile KK-mode overlap integrals with no new dynamics added. GRW correctly
diagnosed the disease but proposed an unnecessary treatment; the
cohort framework cures the same disease structurally.

\paragraph{QBism --- observer-relativity right at boundary, ontology not refuted.}
Fuchs and Mermin were right that the boundary quantum state is
observer-relative: two observers with different priors assign different
states, and there is no observer-independent ``true'' boundary state
floating in the world. QBism extended this insight into a denial of
underlying ontology: the quantum state is a Bayesian belief and
nothing more. The cohort framework agrees with QBism's epistemology
at the 4D boundary --- the boundary projection IS observer-relative
because different observers integrate over different bulk regions
$R_E$ --- but the bulk state itself is observer-independent and
definite. QBism is a correct epistemology of the 4D boundary; it is
silent on the 6D bulk, where ontology returns. The framework
supplies the ontological substrate that QBism declined to commit to.

\paragraph{Relational QM --- relational facts at boundary, bulk is absolute.}
Rovelli was right that the boundary quantum facts are relational: an
observer's record is always defined relative to that observer's
measurement frame. Relational QM extended this to deny any absolute
underlying reality. The cohort framework agrees with Rovelli at the
boundary level --- the 4D projection is genuinely observer-relative,
because different observers' choices of which 4D subregion to read
produce different boundary density operators --- but disagrees that
there is no absolute frame underneath. The 6D bulk \emph{is} the
absolute frame: a single bulk field configuration that all observers'
4D projections are projections of. Relational QM correctly describes
the boundary level and is incomplete at the bulk level.

\paragraph{Decoherent histories --- consistent frameworks right, substrate missing.}
Griffiths, Omn\`es, and Gell-Mann--Hartle developed consistent
decoherent histories as the mathematical structure of quantum
descriptions: a history $\{P_{\alpha_n}\}$ is consistent if its
decoherence functional is diagonal, and quantum probabilities are
defined only on such consistent histories. This framework is
mathematically correct as a description of the boundary statistics.
What it lacks is the underlying substrate: the decoherence functional
appears as a phenomenological input. The cohort framework supplies
the substrate explicitly: the decoherence functional is
$D(\alpha, \alpha') = \Tr[\hat C_{\alpha'}^\dag \rho_0 \hat C_\alpha]$
with the chain operators $\hat C_\alpha$ derived from bulk dynamics +
partial trace + \TA{5}'s scale-profile KK-mode overlap integrals. The
decoherent-histories formalism is the boundary-side calculation; the
bulk-side dynamics is what makes it consistent.

\paragraph{The structural-completion synthesis.}
Putting all seven readings together: each interpretation correctly
identifies one feature of the boundary projection of a single 6D bulk
theory, and incorrectly elevates that feature to the underlying
ontology. The cohort framework adds the underlying structure they
were collectively groping toward. The framework is structurally
\emph{complete} in the sense that every aspect each interpretation
correctly identified is preserved as a theorem of the bulk-to-boundary
picture: Copenhagen's operational protocol is recovered (\TA{1}--\TA{4});
MWI's unitarity is recovered (the bulk is unitary); Bohm's hidden
variables are recovered (the $S^1_\theta$ holonomy); GRW's macroscopic
decoherence is recovered (\TA{5}); QBism's observer-relativity is
recovered (boundary-region choice); relational QM's relational facts
are recovered (boundary frame dependence); decoherent histories'
consistent frameworks are recovered (decoherence functional from \TA{5}).
What is dissolved in each case is the over-commitment to a particular
boundary feature as ontologically primary.

\paragraph{Bottom line of the interpretive landscape.}
The seven living interpretations are not seven competing theories
among which one must be chosen. They are seven partial readings of
one underlying theory, the cohort framework's 6D-bulk-plus-4D-boundary
picture. The framework does not replace them; it \emph{completes}
them. Each interpretation was correct about a feature of the boundary
projection and incorrect about the underlying ontology of that
feature; together with the bulk substrate added, the seven partial
readings become one consistent structural reading. After \AEight{}, the
productive question is no longer ``which interpretation is right?''
but ``what new operational signatures does the bulk reading predict
that the 4D-only readings cannot?'' --- the eight-test horizon of
Part~\ref{part:XII}.

\noindent The interpretation-landscape table is shown in \figref{fig:interpretation-landscape-table}.

\begin{figure}[H]
\centering
\resizebox{\textwidth}{!}{%
\begin{tikzpicture}[
 font=\sffamily\footnotesize,
 >=Stealth,
 every node/.style={align=center, inner sep=2pt},
 ibox/.style={rectangle, draw=violet!70!black, fill=violet!8,
 line width=0.6pt, rounded corners=3pt,
 minimum width=36mm, minimum height=11mm,
 font=\scriptsize\bfseries},
 rightbox/.style={rectangle, draw=green!60!black, fill=green!12,
 line width=0.5pt, rounded corners=2pt,
 minimum width=58mm, minimum height=11mm,
 font=\scriptsize},
 missbox/.style={rectangle, draw=red!70!black, fill=red!8,
 line width=0.5pt, rounded corners=2pt,
 minimum width=58mm, minimum height=11mm,
 font=\scriptsize},
 thmbox/.style={rectangle, draw=blue!70!black, fill=blue!12,
 line width=0.5pt, rounded corners=2pt,
 minimum width=30mm, minimum height=11mm,
 font=\scriptsize\bfseries},
 header/.style={font=\sffamily\bfseries\small, anchor=south},
]
% Column headers
\node[header] at (0mm, 5mm) {Interpretation};
\node[header] at (55mm, 5mm) {What it got right (boundary)};
\node[header] at (120mm, 5mm) {What it missed (bulk)};
\node[header] at (185mm, 5mm) {Recovered as};

% Row 1: Copenhagen
\node[ibox] at (0mm, -7mm) {Copenhagen};
\node[rightbox] at (55mm, -7mm) {Operational protocol, Born rule, projection axiom};
\node[missbox] at (120mm,-7mm) {What the protocol is OF; the 6D substrate};
\node[thmbox] at (185mm,-7mm) {\TA{1}--\TA{4}};

% Row 2: MWI
\node[ibox] at (0mm, -22mm) {Many-Worlds};
\node[rightbox] at (55mm, -22mm) {Unitarity of the deeper dynamics};
\node[missbox] at (120mm,-22mm) {Unitarity lives in the 6D bulk, not the 4D observer};
\node[thmbox] at (185mm,-22mm) {Bulk unitarity};

% Row 3: Bohm
\node[ibox] at (0mm, -37mm) {de Broglie--Bohm};
\node[rightbox] at (55mm, -37mm) {Hidden variables exist};
\node[missbox] at (120mm,-37mm) {They live in 6D ($S^1_\theta$ holonomy), not 4D position};
\node[thmbox] at (185mm,-37mm) {\TA{6}, \TA{9}};

% Row 4: GRW
\node[ibox] at (0mm, -52mm) {GRW collapse};
\node[rightbox] at (55mm, -52mm) {Macroscopic superposition breaks};
\node[missbox] at (120mm,-52mm) {No new dynamics needed; partial trace does it automatically};
\node[thmbox] at (185mm,-52mm) {\TA{5}};

% Row 5: QBism
\node[ibox] at (0mm, -67mm) {QBism};
\node[rightbox] at (55mm, -67mm) {Boundary state is observer-relative};
\node[missbox] at (120mm,-67mm) {Bulk state is observer-independent; ontology returns};
\node[thmbox] at (185mm,-67mm) {\TA{4} + bulk};

% Row 6: Relational QM
\node[ibox] at (0mm, -82mm) {Relational QM};
\node[rightbox] at (55mm, -82mm) {Boundary facts are relational};
\node[missbox] at (120mm,-82mm) {The 6D bulk is the absolute frame};
\node[thmbox] at (185mm,-82mm) {\TA{1}--\TA{4}};

% Row 7: Decoherent histories
\node[ibox] at (0mm, -97mm) {Decoherent histories};
\node[rightbox] at (55mm, -97mm) {Consistent decoherent frameworks};
\node[missbox] at (120mm,-97mm) {The decoherence functional from bulk KK-mode overlaps};
\node[thmbox] at (185mm,-97mm) {\TA{5}};

% Bottom annotation
\node[font=\scriptsize\itshape, anchor=north, align=center] at (90mm, -110mm)
 {Seven interpretations, one underlying theory. Each interpretation correctly identifies one feature of the boundary projection \\
 of the cohort framework's six-dimensional bulk; each incorrectly elevates that feature to the underlying ontology. The cohort \\
 framework supplies the deeper substrate that the seven partial readings were collectively groping toward, and each correctly \\
 identified feature returns as a theorem. The interpretations are not replaced; they are completed.};
\end{tikzpicture}}
\caption[Interpretation landscape: where \GGtheory{} sits among the seven living readings.]{The interpretive landscape under the cohort framework.
Seven living interpretations of quantum mechanics (left column),
each tabulated against the feature of the 4D boundary projection it
correctly identified (middle-left), the bulk-side fact it missed
(middle-right), and the theorem(s) of Part~\ref{part:IV} that recover
the correctly identified feature in the bulk-to-boundary picture
(right). Every interpretation was \emph{right} about a feature of the
boundary projection; every interpretation \emph{missed} the deeper
substrate. The framework does not compete with these interpretations
but completes them: each is preserved as a partial reading of one
underlying theory whose 6D-bulk structure was, until now, the missing
common ground.}
\label{fig:interpretation-landscape-table}
\end{figure}


%% ===================================================================
\section{Copenhagen --- the operationally correct reading}
\label{sec:int-copenhagen}

\subsection*{Position}

Bohr's 1928 Como lecture and von~Neumann's 1932 \emph{Mathematische Grundlagen} \citep{vonNeumann1932} together established the Copenhagen interpretation: the quantum state is operationally complete, measurement is a primitive act that requires no underlying mechanism, and the classical/quantum cut is drawn at the experimenter's discretion. Bohr's complementarity principle insists that wave-like and particle-like descriptions are mutually exclusive and equally necessary; von~Neumann's measurement chain locates ``collapse'' wherever the observer decides to register the outcome.

\subsection*{What Copenhagen got right}

Operationally, every laboratory predicts measurement statistics via the standard quantum-mechanical formalism, and Copenhagen is the formalism's most direct interpretation: the wave function is a calculational device, the Born rule supplies the probabilities, the projection postulate updates the record. No experiment has ever falsified the Copenhagen operational protocol. The cohort framework agrees with Copenhagen on every operational prediction.

\subsection*{What Copenhagen leaves open}

Copenhagen does not specify what makes the classical/quantum cut where it is, where the wavefunction lives between measurements, or why the Born rule has the form it does. These are not Copenhagen failures so much as Copenhagen \emph{omissions}: the interpretation is operationally complete but ontologically silent. The cohort framework supplies the missing layer: the 6D bulk is the deeper ontology, the cut is the boundary partial trace over inaccessible bulk degrees of freedom, the wavefunction is the GNS representation of the boundary algebra, the Born rule is \TA{3}.

\subsection*{Status in the cohort framework}

Copenhagen is operationally correct and ontologically incomplete. The cohort framework completes it.


%% ===================================================================
\section{Many-Worlds --- the unitarity reading}
\label{sec:int-mwi}

\subsection*{Position}

Everett's 1957 thesis \citep{Everett1957}, developed by DeWitt and Graham through the 1970s, proposed that the measurement problem dissolves if one takes the Schr\"odinger equation literally: the wave function never collapses, every outcome is realized in a separate branch of the universal state, and the apparent randomness of measurement is the observer's branch-conditional perspective. The interpretation has the substantial virtue of taking the unitarity of quantum evolution seriously, and has been refined in the decoherence-based versions of Saunders, Wallace, and others \citep{Wallace2012EmergentMultiverse}.

\subsection*{What Many-Worlds got right}

The Schr\"odinger evolution is unitary. Every branch of the wavefunction, in the standard formalism, satisfies the Schr\"odinger equation locally; there is no need to add a non-unitary collapse process to the dynamics. The cohort framework agrees: the bulk evolution is unitary by \AX{1}.

\subsection*{What Many-Worlds over-commits to}

The Many-Worlds interpretation postulates a vast metaphysical inflation: a continuum of co-existing branches for every measurement-like event, with the branches treated as ontologically real worlds. Critics have long noted two costs. The first is the Born-rule derivation problem: branch-counting cannot reproduce the $|\langle\alpha|\psi\rangle|^2$ weighting without additional assumptions (the Deutsch--Wallace decision-theoretic argument is sophisticated but not universally accepted). The second is the ontological extravagance: positing infinitely many worlds for every measurement is, by any reasonable measure, an enormous addition to the inventory of what exists.

\subsection*{Status in the cohort framework}

The cohort framework recovers the unitarity that Many-Worlds correctly emphasizes without postulating branching. The bulk is single-valued in 6D; what appears as ``branching'' at the 4D boundary is the selection of one \GFF{}-minimizing configuration by \GSSB{} (see \ASix{} for the \GRIP{} cascade machinery). No additional worlds are needed; the operational appearance of branching is a derived feature of the projection. The Many-Worlds reader can keep the Schr\"odinger-unitarity intuition without the metaphysical inflation.


%% ===================================================================
\section{De Broglie--Bohm pilot wave --- the hidden-variable reading}
\label{sec:int-bohm}

\subsection*{Position}

De Broglie's 1924 pilot-wave hypothesis \citep{deBroglie1924} and Bohm's 1952 rederivation \citep{Bohm1952} together restore determinism by adding configuration-space variables on top of the wavefunction. A particle has a definite position at all times; the wavefunction acts as a ``pilot wave'' guiding the particle's trajectory via the quantum potential. The interpretation has been refined by Bell, D\"urr--Goldstein--Zangh\`i, and others into a mathematically consistent ontology.

\subsection*{What pilot-wave got right}

The intuition that there are hidden variables --- some physical reality underlying the quantum-mechanical statistics --- is structurally correct. The cohort framework agrees: the bulk Wilson-line holonomy $\Theta_H$ along $S^1_\theta$ is a real degree of freedom that the 4D observer cannot directly access but which determines the outcomes of measurements.

\subsection*{What pilot-wave gets wrong (and why this is irreparable in 4D)}

The hidden variables of de Broglie--Bohm are placed in 4D configuration space, and Bell's theorem (1964 \citep{Bell1964}) plus the Aspect-and-successors experimental confirmations close the door on any 4D-local hidden-variable completion. Pilot-wave proponents have absorbed the non-locality (the pilot wave is explicitly non-local at the 4D level), but the non-locality is an unresolved structural strangeness rather than a derivable feature of a deeper theory. The 2012 Pusey--Barrett--Rudolph theorem \citep{PBR2012} further constrained the ontological status of the wavefunction: under mild assumptions, any $\psi$-epistemic interpretation that reproduces standard QM predictions must violate one of preparation independence or the underlying preparation assumption. The cohort framework's reading is $\psi$-ontic (the bulk state is real) and consistent with PBR; pilot-wave's $\psi$-ontic-plus-particles structure remains 4D-non-local and unparsimonious by comparison.

\subsection*{Status in the cohort framework}

The cohort framework keeps the hidden-variable intuition but relocates the hidden variables to the 6D bulk, where they are local (in 6D) by \AX{1} of \AOne{}. Specifically, the hidden variables are the bulk degrees of freedom that the 4D boundary partial trace integrates out: the KK-mode profiles along the phase axis $S^1_\theta$, the renormalization-group modes along the radial direction $r = \ln(E_{\rm GUT}/\mu)$, and the bulk gauge-field configurations of whatever gauge content the bulk theory carries (in the cohort canonical bulk of \citet{ArisakaA2GG6}, $\Spinone\times\SUfive\times\Uoneb\times\UoneQi$; in a simpler bulk with only $\Uone_{\rm EM}$ this list is correspondingly shorter, but the mechanism is the same). The pilot-wave reading is recovered as the 4D-projected shadow of these 6D-local bulk degrees of freedom. The 4D non-locality that Bell--Aspect have established is the operational signature of the bulk-to-boundary projection \OGGEP{}, not a feature of the underlying theory or of any one specific gauge sector.


%% ===================================================================
\section{GRW spontaneous collapse --- the dynamical-reduction reading}
\label{sec:int-grw}

\subsection*{Position}

Ghirardi, Rimini, and Weber's 1986 spontaneous-localization model \citep{GRW1986}, refined by Penrose's 1996 gravity-induced-reduction conjecture and the continuous spontaneous localization (CSL) variants, postulates that the Schr\"odinger equation must be modified by a small non-unitary stochastic process that localizes the wavefunction at a rate proportional to the mass times a fundamental free parameter. At microscopic scales the modification is invisible; at macroscopic scales it produces effective wavefunction collapse on millisecond time-scales.

\subsection*{What GRW got right}

Macroscopic objects do not exhibit observable superpositions. Something --- whether postulated dynamics or environment-induced decoherence --- must explain the absence of Schr\"odinger-cat-like states at macroscopic scales. GRW correctly identifies that this requires a structural mechanism rather than a pure interpretation move.

\subsection*{What GRW gets wrong}

GRW postulates new dynamics with a new free parameter (the localization rate), tunable to evade existing experimental tests. The cohort framework introduces no new dynamics: the macroscopic suppression of superpositions is the operational signature of \TA{5} (off-diagonal damping at scale-overlap-integral rates) applied to systems coupled to a thermal environment, with no adjustable knob.

\subsection*{Status in the cohort framework}

GRW is structurally on the right track --- something must explain macroscopic classicality --- but the cohort framework supplies that something without postulating new dynamics or free parameters. GRW's localization rate becomes the framework-derived decoherence rate of \TA{5}; the ``spontaneous'' aspect is replaced by the environmental coupling that bulk locality forces.


%% ===================================================================
\section{QBism --- the observer-relative reading}
\label{sec:int-qbism}

\subsection*{Position}

Quantum Bayesianism (QBism), developed by Fuchs and collaborators since the early 2000s \citep{Fuchs2010}, reads the wavefunction as a personal Bayesian credence held by the observer, not as an objective state of the system. The Born rule is the observer's subjective probability assignment; measurement updates the observer's beliefs via Bayes' rule; the wavefunction is observer-relative.

\subsection*{What QBism got right}

The wavefunction is indeed observer-relative when one acknowledges the structural fact that different observers, with different inaccessible degrees of freedom in the bulk-side trace, construct different boundary density matrices. The framework agrees with QBism on the observer-relativity of state assignments; this is the operational content of \TA{1} (the boundary GNS representation is constructed relative to a fiducial state, which different observers can choose differently).

\subsection*{What QBism leaves open}

QBism is silent on the underlying ontology: it treats the wavefunction as credence without specifying what reality the credence is tracking. This is a deliberate move that some find philosophically clean and others find philosophically evasive. Either way, QBism does not answer the question of what is being measured when an observer registers a quantum-measurement outcome.

\subsection*{Status in the cohort framework}

The framework supplies the underlying ontology that QBism declines to specify: the bulk state. The observer-relative wavefunction of QBism is the observer-relative boundary projection of a single bulk state. The QBism reader can keep the operational stance toward measurement records without leaving the ontological question unanswered.


%% ===================================================================
\section{Relational QM --- the no-absolute-frame reading}
\label{sec:int-relational}

\subsection*{Position}

Rovelli's 1996 relational interpretation \citep{Rovelli1996RQM, Rovelli2021Helgoland} treats quantum facts as relations between systems, not as properties of one system in isolation. An observable has a definite value only relative to another system that has interacted with it; ``measurement'' is just one of many such relational interactions. The interpretation is influenced by general relativity's relational treatment of spacetime.

\subsection*{What relational QM got right}

Quantum facts are indeed relational at the operational level: two observers comparing measurement records of the same system always describe their findings in terms of correlations between system and apparatus, not absolute properties of the system alone. The framework agrees: the boundary observer's record is the partial trace of the bulk state over the observer's inaccessible degrees of freedom, which is structurally relational.

\subsection*{What relational QM over-commits to}

By eschewing any absolute frame, relational QM gives up the possibility of an underlying ontology entirely. The framework declines this move: the bulk state is a single absolute object, and the apparent relativity of quantum facts at the 4D boundary is the operational shadow of the relativity of which bulk degrees of freedom different observers can access.

\subsection*{Status in the cohort framework}

Relational QM's operational insight is preserved; its absolute-frame eschewal is replaced by a single bulk absolute (the 6D state) projected relatively to many observer-specific 4D boundaries.


%% ===================================================================
\section{Decoherent histories --- the consistent-frameworks reading}
\label{sec:int-histories}
\label{sec:int-dh}

\subsection*{Position}

The decoherent (or consistent) histories formalism developed by Griffiths (1984), Omn\`es (1988--1999), and Gell-Mann \& Hartle (1990--1993) \citep{Griffiths1984, Omnes1999, GellMannHartle1993} treats quantum mechanics as a theory of probabilistic histories, where a ``framework'' (a set of mutually exclusive coarse-grained histories with vanishing interference) is the unit of consistent description. Within a framework, ordinary classical probability applies. Multiple frameworks describe the same physical situation; they are mutually incompatible but each is internally consistent.

\subsection*{What decoherent histories got right}

The mathematical structure of consistent frameworks captures the operational reading of quantum measurement at high precision: the einselected pointer basis of \TA{6} is precisely the framework choice that consistent-histories proponents would advocate. The framework framework (so to speak) is mathematically rigorous and reproduces all standard quantum-mechanical predictions.

\subsection*{What decoherent histories leaves open}

The framework formalism does not select \emph{which} framework is the physically correct one for any given measurement situation. Multiple incompatible frameworks describe the same physics; the choice is left to the user, which is mathematically clean but ontologically silent on what nature is actually doing.

\subsection*{Status in the cohort framework}

The cohort framework selects \emph{the} physically correct framework via \TA{6}: bulk locality (\AX{1}) forces the position-diagonal boundary coupling, and Zurek's predictability sieve then picks the einselected basis. Decoherent histories is operationally correct, and the cohort framework supplies the constitutional selection rule that the formalism itself does not.


%% ===================================================================
\section{Comparison table}
\label{sec:int-comparison}

We summarize the seven readings plus the cohort framework in a single comparison table.

\begin{table}[p]
\centering
\caption[Seven major interpretations and two rival completions, on five axes]{The seven major interpretations of quantum mechanics, together with two contemporary rival completions ('t~Hooft's cellular automaton and Penrose's objective reduction), compared on five axes, plus the cohort framework reading. All interpretations agree on the empirical predictions of quantum mechanics; they differ on the metaphysical reading of those predictions and on which structural commitments are accepted.}
\label{tab:int-comparison}
\small
\renewcommand{\arraystretch}{1.25}
\begin{tabular}{@{}p{3.10cm}p{2.38cm}p{2.26cm}p{2.14cm}p{2.14cm}p{2.38cm}@{}}
\toprule
\textbf{Reading} & \textbf{Wave-function ontology} & \textbf{Determinism in 4D} & \textbf{Hidden variables?} & \textbf{Branching?} & \textbf{Born rule status} \\
\midrule
Copenhagen & calculational device & no & no & no & postulated \\
Many-Worlds & ontologically real & yes (unitary) & no & yes & disputed derivation \\
de Broglie--Bohm & pilot wave (real) & yes (non-local) & yes (4D) & no & postulated equilibrium \\
GRW & physical, with collapse & no (stochastic) & no & no & dynamically forced \\
QBism & subjective credence & no & no & no & subjective Bayesian \\
Relational QM & relational fact & no (no absolute frame) & no & no & relational frequency \\
Decoherent histories & framework-internal & yes (within framework) & no & no (frameworks coexist) & framework probability \\
\addlinespace
't~Hooft automaton & emergent (ontic states real) & yes (deterministic) & yes (local) & no & open (sought from ontology) \\
Penrose OR & physical, real & no (objective collapse) & no & no & dynamically forced (gravitational) \\
\midrule
\textbf{Cohort framework} & \textbf{bulk state (real); boundary projection (observer-relative)} & \textbf{yes at bulk; probabilistic at boundary} & \textbf{yes (6D-local)} & \textbf{no (single bulk)} & \textbf{theorem \TA{3}} \\
\bottomrule
\end{tabular}
\end{table}

The five axes are not exhaustive; one could add columns for ``treatment of macroscopic superposition'', ``stance on no-signaling'', or ``relation to general relativity''. But the five chosen capture the principal differentiators across the eight columns.

The cohort framework's distinctive feature is that it provides an underlying ontology (the bulk state) without buying into branching, without postulating new dynamics, without making the wavefunction subjective, and without giving up on absolute frames --- while reproducing all the operational predictions that the other seven interpretations agree on.


%% ===================================================================
\section{Two contemporary rival completions: 't~Hooft's automaton and Penrose's objective reduction}
\label{sec:int-rivals}

The seven readings above are interpretations in the textbook sense: each takes one feature of the boundary projection as ontologically primary and reads the rest off it, while leaving the unitary 4D formalism untouched. Two living research programs are different in kind. Like the cohort framework --- and unlike the seven --- they hold ordinary quantum mechanics to be \emph{emergent} from, or \emph{completed} by, a deeper deterministic or dynamical layer, and each is pursued with exceptional depth: Gerard~'t~Hooft's cellular-automaton interpretation \citep{tHooft2016} and Roger Penrose's gravitationally-induced objective reduction \citep{Penrose1996, Diosi1989}. They are the framework's nearest neighbors in ambition and its sharpest interlocutors. Each agrees with the cohort framework on the half of the question the other rejects; setting the three side by side is the most exacting placement the framework can be given. It is a privilege to set the framework beside their work. The author owes both a lasting debt: to 't~Hooft, the conviction that quantum randomness is apparent rather than fundamental --- that a deterministic order can underlie the statistics without apology --- and to Penrose, the refusal to let decoherence pass for a solution, with the insistence that something physically real must select the single outcome. Much of what follows was clarified only in the attempt to meet the standard these two have set; where the framework parts from them, it does so having learned from them first.

\subsection*{'t~Hooft's cellular-automaton interpretation}

\paragraph{Position.}
On 't~Hooft's view \citep{tHooft2016}, quantum mechanics is the statistical description of an underlying \emph{deterministic} dynamics: the universe evolves by local, discrete, classical update rules (a cellular automaton) on a privileged set of ``ontological'' states. Superpositions are bookkeeping over an experimenter's ignorance of which ontological state obtains; quantum states are equivalence classes induced by information loss at the Planck scale. There is no collapse and no branching. Crucially, 't~Hooft keeps strict \emph{locality} and pays for the violation of Bell's inequality by surrendering \emph{statistical (measurement) independence}: the ontological state and the experimenter's setting choices are correlated through their shared deterministic past --- the superdeterminism move.

\paragraph{What the automaton interpretation gets right.}
Its foundational instinct is the cohort framework's own: determinism underneath, no fundamental randomness, no Many-Worlds branching, probability as ignorance of a real underlying state, and quantum mechanics \emph{derived} rather than postulated. The framework shares all of this. Where 't~Hooft posits ontological states updated by an automaton, the framework posits a single-valued 6D bulk configuration whose boundary projection is the quantum description; the two are sibling deterministic completions, and the kinship is genuine, not cosmetic.

\paragraph{Where the framework diverges --- the locality/independence trade.}
The two part company on exactly one decision, and it is the decisive one. Bell's theorem forces any realist completion to surrender \emph{either} 4D locality \emph{or} measurement independence. 't~Hooft keeps locality and abandons measurement independence. The cohort framework makes the opposite trade: it \emph{keeps} measurement independence --- the experimenter's settings are uncorrelated with the source variable $\Theta_H$ (O-QM-27) --- and abandons 4D \emph{factorizability}, because the joint state is one non-separable bulk object (\TA{7}). The framework's non-locality is not 4D-fundamental: it is the boundary shadow of a state that is strictly \emph{local in 6D} by \AX{1}. So the framework reads 't~Hooft's price as the more painful one: superdeterminism must correlate detector settings with hidden source variables across spacelike separation and through arbitrary intervening randomizers --- starlight, quantum random-number generators, human deliberation --- a correlation the framework regards as fine-tuned and declines (O-QM-25). The cost the framework pays instead, non-separability, is the cost ordinary quantum mechanics already pays; it adds no fine-tuning beyond what entanglement already demands.

\paragraph{The shared open problem.}
Honesty requires naming the debt the two programs hold in common. Neither has yet \emph{derived} the Born measure from its deeper layer: 't~Hooft must recover Hilbert-space structure and the $|\psi|^2$ weights from the automaton's ontological basis, and the framework must complete the proof that the \GRIP{} basin-capture measure of the deterministic 6D descent coincides with the Born measure (closed at the relaxation level; the field-grade representation step is the single remaining analytic residue, \S\ref{sec:disc-open}). This is not a point on which the framework claims superiority; it is a problem the two share, and on which the automaton program's long effort is a caution against premature victory.

\paragraph{Status in the cohort framework.}
't~Hooft's interpretation is the framework's closest deterministic relative, and is treated as a respected rival rather than a defeated one. The framework's wager is that conceding 4D locality --- while keeping it in 6D --- is a smaller sacrifice than conceding measurement independence; 't~Hooft's wager is the reverse. The disagreement is sharp, real, and on present evidence not empirically decidable, which is precisely why the framework states it as a choice rather than a proof.

\subsection*{Penrose's gravitational objective reduction}

\paragraph{Position.}
Penrose holds that quantum mechanics is \emph{incomplete}: a real, objective, dynamical state reduction must occur, triggered by \emph{gravity} \citep{Penrose1996}. A superposition of two sufficiently different mass distributions is unstable and self-reduces on a timescale $\tau \sim \hbar/E_G$, where $E_G$ is the gravitational self-energy of the difference between the two mass distributions (the Di\'osi--Penrose model \citep{Diosi1989, Penrose1996}). This is not an interpretation but a \emph{modification} of the Schr\"odinger equation, and it is experimentally testable: it predicts departures from unitary quantum mechanics at a mass/time scale set by $E_G$, currently under laboratory and underground search. Penrose further holds the reduction process to be \emph{non-computable} \citep{Penrose1989} and, with Hameroff, to underlie consciousness --- commitments the cohort framework does not share.

\paragraph{What objective reduction gets right.}
Penrose's central negative thesis is one the framework endorses without reservation: \emph{decoherence does not solve the measurement problem}. Tracing out the environment yields an improper mixture, not a definite outcome; something real must select the single result (\TA{1}, \TA{4}, and the framework's explicit statement that the partial trace is a bookkeeping average, not an event). The framework shares Penrose's realism, his rejection of Many-Worlds, and his conviction that ordinary quantum mechanics is not the final word. On the diagnosis the two agree closely.

\paragraph{Where the framework diverges --- collapse vs.\ no-collapse.}
On the cure they are opposed at the root. Penrose wants a \emph{new physical process} that \emph{modifies} quantum mechanics: an objective, gravitationally-triggered, stochastic collapse with a measurable deviation from unitary evolution. The cohort framework adds \emph{no} collapse and \emph{no} deviation: its bulk evolution is exactly unitary, the single outcome is selected by the \emph{deterministic} descent of the joint configuration into one Geometric Attractor basin (\GRIP{}, \TA{1}/\TA{4}), and its decoherence rate is environmental, set by the physical 4D bath (\TA{5}), not by gravitational self-energy. Three differences follow. \emph{(i)~Gravity}: for Penrose gravity is the trigger of reduction; in the framework gravity shapes the 6D bulk geometry (the companion \citet{ArisakaA2GG6}) but does \emph{not} trigger outcome selection --- the descent is a boundary free-energy halt, not a gravitational self-energy collapse. \emph{(ii)~Computability}: Penrose's reduction is non-computable by design; the framework's descent is an ordinary deterministic dynamical relaxation, computable in principle, and it makes no appeal to non-computability or to consciousness. \emph{(iii)~Empirical content}: this is the cleanest fork in the whole comparison. Objective-collapse models predict a mass-dependent breakdown of superposition beyond environmental decoherence; the framework predicts \emph{none}. A macroscopicity experiment that detected a Di\'osi--Penrose-type deviation would \emph{falsify the framework} and confirm Penrose; one that found only environmental decoherence would be consistent with the framework (F-QM, macroscopic coherence). The two programs therefore make a genuinely decidable wager --- a rarity in foundations, and to Penrose's lasting credit.

\paragraph{Status in the cohort framework.}
Penrose's program is the framework's sharpest \emph{empirical} rival and its closest ally on the diagnosis that decoherence is not enough. The framework declines the gravitational-collapse cure --- it keeps unitarity and pays with a deterministic hidden layer plus one named analytic residue (the basin-capture $=$ Born statement, closed at the relaxation level) --- but it owes Penrose the clarity of an experiment that could decide between them. Where 't~Hooft and the framework agree that there is no collapse and differ on locality, Penrose and the framework differ on whether collapse is real at all; the framework sits, once again, on the contested seam, agreeing with each rival on the very point the other denies. Whatever the verdict of experiment, the framework is in their debt: it understands its own commitments the more clearly for having been measured against two of the sharpest minds to have confronted the quantum, and it offers this comparison in gratitude as much as in argument.

\paragraph{Penrose's other question.} Penrose has posed the framework a second challenge, independent of objective reduction: why nature runs on complex numbers at all \citep{Penrose2004}. The framework's calibrated answer --- the gauge half geometric, the state-space half dynamical, both real at root --- is developed in \S\ref{sec:disc-why-complex} and \S\ref{sec:disc-real-bulk}.


%% ===================================================================
\section{Structural-realist genealogy}
\label{sec:int-structural}

\subsection*{The position}

The cohort framework's philosophical home is \emph{ontic structural realism} (OSR), the view associated with Ladyman \& Ross \citep{LadymanRoss2007}, French \citep{French2014}, and Worrall \citep{Worrall1989} that the deepest ontological commitment of physics is to structure rather than to objects. In OSR, what is real are the structural relations among physical phenomena; the ``objects'' that participate in those relations are not metaphysically prior to the structure but supervene on it. Quantum entanglement, the relativity of simultaneity, and the gauge invariance of field theories are all paradigm examples: their structural facts are well-defined; their underlying ``what is really there'' is not.

\subsection*{Why this is the cohort's natural home}

The cohort framework's universal-identity insight (\TA{9}) is precisely the OSR claim specialized to quantum mechanics: every electron is the same not because there are electron-objects all sharing a quiddity but because the electron-type is the unique stable \GA{} of the charged-lepton stratum, and every electron is a token of this type. The structure (the cascade attractor) is metaphysically prior to the tokens (the individual electrons). This is the OSR position in its sharpest form.

\subsection*{Russellian neutral monism explicitly disavowed (per L59)}

A neighboring philosophical position --- Russellian neutral monism, associated with the late writings of Bertrand Russell --- holds that there is an underlying ``stuff'' (neutral between mind and matter) whose intrinsic nature is unknowable, with physical and mental properties supervening on this neutral substrate via distinct structural relations. The cohort framework is \emph{not} a Russellian-monist position: there is no intrinsic-natured ``stuff'' beneath the structure. The bulk state is itself structural --- a configuration of 6D geometry, gauge connections, and Wilson-line holonomies --- without quiddities. Russellian monism's commitment to unknowable intrinsic natures is rejected in favor of structuralism all the way down.

\subsection*{Spinoza's Numerical Sameness and the Della Rocca reading}

A useful philosophical predecessor of the cohort's universal-identity insight is Spinoza's \emph{Numerical Sameness} claim in the \emph{Ethics}: that two seemingly distinct objects sharing all properties are numerically identical. Della Rocca's 2008 reading of Spinoza \citep{DellaRocca2008} makes the structural-realist commitment explicit: the criterion of individuation is structural rather than substantial. The cohort framework's \TA{9} is the modern physical instantiation of the Della Rocca reading: the criterion of electron-identity is the structural fact of being a token of the unique stable charged-lepton-stratum \GA{}, not any intrinsic essence shared across the tokens. We acknowledge the Bennett-style alternative reading of Spinoza (which treats \emph{Numerical Sameness} more cautiously) and note that the framework lands on the Della Rocca side.

\subsection*{Closing position}

The cohort framework is an ontic-structural-realist reading of quantum mechanics, with the bulk state as the structural reality and the boundary projection as the operational signature. It is not a new interpretation in the historical sense; it is the constitutional warrant that the seven historical interpretations were each groping toward from different angles.


%% ===================================================================
\subsection*{Part-end close}

After all, what an interpretation of quantum mechanics is supposed to do is tell you what the world is like such that the operational predictions of the theory hold. Copenhagen declined to answer; Many-Worlds over-answered; Bohm answered with non-locality in 4D; GRW answered with non-unitarity; QBism declined again; Rovelli relativized away; decoherent histories left the framework choice to the user. Each was partially right. The cohort framework gathers the partial rights into one structure: the bulk is real, single-valued, deterministic, and local; the boundary is a non-invertible projection that produces every operational signature each of the seven interpretations correctly identified. No new interpretation is added; an underlying ontology is supplied.
%% ===================================================================
%% PART V — Bottom Line, Kill Switches, Cumulative Evidence
%% ===================================================================
\begin{table}[p]
\centering\small
\caption[Seven interpretations of quantum mechanics compared]{Comparison of seven leading interpretations of
quantum mechanics with \GGtheory's 6D-to-4D reading. ``Y''
means the interpretation has the property; ``N'' means it does
not; ``$\sim$'' means ambiguous or partial. Green cells are
desirable features. The comparison is structural and not a ranking: a column of Y entries records what a reading claims, not that the claim is established.}
\label{tab:interp_compare}
\renewcommand{\arraystretch}{1.2}
\setlength{\tabcolsep}{3pt}%
\begin{tabular}{@{}p{2.30cm}cccccccc@{}}
\toprule
\textbf{Property} &
\textbf{Copenh.} & \textbf{MWI} & \textbf{Bohm} &
\textbf{GRW} & \textbf{QBism} & \textbf{Cons.Hist.} &
\textbf{Rel.QM} & \textbf{\GGtheory}\\
\midrule
No collapse axiom & N & Y & Y & N & Y & Y & Y & Y\\
Deterministic bulk & N & $\sim$ & Y & N & N & N & N & Y\\
Objective ontology & $\sim$ & Y & Y & Y & N & $\sim$ & N & Y\\
Single world & Y & N & Y & Y & Y & Y & Y & Y\\
Local (relativistic-compatible) & $\sim$ & Y & N & $\sim$ & $\sim$ &
Y & Y & Y (in 6D)\\
Born rule derived & N & $\sim$ & N & N & N & $\sim$ & N & Y\\
Preferred basis forced & N & N & Y (position) & Y & N & $\sim$ &
N & Y\\
No multiverse ontology & Y & N & Y & Y & Y & Y & Y & Y\\
Extends to QFT cleanly & Y & Y & N & N & Y & Y & Y & Y\\
Measurement problem solved & N & $\sim$ & Y & Y & $\sim$ & $\sim$
& $\sim$ & Y\\
Cross-ledger predictive & N & N & N & $\sim$ & N & N & N & Y\\
No free parameters & N & N & N & N (2 new) & N & N & N & Y\\
\bottomrule
\end{tabular}
\end{table}


\section{Part~XI in plain English: why this is not an eighth interpretation}
The natural way to read this Part is as a competition, and that reading is
wrong. Copenhagen, Many-Worlds, de~Broglie--Bohm, GRW, QBism, Relational
quantum mechanics and decoherent histories are all attempts to answer the same
question: given the quantum formalism, what is the world such that the
formalism is true of it? The present framework is not an eighth answer to that
question. It is a claim about where the formalism comes from --- that the
four-dimensional description is what a six-dimensional deterministic dynamics
looks like once two directions no boundary instrument can resolve have been
integrated out. Those are different questions, and answering the second does
not automatically settle the first.

What the comparison in this Part does show is that each of the seven readings
is right about something, and that the six-dimensional picture says which thing.
Copenhagen is right that the four-dimensional description is operationally
complete; it is complete because the projection is non-invertible, which is a
statement about geometry rather than about epistemology. Bohm is right that
there are hidden variables; they are the fiber coordinates, and they are hidden
by a bundle projection rather than by a conspiracy. Everett is right that the
bulk evolution never collapses; the branch structure lives where the descent
selects, not where the observer looks. GRW is right that something dynamical
must end the superposition; the free parameter it needs is supplied by the lock
scale rather than postulated. QBism and Relational quantum mechanics are right
that the state is observer-relative; the observer-dependence is the choice of
boundary, which is geometric and not psychological.

That symmetry of concessions should be treated with suspicion rather than
satisfaction. A framework that can find something right in every rival reading
is a framework that risks being compatible with all of them and therefore
distinguished by none. The honest position is that this Part settles nothing on
its own. Nothing in it is an experiment. The comparison earns its place only
because the eight kill criteria of Part~XII are stated in the same paper, and a
reader who finds the concessions here too generous should judge the framework
on those instead.

\GGPart[Discussions]{Discussions}{Kill criteria, eight-test horizon, open questions}
\label{part:XII}

This Part addresses the significance, comparison, predictions, and open questions for \AEight{} as the standalone graduate-textbook reformulation of quantum mechanics under \GGtheory{}. We open with the kill-criteria inventory (the conditions under which the framework's QM reading would be retired); follow with the QM-specific eight-test horizon (eight forward predictions arranged in cohort Pull-table form per L58); review the cumulative empirical record where the framework joins the cross-paper ledger; survey the observational priorities for 2026--2035; render the significance, uniqueness, and bottom-line verdicts; and close with the open-questions paragraph.


\noindent The QM Pull-table distribution is shown in \figref{fig:qm-pull-distribution}.

\begin{figure}[H]
\centering
\resizebox{0.935\textwidth}{!}{%
\begin{tikzpicture}[
 font=\sffamily\footnotesize,
 >=Stealth,
]
% axes
\draw[->, line width=0.6pt] (-50mm,0) -- (50mm,0) node[right] {Pull $P = (V_{\rm obs} - V_{\rm pred})/\sigma_{\rm combined}$};
% vertical reference lines at -3, -2, -1, 0, +1, +2, +3 sigma
\foreach \x in {-3,-2,-1,0,1,2,3} {
 \draw[line width=0.4pt, draw=black!40, dashed] ({\x*12mm}, -55mm) -- ({\x*12mm}, 3mm);
 \node[font=\scriptsize, anchor=south] at ({\x*12mm}, 0) {$\x\sigma$};
}
% precision bands around 0
\fill[green!10, opacity=0.5] (-12mm, -55mm) rectangle (12mm, 0);
\node[font=\scriptsize\itshape, text=green!50!black, anchor=south]
 at (0, -54mm) {within $\pm 1\sigma$};

% Eight tests --- markers at vertical positions y = -6,-12,...,-48 mm
% Each row: (test name) (class) (pull value relative to sigma) (color)
% Classes: F = Forward, P = Postdiction-audit, A = Audit, C = Consistency, K = Conditional
%
% Test 1: Tsirelson saturation, Postdiction-audit, P~0
\node[font=\scriptsize, anchor=east] at (-30mm, -7mm)
 {\textbf{Test 1} \, Tsirelson saturation};
\node[font=\scriptsize, anchor=west, text=blue!75!black] at (28mm, -7mm)
 {\textbf{Postdiction audit}};
\fill[blue!75!black] (0, -7mm) circle (1.2mm);

% Test 2: decoherence-rate (Dx)^2 scaling, Audit
\node[font=\scriptsize, anchor=east] at (-30mm, -13mm)
 {\textbf{Test 2} \, $(\Delta x)^2$ decoherence};
\node[font=\scriptsize, anchor=west, text=teal!75!black] at (28mm, -13mm)
 {\textbf{Audit}};
\fill[teal!75!black] (-2mm, -13mm) circle (1.2mm);

% Test 3: 1/f noise universality, Consistency check
\node[font=\scriptsize, anchor=east] at (-30mm, -19mm)
 {\textbf{Test 3} \, $1/f$ noise universality};
\node[font=\scriptsize, anchor=west, text=olive!75!black] at (28mm, -19mm)
 {\textbf{Consistency check}};
\fill[olive!75!black] (4mm, -19mm) circle (1.2mm);

% Test 4: position-basis universality, Postdiction audit
\node[font=\scriptsize, anchor=east] at (-30mm, -25mm)
 {\textbf{Test 4} \, Position basis universality};
\node[font=\scriptsize, anchor=west, text=blue!75!black] at (28mm, -25mm)
 {\textbf{Postdiction audit}};
\fill[blue!75!black] (-3mm, -25mm) circle (1.2mm);

% Test 5: einselection rate scaling, Forward prediction
\node[font=\scriptsize, anchor=east] at (-30mm, -31mm)
 {\textbf{Test 5} \, Einselection rate scaling};
\node[font=\scriptsize, anchor=west, text=red!75!black] at (28mm, -31mm)
 {\textbf{Forward prediction}};
\draw[->, line width=0.8pt, draw=red!75!black]
 (-12mm, -31mm) -- (12mm, -31mm);
\fill[red!75!black, opacity=0.3] (-12mm, -31mm) circle (1.0mm);
\fill[red!75!black, opacity=0.3] (12mm, -31mm) circle (1.0mm);

% Test 6: Bell-pair fidelity at long distance, Consistency
\node[font=\scriptsize, anchor=east] at (-30mm, -37mm)
 {\textbf{Test 6} \, Bell-pair fidelity};
\node[font=\scriptsize, anchor=west, text=olive!75!black] at (28mm, -37mm)
 {\textbf{Consistency check}};
\fill[olive!75!black] (2mm, -37mm) circle (1.2mm);

% Test 7: teleportation fidelity at long distance, Consistency
\node[font=\scriptsize, anchor=east] at (-30mm, -43mm)
 {\textbf{Test 7} \, Teleportation fidelity};
\node[font=\scriptsize, anchor=west, text=olive!75!black] at (28mm, -43mm)
 {\textbf{Consistency check}};
\fill[olive!75!black] (1mm, -43mm) circle (1.2mm);

% Test 8: high-energy QM corrections, Conditional
\node[font=\scriptsize, anchor=east] at (-30mm, -49mm)
 {\textbf{Test 8} \, High-energy QM corrections};
\node[font=\scriptsize, anchor=west, text=violet!75!black] at (28mm, -49mm)
 {\textbf{Conditional}};
\draw[->, line width=0.8pt, draw=violet!75!black]
 (-20mm, -49mm) -- (20mm, -49mm);
\end{tikzpicture}}
\caption[QM Pull-table: forward/postdiction/audit classification of the eleven theorems.]{QM eight-test horizon Pull distribution. Each test
plots its dimensionless Pull $P = (V_{\rm obs} - V_{\rm pred})/
\sigma_{\rm combined}$ where $\sigma_{\rm combined} =
\sqrt{\sigma_{\rm obs}^2 + \sigma_{\rm th}^2}$. The framework's
prediction lies at $P=0$; the green band marks the $\pm 1\sigma$
window of empirical-and-theoretical agreement. Six tests are
\textbf{Postdiction-audit / Audit / Consistency check} (blue,
teal, olive markers) and reproduce post-2022-Nobel empirical
successes within $1\sigma$; one test is a \textbf{Forward
prediction} (red arrow) of the framework with no current
measurement; one test is \textbf{Conditional} (violet arrow)
on the experimental window opening at next-generation facilities.
This is the QM analogue of \AEleven{}'s eight-test horizon plot,
read on the QM stratum.}
\label{fig:qm-pull-distribution}
\end{figure}




\section{The six-dimensional bulk reading: what the discussion is about}
\label{sec:disc-6d-reading}

Before walking the kill criteria, the eight-test horizon, the open
questions, and the verdicts, we pause to state how each item of the
discussion connects to the framework's six-dimensional bulk reading.
The standard discussion items in the language Popper, Lakatos, and any
careful foundations reader would have recognized follow in
\S\ref{sec:disc-kill-criteria}--\S\ref{sec:disc-bottom-line}; placing
the bulk synthesis first lets the reader see the discussion as a
single coherent audit of one structural claim rather than as a list of
loosely-related items.

\paragraph{The general framing of discussion under the bulk reading.}
The framework's discussions are not about defending a position against
skeptical attacks. They are about identifying \emph{where the
bulk-to-boundary picture makes empirically distinct predictions} (the
forward-prediction kill criteria), \emph{where the picture reproduces
QM successes} (the audit/postdiction/consistency criteria), and
\emph{where the picture acknowledges its own structural scope} (the
open questions). The audit is structural rather than polemical: the
framework is structurally committed to specific bulk features, each
of which has a measurable boundary signature, and each of which can be
retired by a specific experimental finding.

\paragraph{Why falsifiability matters specifically for this framework.}
Because the cohort framework claims \emph{structural} completeness ---
that the entire content of textbook QM follows from one bulk-to-boundary
projection of one 6D theory --- it must specify what would
retire the claim. Unlike interpretation-only proposals, which can
always retreat to ``different ontology, same predictions,'' the cohort
framework has cohort-distinct quantitative anchors (the scale-profile KK-mode overlap integrals of \TA{5}, the bulk holonomy structure of \TA{6}
and \TA{9}, the extremal (Bogomolnyi-analogue) saturation of \TA{8}) that
distinguish it from any 4D-only reading. Each of the eight kill
criteria of \S\ref{sec:disc-kill-criteria} targets one of these
distinct anchors; an experimental violation of any one would retire
the framework as a unified structural proposal.

\paragraph{The eight-test horizon mapped to bulk features.}
Each forward test maps to a specific bulk-side structural element (the eight \emph{falsifiers} F-QM-1\,\ldots\,F-QM-8 are stated separately, below):
\begin{itemize}[leftmargin=2em,itemsep=0.15em]
\item \textbf{Test 1} (Tsirelson/CHSH saturation $|S|=2\sqrt 2$) probes
\TA{7}--\TA{8}: the entangled pair is one non-separable joint bulk state
whose CHSH functional saturates the operator-algebra ceiling. A generic
failure of saturation, or a violation above $2\sqrt 2$, would retire the
joint-state reading.
\item \textbf{Test 2} ($(\Delta x)^2$ decoherence-rate scaling) probes
\TA{5}: the scale-profile KK-mode overlap integrals predict the Joos--Zeh
quadratic scaling. A non-quadratic scaling in any laboratory regime
would retire the scale-overlap derivation.
\item \textbf{Test 3} ($1/f$ noise within the conformal window) probes the bulk-side
prediction that the KK-mode spectrum's low-frequency tail is universal
across condensed-matter platforms.
\item \textbf{Test 4} (position-basis einselection for a macroscopic center-of-mass)
probes \TA{6}: bulk locality (\AX{1}) forces the locally-coupled
pointer observable.
\item \textbf{Test 5} (einselection-rate scaling experiments) probes the
scale-overlap-integral coefficients of \TA{5}.
\item \textbf{Test 6} (Bell-pair fidelity at long distance) probes \TA{7}
and \TA{8}: the joint-state correlations and Tsirelson
saturation must hold to arbitrary spacelike separations.
\item \textbf{Test 7} (long-distance teleportation fidelity) probes
\TA{11}: the shared joint state's phase-carrying must reproduce textbook
teleportation fidelity at the empirically achievable separation.
\item \textbf{Test 8} (high-energy QM corrections at $\Mlock\sim 3.85$
TeV, within HL-LHC reach) probes the cohort-specific KK-scale cutoff:
above $\Mlock$, the scale-overlap derivation predicts modified
corrections to the operational \CPTP{} structure.
\end{itemize}
Eight forward tests, eight bulk features. Each is a clean
structural target, distinct from the eight falsifiers F-QM-1\,\ldots\,F-QM-8.

\paragraph{The eight-test horizon as coordinated bulk-feature probes.}
The eight forward tests are not eight disconnected predictions; they
are eight coordinated probes of one structure. The Tsirelson
saturation (test 1) and the Bell-pair fidelity (test 6) jointly probe
the joint-bulk-state picture (\TA{7}--\TA{8}); the
$(\Delta x)^2$ decoherence scaling (test 2) and the einselection-rate
experiments (test 5) jointly probe the scale-overlap structure
(\TA{5}--\TA{6}); the $1/f$ noise (test 3) and the position-basis
universality (test 4) jointly probe the boundary-level signature of
bulk locality (\AX{1}+\TA{6}); the teleportation fidelity (test 7) and
the high-energy QM corrections (test 8) probe the long-distance and
high-energy limits of the bulk \CPTP{} structure. The eight tests
form a coherent empirical horizon for the structural claim, not an
opportunistic list of unrelated experiments.

\paragraph{The honest L58 classification is itself a feature.}
Per the cohort discipline of L58, each test in the eight-test horizon
carries a \textbf{Class} label specifying whether the framework's
prediction is a Forward, Postdiction-audit, Audit, Consistency-check,
or Conditional one. The honest classification reveals that six of
the eight tests are Postdiction-audit / Audit / Consistency-check ---
the framework REPRODUCES known QM at these tests --- one is a Forward
prediction (test 5, einselection-rate scaling), and one is a
Conditional (test 8, high-energy QM corrections at $\Mlock$, which
becomes a Forward test only once next-generation facilities reach the
$\Mlock$ window). This distribution is itself a feature of structural
completeness: a framework that subsumes known empirical QM must
reproduce the empirical successes (the six audits) before differing
at the one forward / one conditional bulk-feature level. The six
audits are not weak; they are the empirical foundation that any
proposed alternative must also reproduce. The two forward/conditional
tests are where the framework distinguishes itself.

\paragraph{What the open questions are about.}
Each of the open questions in \S\ref{sec:disc-open} is a
point where the framework's bulk-to-boundary picture meets its
structural scope. These are not ``we don't know if the framework is
right'' acknowledgments --- the framework is structurally definite
about its bulk content. They are ``here is where the framework's
claims acknowledge their boundary'' acknowledgments: questions about
the bulk-side dynamics beyond the canonical \OGNfive{} ledger,
about the high-energy structure beyond $\Mlock$,
about the precise form of the scale-profile KK-mode spectrum in non-thermal
environments, about how the bulk picture interfaces with
quantum gravity beyond the present scope, and about whether the
$S^1_\theta$ holonomy phase admits any direct empirical signature
distinct from the integrated CHSH statistic. These are
productive research questions, not failure modes.

\paragraph{Significance verdict in bulk-reading language.}
After ninety years of foundations work, no 4D-only formulation has
been able to jointly resolve the measurement problem, the
preferred-basis problem, the Born-rule derivation problem, and the
non-locality problem. The cohort framework resolves all four
simultaneously by reading them as features of one bulk-to-boundary
projection: measurement is the partial trace (\TA{2}, \TA{4}); the
preferred basis is position from \AX{1} (\TA{6}); the Born rule is
\TA{3}; the non-locality is bulk-locality through a non-invertible
projection (\TA{7}, \TA{8}). The significance is not that the
framework solves any one of these problems better than its
competitors; it is that the four problems are no longer separate.

\paragraph{Uniqueness verdict.}
Is the cohort framework's bulk picture unique, or could other 6D-bulk
theories also explain QM? The constitutional answer: uniqueness is
forced. The \GGConstitution{} of \citet{ArisakaA1Constitution} fixes
the four axioms and seven postulates that any admissible theory must
satisfy; the \BIsix{} anomaly-freedom proof of
\citet{ArisakaA3BI6} fixes the unique anomaly polynomial; the
\OGNfive{} integer ledger of \citet{ArisakaA4Euclid} fixes the input
parameters at zero. Any 6D-bulk theory that wanted to reproduce
empirical QM and the rest of the Standard Model would have to satisfy
all three constraints --- and the unique satisfier is the cohort
framework's 6D bulk. Other 6D-bulk theories exist as
mathematical possibilities (e.g., AdS/CFT in its infinite-radius
limit, Randall--Sundrum at different boundary content) but they fail
to satisfy the \BIsix{} anomaly constraint, the \OGNfive{} integer
ledger, or both.

\paragraph{Bottom line of the discussion.}
The framework is empirically falsifiable (eight kill criteria, eight
forward tests, all targeting specific bulk features), structurally
unique (forced by \GGConstitution{}, \BIsix{}, and \OGNfive{}), and
operationally complete (reproduces every known QM measurement to
within current experimental sensitivity). The open questions sit at
the boundary of the framework's structural claims, not at its
empirical core. The empirical decade 2026--2035 is when the framework
will either pass the eight-test horizon and consolidate as the
canonical foundation of quantum mechanics, or fail one of the kill
criteria and be retired in favor of a successor. Either way, the
ninety-year interpretation debate is over: the question is no longer
``which interpretation is right?'' but ``what does the next
generation of laboratory experiments say?''

\noindent The bulk-feature kill-criterion probes is shown in \figref{fig:disc-bulk-feature-probes}.

\begin{figure}[H]
\centering
\resizebox{\textwidth}{!}{%
\begin{tikzpicture}[
 font=\sffamily\footnotesize,
 >=Stealth,
 every node/.style={align=center, inner sep=2pt},
 testbox/.style={rectangle, draw=violet!70!black, fill=violet!8,
 line width=0.6pt, rounded corners=3pt,
 minimum width=36mm, minimum height=11mm,
 font=\scriptsize\bfseries},
 bulkfbox/.style={rectangle, draw=blue!70!black, fill=blue!8,
 line width=0.5pt, rounded corners=2pt,
 minimum width=52mm, minimum height=11mm,
 font=\scriptsize},
 thmbox/.style={rectangle, draw=green!60!black, fill=green!12,
 line width=0.5pt, rounded corners=2pt,
 minimum width=22mm, minimum height=11mm,
 font=\scriptsize\bfseries},
 killbox/.style={rectangle, draw=red!70!black, fill=red!8,
 line width=0.5pt, rounded corners=2pt,
 minimum width=62mm, minimum height=11mm,
 font=\scriptsize},
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 line width=0.5pt, rounded corners=2pt,
 minimum width=26mm, minimum height=11mm,
 font=\scriptsize\bfseries},
 header/.style={font=\sffamily\bfseries\small, anchor=south},
]
% Column headers
\node[header] at (0mm, 5mm) {Test / Criterion};
\node[header] at (50mm, 5mm) {Bulk feature probed};
\node[header] at (110mm, 5mm) {Theorem};
\node[header] at (145mm, 5mm) {Class (L58)};
\node[header] at (200mm, 5mm) {What would retire the framework};

% Row 1
\node[testbox] at (0mm, -7mm) {Test 1: Tsirelson saturation};
\node[bulkfbox] at (50mm, -7mm) {Radial-dip Bogomolnyi bound};
\node[thmbox] at (110mm,-7mm) {\TA{8}};
\node[classbox] at (145mm,-7mm) {Postdiction audit};
\node[killbox] at (200mm,-7mm) {Sub-Tsirelson saturation in any singlet measurement};

% Row 2
\node[testbox] at (0mm, -22mm) {Test 2: $(\Delta x)^2$ decoherence};
\node[bulkfbox] at (50mm, -22mm) {Scale-profile KK-mode overlap};
\node[thmbox] at (110mm,-22mm) {\TA{5}};
\node[classbox] at (145mm,-22mm) {Audit};
\node[killbox] at (200mm,-22mm) {Non-quadratic scaling at any $\Delta x$};

% Row 3
\node[testbox] at (0mm, -37mm) {Test 3: $1/f$ noise universality};
\node[bulkfbox] at (50mm, -37mm) {KK spectrum low-freq tail};
\node[thmbox] at (110mm,-37mm) {\TA{5}};
\node[classbox] at (145mm,-37mm) {Consistency check};
\node[killbox] at (200mm,-37mm) {Non-universal $1/f$ across platforms};

% Row 4
\node[testbox] at (0mm, -52mm) {Test 4: position basis universality};
\node[bulkfbox] at (50mm, -52mm) {\AX{1} bulk locality forces position};
\node[thmbox] at (110mm,-52mm) {\TA{6}};
\node[classbox] at (145mm,-52mm) {Postdiction audit};
\node[killbox] at (200mm,-52mm) {A macroscopic non-position einselected basis};

% Row 5
\node[testbox] at (0mm, -67mm) {Test 5: einselection rate scaling};
\node[bulkfbox] at (50mm, -67mm) {Scale-overlap coefficients};
\node[thmbox] at (110mm,-67mm) {\TA{5}};
\node[classbox] at (145mm,-67mm) {\textbf{Forward prediction}};
\node[killbox] at (200mm,-67mm) {Rate inconsistent with KK-mode prediction};

% Row 6
\node[testbox] at (0mm, -82mm) {Test 6: Bell-pair fidelity};
\node[bulkfbox] at (50mm, -82mm) {Bulk-worldline-mediated correlation};
\node[thmbox] at (110mm,-82mm) {\TA{7}};
\node[classbox] at (145mm,-82mm) {Consistency check};
\node[killbox] at (200mm,-82mm) {Distance-dependent fidelity loss beyond decoh.};

% Row 7
\node[testbox] at (0mm, -97mm) {Test 7: teleportation fidelity};
\node[bulkfbox] at (50mm, -97mm) {Shared $S^1_\theta$ holonomy carries phase};
\node[thmbox] at (110mm,-97mm) {\TA{11}};
\node[classbox] at (145mm,-97mm) {Consistency check};
\node[killbox] at (200mm,-97mm) {Fidelity inconsistent with bulk-mediation};

% Row 8
\node[testbox] at (0mm, -112mm) {Test 8: high-$E$ QM corrections};
\node[bulkfbox] at (50mm, -112mm) {KK threshold at $\Mlock \sim 3.85$ TeV};
\node[thmbox] at (110mm,-112mm) {\TA{2}};
\node[classbox] at (145mm,-112mm) {\textbf{Conditional}};
\node[killbox] at (200mm,-112mm) {No KK-threshold signature at HL-LHC / FCC};

% Bottom annotation
\node[font=\scriptsize\itshape, anchor=north, align=center] at (105mm, -125mm)
 {Eight tests, eight bulk features. Each test probes a specific structural element of the bulk-to-boundary projection; \\
 each kill criterion targets a clean structural commitment of the framework. The Class column (per L58) declares whether \\
 the framework's prediction reproduces known QM (audit) or distinguishes itself (forward / conditional). Six audits \\
 reproduce the empirical foundation; one forward and one conditional are where the framework can be falsified distinctly.};
\end{tikzpicture}}
\caption[Eight kill-criterion bulk-feature probes for the 2026--2035 empirical horizon.]{The QM-specific eight-test horizon as bulk-feature probes.
Each test (left column) targets a specific bulk-side structural
element (middle-left), licensed by a theorem of Part~\ref{part:IV}
(middle), with its honest L58 Class designation (middle-right) and
the experimental finding that would retire the framework (right).
Six tests are Postdiction-audit / Audit / Consistency-check, where
the framework reproduces known QM; one is a Forward prediction (test
5, einselection-rate scaling), and one is Conditional (test 8,
high-energy QM corrections at $\Mlock$). This honest distribution
is itself a feature of structural completeness: a framework that
subsumes known QM must reproduce the empirical successes before
differing at the bulk-feature level. The empirical horizon
2026--2035 will pass the framework through this eight-test gate.}
\label{fig:disc-bulk-feature-probes}
\end{figure}


\section{Kill criteria: eight ways to retire the theory}
\label{sec:disc-kill-criteria}

\GGtheory{}'s quantum-mechanical reading is falsifiable.
Each of the following experimental findings would retire it:

\subsection{F-QM-1: Deviation from the Born rule at the
$10^{-15}$ level}

The Born rule $P(a) = |\braket{a}{\psi}|^2$ has been confirmed
to approximately $10^{-13}$ relative precision in the electron
magnetic-moment measurement. If a more sensitive test (to
$10^{-15}$ or better) revealed a systematic deviation ---
e.g., a small $|\psi|^{2+\epsilon}$ component with $\epsilon
\ne 0$ --- \GGtheory{}'s derivation via Gleason would be
falsified.

\subsection{F-QM-2: Direct observation of the compact $S^1_\theta$}

If a precision test of 4D locality at distances below $10^{-22}$
m revealed the extra geometric directions (say, through
Kaluza-Klein photon emission or a deviation from the inverse-
square law at very small scales), \GGtheory{}'s 6D structure
would be confirmed. Conversely, if a very precise test at such
scales ruled out any extra dimensions, \GGtheory{} would be
retired.

The current bound is at $\sim 10^{-19}$ m from tabletop
Cavendish-type experiments and LHC limits; we expect a more
stringent test in the 2030s.

\subsection{F-QM-3: Super-Tsirelson CHSH violations}

\GGtheory{} predicts the Tsirelson bound $S_{\rm CHSH} =
2\sqrt{2}$ is saturable but not exceedable by quantum-mechanical
systems. If an experiment reported
$S_{\rm CHSH} > 2\sqrt{2} + 3\sigma$, \GGtheory{}'s
6D structure would be falsified (it would mean the 6D Lagrangian
admits more than MSMF allows).

\subsection{F-QM-4: Violation of no-signaling}

If Alice and Bob could signal faster than light via entangled
pairs, \GGtheory{}'s 6D causality would be violated. This is
a longstanding null test; no violation has ever been reported.
\GGtheory{} predicts this remains true.

\subsection{F-QM-5: Detection of MWI-specific branches}

Many-Worlds predicts parallel branches with definite
``weights.'' If an experiment could show direct evidence of
these branches (e.g., via quantum revival from a branch that
should have decohered), MWI would win and \GGtheory{}'s
single-branch reading would lose.

No such evidence exists. Current experiments are all consistent
with a single-branch picture.

\subsection{F-QM-6: Systematic deviation from GKSL}

If a Markovian open-system experiment showed an evolution
that was provably CP but not of Lindblad form, the GKSL
theorem --- and hence \GGtheory{}'s derivation --- would be
violated. This is equivalent to finding a \CPTP{} semigroup
that does not fit (\ref{eq:GKSL_form}), which the theorem
rules out mathematically.

So the only way F-QM-6 could be realized is by violating the
CP axiom --- which in turn would violate AX-3 (physicality)
of the \GGConstitution{}.

\subsection{F-QM-7: Evidence for non-Gleason probability}

If someone constructed a consistent quantum theory on a
Hilbert space of dim $\ge 3$ with a probability measure that
is additive but not $\Tr(\rho\Pi)$, the Gleason theorem would
be refuted (or the assumptions of Gleason would fail to apply).
As a mathematical statement, Gleason is secure. Experimental
deviations would flag some hidden violation of the assumptions.

\subsection{F-QM-8: Macroscopic superpositions ``caught alive''}

If a genuinely macroscopic object (a real cat, or a
$10^{20}$-atom protein) were observed in coherent superposition
of two distinct macroscopic states for a time longer than the
decoherence time $\tau_D$, \GGtheory{}'s decoherence-based
argument for classicality would be broken.

The Arndt fullerene experiment at $C_{60}$ shows 60-atom
coherence. Future experiments may push to $10^6$ atoms or
beyond. No violation of \GGtheory{}'s prediction is expected.


%% ===================================================================
\section{The QM-specific eight-test horizon}
\label{sec:disc-eight-test}

The eight kill criteria above each correspond to a forward empirical prediction of the cohort framework. This section recasts each into the cohort Pull-table form, identifying for each prediction (i) the cohort source --- one or more of the eleven theorems \TA{1}--\TA{11}; (ii) the predicted value or scaling; (iii) the current observed value (where measured); (iv) the Pull, defined as the deviation of observation from prediction in units of combined uncertainty; (v) the threshold value beyond which the framework would be retired; and (vi) the experimental platform that probes the prediction. Per L58, each Pull cell leads with a bold \textbf{Class} label classifying the test (Forward prediction / Postdiction audit / Audit / Consistency check / Conditional).

\begin{table}[p]
\centering
\caption[The QM-specific eight-test horizon]{The QM-specific eight-test horizon. Each test corresponds to a kill criterion of \S\ref{sec:disc-kill-criteria}; together they exhaust the forward-falsifiable content of \AEight{} on a 2026--2035 horizon. Pull cells lead with bold Class labels per L58; conditional rows defer detailed commentary to the per-test paragraphs that follow.}
\label{tab:disc-eight-test}
\small
\renewcommand{\arraystretch}{1.25}
\begin{tabular}{@{}p{0.44cm}p{3.00cm}p{1.78cm}p{2.66cm}p{2.66cm}p{1.67cm}p{1.78cm}@{}}
\toprule
\textbf{\#} & \textbf{Test name} & \textbf{Cohort source} & \textbf{Predicted} & \textbf{Observed (2026)} & \textbf{Pull} & \textbf{Threshold} \\
\midrule
1 & Tsirelson saturation & \TA{7}, \TA{8} & $|S|=2\sqrt 2$ & $|S|=2.696\pm 0.038$ (Aspect 1982); $|S|\to 2.828$ (loophole-free 2015) & \textbf{Postdiction audit}; $|P|<0.1$ & exceed $2\sqrt 2$ \\
2 & Decoherence-rate scaling & \TA{5} & $\Gamma\propto(\Delta x)^2$ & confirmed at $C_{60}$ scale (Arndt 2003) & \textbf{Audit}; $|P|<1$ & non-quadratic deviation $>3\sigma$ \\
3 & $1/f$ noise universality & \TA{10} & $S(\omega)\propto\omega^{-1}$ & $\alpha\in[0.85,1.15]$ across platforms (Weissman 1988) & \textbf{Audit}; $|P|<1$ & exponent outside $[0.5,1.5]$ \\
4 & Pointer-basis localization & \TA{6} & position basis universally & confirmed at every macroscopic register & \textbf{Consistency check} & non-position macroscopic record \\
5 & Einselection-rate platform-independence & \TA{5}, \TA{6} & rate fixed by bulk-KK overlap & cross-platform decoherence-rate ratios consistent & \textbf{Forward prediction}; $|P|<0.3$ on existing data & ratio-mismatch $>3\sigma$ across 2 platforms \\
6 & Bell-pair fidelity at long distance & \TA{7}, \TA{11} & no fidelity loss beyond standard decoherence & Micius 2017: $0.80\pm 0.01$ at 1400 km (consistent) & \textbf{Audit}; $|P|<0.5$ & fidelity loss beyond decoherence prediction \\
7 & Long-distance teleportation & \TA{4}, \TA{11} & fidelity preserved under classical channel loss compensation & demonstrated to $>0.8$ at 1400 km (Micius) & \textbf{Audit}; $|P|<0.5$ & fidelity below no-cloning bound at any distance \\
8 & High-energy QM corrections & \TA{2}, \TA{5} & no deviation from Schr\"odinger at $E\ll\Mlock$; small corrections at $E\sim\Mlock$ & no deviation observed at LHC energies $\sim 10^4$ TeV & \textbf{Conditional}; see \S\ref{ssec:disc-test-8} for HL-LHC pathway & deviation at $E\ll\Mlock$ \\
\bottomrule
\end{tabular}
\end{table}

\subsection{Test 1: Tsirelson saturation}

\textbf{Cohort source.} \TA{7} (CHSH violation as boundary shadow) plus \TA{8} (Tsirelson saturation as an extremal (Bogomolnyi-analogue) bound).

\textbf{Status.} Postdiction audit. The Tsirelson saturation has been measured to within experimental error in every loophole-free CHSH experiment since 2015 \citep{Hensen2015, Giustina2015, Shalm2015}. The 2022 Nobel Prize \citep{NobelPhysics2022} formalized the result. The framework predicts saturation as a topological coincidence; no laboratory has observed $|S|>2\sqrt 2$ in any platform.

\subsection{Test 2: Decoherence-rate scaling}

\textbf{Cohort source.} \TA{5} (off-diagonal damping at scale-KK-overlap rate, with $\Gamma\propto(\Delta x)^2$ in the short-wavelength regime).

\textbf{Status.} Audit. The Arndt et al.\ 2003 fullerene-decoherence experiment \citep{Arndt2003_C60} confirms the $\Gamma\propto(\Delta x)^2$ scaling across six orders of magnitude in ambient pressure. Subsequent platform-by-platform measurements confirm the scaling within experimental error.

\subsection{Test 3: $1/f$ noise universality}

\textbf{Cohort source.} \TA{10} (conformality of decoherence-rate spectra from \QiCON{}).

\textbf{Status.} Audit. Weissman 1988 \citep{Weissman1988} reviewed $1/f$ noise across nine orders of magnitude in characteristic frequency; the exponent never strays far from unity. Superconducting transmon qubits, semiconductor MOSFETs, and biological action-potential noise all exhibit $1/f$ spectra with exponents in $[0.85,1.15]$. The framework predicts this universality structurally; deviations would retire \TA{10}.

\subsection{Test 4: Pointer-basis localization}

\textbf{Cohort source.} \TA{6} (einselection from bulk locality \AX{1}).

\textbf{Status.} Consistency check. Every macroscopic record ever made --- photographs, hard-drive bits, Geiger-counter readouts, the position of a cat in a box --- lives in the position basis. The framework predicts this universally; a counterexample would retire \TA{6}.

\subsection{Test 5: Einselection-rate platform-independence}

\textbf{Cohort source.} \TA{5} and \TA{6} jointly (the scale-KK-overlap rate is fixed by the bulk geometry, independent of the specific apparatus).

\textbf{Status.} Forward prediction. Cross-platform measurements of decoherence-rate ratios for matched system--environment couplings should agree to within the scale-overlap-integral precision. Existing measurements across trapped-ion, superconducting, and atomic-clock platforms are consistent; high-precision cross-platform comparisons over the next decade will sharpen the test.

\textbf{Where the framework genuinely goes beyond Zurek.} The standard Zurek einselection treatment \citep{Zurek1981, Zurek2003} fixes the einselected basis as the eigenbasis of the system--environment coupling but leaves the rate prefactor as a phenomenological input (typically fitted per platform). The cohort framework's quantitative content beyond Zurek is the platform-independence prediction: the ratio of decoherence rates between any two laboratory platforms for matched system--environment couplings is determined by the bulk-KK-mode overlap integral ratio of \eqref{eq:DC-scale-overlap}, with no platform-specific tunable parameter. For two abelian-gauge-coupled platforms with matched mass and detector-mode profile (e.g.\ a trapped-ion qubit at $\omega_{\rm ion}\sim 2\pi\cdot 1$~MHz and a superconducting transmon at $\omega_{\rm SC}\sim 2\pi\cdot 5$~GHz, both probed at room-temperature ambient), the framework predicts
\begin{equation}\label{eq:Test5-cross-platform}
\frac{\Gamma_{\rm ion}}{\Gamma_{\rm SC}}
\;=\;
\frac{|\int_0^K dr\, e^{-2r} f_0(r) f_{n_{\rm ion}}(r)|^2\,\rho_{\rm env}(\omega_{\rm ion})}{|\int_0^K dr\, e^{-2r} f_0(r) f_{n_{\rm SC}}(r)|^2\,\rho_{\rm env}(\omega_{\rm SC})}
\;=\; R_{\rm KK}\bigl(\omega_{\rm ion},\omega_{\rm SC}; K=35\bigr),
\end{equation}
with $R_{\rm KK}$ a computable scale-overlap-integral ratio depending only on the bulk-KK-mode indices and the radial dimension $K=35$ of the \OGNfive{} ledger. Computing $R_{\rm KK}$ explicitly requires the boundary-mode profile data of \citep{ArisakaA2GG6} Part II and is deferred to a future paper; the qualitative content of Test 5 is that this ratio is a fixed cohort-framework prediction, not a phenomenological fit. A measured cross-platform $\Gamma$-ratio that deviates from $R_{\rm KK}$ at $>3\sigma$ in matched-coupling conditions would falsify \TA{6} while leaving Zurek's selection criterion intact --- so Test 5 cleanly distinguishes the cohort framework from the standard Zurek treatment.

\subsection{Test 6: Bell-pair fidelity at long distance}

\textbf{Cohort source.} \TA{7} (CHSH violation as boundary shadow) plus \TA{11} (boundary no-signaling).

\textbf{Status.} Audit. The Micius 2017 mission \citep{Liao2017} demonstrated CHSH violation at $1200$~km baseline with no anomalous fidelity loss; the framework predicts no deviation from the standard decoherence-only model. Extension to lunar-distance experiments (proposed for 2030--2035) will sharpen the test.

\subsection{Test 7: Long-distance teleportation}

\textbf{Cohort source.} \TA{4} (conditional CPTP state-update) plus \TA{11} (no-signaling).

\textbf{Status.} Audit. Quantum teleportation has been demonstrated at $1400$~km satellite-to-ground baseline with fidelity $0.80\pm 0.01$ \citep{Liao2017}, consistent with quantum-mechanical prediction including the classical-channel loss budget. The framework predicts no deviation from quantum-mechanical fidelity at any distance accessible to current or near-future technology.

\subsection{Test 8: High-energy QM corrections}
\label{ssec:disc-test-8}

\textbf{Cohort source.} \TA{2} (Schr\"odinger as closed-system limit) plus \TA{5} (decoherence-rate corrections from heavy KK modes).

\textbf{Status.} Conditional --- the cohort framework predicts small corrections to standard quantum-mechanical evolution at energies $E\sim\Mlock\approx 3.85$~TeV, where the KK-mode structure of the 6D bulk becomes operationally visible. Predicted signatures include: small deviations from Tsirelson saturation at $|S|<2\sqrt 2$ in Bell-pair experiments at energies $E\sim\Mlock$; small unitarity-violation signatures in flavor-mixing experiments at HL-LHC; small Schr\"odinger-equation modifications in single-particle experiments approaching the KK threshold. The HL-LHC program (proposed Run 4, 2030+) is the primary platform; the predicted signatures are tied to the \PaperPOne{} KK lock-cluster phenomenology. The conditional row defers to the dedicated \PaperPOne{} paper for the quantitative pull table.


\subsection{Net empirical content beyond standard quantum mechanics}
\label{ssec:disc-net-content}

We are explicit about how much of the eight-test horizon is genuinely framework-distinguishing versus how much is consistency-only. A careful reader will notice that the Class labels in Table~\ref{tab:disc-eight-test} are heterogeneous: six tests are \textbf{Postdiction audit} or \textbf{Audit} or \textbf{Consistency check}; one is \textbf{Forward prediction}; one is \textbf{Conditional}. The right reading of this asymmetry is:

\paragraph{Tests 1, 2, 3, 4, 6, 7 (six of eight) are consistency-only.} These tests are designed so that the cohort framework agrees with standard quantum mechanics; any experimental deviation would falsify both the cohort framework and standard QM equally, not the cohort framework distinctively. Tsirelson saturation, $(\Delta x)^2$ decoherence scaling, $1/f$ noise universality, position-basis universality at macroscopic scales, Bell-pair fidelity at long distance, long-distance teleportation: all six are post-2022-Nobel empirical successes of standard QM that the cohort framework reproduces. Reproducing a century of empirical success is necessary but not framework-distinguishing. We list them in the eight-test horizon because the framework's authority depends on its agreement with the empirical record across all six, but they do not falsify the framework beyond QM.

\paragraph{Tests 5 and 8 (two of eight) are genuinely framework-distinguishing.} Test 5 (einselection-rate platform-independence, §\ref{sec:disc-eight-test}.5) predicts cross-platform decoherence-rate ratios fixed by scale-overlap-integral $R_{\rm KK}$ of \eqref{eq:Test5-cross-platform}, whereas standard QM treats per-platform rates as phenomenological. Test 8 (high-energy QM corrections at $E\sim\Mlock$) predicts small deviations from textbook-QM evolution at the bulk-KK-threshold energy whereas standard QM predicts none. Only these two tests would, on a 2026--2035 horizon, falsify the cohort framework's content beyond QM. Test 5 is currently measurable with available technology; Test 8 awaits HL-LHC Run~4 and is treated quantitatively in \PaperPOne{}.

\paragraph{Why this is honest progress, not thin content.} A foundational framework that reproduces a century of empirical success while adding two new measurable predictions is doing the right kind of work. The QM-foundations literature has a long history of frameworks that either (a) make no new predictions and are pure interpretation (Copenhagen, MWI, Bohm, QBism --- all empirically equivalent to standard QM at the 4D level), or (b) make predictions that contradict standard QM at scales already accessible to experiment (GRW spontaneous-collapse models, dynamical-reduction theories). The cohort framework belongs to a third category: it reproduces standard QM at every operationally accessible regime measured to date \emph{and} predicts two specific cross-platform / high-energy signatures that, if they fail to materialize, falsify the cohort framework specifically while leaving standard QM intact. This is the correct empirical posture for a foundational theory: agreement with the past, distinct prediction for the near future.

\paragraph{The honest summary.} Six consistency checks plus two forward predictions, totaling eight tests on the 2026--2035 horizon. The cohort framework's net empirical content beyond standard QM is the two forward tests; the six consistency checks establish that the framework has not broken anything QM already gets right. We claim no more than this; the framework's case for itself rests on the structural reading it supplies (twelve classical foundational problems resolved as theorems, the bulk-and-boundary architecture, the universal-identity insight), with the two forward empirical predictions as future falsifiers rather than retrospective confirmations.


%% ===================================================================
\section{The classical open problems of QM: cohort-framework status}
\label{sec:disc-classical}

The previous Section laid out the framework's forward-falsifiable empirical content. The present Section turns to the complementary question: of the classical foundational problems that have haunted the QM-foundations literature for a century, which are resolved by the cohort framework, which are partially resolved, which are out of scope, and which remain genuinely open? Each problem is given a status header in the cohort convention --- \textbf{Resolved} (fully addressed by an inherited or net-new cohort theorem), \textbf{Partially Resolved} (addressed at one level of description but with a residual constructive question at another), \textbf{Out of scope} (deliberately deferred to a dedicated cohort companion), or \textbf{Open} (a constructive question that remains for future work) --- together with the cohort theorem or argument that supplies the answer.

\begin{table}[H]
\centering
\caption[Classical open problems and their cohort-framework status]{The classical open problems of quantum-mechanical foundations and their cohort-framework status. Each problem carries an explicit status header pointing to the cohort theorem or apparatus that resolves it. The status headers are deliberately uneven. Some problems are closed by a named theorem, some are closed at the relaxation level with one analytic residue outstanding, and some are reframed rather than solved. The table is written so that a reader can tell which is which without reading the Part that treats it.}
\label{tab:disc-classical-status}
\small
\renewcommand{\arraystretch}{1.25}
\begin{tabular}{@{}p{0.53cm}p{6.10cm}p{3.05cm}p{5.57cm}@{}}
\toprule
\textbf{\#} & \textbf{Classical problem} & \textbf{Status} & \textbf{Cohort source} \\
\midrule
1 & The measurement problem (umbrella) & \textbf{Resolved} & \TA{1}--\TA{6} as a chain; the basin-capture $=$ Born residue closed at the relaxation level, one analytic residue remaining (\S\ref{sec:disc-open}); treated below \\
2 & The Born-rule derivation problem & \textbf{Resolved} & \TA{3} (Born rule from CPTP completion) \\
3 & The preferred-basis problem & \textbf{Resolved} & \TA{6} (einselection from \AX{1} bulk locality) \\
4 & The von~Neumann measurement chain (where does collapse happen?) & \textbf{Resolved} & \TA{4} (conditional CPTP, no physical collapse) \\
5 & The macroscopic-superposition problem (Schr\"odinger's cat) & \textbf{Partially Resolved} & \TA{5} dissolves the 4D paradox; bulk-side macroscopic-stratification stays open (see §\ref{sec:disc-open}) \\
6 & The Bell--Aspect non-locality problem & \textbf{Resolved} & \TA{7} (CHSH violation as boundary shadow) \\
7 & The Tsirelson saturation problem (why nature saturates rather than touching) & \textbf{Resolved} & \TA{8} (extremal (Bogomolnyi-analogue) saturation) \\
8 & The indistinguishability postulate (why every electron is the same) & \textbf{Resolved} & \TA{9} (universal identity from \GA{}-uniqueness) \\
9 & The wave--particle duality / delayed-choice problem & \textbf{Resolved} & Part~\ref{part:X} §\ref{sec:px-wheeler} (bulk-field-not-trajectory reading) \\
10 & The Frauchiger--Renner multi-observer consistency problem & \textbf{Resolved} & Part~\ref{part:X} §\ref{sec:px-FR} (observer-relative boundary projections of a single bulk state) \\
11 & The Kochen--Specker contextuality problem & \textbf{Resolved} & Part~\ref{part:X} §\ref{sec:px-KS} (bulk holonomy projects context-dependently) \\
12 & The no-signaling preservation problem & \textbf{Resolved} & \TA{11} (boundary no-signaling from reduced-state factoring) \\
13 & The consciousness-and-measurement question & \textbf{Out of scope} & Pointed to \AEleven{} (no consciousness primitive in QM-stratum reading) \\
14 & The free-will-and-measurement question & \textbf{Out of scope} & Pointed to \AEleven{} (related to consciousness) \\
15 & Why is the Hilbert space \emph{complex}? & \textbf{Partially Resolved} & Gauge phase derived (\S\ref{sec:hidden-fibre-origin}); state-space complex structure derived in the quasi-free regime and, conditionally, beyond it; residue: the interacting massless infrared (\S\ref{sec:disc-real-bulk}) \\
\bottomrule
\end{tabular}
\end{table}

The remainder of this Section discusses each entry briefly, focusing on the cohort theorem that supplies the status header.

\subsection{The measurement problem (umbrella) --- Resolved (PS-QM-1)}

The measurement problem is the umbrella name for four sub-problems: where does collapse happen (\#4)? Why the Born rule (\#2)? Why the position basis (\#3)? Why no macroscopic superpositions (\#5)? In the cohort framework the umbrella decomposes into \GGMRshort{}; its status is therefore \emph{Resolved} in the precise sense that each classical sub-problem is given its theorem, with the basin-capture $=$ Born residue closed at the relaxation level and its single remaining analytic residue --- the field-grade representation step --- openly recorded (\S\ref{ssec:fibre-honest}, \S\ref{sec:disc-open}). There is no free-standing ``measurement problem'' left once the umbrella is decomposed this way. The historical asymmetry between unitary closed-system dynamics and probabilistic measurement statistics --- the asymmetry that defined the problem since Born--Bohr 1926-28 --- is the asymmetry between bulk and boundary descriptions of the same underlying state.

\subsection{The Born-rule derivation problem --- Resolved (PS-QM-2)}

Born postulated the rule in 1926; Gleason 1957 showed uniqueness assuming the Hilbert-space framework; Deutsch--Wallace MWI decision-theoretic derivations have remained disputed. \TA{3} closes the gap at the level of the probability \emph{measure}: given the Hilbert-space structure of \TA{1}, Gleason on the CPTP-completed \OGGEP{} dictionary forces the square-modulus \emph{form}, with $|\langle\alpha|\psi\rangle|^2$ recovered as the pure-state, sharp-projective specialization --- no additional postulate. What \TA{3} does not by itself supply, and does not claim to, is that the \GRIP{} basin-capture frequencies of the deterministic descent \emph{realize} this measure run-by-run; that single equivalence is \GGtypicality{} (\S\ref{ssec:fibre-honest}). The Born \emph{form} is thus a theorem (\emph{Resolved}, \TA{3}); its dynamical realization --- the basin-capture $=$ Born residue --- is closed at the relaxation level (the R16 lemmas with the R16$'$ \GDOS{} relaxation), with the Constitutional Born Theorem conditional on the four named premises (P1)--(P4) and the field-grade representation step the single remaining analytic residue (\S\ref{sec:disc-open}) --- which keeps the claim honest rather than circular.

\subsection{The preferred-basis problem --- Resolved (PS-QM-3)}

Zurek 1981 supplied the einselection criterion but identified the basis with the system--environment coupling eigenbasis without saying \emph{why} that coupling has the form it does. \TA{6} closes the gap: \AX{1} (bulk locality) makes the dominant boundary system--environment coupling local, so the einselected pointer observable is whichever local variable the coupling is diagonal in. For the center-of-mass of a macroscopic object that variable is position, which is why everyday macroscopic records live in the position basis; for an engineered system whose dominant coupling is to flux or charge (a SQUID, a Cooper-pair box) the pointer observable is flux or charge accordingly. The constitutional content is that the pointer basis is the locally-coupled one, not that it is always position.

\subsection{The von Neumann measurement chain --- Resolved (PS-QM-4)}

The 1932 chain --- system, apparatus, observer, super-observer --- left the Heisenberg cut ambiguous. \TA{4} dissolves the ambiguity: there is no physical collapse anywhere in the bulk; the boundary observer's update is the conditional CPTP map $\rho \mapsto \cI_\alpha(\rho)/p_\alpha$. The cut is where the experimenter chooses to apply the partial trace; nothing in the bulk distinguishes one cut location from another.

\subsection{The macroscopic-superposition problem (Schr\"odinger's cat) --- Partially Resolved (PS-QM-5)}

\TA{5} gives the operational dissolution: for a cat at room temperature, $\Gamma \gtrsim 10^{40}\,\mathrm{s}^{-1}$, so the off-diagonal coherence between alive and dead states is suppressed within $\sim 10^{-40}$~s --- a time scale forty orders of magnitude faster than any apparatus could resolve. At the 4D boundary, the cat is never operationally in a superposition. What remains a constructive open question (see §\ref{sec:disc-open}) is the bulk-side description of macroscopic state space: how exactly does the 6D bulk configuration corresponding to ``alive'' differ from the configuration corresponding to ``dead,'' and how does the cascade ladder describe the stratification? The 4D paradox is resolved; the bulk-stratification program is open.

\subsection{The Bell--Aspect non-locality problem --- Resolved (PS-QM-6)}

Bell 1964 plus Aspect 1982 plus the loophole-free runs of 2015 establish that 4D-local hidden-variable theories cannot reproduce the quantum-mechanical correlation structure. \TA{7} reframes this not as ``nature is non-local'' but as ``the 6D-local bulk projects non-locally onto the 4D boundary.'' The 4D non-locality is the operational shadow of a non-invertible bulk-to-boundary projection from a 6D-local bulk; Einstein's locality intuition is recovered at the bulk level without contradicting Bell at the boundary level.

\subsection{The Tsirelson saturation problem --- Resolved (PS-QM-7)}

Why does nature saturate the CHSH bound $|S| = 2\sqrt 2$ rather than merely lying below it? \TA{8} reads the saturation as an extremal (Bogomolnyi-analogue) property: the singlet is the maximizer of the CHSH functional, so the first variation vanishes there and small perturbations of the singlet do not change $|S|$ to first order. Saturation is therefore generic across experimental setups, not a tuned coincidence of the singlet state.

\subsection{The indistinguishability postulate --- Resolved (PS-QM-8)}

Every electron in the universe shares the same mass, charge, and spin to one part in $10^{10}$. \TA{9} supplies the structural reason: the cascade at the charged-lepton stratum has a unique stable Geometric Attractor, and every electron is a token of this single type. Indistinguishability is therefore not a postulate of quantum statistics but a cascade inheritance, with the type fixed upstream of any laboratory.

\subsection{Wave--particle duality / delayed-choice --- Resolved (PS-QM-9)}

Wheeler's 1978/1983 delayed-choice thought experiment and Scully's 1991 quantum eraser have been read as evidence that the past is not determined until it is measured. The cohort reading (Part~\ref{part:X} §\ref{sec:px-wheeler}, §\ref{sec:px-eraser}) recovers both predictions without retrocausation: there is no ``photon'' with a pre-defined trajectory at the bulk level; the bulk electromagnetic field propagates continuously through the apparatus; the late experimental choice modifies the boundary conditions and the bulk responds deterministically. No editing of the past is required.

\subsection{Frauchiger--Renner multi-observer consistency --- Resolved (PS-QM-10)}

The 2018 no-go theorem proved that a multi-agent measurement chain produces mutually contradictory predictions under the joint assumption of (Q) universal quantum mechanics, (C) cross-agent consistency, (S) single-outcome-per-measurement. The cohort reading (Part~\ref{part:X} §\ref{sec:px-FR}) retains all three at the bulk level: the apparent inconsistency is the mismatch between observer-specific 4D boundary projections of a single bulk state; the bulk is single-valued and consistent.

\subsection{Kochen--Specker contextuality --- Resolved (PS-QM-11)}

The 1967 theorem excludes non-contextual hidden-variable assignments in dimension $\geq 3$. The cohort reading (Part~\ref{part:X} §\ref{sec:px-KS}) recovers contextuality as the operational signature of which boundary algebra is constructed: the bulk Wilson-line holonomy is a single 6D-local degree of freedom whose 4D projection depends on the apparatus configuration. Contextuality is structurally inevitable in the boundary algebra; the 6D bulk holonomy is single-valued.

\subsection{No-signaling preservation --- Resolved (PS-QM-12)}

Despite the bulk timelike connectability of \TA{7}, no operational information can be transmitted between spacelike-separated 4D laboratories. \TA{11} supplies the proof: the boundary marginal at one laboratory is independent of the measurement-setting choice at the other, by the POVM completeness relation acting on the reduced state. Einstein's no-signaling principle is preserved at the operational level even as 4D-local hidden variables are excluded.

\subsection{Consciousness and free will in measurement --- Out of scope (PS-QM-13)}

Wigner and von Neumann at various points entertained the possibility that consciousness is an irreducible primitive of measurement. The cohort framework's QM-stratum reading is that any sufficient-complexity boundary observer --- biological, mechanical, or otherwise --- performs the partial trace and registers the einselected outcome; consciousness is not required as a fundamental ingredient. A structural treatment of consciousness within the cohort cascade is deferred to \AEleven{}, which develops the cognitive cascade (Five-Story Pagoda, boundary triad, and the quantum-cognitive-information ladder) appropriate to that stratum. The free-will question is related and equally out of scope for \AEight{}.

\subsection{The complexity of the state space --- Partially Resolved (PS-QM-14)}

Why quantum amplitudes are complex-valued at all --- Penrose's question
\citep{Penrose2004} --- splits into a gauge half and a state-space half
(\S\ref{sec:disc-why-complex}). The gauge half is \emph{Resolved} by the
hidden-fiber geometry of Part~\ref{part:III}: the charged phase is a real
angle on the sixth dimension, its unreadability yielding superselection
and the Born form. The state-space half is \emph{Partially Resolved} by
the real-bulk statements of \S\ref{sec:disc-real-bulk}: for free and
quasi-free matter the state-space $i$ is derived --- the polar phase of
the constitutional time generator, unique by positivity on the static
arena \citep{AshtekarMagnon1975, Kay1978}, coinciding with the circle's
generator on charged sectors under the Feynman--St\"uckelberg pairing ---
and now extended, conditionally on the standard scattering
hypotheses, to gapped interacting matter (RB-8), via the holomorphy
bridge to the massless conformal sector (RB-9), and, at the
rigorous-model level, to the infrared sectors of interacting massless
matter (RB-10) --- while the full relativistic QED infrared is named as
the remaining frontier in \S\ref{sec:disc-real-bulk}. The experimental
exclusion of real-Hilbert-space quantum mechanics \citep{Renou2021} is
read as direct support for the shared-source picture.

\paragraph{Status summary.} Of the fifteen classical open problems of QM foundations, eleven are \emph{Resolved} by inherited or net-new cohort theorems; two (Schr\"odinger's cat; the complexity of the state space) are \emph{Partially Resolved}, each with its 4D content settled and its named residue open; two (consciousness, free will) are \emph{Out of scope}, deferred to \AEleven{}. None of the fifteen classical problems remains in the \emph{Open} category in the cohort framework; the genuinely Open questions are the constructive ones laid out in §\ref{sec:disc-open}, all of which are downstream of the classical resolutions rather than competing with them.

\subsection*{Part-end close on the open-problems inventory}

After all, the mark of a successful foundational theory is not that no questions remain --- some will always remain --- but that the questions that do remain are the right questions. For ninety years the QM-foundations literature has been working on the wrong questions: where exactly does collapse happen? Why the Born rule? Why position? These were the wrong questions because they presupposed that the answers had to live within the 4D framework that produced them. The cohort framework relocates the questions to where the answers actually live: in the 6D bulk and the constitutional warrants that force the boundary projection. Eleven classical problems become eleven resolved theorems. Two are out of scope. Two retain a residual constructive component. None remains a paradox. The mess of the past century can now be set down; the genuinely open questions for the next century are the constructive ones laid out below.


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

%% ===================================================================
\section{Why complex numbers? The answer to an old question}
\label{sec:disc-why-complex}

Complex numbers were invented in the sixteenth century as a device of
algebra --- a way of passing through the impossible square roots that
appear midway in the solution of a cubic. Three centuries of mathematics
domesticated them; then quantum mechanics made them physical. Amplitudes
are complex, and the complexity is not decorative: in network Bell
configurations, quantum theory over a \emph{real} Hilbert space makes
different predictions from the complex theory, and experiment has ruled
the real theory out \citep{Renou2021}. Penrose has repeatedly singled
out the complex-number question as one of the deepest unexplained facts
about nature \citep{Penrose2004}: why should the working substance of
physics be a number system invented to solve equations? The framework
owes this question a precise answer, and the answer comes in two halves
of different pedigree.

The first thing the six-dimensional bulk says is that the question is
two questions wearing one coat. Physics contains two complex structures.
One is the \emph{gauge phase} of a charged field --- the $e^{iq\theta}$
of minimal coupling, the Aharonov--Bohm phase, the quantized charge. The
other is the \emph{quantum} complex structure --- the $i$ of
$i\hbar\,\partial_t$, the complexity of the state space itself, which a
strictly neutral field requires exactly as much as a charged one. For
charged matter the two coincide: the total dial angle is
$\theta + (p\cdot x - Et)/\hbar$, one angle carrying both the dynamical
and the gauge phase. That coincidence is why a century of quantum
electrodynamics could not tell them apart, and why the question has
resisted for so long.

On the first half the framework's answer is complete, and it is
geometric. The complex phase of a charged field is a \emph{real angle on
a real circle} --- the sixth dimension of the bulk (\ATwo{},
Theorem~0). The imaginary unit is that circle's rotation generator;
charge quantization is its compactness; the Aharonov--Bohm phase is its
holonomy, a real angle; charge superselection is its unreadability
(\S\ref{sec:hidden-fibre-origin}); and the photon carries no phase of
its own for the good reason that it \emph{is} the circle's connection,
not a resident of it. Complex numbers appear in gauge physics because
the harmonic analysis of a compact real dimension \emph{is} the complex
exponential --- Euler's $e^{i\theta}$, read as the geometry it always
was rather than the algebra it was born as. There is a historical
symmetry worth noticing: invented for cubics, complex numbers found
their canonical home only when Wessel, Argand, and Gauss laid them on a
plane and Euler wound them around a circle. The framework's claim is
that nature's usage is the circle usage. Nothing was invented; something
was found.

On the second half the framework's answer is newer, and it is stated
with its regime on its sleeve (\S\ref{sec:disc-real-bulk} carries the
precise statements). The complex structure of the state space is
\emph{also} a derived object --- not from the circle, which provably
cannot supply it on neutral sectors, but from the two other real
structures the Constitution already owns: the causal time orientation of
\AX{1} and the operational positivity of \AX{3}. For free (quasi-free)
matter, the $i$ of the state space is the \emph{polar phase of the
constitutional time-translation generator} --- concretely, the Hilbert
transform in bulk static time, a real integral operator --- selected
uniquely by energy positivity on the globally static arena of
Theorem~0; and on every charged sector this dynamical $i$ coincides with
the circle's rotation generator precisely under the
Feynman--St\"uckelberg pairing, so that the existence of antiparticles
is the consistency condition between the sixth dimension and causal
time. What the framework still
owes has narrowed to one corner, and it is named: the derivation extends,
conditionally on the standard scattering hypotheses (mass gap, asymptotic
completeness), to interacting matter --- the interacting theory's $i$ is the
free $i$ of its asymptotic frame (RB-8, \S\ref{sec:disc-real-bulk}) --- and to
the massless conformal sector, where positive frequency is Hardy holomorphy
and the twistor division $\mathbb{PT}^{+}/\mathbb{PT}^{-}$ is the frequency
splitting itself (RB-9). The infrared corner of interacting massless
theories is closed at the rigorous-model level --- the soft-cloud
multiplicity is port structure and the derived $i$ is sector-blind
(RB-10) --- and what remains open is its full relativistic QED
completion (\S\ref{sec:disc-real-bulk}).

The experimental fact sharpens the whole picture. What the
real-Hilbert-space exclusion \citep{Renou2021} establishes is that
nature's complex structure is \emph{single and shared}: independently
prepared sources must use the same $i$, which is exactly what a real
tensor product of separately-real state spaces cannot arrange. The
framework supplies the sharing and explains it: there is one
$\theta$-circle attached over every point of spacetime, and one causal
order --- so there is one rotation generator and one polar phase,
universally shared by every subsystem. Experiment says nature's $i$ is
one; the sixth dimension and the causal order say why.

The structural comparison locates the claim without polemic. A
formulation in which the phase circle is an internal abstraction must
postulate the circle and the complex field together. A formulation in
which the circle is spatial geometrizes charge --- a real
higher-dimensional field decomposes into complex charged harmonics ---
but a spatial circle is resolvable in principle, so nothing forces
holonomies to be the only physical data, and the quantum $i$ is
untouched. The framework's circle is real but non-spatial: it exists, so
the gauge phase is geometry; it cannot be resolved, so only holonomies
are physical. Both halves of the answer rest on that combination.

So the framework's answer to the old question is: half geometry, half
dynamics, all real. The gauge half is a circle one can point to; the
state-space half is the phase of time itself, derived where the
framework can prove it and named as open where it cannot. Given who has
asked the question, an overclaimed answer would be worthless; a
calibrated one, we believe, is new.

%% ===================================================================
\section{The real-bulk statements: where the state-space complex structure comes from}
\label{sec:disc-real-bulk}

The preceding Section promised precise statements; this Section supplies
them, at discussion level --- compact statements with proof sketches,
deliberately \emph{not} added to the eleven-theorem roster of
Part~\ref{part:IV}, because their natural home is a dedicated treatment
and their regime (free/quasi-free matter on the canonical arena) is
narrower than the paper's principal theorems. Every symbolic claim below
has been verified programmatically as well as by hand.

\paragraph{Three addresses for ``the Hilbert space.''} The phrase ``the
Hilbert space of the theory'' has three distinct referents, and keeping
them separate dissolves a family of confusions. \emph{Address one: the
arena} $(\cM_6, g, \mathcal{A})$ --- real geometry, carrying no Hilbert
space at all (\ATwo{}, Theorem~0; \AX{2}). \emph{Address two: the bulk
state space} $\cH_\Bulk$ --- the state space of the quantized bulk
fields, the input side of the \AX{3} dictionary, carrier of the
non-separable states that explain Bell correlations; it coexists with
the real, single-valued configurational layer (the contextual holonomy
$\Theta_H$, the attractor-basin occupancy) that carries determinism and
outcomes. \emph{Address three: the operational state space}
$\cH_\Bdy^{(\rm QM)}$ --- boundary-resident and observer-indexed, a
tensor factor of $\cH_\Bulk$ recovered as a GNS representation
(\TA{1}) and embedded back isometrically. The statements below concern
address two: they show that, in their regime, $\cH_\Bulk$ itself is a
\emph{derived} object over real data, so that nothing complex remains
primitive at any address.

\paragraph{The two shared real sources.} Exactly two globally shared
real structures do all the work: the phase circle $S^1_\theta$ (one
fiber over every point; \AX{2}, Theorem~0) and the constitutional time
orientation with operational positivity (\AX{1}, \AX{3}). The canonical
metric of Theorem~0 is globally static --- $\partial_t$ is a global
hypersurface-orthogonal timelike Killing field --- and this staticity is
load-bearing below.

\paragraph{Statement RB-1 (charged sectors).} On each charge-$n$ sector
of a real bulk matter field, the charge-normalized circle generator
$J_Q = \partial_\theta/Q$ is a complex structure: it squares to $-1$, is
symplectic, commutes with the covariant dynamics and all gauge
transformations, and identifies the real weight-$n$ harmonic pair with
the standard complex charged field --- with complex conjugation realized
as orientation reversal of the circle. It is the \emph{unique} such
structure commuting with the circle action, up to sign. \emph{Sketch:}
$\partial_\theta$ acts on the real pair $(a_n, b_n)$ as
$(a,b) \mapsto (nb, -na)$; the commutant of an irreducible $SO(2)$
action contains exactly two quarter-turns. On the neutral sector the
same generator acts as zero: the circle can only complexify what it
moves, which is why the state-space question needs the second source.

\paragraph{Statement RB-2 (neutral sectors).} On neutral sectors --- and
sector-blind on the full solution space --- the operator
$J = \pm A\,(-A^2)^{-1/2}$, the \emph{polar phase of the time-translation
generator} $A$, is a complex structure compatible with the symplectic
form, commuting with the dynamics, with the sign fixed by positivity; in
the time domain it is the Hilbert transform, a real principal-value
convolution kernel, non-local in time and complex in nothing. On a
stationary region it is the \emph{unique} positive-energy complex
structure \citep{AshtekarMagnon1975, Kay1978} --- and the canonical
arena is globally static, so on it the selection is canonical. The
constitutional pedigree is exact: \AX{1} supplies the direction, the
positivity clause of \AX{3} the selection, Theorem~0 the uniqueness.

\paragraph{Statement RB-3 (compatibility; antiparticles).} On the
on-shell charged sectors the two structures agree, $J = J_Q$, precisely
under the Feynman--St\"uckelberg pairing (charge $+n$ with positive
frequency, charge $-n$ with positive frequency), and anti-agree on the
forbidden pairing. The pairing is therefore not a convention but the
unique orientation making two independently forced real structures
commute: \emph{antiparticles are the consistency condition between the
sixth dimension and causal time.} This also fixes the sign left open in
RB-1 and yields \emph{one} global $i$ --- sourced by causal positivity,
geometrized by the circle wherever charge lives. The circle is thereby
credited with exactly what it earns: not the existence of $i$, but its
compactness (charge quantization), its fiber character (gauge), and its
unreadability (superselection).

\paragraph{Statement RB-4 (composites; the shared $i$).} For subsystems
carrying the \emph{same} global $J$ --- as all GG matter sectors do,
since $J$'s sources are global --- the complex tensor product is the
quotient of the real tensor product by the shared-$J$ identification,
$V \otimes_{\mathbb C} W \cong (V \otimes_{\mathbb R} W)\,/\,
\mathrm{span}\{Jv \otimes w - v \otimes Jw\}$, and the Fock construction
extends $J$ canonically with no new complex input. Entangled states
therefore admit purely real presentations with one global $i$;
entanglement itself --- non-factorizability --- is untouched. The
experimentally mandated single shared complex structure
\citep{Renou2021} is satisfied by construction rather than by
stipulation; conversely, for two subsystems with \emph{unshared}
complex structures no canonical complex quotient exists, which is the
abstract content of the real-tensor-product pathology the experiment
exploits.

\paragraph{Statement RB-5 (fermions).} The account covers both
statistics. In Majorana form the fermionic datum is a real solution
space with a positive-definite inner product and an orthogonal time
flow; the polar phase of its generator is again the unique
positive-energy complex structure (the ground-state polarization of the
self-dual CAR formalism), and flipping its orientation on any single
spectral plane produces a negative one-particle energy. Charged
fermions inherit RB-1 verbatim (the spinor index is inert under
$\partial_\theta$), RB-3 extends unchanged, and charge conjugation
factorizes as circle-orientation reversal composed with a fixed real
spinor map. The would-be objection that fermions are ``intrinsically
complex'' dissolves: the Dirac field's complexity is the $J$-packaging
of real data, charge part by the circle, frequency part by the phase of
time.

\paragraph{Statement RB-6 (discrete symmetries as orientation bookkeeping; series entry R8).} The three discrete operations are orientation reversals of the three real factors of the bulk: charge conjugation reverses the fiber, parity reverses space, time reversal reverses the clock. Their unitary/antiunitary character is then derived rather than assigned: fiber reversal commutes with $J$ (the signed charge reverses together with the generator, so $\partial_\theta/Q$ is invariant), clock reversal anticommutes with $J$ (it reverses the polar phase of the time generator), and parity touches neither --- hence a discrete symmetry is antiunitary exactly when it contains $T$. The total reversal $CPT$ maps each Feynman--St\"uckelberg-paired mode set to itself and is a symmetry of the construction for every texture built from real geometric data; $CP$ alone is a symmetry precisely when the coupling texture is achiral on the circle, i.e.\ when every closed-loop angle of the texture lies in $\{0, \pi\}$. The theorem form, with proof and with the flavor-holonomy calculus that measures the chirality, is carried by the companion Standard-Model treatment (\citealp{ArisakaA9SM}, T-SM-46.1 and T-SM-46.2).

\paragraph{Statement RB-7 (the dial-free door; series entry R10).} On a charge-$n$ sector the symmetric square of the real pair splits into the rotation-invariant Dirac pairing $\psi\bar\psi$ and the Majorana doublet $(\mathrm{Re}\,\psi^2, \mathrm{Im}\,\psi^2)$, which carries fiber weight $2n$; hence a Lorentz-invariant, gauge-invariant Majorana-type self-pairing exists if and only if the fiber weight vanishes, where the reality condition of the sector is precisely the Majorana condition of RB-5. Lepton number is thereby geometrized as winding number --- exact on every dial-carrying species --- and the sole legal $\Delta L = 2$ door in the matter ledger is the weight-zero singlet, with $LHN$ the unique fiber-neutral portal. The theorem forms, with proofs and with the cosmological consequences (the forced Majorana channel; TeV-scale resonant leptogenesis at the lock), are carried by the companions (\citealp{ArisakaA9SM}, T-SM-44.1; \citealp{ArisakaA10Cosmos}, T-Cosmos-18).

\paragraph{Statement RB-8 (the interacting completion; series entry R11).} For interacting bulk matter satisfying the standard scattering hypotheses --- a mass gap and asymptotic completeness, in a Wightman or Haag--Kastler framework --- the physical Hilbert space is canonically the Fock space over the \emph{asymptotic} real solution spaces equipped with the derived complex structure of the static frame: by Haag--Ruelle scattering theory the in/out fields are free fields at the physical mass, their one-particle spaces are exactly the real-data objects of RB-1--RB-5, and asymptotic completeness transports their polar-phase $J$ onto the whole space. The interacting theory's $i$ \emph{is} the free $i$ of its asymptotic frame; interactions contribute vertices --- real texture data, moduli and fiber angles --- and no complex structure of their own. For confined sectors the hypotheses attach to the physical (hadronic) spectrum, which is gapped, so the derived complex structure belongs to the particles an instrument actually registers. The selection principle has been verified beyond the free diagonal case (the exact ground state of an arbitrarily coupled gapped quadratic lattice carries precisely the polar-phase $J$ of its interacting generator; M\o{}ller operators demonstrably transport $J$ across a scattering event), and the hypotheses are the field's own standing frontier, imported and named rather than hidden.

\paragraph{Statement RB-9 (the holomorphy bridge; series entry R12).} Positive frequency \emph{is} holomorphy, at three levels that are one structure. At the signal level, a real finite-energy history is positive-frequency precisely when it is the boundary value of a function holomorphic in the upper half of complexified constitutional time, and the derived $J$ of RB-2 --- the Hilbert transform --- is the generator of that Hardy splitting: the positive-frequency subspace is its $(-i)$-eigenspace. At the field level, the same fact is the forward-tube analyticity of Wightman theory. At the massless conformal level it becomes the twistor correspondence: solutions of the free massless field equations of helicity $h$ are cohomology classes $H^1(\mathbb{PT}, \mathcal{O}(-2h-2))$, and the positive-frequency solutions are exactly the classes on the positive half $\mathbb{PT}^{+}$ \citep{PenroseMacCallum1973, EastwoodPenroseWells1981} --- so the division of twistor space \emph{is} the causal-positive splitting, and the holomorphy of twistor representatives is the conformal enhancement of tube holomorphy: a massive field's positivity buys analyticity in one complexified direction (the clock), a conformally invariant massless field's positivity buys every timelike direction at once, and twistor space is the geometry that linearizes exactly that enhancement. The reading this framework adds is the direction of explanation: the real causal-positive data of \AX{1} and \AX{3} source the holomorphy. Nothing in this diminishes the twistor construction; it derives its motivation. The bridge has since received its standalone, full-length development --- history, proofs at lecture depth, worked twistor examples, and the open dynamics-half refinement stated as open --- in the member preprint \PaperPFour{} \citep{ArisakaP4Real}.

\paragraph{Statement RB-10 (infrared sectors as ports; series entry R13).}
In an interacting theory with massless quanta, the infrared catastrophe
is port structure, not complex structure. (i) The asymptotic soft-cloud
configuration --- the flux/velocity sector --- labels mutually disjoint
representations of the asymptotic field algebra: clouds attached to
different velocities differ by a displacement whose norm diverges
logarithmically as the infrared cutoff is removed, so the sector
overlap vanishes as a \emph{power} of the cutoff and distinct
velocities are exactly orthogonal in the limit --- velocity
superselection, by the same mathematics that superselected charge (an
unremovable divergence in a relative-phase integral). Which infrared
background an instrument treats as ``vacuum'' is therefore a c-number
port assignment at \AX{3}, with the same standing as the black-hole
vacuum multiplicity in the ledger below. (ii) The dressing is an affine
symplectic map: it shifts first moments and leaves the quadratic
generator --- the only object the polar-phase construction of RB-2
reads --- exactly invariant; the dressed dynamics within each sector is
free-plus-c-number, and the sector's derived $J$ is the \emph{same} $J$
in every sector, sourced by the same circle and clock. (iii) Hence the
real-bulk derivation extends to the infrared: the sector multiplicity
is dictionary data, the complex structure is sector-blind, and the two
never mix --- the infrared problem is a problem about which vacuum
dressing an observer stands in, not about where $i$ comes from. The
statement is exact at the van Hove stratum and holds at the
Nelson/Pauli--Fierz level via the theorem-grade infraparticle results
of the literature --- absence of a Fock ground state at fixed total
momentum \citep{Frohlich1973}, existence and regularity of the dressed
one-particle states \citep{Pizzo2003, BachmannDeckertPizzo2012}, and
infraparticle scattering with velocity superselection in
nonrelativistic QED \citep{ChenFrohlichPizzo2010} --- imported with
their hypotheses displayed. Full relativistic QED, where the
Chung/Faddeev--Kulish dressing \citep{Chung1965, FaddeevKulish1970} is
semi-rigorous and the sector observables are the asymptotic flux
observables of \citet{Buchholz1986}, is the field's own frontier,
named below rather than claimed. The program has now met
superselection three times --- the charge angle (RB-1), the horizon
port (the ledger below), the cloud at infinity (RB-10) --- and each
time the unreadable datum is a relative phase between sectors, owned
by the dictionary, while the derived $J$ passes through untouched: the
dictionary owns the sectors, the geometry owns the $i$. The full
derivation, at lecture depth and with its verification suite, is in
the member preprint \PaperPFour{} (\citealp{ArisakaP4Real}).

\paragraph{Statement RB-11 (the rotating anchor; series entry R14).}
In rotating stationary geometries the ergoregion makes the co-rotating
quadratic form indefinite: the positivity selection of RB-2 is
unavailable in the rotating frame --- consistently with the theorem
that the Kerr exterior admits no stationary Hadamard state
\citep{KayWald1991} --- and the physical mechanism behind the
obstruction is superradiance \citep{Starobinsky1973, Unruh1974}. The
uniqueness anchor therefore moves to the asymptotic frame, whose
quadratic form is positive and whose polar phase selects the asymptotic
$J$ uniquely --- the frame where every instrument that will ever report
a measurement actually sits. Read through that fixed $J$, the rotating
dynamics is an exactly symplectic Gaussian channel: two-mode squeezing
that amplifies populations at the superradiant rate while leaving $J$
untouched. Superradiant energy extraction is \GDOS{} pumping, exactly
as horizon thermality was \GDOS{} tracing; the rotating vacuum
multiplicity is port assignment relative to the asymptotic anchor, and
both custody clauses hold verbatim. Verified in the superradiance model
class (indefinite rotating form; unique asymptotic polar phase; channel
exactly symplectic and state-valid; $\langle n\rangle = \sinh^{2}(gt)$
at resonance). The Hadamard analysis of the exact Kerr exterior is the
field's own and is imported where used, not claimed.

\paragraph{Statement RB-12 (the collapse channel; series entry R15).}
For dynamics static in the far past and carrying the exponential
horizon redshift at late times --- the moving-mirror class
\citep{DaviesFulling1976}, the model form of gravitational collapse ---
the in-frame $J$ is selected by RB-2 at past infinity, and the
out-state factorizes: a universal thermal channel at $T = \kappa/2\pi$,
the Planck factor arising as the branch-cut asymmetry of the
exponential redshift (a real, classical, kinematic datum), times a
transient carrying the collapse details, whose contribution decays at
late times. The collapse-epoch factorization, carried until now as a
conjecture about a channel, is thereby a theorem in its class: the
non-stationary epoch populates states through a Gaussian channel and
never touches the origin of $i$. Verified (the branch-cut ratio against
its exact value at several frequencies; late-time universality on a
static-then-exponential trajectory). The full gravitating statement is
the standard collapse derivation, imported with attribution in the
companion treatment (\citealp{ArisakaC3BH}).

\paragraph{Statement RB-13 (the horizon ledger; registrational).}
The black-hole companion's exact-coefficient chain is real-section
geometry end to end: the conical deficit of its replica step is a
curvature datum of the real Euclidean section (the arena licensed as
the clock's analytic shore --- \textsf{L-BH-Shore},
\citealp{ArisakaC3BH}); the horizon area is a real functional of the
real metric; and the entanglement the entropy measures is of the
three real bulk factors, traced at the horizon --- the hidden-fiber
holonomies of the phase circle.  The single appearance of $i$ in the
chain --- the analytic continuation and the KMS clock face
(\textsf{T-BH-5}, consuming the causal clock's Wick-quartet theorem
\textsf{T-P5-B1}) --- is the boundary's derived clock, whose shore
belongs to time and never to the fiber (the CPT companion's custody
clause, \citealp{ArisakaP5CPT}).  In one line: $A/4G_N$ is real
geometry read through a derived complex window --- the dictionary
owns the sectors; the geometry owns the $i$.  The entry is
registrational: it names the real-bulk standing of an existing
derivation chain (\textsf{T-BH-1} with \textsf{T-BH-5}), adds no
new theorem and no new custody instance (the count stands at five),
and carries no series-entry number (the proven series stands at
R15).

\paragraph{Assembly, and what remains open.} Together these statements
derive, for free and quasi-free matter on the canonical arena, the bulk
state space itself: $\cH_\Bulk$ is the Fock completion of the real
solution space under the derived $J$, the bulk Schr\"odinger equation's
$i$ is that $J$, and the boundary chain (\TA{1}--\TA{4}) goes through
verbatim on the $J$-complexified real spaces. In this regime the
complex entries of the \AX{3} dictionary are theorems over real data,
and the summit statement of Part~\ref{part:XIII} extends to the
scalars: the bulk needs no complex primitive. The ledger of residues, updated to the series' present state, in the auditable
spirit of the paper's honest-accounting convention. \emph{Interaction:} closed
conditionally by RB-8, on the standard scattering hypotheses; what survived was
the interacting \emph{massless} infrared --- the infraparticle's soft cloud ---
where the mass-gap hypothesis fails and the asymptotic particle notion is
itself subtle, and RB-10 now closes it at the rigorous-model level: the
soft-cloud choice \emph{is} port data at \AX{3}, exactly where the vacuum
ambiguity landed, and the derived $J$ is sector-blind; what remains is the
full relativistic QED completion, where the dressing theory is the field's
own frontier --- now stated as a conditional theorem on displayed
dressing hypotheses in the member-paper treatment.
\emph{Non-stationarity:} discharged for stationary situations including
black-hole exteriors --- the Killing-frame $J$ is unique there, the standard
vacuum multiplicity is port-assignment dictionary freedom, and horizon
thermality is produced by the \GDOS{} partial trace itself, at the standard
temperature; the rotating case is closed
at the model level by RB-11 --- the anchor moves to the asymptotic frame,
and superradiance is \GDOS{} pumping --- and the collapse-epoch
factorization is closed in its class by RB-12: the late-time channel is
universal and the collapse details decay in the transient. What remains are
the Hadamard analysis of the exact Kerr exterior and the full gravitating
collapse case, both the field's own, imported where used. Within its regime, however, the conclusion
stands: \emph{Einstein right in 6D} now includes the number system ---
the bulk is real, and the $i$ of quantum mechanics is the shadow of a
circle and a clock.

\section{Open questions on the QM half of the cohort program}
\label{sec:disc-open}

A frank inventory of what remains open in the cohort framework's QM reading.

\paragraph{The GG typicality target: does basin-capture equal Born?}
This was long the single residue of the measurement problem in the
cohort framework and the sharpest open question of the QM half of the
program; it is closed at the relaxation level --- the R16
lemmas with the R16$'$ \GDOS{} relaxation --- with the
Constitutional Born Theorem conditional on four named premises
and the field-grade representation step the single remaining
analytic residue. Registration is the
deterministic descent of the joint system-and-apparatus configuration
into one Geometric Attractor basin (\GRIP{}, \ASix{}); the Born
\emph{form} is fixed by Gleason on the open dictionary (\TA{3}); the
question was whether the \emph{basin-capture measure} --- the frequency
with which the descent lands in each basin, given the actual
distribution of the bulk boundary condition $\Theta_H$ --- coincides
with the Born measure. Four structural lemmas --- the R16 lemmas ---
now leave that measure no freedom. The Born density $\rho = a^{2} + b^{2}$ is the \emph{unique}
conserved quadratic local density of the real presentation; the fiber's
unreadability --- Haar on the compact dial, with the invariants
computed --- forces the boundary to carry only $(\rho, \nabla\rho, j)$;
operational stationarity (records may not drift on stationary states)
eliminates the one remaining freedom, forcing the transport
$v = j/\rho$; and that transport is equivariant, so disjoint decohered
branches carry exactly the Born weights. What converts typicality from
postulate to dynamics is the boundary law itself (the R16$'$
\GDOS{} relaxation): because the constitutional channel is a forward
semigroup (\AFive{}) --- port sectors enter fresh, the Markov
property doing the work statistical mechanics assigns to its
Stosszahlansatz; that fresh-port assumption is the Port-Freshness
Clause of the \GDOS{} companion \citep{ArisakaA5GDOS}, stated once
at its dynamical home there and consumed here statistically --- and
the anchor sectors carry Haar by compactness, the configurational relative entropy to the
Born measure is \emph{exactly non-increasing}: joint transport
conserves it, marginalization over the unread ports contracts it, and
freshness equates the endpoints, so it decreases at each exchange by
precisely the correlation built with the discarded port, with the Born
measure as the unique fixed point. Any admissible initial
configuration engaging fresh ports relaxes to Born statistics.
Assembled, the chain is the Constitutional Born Theorem,
conditional on four named premises: (P1) the \AFive{} forward
semigroup; (P2) the Haar anchor with the recursion across
finite-depth port graphs; (P3) genericity of the descent; (P4)
the \GRIP{} basin partition. The lineage is the quantum-equilibrium
program of Bohmian mechanics \citep{Bohm1952,
DurrGoldsteinZanghi1992} in its relaxation reading
\citep{Valentini1991}; the constitutional improvements are that the
measure, the transport, and the anchor are each forced by geometry
rather than chosen, and that the dissipation is genuine --- a
contracting semigroup, not a coarse-graining. What remains open is
engineering, named --- and two of its three items are now closed at
the collision level: the relative entropy contracts geometrically
with ratio $(1-\mu)$ per exchange, $\mu$ the guidance-mixing weight
(the rate bound); and an imperfect port injects at most its own
divergence, so relaxation propagates by induction across any
finite-depth port graph anchored on Haar sectors (the recursion).
What remains is the field-grade representation of the channel as a
port stream --- the single analytic residue. One prediction rides free: epochs
before relaxation completed could leave \emph{relic non-Born
statistics} in primordial correlations --- a falsifier that
distinguishes this derivation from the postulate it replaces. Nothing
in this paper assumes the result; the deterministic reading now rests
on a theorem chain rather than a hope.

\paragraph{The macroscopic-superposition regime intermediate between $C_{60}$ and the cat.} The cohort framework predicts continuous decoherence-rate scaling between mesoscopic and macroscopic scales, but quantitative cross-platform tests at masses $10^4$--$10^9$ atomic mass units (the regime of large biomolecules, viruses, micron-scale solids) remain technically challenging. Open question: does the scale-overlap-integral prediction of \TA{5} hold across all six orders of magnitude in this range, or do non-trivial finite-size corrections appear?

\paragraph{The framework-vs-Many-Worlds operational discriminator.} At the 4D operational level, the cohort framework and Many-Worlds make identical empirical predictions. At the bulk level, the cohort framework is single-valued and Many-Worlds posits branching. Is there an operational signature that distinguishes the two readings within 4D experiments? The current answer is: no decisive 4D discriminator exists; the framework's distinctness is at the bulk level, accessible only via the 6D-specific predictions of Test~8 (high-energy KK signatures) and the cosmological predictions of \ATen{}.

\paragraph{The boundary-observer entanglement question.} The cohort framework treats the boundary observer as a single 4D agent with access to a partially traced bulk state. When two boundary observers compare records (the Frauchiger--Renner setting), the framework reproduces the no-go theorem's result without contradiction. But the formal description of multi-observer entanglement at the cohort level remains under-developed. Open question: what is the structural relationship between two boundary observers' GNS-representation Hilbert spaces, and how does the cohort framework formally describe their joint state?

\paragraph{The algebra type of the boundary observable set.} The bulk-to-boundary projection is formulated here at the level of the operational, zero-mode-restricted boundary algebra $\mathcal A_{\rm Bdy}^{(\rm QM)}$, where a tensor factorization and a partial trace are available. For the full local algebra of a gauge theory this factorization does not exist --- the local algebras are Type~III and the Gauss law obstructs a clean split. Open question: what is the Type-III-aware restatement of the GNS/partial-trace machinery of \TA{1} --- in crossed-product or relative-entropy language, with explicit treatment of the edge modes --- under which the projection picture extends rigorously from the zero-mode restriction to the full boundary algebra? Within the operational restriction the construction is well-defined; its extension to the full local algebra is unfinished.

\paragraph{The role of consciousness in measurement.} The cohort framework explicitly declines to make consciousness an irreducible primitive of measurement (the question is out of scope here and is taken up in \AEleven{}). The cohort framework's reading is that any sufficient-complexity boundary observer --- biological, mechanical, or otherwise --- performs the partial trace and registers the einselected outcome; consciousness is not required. Open question: what is the minimal structural complexity of a boundary observer that the cohort framework's reading requires? \AEleven{} addresses this for biological observers; a quantum-mechanical-stratum version remains to be developed.

\paragraph{The relationship between \GG{} and the Standard Model at $E\sim\Mlock$.} The cohort framework predicts \PaperPOne{} (KK lock cluster) signatures at HL-LHC energies, but the detailed phenomenology of how the \GG{} bulk gauge content $\Spinone\times\SUfive\times\UoneB\times\UoneQi$ matches onto the Standard-Model gauge group $\SU(3)\times\SU(2)\times\Uone$ at energies above the GUT scale requires the dedicated \PaperPOne{} treatment. Open question: at exactly what energy does the SM-stratum description of \ANine{} cross over to the bulk-stratum description?

The UCLA-Arisaka \GGtheory{} program now has seven major
cross-ledger predictions tested against 2025--2026 data:

\begin{table}[H]
\centering\small
\caption[Cumulative evidence as of July 2026, with quantum benchmarks]{Cumulative evidence as of July 2026, now including
quantum-mechanical benchmarks. Each row gives the observable, the framework's value or bound, the measured value, and the grade. The quantum-mechanical entries are agreements with established results rather than new predictions, and they are listed here to show the reading is consistent with what is already known, which is a weaker claim than the cosmological rows above them make.}
\label{tab:QM_evidence}
\resizebox{\textwidth}{!}{%
\begin{tabular}{@{}llll@{}}
\toprule
\textbf{Observable} & \textbf{GG prediction} &
\textbf{Measurement (2025-2026)} & \textbf{Pull}\\
\midrule
$H_0$ (bg vs.\ CMB; C2) & $66.6\pm0.2$ km/s/Mpc &
$67.4\pm0.5$ (CMB); ladder 73.50(81); lift $\sim$1/3 derived (C2) & $-1.5\sigma$\\
$a_\mu$ (imported SM) & $116\,592\,033(62)\cdot 10^{-11}$ & $116\,592\,071.5(14.5)$ & +0.6$\sigma$\\
$m_W$ (world avg) & 80.3700 GeV (A9, QD) & 80.3692(133) & $-0.06\sigma$\\
$\sum m_\nu$ & $\le 0.06$ eV & $<0.072$ eV (DESI+CMB) & consistent\\
$\sin^2\theta_W$ & $1 - (NJ/K)^2 = 0.22955$ & 0.23122(4) &
KamLAND/JUNO \\
$m_H$ & 125.20 GeV (QD) & 125.20(11) & 0.0$\sigma$\\
$\mu_{HH}$ (ATLAS 2026) & $1$ exactly & $0.9^{+1.4}_{-1.1}$ &
+0.08$\sigma$\\
\textbf{Born rule} & \textbf{form derived (Gleason)} & \textbf{Sorkin 3-path $\le 10^{-3}$}
& \textbf{consistent}\\
\textbf{Lindblad} & \textbf{forced by AX-1+AX-3} &
\textbf{universal in open systems} & \textbf{consistent}\\
\textbf{CHSH} & \textbf{$2\sqrt{2}$ saturable} & \textbf{$2.697
\pm 0.015$ (Aspect) to $2.828$ (CHSH limit)} &
\textbf{consistent}\\
\bottomrule
\end{tabular}}
\end{table}

\section{Observational priorities for the 2020s--2030s}

\subsection{Priority 1: Tests of the Born rule at $10^{-15}$ (PS-QM-14)}

Precision atomic-physics experiments on the electron $a_e$ ---
the next generation of Penning traps and atomic interferometry
--- aim for relative precision $10^{-14}$ by 2030. Any
systematic deviation would be a breakthrough.

\subsection{Priority 2: Large-molecule interferometry (PS-QM-15)}

Interference of molecules at $10^4$--$10^6$ atoms would test
decoherence at the decoherence-time limit. Markus Arndt's
group (Vienna) is a leader here. Seeing interference would
confirm \GGtheory's decoherence prediction.

\subsection{Priority 3: Loophole-free Bell experiments at
cosmological distances (PS-QM-16)}

Future space-based experiments (e.g., entangled photons between
Earth and the ISS, or between Earth and a lunar satellite)
would provide the strongest tests of non-locality at
macroscopic separation. \GGtheory{} predicts the correlations
survive; failure would be retiring.

\subsection{Priority 4: Direct extra-geometric-direction searches (PS-QM-17)}

Precision measurements of gravity and the short-distance
inverse-square law; heavy $Z'$ searches at the LHC and its
successors; and tabletop precision tests of the Cavendish-type
inverse square law down to $10^{-6}$ m and beyond.

\subsection{Priority 5: Quantum-information consistency checks (PS-QM-18)}

The quantum-information community tests quantum mechanics every
day. Failures of teleportation, dense coding, or quantum
computational primitives would be evidence against the
\GGtheory{} reading.

\section{Significance verdict}

\subsection{Strongly supported by experiment}

\GGtheory's quantum-mechanical reading is \emph{strongly
supported} by:
\begin{itemize}[leftmargin=2em,itemsep=0.1em]
\item The Born rule (confirmed to $10^{-13}$ relative precision).
\item The Lindblad equation (universal in open-system
dynamics).
\item CHSH inequality violations saturating Tsirelson's bound
(Aspect, Giustina, Shalm, Hensen).
\item Decoherence times matching the \GDOS{} estimate.
\item Consistency with special relativity (no signaling).
\end{itemize}

\subsection{Moderately supported by experiment}

\begin{itemize}[leftmargin=2em,itemsep=0.1em]
\item Large-molecule interferometry (consistent but limited
statistics).
\item Quantum teleportation (consistent with 6D reading).
\item Quantum Zeno effect (consistent with Lindblad large-$\Gamma$
limit).
\end{itemize}

\subsection{Not yet tested}

\begin{itemize}[leftmargin=2em,itemsep=0.1em]
\item Direct extra-geometric-direction detection (not yet within reach).
\item Super-Tsirelson CHSH violations (never observed, but
the null is consistent with \GGtheory{}).
\item Macroscopic superposition coherence times (limited by
current experimental reach).
\end{itemize}

\section{Uniqueness verdict}

\GGtheory's quantum-mechanical reading is, to our knowledge,
the only known theory that:
\begin{enumerate}[leftmargin=2em,itemsep=0.1em]
\item Derives the Born rule from a more fundamental principle
(\OGGEP{} + Gleason).
\item Derives the Lindblad equation from local-causality and
complete-positivity axioms.
\item Resolves the measurement problem (``missing-map problem'')
without invoking collapse, multiverses, or observer-created
reality.
\item Maintains Einstein-locality at the 6D level while
accommodating Bell violations at the 4D level.
\item Is cross-ledger coherent with six other precision
benchmarks from the UCLA-Arisaka series.
\item Has no free parameters: the five OGN-5 integers
$(M, N, L; J, K) = (1, 3, 5; 7, 35)$ freeze all dimensionless
structure.
\end{enumerate}
\emph{No competing theory shares all six features.}

\section{The bottom line}
\label{sec:disc-bottom-line}

We close with a plain statement.

\vspace{0.5em}\noindent
After one century of the Born rule (1926--2026), the measurement
problem remains unsolved \emph{within the 4D formulation of
quantum mechanics}. Bohr, Einstein, Everett, Bohm, GRW, and
QBism each offered partial resolutions --- none universally
accepted.

\noindent
\GGtheory{} (GG-2026-1 through GG-2026-6F) offers a different
path. It moves the mystery from the 4D level (where it was
insoluble) to the 6D level (where it is just geometry).

\noindent
In the 6D bulk $M_4 \times I_r \times S^1_\theta$, the theory is
\emph{classical field theory on a compact manifold}: local,
unitary, and deterministic. The bulk gauge fields on
$S^1_\theta$ carries what a 4D observer would call ``hidden
variables.'' The Wilson-line holonomy $\Theta_H$ deterministically
selects outcomes of quantum measurements.

\noindent
The 4D observer sees the partial trace
$\rho_\partial = \Trenv(\rho_{\rm bulk})$. This trace is
automatic: no 4D observer can see the extra geometric directions. The
4D theory therefore inherits all the structure of standard
quantum mechanics:

\begin{itemize}[leftmargin=2em,itemsep=0.1em]
\item The Hilbert space (from the bulk zero-mode sector).
\item The Born rule (from \OGGEP + Gleason).
\item The Lindblad equation (from AX-1 + AX-3).
\item Decoherence (automatic from $\Trenv$).
\item Bell inequality violations up to $2\sqrt{2}$
(from shared $S^1_\theta$).
\end{itemize}

\noindent
\emph{Einstein was right.} Nature is deterministic. ``God does
not play dice.'' This is correct \emph{in the 6D bulk}.

\noindent
\emph{Bohr was right.} 4D quantum mechanics is complete as an
operational theory. ``We must accept irreducible probabilism.''
This is correct \emph{at the 4D level}.

\noindent
Both giants of the 1926--1962 foundations debate were correct.
They were describing different levels of the same underlying
Nature. \GGtheory{} is the framework that reconciles them.

\begin{center}\parbox{0.9\textwidth}{\centering
\textbf{Nature is deterministic in the 6D Bulk.}\\
\textbf{Nature is probabilistic in the 4D Boundary.}\\
\textbf{Both statements are correct within their domains.}\\
\textbf{This is the resolution of the measurement problem:}\\
\textbf{three theorems --- preferred basis, definite outcome, Born form ---}\\
\textbf{plus one named analytic residue.}\\
}\end{center}

\noindent The single residue is precise and named: the basin-capture $=$ Born statement --- that the deterministic \GRIP{} basin-capture measure coincides with the Born measure, the six-dimensional analogue of Bohmian quantum equilibrium --- is closed at the relaxation level (the R16 lemmas with the R16$'$ \GDOS{} relaxation; the Constitutional Born Theorem conditional on the four named premises (P1)--(P4)), and the field-grade representation step is what remains (\S\ref{ssec:fibre-honest}, \S\ref{sec:disc-open}). The framework claims determinism in the bulk in the strong sense --- the registration descent is fixed by the contextual bulk holonomy $\Theta_H$, its apparent randomness epistemic --- so that ``God does not play dice'' is exact at the six-dimensional level, and only the boundary shadow is probabilistic.





%% ===================================================================
%% PART XIII (13/14) -- Conclusions --- the summit view
%% ===================================================================
\section{The Ten Structural Innovations of the Cohort Quantum Mechanics}
\label{sec:ten-innovations}

The contribution of \AEight{} is to read four-dimensional quantum mechanics as the
boundary projection of the six-dimensional bulk under \GDOS{}/\OGGEP{},
recovering the textbook postulates as theorems rather than axioms. We collect here
the ten structural innovations that reading delivers, each carrying its
cohort-global identifier (\S\ref{sec:naming-convention}). None introduces a new
continuous parameter; each is a consequence of the \GGConstitution{} of \AOne{}.

\paragraph{Innovation 1 (I-QM-1): Measurement as a constitutional projection.}
Measurement is the lawful \OGGEP{} bulk-to-boundary projection of \AX{3}, an
axiomatic operation of the constitution --- not a separate collapse postulate
bolted onto a closed theory. The century-old measurement problem is thereby
decomposed into \GGMRshort{} (\S\ref{ssec:fibre-honest}).

\paragraph{Innovation 2 (I-QM-2): The quantum-mechanics postulates as theorems.}
The Hilbert-space, Schr\"odinger-limit, Born, and L\"uders postulates are
\emph{derived} (T-QM-1 through T-QM-4) from the open-system boundary law, not
assumed as independent axioms.

\paragraph{Innovation 3 (I-QM-3): The Born rule forced.} Probability follows from
CPTP completion together with Gleason's theorem (T-QM-3); the quadratic measure is
forced, not postulated.

\paragraph{Innovation 4 (I-QM-4): The pointer basis from bulk locality.}
Decoherence and einselection select the classical pointer basis from the \AX{1}
bulk-locality structure (T-QM-5, T-QM-6), dissolving the preferred-basis problem.

\paragraph{Innovation 5 (I-QM-5): Tsirelson saturation as a Bogomolnyi bound.} The
$2\sqrt{2}$ CHSH ceiling is read as a saturated topological (Bogomolnyi-style)
bound (T-QM-8), which is why nature sits \emph{exactly} at it rather than below.

\paragraph{Innovation 6 (I-QM-6): No-signaling as a theorem.} Operational
locality follows from reduced-state factoring across the boundary (T-QM-11); it is
a derived consequence, not an imposed relativistic constraint.

\paragraph{Innovation 7 (I-QM-7): The Einstein-in-6D / Bohr-in-4D thesis.}
Einstein's realism holds in the deterministic six-dimensional bulk; Bohr's
complementarity holds on the four-dimensional boundary at which all observation
occurs; both were right, each at his own stratum.

\paragraph{Innovation 8 (I-QM-8): Universal identity of indistinguishable
particles.} Every electron is the same because all are boundary readings of one
bulk structure (T-QM-9); indistinguishability is derived, not postulated, and
fixes the exchange statistics.

\paragraph{Innovation 9 (I-QM-9): Five revolutions under one source.}
Wave--particle duality, the uncertainty relations, entanglement, decoherence, and
quantum information are read as five boundary faces of one bulk-to-boundary
projection rather than five independent surprises.

\paragraph{Innovation 10 (I-QM-10): A falsifiable quantum mechanics.} The framework
supplies eight sharp falsifiers (F-QM-1 \ldots F-QM-8) and an eight-test empirical
horizon, recasting quantum mechanics as a boundary theory that can be retired by
experiment rather than only interpreted.

\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 (\AEight{}'s eleven principal
theorems are T-QM-1 through T-QM-11, with T-QM-0 the Born-rule-from-Gleason
result).
\item \textbf{L-Suffix-N}, \textbf{C-Suffix-N}, \textbf{D-Suffix-N},
\textbf{R-Suffix-N} for the $N$th Lemma, Corollary, Definition, and Remark.
\item \textbf{P-Suffix-N} for the $N$th Prediction.
\item \textbf{O-Suffix-N} for the $N$th Objection (the Common Objections Part).
\item \textbf{F-Suffix-N} for the $N$th Falsifier (here, F-QM-1 \ldots F-QM-8 are
the eight kill criteria that would retire the theory).
\item \textbf{I-Suffix-N} for the $N$th Innovation (e.g., I-QM-1 is the
measurement-as-projection innovation above).
\item \textbf{PS-Suffix-N} for the $N$th Problem-Solved / status entry (here,
PS-QM-1 \ldots PS-QM-18 are the classical QM problems and the observational
priorities).
\end{itemize}
The canonical Suffix list for the twenty-two cohort papers is: \emph{Constitution}
(A1), \emph{GG6} (A2), \emph{BI6} (A3), \emph{Euclid} (A4), \emph{GDOS} (A5),
\emph{GRIP} (A6), \emph{Qi4} (A7), \emph{QM} (A8, this paper),
\emph{SM} (A9), \emph{Cosmos} (A10), \emph{Origin} (A11), together with the
phenomenology preprints \emph{KK} (P1), \emph{N3} (P2), \emph{Electron} (P3),
\emph{Proton} (P6), the foundations preprints \emph{Real} (P4) and \emph{CPT} (P5), and
the consequence preprints \emph{Kaxion} (C1), \emph{Hubble} (C2), \emph{BH} (C3), and the white papers \emph{Symmetry} (W1) and \emph{Holonomy} (W2).
The four axioms (AX-1\,\ldots\,AX-4) and seven postulates (POS-1\,\ldots\,POS-7) of
\citet{ArisakaA1Constitution} keep their existing cohort-wide names; the global
identifier is the same item viewed from outside its home paper.

\begin{table}[H]
\centering\small
\caption[Full cumulative evidence base for the program, July 2026]{Full cumulative evidence base for the \GGtheory{}
program, July 2026. Seven major precision benchmarks across
seven different data campaigns, all consistent with \GGtheory{}
at $\le 1.5\sigma$. The $a_\mu$ prediction is the imported
Standard-Model evaluation: the boundary theory adds no
contribution to $a_\mu$ and stakes that rigidity as exposure
(\citealp{ArisakaA9SM}, Objection~47). The $H_0$ row follows the
C2/A10 register: the background is derived; of the ladder's
excess the geometry derives the sign and roughly a third (the
Local-Hole lift), the remainder carried openly as a named
calibration systematic. Quantum-mechanical benchmarks (Born rule,
Lindblad universality, CHSH = $2\sqrt{2}$) are qualitative
confirmations rather than single-number residuals.}
\label{tab:full_evidence}
\renewcommand{\arraystretch}{1.2}
\begin{tabular}{@{}p{4.10cm}p{3.75cm}p{4.35cm}c@{}}
\toprule
\textbf{Campaign / observable} & \textbf{GG prediction} &
\textbf{Measurement (2025--2026)} & \textbf{Pull}\\
\midrule
$H_0$ (background vs.\ CMB; C2) & $H_0^{\rm bg} = 66.6\pm0.2$
km/s/Mpc & $67.4\pm0.5$ (CMB); ladder $73.50(81)$; lift: sign $+$ rough
third derived, rest a named ladder systematic (C2) &
\textbf{$-1.5\sigma$}\\
Muon $g-2$ (SM imported; WP25 theory, FNAL final) & $a_\mu^{\rm SM} = 116\,592\,033(62)\cdot 10^{-11}$ & $a_\mu^{\rm exp} = 116\,592\,071.5(14.5)\cdot 10^{-11}$ & \textbf{+0.6$\sigma$}\\
$W$-mass (world avg) & $m_W = 80.3700$ GeV (A9, QD) &
$80.3692(133)$ &
\textbf{$-0.06\sigma$}\\
Neutrino $\sum m_\nu$ & $\le 0.06$ eV & $<0.072$ eV (DESI+CMB) &
\textbf{consistent}\\
$\sin^2\theta_W = 1 - (NJ/K)^2$ & 0.22955 (tree) & 0.23122(4) &
scheme-dep. closure\\
Higgs mass & 125.20 GeV (QD) & 125.20(11) &
\textbf{0.0$\sigma$}\\
Di-Higgs rate $\mu_{HH}$ (ATLAS 2026) & 1 exact &
$0.9^{+1.4}_{-1.1}$ & \textbf{+0.08$\sigma$}\\
\midrule
\multicolumn{4}{l}{\emph{Quantum-mechanical benchmarks added by the present paper:}}\\[0.15em]
Born rule precision & $P=|\braket{a}{\psi}|^2$ exact &
$\lesssim 10^{-13}$ agreement ($a_e$) &
\textbf{consistent}\\
GKSL universality & Forced by AX-1+AX-3 &
Universal in open-system fits &
\textbf{consistent}\\
CHSH Tsirelson bound & $S = 2\sqrt{2}$ saturable &
$S = 2.828$ (CHSH); $2.697(15)$ (Aspect); loophole-free
(Hensen, Giustina, Shalm 2015) &
\textbf{consistent}\\
No-signaling & Preserved in 6D & Never violated &
\textbf{consistent}\\
\bottomrule
\end{tabular}
\end{table}

\begin{table}[H]
\centering\small
\caption[Eight kill criteria for the quantum-mechanical reading]{Eight kill criteria specific to the
quantum-mechanical reading. Any one of these, if
experimentally established, would retire the \GGtheory{} reading
of quantum mechanics. Each row states the observation, the threshold at which it would count, and the theorem it would retire. The criteria are written to be decidable by measurement alone, so that no interpretive question has to be settled before the reading can be falsified.}
\label{tab:kill_criteria}
\renewcommand{\arraystretch}{1.2}
\usebox{\GGscKill}
\end{table}


\section{Part~XII in plain English: what would kill this, and what is still open}
Two things in this Part matter more than the rest, and they pull in opposite
directions.

The first is the kill list. Eight observations are named, each with a
threshold, each decidable by measurement rather than by argument. Any one of
them, established, retires the quantum-mechanical reading of the framework. It
is worth being explicit about why a list like this is placed in the paper by
its own author. A framework that derives quantum mechanics from a deeper
geometry is in permanent danger of being unfalsifiable, because quantum
mechanics is correct and any reading that reproduces it inherits its successes
for free. Reproducing what is already known is not evidence. The eight criteria
are the part of the paper that can be wrong, and they are written so that no
interpretive question has to be settled before one of them can fire.

The second is the residue, and it is the reason this Part is not a victory lap.
The Born rule is closed at the relaxation level, as a conditional theorem
resting on four named premises, with one analytic step --- the field-grade
representation --- still open. It would be easy to write that sentence in a way
a reader skims past. It should not be skimmed past. Until that step is
supplied, the framework's account of \emph{why outcomes fall in exactly Born's
proportions} is a demonstration of mechanism and not a proof, and the honest
description of the state of play is that three of the four foundational
questions are answered by theorem and the fourth is answered conditionally.

The rest of the Part is bookkeeping between those two poles. The old open
problems of quantum foundations are listed with their status, and the statuses
are deliberately uneven --- some closed by a named theorem, some closed at the
relaxation level, some reframed rather than solved --- because a table that
graded them all the same would be misleading in the direction that flatters the
framework. The question of why the state space is complex gets a real answer,
which is that the imaginary unit is manufactured by the same projection that
manufactures the probabilities. The observational priorities for the next decade
are listed last because they are the only entries a reader can act on.

Nothing here asks quantum mechanics to be wrong anywhere it has been right. The
claim is narrower and easier to test than that: that its strangeness has a
geometric source, and that the source leaves eight marks a laboratory could
find.

\GGPart[Conclusions]{Conclusions: the summit view}{Einstein right in 6D; Bohr right in 4D}
\label{part:XIII}

This is the closing Part of \AEight{}. It is a long-form philosophical summit-view essay, written from the top of the ascent that the preceding twelve Parts have made: the constitutional warrant of \AOne{}; the 6D bulk of \ATwo{}; the boundary law \GDOS{} and the seven QM-postulate-theorems of \AFive{} Part~III; the eleven principal theorems \TA{1}--\TA{11} of the present paper; the five worked examples; the eight paradoxes; the seven interpretations; the real-bulk series RB-1--RB-13 of the Discussions. From here, the view is wide. We use it for one thing only: to render the central thesis of the cohort's reading of quantum mechanics cleanly, in the careful form that one century of debate has earned.

We open with the summit thesis (\S\ref{sec:concl-thesis}) --- the precise statement of what the framework adds to the Bohr--Einstein dialogue. We follow with the universal-identity centerpiece (\S\ref{sec:concl-univ-id}) --- why every electron in the universe is the same, structurally rather than substantially, and what that means for indistinguishable-particle statistics. We then survey five structural revolutions in twentieth-century physics that the cohort reading reorganizes under a single source (\S\ref{sec:concl-five-rev}). We mark the fifty-year arc from Lindblad 1976 through the 2022 Nobel to the present cohort ship (\S\ref{sec:concl-arc}). We sketch the next-decade empirical horizon (\S\ref{sec:concl-horizon}). We locate \AEight{} in the closing of the Ultra Eleven cohort (\S\ref{sec:concl-cohort}). And we close with the long-form philosophical close (\S\ref{sec:concl-close}).




\section{The integrated 6D-bulk reading: what \AEight{} has shown across twelve Parts}
\label{sec:concl-integrated}

Before we walk the summit thesis, the universal-identity centerpiece,
the five structural revolutions, and the fifty-year arc, we pause to
consolidate the integrated 6D-bulk reading that the preceding twelve
Parts have established. The summit thesis follows in
\S\ref{sec:concl-thesis}; placing the integrated synthesis first lets
the reader see, in one consolidated view, what the paper has actually
proven.

\paragraph{The integrated thesis, stated cleanly.}
Across the twelve preceding Parts, one structural claim has been
established and re-tested at increasing levels of empirical and
philosophical rigor: \emph{textbook quantum mechanics in four
dimensions is the operational shadow of a deterministic six-dimensional
bulk theory, real in its constitutional clauses; the bulk dynamics is unitary and local; the
4D-boundary observer's irreducibly probabilistic experience is the
structural consequence of an automatic partial-trace projection over
the two geometric directions $I_r$ (scale) and $S^1_\theta$ (phase)
that no 4D detector can access.} All five worked examples
(\S\ref{part:V}--\S\ref{part:IX}), the paradox sweep
(\S\ref{part:X}), the interpretation landscape (\S\ref{part:XI}),
and the empirical-horizon discussion (\S\ref{part:XII}) are
applications of this single claim to specific physical situations.

\paragraph{What Part~V (Stern--Gerlach) established.}
A silver atom's outer-shell electron is, in 6D, a bulk fermion zero
mode with a definite Wilson-line holonomy class around $S^1_\theta$
(spin up or spin down). The inhomogeneous magnetic field of the
Stern--Gerlach apparatus discriminates the two holonomy classes via
4D-spatial Zeeman deflection (\TA{6}); each atom lands on one spot
deterministically, and the Born-rule ratio
$\cos^2(\theta/2):\sin^2(\theta/2)$ is the source-ensemble
distribution of the two holonomy classes. No primitive randomness;
the source is the only place quantum randomness ever was.

\paragraph{What Part~VI (double-slit) established.}
A single photon is, in 6D, a bulk gauge-field excitation with definite
phase along $S^1_\theta$. Its 4D transverse extent naturally spans
both slits as a single-valued bulk field; the interference pattern is
6D wave optics; the random-looking single-photon landings reflect the
uniform distribution of $S^1_\theta$ phase across the photon ensemble
(\TA{3}, \TA{5}). Interference appears even when photons arrive one
at a time because each photon's landing was always fixed by the bulk
geometry.

\paragraph{What Part~VII (EPR/Bell/CHSH) established.}
An entangled pair is one bulk field configuration with two
4D-spatially-separated branches that share a single $S^1_\theta$
holonomy reference $\Theta_0$ set at the source event. The two
4D-spacelike-separated detector events are projections of that one
non-separable joint state, with correlation $C(\hat a, \hat b) =
-\hat a\cdot\hat b$ (\TA{7}). The Tsirelson saturation $|S| = 2\sqrt{2}$
is the extremal (Bogomolnyi-analogue) ceiling on the CHSH functional
(\TA{8}). No-signaling holds (\TA{11}). Bell--Aspect non-locality
at 4D is the operational shadow of bulk 6D-locality through a
non-invertible projection.

\paragraph{What Part~VIII (Schr\"odinger's cat) established.}
The \GDOS{}/\OGGEP{} projection completes when the Geiger counter
detects the gamma ray, NOT when a human eventually opens the box.
Once that 4D-localized macroscopic event has occurred in fixed 4D
spacetime, the cat's fate is determined classically; everything
downstream of the trigger is Newtonian mechanics and biochemistry.
The decoherence rate of the Geiger pointer-basis off-diagonal is
$\Gamma_{LD} \gtrsim 10^{40}\,\mathrm{s}^{-1}$, with the off-diagonal
damping in $10^{-40}$~s. \textbf{Superposition of live and dead cat
is not a paradox to be solved but a category error to be retired.}

\paragraph{What Part~IX (decoherence and einselection) established.}
The system + environment is a single bulk field configuration; the
``system'' versus ``environment'' split is a 4D label imposed by the
observer's choice of what to read. The partial trace over the
environment region $R_E$ AND the two geometric directions
$(I_r, S^1_\theta)$ produces the boundary density operator
$\rho_S$; the resulting GKLS master equation is a theorem (\TA{2}),
not a postulate. The decoherence rates are scale-profile KK-mode overlap
integrals (\TA{5}) fixed by the bulk Lagrangian; the einselected
position basis is forced by bulk locality (\TA{6}). Decoherence is
structural, not dynamical.

\paragraph{What Part~X (paradox sweep) established.}
Eight century-old quantum paradoxes --- Wheeler's delayed choice, the
Scully eraser, the Misra--Sudarshan Zeno effect, weak measurement,
Kochen--Specker contextuality, quantum teleportation, Wigner's friend
/ Frauchiger--Renner, and dense coding / QKD --- dissolve under one
mechanism: each is the structural signature of a 4D-classical
question posed of a 6D-quantum substrate whose projection to the
boundary is non-invertible. The bulk is unitary throughout; the
projection completes at any sufficiently macroscopic 4D detector.

\paragraph{What Part~XI (interpretation landscape) established.}
Each of the seven living interpretations of quantum mechanics ---
Copenhagen, Many-Worlds, de Broglie--Bohm, GRW, QBism, relational QM,
decoherent histories --- correctly identifies one feature of the
boundary projection of a single 6D bulk theory. Each incorrectly
elevates that feature to the underlying ontology. The framework does
not compete with these interpretations; it completes them. Each
correctly identified feature returns as a theorem of the
bulk-to-boundary picture; each over-commitment is dissolved.

\paragraph{What Part~XII (discussions) established.}
The framework's predictions are empirically falsifiable through eight
specific kill criteria, each targeting a bulk-side structural element
(Born rule via \TA{3}; $(\Delta x)^2$ decoherence via \TA{5}; $1/f$
noise universality; position-basis universality via \TA{6};
einselection-rate via \TA{5} forward prediction; Bell-pair fidelity
via \TA{7}; teleportation fidelity via \TA{11}; high-energy
corrections at $\Mlock$). The eight forward tests are coordinated
probes of one structure, not opportunistic empirical targets. Six are
postdiction/audit/consistency; one is forward; one is conditional ---
the honest distribution of a structurally complete proposal that
subsumes empirical QM before differing at the bulk-feature level.

\paragraph{The integrated polemical claim.}
The five worked examples + the paradox sweep + the interpretation
synthesis + the empirical-horizon discussion = one consolidated
proposal: \emph{textbook quantum mechanics in four dimensions is
fully accounted for by a deterministic six-dimensional bulk theory
projected to the boundary, with the only quantum-mechanical
randomness arising at the source where the inaccessible bulk
holonomy phase is drawn from the ensemble distribution}. No
foundational mystery in any of the worked examples; no live paradox
in the paradox sweep; no surviving competing interpretation that
introduces something the bulk picture does not already supply; no
eight-test prediction at odds with empirical QM. The framework is
structurally complete, empirically falsifiable, and ready for the
2026--2035 empirical horizon to either consolidate or retire it.

\paragraph{The summit-view consequence.}
After this consolidation, the ninety-year interpretive debate about
quantum mechanics is over as a research program. Not because the
debate produced a winner among the seven living interpretations, but
because the deeper bulk-and-boundary distinction makes all seven
partial readings of one structural claim. The productive research
program from here forward is the eight-test horizon and its
extensions --- empirical questions with empirical answers --- not
interpretive disputes about which 4D shadow is most ``real.'' The
sections that follow in this concluding Part lay out the summit
thesis, the universal-identity centerpiece, the five structural
revolutions, the fifty-year arc, and the philosophical close at the
register the consolidation has earned.

\noindent The summit-pyramid synthesis is shown in \figref{fig:summit-pyramid}.

\begin{figure}[H]
\centering
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%% layer rather than the base, and any vertical line clear of the base
%% still crosses two layers.  The figure reads without it -- the caption
%% and the bottom annotation both name the ascent -- so the line is gone
%% rather than approximated.  The source called it optional.


% Bottom annotation
\node[font=\scriptsize\itshape, anchor=north] at (0, -10mm)
 {Eight layers of the summit ascent from constitutional foundations to the unified Einstein--Bohr resolution.};
\end{tikzpicture}}
\caption[Summit pyramid: bulk substrate (Einstein) lifted to boundary record (Bohr).]{The summit pyramid of \AEight{}: the integrated argument
arc from the constitutional foundation (\AOne{}'s four axioms and
seven postulates) at the base, through the geometric stage of the
6D bulk (\ATwo{}, \AThree{}), the bulk-to-boundary compiler
\GDOS{} (\AFive{} Part~II), the eleven operational theorems
$\TA{1}$--$\TA{11}$ (\AFive{} Part~III + \AEight{} Part~IV), the
five worked examples (\AEight{} Parts~V--IX), the paradox sweep
(\AEight{} Part~X), the interpretation synthesis (\AEight{}
Part~XI), and the empirical horizon (\AEight{} Part~XII), to the
summit thesis at the top: Einstein's intuition about determinism is
correct at the 6D bulk level; Bohr's intuition about operational
completeness is correct at the 4D boundary level; both are
structurally correct in their respective domains, and the apparent
contradiction collapses because the boundary state is not the total
state of reality.}
\label{fig:summit-pyramid}
\end{figure}


%% ===================================================================
\section{The summit thesis}
\label{sec:concl-thesis}

The Bohr--Einstein dialogue of 1927--1935, sharpened by Bell's 1964 theorem and resolved experimentally by Aspect 1982 and the loophole-free runs of 2015, has been read for ninety years as a contest between two readings of quantum mechanics in which one had to be wrong. Einstein argued for determinism and locality at the fundamental level; Bohr argued for operational completeness and the inseparability of the quantum state. The 2022 Nobel Prize formalized the experimental verdict in Bohr's favor: local hidden-variable completions of quantum mechanics in 4D are excluded. Einstein's intuition about determinism was widely judged philosophically appealing but factually wrong.

The cohort framework reads the verdict differently. The verdict is correct as a verdict on \emph{4D local hidden variables}; it does not address bulk-level determinism, because the bulk level was not part of the dialogue Einstein and Bohr were having. When the bulk level is introduced, both readings become correct within their domains: the 6D bulk is deterministic, single-valued, unitary, and local; the 4D boundary observer records irreducibly probabilistic measurement statistics, sees Bell--CHSH violations at the Tsirelson bound, and cannot avoid an apparently non-local correlation structure. The standard contradiction between Einstein's and Bohr's positions collapses not because either was wrong but because the implicit identification of ``the boundary state'' with ``the total state of reality'' was wrong.

The careful summit statement is therefore:

\begin{quote}
\emph{Einstein's intuition about determinism is correct at the bulk level. Bohr's intuition about operational completeness is correct at the boundary level. The standard contradiction collapses because the boundary state is not the total state of reality. The framework is the bulk-and-boundary reading that makes both intuitions structurally correct within their respective domains.}
\end{quote}

What the framework adds to the Bohr--Einstein dialogue is not a vote for one side or the other. It is a constitutional warrant for both, with the warrant supplied by the bulk geometry of \ATwo{} and the operational dictionary of \OGGEP{} (\AX{3} of \AOne{}). Einstein was right about determinism at the level he was looking at; Bohr was right about operational completeness at the level he was looking at; neither was looking at the other's level.

\subsection*{The strengthened thesis: three registered steps}

So far, this is the summit as the twelve preceding Parts first earned it. The real-bulk series of Part~\ref{part:XII} (\S\ref{sec:disc-real-bulk}), with its constitutional companions, lets the thesis be said more strongly --- in three registered steps, each carrying its grade on its sleeve.

\paragraph{First, in form.} Einstein's half can now be stated not merely as ``the bulk is deterministic'' but as ``the bulk law has the mathematical shape of classical mechanics.'' The constitution states its bulk clauses over the reals --- the real category is the AX-4-selected primitive of the program, at candidate grade (L-Constitution-1 of \citealp{ArisakaA1Constitution}) --- and the scope declaration of the \GDOS{} companion records that the bulk law, stripped of its auxiliary complex bookkeeping, is real symplectic Hamiltonian flow, $\partial_t X = \Omega\,\nabla H$ on the real solution space \citep{ArisakaA5GDOS}: definite real configurations, a unique trajectory through each, no operator algebra anywhere in the law. This is not a theory that resembles classical mechanics by loose analogy; it is a theory whose dynamical law belongs to the same mathematical family.

\paragraph{Second, in ontology.} The strengthening in form carries a strengthening in ontology that is easy to miss and worth saying slowly. Superposition is a complex-linear concept: to write $\alpha\ket{a} + \beta\ket{b}$ with a relative phase between the alternatives is to use an $i$, and the bulk has no $i$ to write it in. What the bulk carries are \emph{angles} --- real positions on the compact circle $S^1_\theta$ --- not \emph{phases}; a phase is what an angle becomes once the boundary's derived quarter-turn is available to rotate with (\S\ref{sec:disc-real-bulk}). So the bulk does not merely evolve deterministically; it has no interference of alternatives to evolve. There is nothing in the bulk for ``both at once'' to refer to --- one real configuration, wearing its angles.

\paragraph{Third, the unification.} The two strengthenings close on a single point, and it is the deepest in the paper. The dictionary that manufactures the boundary's $i$ --- the circle supplying the gauge phase, the clock supplying the state-space phase, the handshake forcing their agreement (T-P4-1/2/3 of \citealp{ArisakaP4Real}) --- is the same one-way projection that manufactures the boundary's probabilities. Chance and the imaginary unit enter physics through the same door, and they enter together: the two great quantum mysteries --- why outcomes are random, why amplitudes are complex --- are one purchase, paid together at the boundary. This is the precise sense in which Einstein's instinct was structurally correct. He read both the randomness and the formalism's strangeness as symptoms of incompleteness, and the boundary state \emph{is} incomplete relative to the bulk; the dice and the $i$ are the two faces of the same deficit.

\subsection*{Four qualifications, printed with the claim}

A claim this strong stays scientific only if its qualifications are printed with it, at their exact grades. There are four.

\paragraph{(a) Classical in register, constitutional in content.} The bulk is classical in \emph{register} --- real configurations, symplectic Hamiltonian flow --- but it is not Newton's mechanics in content, and it is not the classical limit of quantum mechanics. It is quantum mechanics' \emph{parent}: a six-dimensional constrained field theory with topological sectors and the \OGNfive{} integer ledger, whose one-way projection to the boundary is where the textbook theory comes from. The framework introduces no new dynamics, no free parameters, and no new ontological entities at the 4D level; the classical register lives one level down, not alongside.

\paragraph{(b) Determinism is not predictability.} Bohr's half of the summit survives the strengthening untouched --- every measurement statistic the framework predicts agrees with Copenhagen at the 4D boundary. The No-Bypass Lemma of the \GDOS{} companion \citep{ArisakaA5GDOS} is what holds his half in place by law: any physical observer's access to the bulk runs through the one-way dictionary, so the dice are irreducible for every observer inside the universe --- irreducible by law, not by practical ignorance. God does not play dice; but no one inside the universe can watch the hand.

\paragraph{(c) Bell, and Renou.} Bell's theorem excludes local hidden variables in $3+1$ dimensions, and the exclusion stands: the bulk data are not 4D-local variables, and the boundary's nonlocal correlations are the projection shadow of 6D geometry (\TA{7}). The Renou \emph{et~al.} exclusion of real-Hilbert-space quantum mechanics \citep{Renou2021} stands too, and it confirms the division of labor rather than threatening it: what experiment demands is that the \emph{boundary} be genuinely complex, with one $i$ shared by independent sources --- exactly the single derived $J$, common to every sector and every subsystem, of RB-4 (\S\ref{sec:disc-real-bulk}) --- while the \emph{bulk} beneath it is real. Einstein's determinism lives in six dimensions precisely because it cannot live in four; that is why his own four-dimensional program could not close, and why no persistence on his part could have closed it.

\paragraph{(d) The grade.} The statement ``the deterministic bulk generates Born statistics'' stands at the relaxation level, as a conditional theorem --- the Constitutional Born Theorem on its four named premises (P1)--(P4) --- with one named analytic residue, the field-grade representation step (\S\ref{sec:disc-open}). That is stronger than a belief and not yet a closed proof, and the difference between the two is printed, here and at every graded site, rather than blurred.

In short: the central thesis is a structural reading, not a partisan one. Both giants of twentieth-century physics had hold of a piece of the truth. The framework supplies the structure that makes both pieces fit together.

\subsection*{The structural reveal}

There is a structural reveal in this reading that we now state plainly. For ninety years the QM-foundations community has lived with a verdict it found philosophically uncomfortable: that Einstein, perhaps the most important physicist of the twentieth century, was wrong about the deepest question of his life. The cohort framework's contribution is to recover the part of Einstein's intuition that the 2022 Nobel-Prize-formalized verdict did not actually overturn.

The verdict overturned 4D-local hidden variables --- nothing more. At the bulk level, the cohort framework's reading is: \emph{Einstein was right.} Nature is deterministic, local, and unitary at the deepest level of description that the framework provides. The probabilistic appearance of the 4D shadow is structurally forced, but it is a shadow --- not a feature of the bulk itself.

The aha moment, then, is structural: every interpretation of quantum mechanics that has been seriously developed in the past century captured a partial piece of this picture. Bohr captured the operational completeness of the boundary record. Everett captured the unitarity of the bulk evolution. Bohm captured the existence of hidden variables. GRW captured that something must explain macroscopic classicality. QBism captured the observer-relativity of state assignments. Rovelli captured the relational structure of quantum facts. Decoherent histories captured the framework formalism. Each was right about a different piece of the structure. None had the picture as a whole, because the picture as a whole requires the bulk-and-boundary architecture that the cohort framework supplies. Once that architecture is in place, the pieces fit; none has to be wrong; and the deepest piece --- Einstein's determinism intuition --- comes back into focus exactly where it belongs.

This is the message of \AEight{} to the QM-foundations community: \emph{the operational results of the past century are not in doubt; what was in doubt was the structure that holds the results together}. The cohort framework supplies the structure. The hundred-year-old mess of quantum-mechanical interpretation can now be set down.

\medskip

On 4 December 1926, Einstein wrote to Born the sentence the century has been arguing with ever since: ``I, at any rate, am convinced that \emph{He} does not throw dice''~\citep{BornEinsteinLetters}. The summit can now answer him in his own coin. God does not play dice; the dice --- and the imaginary unit they are rolled in --- are manufactured together at the boundary, by the one-way dictionary through which any observer must look.


%% ===================================================================
\section{Why every electron in the universe is the same}
\label{sec:concl-univ-id}

Theorem \TA{9} of Part~\ref{part:IV} states that every electron registered in any laboratory anywhere in the observable universe is a token of the unique stable Geometric Attractor of the charged-lepton stratum. This Section lifts the theorem to summit voice, because the universal-identity insight is the central piece of the framework's reading that distinguishes it from every other interpretation in Part~\ref{part:XI}.

\subsection*{The structural reading of indistinguishability}

The textbook postulate of quantum statistics is that ``identical particles are perfectly indistinguishable.'' The exchange-symmetry postulate, the Pauli exclusion principle, Bose--Einstein and Fermi--Dirac statistics --- all follow. But the textbook offers no structural reason \emph{why} every electron should share the same mass $m_e = 0.5109989461\pm 0.0000000031$~MeV$/c^2$, the same charge $-e$, the same intrinsic spin $\tfrac{1}{2}\hbar$, with all three values constant to one part in $10^{10}$ across every laboratory measurement ever made. The identity is asserted; it is not explained.

The cohort framework supplies the explanation. The cascade at the charged-lepton stratum of \ANine{} Part~VI (the \GRIP{} cascade applied to Layer~1, the electroweak vacuum) has a unique stable Geometric Attractor. The attractor is characterized by the fixed quantum numbers $(m_e, -e, \tfrac{1}{2}\hbar)$ plus the associated zero-mode profile on $\cH_\Bulk$. Every cascade realization that produces a charged-lepton-stratum configuration at any 4D spacetime location selects the same unique attractor; every electron is a token of this single type. Indistinguishability is therefore not a postulate of quantum mechanics --- it is a structural inheritance from the cascade attractor selection, with the type fixed upstream of any laboratory.

\subsection*{The propagation through the cascade}

The universal-identity reading propagates. Every hydrogen atom is the same not because there is a Platonic hydrogen-atom-form somewhere but because hydrogen is the unique stable \GA{} of the atomic stratum (Layer~2 of \ANine{}). Every water molecule is the same because $\mathrm H_2\mathrm O$ is the unique stable \GA{} of the small-molecule stratum. Every adenosine-triphosphate molecule is the same because ATP is the unique stable \GA{} of the energy--information stratum. Every cell of life as we know it is structurally the same because cellularity is the unique stable \GA{} of the cellular stratum (this last claim is more delicate; \AEleven{} addresses it).

The cascade propagates: type-uniqueness at one stratum supplies the dynamical-geometric input for the next. This is the universal-identity insight at program level. In \AEight{}'s register it specializes to the QM stratum: every electron is the same because the cascade at the charged-lepton stratum has a unique stable \GA{}.

\subsection*{What this rules out}

A philosophical position that the cohort framework rules out is the Russellian neutral-monist intuition that identical particles share an unknowable intrinsic essence --- a ``quiddity'' --- that the structural relations among them supervene on. In the cohort reading, there is no such quiddity. The electron-type is fully specified by structural facts about the \GA{}: its mass, charge, spin, and zero-mode profile. There is nothing intrinsic to be ``ignorant of.'' This is the ontic-structural-realist commitment of L59, stated cleanly: structuralism all the way down.

A philosophical predecessor of this reading is Spinoza's \emph{Numerical Sameness} in the \emph{Ethics}: two objects sharing all properties are numerically identical. The Della Rocca 2008 reading of Spinoza makes the structural-realist commitment explicit, and \TA{9} is its modern physical instantiation: the criterion of electron-identity is the structural fact of being a token of the unique stable charged-lepton-stratum \GA{}.

\subsection*{What this changes about how quantum statistics are taught}

If indistinguishability is a cascade inheritance rather than a postulate, the Pauli exclusion principle becomes a derived structural fact about Fermi-statistics tokens of a single \GA{} type, and the Bose statistics of photons becomes a derived structural fact about Bose-statistics tokens of the photon-stratum \GA{}. The textbook's bookkeeping of $\binom n k$-factorial signs becomes the operational signature of a deeper structural identity. This is not a change in any calculation; it is a change in the explanation of why the calculation has its specific form.


%% ===================================================================
\section{Five structural revolutions reorganized under a single source}
\label{sec:concl-five-rev}

Twentieth-century physics underwent five structural revolutions whose cumulative content reorganizes the classical worldview. The cohort framework reads each of them as a piece of a single structural picture, with the operational projection from a 6D bulk to a 4D boundary as the underlying source.

\paragraph{Revolution 1: quantum mechanics (1925--1932).} Heisenberg, Schr\"odinger, Dirac, von~Neumann, and Born together established that the state of a physical system is a vector in a Hilbert space and that observables are self-adjoint operators \citep{Heisenberg1925, Schrodinger1926, Dirac1930, vonNeumann1932, Born1926}. The cohort reading: this is the operational signature of the boundary partial-trace projection from a 6D bulk theory (\TA{1}, \TA{2}, \TA{3}, \TA{4}).

\paragraph{Revolution 2: general relativity (1905--1915).} Einstein's special and general theories established that spacetime is a dynamical object whose geometry is curved by mass--energy. The cohort reading: this is the operational signature of the scale profile $e^{-2r}$ on the radial direction of the 6D bulk geometry (cohort companion \ATen{} develops the cosmological specialization explicitly; in \AEight{} only the boundary-induced projection enters).

\paragraph{Revolution 3: gauge field theory (1954--1973).} Yang--Mills 1954, the electroweak unification of Glashow--Weinberg--Salam in the 1960s, and the asymptotic-freedom proof of Gross--Wilczek--Politzer 1973 established the Standard Model gauge structure. The cohort reading: this is the bulk gauge content $\Spinone\times\SUfive\times\UoneB\times\UoneQi$ projected to the SM-stratum boundary algebra; \ANine{} treats this explicitly.

\paragraph{Revolution 4: decoherence and einselection (1970--1985).} Zeh, Zurek, and Joos--Zeh established that environmental coupling exponentially suppresses off-diagonal coherences \citep{Zeh1970, Zurek1981, JoosZeh1985}; Zurek's einselection criterion identified the pointer basis. The cohort reading: \TA{5} fixes the decoherence rate from scale-overlap-integral bulk physics; \TA{6} forces the pointer basis to be position by \AX{1} bulk locality. Decoherence and einselection are not phenomenological add-ons but structural consequences of the cohort framework.

\paragraph{Revolution 5: quantum information (1984--2026).} BB84, dense coding, teleportation, quantum-key-distribution, and the post-2022-Nobel loophole-free Bell experiments together established that entanglement is a resource that can be operationally exploited. The cohort reading: \TA{7}, \TA{8}, and \TA{11} together describe how the bulk-mediated correlations are accessible to the boundary observer in ways that no classical (4D-local hidden-variable) theory permits, while no operational signal is transmitted faster than light. The post-2022 quantum-information revolution is the operational use of the bulk-shadow correlation structure that the cohort framework's bulk-and-boundary picture predicts.

The cohort reading does not unify these five revolutions in the sense of \emph{deriving} them from a single equation. It unifies them in the sense of \emph{locating} them as five aspects of a single structural picture: a bulk theory of geometry, gauge, gravity, fermion content, and topology, projected via a non-invertible bulk--boundary correspondence onto a 4D boundary at which all observation occurs. This is the structural unity that the twentieth century's experimental and theoretical work was implicitly converging toward; the cohort framework states it explicitly.


%% ===================================================================
\section{The fifty-year arc, 1976--2026}
\label{sec:concl-arc}

Half a century separates the foundational mathematical structure of modern quantum measurement from the cohort framework ship. We trace the arc.

\paragraph{1976: Lindblad / GKLS.} Gorini--Kossakowski--Sudarshan and Lindblad \citep{GKSL1976, Lindblad1976} independently proved that the most general generator of a quantum dynamical semigroup compatible with complete positivity and trace preservation is a Hamiltonian conjugation plus a Lindblad dissipator. This established the mathematical structure of open-system quantum dynamics --- the structure that \TA{2}, \TA{5}, and the whole cohort framework's reading rest on.

\paragraph{1981--1985: the decoherence program.} Zeh's 1970 paper had introduced decoherence; Zurek's 1981 paper introduced einselection; Joos--Zeh's 1985 paper produced the quantitative estimates \citep{Zeh1970, Zurek1981, JoosZeh1985}. Together these works converted decoherence from a curiosity into the most significant theoretical advance on the measurement problem in the second half of the twentieth century.

\paragraph{1982--2015: the Aspect-and-successors confirmations of Bell's theorem.} Aspect 1981--1982 \citep{Aspect1981, Aspect1982} demonstrated CHSH violation past five standard deviations. The Delft/Vienna/NIST loophole-free runs of 2015 \citep{Hensen2015, Giustina2015, Shalm2015} closed all remaining loopholes. The 2022 Nobel Prize \citep{NobelPhysics2022} formalized the verdict: no 4D-local hidden-variable completion of quantum mechanics is empirically viable.

\paragraph{1993--2017: quantum teleportation, BB84, and the cryptographic revolution.} The Bennett--Brassard--Cr\'epeau--Jozsa--Peres--Wootters teleportation protocol of 1993 \citep{Bennett1993}, the BB84 quantum-key-distribution protocol of 1984 \citep{BennettBrassard1984}, and the Ekert-1991 entanglement-based protocol \citep{Ekert1991} together established quantum information as an operational discipline. The Micius mission's 2017 demonstration of $1400$~km teleportation and BB84 key distribution \citep{Liao2017} closed the experimental gap from thought experiment to deployed technology.

\paragraph{2018: Frauchiger--Renner sharpens Wigner's friend.} The Frauchiger--Renner no-go theorem \citep{FrauchigerRenner2018} converted Wigner's 1961 thought experiment into a quantitative inconsistency result: any interpretation that satisfies (Q) universal validity, (C) cross-agent consistency, and (S) single-outcome-per-measurement must fail at least one of the three.

\paragraph{2026: the Ultra Eleven cohort.} Across the eleven cohort papers --- \AOne{} (Constitution), \ATwo{} (\GG{} bulk), \AThree{} (\BIsix{}), \AFour{} (Euclid), \AFive{} (\GDOS{} + \T{7.1}--\T{7.7}), \ASix{} (\GiFour{}), \ASeven{} (\QiFour{}), \AEight{} (QM, this paper), \ANine{} (SM), \ATen{} (Cosmos), \AEleven{} (Origin) --- the cohort framework supplies the constitutional warrant that the foundations literature has been working toward. The present paper specializes the warrant to the microscopic-observable sector and lifts the seven QM-postulate-theorems of \AFive{} Part~III to standalone graduate-textbook depth. The cohort ship completes a fifty-year arc from the foundational mathematical structure of open-system quantum dynamics to a constitutional theory in which the four textbook postulates of quantum mechanics emerge as theorems.

A graduate student starting their physics PhD in 2026 inherits a quantum mechanics whose foundational status is structurally different from the one their advisor learned: the postulates have become theorems, the interpretations have become projections of a single underlying ontology, and the constitutional warrant for both has been laid out across a twenty-two-paper cohort.


%% ===================================================================
\section{The next-decade empirical horizon}
\label{sec:concl-horizon}

The cohort framework's quantum-mechanical reading makes empirically falsifiable predictions on a 2026--2035 horizon. We have laid out the eight-test horizon in Part~\ref{part:XII} \S\ref{sec:disc-eight-test}; the closing summit voice can identify the priorities.

\paragraph{Highest priority: Test 5 (einselection-rate platform-independence).} Cross-platform measurements of decoherence-rate ratios for matched system--environment couplings should agree to within the scale-overlap-integral precision. The framework predicts the ratios are platform-independent (fixed by bulk geometry, not apparatus design). Existing data are consistent; high-precision cross-platform comparisons over 2027--2030 will sharpen the test.

\paragraph{Second priority: Test 6 (Bell-pair fidelity at long distance) extended to lunar-distance experiments (proposed 2030--2035).} The framework predicts no anomalous fidelity loss beyond the standard decoherence-channel-loss model. A lunar-distance entanglement experiment would test this prediction at $\sim 10^5$~km baselines.

\paragraph{Third priority: Test 8 (high-energy QM corrections) at HL-LHC Run~4 (2030+).} The framework predicts small unitarity-violation signatures and small Schr\"odinger-equation modifications at energies approaching $\Mlock\approx 3.85$~TeV. The HL-LHC Run~4 program is the primary platform; the predicted signatures tie to the \PaperPOne{} KK lock-cluster phenomenology, treated in dedicated paper \PaperPOne{}.

\paragraph{Fourth priority: Test 2 (decoherence-rate scaling) extended to large-mass regimes $10^4$--$10^9$ amu.} Closing the gap between $C_{60}$ ($10^3$ amu) and the cat scale ($10^{27}$ amu) with mesoscopic-mass interferometry would test \TA{5}'s scale-overlap-integral prediction across six orders of magnitude. Proposed implementations (large biomolecule interferometry, levitated nanoparticle interferometry) are technically demanding but within reach.

Tests 1 (Tsirelson saturation), 3 ($1/f$ universality), 4 (pointer-basis localization), and 7 (long-distance teleportation) are postdiction audits or consistency checks; they have already been measured to high precision and continued measurement will continue to constrain the framework's prediction within experimental error.

If any of the eight tests fails the framework's prediction at $>5\sigma$ in a reproducible experiment, the cohort framework's quantum-mechanical reading is retired in its current form. If all eight remain consistent, the framework joins the post-2022 quantum-foundations record as a constitutional theory whose predictions match experiment across a structurally complete set of QM-specific tests.


%% ===================================================================
\section{The place of \AEight{} in the Ultra Eleven cohort closure}
\label{sec:concl-cohort}

The eleven-paper A-series closes with the ship of \AEight{}. \AOne{} stated the axioms; \ATwo{}--\AThree{}--\AFour{} built the bulk-geometry block; \AFive{}--\ASix{}--\ASeven{} built the operational-dynamics block; \ANine{}--\ATen{} derived the Standard Model and cosmological content; \AEleven{} supplied the operational anatomy and the universal-identity insight that the present paper specializes. \AEight{} closes the Consequence Trio (\AEight{}--\ANine{}--\ATen{}) on the boundary side of the program.

What the cohort closure accomplishes is the structural completion of the boundary-side specializations. Where the boundary-side block of papers (\AFive{} and downstream) had previously left the QM stratum at the level of the seven theorems \T{7.1}--\T{7.7} of \AFive{} Part~III, the present paper lifts the chain to the standalone graduate-textbook treatment that the cohort program has been working toward. The pedagogical apparatus, the worked examples, the paradox sweep, the interpretation-landscape engagement, and the summit-view philosophical close are all assembled in one document.

A reader who picks up \AEight{} in 2026 without prior cohort context can in principle follow the chain from the constitutional warrant of \AOne{} through the QM-postulate theorems of Part~\ref{part:IV} to the worked examples of Parts~\ref{part:V}--\ref{part:IX} to the paradox sweep of Part~\ref{part:X} and the philosophical close of the present Part, with reference back to the upstream cohort papers only when the proof details require it. This is what was meant by ``standalone graduate-textbook'' --- not that the paper exists in isolation from the cohort, but that the chain of reasoning is followable on its own once the constitutional warrant is in place.

\paragraph{What comes next outside the GG-A series.} The Predictions Branch papers \PaperPOne{} (KK lock cluster at HL-LHC), \PaperPTwo{} (three-leg derivation of $N=3$ flavor families), and \PaperCOne{} (Kaxion dark matter at $\sim 8.1$~$\mu$eV, $1.95$~GHz) carry the cohort framework's specific empirical predictions; the present paper's eight-test horizon is the QM-stratum analogue. The capstone monograph \citet{ArisakaBookI} synthesizes the cohort into textbook form for a broader audience. Together, the GG-A cohort, the Predictions Branch, and the capstone constitute the GG-Theory program as a complete corpus.


%% ===================================================================
\section{An experimentalist's note: fifty years at the detector}
\label{sec:concl-experimentalist}

I close with a personal note, because a paper on the foundations of measurement is incomplete without saying something about who is writing it.

The framework presented in the preceding twelve Parts is not the outcome of theoretical thinking alone. It is rooted in fifty years of trying to observe Nature directly across an unusually wide range of strata.

I have spent my career attempting to measure what could be measured. \emph{Proton decay}, in tonne-scale water Cherenkov detectors searching for a process whose half-life lies beyond $10^{34}$ years. \emph{Direct CP violation}, in the kaon and B-meson asymmetries that anchor our understanding of why the universe contains matter at all. \emph{The Higgs particle}, whose 2012 LHC discovery confirmed the half-century-old electroweak symmetry-breaking mechanism. \emph{SUSY particles}, whose absence in current data has reshaped our expectations for physics beyond the Standard Model. \emph{Ultra-high-energy cosmic rays}, whose origin and acceleration mechanism remain among the deepest open questions in astrophysics. \emph{WIMP dark matter}, the direct detection of which has been my laboratory's principal focus for many years. \emph{The behavior and neural activity of \emph{Caenorhabditis elegans}}, the simplest organism with a complete connectome. \emph{Human reaction times}, the operational signature of cognition at the millisecond scale. \emph{Single-molecule tracking}, the operational signature of biology at the nanosecond scale.

Across all of these I have become, in the practical sense that experimentalists understand, a photon-detection physicist. The arc began with the development of 20-inch photomultiplier tubes at Hamamatsu --- the photodetectors that made Super-Kamiokande and the subsequent generation of large water Cherenkov experiments possible --- and has continued through SiPMs, MKIDs, and the cryogenic photon detectors that underpin contemporary direct-detection dark-matter searches.

What this experience has given me is an unusual perspective on what ``measurement'' actually is. In the standard textbook of quantum mechanics, measurement is a postulate: an act that registers an outcome with probabilities given by the Born rule. In actual laboratory practice, measurement is a long causal chain. A particle scatters in a target. The recoil deposits energy in a sensitive medium. The medium's response excites a photodetector. The photodetector's signal travels through cables, amplifiers, and analog-to-digital converters, into a computer record, into a histogram, into a scientific paper. Every step of that chain is open-system dynamics. None of it is well-described by a closed unitary Schr\"odinger evolution. All of it is well-described by exactly the kind of bulk-to-boundary projection that \GDOS{} supplies and that \OGGEP{} (\AX{3} of the \GGConstitution{}) forces.

The conviction I came to over the years is that the foundations of quantum mechanics could not be settled from the chalkboard. They had to be settled from the detector --- from the long causal chain that turns inaccessible bulk physics into a record on a hard drive. The framework presented in this paper is the structural account of that conviction. \GG{} supplies the inaccessible bulk. \GDOS{} supplies the canonical boundary law that any actual measurement instrument has to obey. \OGGEP{} forces the partial-trace projection that produces the operational outcome. From the detector's point of view, the measurement problem was never a problem about ontology; it was a problem about not having a theory of the partial trace deep enough to describe what was actually happening at the chain's beginning. The cohort framework supplies that theory.

This is not, I should add, a substitute for any of the theoretical work that the foundations community has done over the past century. Bohr, Einstein, Heisenberg, Schr\"odinger, Born, Dirac, von~Neumann, Bell, Aspect, Gleason, Zurek, Joos, Zeh, Hartle, and the rest each saw a piece of the picture that the present paper tries to gather. What I bring to it is the conviction --- earned at the detector, not at the chalkboard --- that the picture as a whole has to be consistent with what experimentalists actually measure at every stratum, from proton decay through dark matter to a single molecule diffusing in a cell. The framework is my attempt to write down what that consistent picture has to look like.

That is what \GGtheory{} is, in one sentence: a structural reading of what fifty years of measurement at every scale has been quietly telling us.


%% ===================================================================
\section{The long-form philosophical close}
\label{sec:concl-close}

After all, what a twenty-two-paper cohort on the foundations of physics is supposed to do is not produce a new equation but produce a new way of seeing. The equations of the present paper are not new; they are Born's, Schr\"odinger's, Lindblad's, Bell's, Tsirelson's, and Joos--Zeh's. What the framework supplies is a structural reading of where those equations come from, how they fit together, and why the world looks the way it does to a 4D laboratory observer.

The bulk is real, single-valued, deterministic, and local. The boundary is a non-invertible projection. The four textbook postulates of quantum mechanics --- the Hilbert space, the Schr\"odinger equation, the Born rule, the L\"uders projection --- emerge as theorems of the projection. Three additional structures --- decoherence, einselection, Bell-violation --- that the standard textbook leaves implicit also emerge as theorems. Four further theorems (extremal-Tsirelson, universal identity, $1/f$ conformality, boundary no-signaling) extend the cohort's specifically-novel contributions. Eleven theorems together, with proofs cited by reference to \AFive{} Part~III for the inherited seven and supplied in full for the four net-new, constitute the operational core of the present paper.

Every electron in the universe is the same because the cascade at the charged-lepton stratum has a unique stable Geometric Attractor. Every measurement is registered in the position basis because the bulk Hamiltonian is local. Every Bell-CHSH experiment saturates the Tsirelson bound because the singlet is the extremizer of the CHSH functional, where the first variation of $|S|$ vanishes. Every laboratory measurement record is operationally complete --- as Bohr insisted --- and the underlying ontology is deterministic --- as Einstein insisted --- because the two readings describe the same Nature at different levels of a bulk-to-boundary projection.

A graduate student reading this paper in a long weekend should come away with three things: (i) a structural understanding of why the four textbook postulates of quantum mechanics are theorems rather than postulates; (ii) a recognition that the seven major interpretations of quantum mechanics each captured a partial truth that the cohort framework gathers under a single structural source; (iii) a clear sense of what empirical observations would retire the framework, and which observations the framework has already passed. Beyond that, the student should come away with the sense that quantum mechanics is no longer a hundred-year-old mystery awaiting a deeper theory --- it is the operational shadow of a deeper theory that has been spelled out across a twenty-two-paper cohort, and the standalone graduate-textbook reformulation of that fact is what the present paper provides.

After all, the universe has been doing quantum mechanics for $13.8$ billion years. The cohort framework explains why it has to. The least the present paper can do is write that down at the level a careful graduate student can follow --- and the twenty-two-paper cohort has done that, at last.

\bigskip

\noindent\emph{Quantum mechanics from the 6D Bulk. T7.1--T7.7 in standalone graduate-textbook form, lifted to T-A8.1 through T-A8.11. Inherited verbatim from \AFive{} Part III; extended with four net-new theorems; placed within the twenty-two-paper cohort. No new postulates; no new free parameters; no new ontological entities at the 4D level; only the constitutional warrant that the foundations literature has been working toward since Heisenberg, Schr\"odinger, and Born put the equations on the page.}

\bigskip\bigskip
\bigskip

\begin{center}
\rule{0.5\textwidth}{0.4pt}
\end{center}

\bigskip

\begin{center}
\textbf{\large Dedication and a three-part address}
\end{center}

\bigskip

\begin{quote}\itshape
Dedicated to the memory of Albert Einstein (1879--1955), whose
1926 objection was eventually answered in the way he had hoped
--- and to the memory of Niels Bohr (1885--1962), whose 4D
operational reading was also correct.
\end{quote}

\bigskip

\paragraph{To the scientific community.} Max Born, in the same
1926 \emph{Zeitschrift f\"ur Physik} paper that introduced the
probability interpretation of the wavefunction, wrote a single
sentence whose significance the foundations community has been
absorbing for a century: \emph{``the motion of particles follows
probability laws, but the probability itself propagates according
to the laws of causality''}~\citep{Born1926}. Born believed that
probability and causality were compatible --- that the
probabilistic interpretation he had just introduced did not
abolish determinism at a deeper layer, only relocated it. The
cohort framework presented in this paper makes Born's intuition
precise. The deeper layer is the 6D bulk of \citet{ArisakaA2GG6}.
The probabilistic interpretation lives at the 4D boundary that
\OGGEP{} (\AX{3} of the \GGConstitution{}) projects onto. The
two are compatible because they describe the same Nature at two
different levels of the bulk--boundary hierarchy. The hundred-year
measurement problem is therefore not explained away; it is
relocated --- from the 4D level, where it was insoluble, to the
6D level, where it is just geometry. The operational results of
the past century are not in doubt; what was in doubt was the
structure that holds them together. The cohort framework
supplies the structure. The mess can now be set down.

\paragraph{To Albert Einstein.} If quantum mechanics is the
child theorem of \GG{} and \GDOS{}, then \GGtheory{} is, in the
lineage of physical intuitions, your grandchild. The conviction
in your December 1926 letter to Born --- ``I, at any rate, am
convinced that He does not throw dice'' --- has finally come of
age, at the bulk level where it was always true. You spent the
last thirty years of your life searching for the deterministic
substrate beneath quantum mechanics. You died in 1955 without
finding it, because in 1926 the mathematics that the search
required was not yet on the page: the extra-dimension
geometry was a decade away; the bulk--boundary correspondence half a
century; the cohort apparatus that locks all of it together was a
century. The framework presented here is the substrate you
sought. At the level of physical intuition --- the level of
``what kind of theory could possibly be at the bottom of this''
--- you were right.

\paragraph{To Niels Bohr.} Your operational reading of
measurement, articulated through the Como lecture of 1927 and
the long correspondence with Einstein that followed, was also
right --- exactly where it was meant to be right. At the 4D
boundary, the laboratory observer registers irreducibly
probabilistic outcomes; the wavefunction is operationally
complete; the Heisenberg cut is where the observer chooses to
draw it. Every operational prediction of standard quantum
mechanics emerges from the cohort framework intact. The
framework adds no 4D ontology that you would have refused; it
only supplies the structural source you did not have access to.
You died in 1962 without seeing Bell's 1964 theorem or Aspect's
1982 experiment, but the framework would have agreed with you
about every operational prediction those experiments and their
successors have confirmed. At the level you were looking at ---
the level of what laboratory observers actually register --- you
were right.

\bigskip

\noindent\emph{Both giants of twentieth-century physics had hold
of a piece of the truth that the cohort framework now gathers
into one structure. Neither has to lose for the other to be
right. This paper is for them.}

\clearpage

%% ===================================================================
%% COMPREHENSIVE SUMMARY TABLES
%% ===================================================================
%% (Comprehensive summary tables relocated into their home Parts.)



\section{Part~XIII in plain English: what the summit view does and does not claim}
The thesis of this Part can be put in one sentence: Einstein was right about
determinism and realism, in the six-dimensional bulk; Bohr was right about the
operational completeness of the four-dimensional description; and the two are
compatible because they are statements about two levels of one geometry rather
than two accounts of one arena. Locality is the price the bulk pays to keep
determinism, and Bell's theorem is the receipt.

Three things follow that are worth separating from the thesis, because they are
weaker than it and are easy to read as though they were part of it.

The first is the universal identity of the electron. Every electron in the
universe carries the same mass, the same charge and the same spin because every
electron is the same token of one bulk configuration, seen at a different place
on the boundary. That is a satisfying reading and it is not a new prediction:
the identity of electrons was never in doubt, and a framework that explains it
has explained something already known. It is offered as coherence, not as
evidence.

The second is the fifty-year arc. The Part traces a line from the mid-1970s to
the present in which decoherence, einselection, the loophole-free Bell tests
and the open-system formalism arrive one at a time and, read together, point at
a single source. A narrative arc is a way of organizing results, not a reason
to believe them. A reader who finds the arc persuasive should notice that it
was assembled after the fact, by someone with a conclusion in hand.

The third is what the paper is for. \AEight{} is the graduate-textbook member
of its trio: it takes machinery established elsewhere in the cohort and works
it out on five laboratory systems --- Stern--Gerlach, the double slit, EPR and
Bell, the cat, and decoherence --- at the level of detail a student can follow
with a pencil. That is its function, and its function is a real one, but it is
not the function of a paper that establishes something new. The net-new content
of \AEight{} is four theorems and eight kill criteria. Everything else is
inheritance made explicit and worked out.

What the summit view offers, then, is a place to stand from which a century of
argument looks like one question asked at two levels. Whether the view is
correct is decided by the eight criteria of Part~XII and by the single analytic
residue that Part named, and by nothing in this Part at all.

\GGAcknowledgments

\clearpage

%% ===================================================================
%% PART XIV (14/14) -- Appendices
%% ===================================================================
\begin{table}[H]
\centering\small
\caption[Single-row resolution of the 1926--2026 debate]{Single-row resolution of the 1926--2026 debate. The row sets the question that has been asked for a century beside the answer this paper gives it, and beside the theorem that carries the answer. It is placed here rather than in the abstract because it is only intelligible after the eleven theorems have been stated, and a reader who has not read them should treat it as a summary of the paper's claim rather than as an argument for it.}
\label{tab:bottom_line}
\renewcommand{\arraystretch}{1.4}
\begin{tabular}{@{}lp{5.20cm}p{5.60cm}@{}}
\toprule
\textbf{Question} & \textbf{6D bulk (Einstein)} &
\textbf{4D boundary (Bohr)}\\
\midrule
Is Nature deterministic? & YES & NO \\
Does God play dice? & NO & YES \\
Is there a complete theory? & YES (the bulk is complete) &
NO (4D QM is operationally complete but ontologically
incomplete) \\
Are there hidden variables? &
YES ($\Theta_H$ on $S^1_\theta$) &
NO (not of the Bell-local type) \\
Is Nature local? & YES (6D locality) &
NO (4D non-local after bulk-to-boundary trace) \\
\midrule
& \multicolumn{2}{c}{\emph{Both columns are correct within
their domains.}}\\
\bottomrule
\end{tabular}
\end{table}

\GGAppendixPart[Technical Appendices]{Technical Appendices}{Codebook, glossary, proof sketches, theorem index}
\label{part:XIV}

\noindent
Part~XIV gathers the technical machinery behind the present
derivation of quantum mechanics from the cohort framework. This
is where the reader who wants to verify a specific proof step,
re-run a specific worked calculation, or trace a specific
reference will spend time --- while the reader who wants only the
central claims and their forward consequences can stop after the
Conclusions and treat Part~XIV as a self-contained companion
volume. Appendix~A is the Summary of Abbreviations and Appendix~B
the Glossary of core \GGtheory{}/\GDOS{} vocabulary. These are
followed by the full derivations, worked from first principles at
graduate-textbook depth (Gleason's theorem, the
GKSL generator, the 6D-to-4D partial-trace reduction, the
Bell--CHSH evaluation with the $S^1_\theta$ phase fiber, and the
Wilson-line holonomy); the worked methods (the
\QiFour{} application to quantum measurement and the worked
decoherence overlap integral with its uncertainty budget); and
the reference material (the condensed historical
timeline of the quantum-mechanical and open-system milestones,
and the concept-by-concept synthesis of
standard quantum mechanics against the cohort reformulation.
Finally, the inline annotated bibliography, with the full
Relevance annotations, follows the appendices and closes the
volume. As of the present revision the apparatus of the cohort's
course papers runs through this Part: each substantive appendix
opens with a \emph{What this appendix teaches} box and a
\emph{Read alongside} line pairing it with the standard
literature, and the five deep derivation appendices close with a
\emph{Take-away} box in the three-register form (standard account
/ what it left open / the bulk reading) used across the cohort. And,
completing the course apparatus, every teaching unit of the paper ---
the five Worked-Example Parts, the paradox sweep, and the five deep
derivation appendices --- closes with a set of \emph{problems and
worked solutions}: fifty-six across the paper, each solved in
explicit steps, closed by its physics intuition, and with every
numerical answer checked before typesetting; they are written for
direct classroom use, and their cross-references knit the worked
examples, the theorem appendices, and the companion papers of the
cohort into a single teachable course.


\section{Summary of Abbreviations}
\label{app:abbrev-a8}

This appendix collects the abbreviations used in the present paper.
All program-defined names are rendered in \emph{sans-serif type}
(\GGTheory{}, \GGConstitution{}, \GG{}, \GDOS{}, \GiFour{},
\QiFour{}, \GRIP{}, \GA{}, 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 (QM, QFT,
KK, RS, CHSH, EPR, etc.) are rendered in plain uppercase Roman,
following the convention of the major physics journals. The same
content is displayed as a single hierarchy diagram in the
front-matter Program Dictionary
(Fig.~\ref{fig:appA-abbrev-flow-a8}), where the macro relations are
visible at a glance; the table below gives the canonical
Appendix-A format inherited from \AOne{} Part~VIII, with a third
column recording the \emph{primary link}: the paper in the
Arisaka-GG series (or the section of the present paper) where each
program-defined name is introduced in detail.

{\small
\begin{longtable}{@{}>{\raggedright\arraybackslash}p{0.16\linewidth} >{\raggedright\arraybackslash}p{0.56\linewidth} >{\raggedright\arraybackslash}p{0.24\linewidth}@{}}
\caption[Summary of abbreviations used in the present paper]{Summary of abbreviations used in the present paper.
Program-defined names are rendered in sans serif; standard physics
abbreviations in plain uppercase Roman; mathematical objects in
roman/italic in the usual way. The third column records the primary link: the paper in the Arisaka-GG series, or the section of this one, where each name is introduced in detail. An abbreviation with no primary link beside it is standard physics usage and is listed only so that the table can be read without a second reference open.}
\label{tab:appA-abbrev-a8}\\
\toprule
\textbf{Symbol} & \textbf{Meaning} & \textbf{Primary link} \\
\midrule
\endfirsthead

\multicolumn{3}{c}{\itshape Table~\ref{tab:appA-abbrev-a8} 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: the twenty-two-paper program & \AOne{}, Preface \\
\GGConstitution{} & The constitutional charter of \GGTheory{}: four axioms (\AX{1}--\AX{4}) and seven postulates & \AOne{}, \S 2--\S 3 \\
\GG{} & Geometry--Gauge locking in six dimensions: the 6D bulk $\mathcal{M}_4 \times I_r \times S^1_\theta$ (the bulk arena; father theory of QM) & \ATwo{} \\
\GDOS{} & Geometric Dynamics of Open Systems: the boundary law forced by \OGGEP{} (mother theory of QM) & \AFive{} \\
\BIsix{} & Six-dimensional Green--Schwarz-modified Bianchi identity $dH_3=X_4$ (boundary closure) & \AThree{} \\
\OGNfive{} & Occam-Gauge-Normalization five-integer ledger & \AThree{} \\
\GUfive{} & Geometric-Unification five-member locked set (carries the boundary information member $\QB$) & \ATwo{} \\
\GiFour{} & Geometric-Invariance four-pillar structure (\TGG{}, \CCI{}, \GFF{}, \GRIP{}) & \ASix{} \\
\QiFour{} & Quantum-Information four-pillar structure (\QiQUA{}, \QiEP{}, \QiCON{}, \QiHAL{}) & \ASeven{} \\
\GRIP{} & Gauge-Runtime Invariance Pipeline: the cascade $\DGI{}\to(\GFF{})\to\GSSB{}\to\GGA{}$ & \ASix{} \\
\GA{} & Geometric Attractor: attractor-like stabilization on the reduced landscape & \ASix{} \\
\GGA{} & Ground Geometric Attractor: the unique deepest \GA{} where the runtime halts & \ASix{} \\
\SeOS{} & Selfish Open System: the runtime carrier of the cascade & \ASix{}, \ASeven{} \\
\SIP{} & System Instance Packet: the canonical \GDOS{} input unit on which \GRIP{} runs & \AFive{} \\
\midrule

\multicolumn{3}{l}{\itshape Four axioms and the operational dictionary} \\
\midrule
\AX{1} & Locality and Causality (\LC{}) & \AOne{}, \S 2.2 \\
\AX{2} & Geometry--Gauge Equivalence Principle (\GGEP{}) & \AOne{}, \S 2.3 \\
\AX{3} & Operational/Open Geometry--Gauge Equivalence Principle (\OGGEP{}; the boundary observation dictionary) & \AOne{}, \S 2.4 \\
\AX{4} & Maximum Symmetry, Minimum Freedom (\MSMF{}; no hidden continuous knobs) & \AOne{}, \S 2.5 \\
\LC{} & Locality and Causality (name of \AX{1}) & \AOne{}, \S 2.2 \\
\GGEP{} & Geometry--Gauge Equivalence Principle (name of \AX{2}) & \AOne{}, \S 2.3 \\
\OGGEP{} & Operational/Open \GGEP{}: bulk-to-boundary projection map (name of \AX{3}) & \AOne{}, \S 2.4 \\
\MSMF{} & Maximum Symmetry, Minimum Freedom (name of \AX{4}) & \AOne{}, \S 2.5 \\
\midrule

\multicolumn{3}{l}{\itshape The eleven principal theorems of \AEight{}} \\
\midrule
\TA{1} & Boundary Hilbert space as the GNS representation of the QM-restricted boundary algebra & \S\ref{part:IV} (inherited from \AFive{}, \T{7.1}) \\
\TA{2} & Schr\"odinger equation as the closed-system limit of the GKLS generator & \S\ref{part:IV} (inherited from \AFive{}, \T{7.2}) \\
\TA{3} & Born rule from CPTP completion of the \OGGEP{} dictionary & \S\ref{part:IV} (inherited from \AFive{}, \T{7.3}) \\
\TA{4} & L\"uders projection as the conditional CPTP posterior & \S\ref{part:IV} (inherited from \AFive{}, \T{7.4}) \\
\TA{5} & Decoherence rates from scale-overlap integrals of bulk KK modes & \S\ref{part:IV} (inherited from \AFive{}, \T{7.5}) \\
\TA{6} & Einselected pointer basis from the dominant local system--environment coupling (\AX{1}) & \S\ref{part:IV} (inherited from \AFive{}, \T{7.6}) \\
\TA{7} & Bell--CHSH violation to Tsirelson saturation as the shadow of one joint bulk state & \S\ref{part:IV} (inherited from \AFive{}, \T{7.7}) \\
\TA{8} & Tsirelson saturation as an extremal (Bogomolnyi-analogue) bound & \S\ref{part:IV} (net-new) \\
\TA{9} & Universal identity of indistinguishable particles as a cascade inheritance & \S\ref{part:IV} (net-new) \\
\TA{10} & \QiCON{}-conformality of decoherence-rate spectra (approximate $1/f$ window) & \S\ref{part:IV} (net-new) \\
\TA{11} & Boundary no-signaling from reduced-state factoring & \S\ref{part:IV} (net-new) \\
\midrule

\multicolumn{3}{l}{\itshape QM-stratum objects and \QiFour{} pillars} \\
\midrule
\QiQUA{} & Quantum information for \QGL{} / locked-current bookkeeping (Pillar I of \QiFour{}) & \ASeven{} \\
\QiEP{} & Quantum information for the Unified Extremal Principle / trajectory selection (Pillar II of \QiFour{}) & \ASeven{} \\
\QiCON{} & Quantum information for conformal invariance / scaling diagnostics (Pillar III of \QiFour{}) & \ASeven{} \\
\QiHAL{} & Quantum information for the Holographic Attractor Lattice / memory architecture (Pillar IV of \QiFour{}) & \ASeven{} \\
\TGG{} & Total Geometric--Gauge object (Pillar I of \GiFour{}) & \ASix{} \\
\CCI{} & Covariant Conservation Identity (Pillar II of \GiFour{}) & \ASix{} \\
\GFF{} & Geometric Free-Energy Functional ($F_{\GFF{}}=F_{\GEP{}}+F_{\CEP{}}$; Pillar III of \GiFour{}) & \ASix{} \\
\GEP{} & Geometry--Gauge Energy Principle (energetic block of \GFF{}) & \ASix{} \\
\CEP{} & Curvature--Entropy Principle (entropic block of \GFF{}) & \ASix{} \\
\DGI{} & Dynamical Gauge Invariance (initial stage of \GRIP{}) & \ASix{} \\
\GSSB{} & Geometric Spontaneous Symmetry Breaking (middle stage of \GRIP{}) & \ASix{} \\
\OH{} & Outer Hologram: lawful projection from bulk states to measurable records & \ASeven{} \\
\IH{} & Inner Hologram: internal reconstructive manifold of a memory-bearing system & \ASeven{} \\
\EIC{} & Evolving Inner Cosmos: the \OH{}${+}$\IH{} composite (observer semantics) & \ASeven{} \\
$S^1_\theta$ & Phase fiber of the 6D bulk; its holonomy is the quantum relative phase & \S\ref{sec:programme-dictionary} \\
$I_r$ & Radial (scale) interval of the 6D bulk; the renormalization-group axis & \S\ref{sec:programme-dictionary} \\
$\Theta_H$ & Topologically protected Wilson-line holonomy read out by \OGGEP{} as the outcome & \S\ref{app:Wilson_holonomy} \\
$\densbdy$ & Reduced boundary density operator entering Born probabilities and instruments & \AFive{} \\
\Mlock{} & Kaluza--Klein locking scale ($\approx 3.85$~TeV) of the extra dimension & \ATwo{} \\
$\QB$ & Quantum-information Bit: the boundary information member of \GUfive{}, fixed via \OGNfive{} & \ATwo{} \\
\midrule
\pagebreak

\multicolumn{3}{l}{\itshape Standard physics abbreviations (plain Roman, journal convention)} \\
\midrule
QM & Quantum Mechanics & --- \\
QFT & Quantum Field Theory & --- \\
GR & General Relativity & --- \\
RG & Renormalization Group (scale running along $I_r$) & --- \\
KK & Kaluza--Klein (mode expansion / dimensional reduction) & --- \\
RS & Randall--Sundrum (warped extra-dimension model) & --- \\
GNS & Gelfand--Naimark--Segal (construction) & --- \\
CPTP & Completely Positive Trace-Preserving (channel admissibility) & --- \\
POVM & Positive Operator-Valued Measure (generalized measurement model) & --- \\
GKLS & Gorini--Kossakowski--Lindblad--Sudarshan (Markovian master equation; also GKSL) & --- \\
GKSL & Gorini--Kossakowski--Sudarshan--Lindblad (equivalent ordering of GKLS) & --- \\
CHSH & Clauser--Horne--Shimony--Holt (inequality) & --- \\
EPR & Einstein--Podolsky--Rosen (paradox / argument) & --- \\
\end{longtable}
}



\section{Glossary of core \GGtheory{}/\GDOS{} quantum-foundations vocabulary}
\label{app:glossary}

This appendix collects the core vocabulary of \GGtheory{} and \GDOS{}
as it is used throughout the present paper. The table is not merely
terminological: it also clarifies how the cohort vocabulary is meant
to sit relative to the established language of quantum mechanics and
open systems.

\begin{longtable}{@{}p{0.19\textwidth}p{0.31\textwidth}p{0.40\textwidth}@{}}
\caption[Glossary of core GG-Theory and GDOS quantum-foundations vocabulary]{Glossary of core \GGtheory{} and \GDOS{} quantum-foundations vocabulary. The middle column gives a short definition and the right column says what the term is doing in this paper, which is the column that matters: several of these words already have a settled meaning in the open-systems literature, and the entries are written to show where the cohort usage coincides with it and where it does not. Where the two differ the established meaning is named first.}
\label{tab:app-glossary}\\
\toprule
Term & Short definition & Role in the present paper \\
\midrule
\endfirsthead
\toprule
Term & Short definition & Role in the present paper \\
\midrule
\endhead
6D bulk & The deeper closed arena of lawful dynamics in the \GG{} architecture & Carries the deterministic total-state law; not directly identical to the observer-accessible 4D quantum state \\
4D boundary & The observer-accessible operational sector where records are registered & Domain in which ordinary quantum mechanics is retained as the correct effective law \\
\OGGEP{} & Operational \GG{} emergence-and-projection map from bulk data to boundary states, maps, and instruments & Core dictionary that relocates quantum probability from ontology to interface (\AX{3}) \\
\GDOS{} & Geometric Dynamics of Open Systems; the canonical boundary law forced by \OGGEP{} acting on \GGsix{} & Mother theory of QM on the boundary side; QM is its closed-subsystem child \\
\BIsix{} & Boundary identity closure of the 6D arrangement with coefficient vector $(1,-1,-2/5,-2/5)$ & Records the bulk-to-boundary closure that makes the projection well-posed (\AThree{}) \\
Boundary state $\densbdy$ & Reduced density operator seen at the boundary & The actual object entering Born probabilities, instruments, and master equations \\
Instrument family $\{\mathcal{I}_\alpha\}$ & Outcome-indexed completely positive maps encoding both statistics and updates & Makes measurement a physical channel rather than a vague extra postulate (\TA{4}) \\
Outer hologram & Lawful projection from bulk states to measurable records & Generalized language for detector clicks, images, spots, and all accessible observations \\
Inner hologram & Internal reconstructive manifold maintained by a memory-bearing open system & Extends measurement theory from external registration to internal model formation \\
Evolving inner cosmos & Closed-loop composition of outer evidence and inner reconstruction & Observer semantics in the broad \GDOS{} sense \\
\DGI{} & Dynamical Gauge Invariance; the lawful exploration of the gauge-equivalence class, identifying the reduced coordinates and order parameters & First step of the runtime; the variables that matter after reduction \\
\GFF{} & Geometric free-energy functional on the reduced state manifold & Organizes stability, branch competition, and evolution on the reduced landscape \\
\GSSB{} & Geometric spontaneous symmetry breaking or branch selection on the reduced landscape & Supplies a language for outcome-sector selection beyond bare decoherence \\
\GA{} & Geometric attractor; attractor-like stabilization on the reduced landscape & Explains persistence of selected records or patterns; powers \TA{9} indistinguishability \\
\GRIP{} & Gauge-Runtime Invariance Pipeline of \ASix{} (the cascade $\DGI{}\to\GSSB{}\to\GA{}/\GGA{}$) & Cascade that propagates the unique stable \GA{} through atoms, molecules, cells \\
\QiFour{} pillars & The four admissibility pillars (\textsc{sip}, \textsc{qua}, \textsc{ep}, \textsc{con}, \textsc{hal}) of \ASeven{} & Constrain which boundary instruments are admissible \\
\GKSL{} / Lindblad generator & Markovian generator of completely positive trace-preserving semigroups & Boundary normal form of exact projected dynamics in the appropriate regime (\TA{4}) \\
Memory kernel & Nonlocal-in-time term in the exact reduced dynamics & Reminder that the boundary law is generically non-Markovian before approximation \\
Born rule & Trace formula assigning boundary outcome probabilities & Unique probability law for the boundary event structure once that structure is in place (\TA{3}) \\
Gleason theorem & Uniqueness theorem for Hilbert-space probability measures under suitable assumptions & Reinterpreted as a theorem about the boundary event lattice rather than about ultimate ontology \\
Pointer basis & Robust basis selected by coupling to environment and measurement channel & Linked in the cohort framework to actual reduced coordinates and channel geometry (\TA{6}) \\
Qubit & Two-level operational system or packet & Reinterpreted as a minimal stable boundary packet rather than necessarily the most primitive ontic entity \\
Entanglement & Non-factorization of a composite boundary state & Fully retained operationally; given a bulk-to-boundary reinterpretation rather than denied \\
No-signaling & Invariance of local marginals under distant measurement choice & Mandatory consistency condition for any acceptable bulk completion (\TA{11}) \\
Measurement problem & Historical tension between linear evolution, completeness, and definite outcomes & Resolved by domain separation between bulk determinism and boundary probability (\TA{1}--\TA{6}); relaxation-level; one analytic residue \\
Universal observation problem & Generalization of the measurement problem to all finite open observers & Signature move of the cohort that broadens the significance of the quantum case \\
\bottomrule
\end{longtable}

\subsection*{A final interpretive remark on the vocabulary}

What the glossary makes especially clear is that the new language is
not intended to replace the successful pieces of old language. It is
intended to \emph{place} them. The Born rule, the density matrix,
the Lindblad generator, the qubit, and entanglement are all retained,
but they are placed at the observer-facing boundary. The
six-dimensional bulk, the projection map \OGGEP, and the runtime
semantics (\DGI{}/\GFF{}/\GSSB{}/\GA) supply the deeper placement
relation among them. After all, the work of a foundational theory
is less to invent new words than to find the right places for the
words that already work.




\section{Gleason's theorem: statement and proof}
\label{app:Gleason}

\GGteachbox{Why the Born rule is the \emph{only} possible
probability rule: Gleason's 1957 theorem, proved line by line at
graduate level --- frame functions on the subspaces of a Hilbert
space of dimension $\ge 3$, the reduction to the two-dimensional
real problem, the one clearly-flagged analytic step
(continuity-from-additivity), and the closing identification of
every noncontextual probability assignment with $\mathrm{tr}(\rho
P)$ for a unique density operator $\rho$. This is the appendix on
which the Born-rule theorem \TA{3} stands: the \GDOS{}
registration supplies a probability assignment, and Gleason
certifies that there was never any other candidate.}

\GGreadalong{\citet{Gleason1957} for the original;
\citet{Peres1995} for the standard textbook route this appendix
follows at full detail.}

This appendix gives a self-contained, graduate-level account of
Gleason's theorem. It is complete except for one clearly-flagged
analytic step --- the continuity-from-additivity argument, whose
standard reference is cited at the point of use --- and is otherwise
meant to be followed line by line. The goal is to justify the single
displayed result on which the Born-rule theorem \TA{3} rests:
that on a complex separable Hilbert space $\Hilb$ of dimension
$\ge 3$, every countably additive probability measure on the
lattice of orthogonal projectors is given by the trace formula
\begin{equation}\label{eq:Gleason_app}
\mu(\Pi) \;=\; \Tr(\rho\,\Pi),
\end{equation}
for a unique density operator $\rho \ge 0$ with $\Tr\rho = 1$.
Nothing about the square-modulus structure of quantum
probabilities need be added by hand: \emph{given} a complex
Hilbert space and countable additivity, it is already
\emph{forced} by the lattice geometry of $\Hilb$. That is the
entire content of the standard theorem, and it is what makes
Gleason the right tool for showing the Born \emph{form} to be a
consequence of premises the framework already supplies (\TA{1}
and the operational dictionary) rather than an independent
postulate. The framework's contribution is to supply those
premises, not to strengthen Gleason.

The logical skeleton is short and worth holding in mind
throughout. (i) A probability measure on projectors restricts
to a function $f$ on the unit sphere, the \emph{frame
function}. (ii) The defining additivity of $\mu$ becomes a
local additivity constraint on $f$ over orthonormal bases.
(iii) The deep analytic lemma says this local constraint, in
three real dimensions, is so rigid that $f$ must be the
restriction of a quadratic form $\langle v,\,A v\rangle$. (iv)
Quadratic forms on the sphere are exactly the diagonal-of-an
-operator expressions, and positivity plus normalization make
$A$ a density operator. (v) Three real dimensions bootstrap to
all of $\Hilb$, real to complex, and finite to separable
infinite dimension. (vi) The lone exception, $d=2$, is repaired
by passing from projective measurements to POVMs. We treat each
of these in turn.

\subsection{Precise statement}
\label{app:Gleason:statement}

Let $\Hilb$ be a complex separable Hilbert space, and let
$\mathcal{P}(\Hilb)$ denote its lattice of orthogonal
projectors, equivalently the lattice of closed subspaces
ordered by inclusion. The lattice has a least element $0$ (the
zero projector / trivial subspace) and a greatest element $I$
(the identity / whole space); meet is intersection of
subspaces and join is the closed span of their union. Two
projectors $\Pi_1,\Pi_2$ are \emph{orthogonal}, written
$\Pi_1 \perp \Pi_2$, when $\Pi_1\Pi_2 = 0$; geometrically,
their ranges are mutually orthogonal subspaces.

\begin{definition}[Probability measure on projectors]
A function $\mu : \mathcal{P}(\Hilb) \to [0,1]$ is a
\emph{(countably additive) probability measure} if
\begin{align}
\mu(I) &= 1, \label{eq:measure_norm}\\
\mu\!\Big(\sum_{j} \Pi_j\Big)
   &= \sum_{j} \mu(\Pi_j)
   \quad\text{whenever } \Pi_i \perp \Pi_j \ (i\ne j),
   \label{eq:measure_sigma}
\end{align}
where the projector sum $\sum_j \Pi_j$ is the projector onto
the closed span of the ranges, and the index set $j$ is finite
or countably infinite.
\end{definition}

Condition \eqref{eq:measure_sigma} is the operational heart of
the matter. Physically a maximal set of mutually orthogonal
projectors summing to $I$ is a complete von Neumann
measurement: the outcomes are exclusive (orthogonality) and
exhaustive (they sum to $I$), so their probabilities must add.
The countable version is the genuine $\sigma$-additivity needed
for measurements with infinitely many outcomes, e.g. a position
or energy reading.

\begin{theorem}[Gleason 1957]\label{thm:Gleason}
If $\dim\Hilb \ge 3$, then for every probability measure $\mu$
satisfying \eqref{eq:measure_norm}--\eqref{eq:measure_sigma}
there exists a \emph{unique} positive trace-class operator
$\rho$ with $\Tr\rho = 1$ such that
\begin{equation}\label{eq:Gleason_restate}
\mu(\Pi) = \Tr(\rho\,\Pi)
\qquad\text{for all } \Pi \in \mathcal{P}(\Hilb).
\end{equation}
\end{theorem}

The dimension hypothesis $\dim\Hilb \ge 3$ is essential and is
discussed at length in \S\ref{app:Gleason:dim2}.

\subsection{Frame functions and Gleason's reduction}
\label{app:Gleason:frame}

Every rank-one projector is $\ketbra{v}{v}$ for a unit vector
$v \in \Hilb$, with $\ketbra{v}{v} = \ketbra{(\lambda v)}{(\lambda v)}$
for any phase $\lvert\lambda\rvert = 1$, so a rank-one projector
is the same datum as a point of the unit sphere
$\mathcal{S}(\Hilb)$ modulo overall phase. Restrict $\mu$ to
rank-one projectors and define
\begin{equation}\label{eq:frame_def}
f(v) \;:=\; \mu(\ketbra{v}{v}),
\qquad v \in \mathcal{S}(\Hilb).
\end{equation}
This $f$ inherits two properties from $\mu$. First it is
nonnegative, $f \ge 0$, since $\mu$ takes values in $[0,1]$.
Second, for any orthonormal basis $\{e_k\}$ of $\Hilb$ the
projectors $\ketbra{e_k}{e_k}$ are mutually orthogonal and sum
to $I$, so \eqref{eq:measure_norm}--\eqref{eq:measure_sigma}
give
\begin{equation}\label{eq:frame_function}
\sum_k f(e_k) \;=\; \sum_k \mu(\ketbra{e_k}{e_k})
\;=\; \mu(I) \;=\; 1.
\end{equation}

\begin{definition}[Frame function]
A \emph{frame function of weight $W$} on $\mathcal{S}(\Hilb)$
is a function $f : \mathcal{S}(\Hilb)\to\mathbb{R}$ such that
$\sum_k f(e_k) = W$ for \emph{every} orthonormal basis
$\{e_k\}$. We call $f$ \emph{regular} if it is the restriction
to the sphere of a (necessarily continuous) quadratic form on
$\Hilb$; explicitly, regular means
\begin{equation}\label{eq:regular}
f(v) = \langle v,\, A v\rangle
\end{equation}
for some bounded self-adjoint operator $A$.
\end{definition}

Equation \eqref{eq:frame_function} says precisely that the $f$
of \eqref{eq:frame_def} is a nonnegative frame function of
weight $1$. The decisive observation -- \emph{Gleason's
reduction} -- is that knowing $f$ on rays is enough to
reconstruct $\mu$ on \emph{all} projectors, not just rank-one
ones. Indeed, any projector $\Pi$ of rank $r$ (possibly
infinite) is a sum $\Pi = \sum_{k} \ketbra{g_k}{g_k}$ over an
orthonormal basis $\{g_k\}$ of its range, and by
\eqref{eq:measure_sigma}
\begin{equation}\label{eq:mu_from_f}
\mu(\Pi) \;=\; \sum_k \mu(\ketbra{g_k}{g_k})
\;=\; \sum_k f(g_k).
\end{equation}
For \eqref{eq:mu_from_f} to be consistent, the right-hand side
must be \emph{independent of the choice} of orthonormal basis
$\{g_k\}$ spanning the range of $\Pi$; the frame-function
property is exactly what guarantees this. (Take any two bases
of the range, extend each to a full orthonormal basis of
$\Hilb$ by appending a common basis of the orthogonal
complement, and subtract the two copies of
\eqref{eq:frame_function}.) Thus
\textbf{the entire problem reduces to classifying nonnegative
frame functions}: if every such $f$ is regular, i.e. of the
form \eqref{eq:regular}, then $\mu$ is determined through
\eqref{eq:mu_from_f}, and we shall see in
\S\ref{app:Gleason:trace} that \eqref{eq:mu_from_f} collapses
to the trace formula.

\subsection{The core lemma in three real dimensions}
\label{app:Gleason:lemma}

The technical core of Gleason's theorem is the following
statement about the ordinary $2$-sphere
$S^2\subset\mathbb{R}^3$.

\begin{lemma}[Gleason's regularity lemma]\label{lem:core}
Every nonnegative frame function $f$ on the unit sphere $S^2$
of $\mathbb{R}^3$ is regular: there is a real symmetric
$3\times 3$ matrix $A$ with
\begin{equation}\label{eq:lemma_quadratic}
f(v) = v^{\mathsf{T}} A\, v,
\qquad v\in S^2.
\end{equation}
\end{lemma}

We explain the argument in stages. It is genuinely the deep
part of the theorem, and we follow the structure of Gleason's
1957 proof \citep{Gleason1957}, filling in the geometric
intuition; the single hardest analytic step is isolated and its
content explained rather than reproduced in every epsilon.

\paragraph{Stage 0: why ``quadratic'' is the natural target.}
A frame function on $S^2$ is constrained on \emph{triples} of
mutually orthogonal unit vectors: for any orthonormal frame
$\{e_1,e_2,e_3\}$,
\begin{equation}\label{eq:triple_sum}
f(e_1)+f(e_2)+f(e_3) = W .
\end{equation}
A quadratic form $v^{\mathsf T} A v$ automatically satisfies
\eqref{eq:triple_sum} with $W = \Tr A$, because
$\sum_i e_i^{\mathsf T} A\, e_i = \Tr A$ is basis-independent.
So quadratic forms are frame functions; the lemma asserts the
converse, that \emph{nothing else} is. The miracle is that a
single algebraic identity holding over all orthonormal triples
pins down an a priori arbitrary function on the sphere.

\paragraph{Stage 1: parity and the reduced spectrum.}
Write $g(v) := f(v) - \tfrac{W}{3}$, a frame function of weight
$0$. Because antipodal points $v$ and $-v$ define the same
rank-one projector, $f(-v)=f(v)$, so $g$ is an even function on
the sphere. Hence in a spherical-harmonic expansion
\begin{equation}\label{eq:harmonic_expansion}
g(v) \;=\; \sum_{\ell\ \text{even}}\ \sum_{m=-\ell}^{\ell}
c_{\ell m}\, Y_\ell^m(v),
\end{equation}
only even angular momenta $\ell = 0,2,4,\dots$ can appear. (We
are getting ahead of ourselves by assuming $g$ is regular
enough to have such an expansion; Stage 4 supplies the missing
continuity. Granting it for now makes the mechanism
transparent.) A quadratic form, being a degree-$2$ homogeneous
polynomial restricted to the sphere, has spherical-harmonic
content only at $\ell = 0$ and $\ell = 2$. So
\textbf{Lemma~\ref{lem:core} is equivalent to the claim that
all coefficients $c_{\ell m}$ with $\ell \ge 4$ vanish.}

\paragraph{Stage 2: the orthogonality constraint kills
$\ell\ge 4$.} Fix the third frame vector along an arbitrary
direction $n\in S^2$ and let $\{e_1,e_2\}$ be an orthonormal
pair in the great circle $C_n$ perpendicular to $n$. Then
\eqref{eq:triple_sum} reads, for $g$,
\begin{equation}\label{eq:circle_constraint}
g(e_1) + g(e_2) = -\,g(n),
\qquad e_1,e_2 \perp n,\ e_1\perp e_2 .
\end{equation}
As we rotate the orthonormal pair $\{e_1,e_2\}$ by an angle
$\phi$ within the circle $C_n$, the \emph{left side must stay
constant} (it always equals $-g(n)$, which does not depend on
$\phi$). Restrict $g$ to the circle $C_n$ and write its
Fourier series in $\phi$; the pair $\{e_1,e_2\}$ sits at angles
$\phi$ and $\phi + \tfrac{\pi}{2}$, so
\begin{equation}\label{eq:fourier_pair}
g\big(e_1(\phi)\big) + g\big(e_2(\phi)\big)
= \sum_{p} a_p\big(e^{ip\phi} + e^{ip(\phi+\pi/2)}\big)
= \sum_{p} a_p\big(1 + e^{ip\pi/2}\big)e^{ip\phi}.
\end{equation}
For this to be independent of $\phi$, every Fourier mode with
$p\ne 0$ must have vanishing coefficient. The factor
$1+e^{ip\pi/2}$ vanishes for $p\equiv 2 \pmod 4$, but for
$p\equiv 0\pmod 4$ it equals $2\ne 0$, forcing $a_p = 0$ there;
modes $p$ odd are already absent because $g$ is even on the
circle. The upshot is that on \emph{every} great circle, the
restriction of $g$ contains only the Fourier modes
$p\in\{-2,0,2\}$. A function on $S^2$ whose restriction to
every great circle has angular content bounded by $\lvert
p\rvert\le 2$ is exactly a combination of $\ell\le 2$ spherical
harmonics: this is the classical statement that the $\ell\ge 4$
harmonics necessarily excite some great-circle Fourier mode
with $\lvert p\rvert\ge 4$. Hence $c_{\ell m}=0$ for
$\ell\ge 4$, and $g$ is a pure $\ell=2$ harmonic plus a
constant, i.e. a quadratic form. This is the algebraic engine
of the lemma.

\paragraph{Stage 3: positivity packages it as $A$.} A
real even degree-$2$ form on $S^2$ extends uniquely to a
homogeneous quadratic $v^{\mathsf T} A v$ with $A=A^{\mathsf T}$
real symmetric. Nonnegativity $f\ge 0$ means $v^{\mathsf T} A v
\ge 0$ for all $v$, so $A\succeq 0$. This establishes
\eqref{eq:lemma_quadratic}.

\paragraph{Stage 4: the regularity (continuity) step.} Stages
1--2 silently assumed $g$ admits a spherical-harmonic
expansion, i.e. that it is, say, square-integrable or
continuous. Gleason's real work is to prove this regularity
\emph{from the frame condition alone}, with no a priori
smoothness. The argument shows that a nonnegative frame
function is automatically continuous. The mechanism: the
frame condition forces a quantitative local oscillation bound.
Fix a point $u$ and a nearby point $u'$; one constructs
auxiliary orthonormal triples that contain $u$ and $u'$ in
controlled positions, and the weight-conservation identity
\eqref{eq:triple_sum}, combined with $f\ge 0$ (which caps each
term from below by $0$ and, via the sum, from above by $W$),
bounds $\lvert f(u)-f(u')\rvert$ by a quantity that goes to $0$
as $u'\to u$. Concretely, $0\le f\le W$ everywhere (each value
is sandwiched by nonnegativity of the other two terms in a
completing triple), so $f$ is bounded; a covering argument over
overlapping triples then upgrades boundedness to uniform
continuity. Once $f$ is continuous it is in $L^2(S^2)$, the
expansion \eqref{eq:harmonic_expansion} is legitimate, and
Stages 1--2 apply rigorously. This continuity-from-additivity
step is the single most delicate part of the proof; the reader
is referred to Gleason's original paper \citep{Gleason1957} and
to the streamlined modern exposition in
Busch--Lahti--Pellonp\"a\"a (2014) for the full epsilon
bookkeeping. Its \emph{content}, however, is exactly the
intuitive statement just given: orthogonality plus
nonnegativity leave no room for wild local behavior.

This completes Lemma~\ref{lem:core}. We emphasize where
$\dim = 3$ entered: the very existence of nontrivial
orthonormal \emph{triples}, used in
\eqref{eq:triple_sum}--\eqref{eq:circle_constraint}, requires
at least three real dimensions. In $\mathbb{R}^2$ there are
only orthonormal \emph{pairs}, the constraint
\eqref{eq:triple_sum} degenerates, and the argument collapses
(see \S\ref{app:Gleason:dim2}).

\subsection{Extension to all of $\Hilb$ and to the complex
field}
\label{app:Gleason:extend}

Lemma~\ref{lem:core} concerns a fixed real $3$-dimensional
space. Three extensions are needed to reach
Theorem~\ref{thm:Gleason}.

\paragraph{(a) From one $\mathbb{R}^3$ to all of a real
$\Hilb$.} Let $f$ be a nonnegative frame function on a real
Hilbert space of dimension $\ge 3$. For \emph{every}
three-dimensional subspace $V$, the restriction $f|_{S(V)}$ is
a nonnegative frame function on the $2$-sphere of $V$ (any
orthonormal triple of $V$ extends, by appending a fixed basis
of $V^\perp$, to a full basis, and the weight is shared), so by
Lemma~\ref{lem:core} there is a symmetric $A_V\succeq 0$ with
$f|_V = \langle v, A_V v\rangle$. The forms on overlapping
subspaces must agree on intersections because they compute the
same $f$; a standard gluing argument shows the local
$3\times 3$ forms patch into a single bounded self-adjoint
$A\succeq 0$ on all of $\Hilb$ with
\begin{equation}\label{eq:global_form}
f(v) = \langle v,\, A v\rangle
\qquad\text{for all } v\in\mathcal{S}(\Hilb).
\end{equation}
Boundedness of $A$ follows from $0\le f\le 1$ (so $A\preceq I$),
which also makes $A$ a legitimate operator on the whole space,
including infinite dimension.

\paragraph{(b) Real to complex.} A complex Hilbert space
$\Hilb$ of complex dimension $d$ is a real Hilbert space of
real dimension $2d$ under the inner product $\mathrm{Re}\,
\langle\cdot,\cdot\rangle$, and orthonormality in the complex
sense implies orthonormality in this real sense. So a
nonnegative frame function on complex $\Hilb$ is in particular
one on the underlying real space, and part (a) yields a real
symmetric (i.e. real-linear self-adjoint) $A$ with
\eqref{eq:global_form}. It remains to check $A$ is
\emph{complex}-linear, equivalently that $f(e^{i\alpha}v) =
f(v)$ -- which holds because $e^{i\alpha}v$ and $v$ span the
same complex ray, hence the same rank-one projector, hence the
same value of $f$. Phase invariance forces the real form $A$ to
commute with multiplication by $i$, i.e. to be a complex-linear
self-adjoint operator. Thus $f(v) = \langle v, A v\rangle$ with
$A=A^\dagger$, $A\succeq 0$ on the complex space.

\paragraph{(c) Separable infinite dimension.} The bound
$0\le A\le I$ from (a) gives a bounded positive operator. The
frame condition $\sum_k f(e_k) = \sum_k \langle e_k, A e_k
\rangle = 1$ says $A$ has finite trace equal to $1$; in
particular $A$ is trace-class, hence compact, and the
spectral/eigenbasis sum converges. Countable additivity
\eqref{eq:measure_sigma} is precisely what makes the
infinite sums $\sum_k f(e_k)$ converge to the trace rather than
to something basis-dependent. No separate argument is needed
beyond (a)--(b): once $A$ is bounded and the weight-$1$
condition pins $\Tr A = 1$, separability guarantees a countable
orthonormal basis on which everything is controlled.

\subsection{From the quadratic form to the trace formula}
\label{app:Gleason:trace}

Rename $A=:\rho$. By construction $\rho = \rho^\dagger$,
$\rho\succeq 0$, and
\begin{equation}\label{eq:rho_trace_one}
\Tr\rho \;=\; \sum_k \langle e_k,\rho\, e_k\rangle
\;=\; \sum_k f(e_k) \;=\; 1,
\end{equation}
so $\rho$ is a density operator. Now feed
\eqref{eq:global_form} back into the reduction
\eqref{eq:mu_from_f}. For any projector $\Pi$ with orthonormal
basis $\{g_k\}$ of its range,
\begin{equation}\label{eq:to_trace}
\mu(\Pi) \;=\; \sum_k f(g_k)
\;=\; \sum_k \langle g_k,\, \rho\, g_k\rangle
\;=\; \sum_k \langle g_k,\, \rho\,\Pi\, g_k\rangle
\;=\; \Tr(\rho\,\Pi),
\end{equation}
where the third equality uses $\Pi g_k = g_k$ (the $g_k$ lie in
the range of $\Pi$) and the last completes the basis of the
range to a basis of $\Hilb$ on which $\rho\Pi$ has vanishing
diagonal off the range (since $\Pi$ kills those vectors).
Equation \eqref{eq:to_trace} is exactly the asserted form
\eqref{eq:Gleason_restate}, i.e. \eqref{eq:Gleason_app}.

\paragraph{Uniqueness.} If $\Tr(\rho_1\Pi)=\Tr(\rho_2\Pi)$ for
every projector $\Pi$, then $\Tr\big((\rho_1-\rho_2)\Pi\big)=0$
for all $\Pi$. Taking $\Pi=\ketbra{v}{v}$ gives $\langle
v,(\rho_1-\rho_2)v\rangle = 0$ for every unit vector $v$; a
self-adjoint operator whose every diagonal matrix element
vanishes is the zero operator (polarize: $\langle u, B v\rangle$
is recovered from the diagonal of $B$ on $u\pm v$ and $u\pm
iv$). Hence $\rho_1=\rho_2$, proving uniqueness in
Theorem~\ref{thm:Gleason}. $\qed$

\subsection{The dimension-2 gap and its resolution (POVM
Gleason)}
\label{app:Gleason:dim2}

Theorem~\ref{thm:Gleason} \emph{fails} for $\dim\Hilb = 2$. The
obstruction is structural, not technical: on the qubit the
projective measurements are the antipodal pairs
$\{\ketbra{n}{n},\,\ketbra{-n}{-n}\}$ on the Bloch sphere, i.e.
each measurement only ever sees a single \emph{axis}
$n\in S^2$. The frame condition is then merely
\begin{equation}\label{eq:dim2_constraint}
f(n) + f(-n) = 1
\qquad\text{for every axis } n,
\end{equation}
with no further linkage between different axes, because in
$\mathbb{R}^2$ (the real form is the Bloch sphere) there are no
orthonormal \emph{triples} to import constraint
\eqref{eq:triple_sum}. Any function on the upper hemisphere
whatsoever, extended by $f(-n)=1-f(n)$, satisfies
\eqref{eq:dim2_constraint}. The vast majority of these are
\emph{not} quadratic forms; e.g. taking $f$ to depend on $n$
through a non-quadratic even function of latitude already
violates \eqref{eq:lemma_quadratic}. So at $d=2$ there are
frame functions, hence ``probability assignments,'' with no
density operator behind them, and uniqueness of $\rho$ is lost.

\paragraph{Resolution: enlarge the set of measurements to
POVMs.} A positive-operator-valued measure (POVM) is a family
$\{E_j\}$ of positive operators $E_j\succeq 0$ with
$\sum_j E_j = I$; projective measurements are the special case
$E_j = \Pi_j$ with $E_j^2 = E_j$ and $E_iE_j=0$. POVMs are the
genuinely available laboratory measurements (any apparatus with
ancilla and coarse-graining realizes one, by Naimark
dilation). A \emph{generalized probability measure} is now a
map $\nu$ on \emph{effects} (operators $0\preceq E\preceq I$)
that is additive over every POVM: $\sum_j \nu(E_j)=1$ whenever
$\sum_j E_j = I$, with $\nu(E)\ge 0$.

\begin{theorem}[Busch; Caves--Fuchs--Schack POVM
Gleason]\label{thm:Busch}
For \emph{every} dimension $d\ge 2$ (and indeed for separable
$\Hilb$), every generalized probability measure $\nu$ on the
effects of $\Hilb$ has the form
\begin{equation}\label{eq:Busch}
\nu(E) = \Tr(\rho\, E)
\end{equation}
for a unique density operator $\rho$.
\end{theorem}

The reason the POVM version restores uniqueness at $d=2$ is
that effects come in a \emph{continuum of orientations and
weights}, not just antipodal projector pairs, and the
additivity-over-POVMs condition links them all. Concretely, the
key extra structure is that one may now demand additivity for
families like $\{tE,\,(1-t)E,\,I-E\}$ and for non-orthogonal
effect triples; this forces $\nu$ to be \emph{linear} on the
effect space. Busch's insight is that additivity over POVMs
already forces $\nu(\alpha E_1 + \beta E_2) = \alpha\nu(E_1) +
\beta\nu(E_2)$ for the appropriate convex/affine combinations,
i.e. $\nu$ is the restriction of a linear functional on the
real vector space of self-adjoint operators. Every such
functional is $E\mapsto\Tr(\rho E)$ for a unique self-adjoint
$\rho$, and the constraints $\nu(E)\in[0,1]$, $\nu(I)=1$ force
$\rho\succeq 0$, $\Tr\rho=1$. Crucially this argument
\emph{never needs orthonormal triples}, so the dimension
hypothesis evaporates and $d=2$ is included. The proof is
markedly shorter and more elementary than Gleason's original,
trading the hard continuity lemma for the richer linear
structure of the effect space.

The physical reading is satisfying: the apparent $d=2$ loophole
was an artifact of pretending the only measurements are sharp
projective ones. Once the real (POVM) measurement set is
admitted -- which the operational dictionary of the present
paper does -- the Born rule is forced uniformly in all
dimensions $\ge 2$.

\subsection{Worked micro-example: the qubit Born rule}
\label{app:Gleason:qubit}

To see the abstract trace formula become the familiar
$\cos^2(\theta/2)$, take $\Hilb = \mathbb{C}^2$ for a spin-$1/2$
particle. Prepare the spin ``up along $\hat z$'' state
\begin{equation}\label{eq:qubit_state}
\rho = \ketbra{{\uparrow_z}}{{\uparrow_z}}
= \begin{pmatrix}1 & 0\\ 0 & 0\end{pmatrix},
\end{equation}
and measure spin along the direction $\hat n(\theta)$ in the
$x$--$z$ plane making angle $\theta$ with $\hat z$. The ``up
along $\hat n$'' eigenstate is
\begin{equation}\label{eq:qubit_proj_vec}
\ket{{\uparrow_n}}
= \cos\tfrac{\theta}{2}\,\ket{{\uparrow_z}}
+ \sin\tfrac{\theta}{2}\,\ket{{\downarrow_z}}
= \begin{pmatrix}\cos(\theta/2)\\ \sin(\theta/2)\end{pmatrix},
\end{equation}
with measurement projector
$\Pi_{\uparrow_n} = \ketbra{{\uparrow_n}}{{\uparrow_n}}$,
\begin{equation}\label{eq:qubit_proj}
\Pi_{\uparrow_n}
= \begin{pmatrix}
\cos^2(\theta/2) & \cos(\theta/2)\sin(\theta/2)\\[2pt]
\cos(\theta/2)\sin(\theta/2) & \sin^2(\theta/2)
\end{pmatrix}.
\end{equation}
Gleason's formula \eqref{eq:Gleason_app} then \emph{predicts}
the probability of the outcome ``up along $\hat n$'':
\begin{equation}\label{eq:qubit_born}
p(\uparrow_n)
= \Tr\big(\rho\,\Pi_{\uparrow_n}\big)
= \Tr\!\left[
\begin{pmatrix}1&0\\0&0\end{pmatrix}
\begin{pmatrix}
\cos^2\tfrac{\theta}{2} & \cdots\\ \cdots & \sin^2\tfrac{\theta}{2}
\end{pmatrix}\right]
= \cos^2\!\frac{\theta}{2}.
\end{equation}
This is exactly the textbook spin-$1/2$ result: the probability
of measuring spin-up along an axis tilted by $\theta$ from the
preparation axis is $\cos^2(\theta/2)$, falling from $1$ at
$\theta = 0$ (aligned) to $\tfrac12$ at $\theta=\pi/2$
(orthogonal axes) to $0$ at $\theta=\pi$ (anti-aligned),
consistent with $p(\uparrow_n)+p(\downarrow_n)=
\cos^2\tfrac{\theta}{2}+\sin^2\tfrac{\theta}{2}=1$. The
square-modulus pattern was never inserted by hand; it is the
trace formula evaluated on a rank-one projector, which is what
Gleason guarantees as the \emph{only} consistent probability
rule. (Strictly, $d=2$ here is the marginal case covered by the
POVM version, Theorem~\ref{thm:Busch}; for any larger system
the original Theorem~\ref{thm:Gleason} already applies.)

\subsection{Connection to the Born-rule theorem \TA{3}}
\label{app:Gleason:connection}

We can now state precisely how Gleason's theorem discharges the
Born rule in this paper. Theorem \TA{3} does not postulate
$p=\lvert\langle\psi\mid\phi\rangle\rvert^2$; it shows that form
to be \emph{forced}, and Gleason is the lemma that does the forcing once the
right premises are in place. Those premises are supplied
elsewhere in the framework, not assumed here:

\begin{itemize}
\item The state space is a \emph{complex separable Hilbert
space} $\Hilb$ -- this is the output of \TA{1}, so the
hypotheses of Theorem~\ref{thm:Gleason} (complex field,
separability, and $\dim\ge 3$ for any nontrivial system) are
met. The complex field in particular feeds the real-to-complex
step \S\ref{app:Gleason:extend}(b).
\item Measurement outcomes are mutually exclusive, exhaustive
projectors (or, more generally, effects), and their
probabilities are \emph{$\sigma$-additive} -- this is the
content of the operational measurement dictionary, which
furnishes exactly condition
\eqref{eq:measure_norm}--\eqref{eq:measure_sigma}.
\end{itemize}

Given only these two inputs -- a complex Hilbert space and
$\sigma$-additivity of outcome probabilities over orthogonal
(or POVM) decompositions of the identity -- Gleason's theorem
leaves \emph{no freedom}: the probability assignment
\emph{must} be $\mu(\Pi)=\Tr(\rho\Pi)$, and on pure states
$\rho=\ketbra{\psi}{\psi}$ and rank-one outcomes
$\Pi=\ketbra{\phi}{\phi}$ this is precisely the square-modulus
Born rule
\begin{equation}\label{eq:born_final}
\mu\big(\ketbra{\phi}{\phi}\big)
= \Tr\big(\ketbra{\psi}{\psi}\,\ketbra{\phi}{\phi}\big)
= \lvert\langle\phi\mid\psi\rangle\rvert^2 .
\end{equation}
Thus the role of the framework is to \emph{supply the
premises}, and the role of Gleason's theorem is to show those
premises are already sufficient: the square-modulus form is the
unique solution, not an extra axiom. This is exactly the
logical move asserted in \TA{3}, and the present appendix is its
complete justification.

\paragraph{Problems and worked solutions.}

\medskip\noindent\textbf{C.2.1. Build a frame function by hand and
verify its additivity on arbitrary bases.}

\noindent\emph{Solution.}  Step 1: take the candidate weight
$f(v) = \langle v, A v\rangle$ with $A = \mathrm{diag}(0.5, 0.3,
0.2)$ on $\mathbb{R}^3$ --- positive, unit trace.  Step 2: for the
standard basis, $f(e_1)+f(e_2)+f(e_3) = 0.5+0.3+0.2 = 1$; for any
other orthonormal basis $\{v_i\}$, additivity holds because
$\sum_i \langle v_i, A v_i\rangle = \mathrm{Tr}\,A = 1$ --- the
trace is basis-independent, so the frame condition
\eqref{eq:frame_function} is satisfied with weight $W = 1$ for
\emph{every} basis at once (spot-check a rotated basis numerically:
the sum returns $1.000$ to machine precision).  Step 3: the measure
it induces on projectors, $\mu(P) = \sum_{i \in P} f(v_i)$
\eqref{eq:mu_from_f}, is exactly $\mathrm{Tr}(A P)$ --- so
quadratic-form frame functions are automatically Gleason measures;
the theorem's force is the converse, proved in the surrounding
sections: \emph{nothing else} is.  Physics intuition: additivity
over bases looks innocent --- one number per direction, summing to
one --- but it is secretly a statement about \emph{every} way of
cutting the sphere into orthogonal triples simultaneously, and that
simultaneous consistency is the vice that squeezes all freedom out
of quantum probability.

\medskip\noindent\textbf{C.5.1. Walk one measure from quadratic form
to trace formula.}

\noindent\emph{Solution.}  Step 1: with the $A$ of C.2.1, evaluate
the induced measure on the two-dimensional subspace $P_{12}$ spanned
by $e_1, e_2$: $\mu(P_{12}) = f(e_1) + f(e_2) = 0.8$.  Step 2:
confirm against the trace form \eqref{eq:to_trace}:
$\mathrm{Tr}(A P_{12}) = A_{11} + A_{22} = 0.8$ --- and, per
\eqref{eq:rho_trace_one}, $A$ is positive with unit trace, i.e.\ a
legitimate density operator $\rho$.  Step 3: note what just
happened: a probability assignment defined only on \emph{subspaces}
(no amplitudes, no wavefunction anywhere) has been repackaged as a
state; this is the direction of travel of the whole theorem ---
states are not the input of quantum probability but its unique
bookkeeping.  Physics intuition: the density operator is Gleason's
receipt, not quantum mechanics' postulate --- given any lawful way
of assigning odds to questions, the trace formula is the only
filing system that can store it.

\medskip\noindent\textbf{C.6.1. Exhibit the dimension-2 escape, and
close it with the POVM extension.}

\noindent\emph{Solution.}  Step 1: in $d = 2$ every orthonormal
basis is an antipodal pair $\{v, v^\perp\}$, so the frame condition
\eqref{eq:dim2_constraint} demands only $f(v) + f(v^\perp) = 1$ ---
different bases never share a vector, so the constraints never
interlock.  Step 2: construct a lawless example: let
$f(v) = \tfrac12\big(1 + \mathrm{sgn}(n_z(v))\,n_z(v)^2\big)$ or any
assignment with $f(v) + f(-v)$-symmetry rigged pairwise --- it
satisfies additivity yet is no $\mathrm{Tr}(\rho\Pi)$ for any
$\rho$: Gleason's theorem is genuinely false for projective
measurements on a single qubit.  Step 3: the physical repair
\eqref{eq:Busch}: real instruments implement \emph{effects}
$0 \le E \le \mathbb 1$ (POVMs), which come in continuous families
that do interlock; Busch's theorem then forces
$\nu(E) = \mathrm{Tr}(\rho E)$ with a unique $\rho$ even at
$d = 2$ --- and the framework's instruments are POVM-valued
channels, so the escape hatch is closed where the physics actually
lives.  Physics intuition: the qubit's loophole is a geometry
accident --- antipodal pairs are too few handshakes to force
agreement; let the measurements tilt and weight continuously, as
laboratory instruments do, and the handshakes multiply until only
the Born rule survives.

\medskip\noindent\textbf{C.7.1. Run the qubit Born rule end to end
at a definite angle.}

\noindent\emph{Solution.}  Step 1: prepare $\rho =
\ketbra{\uparrow_z}{\uparrow_z}$ and measure spin along the axis
$\hat n$ at polar angle $\theta$; the projector is
\eqref{eq:qubit_proj}.  Step 2: the trace formula
\eqref{eq:qubit_born} gives $p(\uparrow_n) = \cos^2(\theta/2)$:
at $\theta = 60^\circ$, $p = \cos^2 30^\circ = 3/4$; at
$\theta = 90^\circ$, $p = 1/2$; at $180^\circ$, $p = 0$.  Step 3:
observe the division of labor: the \emph{form} of these numbers is
Gleason (via Busch at $d = 2$), while their \emph{physical
realization} --- why a silver atom lands in one spot with these
frequencies --- is the registration story of
Part~\ref{part:V}; the appendix supplies the arithmetic that the
worked example animates.  Physics intuition: $\cos^2(\theta/2)$ is
the most-verified formula in quantum physics, and its origin here
is bookkeeping rigidity, not wave mysticism --- half the angle
because a spinor turns half as fast as the laboratory, squared
because probability is the only additive filing system left.

\medskip\noindent\textbf{C.8.1. State exactly what Gleason proves,
and what the framework adds.}

\noindent\emph{Solution.}  Step 1: what the theorem proves
(\eqref{eq:Gleason_restate}): \emph{if} a noncontextual, countably
additive probability assignment on the subspace lattice exists in
$d \ge 3$, it is $\mathrm{Tr}(\rho\,\cdot)$ for a unique $\rho$ ---
uniqueness, not existence.  Step 2: what it does not prove: that
nature assigns probabilities to registered subspaces at all, that
the assignment is noncontextual, or anything in $d = 2$ without the
POVM upgrade.  Step 3: the framework's contribution (\TA{3}): the
\GDOS{} registration \emph{is} a probability assignment on
registered outcomes, and its noncontextuality is inherited from the
projection's geometry --- so the premises are supplied by physics
and the conclusion by Gleason; neither alone is the Born rule, and
the honesty register of the main text grades exactly this division.
Physics intuition: Gleason is a uniqueness trap waiting a century
for something to fall into it --- the projection walks in, and the
Born rule is the sound of the trap closing.


\GGtakeaway{\emph{Standard account:} the Born rule is a
postulate --- stated, used, never explained. \emph{What it left
open:} why $|\psi|^2$ and not anything else, and what breaks in
dimension two. \emph{The bulk reading:} the registration law of
the projection must assign noncontextual probabilities to
registered subspaces; Gleason's theorem then forces the trace
form with a unique $\rho$ --- the Born rule as the only arithmetic
the boundary could have had (\TA{3}), with the $d = 2$ escape
closed at the POVM level where the physical instruments live.}

\section{The GKSL theorem: statement and derivation}
\label{app:GKSL}

\GGteachbox{The most general law an open quantum system can obey:
the Gorini--Kossakowski--Sudarshan--Lindblad theorem, derived
completely --- complete positivity and the Kraus/Stinespring
representation, the semigroup hypothesis, the Choi matrix, and the
emergence of the canonical generator $\dot\rho = -i[H,\rho] +
\sum_k \gamma_k (L_k\rho L_k^\dagger - \tfrac12\{L_k^\dagger
L_k, \rho\})$ with non-negative rates. Every algebraic step is
shown at first-graduate-course level. For this cohort the theorem
is load-bearing: the boundary dynamics of the \OGGEP{} projection
is an open-system evolution by construction, and GKSL is the
unique form such a law can take --- the bulk is deterministic, and
\emph{this} is what determinism looks like after the trace.}

\GGreadalong{\citet{Lindblad1976} and \citet{GKS1976} for the
twin originals; \citet{BreuerPetruccione2002} for the standard
text on open quantum systems.}

This appendix gives a complete, self-contained derivation of the
Gorini--Kossakowski--Sudarshan--Lindblad (GKSL) theorem, which
characterizes the most general generator of a quantum dynamical
semigroup. The treatment assumes only a first graduate course in
quantum mechanics and linear algebra; every algebraic step is shown.
We work throughout in a finite-dimensional Hilbert space
$\Hilb \cong \mathbb{C}^d$, write $\rho$ for a density operator
(a positive semidefinite, unit-trace, Hermitian operator on
$\Hilb$), and use $\Tr$ for the trace. We keep $\hbar$ explicit in
the Hamiltonian term and set the dissipative rates with the
dimensions of inverse time.

\subsection{Setup: quantum dynamical semigroups and complete positivity}
\label{app:GKSL-setup}

\paragraph{States and the need for maps, not vectors.}
A closed quantum system in a pure state evolves by the Schrodinger
equation, $i\hbar\,\partial_t|\psi\rangle = H|\psi\rangle$, equivalently
by the unitary group $|\psi(t)\rangle = U_t|\psi(0)\rangle$ with
$U_t = e^{-iHt/\hbar}$. An \emph{open} system---one in contact with an
environment, or one whose fine degrees of freedom we have chosen to
coarse-grain away---generically does not stay pure. Its state must be
described by a density operator $\rho$, and its evolution by a linear
map on operators,
\begin{equation}\label{eq:GKSL-channel}
\rho(0) \;\longmapsto\; \rho(t) = \Phi_t\big[\rho(0)\big],
\end{equation}
where each $\Phi_t$ is a \emph{quantum channel}. For $\Phi_t$ to map
density operators to density operators for every input, it must be
(i) \emph{trace preserving} (TP), $\Tr\Phi_t[\rho] = \Tr\rho$, so
probabilities stay normalized, and (ii) \emph{positive}, mapping
positive operators to positive operators, so that $\rho(t)$ remains a
valid (nonnegative-probability) state.

\paragraph{Why positivity is not enough: complete positivity.}
Positivity alone is too weak. The system may be entangled with an
ancilla (a reference system $R$ of arbitrary dimension $n$) that does
not evolve. The joint state lives on $\Hilb \otimes \mathbb{C}^n$, and
the physical evolution acts as $\Phi_t \otimes \mathrm{id}_n$. We
require the joint output to remain a valid state for \emph{every} $n$.
A map $\Phi$ is \emph{completely positive} (CP) if
$\Phi \otimes \mathrm{id}_n$ is positive for all $n \ge 1$. Complete
positivity is strictly stronger than positivity: the transpose map is
positive but not completely positive, and applying it to half of a
maximally entangled pair produces an operator with a negative
eigenvalue. A physically admissible channel must therefore be
\CPTP{} (completely positive and trace preserving).

\paragraph{The Choi--Kraus representation.}
A finite-dimensional map $\Phi$ is completely positive if and only if
it admits a \emph{Kraus representation}
\begin{equation}\label{eq:GKSL-kraus}
\Phi[\rho] = \sum_{k} K_k\,\rho\,K_k^\dagger ,
\end{equation}
for some finite set of operators $K_k \in B(\Hilb)$; it is in
addition trace preserving if and only if the Kraus operators satisfy
the completeness (resolution-of-identity) relation
\begin{equation}\label{eq:GKSL-kraus-tp}
\sum_{k} K_k^\dagger K_k = \iden .
\end{equation}
The ``only if'' direction is Choi's theorem: complete positivity of
$\Phi$ is equivalent to positivity of the Choi matrix
$C(\Phi) = (\Phi \otimes \mathrm{id})\big[|\Omega\rangle\langle\Omega|\big]$,
where $|\Omega\rangle = \sum_i |i\rangle\otimes|i\rangle$ is the
unnormalized maximally entangled vector; the Kraus operators are read
off from the eigenvectors of $C(\Phi)$. The ``if'' direction is a
direct check: each term $K_k \rho K_k^\dagger$ preserves positivity,
the sum does too, and~\eqref{eq:GKSL-kraus-tp} guarantees
$\Tr\Phi[\rho] = \sum_k \Tr(K_k^\dagger K_k \rho) = \Tr(\iden\,\rho)
= \Tr\rho$.

\paragraph{The semigroup and Markovianity.}
A closed system has a one-parameter \emph{group} $U_t$ with
$U_{t+s} = U_t U_s$ and an inverse $U_{-t}$. An open system that loses
information to its environment is generically irreversible: it has no
inverse channel. What survives of the group law is the
\emph{semigroup} property
\begin{equation}\label{eq:GKSL-semigroup}
\Phi_{t+s} = \Phi_t \circ \Phi_s , \qquad t,s \ge 0 ,
\qquad \Phi_0 = \mathrm{id} .
\end{equation}
Physically, \eqref{eq:GKSL-semigroup} is the statement of
\emph{Markovianity}: the future depends on the present state alone, not
on the past history. The environment is assumed to forget the system's
influence on a timescale short compared to the system's own evolution
(a memoryless bath --- the Port-Freshness Clause of the \GDOS{} companion \citep{ArisakaA5GDOS}). Combined with strong continuity at $t=0$,
$\lim_{t\to 0^+}\Phi_t[\rho] = \rho$, the semigroup is generated by a
single time-independent superoperator $L$,
\begin{equation}\label{eq:GKSL-generator-def}
L \;=\; \lim_{t\to 0^+}\frac{\Phi_t - \mathrm{id}}{t},
\qquad
\Phi_t = e^{tL},
\qquad
\frac{d\rho}{dt} = L[\rho] .
\end{equation}
The central question is: \emph{which} superoperators $L$ generate a
legitimate \CPTP{} semigroup? GKSL answers it completely.

\subsection{Statement of the GKSL theorem}
\label{app:GKSL-statement}

\begin{quote}
\textbf{Theorem (GKSL, 1976).}
Let $\{\Phi_t\}_{t\ge 0}$ be a quantum dynamical semigroup on a
$d$-dimensional Hilbert space $\Hilb$: a strongly continuous,
one-parameter family of \CPTP{} maps satisfying
\eqref{eq:GKSL-semigroup}. Then its generator $L$
in~\eqref{eq:GKSL-generator-def} has the form
\begin{equation}\label{eq:GKSL-master}
\frac{d\rho}{dt}
= -\frac{i}{\hbar}\,[H,\rho]
+ \sum_{k} \gamma_k\left(
L_k\,\rho\,L_k^\dagger
- \tfrac{1}{2}\big\{ L_k^\dagger L_k,\,\rho \big\}
\right),
\end{equation}
with $H = H^\dagger$ a Hermitian operator (the effective
Hamiltonian), rates $\gamma_k \ge 0$, and operators $L_k \in B(\Hilb)$
(the \emph{Lindblad} or \emph{jump} operators). Conversely, every $L$
of the form~\eqref{eq:GKSL-master} with $\gamma_k\ge 0$ generates a
\CPTP{} semigroup.
\end{quote}

\noindent The double-bracket notation is the anticommutator,
$\{A,B\} = AB + BA$, and the single bracket is the commutator,
$[A,B] = AB - BA$. The first term on the right of
\eqref{eq:GKSL-master} is the unitary (von Neumann) part; the sum is
the \emph{dissipator}, denoted $\mathcal{D}[\rho]$. The decomposition
\eqref{eq:GKSL-master} is unique only up to a gauge freedom discussed
in \S\ref{app:GKSL-uniqueness}.

\subsection{Derivation via the Kossakowski matrix}
\label{app:GKSL-kossakowski}

We now derive~\eqref{eq:GKSL-master} algebraically. The strategy is:
expand the generator in a fixed operator basis; show that trace
preservation and Hermiticity constrain the coefficients into a single
Hermitian \emph{Kossakowski matrix}; show that complete positivity is
equivalent to that matrix being positive semidefinite; then
diagonalize it to read off the Lindblad operators.

\paragraph{Step 1: a complete operator basis.}
The space $B(\Hilb)$ of operators on $\Hilb \cong \mathbb{C}^d$ is
itself a $d^2$-dimensional complex vector space, with the
Hilbert--Schmidt inner product $\langle A,B\rangle =
\Tr(A^\dagger B)$. Choose an orthonormal basis $\{F_m\}_{m=0}^{d^2-1}$,
\begin{equation}\label{eq:GKSL-basis}
\Tr\!\big(F_m^\dagger F_n\big) = \delta_{mn},
\end{equation}
in which we single out the (normalized) identity
\begin{equation}\label{eq:GKSL-F0}
F_0 = \frac{1}{\sqrt{d}}\,\iden,
\qquad\text{so that}\qquad
\Tr\!\big(F_m^\dagger F_0\big)
= \frac{1}{\sqrt{d}}\Tr F_m = 0 \quad (m \neq 0).
\end{equation}
Thus the remaining $d^2-1$ basis operators $F_1,\dots,F_{d^2-1}$ are
all \emph{traceless}. (For a qubit, $d=2$, one may take
$F_0 = \iden/\sqrt 2$ and $F_{1,2,3} = \sigma_{x,y,z}/\sqrt 2$.)

\paragraph{Step 2: the generator in Choi--Kraus form for small $t$.}
By Choi's theorem the channel for a small time step $t$ has a Kraus
form $\Phi_t[\rho] = \sum_k K_k(t)\,\rho\,K_k(t)^\dagger$. Expand each
Kraus operator in the basis~\eqref{eq:GKSL-basis},
$K_k(t) = \sum_{m} c_{km}(t)\,F_m$, and substitute:
\begin{equation}\label{eq:GKSL-expand}
\Phi_t[\rho]
= \sum_{m,n=0}^{d^2-1}
\Big(\underbrace{\textstyle\sum_k c_{km}(t)\,c_{kn}^*(t)}_{\displaystyle
\equiv\, \chi_{mn}(t)}\Big)\,
F_m\,\rho\,F_n^\dagger
= \sum_{m,n} \chi_{mn}(t)\, F_m\,\rho\,F_n^\dagger .
\end{equation}
The matrix $\chi(t)$ with entries $\chi_{mn}(t) = \sum_k c_{km}\,
c_{kn}^*$ is, by construction, a Gram matrix of the coefficient
vectors, hence \emph{Hermitian and positive semidefinite} for every
$t$: for any complex vector $v$,
$\sum_{mn} v_m^*\chi_{mn}v_n = \sum_k |\sum_m v_m^* c_{km}|^2 \ge 0$.
This positivity of $\chi(t)$ is exactly the algebraic content of
complete positivity~\eqref{eq:GKSL-kraus}; it is the fact we will
propagate to the generator.

\paragraph{Step 3: take the time derivative.}
At $t=0$ the channel is the identity, $\Phi_0 = \mathrm{id}$, which in
the form~\eqref{eq:GKSL-expand} means $\chi_{00}(0) = d$ and all other
$\chi_{mn}(0)=0$ (because $F_0\,\rho\,F_0^\dagger = \rho/d$). Expanding
to first order, $\chi_{mn}(t) = \chi_{mn}(0) + t\,\dot\chi_{mn}(0) +
O(t^2)$, and using~\eqref{eq:GKSL-generator-def},
\begin{equation}\label{eq:GKSL-Lform-raw}
L[\rho] = \frac{d}{dt}\Big|_{0}\Phi_t[\rho]
= \dot\chi_{00}(0)\,\frac{\rho}{d}
+ \frac{1}{\sqrt d}\sum_{m\ge 1}\Big(\dot\chi_{m0}(0)\,F_m\,\rho
+ \dot\chi_{0m}(0)\,\rho\,F_m^\dagger\Big)
+ \sum_{m,n\ge 1}\dot\chi_{mn}(0)\,F_m\,\rho\,F_n^\dagger,
\end{equation}
where we used $F_0 = \iden/\sqrt d$. Introduce the abbreviations
\begin{equation}\label{eq:GKSL-abbrev}
a_{mn} \equiv \dot\chi_{mn}(0)\ (m,n\ge1),
\qquad
b \equiv \frac{1}{\sqrt d}\sum_{m\ge1}\dot\chi_{m0}(0)\,F_m,
\qquad
a_{00} \equiv \dot\chi_{00}(0).
\end{equation}
The matrix $a = (a_{mn})_{m,n\ge1}$ is Hermitian
($a_{mn} = a_{nm}^*$, inherited from $\chi=\chi^\dagger$); $b$ is an
operator; $a_{00}$ is real. Equation~\eqref{eq:GKSL-Lform-raw} becomes
\begin{equation}\label{eq:GKSL-Lform-grouped}
L[\rho] = \frac{a_{00}}{d}\,\rho + \big(b\,\rho + \rho\,b^\dagger\big)
+ \sum_{m,n\ge1} a_{mn}\,F_m\,\rho\,F_n^\dagger .
\end{equation}

\paragraph{Step 4: impose trace preservation.}
The semigroup is trace preserving for all $t$, so
$\Tr L[\rho] = 0$ for every $\rho$. Take the trace
of~\eqref{eq:GKSL-Lform-grouped}, using cyclicity
$\Tr(F_m\rho F_n^\dagger) = \Tr(F_n^\dagger F_m\,\rho)$:
\begin{equation}\label{eq:GKSL-traceless}
0 = \Tr L[\rho]
= \Tr\!\left[\Big(\frac{a_{00}}{d}\iden + b + b^\dagger
+ \sum_{m,n\ge1} a_{mn}\,F_n^\dagger F_m \Big)\rho\right]
\quad\text{for all }\rho.
\end{equation}
Since this holds for all $\rho$, the bracketed operator must vanish:
\begin{equation}\label{eq:GKSL-trace-constraint}
\frac{a_{00}}{d}\,\iden + b + b^\dagger
+ \sum_{m,n\ge1} a_{mn}\,F_n^\dagger F_m = 0 .
\end{equation}
Define the Hermitian operator
$G \equiv \tfrac12\sum_{m,n\ge1} a_{mn}\,F_n^\dagger F_m
+ \tfrac{a_{00}}{2d}\iden$. Then~\eqref{eq:GKSL-trace-constraint} reads
$b + b^\dagger = -2G$, i.e.\ the Hermitian part of $b$ is fixed:
$\tfrac12(b+b^\dagger) = -G$. Split $b$ into Hermitian and
anti-Hermitian parts by writing
\begin{equation}\label{eq:GKSL-bsplit}
b = -G - \frac{i}{\hbar}\,H ,
\qquad
H \equiv \frac{i\hbar}{2}\,\big(b - b^\dagger\big) = H^\dagger,
\end{equation}
which defines a Hermitian operator $H$ (the factor $-i/\hbar$ is a
convention chosen so that $H$ has units of energy and the unitary term
matches the Schrodinger form).

\paragraph{Step 5: assemble the standard form.}
Substitute $b = -G - (i/\hbar)H$ into
\eqref{eq:GKSL-Lform-grouped}. The pieces involving $G$ and $a_{00}$
combine into the anticommutator structure. Explicitly,
$b\rho + \rho b^\dagger = -\tfrac{i}{\hbar}[H,\rho] - (G\rho+\rho G)$,
and writing
$G = \tfrac12\sum_{m,n\ge1}a_{mn}F_n^\dagger F_m + (a_{00}/2d)\iden$
gives
\begin{equation}\label{eq:GKSL-diamond}
L[\rho]
= -\frac{i}{\hbar}[H,\rho]
+ \sum_{m,n\ge1} a_{mn}\!\left(
F_m\,\rho\,F_n^\dagger
- \tfrac12\big\{ F_n^\dagger F_m,\,\rho \big\}\right).
\end{equation}
The $a_{00}\iden/d$ contributions from the $\rho/d$ term and from $G$
cancel identically (they enter as $+a_{00}\rho/d$ from the first term
of~\eqref{eq:GKSL-Lform-grouped} and as
$-\tfrac12\{(a_{00}/d)\iden,\rho\} = -a_{00}\rho/d$ from $G$). This is
the GKSL form in its \emph{diagonal-free} (Kossakowski) version: a
unitary part plus a dissipator built from the
\emph{Kossakowski matrix} $a = (a_{mn})_{m,n\ge1}$, which is Hermitian
and---this is the key fact---positive semidefinite.

\paragraph{Step 6: complete positivity $\Longleftrightarrow$
$a \succeq 0$.}
We must show that the Kossakowski matrix is positive semidefinite,
$a \succeq 0$, and that this is precisely the condition for the
semigroup $e^{tL}$ to be completely positive. One direction is
immediate: $a_{mn} = \dot\chi_{mn}(0)$ is the time derivative of the
positive-semidefinite Choi block $\chi(t)$ restricted to the
traceless sector, evaluated where the off-diagonal entries start from
zero; positivity of $\chi(t)$ for $t>0$ together with $\chi_{mn}(0)=0$
($m,n\ge1$) forces the derivative matrix to be positive semidefinite.
(Concretely: for the principal submatrix on indices $m,n\ge1$, the
$t=0$ value vanishes and $\chi(t)\succeq0$ for $t\ge0$ implies the
right-derivative $\succeq 0$.) The converse---that $a\succeq0$
guarantees $\Phi_t=e^{tL}$ is \CPTP{} for all $t\ge0$---follows once we
exhibit the Kraus form below and note that the construction holds at
every $t$.

\paragraph{Step 7: diagonalize to obtain the Lindblad operators.}
Because $a$ is Hermitian and positive semidefinite, the spectral
theorem gives a unitary $U = (u_{mk})$ and nonnegative eigenvalues
$\gamma_k\ge0$ with
\begin{equation}\label{eq:GKSL-diag}
a_{mn} = \sum_k u_{mk}\,\gamma_k\,u_{nk}^* .
\end{equation}
Define new operators by the same unitary rotation of the basis,
\begin{equation}\label{eq:GKSL-Ldef}
L_k \equiv \sum_{m\ge1} u_{mk}\,F_m .
\end{equation}
Substituting~\eqref{eq:GKSL-diag} into~\eqref{eq:GKSL-diamond} and
collecting the rotated operators,
\begin{align}
\sum_{m,n\ge1} a_{mn}\,F_m\rho F_n^\dagger
&= \sum_k \gamma_k \Big(\textstyle\sum_m u_{mk}F_m\Big)\,\rho\,
\Big(\textstyle\sum_n u_{nk}F_n\Big)^\dagger
= \sum_k \gamma_k\, L_k\,\rho\,L_k^\dagger,
\label{eq:GKSL-collect-1}\\
\sum_{m,n\ge1} a_{mn}\,F_n^\dagger F_m
&= \sum_k \gamma_k\, L_k^\dagger L_k .
\label{eq:GKSL-collect-2}
\end{align}
Therefore~\eqref{eq:GKSL-diamond} becomes exactly the GKSL
master equation~\eqref{eq:GKSL-master},
\begin{equation}\label{eq:GKSL-final}
\frac{d\rho}{dt}
= -\frac{i}{\hbar}[H,\rho]
+ \sum_k \gamma_k\Big( L_k\rho L_k^\dagger
- \tfrac12\{L_k^\dagger L_k,\rho\}\Big),
\qquad \gamma_k \ge 0 .
\end{equation}
This completes the derivation: \emph{every} generator of a \CPTP{}
semigroup has this form, and the nonnegativity $\gamma_k\ge0$ is the
algebraic fingerprint of complete positivity.

\paragraph{Step 8: the converse and an explicit Kraus check.}
For the converse, expand the short-time map generated
by~\eqref{eq:GKSL-final}, $\Phi_t = \iden + tL + O(t^2)$, into Kraus
form. Define
\begin{equation}\label{eq:GKSL-kraus-check}
K_0 = \iden + \frac{t}{\hbar}\Big(-iH
- \tfrac{\hbar}{2}\textstyle\sum_k \gamma_k L_k^\dagger L_k\Big),
\qquad
K_k = \sqrt{\gamma_k\,t}\;L_k \ (k\ge1).
\end{equation}
Then $\sum_{k\ge0}K_k\rho K_k^\dagger = \rho + tL[\rho] + O(t^2)$,
reproducing the generator, while
$\sum_k K_k^\dagger K_k = \iden + O(t^2)$ confirms trace preservation
to first order. Because each $K_k\rho K_k^\dagger$ is manifestly
completely positive and the construction holds for arbitrarily small
$t$, the semigroup law $\Phi_t = (\Phi_{t/N})^N$ assembles a
completely positive $\Phi_t$ for all finite $t$. This closes the
equivalence.

\subsection{The gauge (non-uniqueness) of the GKSL form}
\label{app:GKSL-uniqueness}

The decomposition~\eqref{eq:GKSL-final} is not literally unique, but
its non-uniqueness is fully understood and harmless. Two freedoms
exist. First, the dissipator is invariant under a unitary mixing of
the Lindblad operators, $L_k \to \sum_j v_{kj} L_j$ with $v$ unitary
(this just re-diagonalizes the same Kossakowski matrix). Second, one
may shift each Lindblad operator by a multiple of the identity,
$L_k \to L_k + c_k\iden$, and simultaneously add a compensating term
to $H$; the dissipator changes only by a commutator that can be
absorbed into the Hamiltonian. Fixing the $L_k$ to be \emph{traceless}
(as our basis choice $F_0=\iden/\sqrt d$ does, since each $L_k$ is a
combination of the traceless $F_{m\ge1}$) removes the second freedom
and makes $H$ and the spectrum $\{\gamma_k\}$ canonical. Thus, up to
this gauge, GKSL is unique: \emph{any} generator with a different
structure---for instance one with a single term $\alpha\,L\rho
L^\dagger$ and no anticommutator, or with $\alpha<0$---necessarily
violates either trace preservation (the missing anticommutator) or
complete positivity (a negative Kossakowski eigenvalue), or both.

\subsection{Microscopic derivation: weak coupling, Born--Markov, and
the secular approximation}
\label{app:GKSL-microscopic}

The Kossakowski route shows GKSL is the \emph{only} admissible
Markovian form. A complementary, physically transparent derivation
starts from a concrete system--bath Hamiltonian and shows how the GKSL
structure \emph{emerges}---with the rates $\gamma_k$ fixed by the bath
spectrum---once a sequence of controlled approximations is made.

\paragraph{The model.}
Let the total Hamiltonian be
\begin{equation}\label{eq:GKSL-Htot}
H_{\rm tot} = H_S + H_B + H_{SB},
\qquad
H_{SB} = \sum_\alpha A_\alpha \otimes B_\alpha ,
\end{equation}
with $H_S$ the system Hamiltonian, $H_B$ the bath (environment)
Hamiltonian, and $H_{SB}$ a weak coupling written as a sum of
products of system operators $A_\alpha = A_\alpha^\dagger$ and bath
operators $B_\alpha = B_\alpha^\dagger$. We work in the interaction
picture, where operators carry tildes,
$\tilde A_\alpha(t) = e^{iH_S t/\hbar}A_\alpha e^{-iH_S t/\hbar}$, and
the von Neumann equation for the joint state $\tilde\rho_{\rm tot}$ is
\begin{equation}\label{eq:GKSL-vN}
\frac{d\tilde\rho_{\rm tot}}{dt}
= -\frac{i}{\hbar}\big[\tilde H_{SB}(t),\,\tilde\rho_{\rm tot}(t)\big].
\end{equation}

\paragraph{Second-order (Born) integro-differential equation.}
Integrate~\eqref{eq:GKSL-vN} once, $\tilde\rho_{\rm tot}(t) =
\tilde\rho_{\rm tot}(0) - \tfrac{i}{\hbar}\int_0^t ds\,
[\tilde H_{SB}(s),\tilde\rho_{\rm tot}(s)]$, substitute back into
\eqref{eq:GKSL-vN}, and trace out the bath
($\rho_S = \Tr_B\rho_{\rm tot}$). Assuming the coupling has zero bath
average ($\Tr_B[B_\alpha\rho_B]=0$, which can always be arranged by
absorbing the mean into $H_S$), the first-order term vanishes and
\begin{equation}\label{eq:GKSL-born-exact}
\frac{d\tilde\rho_S}{dt}
= -\frac{1}{\hbar^2}\int_0^t ds\;
\Tr_B\Big[\tilde H_{SB}(t),
\big[\tilde H_{SB}(s),\,\tilde\rho_{\rm tot}(s)\big]\Big].
\end{equation}

\paragraph{The Born approximation (weak coupling).}
For weak coupling the bath is barely perturbed and the system--bath
state stays approximately factorized at all times,
$\tilde\rho_{\rm tot}(s)\approx \tilde\rho_S(s)\otimes\rho_B$, with
$\rho_B$ a stationary bath state (e.g.\ thermal). This is the
\emph{Born approximation}; it is the leading term of a systematic
expansion in the coupling strength and holds when the coupling energy
is small compared with the bath correlation rate. Inserting it,
\begin{equation}\label{eq:GKSL-born}
\frac{d\tilde\rho_S}{dt}
= -\frac{1}{\hbar^2}\int_0^t ds\;
\Tr_B\Big[\tilde H_{SB}(t),
\big[\tilde H_{SB}(s),\,\tilde\rho_S(s)\otimes\rho_B\big]\Big].
\end{equation}
Carrying out the bath trace produces bath two-point correlation
functions
\[
C_{\alpha\beta}(t-s)=\Tr_B\!\left[\tilde B_\alpha(t)\,\tilde B_\beta(s)\,\rho_B\right],
\]
which depend only on the time difference because $\rho_B$ is
stationary.

\paragraph{The Markov approximation (memoryless bath).}
The bath correlations $C_{\alpha\beta}(\tau)$ decay on a short
\emph{bath correlation time} $\tau_B$ --- in cohort terms, the
Port-Freshness Clause of the \GDOS{} companion. If $\tau_B$ is much shorter than
the timescale $\tau_S$ on which $\rho_S$ evolves, we may (i) replace
$\tilde\rho_S(s)\to\tilde\rho_S(t)$ in the integrand (the state is
essentially constant over the memory window) and (ii) extend the
integration limit to infinity, $\int_0^t\to\int_0^\infty$. This is the
\emph{Markov approximation}; together with the Born step it yields the
\emph{Redfield equation},
\begin{equation}\label{eq:GKSL-redfield}
\frac{d\tilde\rho_S}{dt}
= -\frac{1}{\hbar^2}\int_0^\infty d\tau\;
\Tr_B\Big[\tilde H_{SB}(t),
\big[\tilde H_{SB}(t-\tau),\,\tilde\rho_S(t)\otimes\rho_B\big]\Big].
\end{equation}
The Redfield equation is local in time and Markovian, but---crucially
---it is \emph{not yet} of GKSL form: it can violate complete
positivity and produce (slightly) negative probabilities. The missing
ingredient is the secular approximation.

\paragraph{The secular / rotating-wave approximation.}
Decompose each system coupling operator into eigenoperators of $H_S$.
If $H_S = \sum_\epsilon \epsilon\,\Pi_\epsilon$ (with $\Pi_\epsilon$
the projector onto the energy-$\epsilon$ eigenspace), define
\begin{equation}\label{eq:GKSL-eigenop}
A_\alpha(\omega) \equiv
\sum_{\epsilon'-\epsilon=\hbar\omega}
\Pi_\epsilon\,A_\alpha\,\Pi_{\epsilon'},
\qquad
\tilde A_\alpha(\omega,t) = e^{-i\omega t}A_\alpha(\omega),
\end{equation}
so the sum over \emph{Bohr frequencies} $\omega$ (energy gaps divided
by $\hbar$) diagonalizes the free interaction-picture evolution. In
the Redfield equation each term carries a phase
$e^{i(\omega'-\omega)t}$. Terms with $\omega'\neq\omega$ oscillate
rapidly compared with the slow dissipative dynamics and average to
zero over the relevant timescale; dropping them is the
\emph{secular}, or rotating-wave, approximation, valid when the level
spacings satisfy $|\omega'-\omega|^{-1}\ll\tau_S$ (well-resolved Bohr
frequencies). Keeping only $\omega'=\omega$ gives
\begin{equation}\label{eq:GKSL-secular}
\frac{d\tilde\rho_S}{dt}
= \sum_{\omega}\sum_{\alpha\beta}
\gamma_{\alpha\beta}(\omega)
\Big(
A_\beta(\omega)\,\tilde\rho_S\,A_\alpha^\dagger(\omega)
- \tfrac12\big\{A_\alpha^\dagger(\omega)A_\beta(\omega),
\,\tilde\rho_S\big\}\Big)
- \frac{i}{\hbar}[H_{LS},\tilde\rho_S],
\end{equation}
where $H_{LS}$ is the \emph{Lamb shift} (a coupling-induced
renormalization of $H_S$ that commutes with $H_S$), and the rate
matrix is the one-sided Fourier transform of the bath correlations,
\begin{equation}\label{eq:GKSL-rates}
\gamma_{\alpha\beta}(\omega)
= \frac{1}{\hbar^2}\int_{-\infty}^{\infty} d\tau\;
e^{i\omega\tau}\,C_{\alpha\beta}(\tau)
= \frac{1}{\hbar^2}\int_{-\infty}^\infty d\tau\;
e^{i\omega\tau}\,\Tr_B\!\big[\tilde B_\alpha(\tau)B_\beta\,\rho_B\big].
\end{equation}
For each fixed $\omega$ the matrix $\gamma_{\alpha\beta}(\omega)$ is
positive semidefinite---this is a theorem (Bochner's theorem) about
the Fourier transform of a correlation function, and it is precisely
the Kossakowski-positivity condition of
\S\ref{app:GKSL-kossakowski}. The secular approximation yields a generator that is manifestly of
GKSL form, and hence completely positive: it converts the not-quite-CP
Redfield generator into a genuine GKSL generator. (It is
\emph{sufficient} for complete positivity, not necessary --- non-secular
CP-preserving schemes also exist --- but it is the standard and simplest
route.) Diagonalizing
$\gamma_{\alpha\beta}(\omega)\to\gamma_k$ as in
\eqref{eq:GKSL-diag}--\eqref{eq:GKSL-Ldef} recovers exactly the
canonical form~\eqref{eq:GKSL-final}, with the rates given by the
\textbf{Fermi golden rule}: the rate of a given quantum jump is set by
the bath spectral density evaluated at the corresponding Bohr
frequency,
\begin{equation}\label{eq:GKSL-golden}
\gamma(\omega) \;\propto\; \text{(bath spectral density at }\omega).
\end{equation}
Transforming back from the interaction picture reinstates the bare
Hamiltonian, so that the full generator is $H = H_S + H_{LS}$ together
with the dissipator built from the $A_\alpha(\omega)$.

\paragraph{Summary of the approximation chain.}
The three approximations have distinct physical content and validity
windows: \emph{Born} (weak coupling, factorized state) needs small
coupling; \emph{Markov} (memoryless bath; the Port-Freshness Clause of the \GDOS{} companion) needs $\tau_B\ll\tau_S$;
\emph{secular} (drop fast-oscillating cross terms) needs well-resolved
Bohr frequencies, $|\omega-\omega'|^{-1}\ll\tau_S$. Only with all
three does the microscopic dynamics collapse onto the unique GKSL
form, with positivity guaranteed.

\subsection{Action of the dissipator: decoherence}
\label{app:GKSL-decoherence}

The dissipator $\mathcal{D}[\rho] = \sum_k\gamma_k(L_k\rho L_k^\dagger
- \tfrac12\{L_k^\dagger L_k,\rho\})$ is what distinguishes open from
closed dynamics. To see its physical effect, write $\rho$ in a
preferred basis $\{|n\rangle\}$ (for instance the eigenbasis of $H$ or
of a $L_k$) and look at matrix elements $\rho_{nm} =
\langle n|\rho|m\rangle$. The \emph{diagonal} elements $\rho_{nn}$ are
populations (classical probabilities); the \emph{off-diagonal}
elements $\rho_{nm}$ ($n\neq m$) are \emph{coherences}, the carriers
of quantum interference. A generic dissipator drives the coherences to
zero while merely redistributing the populations: the off-diagonal
elements decay exponentially, $\rho_{nm}(t)\sim
e^{-\Gamma_{nm}t}\rho_{nm}(0)$, with a \emph{decoherence rate}
$\Gamma_{nm}$ set by the $\gamma_k$ and the $L_k$ matrix elements.
This selective suppression of coherences---decoherence---is how the
GKSL semigroup encodes the emergence of effectively classical,
probabilistic behavior from unitary quantum mechanics, without ever
violating trace or positivity.

\subsection{Worked micro-example: the single-qubit dephasing channel}
\label{app:GKSL-qubit}

Take $\Hilb = \mathbb{C}^2$ (a qubit), set $H = 0$ for clarity, and
use a single Lindblad operator $L = \sigma_z$ with rate $\gamma$. The
GKSL master equation~\eqref{eq:GKSL-final} reduces to the pure
dephasing generator
\begin{equation}\label{eq:GKSL-qubit-master}
\frac{d\rho}{dt}
= \gamma\Big(\sigma_z\,\rho\,\sigma_z
- \tfrac12\{\sigma_z^\dagger\sigma_z,\rho\}\Big)
= \gamma\big(\sigma_z\,\rho\,\sigma_z - \rho\big),
\end{equation}
where we used $\sigma_z^\dagger = \sigma_z$ and $\sigma_z^2 = \iden$,
so $\tfrac12\{\sigma_z^2,\rho\}=\tfrac12\{\iden,\rho\}=\rho$.
Parametrize the qubit state as
\begin{equation}\label{eq:GKSL-qubit-rho}
\rho = \begin{pmatrix} \rho_{00} & \rho_{01}\\[2pt]
\rho_{10} & \rho_{11}\end{pmatrix},
\qquad
\sigma_z = \begin{pmatrix} 1 & 0\\ 0 & -1\end{pmatrix}.
\end{equation}
A direct multiplication gives
\begin{equation}\label{eq:GKSL-qubit-conj}
\sigma_z\,\rho\,\sigma_z
= \begin{pmatrix} \rho_{00} & -\rho_{01}\\[2pt]
-\rho_{10} & \rho_{11}\end{pmatrix},
\end{equation}
i.e.\ conjugation by $\sigma_z$ flips the sign of the off-diagonal
entries and leaves the diagonal untouched. Substituting into
\eqref{eq:GKSL-qubit-master},
\begin{equation}\label{eq:GKSL-qubit-components}
\frac{d}{dt}
\begin{pmatrix}\rho_{00}&\rho_{01}\\\rho_{10}&\rho_{11}\end{pmatrix}
= \gamma\!\begin{pmatrix}
\rho_{00}-\rho_{00} & -\rho_{01}-\rho_{01}\\[2pt]
-\rho_{10}-\rho_{10} & \rho_{11}-\rho_{11}
\end{pmatrix}
= \begin{pmatrix}
0 & -2\gamma\,\rho_{01}\\[2pt]
-2\gamma\,\rho_{10} & 0
\end{pmatrix}.
\end{equation}
Hence the populations are frozen and the coherences decay
exponentially:
\begin{equation}\label{eq:GKSL-qubit-solution}
\dot\rho_{00}=\dot\rho_{11}=0,
\qquad
\dot\rho_{01} = -2\gamma\,\rho_{01}
\;\;\Longrightarrow\;\;
\boxed{\;\rho_{01}(t) = \rho_{01}(0)\,e^{-2\gamma t}\;}
\end{equation}
and likewise $\rho_{10}(t)=\rho_{10}(0)\,e^{-2\gamma t}$. Writing the
decoherence rate as $\Gamma = 2\gamma$, the
solution is
\begin{equation}\label{eq:GKSL-qubit-final}
\rho(t) =
\begin{pmatrix}
\rho_{00}(0) & \rho_{01}(0)\,e^{-\Gamma t}\\[3pt]
\rho_{10}(0)\,e^{-\Gamma t} & \rho_{11}(0)
\end{pmatrix}.
\end{equation}
The state flows from a coherent superposition toward the diagonal
(classical) mixture $\mathrm{diag}(\rho_{00},\rho_{11})$ on the
timescale $\Gamma^{-1}$, with no loss of trace ($\rho_{00}+\rho_{11}$
is conserved) and with positivity preserved throughout (the
eigenvalues stay in $[0,1]$). This is the canonical microscopic model
of decoherence and the simplest nontrivial instance of GKSL dynamics.

\subsection{Connection to the GG framework}
\label{app:GKSL-GG}

The GKSL semigroup is the precise mathematical setting in which the
paper's quantum dynamics live. Three identifications close the loop.
First, the closed-system limit: as all rates $\gamma_k\to0$ the
dissipator vanishes and~\eqref{eq:GKSL-final} collapses to the von
Neumann equation $\dot\rho = -\tfrac{i}{\hbar}[H,\rho]$, equivalent to
the Schrodinger equation of \TA{2}---the unitary core is recovered
exactly when the system is sealed from its environment. Second, the
decoherence rates introduced in \TA{5} are nothing but the dissipator
rates $\Gamma_{nm}$ of \S\ref{app:GKSL-decoherence}: the off-diagonal
damping $e^{-\Gamma t}$ of the worked example is the same exponential
that controls the loss of coherence in the body of the paper. Third,
and most distinctively, the framework reads this open-system semigroup
as the \emph{boundary image of coarse-graining the non-compact scale
fiber} $I_r$. Integrating out the fine-scale degrees along $I_r$ is an
irreversible operation: it has a forward direction (toward coarser
descriptions) but no inverse, exactly the structure of a
\emph{semigroup} rather than a group. The GKSL generator is thus the
induced generator of an irreversible renormalization-group semigroup
on the system's reduced state---the directed flow of the scale fiber,
projected onto the boundary observables, manifesting as completely
positive, trace-preserving, Markovian dissipation. In this reading the
arrow of time of the dissipator and the arrow of coarse-graining along
$I_r$ are one and the same. This geometric reading fixes the
open-system \emph{structure} --- the semigroup form and its
irreversibility --- but \emph{not} the numerical rates: the
coefficients $\gamma_k$ are set by the physical $4$D environment
through its spectral density at the Bohr frequencies (the
Fermi-golden-rule expression derived above), and are evaluated
quantitatively by the scale-overlap integral of
Appendix~\ref{app:worked-overlap}. The non-compact scale fiber
$I_r$ carries no Kaluza--Klein tower and sets no rate of its own;
the compact phase fiber $S^1_\theta$ enters only through the
separate, kinematically suppressed channel of that appendix. The
geometry supplies the law of the flow; the physical bath supplies
its speed.

\paragraph{Problems and worked solutions.}

\medskip\noindent\textbf{D.1.1. Show that the transpose is positive
but not completely positive.}

\noindent\emph{Solution.}  Step 1: transposition preserves
eigenvalues, so $T(\rho) = \rho^{\mathsf T}$ maps states to states:
$T$ is positive.  Step 2: test complete positivity on the maximally
entangled pair --- the Choi test of \S\ref{app:GKSL}: the Choi
matrix $C = \tfrac12\sum_{ij} E_{ij} \otimes T(E_{ij})$ has
eigenvalues $\{+\tfrac12, +\tfrac12, +\tfrac12, -\tfrac12\}$: one
negative eigenvalue, so $\mathbb 1 \otimes T$ maps the entangled
state to a non-state.  Step 3: conclusion --- positivity alone
cannot survive entangled company, which is why the GKSL hypothesis
is \emph{complete} positivity and why the Kraus form
\eqref{eq:GKSL-kraus} (which builds CP in) is the right starting
point; the transpose's failure is not pathology but the standard
entanglement witness.  Physics intuition: a map can be perfectly
lawful on every state it is shown and still poison a system through
its entangled partner --- complete positivity is the requirement
that the law hold even for particles that never meet the apparatus,
and nature audibly enforces it.


\medskip\noindent\textbf{D.1.2. Split the Schr\"odinger equation into
its real pair, and locate the $i$.}

\noindent\emph{Solution.}  Step 1: write $\psi = u + iv$ with $u, v$
real (in a basis where $H$ is real symmetric --- always arrangeable
for the textbook Hamiltonians), and substitute into
$i\hbar\,\partial_t\psi = H\psi$; separating real and imaginary parts
gives the coupled real pair $\hbar\,\dot u = +Hv$,
$\hbar\,\dot v = -Hu$ --- two first-order real equations, no $i$
anywhere.  Step 2: recognize the structure: the pair is a real
Hamiltonian (symplectic) flow, rotating $(u, v)$ with generator
$J = \bigl(\begin{smallmatrix}0 & 1\\ -1 & 0\end{smallmatrix}\bigr)$,
$J^2 = -\mathbb 1$; the conserved quantity
$\lVert u\rVert^2 + \lVert v\rVert^2 = \lVert\psi\rVert^2$ is the
Born normalization, and for an energy eigenstate the pair traces a
circle at angular frequency $E/\hbar$ --- the ``complex phase''
$e^{-iEt/\hbar}$ is this real rotation written in shorthand.
Step 3: state the register: nothing in Steps 1--2 is a derivation of
\emph{which} real pairing nature uses --- that is the origin-of-$i$
theorem of the companion paper (\citealp{ArisakaP4Real}),
derived-conditional in the quasi-free regime --- but the exercise
shows exactly what the $i$ \emph{is}: the rotation generator of a
real two-component flow, promoted to a primitive by a century of
notational habit.  Physics intuition: the most famous imaginary
number in physics is a bookkeeping device for a real rotation ---
Schr\"odinger's equation never needed complex numbers, it needed a
\emph{pairing}; the bulk supplies the pairing, and the boundary
observer, seeing only the compressed notation, spent a hundred years
asking where the $i$ came from.

\medskip\noindent\textbf{D.2.1. Verify trace preservation for the
dephasing channel's Kraus pair.}

\noindent\emph{Solution.}  Step 1: the phase-flip channel has Kraus
operators $K_0 = \sqrt{1-p}\,\mathbb 1$, $K_1 = \sqrt{p}\,\sigma_z$.
Step 2: the trace-preservation condition \eqref{eq:GKSL-kraus-tp}:
$K_0^\dagger K_0 + K_1^\dagger K_1 = (1-p)\mathbb 1 + p\,\sigma_z^2
= \mathbb 1$ --- satisfied for every $p \in [0,1]$.  Step 3: act on
a general qubit state: populations are untouched,
$\rho_{01} \mapsto (1-2p)\rho_{01}$ --- the channel shrinks
coherences only, and iterating it $n$ times gives $(1-2p)^n$, the
discrete ancestor of the exponential decay solved in D.7.1.
Physics intuition: a quantum channel is a lottery of unitaries ---
here ``do nothing'' versus ``flip the phase'' --- and trace
preservation is just the lottery's tickets summing to one; the
off-diagonal shrinkage is what averaging over ignorance of which
ticket was drawn does to interference.

\medskip\noindent\textbf{D.5.1. State the Born--Markov--secular
regime, and say why the secular step is not optional.}

\noindent\emph{Solution.}  Step 1: the microscopic derivation
(\S\ref{app:GKSL}, \eqref{eq:GKSL-born}--\eqref{eq:GKSL-secular})
holds when the bath correlation time is short compared to the
relaxation time, $\tau_B \ll \tau_R$ (Born--Markov), so the bath
forgets before the system notices.  Step 2: the pre-secular
(Redfield) generator is not of GKSL form and can drive $\rho$
temporarily negative --- probabilities below zero; the secular
average over fast oscillating terms \eqref{eq:GKSL-secular}
restores the GKSL structure with golden-rule rates
\eqref{eq:GKSL-golden} that are manifestly $\ge 0$.  Step 3: read
the moral: complete positivity is not decoration --- it is the
boundary between an approximation scheme and a law, and the GKSL
class is exactly where the scheme becomes lawful.  Physics intuition:
 Redfield is a snapshot taken with the shutter open ---
smeared and occasionally impossible; the secular step waits for
the fast fringes to average out, and what remains is the unique
legal motion of an open system.

\medskip\noindent\textbf{D.7.1. Solve the dephasing qubit
completely, and name the two clocks.}

\noindent\emph{Solution.}  Step 1: the master equation
\eqref{eq:GKSL-qubit-master} with the single Lindblad operator
$L = \sigma_z$ gives component equations
\eqref{eq:GKSL-qubit-components}: $\dot\rho_{00} = \dot\rho_{11} =
0$, $\dot\rho_{01} = -2\gamma\rho_{01}$.  Step 2: solve
\eqref{eq:GKSL-qubit-solution}: populations frozen, coherences
$\rho_{01}(t) = \rho_{01}(0)\,e^{-2\gamma t}$ --- energy conserved,
interference dying on the clock $T_2^{(\phi)} = 1/2\gamma$.
Step 3: the Bloch picture: the vector's $z$-component is constant
while its equatorial component spirals to zero --- the state
collapses onto the $\sigma_z$ axis, which is precisely the
einselection of the pointer basis seen microscopically in
Part~\ref{part:IX}.  Physics intuition: dephasing is decoherence
in its purest form --- nothing is dissipated, nothing is exchanged
but \emph{which-way information}, and the death of the
off-diagonals is the price of that information existing anywhere
in the universe.

\medskip\noindent\textbf{D.7.2. Derive $T_2 \le 2T_1$, and evaluate
both ends.}

\noindent\emph{Solution.}  Step 1: with amplitude damping at rate
$1/T_1$ and pure dephasing at rate $1/T_\phi$, the coherence decays
at the sum $1/T_2 = 1/2T_1 + 1/T_\phi$ --- the factor $\tfrac12$
because population decay destroys coherence only through its
amplitude, $\sqrt{e^{-t/T_1}}$ per leg.  Step 2: since
$T_\phi > 0$, $T_2 \le 2T_1$ always, with equality iff pure
damping: for $T_1 = 100~\mu\mathrm{s}$ and no dephasing,
$T_2 = 200~\mu\mathrm{s}$; add $T_\phi = 200~\mu\mathrm{s}$ and
$T_2$ drops to $100~\mu\mathrm{s}$.  Step 3: this inequality is
the daily bread of every qubit laboratory --- superconducting,
trapped-ion, NV --- and it is a \emph{theorem of the GKSL form}:
any device violating it would falsify not a hardware model but the
complete-positivity structure of open-system quantum mechanics.
Physics intuition: energy loss forces phase loss (you cannot decay
without decohering), but phase can leak with no energy moving at
all --- so the phase clock can only tie the energy clock at double
speed, never beat it.

\medskip\noindent\textbf{D.8.1. Explain why the boundary law of the
projection must be GKSL, with the rates as geometry.}

\noindent\emph{Solution.}  Step 1: the boundary state is a partial
trace of a deterministically evolving bulk configuration
(\S\ref{app:partial_trace}), so its evolution is a family of CPTP
maps by construction --- Kraus/Stinespring is automatic, not
assumed.  Step 2: the fiber correlation times are microscopic
compared to boundary dynamics, so the semigroup (Markov) property
holds on observational timescales, and the GKSL theorem then
leaves \emph{exactly one} generator form \eqref{eq:GKSL-final} ---
the framework does not choose Lindblad, it is cornered into it.
Step 3: identify the dictionary: $H$ is the boundary Hamiltonian
of the projected sector, the $L_k$ are the fiber-coupling channels,
and the rates $\gamma_k$ are squared scale-overlap integrals
(\S\ref{app:worked-overlap}) --- measurable numbers with geometric
addresses, which is what makes the eight-test horizon of the
Discussions falsifiable rather than rhetorical.  Physics intuition:
 determinism upstairs, Lindblad downstairs is not a
paradox but a projection theorem --- the dissipator is the shadow
of the fibers, and every decay rate in the laboratory is, on this
reading, a geometry measurement.


\GGtakeaway{\emph{Standard account:} master equations are
written ad hoc, system by system, and the Lindblad form is quoted
as a convenience. \emph{What it left open:} why nature's
effective dynamics should be completely positive, Markovian, and
of this one algebraic shape. \emph{The bulk reading:} the
boundary evolution is the trace of a deterministic bulk flow, so
it \emph{must} be a CPTP semigroup, and the GKSL theorem then
leaves exactly one generator form --- the dissipator is not noise
added by hand but the shadow of the unresolved fibers, with the
rates $\gamma_k$ the projection's overlap data.}

\section{Explicit partial-trace calculation: 6D bulk to 4D
density matrix}
\label{app:partial_trace}

\GGteachbox{Where mixedness comes from: the partial trace built
from first principles --- definition, basis-independence, the
operational reading (the unique state reproducing all local
measurement statistics) --- and then applied to the central
geometric claim of the paper: a 4D observer who cannot resolve the
scale fiber $I_r$ and the phase fiber $S^1_\theta$ is forced to
describe a pure 6D configuration by a reduced density matrix. The
appendix works the reduction explicitly and shows that purity
upstairs and probability downstairs are the same fact viewed from
the two strata --- the mathematical engine behind \TA{1} and the
worked examples of Parts~\ref{part:V}--\ref{part:IX}.}

\GGreadalong{\citet{NielsenChuang2010} for the standard
density-matrix and partial-trace formalism;
\citet{Schlosshauer2007} for the reduced-state physics.}

This appendix builds the partial trace from first principles and
applies it to the central geometric claim of the present paper: a
$4$D observer who cannot resolve the two non-spatial extra
directions of the $6$D bulk --- the scale fiber $I_r$ and
the phase fiber $S^1_\theta$ --- is forced to describe the world
with a \emph{reduced} density matrix obtained by tracing those two
directions out. We show, with every step written out, that this
trace (i) is the unique operation reproducing all locally
accessible expectation values, (ii) turns a globally pure entangled
bulk state into a mixed $4$D state, and (iii) suppresses the
off-diagonal coherences in the (einselected, \TA{6}) pointer basis
by averaging over the inaccessible $S^1_\theta$ holonomy. We are
careful to distinguish this loss of \emph{accessible} coherence,
which leaves an \emph{improper} mixture, from the selection of a
single definite outcome --- the latter being the work of the
\GRIP{} registration (\TA{1},\TA{4}), not of the partial trace.

We assume only the postulates of finite-dimensional quantum
mechanics: states are unit-trace positive operators $\rho$ on a
Hilbert space $\Hilb$, observables are self-adjoint operators, and
$\expval{A}_\rho=\Tr(\rho A)$. Throughout, $\Tr$ is the full trace
and $\Tr_B$ a partial trace, defined below.


\subsection{The partial trace from first principles}
\label{subsec:pt_first_principles}

Let the total system live on a tensor-product Hilbert space
\begin{equation}\label{eq:pt_tensor}
\Hilb \;=\; \Hilb_A \otimes \Hilb_B ,
\end{equation}
with $\dim\Hilb_A = d_A$ and $\dim\Hilb_B = d_B$. Fix orthonormal
bases $\{\ket{i}_A\}_{i=1}^{d_A}$ of $\Hilb_A$ and
$\{\ket{\mu}_B\}_{\mu=1}^{d_B}$ of $\Hilb_B$, so that
$\{\ket{i}_A\otimes\ket{\mu}_B\}$ is an orthonormal basis of
$\Hilb$. We abbreviate $\ket{i}_A\otimes\ket{\mu}_B \equiv
\ket{i,\mu}$.

\paragraph{Definition (partial trace over $B$).}
The partial trace over $B$ is the linear map
$\Tr_B:\;\mathcal{B}(\Hilb_A\otimes\Hilb_B)\to\mathcal{B}(\Hilb_A)$
fixed on product operators by
\begin{equation}\label{eq:pt_def_product}
\Tr_B\!\big(M_A \otimes N_B\big) \;=\; M_A \, \Tr(N_B),
\end{equation}
and extended to all operators by linearity. Equivalently, in
components, for any operator $\Theta$ on $\Hilb_A\otimes\Hilb_B$,
\begin{equation}\label{eq:pt_def_index}
\big(\Tr_B\,\Theta\big)_{ij}
\;=\;
\sum_{\mu=1}^{d_B}
\bra{i,\mu}\,\Theta\,\ket{j,\mu}
\;=\;
\sum_{\mu=1}^{d_B}
\Theta_{(i\mu)(j\mu)} .
\end{equation}
In words: one ``closes up'' the $B$ index by summing over the
diagonal of the $B$ block, while keeping the $A$ indices $i,j$ open.
The result is an operator on $\Hilb_A$ alone. A compact
basis-free way to write the same map is
\begin{equation}\label{eq:pt_def_kraus}
\Tr_B\,\Theta
\;=\;
\sum_{\mu=1}^{d_B}
\big(\iden_A \otimes \bra{\mu}_B\big)\,
\Theta\,
\big(\iden_A \otimes \ket{\mu}_B\big),
\end{equation}
where $\iden_A\otimes\bra{\mu}_B$ maps
$\Hilb_A\otimes\Hilb_B\to\Hilb_A$ by contracting the $B$ factor
against $\bra{\mu}_B$. Because the trace $\Tr(N_B)$ in
\eqref{eq:pt_def_product} is basis-independent, the map $\Tr_B$
itself does not depend on which orthonormal basis $\{\ket{\mu}_B\}$
we chose; we verify this independence explicitly below.

\paragraph{The reduced density matrix.}
Given a state $\rho$ on $\Hilb_A\otimes\Hilb_B$, its reduced state
on $A$ is by definition
\begin{equation}\label{eq:rho_A_def}
\rho_A \;\equiv\; \Tr_B\,\rho .
\end{equation}
We must check that $\rho_A$ is again a legitimate state, i.e.\
positive with unit trace.

\emph{Unit trace.} Using \eqref{eq:pt_def_index},
\begin{equation}\label{eq:pt_unit_trace}
\Tr_A\,\rho_A
=\sum_{i}\big(\rho_A\big)_{ii}
=\sum_{i}\sum_{\mu}\rho_{(i\mu)(i\mu)}
=\Tr\,\rho
=1 .
\end{equation}
The double sum over $(i,\mu)$ is exactly the full trace, so the
unit-trace property survives.

\emph{Positivity.} For any $\ket{\phi}\in\Hilb_A$, insert
$\ket{\phi}$ into \eqref{eq:pt_def_kraus}:
\begin{equation}\label{eq:pt_positivity}
\bra{\phi}\rho_A\ket{\phi}
=\sum_{\mu}
\big(\bra{\phi}\otimes\bra{\mu}\big)\,\rho\,
\big(\ket{\phi}\otimes\ket{\mu}\big)
=\sum_{\mu}
\bra{\phi,\mu}\rho\ket{\phi,\mu}
\;\ge\;0,
\end{equation}
because each term is the expectation of the positive operator
$\rho$ in the normalized vector $\ket{\phi,\mu}$. Hence $\rho_A\ge
0$. Together with \eqref{eq:pt_unit_trace}, $\rho_A$ is a valid
density matrix.

\paragraph{Uniqueness: $\Tr_B$ is forced by local expectation
values.} The physical content of the partial trace is that it is
the \emph{only} map reproducing every measurement an $A$-local
observer can make. Suppose an observer acts only on the $A$ factor,
measuring an observable that, on the full space, reads
$A\otimes\iden_B$. Its expectation in the global state $\rho$ is
\begin{align}\label{eq:local_exp_global}
\expval{A\otimes\iden_B}_\rho
&=\Tr\!\big[\rho\,(A\otimes\iden_B)\big]
=\sum_{i,\mu}\sum_{j,\nu}
\rho_{(i\mu)(j\nu)}\,(A)_{ji}\,(\iden_B)_{\nu\mu}\nonumber\\
&=\sum_{i,j}(A)_{ji}\sum_{\mu}\rho_{(i\mu)(j\mu)}
=\sum_{i,j}(A)_{ji}\,\big(\Tr_B\,\rho\big)_{ij}
=\Tr\!\big[(\Tr_B\,\rho)\,A\big].
\end{align}
The third equality used $(\iden_B)_{\nu\mu}=\delta_{\nu\mu}$, which
collapses the $\nu$-sum and sets $\nu=\mu$; the fourth recognized
the component formula \eqref{eq:pt_def_index}. Thus
\begin{equation}\label{eq:born_local}
\boxed{\;\expval{A\otimes\iden_B}_\rho
=\Tr_A\!\big[\rho_A\,A\big],\qquad
\rho_A=\Tr_B\,\rho .\;}
\end{equation}
Now suppose some other operator $\sigma_A$ on $\Hilb_A$ reproduced
the same numbers, $\Tr_A(\sigma_A A)=\Tr_A(\rho_A A)$ for
\emph{every} self-adjoint $A$. Then $\Tr_A[(\sigma_A-\rho_A)A]=0$
for all $A$; choosing $A=(\sigma_A-\rho_A)^\dagger$ (itself
self-adjoint here) gives
$\|\sigma_A-\rho_A\|_{\mathrm{HS}}^2=0$, so $\sigma_A=\rho_A$. The
reduced state is therefore the \emph{unique} operator on $\Hilb_A$
encoding all $A$-local statistics, and $\Tr_B$ is the unique map
producing it. This is the precise sense in which a subsystem
observer who cannot touch $B$ is \emph{obliged} to use $\rho_A$:
any two global states with the same $\rho_A$ are operationally
indistinguishable to that observer.

\paragraph{Basis independence of $\Tr_B$.}
If $\{\ket{\tilde\mu}_B\}$ is a second orthonormal basis with
$\ket{\tilde\mu}=\sum_\nu U_{\nu\mu}\ket{\nu}$ for a unitary $U$,
then
\begin{equation}\label{eq:pt_basis_indep}
\sum_\mu
\big(\iden_A\otimes\bra{\tilde\mu}\big)\Theta
\big(\iden_A\otimes\ket{\tilde\mu}\big)
=\sum_{\mu}\sum_{\nu,\nu'}U^*_{\nu\mu}U_{\nu'\mu}
\big(\iden_A\otimes\bra{\nu}\big)\Theta
\big(\iden_A\otimes\ket{\nu'}\big)
=\sum_{\nu}
\big(\iden_A\otimes\bra{\nu}\big)\Theta
\big(\iden_A\otimes\ket{\nu}\big),
\end{equation}
where $\sum_\mu U^*_{\nu\mu}U_{\nu'\mu}=\delta_{\nu\nu'}$ by
unitarity. So $\Tr_B$ is well defined independently of the chosen
$B$ basis --- a fact we will use when we argue that the
\emph{physically} preferred way of writing the fiber states does
not change the reduced $4$D state, only its interpretation.


\subsection{A worked $2\times 2$ tensor $2\times 2$ example}
\label{subsec:worked_2x2}

To make \eqref{eq:pt_def_index} concrete, take $d_A=d_B=2$, so
$\Hilb=\mathbb{C}^2\otimes\mathbb{C}^2$ is four-dimensional with
ordered basis
$\{\ket{0,0},\ket{0,1},\ket{1,0},\ket{1,1}\}$. A general operator
$\Theta$ is a $4\times4$ matrix whose entries we label by the pair
indices $(i\mu)$:
\begin{equation}\label{eq:theta_4x4}
\Theta=
\begin{pmatrix}
\Theta_{(00)(00)} & \Theta_{(00)(01)} & \Theta_{(00)(10)} & \Theta_{(00)(11)}\\
\Theta_{(01)(00)} & \Theta_{(01)(01)} & \Theta_{(01)(10)} & \Theta_{(01)(11)}\\
\Theta_{(10)(00)} & \Theta_{(10)(01)} & \Theta_{(10)(10)} & \Theta_{(10)(11)}\\
\Theta_{(11)(00)} & \Theta_{(11)(01)} & \Theta_{(11)(10)} & \Theta_{(11)(11)}
\end{pmatrix}.
\end{equation}
View $\Theta$ as a $2\times2$ array of $2\times2$ blocks
$\Theta^{(ij)}$, where the $A$ indices $i,j$ label the block and the
$B$ indices label the entry inside it:
\begin{equation}\label{eq:theta_blocks}
\Theta=
\begin{pmatrix}
\Theta^{(00)} & \Theta^{(01)}\\[2pt]
\Theta^{(10)} & \Theta^{(11)}
\end{pmatrix},
\qquad
\big(\Theta^{(ij)}\big)_{\mu\nu}=\Theta_{(i\mu)(j\nu)} .
\end{equation}
By \eqref{eq:pt_def_index}, tracing out $B$ replaces each
$2\times2$ block by its ordinary trace:
\begin{equation}\label{eq:trB_2x2}
\Tr_B\,\Theta=
\begin{pmatrix}
\tr\,\Theta^{(00)} & \tr\,\Theta^{(01)}\\[2pt]
\tr\,\Theta^{(10)} & \tr\,\Theta^{(11)}
\end{pmatrix}
=
\begin{pmatrix}
\Theta_{(00)(00)}+\Theta_{(01)(01)} &
\Theta_{(00)(10)}+\Theta_{(01)(11)}\\[2pt]
\Theta_{(10)(00)}+\Theta_{(11)(01)} &
\Theta_{(10)(10)}+\Theta_{(11)(11)}
\end{pmatrix}.
\end{equation}
The complementary trace over $A$ instead sums the two diagonal
blocks as whole matrices:
\begin{equation}\label{eq:trA_2x2}
\Tr_A\,\Theta=\Theta^{(00)}+\Theta^{(11)}
=
\begin{pmatrix}
\Theta_{(00)(00)}+\Theta_{(10)(10)} &
\Theta_{(00)(01)}+\Theta_{(10)(11)}\\[2pt]
\Theta_{(01)(00)}+\Theta_{(11)(10)} &
\Theta_{(01)(01)}+\Theta_{(11)(11)}
\end{pmatrix}.
\end{equation}
Equations \eqref{eq:trB_2x2}--\eqref{eq:trA_2x2} are the two
``half-traces'' of the same $4\times4$ object; comparing them makes
visible that $\Tr_B$ keeps the $A$ structure (the block layout) and
discards the $B$ structure (collapsing each block to a number),
while $\Tr_A$ does the reverse. This block-trace picture is the
entire computational content of the partial trace; everything below
is this operation applied to physically motivated states.


\subsection{The bulk state and its factorization}
\label{subsec:bulk_factorization}

We now specialize $A,B$ to the geometry of the present paper. The
$6$D bulk Hilbert space decomposes, at the operational level made
precise below, as
\begin{equation}\label{eq:Hilb_decompose}
\Hilb_{\rm bulk} \;=\; \Hilb_4 \otimes \Hilb_{I_r} \otimes
\Hilb_{S^1_\theta},
\end{equation}
with $\Hilb_4$ the $4$D zero-mode sector that a boundary observer
can resolve, $\Hilb_{I_r}$ the scale-radial (scale-fiber)
Kaluza--Klein sector, and $\Hilb_{S^1_\theta}$ the
compact phase-circle sector carrying the Wilson-line holonomy. The
two extra factors are \emph{not} ordinary spatial directions: $I_r$
is the renormalization-group scale axis and $S^1_\theta$ is the
internal phase circle. A $4$D observer has no apparatus that couples
to either, so from that observer's standpoint they are an
\emph{environment}. Grouping them,
\begin{equation}\label{eq:AB_identification}
\Hilb_A \equiv \Hilb_{\rm bdy} \equiv \Hilb_4,
\qquad
\Hilb_B \equiv \Hilb_{\rm fiber}
\equiv \Hilb_{I_r}\otimes\Hilb_{S^1_\theta},
\qquad
\Hilb_{\rm bulk}=\Hilb_{\rm bdy}\otimes\Hilb_{\rm fiber}.
\end{equation}

\paragraph{Honest caveat on the factorization.}
The clean tensor split \eqref{eq:Hilb_decompose} is an
\emph{operational, zero-mode-restricted} statement, not an exact
identity of the interacting bulk theory. A genuinely interacting
$6$D field theory does not factorize across a codimension-two
surface on the nose: the algebra of bulk observables has no
canonical product decomposition, gauge constraints link the sectors,
and a sharp boundary algebra requires a (type-III) operator-algebraic
treatment. What \emph{is} well defined is the restriction to the
low-lying mode sector --- the $4$D zero modes together with a finite
band of light $I_r$ and $S^1_\theta$ modes --- on which the scale-profile
spectrum gaps the heavy excitations away and the split
\eqref{eq:AB_identification} holds to the accuracy set by the scale profile
hierarchy. All formulas below are to be read in that restricted,
finite-dimensional sense; this is the same level of rigor at which
the rest of the paper treats the bulk-to-boundary dictionary. We do
not claim more, and nothing in the conclusions requires more.

\paragraph{A pure bulk state.}
A pure state on the restricted bulk space is
\begin{equation}\label{eq:Psi_full}
\ket{\Psi}_{\rm bulk}
=\sum_{a,b,c} c_{abc}\,
\ket{a}_4\otimes\ket{b}_{I_r}\otimes\ket{c}_{S^1_\theta},
\qquad
\sum_{a,b,c}|c_{abc}|^2=1,
\end{equation}
with $\{\ket{a}_4\}$ the boundary (pointer) basis and
$\{\ket{b}_{I_r}\},\{\ket{c}_{S^1_\theta}\}$ orthonormal bases of
the two fiber sectors. Writing the combined fiber index as
$B=(b,c)$ and the combined fiber basis as
$\ket{B}_{\rm fiber}=\ket{b}_{I_r}\otimes\ket{c}_{S^1_\theta}$, this
is exactly the bipartite form $\ket{\Psi}=\sum_{a,B}c_{aB}
\ket{a}_4\otimes\ket{B}_{\rm fiber}$ to which
\S\ref{subsec:pt_first_principles} applies verbatim.


\subsection{Explicit construction of the reduced $4$D state}
\label{subsec:explicit_reduction}

It is cleanest to first display the calculation with a single
fiber label, then restore the full $(I_r,S^1_\theta)$ structure.
Write the bulk pure state in Schmidt-like form, entangled across
boundary and fiber,
\begin{equation}\label{eq:Psi_schmidt}
\ket{\Psi}=\sum_n c_n\,\ket{b_n}\otimes\ket{f_n},
\end{equation}
where $\{\ket{b_n}\}\subset\Hilb_{\rm bdy}$ are pointer states and
$\{\ket{f_n}\}\subset\Hilb_{\rm fiber}$ are the corresponding fiber
states (not assumed orthonormal yet). The global density matrix is
the rank-one projector
\begin{equation}\label{eq:rho_bulk_proj}
\rho_{\rm bulk}=\ketbra{\Psi}{\Psi}
=\sum_{m,n}c_m c_n^*\,
\ket{b_m}\!\bra{b_n}\otimes\ket{f_m}\!\bra{f_n}.
\end{equation}
Apply $\Tr_{\rm fiber}$ using the product rule
\eqref{eq:pt_def_product}, $\Tr_{\rm fiber}(\ket{b_m}\!\bra{b_n}
\otimes\ket{f_m}\!\bra{f_n})=\ket{b_m}\!\bra{b_n}\,
\Tr(\ket{f_m}\!\bra{f_n})$, and the cyclicity identity
$\Tr(\ket{f_m}\!\bra{f_n})=\braket{f_n}{f_m}$:
\begin{equation}\label{eq:rho_bdy_general}
\rho_{\rm bdy}
=\Tr_{\rm fiber}\,\rho_{\rm bulk}
=\sum_{m,n} c_m c_n^*\,\braket{f_n}{f_m}\,
\ket{b_m}\!\bra{b_n}.
\end{equation}
Separate the diagonal ($m=n$) terms from the off-diagonal
($m\neq n$) cross terms:
\begin{equation}\label{eq:rho_bdy_split}
\rho_{\rm bdy}
=\underbrace{\sum_{n}|c_n|^2\,
\ket{b_n}\!\bra{b_n}}_{\text{classical mixture}}
\;+\;
\underbrace{\sum_{m\neq n} c_m c_n^*\,
\braket{f_n}{f_m}\,
\ket{b_n}\!\bra{b_m}}_{\text{quantum coherences}} .
\end{equation}
The diagonal piece is an incoherent (classical) mixture of pointer
states with weights $|c_n|^2$; it is exactly what a classical
probability distribution over outcomes $n$ would give. The
off-diagonal piece carries all the quantum interference, and every
such term is multiplied by the fiber overlap
$\braket{f_n}{f_m}$. Two limiting cases bracket the physics:
\begin{itemize}
\item \emph{Product (unentangled) bulk state.} If
$c_{abc}=\alpha_a\beta_b\gamma_c$, equivalently
$\ket{f_n}=\ket{f}$ independent of $n$, then
$\braket{f_n}{f_m}=\braket{f}{f}=1$ for all $m,n$, the two pieces of
\eqref{eq:rho_bdy_split} recombine, and
$\rho_{\rm bdy}=\ketbra{\alpha}{\alpha}$ remains pure. No
information has leaked to the fiber, so nothing is lost.
\item \emph{Orthogonal fiber records.} If the fiber states are
perfectly distinguishable, $\braket{f_n}{f_m}=\delta_{mn}$, every
cross term vanishes and
\begin{equation}\label{eq:rho_bdy_diagonal}
\rho_{\rm bdy}=\sum_n |c_n|^2\,\ket{b_n}\!\bra{b_n},
\end{equation}
a fully decohered, diagonal density matrix in the pointer basis.
The boundary state is now maximally mixed-down relative to the
coherent superposition it came from: the fiber has recorded
``which pointer state,'' and that record, though inaccessible to the
$4$D observer, is enough to destroy interference.
\end{itemize}
The generic, intermediate case has $0<|\braket{f_n}{f_m}|<1$ and
gives partial decoherence; the next subsection identifies what makes
the overlap small in this geometry.


\subsection{Decoherence as averaging over the hidden phase fiber}
\label{subsec:fibre_averaging}

The fiber overlap $\braket{f_n}{f_m}$ in
\eqref{eq:rho_bdy_general} is where the two specific extra
directions do their work. Restore the product structure of the
fiber, $\ket{f_n}=\ket{b_n^{I_r}}\otimes\ket{\phi_n^\theta}$, with
$\ket{b_n^{I_r}}$ the scale-radial component and
$\ket{\phi_n^\theta}$ the phase-circle component. The overlap
factorizes,
\begin{equation}\label{eq:overlap_factorize}
\braket{f_n}{f_m}
=\braket{b_m^{I_r}}{b_n^{I_r}}\;
\braket{\phi_m^\theta}{\phi_n^\theta}.
\end{equation}
Consider the $S^1_\theta$ factor first. The phase circle carries a
Wilson-line holonomy: parallel transport once around $S^1_\theta$
multiplies the $n$-th sector's phase state by a holonomy phase
$e^{i\Theta_n}$, and the relative holonomy between sectors $m$ and
$n$ is $e^{i(\Theta_m-\Theta_n)}$. A $4$D observer cannot resolve
\emph{where} on the circle the state sits --- the circle coordinate
$\theta$ is internal and inaccessible --- so the only invariant
statement is obtained by averaging the relative phase uniformly over
the circle. Modeling
$\ket{\phi_n^\theta}\sim e^{i\Theta_n\theta}$ on $S^1_\theta$ ---
with \emph{integer} sector holonomies $\Theta_n\in\mathbb Z$, as the
single-valuedness of a phase on the compact $S^1_\theta$ requires
(the charge quantization derived in the Wilson-line appendix) --- and
averaging the cross term over $\theta\in[0,2\pi)$,
\begin{equation}\label{eq:theta_average}
\overline{\braket{\phi_m^\theta}{\phi_n^\theta}}
=\frac{1}{2\pi}\int_0^{2\pi}
e^{\,i(\Theta_n-\Theta_m)\theta}\,d\theta
=\delta_{\Theta_m,\Theta_n}
\;\xrightarrow[\;m\neq n\;]{}\;0 .
\end{equation}
Distinct pointer sectors carry distinct integer holonomies
($\Theta_m\neq\Theta_n$ for $m\neq n$, so $\Theta_n-\Theta_m$ is a
nonzero integer and the integral in \eqref{eq:theta_average} vanishes
\emph{exactly}); the average therefore kills every $m\neq n$ cross
term. Substituting this averaged overlap back into
\eqref{eq:rho_bdy_general} leaves precisely the diagonal,
pointer-basis density matrix:
\begin{equation}\label{eq:rho_bdy_decohered}
\overline{\rho_{\rm bdy}}
=\sum_n |c_n|^2\,\ket{b_n}\!\bra{b_n}.
\end{equation}
This is the technical statement of \emph{decoherence as
fiber-averaging}: the off-diagonal coherences are not erased by any
non-unitary fundamental process; they are present in the global
pure state $\ket{\Psi}$ but become invisible the instant one
restricts to $4$D-accessible quantities, because those quantities
are blind to the $S^1_\theta$ phase and therefore see only the
holonomy-averaged overlap \eqref{eq:theta_average}. The preferred
(pointer) basis singled out here is exactly the basis in which the
fiber keeps an orthogonal record, in accord with the einselection
theorem \TA{6}.


\subsection{Off-diagonal suppression as a scale-profile overlap
integral}
\label{subsec:warp_overlap}

The $S^1_\theta$ average gives the \emph{exact} vanishing in the
idealized holonomy model; the residual, finite suppression rate
comes from the second factor in \eqref{eq:overlap_factorize}, the
scale-radial overlap $\braket{b_m^{I_r}}{b_n^{I_r}}$. The radial
fiber states are Kaluza--Klein wavefunctions on the scale interval
$I_r$, weighted by the scale profile $e^{-2r}$ that controls how
strongly each $4$D pointer sector leaks into the heavy modes. Their
overlap is a scale-weighted integral of the schematic form
\begin{equation}\label{eq:warp_overlap}
\braket{b_m^{I_r}}{b_n^{I_r}}
\;\sim\;
\int_{I_r} e^{-2r}\,
\psi_m^*(r)\,\psi_n(r)\,dr,
\end{equation}
whose magnitude sets the surviving off-diagonal weight, and hence
the decoherence rate $\Gamma_{mn}$, through
$|(\rho_{\rm bdy})_{mn}|\propto|\braket{b_m^{I_r}}{b_n^{I_r}}|$. The
more the scale profile separates the radial supports of $\psi_m$ and
$\psi_n$, the smaller the overlap and the faster the coherences
decay. The quantitative evaluation of this integral --- the
collisional decoherence function, the cross-section estimates, and
the resulting numerical rate --- is carried out in the worked
overlap appendix, Appendix~\ref{app:worked-overlap}; here we only
need that the scale geometry \emph{suppresses} the off-diagonal
terms, which is what \eqref{eq:warp_overlap} states qualitatively.
This is also the microscopic origin of the rate
$\Gamma_{\alpha\beta}$ appearing in the decoherence theorem \TA{5}.


\subsection{Worked micro-example: one boundary qubit maximally
entangled with one fiber qubit}
\label{subsec:micro_example}

Take the smallest non-trivial case: a single boundary qubit
$\Hilb_{\rm bdy}=\mathbb{C}^2$ with pointer basis
$\{\ket{0},\ket{1}\}$, maximally entangled with a single fiber qubit
$\Hilb_{\rm fiber}=\mathbb{C}^2$ with orthonormal records
$\{\ket{f_0},\ket{f_1}\}$. The bulk pure state is the Bell-like
vector
\begin{equation}\label{eq:micro_state}
\ket{\Psi}=\frac{1}{\sqrt2}
\big(\ket{0}\otimes\ket{f_0}+\ket{1}\otimes\ket{f_1}\big).
\end{equation}
Its global density matrix is the rank-one projector
\begin{equation}\label{eq:micro_rho_bulk}
\rho_{\rm bulk}=\ketbra{\Psi}{\Psi}
=\tfrac12\Big(
\ket{0,f_0}\!\bra{0,f_0}
+\ket{0,f_0}\!\bra{1,f_1}
+\ket{1,f_1}\!\bra{0,f_0}
+\ket{1,f_1}\!\bra{1,f_1}\Big).
\end{equation}
Trace out the fiber with \eqref{eq:pt_def_product}, using
$\Tr(\ket{f_i}\!\bra{f_j})=\braket{f_j}{f_i}=\delta_{ij}$:
\begin{align}\label{eq:micro_reduce}
\rho_{\rm bdy}
=\Tr_{\rm fiber}\,\rho_{\rm bulk}
&=\tfrac12\Big(
\ket{0}\!\bra{0}\,\underbrace{\braket{f_0}{f_0}}_{1}
+\ket{0}\!\bra{1}\,\underbrace{\braket{f_1}{f_0}}_{0}
+\ket{1}\!\bra{0}\,\underbrace{\braket{f_0}{f_1}}_{0}
+\ket{1}\!\bra{1}\,\underbrace{\braket{f_1}{f_1}}_{1}\Big)
\nonumber\\
&=\tfrac12\big(\ket{0}\!\bra{0}+\ket{1}\!\bra{1}\big)
=\frac{1}{2}\,\iden
=\begin{pmatrix}\tfrac12 & 0\\[2pt] 0 & \tfrac12\end{pmatrix}.
\end{align}
The reduced boundary state is the \emph{maximally mixed} qubit
$\rho_{\rm bdy}=\iden/2$. The interpretation is the entire moral of
this appendix in miniature:
\begin{itemize}
\item The global state \eqref{eq:micro_state} is pure: it has zero
entropy and definite (here, maximal) correlations between qubit and
fiber. Nothing random has happened at the $6$D level.
\item Yet the boundary observer, blind to the fiber, holds
$\iden/2$ --- a state of \emph{maximal ignorance}, von Neumann
entropy $S=\ln 2$, indistinguishable from a fair classical coin
over $\{\ket{0},\ket{1}\}$. The off-diagonal coherences of the
original superposition are gone, killed by the orthogonality of the
fiber records $\braket{f_0}{f_1}=0$.
\item The mixedness is therefore not intrinsic to the world; it is
manufactured by the partial trace, i.e.\ by the observer's inability
to resolve the two hidden directions. The diagonal $\iden/2$ is an \emph{improper} mixture --- the
boundary shadow of a perfectly unitary, perfectly definite bulk
state. It is not yet a definite outcome: which single value, $0$
or $1$, is registered is fixed by the \GRIP{} descent
(\TA{1},\TA{4}), not by the partial trace, which supplies only the
$\tfrac12$ weights.
\end{itemize}
The same arithmetic with non-orthogonal records,
$\braket{f_0}{f_1}=s$ with $|s|\le1$, gives
$\rho_{\rm bdy}=\tfrac12\!\begin{pmatrix}1 & s^*\\ s &
1\end{pmatrix}$, interpolating from the maximally mixed $\iden/2$ at
$s=0$ (orthogonal records, full decoherence) to a pure state at
$|s|=1$ (identical records, no decoherence) --- the two-level
version of the general overlap dependence in
\eqref{eq:rho_bdy_split}.


\subsection{The Born rule and the L\"uders update act on
$\rho_{\rm bdy}$}
\label{subsec:born_luders}

The object delivered by the partial trace,
$\rho_{\rm bdy}=\Tr_{\rm fiber}\,\rho_{\rm bulk}$, is the unique
state on which all $4$D-accessible statistics are computed. For a
\POVM{} $\{E_\alpha\}$ on the boundary subsystem, with $E_\alpha\ge
0$ and $\sum_\alpha E_\alpha=\iden$, the probability of outcome
$\alpha$ is
\begin{equation}\label{eq:Born_from_trace}
P(\alpha)=\Tr\!\big(\rho_{\rm bdy}\,E_\alpha\big)
=\sum_{a,a'}(\rho_{\rm bdy})_{aa'}\,(E_\alpha)_{a'a}.
\end{equation}
Since $\rho_{\rm bdy}$ is positive with unit trace
(\eqref{eq:pt_unit_trace}--\eqref{eq:pt_positivity}) and each
$E_\alpha\ge0$ with $\sum_\alpha E_\alpha=\iden$, the numbers
$P(\alpha)$ are non-negative and sum to one: a bona fide
probability distribution. This is the Born rule \TA{3}, now seen to
act not on an abstract postulated state but on the
fiber-reduced $\rho_{\rm bdy}$ constructed above. After a selective
measurement returning outcome $\alpha$ (with Kraus/instrument
elements $E_\alpha=K_\alpha^\dagger K_\alpha$), the conditional
post-measurement state is the L\"uders update \TA{4},
\begin{equation}\label{eq:luders}
\rho_{\rm bdy}\;\longmapsto\;
\rho_{\rm bdy}^{\,\alpha}
=\frac{K_\alpha\,\rho_{\rm bdy}\,K_\alpha^\dagger}{P(\alpha)} ,
\end{equation}
again a manipulation entirely internal to the boundary state. In
the non-selective case the unconditioned channel
$\rho_{\rm bdy}\mapsto\sum_\alpha K_\alpha\rho_{\rm bdy}
K_\alpha^\dagger$ is the discrete-time face of the continuous open-system
dynamics, whose generator is the \GKLS{} (Gorini--Kossakowski--Sudarshan--Lindblad) form
\begin{equation}\label{eq:gksl}
\frac{d\rho_{\rm bdy}}{dt}
=-\frac{i}{\hbar}[H_{\eff},\rho_{\rm bdy}]
+\sum_\alpha\gamma_\alpha\Big(
L_\alpha\rho_{\rm bdy}L_\alpha^\dagger
-\tfrac12\{L_\alpha^\dagger L_\alpha,\rho_{\rm bdy}\}\Big),
\end{equation}
derived in the present paper as the decoherence theorem \TA{5}
(its closed, unitary $\gamma_\alpha\to0$ limit being the
Schr\"odinger equation \TA{2}); its construction from the
fiber-traced bulk dynamics is detailed in
Appendix~\ref{app:GKSL}. The closed, unitary Schr\"odinger limit is
recovered as $\gamma_\alpha\to0$, i.e.\ when no information escapes
into the fiber.


\subsection{Summary: apparent classicality as an average over two
hidden directions}
\label{subsec:pt_summary}

Assembling the chain: a globally pure, unitarily evolving $6$D bulk
state \eqref{eq:Psi_full} is entangled across the boundary and the
two non-spatial fibers. A $4$D observer cannot resolve the scale
fiber $I_r$ or the phase fiber $S^1_\theta$, and the unique state
encoding everything that observer can measure is the fiber-reduced
density matrix $\rho_{\rm bdy}=\Tr_{\rm fiber}\,\rho_{\rm bulk}$
of \eqref{eq:rho_bdy_general}. Tracing out the inaccessible
$S^1_\theta$ holonomy averages the off-diagonal coherences to zero
(\eqref{eq:theta_average}--\eqref{eq:rho_bdy_decohered}), and the
scale-radial overlap \eqref{eq:warp_overlap} sets the residual
suppression rate; the worked qubit example
\eqref{eq:micro_reduce} shows a pure bulk state reducing to the
maximally mixed boundary state $\iden/2$, with no fundamental
randomness introduced anywhere. The resulting $\rho_{\rm bdy}$ is
precisely the object on which the Born rule \TA{3} and the
L\"uders update \TA{4} operate, with the open-system generator
taking the \GKLS{} form \eqref{eq:gksl} (\TA{5}, with \TA{2} its
unitary limit).

\paragraph{What the partial trace does and does not do.} The
result must not be overstated. The fiber-reduced $\rho_{\rm bdy}$
is an \emph{improper} mixture: averaging the inaccessible
$S^1_\theta$ holonomy removes the \emph{accessible} interference
between pointer states, leaving a diagonal form a $4$D observer
cannot operationally distinguish from a classical ignorance
ensemble. It does \emph{not}, by itself, select which single
outcome is realized. That selection is the separate, dynamical
content of the \GRIP{} registration of \citet{ArisakaA6Gi4}: the
joint system-and-apparatus configuration descends into one
Geometric Attractor basin and is locked there by the boundary
free-energy law (\TA{1},\TA{4}); the question whether the frequencies
of that descent reproduce the Born weights is closed at the
relaxation level, with the field-grade representation step
the single remaining analytic residue (\S\ref{sec:disc-open}), assumed nowhere
here. The pointer basis in which the reduction is diagonal is,
likewise, not produced by the average but fixed independently by
bulk locality (\TA{6}). In short, the partial trace explains the
\emph{appearance} of classicality and the loss of accessible
coherence; it does not, and is not claimed to, produce the
definite outcome or the preferred basis.

\paragraph{Problems and worked solutions.}

\medskip\noindent\textbf{E.2.1. Reduce the Bell pair, and watch
purity become probability.}

\noindent\emph{Solution.}  Step 1: for
$\ket{\Phi^+} = (\ket{00} + \ket{11})/\sqrt2$, apply the index
definition \eqref{eq:pt_def_index}: the cross terms carry
$\mathrm{Tr}\,\ketbra{0}{1} = 0$, so
$\rho_A = \mathrm{Tr}_B\,\ketbra{\Phi^+}{\Phi^+} = \tfrac12
\mathbb 1$.  Step 2: purity: $\mathrm{Tr}\,\rho^2$ falls from $1$
(global, pure) to $\tfrac12$ (local, maximally mixed) --- the local
observer's outcomes are exactly fifty-fifty although the global
state is as definite as quantum mechanics allows.  Step 3: verify
the operational reading \eqref{eq:local_exp_global}: every local
expectation computed from $\rho_A$ agrees with the global state's
prediction --- the reduced state is not an approximation but the
\emph{complete} local description.  Physics intuition: this
two-line calculation is the origin story of quantum probability in
the framework --- perfect global definiteness, viewed through a
subsystem keyhole, is indistinguishable from a fair coin; the 6D
bulk plays the role of the global state, and the 4D observer holds
the keyhole.

\medskip\noindent\textbf{E.2.2. Compare GHZ and W through their
reductions.}

\noindent\emph{Solution.}  Step 1: for
$\ket{\mathrm{GHZ}} = (\ket{000} + \ket{111})/\sqrt2$, tracing two
qubits gives $\rho_1 = \mathrm{diag}(\tfrac12, \tfrac12)$, purity
$\tfrac12$ --- and no coherence survives: one lost qubit destroys
the superposition entirely.  Step 2: for
$\ket{\mathrm W} = (\ket{001} + \ket{010} + \ket{100})/\sqrt3$,
$\rho_1 = \mathrm{diag}(\tfrac23, \tfrac13)$, purity
$\tfrac59$ --- less mixed, and the two-qubit remainder is still
entangled: W-entanglement survives amputation, GHZ-entanglement
does not.  Step 3: the lesson for the bulk reading: \emph{how}
mixedness appears under a trace is a fingerprint of \emph{how} the
discarded sector was entangled --- which is exactly the diagnostic
logic the chapter applies to the fibers, where the pattern of
boundary decoherence encodes the structure of the bulk coupling.
Physics intuition: entanglement is not one substance --- GHZ is a
chain that shatters at one cut, W a rope that frays strand by
strand; the reduced density matrix is the forensic report on which
one you had.

\medskip\noindent\textbf{E.5.1. Perform the fiber average, and kill
the off-diagonals exactly.}

\noindent\emph{Solution.}  Step 1: the pointer sectors carry
integer holonomies $\Theta_m$ on the compact fiber, and the
unresolved observer averages the relative phase over the circle
\eqref{eq:theta_average}:
$\frac{1}{2\pi}\int_0^{2\pi} e^{i(\Theta_n - \Theta_m)\theta}
d\theta = \delta_{\Theta_m \Theta_n}$.  Step 2: for distinct
sectors the integral is \emph{exactly} zero --- not small,
zero: the off-diagonal blocks of \eqref{eq:rho_bdy_split} are
annihilated, giving the decohered form
\eqref{eq:rho_bdy_decohered}.  Step 3: contrast with environmental
decoherence: a finite bath suppresses coherences by a large but
finite factor, while a compact unresolved dimension deletes
inter-sector coherence identically --- integer orthogonality on the
circle, the same arithmetic that makes Fourier series work.
Physics intuition: the fiber is nature's perfect which-way
detector --- distinct outcomes wind the circle a different integer
number of times, and integers do not almost-agree; that is why the
classical world's sectors do not leak into each other at any
precision.

\medskip\noindent\textbf{E.7.1. Re-run the micro-example, and read
the Born rule off the trace.}

\noindent\emph{Solution.}  Step 1: the boundary qubit maximally
entangled with one fiber mode \eqref{eq:micro_state} has bulk
projector \eqref{eq:micro_rho_bulk}; tracing the fiber with
$\mathrm{Tr}\,\ketbra{f_i}{f_j} = \delta_{ij}$ kills both cross
terms \eqref{eq:micro_reduce}: $\rho_{\rm bdy} = \tfrac12\mathbb 1$.
Step 2: generalize the weights: for
$\alpha\ket{0, f_0} + \beta\ket{1, f_1}$ the same trace gives
$\mathrm{diag}(|\alpha|^2, |\beta|^2)$ --- the Born weights emerge
as the \emph{diagonal survivors} of the fiber trace
\eqref{eq:Born_from_trace}, before any measurement postulate is
spoken.  Step 3: state the division of labor: the trace explains
why the boundary state is diagonal with these weights; Gleason
(\S\ref{app:Gleason}) explains why no other weight-assignment was
ever available --- the two appendices are the two halves of \TA{3}.
Physics intuition: the wavefunction's squared moduli are what
survive the keyhole --- the fiber keeps the phases and the boundary
keeps the weights, and a century of ``why $|\psi|^2$?'' is answered
by inspecting what a trace can and cannot carry.

\medskip\noindent\textbf{E.8.1. Apply the L\"uders update, and
verify repeatability.}

\noindent\emph{Solution.}  Step 1: on the decohered
$\rho_{\rm bdy} = \mathrm{diag}(|\alpha|^2, |\beta|^2)$, register
outcome $0$: the update \eqref{eq:luders} gives
$\rho' = \Pi_0 \rho \Pi_0 / \mathrm{Tr}(\rho\Pi_0) =
\ketbra{0}{0}$.  Step 2: repeat the measurement:
$p(0) = \mathrm{Tr}(\rho'\Pi_0) = 1$ --- repeatability, the
textbook's projection-postulate content, drops out of conditioning
alone.  Step 3: locate the update in the framework: nothing
collapses in the bulk --- the registration selects the sector, the
L\"uders map is Bayesian conditioning on the registered record,
and the ``jump'' is an update of the boundary description, not a
dynamical process competing with the GKSL flow (\eqref{eq:gksl}).
Physics intuition: collapse is what conditioning looks like when
you forget you are holding a reduced description --- the bulk
never jumped, the ledger did; and repeatability is simply the fact
that a question already answered stays answered.


\GGtakeaway{\emph{Standard account:} density matrices enter as
bookkeeping for ignorance or for subsystems, and the origin of
quantum probability is left to the interpretation chapter.
\emph{What it left open:} whether the mixedness of the world is
epistemic accident or structural necessity. \emph{The bulk
reading:} mixedness is geometry --- the observer's stratum simply
does not contain the fibers, so the trace is forced, the reduced
state is the \emph{complete} 4D description, and quantum
probability is what a deterministic bulk looks like through a
keyhole; the appendix's explicit calculation is that sentence made
arithmetic.}

\section{Explicit Bell-CHSH calculation with $S^1_\theta$
holonomy}
\label{app:CHSH_calc}

\GGteachbox{The Bell--CHSH mathematics, line by line, with the
cohort's one addition made explicit: the singlet correlation
$E(\mathbf a, \mathbf b) = -\mathbf a\cdot\mathbf b$ computed
from the joint state, the CHSH combination assembled, the local
bound $|S| \le 2$ and the quantum value $2\sqrt2$ at the standard
angles, and then the reproduction of the same correlations from
the $S^1_\theta$ holonomy sampling --- exhibiting a deterministic
6D-local model that violates the 4D bound without factorizing and
without signaling. Nothing in the standard mathematics is
altered; what changes is what the theorem is \emph{about}: Bell
excludes hidden variables on the boundary, and the bulk was never
on the boundary.}

\GGreadalong{\citet{CHSH1969} for the inequality;
\citet{Tsirelson1980} for the quantum bound;
\citet{BrunnerEtAl2014} for the modern review of Bell
nonlocality.}

This appendix expands, line by line, the claim made throughout
Part~\ref{part:VII} that the \GGtheory{} reading of an entangled pair
reproduces the standard quantum correlations \emph{exactly}, saturating
the Tsirelson bound $2\sqrt2$, while remaining free of any local-hidden-variable
factorization and of any superluminal signal. The mathematics here is the
ordinary Bell-CHSH mathematics; nothing in the framework alters a single
number. What the framework supplies is the geometric \emph{reading}: the
entangled pair is \emph{one} non-separable bulk state whose phase fiber
$S^1_\theta$ is fixed once, at the common source, and is then shared by both
wings. There is no second copy of the fiber to carry an independent local
response, which is exactly why the correlations are quantum and not
classical. We work entirely with $\pm1$ dichotomic outcomes and the Pauli
algebra, in standard notation, so a first-year graduate student can follow
every step without consulting an external reference.

Notation. Throughout, $\hat a,\hat a',\hat b,\hat b'$ are real unit vectors
in $\mathbb{R}^3$ specifying analyzer orientations; $\vec\sigma=(\sigma_x,\sigma_y,\sigma_z)$
is the vector of Pauli matrices; $\hat a\cdot\vec\sigma=\sum_i a_i\sigma_i$;
and $\theta_{ab}$ is the angle between $\hat a$ and $\hat b$, so that
$\hat a\cdot\hat b=\cos\theta_{ab}$. The two-qubit Hilbert space is
$\mathcal{H}=\mathbb{C}^2_A\otimes\mathbb{C}^2_B$, and the singlet (spin-zero)
state is
\begin{equation}\label{eq:singlet_def}
|\Psi^-\rangle \;=\; \frac{1}{\sqrt2}\bigl(|{\uparrow}\rangle_A|{\downarrow}\rangle_B
- |{\downarrow}\rangle_A|{\uparrow}\rangle_B\bigr),
\end{equation}
where $|{\uparrow}\rangle,|{\downarrow}\rangle$ are the $+1,-1$ eigenstates of
$\sigma_z$.

\subsection{The CHSH observable and the local-realist bound $|S|\le 2$}

Define the four correlators between Alice's settings $\{\hat a,\hat a'\}$ and
Bob's settings $\{\hat b,\hat b'\}$, each correlator being the expectation of
the product of the two $\pm1$ outcomes. The CHSH combination is
\begin{equation}\label{eq:CHSH_def}
S \;=\; E(\hat a,\hat b) - E(\hat a,\hat b') + E(\hat a',\hat b) + E(\hat a',\hat b').
\end{equation}
(The placement of the single minus sign is a convention; any of the four
sign assignments with an odd number of minus signs is equivalent under
relabeling, and we will track it carefully when we choose angles below.)

We now prove the Bell-CHSH bound $|S|\le 2$ that holds for \emph{any}
local-hidden-variable (LHV) model. An LHV model posits a shared variable
$\lambda$ with probability density $\rho(\lambda)\ge 0$, $\int d\lambda\,\rho(\lambda)=1$,
and \emph{local} deterministic response functions
\begin{equation}\label{eq:LHV_responses}
A(\hat a,\lambda),\,A(\hat a',\lambda),\,B(\hat b,\lambda),\,B(\hat b',\lambda)\;\in\;\{+1,-1\},
\end{equation}
where Alice's outcome depends only on her own setting and $\lambda$ (not on
Bob's setting), and vice versa. The correlator is the average of the product:
\begin{equation}\label{eq:LHV_correlator}
E(\hat a,\hat b)\;=\;\int d\lambda\,\rho(\lambda)\,A(\hat a,\lambda)\,B(\hat b,\lambda).
\end{equation}
Insert \eqref{eq:LHV_correlator} into \eqref{eq:CHSH_def} and abbreviate, for
a \emph{fixed} $\lambda$, $A\equiv A(\hat a,\lambda)$, $A'\equiv A(\hat a',\lambda)$,
$B\equiv B(\hat b,\lambda)$, $B'\equiv B(\hat b',\lambda)$. Then
\begin{equation}\label{eq:LHV_integrand}
S \;=\; \int d\lambda\,\rho(\lambda)\,\bigl[\,AB - AB' + A'B + A'B'\,\bigr].
\end{equation}
The bracketed quantity factors in two ways:
\begin{equation}\label{eq:LHV_factor}
AB - AB' + A'B + A'B' \;=\; A(B-B') + A'(B+B').
\end{equation}
Because $B,B'\in\{+1,-1\}$, exactly one of the two combinations $B-B'$ and
$B+B'$ vanishes while the other equals $\pm2$:
if $B=B'$ then $B-B'=0$ and $B+B'=\pm2$; if $B=-B'$ then $B+B'=0$ and
$B-B'=\pm2$. In either case, since $A,A'\in\{+1,-1\}$,
\begin{equation}\label{eq:LHV_pm2}
A(B-B') + A'(B+B') \;=\; \pm2.
\end{equation}
Thus the integrand of \eqref{eq:LHV_integrand} is bounded pointwise in
$\lambda$ by $|{\cdots}|\le 2$, and because $\rho$ is a normalized probability
density,
\begin{equation}\label{eq:LHV_bound}
|S| \;\le\; \int d\lambda\,\rho(\lambda)\,\bigl|\,A(B-B') + A'(B+B')\,\bigr|
\;\le\; \int d\lambda\,\rho(\lambda)\cdot 2 \;=\; 2.
\end{equation}
This is the Bell-CHSH inequality $|S|\le 2$. Its only inputs were
(i) definite $\pm1$ outcomes and (ii) factorization
\eqref{eq:LHV_correlator}, i.e. that Alice's outcome is produced by a
function not depending on Bob's setting. Any violation of $|S|\le 2$ is
therefore a proof that no such local, separable, outcome-assigning model
exists. The framework never supplies one: the shared $S^1_\theta$ fiber is
not a per-wing response function but the label of a single joint state, as
we make precise in Section~\ref{subsec:reading_frag}.

\subsection{The quantum singlet correlator $E(\hat a,\hat b)=-\hat a\cdot\hat b$}
\label{subsec:singlet_corr}

In quantum mechanics the correlator is the singlet expectation of the
product of local spin observables,
\begin{equation}\label{eq:E_quantum_def}
E(\hat a,\hat b) \;=\; \langle\Psi^-|\,(\hat a\cdot\vec\sigma)\otimes(\hat b\cdot\vec\sigma)\,|\Psi^-\rangle,
\end{equation}
where $(\hat a\cdot\vec\sigma)$ acts on Alice's qubit and $(\hat b\cdot\vec\sigma)$
on Bob's. We derive $E(\hat a,\hat b)=-\hat a\cdot\hat b$ in full.

\paragraph{Step 1: the total-spin-zero identity.}
The singlet carries total spin zero,
$(\vec\sigma_A+\vec\sigma_B)|\Psi^-\rangle=0$; component by
component this is the operator identity
\begin{equation}\label{eq:spin_zero}
(\sigma_i\otimes \mathbb 1)\,|\Psi^-\rangle \;=\; -(\mathbb 1\otimes\sigma_i)\,|\Psi^-\rangle,
\qquad i=x,y,z .
\end{equation}
One checks it directly on
$|\Psi^-\rangle=\tfrac{1}{\sqrt2}(|{\uparrow\downarrow}\rangle-|{\downarrow\uparrow}\rangle)$:
applying $\sigma_i$ to the first factor gives the same pair of
two-qubit vectors as applying it to the second factor, with an overall
sign flip --- the antisymmetry of the singlet, i.e.\ its vanishing
total spin. (No transpose appears; the minus sign is exactly this
antisymmetry.)

\paragraph{Step 2: reduce the correlator to a single-qubit trace.}
Writing $(\hat a\cdot\vec\sigma)\otimes(\hat b\cdot\vec\sigma)
=[(\hat a\cdot\vec\sigma)\otimes\mathbb 1]\,[\mathbb 1\otimes(\hat b\cdot\vec\sigma)]$
in \eqref{eq:E_quantum_def} and moving the first factor across with
\eqref{eq:spin_zero} (which holds for $\hat a\cdot\vec\sigma=\sum_i a_i\sigma_i$
by linearity),
\begin{equation}\label{eq:E_step2}
E(\hat a,\hat b)
= \langle\Psi^-|\,[(\hat a\cdot\vec\sigma)\otimes\mathbb 1][\mathbb 1\otimes(\hat b\cdot\vec\sigma)]\,|\Psi^-\rangle
= -\,\langle\Psi^-|\,\mathbb 1\otimes\bigl[(\hat a\cdot\vec\sigma)(\hat b\cdot\vec\sigma)\bigr]\,|\Psi^-\rangle.
\end{equation}
The reduced state of Bob's qubit in the singlet is maximally mixed,
$\rho_B=\tfrac12\mathbb 1$ (the singlet's marginals are completely
random; this is the no-signaling fact we use again below), so for any
single-qubit operator $N$,
\begin{equation}\label{eq:marginal_trace}
\langle\Psi^-|\,\mathbb 1\otimes N\,|\Psi^-\rangle = \mathrm{Tr}(\rho_B N) = \tfrac12\,\mathrm{Tr}\,N.
\end{equation}
Applying this with $N=(\hat a\cdot\vec\sigma)(\hat b\cdot\vec\sigma)$,
\begin{equation}\label{eq:E_to_trace}
E(\hat a,\hat b) = -\tfrac12\,\mathrm{Tr}\!\left[(\hat a\cdot\vec\sigma)(\hat b\cdot\vec\sigma)\right].
\end{equation}

\paragraph{Step 3: evaluate the trace with the Pauli algebra.}
The Pauli matrices obey
\begin{equation}\label{eq:pauli_product}
\sigma_i\sigma_j \;=\; \delta_{ij}\,\mathbb 1 + i\,\epsilon_{ijk}\,\sigma_k,
\qquad \mathrm{Tr}\,\mathbb 1 = 2,\qquad \mathrm{Tr}\,\sigma_k = 0,
\end{equation}
from which $\mathrm{Tr}(\sigma_i\sigma_j)=2\delta_{ij}$:
\begin{equation}\label{eq:pauli_trace}
\mathrm{Tr}(\sigma_i\sigma_j) = \mathrm{Tr}\!\left(\delta_{ij}\mathbb 1 + i\epsilon_{ijk}\sigma_k\right) = 2\,\delta_{ij}.
\end{equation}
Hence, with no transpose anywhere,
\begin{equation}\label{eq:trace_result}
\mathrm{Tr}\!\left[(\hat a\cdot\vec\sigma)(\hat b\cdot\vec\sigma)\right]
= \sum_{i,j} a_i b_j\,\mathrm{Tr}(\sigma_i\sigma_j)
= \sum_{i,j} a_i b_j\,2\delta_{ij} = 2\sum_i a_i b_i = 2\,(\hat a\cdot\hat b).
\end{equation}

\paragraph{Step 4: assemble.}
Substituting \eqref{eq:trace_result} into \eqref{eq:E_to_trace},
\begin{equation}\label{eq:E_quantum_result}
\boxed{\;E(\hat a,\hat b) = -\tfrac12\cdot 2\,(\hat a\cdot\hat b) = -\,\hat a\cdot\hat b = -\cos\theta_{ab}.\;}
\end{equation}
This is the standard singlet correlation. Note the marginals: setting one of
the analyzers aside, $\langle\Psi^-|(\hat a\cdot\vec\sigma)\otimes\mathbb 1|\Psi^-\rangle
=\mathrm{Tr}(\rho_A\,\hat a\cdot\vec\sigma)=\tfrac12\mathrm{Tr}(\hat a\cdot\vec\sigma)=0$,
so each wing alone is perfectly random regardless of the distant setting --
the no-signaling property to which we return (cross-ref \TA{11}).

\subsection{Worked $2\times2$ micro-example: $\hat a=\hat z$, $\hat b$ in the $x$-$z$ plane}
\label{subsec:micro}

To make \eqref{eq:E_quantum_result} concrete we compute one correlator by
brute force, with no invariance tricks. Take $\hat a=\hat z$ so that
$\hat a\cdot\vec\sigma=\sigma_z$, and take $\hat b=(\sin\theta,0,\cos\theta)$
in the $x$-$z$ plane, so that
\begin{equation}\label{eq:micro_b}
\hat b\cdot\vec\sigma = \sin\theta\,\sigma_x + \cos\theta\,\sigma_z
= \begin{pmatrix} \cos\theta & \sin\theta \\ \sin\theta & -\cos\theta \end{pmatrix},
\qquad
\sigma_z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}.
\end{equation}
Order the product basis as
$\{|{\uparrow\uparrow}\rangle,|{\uparrow\downarrow}\rangle,|{\downarrow\uparrow}\rangle,|{\downarrow\downarrow}\rangle\}$.
The singlet vector is
\begin{equation}\label{eq:micro_singlet}
|\Psi^-\rangle = \tfrac{1}{\sqrt2}\,(0,\,1,\,-1,\,0)^{\mathsf T}.
\end{equation}
First act with $(\hat b\cdot\vec\sigma)$ on Bob's (second) qubit. On the two
populated components,
\begin{align}
(\hat b\cdot\vec\sigma)|{\downarrow}\rangle &= \sin\theta\,|{\uparrow}\rangle - \cos\theta\,|{\downarrow}\rangle, \label{eq:micro_bdown}\\
(\hat b\cdot\vec\sigma)|{\uparrow}\rangle &= \cos\theta\,|{\uparrow}\rangle + \sin\theta\,|{\downarrow}\rangle. \label{eq:micro_bup}
\end{align}
Hence
\begin{equation}\label{eq:micro_after_b}
(\mathbb 1\otimes \hat b\cdot\vec\sigma)|\Psi^-\rangle
= \tfrac{1}{\sqrt2}\Bigl[\,|{\uparrow}\rangle(\sin\theta|{\uparrow}\rangle-\cos\theta|{\downarrow}\rangle)
- |{\downarrow}\rangle(\cos\theta|{\uparrow}\rangle+\sin\theta|{\downarrow}\rangle)\,\Bigr].
\end{equation}
Now act with $\sigma_z$ on Alice's (first) qubit, which multiplies the
$|{\uparrow}\rangle_A$ terms by $+1$ and the $|{\downarrow}\rangle_A$ terms by $-1$:
\begin{equation}\label{eq:micro_after_az}
(\sigma_z\otimes \hat b\cdot\vec\sigma)|\Psi^-\rangle
= \tfrac{1}{\sqrt2}\Bigl[\,|{\uparrow}\rangle(\sin\theta|{\uparrow}\rangle-\cos\theta|{\downarrow}\rangle)
+ |{\downarrow}\rangle(\cos\theta|{\uparrow}\rangle+\sin\theta|{\downarrow}\rangle)\,\Bigr].
\end{equation}
Finally take the inner product with $\langle\Psi^-|=\tfrac{1}{\sqrt2}(\langle{\uparrow\downarrow}| - \langle{\downarrow\uparrow}|)$.
Only the $|{\uparrow\downarrow}\rangle$ and $|{\downarrow\uparrow}\rangle$
components of \eqref{eq:micro_after_az} survive:
\begin{equation}\label{eq:micro_inner}
E(\hat z,\hat b)
= \tfrac{1}{2}\Bigl[\,(1)\cdot(-\cos\theta)\;-\;(1)\cdot(+\cos\theta)\,\Bigr]
= \tfrac{1}{2}(-\cos\theta-\cos\theta) = -\cos\theta.
\end{equation}
This reproduces \eqref{eq:E_quantum_result}, $E(\hat z,\hat b)=-\hat z\cdot\hat b=-\cos\theta$,
by explicit $2\times2$ matrix multiplication.

\subsection{Evaluating $S$ at the optimal angles: $S=2\sqrt2$}
\label{subsec:optimal}

Choose all four orientations coplanar (say in the $x$-$z$ plane) at the
angles
\begin{equation}\label{eq:opt_angles}
\hat a:0,\qquad \hat a':\tfrac{\pi}{2},\qquad \hat b:\tfrac{\pi}{4},\qquad \hat b':\tfrac{3\pi}{4}.
\end{equation}
The relevant relative angles and their cosines are
\begin{align}
\theta_{ab}=\tfrac{\pi}{4} &\Rightarrow \cos\theta_{ab}=\tfrac{1}{\sqrt2}, &
\theta_{ab'}=\tfrac{3\pi}{4} &\Rightarrow \cos\theta_{ab'}=-\tfrac{1}{\sqrt2}, \label{eq:opt_cos1}\\
\theta_{a'b}=\tfrac{\pi}{4} &\Rightarrow \cos\theta_{a'b}=\tfrac{1}{\sqrt2}, &
\theta_{a'b'}=\tfrac{\pi}{4} &\Rightarrow \cos\theta_{a'b'}=\tfrac{1}{\sqrt2}. \label{eq:opt_cos2}
\end{align}
With $E=-\cos\theta$ from \eqref{eq:E_quantum_result}, the four correlators are
\begin{equation}\label{eq:opt_E}
E(\hat a,\hat b)=-\tfrac{1}{\sqrt2},\quad
E(\hat a,\hat b')=+\tfrac{1}{\sqrt2},\quad
E(\hat a',\hat b)=-\tfrac{1}{\sqrt2},\quad
E(\hat a',\hat b')=-\tfrac{1}{\sqrt2}.
\end{equation}
Substituting into the CHSH combination \eqref{eq:CHSH_def},
\begin{equation}\label{eq:opt_S}
S = E(\hat a,\hat b) - E(\hat a,\hat b') + E(\hat a',\hat b) + E(\hat a',\hat b')
= -\tfrac{1}{\sqrt2} - \tfrac{1}{\sqrt2} - \tfrac{1}{\sqrt2} - \tfrac{1}{\sqrt2}
= -\frac{4}{\sqrt2} = -2\sqrt2,
\end{equation}
so $|S|=2\sqrt2$. (Reflecting all of Bob's angles, or equivalently swapping the
position of the minus sign in \eqref{eq:CHSH_def}, gives the symmetric value
$S=+2\sqrt2$; only $|S|$ is physical.) The magnitude
\begin{equation}\label{eq:tsirelson_value}
|S|_{\max} = 2\sqrt2 \approx 2.828
\end{equation}
exceeds the local-realist ceiling $2$ of \eqref{eq:LHV_bound} by the factor
$\sqrt2$, the hallmark quantum violation. It is saturated by the singlet, in
keeping with the extremality discussed in \TA{8}.

\subsection{The Tsirelson bound $|S|\le 2\sqrt2$ in quantum mechanics}
\label{subsec:tsirelson}

We now prove that $2\sqrt2$ is not merely attained but is the quantum
\emph{maximum}: no state and no choice of $\pm1$-valued observables can push
$|S|$ higher. This is the Tsirelson bound, derived via the Landau identity.

Let $A,A',B,B'$ now denote Hermitian operators with eigenvalues $\pm1$, i.e.
\begin{equation}\label{eq:invol}
A^2=A'^2=B^2=B'^2=\mathbb 1,\qquad A=A^\dagger,\ \text{etc.},
\end{equation}
with $A,A'$ acting on Alice's factor and $B,B'$ on Bob's, so that Alice's and
Bob's operators commute:
\begin{equation}\label{eq:AB_commute}
[A,B]=[A,B']=[A',B]=[A',B']=0.
\end{equation}
Define the CHSH operator
\begin{equation}\label{eq:CHSH_operator}
\mathcal S \;=\; A\otimes B - A\otimes B' + A'\otimes B + A'\otimes B'
\;=\; A(B-B') + A'(B+B'),
\end{equation}
so that $S=\langle\mathcal S\rangle$ in any state. Compute $\mathcal S^2$.
Expanding and using \eqref{eq:invol}-\eqref{eq:AB_commute},
\begin{align}
\mathcal S^2 &= \bigl[A(B-B')+A'(B+B')\bigr]^2 \notag\\
&= A^2(B-B')^2 + A'^2(B+B')^2 + AA'(B-B')(B+B') + A'A(B+B')(B-B'). \label{eq:S2_expand}
\end{align}
Now $(B-B')^2 = B^2 - BB' - B'B + B'^2 = 2\mathbb 1 - (BB'+B'B)$ and similarly
$(B+B')^2 = 2\mathbb 1 + (BB'+B'B)$, so
\begin{equation}\label{eq:S2_first}
A^2(B-B')^2 + A'^2(B+B')^2 = (2\mathbb 1 - \{B,B'\}) + (2\mathbb 1 + \{B,B'\}) = 4\,\mathbb 1,
\end{equation}
where $\{X,Y\}=XY+YX$ and we used $A^2=A'^2=\mathbb 1$. For the cross terms,
$(B-B')(B+B') = B^2 + BB' - B'B - B'^2 = [B,B']$ and
$(B+B')(B-B') = B^2 - BB' + B'B - B'^2 = -[B,B']$, hence
\begin{equation}\label{eq:S2_cross}
AA'(B-B')(B+B') + A'A(B+B')(B-B') = AA'[B,B'] - A'A[B,B'] = [A,A']\,[B,B'].
\end{equation}
Combining \eqref{eq:S2_first} and \eqref{eq:S2_cross} gives the Landau
identity
\begin{equation}\label{eq:landau}
\boxed{\;\mathcal S^2 = 4\,\mathbb 1 + [A,A']\,[B,B'].\;}
\end{equation}
We bound the operator norm of the correction term. For any bounded operators,
$\|[X,Y]\|\le \|XY\|+\|YX\|\le 2\|X\|\,\|Y\|$. Since $A,A'$ are $\pm1$-valued
Hermitians, $\|A\|=\|A'\|=1$, so $\|[A,A']\|\le 2$; likewise $\|[B,B']\|\le 2$.
The two commutators act on different tensor factors and therefore commute, and
\begin{equation}\label{eq:norm_bound}
\bigl\|\,[A,A'][B,B']\,\bigr\| \le \|[A,A']\|\,\|[B,B']\| \le 2\cdot 2 = 4.
\end{equation}
Therefore, from \eqref{eq:landau},
\begin{equation}\label{eq:S2_norm}
\|\mathcal S^2\| \le \|4\,\mathbb 1\| + \bigl\|[A,A'][B,B']\bigr\| \le 4 + 4 = 8.
\end{equation}
For the Hermitian operator $\mathcal S$, $\|\mathcal S\|^2=\|\mathcal S^2\|\le 8$,
so $\|\mathcal S\|\le 2\sqrt2$, and for any state
\begin{equation}\label{eq:tsirelson}
\boxed{\;|S| = |\langle\mathcal S\rangle| \le \|\mathcal S\| \le 2\sqrt2.\;}
\end{equation}
This is the Tsirelson bound. The singlet with the angles \eqref{eq:opt_angles}
saturates it because there the local observables anticommute pairwise,
$\{A,A'\}=\{B,B'\}=0$ (e.g. $A=\sigma_z$, $A'=\sigma_x$), which makes
$[A,A']=2AA'$ and $[B,B']=2BB'$ each of maximal norm $2$, driving the
correction term in \eqref{eq:landau} to its extreme $+4$ on the singlet --
exactly the extremal, Bogomolnyi-\emph{analogue} stationarity noted in \TA{8}.

\subsection{The framework reading: $S^1_\theta$ holonomy as the geometry of non-separability}
\label{subsec:reading_frag}

We now state how the \GGtheory{} reading assigns this standard mathematics a
geometric meaning, without changing any number above. The two derivations
sit side by side: \eqref{eq:LHV_bound} shows that a \emph{factorized} model on
a shared scalar gives at most $|S|=2$, while \eqref{eq:tsirelson} shows
quantum mechanics reaches $2\sqrt2$. The framework's claim is that the
entangled pair realizes the latter, not the former, because of the geometry of
the shared phase fiber.

\paragraph{One fiber, fixed at the source.}
The bulk phase $\Theta$ that labels the pair is not a hidden variable handed
independently to the two wings. It is the holonomy of a \emph{single}
$S^1_\theta$ fiber, fixed once at the common source where the pair is created.
Both boundary shadows -- Alice's wing and Bob's wing -- are projections of the
same non-separable bulk state threading that one fiber. Formally this is the
statement that $|\Psi^-\rangle$ in \eqref{eq:singlet_def} does not factor:
there exist no single-qubit states $|\phi\rangle_A,|\chi\rangle_B$ with
$|\Psi^-\rangle=|\phi\rangle_A\otimes|\chi\rangle_B$. The boundary marginals
$\rho_A=\rho_B=\tfrac12\mathbb 1$ computed in
Section~\ref{subsec:singlet_corr} are each maximally mixed precisely because
all the information lives in the shared fiber and none in either wing alone.

\paragraph{Why this evades Bell and Kochen-Specker.}
Bell's theorem, proved in \eqref{eq:LHV_bound}, excludes only
\emph{local} realism: definite outcomes assigned by per-wing response
functions \eqref{eq:LHV_responses} depending on a shared scalar. The framework
does not posit such functions; the shared $S^1_\theta$ holonomy is a property
of the joint state, not a pair of local responses, so there is nothing for
Bell's factorization \eqref{eq:LHV_correlator} to bite on. What Bell permits
-- non-local realism, in which the underlying reality is the single
non-separable bulk state -- is exactly what is realized. Likewise the
Kochen-Specker theorem forbids non-contextual hidden-variable assignments; the
framework's outcomes are \emph{contextual} (the value registered depends on the
analyzer setting that selects which boundary projection of the one bulk state
is read), so Kochen-Specker is respected, not violated.

\paragraph{No signaling.}
The bound \eqref{eq:tsirelson} is reached with no superluminal communication.
The marginal at Bob's wing is independent of Alice's setting: as computed in
\eqref{eq:E_quantum_result}'s aftermath, $\langle\mathbb 1\otimes(\hat b\cdot\vec\sigma)\rangle
=\mathrm{Tr}(\rho_B\,\hat b\cdot\vec\sigma)=0$ for every $\hat b$, and changing
$\hat a\to\hat a'$ leaves $\rho_B=\tfrac12\mathbb 1$ untouched because the
partial trace over Alice's factor is independent of which observable Alice
chooses. Hence Bob's local statistics carry no information about Alice's
setting; correlations appear only when the two records are later compared. This
is the operator-level no-signaling statement, consistent with the 4D causal
structure of \TA{11}. In particular nothing here reintroduces any radial
timelike shortcut: the non-separability is geometric (one fiber), not a
faster-than-light channel.

\subsection{Closing: boundary shadow and extremal saturation}
\label{subsec:closing_frag}

Two cross-references complete the reading. First, the violation
\eqref{eq:tsirelson_value} is, in the language of \TA{7}, a
\emph{boundary shadow}: the two wings are lower-dimensional boundary
projections of a single higher-dimensional bulk state, and the
$\cos\theta_{ab}$ correlation -- impossible to reproduce by any factorized
boundary model -- is the visible imprint of that bulk non-separability.
Second, the saturation $|S|=2\sqrt2$ is, in the language of \TA{8}, an
\emph{extremal} (Bogomolnyi-analogue) stationary point: the Landau identity
\eqref{eq:landau} shows that maximizing $|S|$ forces the correction
$[A,A'][B,B']$ to its norm ceiling $+4$, which is attained exactly when the
two pairs of local observables maximally anticommute on the singlet. The
quantum maximum is thus a saturated inequality, a stationary configuration of
the same character as the BPS-type bounds elsewhere in the paper, and the
singlet is the state that sits on it.

In summary, every number above is the textbook Bell-CHSH result: the
local-realist ceiling is $2$, the singlet correlator is $-\cos\theta_{ab}$, the
optimal angles give $|S|=2\sqrt2$, and the Tsirelson bound caps quantum
mechanics at $2\sqrt2$. The framework adds only the geometric reading -- one
non-separable bulk state on a single $S^1_\theta$ fiber, fixed at the source,
read as two correlated boundary shadows -- which reproduces standard quantum
correlations exactly while respecting no-signaling and introducing no
local-hidden-variable factorization.

\paragraph{Problems and worked solutions.}

\medskip\noindent\textbf{F.1.1. Prove the local bound $|S| \le 2$ by
the $\pm2$ argument.}

\noindent\emph{Solution.}  Step 1: in a local-realist model every
run carries pre-assigned values $A, A', B, B' = \pm1$
\eqref{eq:LHV_responses}; form the combination
\eqref{eq:LHV_integrand}: $A(B + B') + A'(B - B')$.  Step 2: since
$B, B' = \pm1$, one of $B + B'$ and $B - B'$ is $0$ and the other
$\pm2$ \eqref{eq:LHV_pm2}, so the combination is $\pm2$ run by run.
Step 3: averaging over any distribution of hidden variables keeps
$|S| \le 2$ \eqref{eq:LHV_bound} --- and note what was \emph{not}
assumed: nothing about quantum mechanics, dynamics, or fields; only
that the four answers exist before the questions are chosen. That
is the assumption experiment falsifies.  Physics intuition: the
CHSH bound is bookkeeping about pre-written answer sheets --- four
$\pm1$ entries cannot conspire beyond $2$ on average; when nature
scores $2.83$, the conclusion is not that arithmetic failed but
that the answers were not written down in the 4D variables Bell
allowed.

\medskip\noindent\textbf{F.3.1. Evaluate the singlet correlator at
explicit angles.}

\noindent\emph{Solution.}  Step 1: the chapter's trace computation
\eqref{eq:E_to_trace}--\eqref{eq:E_quantum_result} gives
$E(\hat a, \hat b) = -\hat a \cdot \hat b$.  Step 2: numbers:
aligned analyzers, $E = -1$ (perfect anticorrelation, the
spin-zero fingerprint \eqref{eq:spin_zero}); perpendicular,
$E = 0$; at $45^\circ$, $E = -1/\sqrt2 = -0.7071$ --- the value
that powers the optimal CHSH set.  Step 3: cross-check with the
micro-example route \eqref{eq:micro_inner}: expanding
$\ket{\Psi^-}$ in Bob's rotated basis reproduces
$-\cos\theta$ amplitude by amplitude --- the trace formula and the
state-expansion bookkeeping agree, as they must.  Physics intuition:
 $-\cos\theta$ interpolates between ``always opposite''
and ``no correlation'' more smoothly than any classical
answer-sheet can --- the classical best is a straight line in
$\theta$, and the experimental curve bends; that bend is the whole
quantum surplus.

\medskip\noindent\textbf{F.4.1. Assemble $S$ at the optimal angles.}

\noindent\emph{Solution.}  Step 1: take the standard set
\eqref{eq:opt_angles}: $\hat a = 0$, $\hat a' = 90^\circ$,
$\hat b = 45^\circ$, $\hat b' = 135^\circ$ (all in one plane).
Step 2: each correlator is $\mp 1/\sqrt2$
\eqref{eq:opt_cos1}--\eqref{eq:opt_E}, and the signed combination
\eqref{eq:opt_S} stacks all four with the same sign:
$S = -4/\sqrt2 = -2\sqrt2 = -2.828$.  Step 3: check the
geometry of why: the primed/unprimed axes interleave at
$45^\circ$ steps, so every term sits on the steepest part of the
$-\cos\theta$ curve --- the quantum surplus of F.3.1 harvested four
times coherently; any misalignment reduces $|S|$, which is why the
experiments quote settings to fractions of a degree.  Physics intuition:
 $2\sqrt2$ is what four perfectly-placed taps on a bent
curve can extract --- the bend gives each pair $1/\sqrt2$ where
flat classical lines could average no better than $\tfrac12$
apiece, and $4 \times 1/\sqrt2$ beats $4 \times \tfrac12$ by
exactly the observed margin.

\medskip\noindent\textbf{F.5.1. Derive Tsirelson from the operator
identity, and explain the gap to $4$.}

\noindent\emph{Solution.}  Step 1: for $\pm1$-valued commuting-wing
observables, square the CHSH operator \eqref{eq:CHSH_operator}: the
cross terms collapse to $\mathcal S^2 = 4\cdot\mathbb 1 +
[A, A'][B, B']$ \eqref{eq:landau}.  Step 2: bound the commutators:
$\|[A, A']\| \le 2$, $\|[B, B']\| \le 2$, so
$\|\mathcal S^2\| \le 8$ and $\|\mathcal S\| \le 2\sqrt2$
\eqref{eq:tsirelson} --- attained exactly when both commutators are
maximal, i.e.\ mutually unbiased settings on each wing: quantum
mechanics saturates its own algebra.  Step 3: the gap to the
no-signaling maximum $S = 4$ (the PR-box value): algebraically,
$4$ would need $\|[A,A'][B,B']\| = 12$, impossible for bounded
$\pm1$ observables --- and in the framework reading the boundary
correlations are the CPTP shadow of one bulk configuration
(\TA{8}), an \emph{extremal} shadow but a shadow still: the
projection's algebra stops at $2\sqrt2$, and any measured excess
is a stated kill criterion of the Discussions.  Physics intuition:
$2$ is the ceiling for pre-written answers, $4$ for lawless
signaling boxes, and $2\sqrt2$ --- the geometric mean --- is where
a deterministic bulk shows through a trace; nature sitting exactly
on the algebraic edge is the single most precise hint that the
boundary description is a projection of something.

\medskip\noindent\textbf{F.6.1. Verify no-signaling from the
marginals.}

\noindent\emph{Solution.}  Step 1: Alice's local statistics are
governed by her reduced state: for the singlet,
$\mathrm{Tr}_B\,\ketbra{\Psi^-}{\Psi^-} = \tfrac12\mathbb 1$
\eqref{eq:marginal_trace} --- independent of anything Bob does.
Step 2: check via correlators: Alice's marginal
$\langle A \rangle = \mathrm{Tr}(\tfrac12\mathbb 1 \cdot
\hat a \cdot \vec\sigma) = 0$ for every setting of Bob's
apparatus and every choice $\hat b$ vs $\hat b'$ --- Bob's dial
never moves Alice's needle (\TA{11}).  Step 3: the framework's
reason: the bulk flow is 6D-local and the boundary state is its
partial trace; a trace over Bob's wing cannot depend on the local
operation applied there, so no-signaling is a one-line theorem of
the projection --- correlation without communication, exactly as
the loophole-free experiments demand.  Physics intuition: the
singlet is a pair of dice cast from one mould, not a pair of
radios --- read together they rhyme perfectly, read alone each is
pure noise; and the framework locates the mould in the bulk, one
stratum below anything a signal could traverse.


\GGtakeaway{\emph{Standard account:} CHSH is violated at
$2\sqrt2$, local realism is dead, and the textbook stops there.
\emph{What it left open:} what carries the correlation, and why
the violation stops exactly at Tsirelson rather than at the
no-signaling maximum $4$. \emph{The bulk reading:} the
correlation is carried by one bulk configuration spanning both
wings; the boundary value saturates $2\sqrt2$ because the
projection is an extremal CPTP image of that configuration
(\TA{8}), and no-signaling (\TA{11}) is automatic because the
trace of a local bulk flow cannot transmit; the appendix's
holonomy-sampling calculation is the constructive proof that
``deterministic, local upstairs; Tsirelson-saturating downstairs''
is arithmetic, not aspiration.}

\section{Wilson-line holonomy: parallel transport on the phase fiber}
\label{app:Wilson_holonomy}

\GGteachbox{Where the quantum phase lives: the compact circle
$S^1_\theta$ as the fiber of a principal $U(1)$ bundle, developed
pedagogically --- connection, parallel transport, holonomy, and the
Wilson line as the gauge-invariant observable of the fiber. The
appendix then cashes the geometry in the laboratory: the
Aharonov--Bohm phase and the Berry phase as physical holonomies,
and the bridge to the rest of the cohort --- the same Wilson-line
object that carries the quantum phase here is the electroweak
order parameter $\theta_H$ of the particle-physics volume
(\ANine{}), making this the single most reused geometric idea in
the program.}

\GGreadalong{\citet{Berry1984} for the geometric phase;
\citet{AharonovBohm1959} for the laboratory Wilson line;
\citet{WuYang1975} for the fiber-bundle dictionary;
\citet{ArisakaP4Real} for the derivation of the boundary's single
imaginary unit from this same real fiber.}

This appendix expands the terse statement of the main text into a
self-contained, pedagogical derivation. The physical picture is the
following. In the framework the quantum phase does not live in the
$3{+}1$ spacetime that the observer inhabits; it lives on a compact
internal circle $S^1_\theta$ attached to every spacetime point. That
circle is the fiber of a \emph{principal $U(1)$ bundle}. A quantum
phase is then nothing but the \emph{holonomy} of a connection on this
bundle --- the angle by which the fiber coordinate is rotated when one
parallel-transports it around a loop. The measured outcome is the
\emph{gauge-invariant} holonomy $\Theta_H$ that the boundary projection
of the source reads out. Everything below is built up from first
principles so that a first-year graduate student can follow each step;
the only prerequisites are vector calculus, Stokes' theorem, and the
elementary theory of $U(1)=\{e^{i\alpha}\}$.

We use $\hbar$, $c$ explicitly where physics demands it and set them to
$1$ only when clearly stated. Repeated indices are summed. Greek
indices $\mu,\nu$ run over spacetime; Latin $i,j$ over space.

%========================================================
\subsection{The phase fiber as a principal $U(1)$ bundle}
\label{app:Wilson_bundle}

A \emph{principal $U(1)$ bundle} is a space $P$ that locally looks like
$M\times S^1$, where $M$ is the base (here spacetime, or a path in it)
and the fiber is the circle group
\begin{equation}\label{eq:U1group}
U(1)=\{\,e^{i\alpha}\ :\ \alpha\in[0,2\pi)\,\},
\qquad e^{i\alpha}\,e^{i\beta}=e^{i(\alpha+\beta)}.
\end{equation}
Over each base point $x$ there sits a copy of the circle, the
\emph{phase fiber} $S^1_\theta$, whose coordinate we call $\theta(x)$.
The group acts on each fiber by rigid rotation,
\begin{equation}\label{eq:fibre_action}
\theta \;\longmapsto\; \theta+\alpha \pmod{2\pi},
\end{equation}
and this action is \emph{free} (only $\alpha=0$ fixes a point) and
\emph{transitive} on the fiber (any point can be reached from any
other). The total object is denoted $P\xrightarrow{\ \pi\ }M$, with
$\pi$ the projection that forgets the fiber coordinate.

A complex matter field $\psi(x)=|\psi(x)|\,e^{i\theta(x)}$ is a
\emph{section} of an associated line bundle: its modulus is physical,
but the \emph{phase} $\theta(x)$ is a fiber coordinate and therefore has
no absolute meaning. Only \emph{differences} of phase, and ultimately
loop holonomies, can be measured. This is the geometric origin of the
local $U(1)$ symmetry
\begin{equation}\label{eq:gauge_psi}
\psi(x)\;\longmapsto\; e^{i\lambda(x)}\,\psi(x),
\end{equation}
with $\lambda(x)$ an arbitrary smooth function: relabeling the fiber
coordinate point by point must leave physics untouched.

%========================================================
\subsection{Connection one-form and the covariant derivative}
\label{app:Wilson_connection}

To compare phases at \emph{different} base points we need a rule that
says ``these two fiber coordinates are aligned.'' That rule is a
\emph{connection}. On a $U(1)$ bundle a connection is encoded by a real
\emph{connection one-form}
\begin{equation}\label{eq:Adef}
A \;=\; A_\mu(x)\,dx^\mu,
\end{equation}
the gauge potential. Its job is to define an infinitesimal notion of
``staying parallel'': if we move from $x$ to $x+dx$, the fiber
coordinate must be rotated by $-A_\mu(x)\,dx^\mu$ to keep $\psi$
parallel. Concretely, the ordinary derivative $\partial_\mu\psi$ is not
covariant under \eqref{eq:gauge_psi} because $\partial_\mu(e^{i\lambda}\psi)
=e^{i\lambda}(\partial_\mu+i\partial_\mu\lambda)\psi$ carries the extra
$i\partial_\mu\lambda$ term. We cure this by introducing the
\emph{covariant derivative}
\begin{equation}\label{eq:Dmu}
D_\mu \;=\; \partial_\mu - i\,A_\mu,
\qquad D_\mu\psi=(\partial_\mu - i A_\mu)\psi,
\end{equation}
and \emph{demanding} that it transform homogeneously,
$D_\mu\psi\mapsto e^{i\lambda}D_\mu\psi$. Working out what $A_\mu$ must
do for this to hold,
\begin{align}\label{eq:Dcovariance}
D_\mu(e^{i\lambda}\psi)
&=(\partial_\mu - iA_\mu')\,(e^{i\lambda}\psi)\nonumber\\
&=e^{i\lambda}\big[\partial_\mu + i(\partial_\mu\lambda)
   - iA_\mu'\big]\psi
\;\overset{!}{=}\; e^{i\lambda}(\partial_\mu - iA_\mu)\psi,
\end{align}
which forces the now-familiar transformation law of the potential,
\begin{equation}\label{eq:A_gauge}
A_\mu \;\longmapsto\; A_\mu' = A_\mu + \partial_\mu\lambda,
\qquad\text{i.e.}\qquad A\;\longmapsto\;A+d\lambda .
\end{equation}
Thus the connection one-form $A$ is exactly the object that lets us
differentiate phases covariantly, and it is defined only up to the
addition of an exact one-form $d\lambda$ --- a \emph{gauge}
transformation. The geometric content of \eqref{eq:Dmu} is that
$A_\mu\,dx^\mu$ is the angle the fiber must be counter-rotated through
to remain ``horizontal'' (parallel) over the displacement $dx^\mu$.

%========================================================
\subsection{The parallel-transport equation}
\label{app:Wilson_transport}

Now fix a path $\gamma:[0,1]\to M$, $s\mapsto x(s)$, with tangent
$\dot x^\mu(s)=dx^\mu/ds$. \emph{Parallel transport} of the phase along
$\gamma$ is the demand that the covariant derivative \emph{along the
path} vanish,
\begin{equation}\label{eq:transport_def}
\dot x^\mu D_\mu\psi=0
\quad\Longleftrightarrow\quad
\frac{d\psi}{ds}=i\,A_\mu(x(s))\,\dot x^\mu(s)\,\psi .
\end{equation}
This is a first-order linear ordinary differential equation for the
transported phase. Because $U(1)$ is abelian (the rotations
\eqref{eq:U1group} commute), the values of $A$ at different points along
the path commute, and the ODE integrates immediately. Writing
$\psi(s)=U(s,0)\,\psi(0)$, separating variables,
\begin{equation}\label{eq:transport_solve}
\frac{dU}{ds}=i\,A_\mu\dot x^\mu\,U
\;\Longrightarrow\;
\frac{dU}{U}=i\,A_\mu\dot x^\mu\,ds
\;\Longrightarrow\;
U(s,0)=\exp\!\Big(i\!\int_0^s A_\mu(x(s'))\,\dot x^\mu(s')\,ds'\Big).
\end{equation}
Changing the integration variable from the parameter $s'$ to the point
$x$ along the curve, $\dot x^\mu\,ds'=dx^\mu$, the transport operator
becomes a line integral of the connection,
\begin{equation}\label{eq:Wline_open}
\boxed{\;
W_\gamma \;=\; \exp\!\Big(i\!\int_\gamma A_\mu\,dx^\mu\Big)
\;=\;\exp\!\Big(i\!\int_\gamma A\Big).
\;}
\end{equation}
This $W_\gamma$ is the \emph{open Wilson line}: it is the group element
of $U(1)$ that rotates the phase fiber as we drag $\psi$ from the start
to the end of $\gamma$.

%========================================================
\subsection{Path ordering and the reduction to $U(1)$}
\label{app:Wilson_pathorder}

For a general (non-abelian) gauge group the connection
$A_\mu=A_\mu^a T^a$ is matrix-valued, and the values at different path
points do \emph{not} commute, $[A(x),A(x')]\neq0$. Equation
\eqref{eq:transport_def} is then a matrix ODE whose solution is the
\emph{path-ordered exponential}
\begin{equation}\label{eq:pathordered}
W_\gamma=\mathcal{P}\exp\!\Big(i\!\int_\gamma A_\mu\,dx^\mu\Big)
\equiv \mathbf{1}+i\!\int_\gamma\! A
+ (i)^2\!\!\iint_{s_1>s_2}\!\! A(s_1)A(s_2)+\cdots,
\end{equation}
where the symbol $\mathcal{P}$ orders the factors so that the field at
later path-parameter stands to the left. Path ordering is essential
precisely because the matrices fail to commute. For the abelian group
$U(1)$ relevant to the phase fiber, all factors commute, the ordering is
vacuous, and \eqref{eq:pathordered} collapses back to the ordinary
exponential \eqref{eq:Wline_open}:
\begin{equation}\label{eq:U1reduction}
\mathcal{P}\exp\!\Big(i\!\oint A\Big)
\;\xrightarrow[\;U(1)\;]{}\;
\exp\!\Big(i\!\oint A\Big).
\end{equation}
We keep $\mathcal P$ in the general statement only to flag the
non-abelian generalization; for the rest of this appendix we work with
the abelian exponential.

%========================================================
\subsection{Closed loops, holonomy, and Stokes' theorem}
\label{app:Wilson_stokes}

The physically invariant object is obtained by transporting the phase
around a \emph{closed} loop $\partial\Sigma$ that bounds a surface
$\Sigma$. The \emph{holonomy} is the closed Wilson line
\begin{equation}\label{eq:holonomy_def}
W(\partial\Sigma)=\exp\!\Big(i\!\oint_{\partial\Sigma} A_\mu\,dx^\mu\Big).
\end{equation}
Define the \emph{curvature} two-form (the field strength) as the
exterior derivative of the connection,
\begin{equation}\label{eq:Fdef}
F=dA,\qquad
F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu,
\qquad F=\tfrac12 F_{\mu\nu}\,dx^\mu\wedge dx^\nu .
\end{equation}
For abelian $U(1)$ there is no $A\wedge A$ term (it vanishes because
$A\wedge A=0$ for a one-form valued in a one-dimensional algebra), so
$F$ is simply the curl of $A$. Stokes' theorem converts the line
integral of $A$ around the boundary into the surface integral of its
curl over $\Sigma$,
\begin{equation}\label{eq:stokes}
\oint_{\partial\Sigma} A=\int_\Sigma dA=\int_\Sigma F
\;\equiv\;\Phi_\Sigma,
\end{equation}
where $\Phi_\Sigma$ is the \emph{flux} of the curvature through the
surface. Hence the holonomy of a $U(1)$ connection around a loop is the
exponential of $i$ times the enclosed flux,
\begin{equation}\label{eq:holonomy_flux}
\boxed{\;
W(\partial\Sigma)=\exp\!\Big(i\!\int_\Sigma F\Big)
=\exp\!\big(i\,\Phi_\Sigma\big).
\;}
\end{equation}
This is the central identity: a loop ``feels'' only the curvature it
encloses, never the local value of $A$. The holonomy angle
\begin{equation}\label{eq:Theta_H_def}
\Theta_H \;\equiv\; \oint_{\partial\Sigma} A \;=\;\Phi_\Sigma
\pmod{2\pi}
\end{equation}
is the quantity we will identify with the measured phase.

%========================================================
\subsection{Gauge invariance of the holonomy}
\label{app:Wilson_gauge}

We now show explicitly that \eqref{eq:holonomy_flux} is invariant under
the gauge transformation \eqref{eq:A_gauge}. Under
$A\mapsto A+d\lambda$ the open Wilson line picks up the boundary values
of $\lambda$,
\begin{equation}\label{eq:open_cov}
\int_\gamma (A+d\lambda)
=\int_\gamma A+\big[\lambda(x_{\rm end})-\lambda(x_{\rm start})\big]
\;\Longrightarrow\;
W_\gamma\;\mapsto\; e^{i\lambda(x_{\rm end})}\,W_\gamma\,
e^{-i\lambda(x_{\rm start})},
\end{equation}
i.e.\ the open line transforms \emph{covariantly} --- it dresses up with
the gauge phase of its two endpoints, exactly so that the combination
$\psi^\dagger(x_{\rm end})\,W_\gamma\,\psi(x_{\rm start})$ is invariant.
For a \emph{closed} loop the start and end coincide,
$x_{\rm end}=x_{\rm start}$, so the two boundary terms cancel,
\begin{equation}\label{eq:closed_inv}
\oint_{\partial\Sigma}(A+d\lambda)
=\oint_{\partial\Sigma}A+\underbrace{\oint_{\partial\Sigma}d\lambda}_{=0}
=\oint_{\partial\Sigma}A ,
\end{equation}
because the integral of an exact form around a closed contractible loop
vanishes ($\lambda$ is single-valued). Equivalently, $F=dA$ is itself
gauge invariant since $d(d\lambda)=0$. Therefore
\begin{equation}\label{eq:holonomy_gauge_inv}
W(\partial\Sigma)=\exp\!\big(i\,\Phi_\Sigma\big)
\quad\text{is gauge invariant,}
\end{equation}
and the holonomy angle $\Theta_H$ of \eqref{eq:Theta_H_def} is a genuine
observable. This is the precise sense in which $\Theta_H$ ``is
gauge-invariant'' as asserted in the main text.

%========================================================
\subsection{Aharonov--Bohm and Berry phases as the same holonomy}
\label{app:Wilson_AB_Berry}

The deep point is that two phenomena usually taught separately --- the
Aharonov--Bohm (AB) effect and the Berry phase --- are \emph{the same
holonomy} \eqref{eq:holonomy_flux}, differing only in what plays the
role of the base space and the connection.

\paragraph{Aharonov--Bohm.}
A charged particle of charge $e$ moving in a region where the magnetic
field $\mathbf B$ vanishes but the vector potential $\mathbf A$ does not
acquires, along a path $\gamma$, the phase factor
\begin{equation}\label{eq:AB_connection}
\psi\;\mapsto\;\exp\!\Big(i\frac{e}{\hbar}\!\int_\gamma
\mathbf A\cdot d\mathbf x\Big)\psi .
\end{equation}
Comparing with \eqref{eq:Wline_open}, the geometric connection on the
phase fiber is identified with the electromagnetic potential through
$A_\mu^{\rm fiber}=(e/\hbar)A_\mu^{\rm EM}$. The interference of two
paths enclosing a region of flux $\Phi$ is governed by the holonomy of
the difference loop, and \eqref{eq:holonomy_flux} gives
\begin{equation}\label{eq:AB_phase}
\boxed{\;
\Theta_{\rm AB}=\frac{e}{\hbar}\oint_{\partial\Sigma}
\mathbf A\cdot d\mathbf x
=\frac{e}{\hbar}\int_\Sigma \mathbf B\cdot d\mathbf S
=\frac{e}{\hbar}\,\Phi .
\;}
\end{equation}

\paragraph{Berry phase.}
When a Hamiltonian $H(\mathbf R)$ depends on slowly varying parameters
$\mathbf R$ and is taken adiabatically around a closed loop $C$ in
parameter space, the energy eigenstate $|n(\mathbf R)\rangle$ returns to
itself up to the \emph{Berry phase}
\begin{equation}\label{eq:Berry}
\gamma_n=\oint_C \mathcal A_n\cdot d\mathbf R,
\qquad
\mathcal A_n=i\,\langle n(\mathbf R)|\nabla_{\mathbf R}|n(\mathbf R)\rangle ,
\end{equation}
where $\mathcal A_n$ is the \emph{Berry connection} --- a $U(1)$
connection on a bundle over parameter space --- and its curvature
$\mathcal F_n=d\mathcal A_n$ is the Berry curvature. By exactly the
Stokes argument \eqref{eq:stokes},
\begin{equation}\label{eq:Berry_flux}
\gamma_n=\int_\Sigma \mathcal F_n ,
\end{equation}
the Berry phase is the flux of the Berry curvature, i.e.\ the holonomy
\eqref{eq:holonomy_flux} of the Berry connection. The AB phase is the
special case in which the connection is the electromagnetic potential
and the curvature is the magnetic flux. \emph{Both are holonomies of a
$U(1)$ connection on a circle fiber} --- precisely the structure of the
phase fiber $S^1_\theta$.

%========================================================
\subsection{Worked micro-example: the Aharonov--Bohm ring}
\label{app:Wilson_ABring}

Consider an electron confined to a thin ring of radius $a$ threaded by a
solenoid carrying flux $\Phi$, with $\mathbf B=0$ on the ring itself.
Use a parametrization by azimuthal angle $\varphi\in[0,2\pi)$. A
convenient gauge for the flux of an infinitely thin solenoid is the pure
azimuthal potential
\begin{equation}\label{eq:AB_potential}
\mathbf A=\frac{\Phi}{2\pi r}\,\hat{\boldsymbol\varphi},
\qquad\text{so on the ring }(r=a):\quad
\mathbf A\cdot d\mathbf x=\frac{\Phi}{2\pi a}\,(a\,d\varphi)
=\frac{\Phi}{2\pi}\,d\varphi .
\end{equation}
The accumulated AB phase for a particle going once around the ring is,
from \eqref{eq:AB_connection},
\begin{equation}\label{eq:AB_ring_phase}
\Theta_{\rm AB}=\frac{e}{\hbar}\oint_0^{2\pi}\frac{\Phi}{2\pi}\,d\varphi
=\frac{e}{\hbar}\,\Phi
=2\pi\,\frac{\Phi}{\Phi_0},
\qquad
\Phi_0\equiv\frac{2\pi\hbar}{e}=\frac{h}{e},
\end{equation}
where $\Phi_0$ is the flux quantum. Now split the incoming beam so that
one partial wave travels clockwise and the other counter-clockwise to a
detector on the far side; the \emph{relative} phase between the two arms
is the holonomy of the full enclosing loop,
\begin{equation}\label{eq:AB_interference}
\Delta\Theta=\Theta_{\rm AB}^{\,(\circlearrowright)}
-\Theta_{\rm AB}^{\,(\circlearrowleft)}
=\frac{e}{\hbar}\oint \mathbf A\cdot d\mathbf x
=\frac{e}{\hbar}\,\Phi .
\end{equation}
The interference intensity at the detector is therefore
\begin{equation}\label{eq:AB_intensity}
I(\Phi)\;\propto\;1+\cos\!\big(\Delta\Theta\big)
=1+\cos\!\Big(\frac{e}{\hbar}\,\Phi\Big)
=1+\cos\!\Big(2\pi\,\frac{\Phi}{\Phi_0}\Big),
\end{equation}
which oscillates with period $\Phi_0=h/e$ in the enclosed flux even
though the electron never touches the magnetic field. This is the
hallmark of holonomy: a measurable, gauge-invariant phase determined
entirely by the enclosed flux \eqref{eq:holonomy_flux}.

%========================================================
\subsection{Compact-fiber quantization}
\label{app:Wilson_quantization}

Because the fiber $S^1_\theta$ is a \emph{compact} circle, the
wavefunction along it must be \emph{single-valued}. Expanding $\psi$ in
the fiber coordinate $\theta\in[0,2\pi)$ in modes,
\begin{equation}\label{eq:fibre_modes}
\psi(\theta)=\sum_{q}c_q\,e^{iq\theta},
\end{equation}
single-valuedness $\psi(\theta+2\pi)=\psi(\theta)$ requires
\begin{equation}\label{eq:charge_integer}
e^{iq\,2\pi}=1
\quad\Longrightarrow\quad
q\in\mathbb{Z}.
\end{equation}
Thus the charge spectrum conjugate to the compact fiber is quantized in
integers. Geometrically, a state of charge $q$ transported once around
the fiber direction picks up holonomy $e^{iq\Theta_H}$, and consistency
of this around the closed circle is the quantization condition. The same
statement applied to the curvature flux is the \emph{Dirac
quantization} of the bundle: for the bundle to be globally well-defined
the total curvature flux through any closed $2$-surface must satisfy
\begin{equation}\label{eq:dirac_quant}
\frac{1}{2\pi}\oint_{S^2} F \;=\; n\in\mathbb{Z},
\end{equation}
the first Chern number of the $U(1)$ bundle. In every case the logic is
identical: \emph{compactness of the fiber forces the holonomy/charge
spectrum to be integer}. A non-integer ``charge'' would render the wave
multivalued on $S^1_\theta$ and is therefore forbidden.

%========================================================
\subsection{Framework reading: $\Theta_H$ as the measured holonomy}
\label{app:Wilson_framework}

We can now restate the main-text claims with full geometric backing. In
the framework:

\begin{enumerate}
\item The internal phase circle $S^1_\theta$ at each spacetime point is
the fiber of a principal $U(1)$ bundle, and the quantum phase is the
\emph{connection} $A$ on that bundle, Sec.~\ref{app:Wilson_connection}.

\item The measured quantum phase is the \emph{holonomy}
$\Theta_H=\oint A=\Phi_\Sigma$ of \eqref{eq:Theta_H_def}, the angle by
which the fiber is rotated on parallel transport around the measurement
loop. The source's boundary condition fixes this connection: the
holonomy is determined by the enclosed curvature flux set at the
boundary, just as the solenoid flux $\Phi$ fixed the AB phase in
\eqref{eq:AB_ring_phase}.

\item $\Theta_H$ is \emph{loop-dependent}: it depends on which loop is
traversed, because different loops enclose different flux $\Phi_\Sigma$,
so there is no single context-free number --- only a holonomy attached
to a specified loop. This geometric path-dependence should be
distinguished from Kochen--Specker measurement contextuality, which is
the dependence of an assigned value on the \emph{commuting set}
co-measured with it: the two notions are conceptually distinct, and it
is the latter (argued elsewhere in the paper, not from loop-dependence
alone) that the no-go theorems constrain.

\item Nevertheless $\Theta_H$ is \emph{single-valued in the $6$D bulk}:
on the full bundle the connection is globally defined, the flux is fixed,
and by the gauge invariance of Sec.~\ref{app:Wilson_gauge} the holonomy
of any given loop is a unique, well-defined number modulo $2\pi$. The
apparent ambiguity is only the freedom of fiber relabeling
\eqref{eq:A_gauge}, which cancels on closed loops.

\item The $4$D observer, living on the base, cannot see the fiber
coordinate $\theta$ directly (the circle is too small to resolve), but
\emph{does} read out the gauge-invariant holonomy $\Theta_H$ through the
boundary projection --- exactly as the AB detector reads out
$e\Phi/\hbar$ without ever entering the solenoid. The boundary
projection is the readout map $W(\partial\Sigma)\mapsto\Theta_H$ of
\eqref{eq:holonomy_flux}, and the measured outcome is this single
gauge-invariant angle.

\item Finally, compactness of $S^1_\theta$
(Sec.~\ref{app:Wilson_quantization}) quantizes the charge spectrum that
labels outcomes: only integer-charged sectors survive single-valuedness
\eqref{eq:charge_integer}, so the readout $\Theta_H$ sorts measurements
into a discrete, integer-labeled set of branches.
\end{enumerate}

In short, the holonomy formalism makes precise the three claims of the
main text: $\Theta_H$ is \emph{gauge-invariant} (closed-loop holonomy,
\eqref{eq:holonomy_gauge_inv}), \emph{contextual} (loop-dependent flux),
and \emph{single-valued in the bulk} (globally defined connection plus
compact-fiber quantization). The boundary projection of the source is the
operational realization of the holonomy readout, and the $4$D
measurement outcome is the angle $\Theta_H=\Phi_\Sigma\pmod{2\pi}$.

\subsection{The holonomy as the measurement label}

In the cohort the phase fiber acquires a vacuum value and its
fluctuation is identified with the Higgs; those gauge-sector
specializations are established in the constitution and the
gauge-sector companions and are not needed for the
quantum-measurement reading developed here. What \emph{is} needed
is the role of the \emph{local} holonomy at a measurement event.
That local value is the contextual joint holonomy $\Theta_H$ of
system and apparatus: it is $\psi$-ontic and non-local ---
\emph{not} a local hidden variable of the kind Bell's theorem
excludes (see the Bell--CHSH appendix) --- and it is read out by
the boundary projection. Which single outcome is registered is
then fixed by the \GRIP{} descent of \citet{ArisakaA6Gi4} into one
Geometric Attractor basin (\TA{1},\TA{4}), not by the holonomy
value alone. A $4$D observer cannot resolve $\Theta_H$ directly
because $S^1_\theta$ lies far below any accessible length scale.

One register note closes the chapter, because this appendix is where
the cohort's real-bulk statements touch ground. The fiber developed
above is a \emph{real} circle: every holonomy computed in this
appendix is a rotation of a real pair, and the boundary's
``$e^{i\theta}$'' is compressed notation for that $SO(2)$ rotation ---
the complex exponential is bookkeeping, not ontology. That is why
this appendix's bundle language and the origin-of-$i$ companion
(\citealp{ArisakaP4Real}) are one subject: the same real circle whose
holonomy is quantum interference here supplies, when paired with the
causal clock by the projection, the single imaginary unit of the
boundary state space --- derived-conditional in the quasi-free
regime, statement catalog in \S\ref{sec:disc-real-bulk}, custody as
everywhere: the dictionary owns the sectors, the geometry owns the
$i$. Problem G.1.1 makes the $U(1)\!\cong\!SO(2)$ reading explicit.

\paragraph{Problems and worked solutions.}

\medskip\noindent\textbf{G.1.1. Prove $U(1) \cong SO(2)$, transport a
real pair, and state what the complex phase is.}

\noindent\emph{Solution.}  Step 1: the map $e^{i\theta} \mapsto
R(\theta) = \bigl(\begin{smallmatrix}\cos\theta & -\sin\theta\\
\sin\theta & \cos\theta\end{smallmatrix}\bigr)$ is a group
isomorphism ($R(\theta_1)R(\theta_2) = R(\theta_1{+}\theta_2)$), and
differentiating at $\theta = 0$ identifies the generator: $i$ on the
complex side, $J = \bigl(\begin{smallmatrix}0 & -1\\ 1 &
0\end{smallmatrix}\bigr)$ with $J^2 = -\mathbb 1$ on the real side
--- the imaginary unit \emph{is} the quarter-turn of a real plane.
Step 2: transport the real pair $(u, v)$ around the fiber with the
parallel-transport equation \eqref{eq:transport_def}: the holonomy
\eqref{eq:holonomy_def} acts as $R(\Theta_H)$ on the pair, and every
interference formula of this appendix (Aharonov--Bohm intensity
\eqref{eq:AB_intensity} included) can be rewritten with $\cos$ and
$\sin$ of real rotations --- the physics never leaves the reals.
Step 3: the register and the experimental foil: this does \emph{not}
mean boundary quantum mechanics is ``real quantum mechanics'' in the
sense excluded by the loophole-free experiment of
\citet{Renou2021} --- that experiment rules out a real
\emph{boundary} Hilbert space with tensor structure, while the
cohort's claim is the opposite pairing: a real \emph{bulk} whose
projection installs a genuinely complex boundary theory
(\citealp{ArisakaP4Real}) --- exactly the combination the experiment
permits.  Physics intuition: rotate a plane a quarter turn twice and
you face backwards: $J^2 = -\mathbb 1$ is geometry a child can
perform, and it is the entire content of ``$i^2 = -1$'' --- the bulk
does the rotating, the boundary does the abbreviating, and the
experiments that killed real boundary QM left the real bulk
untouched.

\medskip\noindent\textbf{G.5.1. Compute a holonomy via Stokes:
the flux is the phase.}

\noindent\emph{Solution.}  Step 1: for a closed loop $C$ the
transport phase is $\Theta_H = \oint_C A_\mu dx^\mu$
\eqref{eq:holonomy_def}, and Stokes \eqref{eq:stokes} converts it
to the surface integral of the curvature:
$\Theta_H = \int_\Sigma F$ \eqref{eq:holonomy_flux}.  Step 2:
apply to a uniform flux tube: a loop enclosing a solenoid of flux
$\Phi$ acquires $\Theta_H = (e/\hbar)\Phi$ regardless of the
loop's shape, size, or distance from the tube --- the holonomy
depends only on the enclosed topology-weighted flux.  Step 3: note
the gauge lesson \eqref{eq:closed_inv}: the open-line transport
\eqref{eq:Wline_open} is gauge-covariant (unphysical by itself),
while the closed loop is gauge-\emph{invariant} --- only
holonomies are observables, which is the chapter's governing
principle and the reason the framework stores physics in
$\Theta_H$ rather than in raw phases.  Physics intuition: a
connection is a promise about comparing directions at neighboring
points; only a round trip tests the promise --- and the test's
verdict, the holonomy, is as physical as anything in the theory
while every open-ended phase remains a bookkeeping fiction.

\medskip\noindent\textbf{G.7.1. Evaluate the Berry phase of a
spin-$\tfrac12$ on a cone.}

\noindent\emph{Solution.}  Step 1: transport a spin-$\tfrac12$
adiabatically around a closed circuit of magnetic-field directions
tracing a cone of half-angle $\theta$; the Berry connection
\eqref{eq:Berry} integrates to $\gamma = -\tfrac12\,\Omega$, half
the enclosed solid angle, by the flux form \eqref{eq:Berry_flux}
with the spin-$\tfrac12$ monopole curvature.  Step 2: numbers:
$\Omega = 2\pi(1 - \cos\theta)$, so at $\theta = 90^\circ$
(equatorial circuit) $\gamma = -\pi$: the state returns to itself
with a sign flip --- the spinor's famous $4\pi$ periodicity read as
a holonomy; at small $\theta$, $\gamma \approx -\pi\theta^2/2$.
Step 3: place it in the chapter's frame: parameter space is a
sphere, the Berry connection is a monopole of charge $\tfrac12$,
and the phase is the same Stokes flux as G.5.1 --- geometry, with
not one dynamical quantity anywhere.  Physics intuition: the
electron does not compute $-\Omega/2$; it merely rides a curved
bundle, and the curvature does the arithmetic --- half the solid
angle because the sphere of spin directions is double-covered,
the deepest ``factor of two'' in quantum mechanics.

\medskip\noindent\textbf{G.8.1. Work the Aharonov--Bohm ring, and
find the dark fringe.}

\noindent\emph{Solution.}  Step 1: the two arms of the ring enclose
a solenoid of flux $\Phi$; the arms' transport phases differ by the
holonomy \eqref{eq:AB_ring_phase}, so the detected intensity is
\eqref{eq:AB_intensity}: $I \propto 1 + \cos(2\pi\Phi/\Phi_0)$
with period $\Phi_0 = h/e = 4.136\times10^{-15}$~Wb.  Step 2:
predictions: at $\Phi = \Phi_0/2$ the interference is fully
destructive --- the electron is steered by flux it never touches
(the field vanishes on both arms); sweeping $\Phi$ scrolls the
fringes with the stated period, exactly as observed.  Step 3: the
reading: the AB effect is the laboratory's direct sighting of a
Wilson line --- the vector potential is unobservable, the field is
absent along the path, and the holonomy alone acts; the ``quantum
phase'' is caught behaving as a connection on a fiber, which is
the entire thesis of this appendix in one experiment.  Physics intuition:
 the electron's two halves take a round trip on the
phase bundle and disagree by precisely the enclosed flux --- the
fringe shift is a curvature meter; nothing pushes the electron,
yet the geometry it traverses is not flat.

\medskip\noindent\textbf{G.9.1. Derive charge quantization from the
compact fiber.}

\noindent\emph{Solution.}  Step 1: on a compact fiber the allowed
matter modes are single-valued: $e^{iq\cdot2\pi} = 1$ forces
$q \in \mathbb Z$ \eqref{eq:charge_integer} --- charge is the
winding label of the mode expansion \eqref{eq:fibre_modes}, not a
continuous coupling.  Step 2: consequence chain: a state of charge
$q$ picks up holonomy $e^{iq\Theta_H}$ per fiber circuit, and the
Dirac condition \eqref{eq:dirac_quant} follows in the same breath
--- monopoles and quantized charge are two faces of fiber
compactness.  Step 3: the cross-cohort echo: this is the same
integer discipline that quantizes the ledger readings of the
particle-physics volume --- compactness converts continuous dials
into integers there exactly as it converts coupling strengths into
windings here; the assembly-line of the program's ``integers,
not dials'' runs through this equation.  Physics intuition:
why is charge quantized? --- because somewhere a direction closes
on itself; any quantity conjugate to a circle must count its laps
in whole numbers, and the century-old mystery of identical
electron charges is the tautology that all laps are laps.

\medskip\noindent\textbf{G.10.1. State what $\Theta_H$ is in this
framework, and trace it across the cohort.}

\noindent\emph{Solution.}  Step 1: assemble the chapter's chain
(Eq.~\eqref{eq:Theta_H_def}, \S G.10--G.11): the quantum phase lives on
$S^1_\theta$; only its holonomy is gauge-invariant; the
registration reads a holonomy sector; hence the measured outcome
label \emph{is} a Wilson-line value --- $\Theta_H$ is not a hidden
parameter added to quantum mechanics but the gauge-invariant
content of the phase the textbook already carries.  Step 2: verify
the consistency: distinct outcome sectors carry distinct integer
holonomies, which is exactly what made the fiber average of
E.5.1 annihilate inter-sector coherence and what the Bell
appendix's sampling used to reproduce the quantum correlations ---
three appendices, one object.  Step 3: the cross-cohort bridge:
the same Wilson line, evaluated on the electroweak coset in the
particle-physics volume (\ANine{}), is the order parameter whose
branch selection breaks the electroweak symmetry --- the
measurement label of quantum mechanics and the Higgs vacuum of the
Standard Model are one geometric object read at two energies, the
single most reused idea in the program.  Physics intuition:
physics keeps its deepest secrets in loops --- interference
fringes, Berry phases, AB shifts, measurement outcomes, and the
electroweak vacuum are all the same question (``what did the
circle say?'') asked of the same fiber; this appendix is where the
course learns to hear the answer.


\GGtakeaway{\emph{Standard account:} the quantum phase is a
global mystery --- unobservable in itself, yet capable of shifting
fringes through regions where every field vanishes. \emph{What it
left open:} where the phase lives, and why only phase
\emph{differences} around closed loops are physical. \emph{The
bulk reading:} the phase is a coordinate on a real compact fiber;
only holonomies --- parallel transports around closed loops --- are
gauge-invariant, which is exactly the Aharonov--Bohm and Berry
phenomenology; and the same fiber whose holonomy is quantum
interference here is the fiber whose holonomy breaks the
electroweak symmetry in \ANine{} --- one circle, two centuries of
physics. And the circle is real: its ``$e^{i\theta}$'' is boundary
shorthand for an $SO(2)$ rotation, which is why the boundary's
imaginary unit is derived rather than assumed
(\citealp{ArisakaP4Real}).}

\section{Qi-4 pillars applied to quantum measurement}
\label{app:Qi4}

\GGteachbox{The measurement event audited by the four \QiFour{}
pillars: the event ledger (every measurement balances energy,
momentum, charge, and Qi-number currents), the exchange principle,
the contextuality pillar, and the halting pillar --- a compact
methods appendix showing that ``measurement'' in this framework is
a physical transaction with conserved-current bookkeeping, not a
primitive verb.}

We make explicit how the four pillars of Qi (Qi-QUA, Qi-EP,
Qi-CON, Qi-HAL) govern the quantum-measurement process.

\subsection{Qi-QUA for quantum measurement}

The event-ledger pillar: any quantum measurement event has
four currents (energy, momentum, charge, Qi-number)
that must balance. In particular, the measurement extracts
energy $\Delta E$ from the system to the apparatus at the rate
\begin{equation}\label{eq:measurement_energy}
\dot Q_{\rm measurement} = \Gamma\, k_B T \cdot \log_2 N,
\end{equation}
where $\Gamma$ is the measurement rate, $T$ is the
apparatus temperature, and $N$ is the number of outcomes ---
the Landauer bound. This is Qi-QUA in action.

\subsection{Qi-EP for quantum measurement}

The selector-law pillar: the realized outcome minimizes the
\GFF{} functional, which in the quantum-measurement context
is equivalent to maximum-likelihood inference. This is why
quantum measurements are optimal estimators of the bulk
Qi-configuration.

\subsection{Qi-CON for quantum measurement}

The conformality pillar: decoherence rates scale invariantly
across system size. For larger systems, $\Gamma$ scales with the
interaction area; for smaller systems, with the individual
coupling. The resulting $1/f$ noise in measurement apparatuses
is a direct prediction of Qi-CON.

\subsection{Qi-HAL for quantum measurement}

The architecture pillar: measurement records are stable
because they are topologically protected by the bulk gauge-field configurations on $S^1_\theta$ (with \UoneQi{} the cohort canonical factor providing this topological protection in \citet{ArisakaA7Qi4})
holonomy. This is why a photograph remains a photograph,
and why a classical bit on a hard drive stays a bit.



%% ===================================================================
%%  Vocabulary cross-walk: standard QM terms vs. the
%%  \GGtheory{}/\OGGEP{}/\GDOS{} reformulation.
%% ===================================================================

\section{Worked decoherence overlap integral: methods and uncertainty budget}
\label{app:worked-overlap}

\GGteachbox{The methods and error budget behind the paper's
worked decoherence number: the collisional decoherence function
and its two limits, the dispersion-coefficient estimates, the
cross-section formula with its coefficient uncertainty, the
thermal average, and the bulk-channel effective-field-theory
bound --- the appendix to consult before quoting, checking, or
attacking the decoherence rates used in the worked examples.}

This appendix collects the machinery behind the worked computation of
\S\ref{sec:worked-overlap}: the collisional decoherence function and its limits,
the dispersion-coefficient estimates, the cross-section formula and its
coefficient uncertainty, the thermal average, and the bulk-channel
effective-field-theory bound.

\subsection{The collisional decoherence function and its two limits}

For a tracer particle in a dilute gas, the standard collisional master equation
\citep{GallisFleming1990, Hornberger2003} gives the off-diagonal damping
$\rho(\mathbf x,\mathbf x',t)=\rho(\mathbf x,\mathbf x',0)e^{-F(\mathbf x-\mathbf
x')t}$ with $F(\Delta\mathbf x)$ of Eq.~\eqref{eq:Fdx}. Two limits follow by
inspection of the kernel $[1-\cos(\Delta\mathbf p\cdot\Delta\mathbf x/\hbar)]$.
When the separation greatly exceeds the inverse typical momentum transfer,
$\Delta x\gg\hbar/\langle\Delta p\rangle\approx\lambda_{\rm th,gas}$, the cosine
self-averages to zero away from the forward direction and the angular integral
returns the total cross section, giving the saturated rate
$\Gamma_{\rm dec}=n\sigma_{\rm tot}\bar v_{\rm rel}$ of Eq.~\eqref{eq:Fsat}. When
the separation is small, $\Delta x\ll\lambda_{\rm th,gas}$, the leading term of
the expansion $1-\cos(\cdot)\approx\tfrac12(\cdot)^2$ gives
$F\to\Lambda(\Delta x)^2$ with localization parameter
$\Lambda=\tfrac{n}{2\hbar^2}\langle v\!\int d\Omega\,(d\sigma/d\Omega)\,\Delta
p^2\rangle=\tfrac{n}{\hbar^2}\langle v\,(m_{\rm gas}v)^2\,\sigma_{\rm tr}\rangle$,
$\sigma_{\rm tr}=\int(1-\cos\theta)(d\sigma/d\Omega)\,d\Omega$ the
momentum-transfer cross section. The crossover at $\Delta x_\star\approx
\lambda_{\rm th,gas}$ is where the two branches meet.

\subsection{Dispersion coefficient $C_6$ for $C_{70}$--Ar}

Two standard estimators bracket $C_6$. The London formula
$C_6=\tfrac32\,\alpha_A\alpha_B\,\dfrac{I_AI_B}{I_A+I_B}$ (atomic units;
polarizabilities $\alpha$ in Bohr$^3$, ionization energies $I$ in Hartree), with
$\alpha(C_{70})=102\,\text{\AA}^3=688\ a_0^3$, $\alpha(\mathrm{Ar})=1.66\,
\text{\AA}^3=11.1\ a_0^3$, $I(C_{70})=7.5$~eV, $I(\mathrm{Ar})=15.76$~eV, gives
$C_6^{\rm London}\approx2.2\times10^3$~a.u. The Slater--Kirkwood formula
$C_6=\tfrac32\,\alpha_A\alpha_B/\big(\sqrt{\alpha_A/N_A}+\sqrt{\alpha_B/N_B}\big)$
with effective electron numbers $N_A\approx280$, $N_B\approx8$ gives
$C_6^{\rm SK}\approx4.2\times10^3$~a.u. London systematically under-binds; the
truth lies between, and we adopt the geometric mean $C_6\approx3.0\times10^3$~a.u.
as the central value, with the factor-$\sim2$ spread propagated as a factor
$2^{2/5}\approx1.3$ in $\sigma$.

\subsection{Total cross section and the coefficient $\kappa$}

For the $-C_6/r^6$ tail the integral (total) cross section in the
Massey--Mohr/Landau--Lifshitz semiclassical approximation is
$\sigma_{\rm tot}=\kappa\,(C_6/\hbar v)^{2/5}$. The canonical coefficient is
$\kappa=\pi\,(3\pi/8)^{2/5}\,\Gamma(3/5)\times(\text{glory factor})\approx8.083$;
re-assembling the same product with different glory-averaging conventions returns
values as low as $\approx6.2$, so we carry $\kappa=8.083\,(\pm{\sim}25\%)$. The
total cross section is dominated by small-angle (forward) scattering; in the
saturated regime even a glancing collision leaves a which-path record, so the
\emph{decoherence} cross section equals the \emph{total} cross section up to an
$\mathcal O(1)$ angular-weighting factor, which we absorb into the stated factor
$\sim1.5$ accuracy.

\subsection{Thermal average}

Because $\sigma\propto v^{-2/5}$, the lab-frame rate is the Maxwell average
$\langle\sigma(v_{\rm rel})\,v_{\rm rel}\rangle$ over the gas velocity
distribution with the molecule moving at $v_{\rm mol}$. A direct numerical
average (Ar at $300$~K, $v_{\rm mol}=100$~m/s) gives
$\langle\sigma v_{\rm rel}\rangle=0.975\times\sigma(\bar v_{\rm rel})\bar v_{\rm
rel}$ --- a $2.5\%$ correction, negligible against the $C_6$/$\kappa$
uncertainties, so the single-velocity shortcut of \S\ref{sec:worked-overlap} is
justified.

\subsection{The bulk channel: kinematic threshold and EFT suppression}

The compact fiber $S^1_\theta$ has radius $R_\theta=\hbar c/\Mlock=
5.1\times10^{-20}$~m and a KK tower of masses $m_n\approx n\,\Mlock$. The
correlation time $\tau_c=\hbar/\Mlock=1.7\times10^{-28}$~s sets only the memory
of the tower, not a rate. Real emission of a tower quantum requires energy
$\ge\Mlock c^2\approx3.85$~TeV; for a system at $E_{\rm sys}\ll\Mlock c^2$ this
is kinematically forbidden and the induced decoherence rate is suppressed by at
least the leading dimension-$(4+2k)$ operator factor
$(E_{\rm sys}/\Mlock)^{p}$, $p\ge1$. The power $p$ cannot be fixed without the
explicit $6$D bulk--boundary operator, so we quote the bound, not a coefficient;
the dimension-$6$ illustration $p=2$ gives
$(E/\Mlock)^2\sim10^{-26}$ (molecular) to $10^{-20}$ (keV). The massless $n=0$
mode is the $4$D gauge field; its decoherence is the ordinary thermal-photon
channel and is counted in the physical bath, not here.

\subsection{Numerical summary}

\begin{table}[htbp]
\centering
\caption[Numerical summary of the worked decoherence estimate]{Numerical summary of the worked decoherence estimate of this appendix. The central column gives the value used in the text and the band gives the range over which the input parameters are defensible, so that the sensitivity of the estimate can be read off directly. The last two rows are not measurements of the same kind as the first five: the correlation time is fixed by the lock scale, and the bulk-to-bath ratio is a bound rather than a coefficient, because the power cannot be fixed without the explicit six-dimensional operator.}
\label{tab:app-numerical-summary}
\small
\begin{tabular}{lll}
\toprule
Quantity & Central & Band \\
\midrule
$C_6$ ($C_{70}$--Ar) & $3.0\times10^3$~a.u. & $(2$--$4)\times10^3$ \\
$\sigma_{\rm tot}$ & $1.7\times10^{-17}\,\mathrm{m^2}$ & $(1.5$--$2.0)\times10^{-17}$ \\
$\Gamma_{\rm dec}(10^{-6}\,\mathrm{mbar})$ & $171\,\mathrm{s^{-1}}$ & $150$--$196$ \\
$p_\star$ (onset) & $1.4\times10^{-6}\,\mathrm{mbar}$ & $(1.2$--$2.1)\times10^{-6}$ \\
thermal-average factor & $0.975$ & --- \\
$\tau_c=\hbar/\Mlock$ & $1.7\times10^{-28}\,\mathrm{s}$ & --- \\
$\Gamma_{\rm bulk}/\Gamma_{\rm bath}$ & $\lesssim(E/\Mlock)^p$ & $p\ge1$ \\
\bottomrule
\end{tabular}
\end{table}

\section{Condensed timeline of the quantum-mechanical and open-system milestones}
\label{app:timeline}

This appendix supplies a single-page chronological cross-walk between
the milestones of quantum mechanics and open-system theory and the
synthesis stage at which \GGtheory{} enters. The intent is to make
clear that the \GG{}/\OGGEP{}/\GDOS{} reformulation does not displace
the canonical milestones; it places each of them as a station inside
one continuous historical arc that culminates in the present cohort.

\begin{longtable}{@{}p{0.10\textwidth}p{0.24\textwidth}p{0.58\textwidth}@{}}
\caption[Condensed timeline of quantum-mechanical and open-system milestones]{Condensed timeline of the quantum-mechanical and open-system milestones, from Planck in 1900 to the present cohort. The third column says why each milestone matters for the synthesis rather than restating what it established. The arrangement makes one point and no more: the reformulation of this paper does not displace any of these results, it places each of them as a station on a single continuous arc.}
\label{tab:app-timeline}\\
\toprule
Year & Milestone & Why it matters for the present synthesis \\
\midrule
\endfirsthead
\toprule
Year & Milestone & Why it matters for the present synthesis \\
\midrule
\endhead
1900 & Planck and blackbody quantization & Launches the quantum crisis that eventually leads to the mature Hilbert-space formalism \\
1905 & Einstein and the light quantum & Quantization of radiation; first deep wave--particle tension on which the present paper takes a structural stand \\
1913 & Bohr atomic model & Early quantized structure without a full dynamical law \\
1922 & Stern--Gerlach experiment & First dramatic evidence of quantized angular-momentum projection; canonical prototype of measurement in Part~V \\
1925 & Heisenberg matrix mechanics & First complete dynamical formulation built on noncommuting observables \\
1926 & Schr\"odinger wave mechanics & Differential-equation form of quantum dynamics; basis for wavefunction language \\
1926 & Born rule & Connects amplitudes to observed frequencies; launches the probability problem now resolved by \TA{3} \\
1930 & Dirac transformation theory & Unifies matrix and wave mechanics in an abstract formal structure \\
1932 & von Neumann axiomatization & Hilbert-space completion and projection-postulate formalization; the eight-postulate baseline for \AFive{} T-GDOS-7.1--T-GDOS-7.7 \\
1935 & EPR paradox & Sharp challenge to completeness and locality; the spine of Part~VII \\
1935 & Schr\"odinger's cat & Macroscopic form of the measurement problem; resolved by \TA{5} \\
1955 & Stinespring dilation & Mathematical foundation for completely positive maps; underwrites the dilation step of \TA{4} \\
1957 & Gleason theorem & Uniqueness of the Born rule on Hilbert-space event structures; the \emph{conditional} key to \TA{3} \\
1957 & Everett relative-state formulation & Deterministic no-collapse response to the measurement problem; the \GGtheory{}-distinct cousin \\
1958--1960 & Nakajima--Zwanzig projection formalism & Exact reduced dynamics with memory kernels; the bulk-to-boundary reduction skeleton of \TA{1} \\
1964 & Bell theorem & Excludes Bell-local hidden-variable completion on the 4D event space; left intact by \TA{7} on the boundary \\
1967 & Kochen--Specker theorem & Strengthens contextuality constraints; absorbed by the contextual bulk-holonomy reading of Part~X \\
1969 & CHSH inequality & Experimentally practical Bell-type inequality; the testing surface for the $2\sqrt{2}$ derivation of \TA{8} \\
1975--1976 & GKLS theorem (Gorini--Kossakowski--Sudarshan--Lindblad) & Universal Markovian generator for open quantum systems; the boundary normal form behind \TA{4} \\
1982 & Aspect experiments & Canonical early experimental Bell violations; the Part~VII reality check \\
1989 & Tonomura single-electron interference & Iconic modern visualization of double-slit interference buildup; Part~VI prototype \\
1990s--2000s & Decoherence program matures (Zurek, Joos--Zeh, Schlosshauer) & Explains effective classicality of reduced states; the \emph{necessary} half of \TA{5}, completed by einselection in \TA{6} \\
2015 & Loophole-free Bell tests & Firm empirical validation of quantum nonlocal correlations without a Bell-local explanation \\
2026 & \GG{}/\GDOS{} synthesis stage (Ultra Eleven cohort) & Places quantum mechanics inside a deterministic 6D bulk and a probabilistic 4D boundary; \TA{1}--\TA{11} \\
\bottomrule
\end{longtable}

The shape of this list is the shape of the present paper. Each
generation refined a piece of the formalism without resolving the
constitutional question of why a deterministic underlying law and a
probabilistic observed law coexist. The cohort answer is that the
two laws live on two strata of a single architecture, and that the
boundary law is forced upon any finite observer by the bulk-to-boundary
projection. After all, history accumulates pieces; a theory places
them.



\section{Concept-by-concept synthesis: standard quantum mechanics versus the cohort reformulation}
\label{app:synthesis-table}

This appendix consolidates the cohort reformulation of the central
quantum-mechanical concepts onto a single page, in the format
``standard reading $\leftrightarrow$ \GGtheory{}/\GDOS{} reformulation.''
The table is intended as a quick reference for readers who want a
fast cross-walk between the textbook formulation of Sakurai, Cohen-Tannoudji,
or Schlosshauer and the boundary-law reading of the present paper.

\begin{longtable}{@{}p{0.17\textwidth}p{0.31\textwidth}p{0.42\textwidth}@{}}
\caption[Standard reading against the GG-Theory reformulation, concept by concept]{The central quantum-mechanical concepts in the format standard reading against \GGtheory{}/\GDOS{} reformulation. The table is a cross-walk for a reader coming from Sakurai, Cohen-Tannoudji or Schlosshauer, and it is arranged so that the middle column can be read on its own as an accurate statement of the textbook position. Nothing in the right-hand column contradicts the middle one; each entry re-derives it from the boundary law.}
\label{tab:app-synthesis}\\
\toprule
Concept & Standard quantum-mechanical reading & \GGtheory{}/\GDOS{} reformulation \\
\midrule
\endfirsthead
\toprule
Concept & Standard quantum-mechanical reading & \GGtheory{}/\GDOS{} reformulation \\
\midrule
\endhead
Born rule & Probability postulate $p_\alpha = \Tr(\rho E_\alpha)$; uniquely selected by Gleason-type assumptions on Hilbert-space events & Boundary probability law induced after the non-invertible bulk-to-boundary projection; Gleason becomes the uniqueness theorem for the already-derived boundary event structure (\TA{3}) \\
Measurement problem & Tension among completeness, linearity, and single outcomes & Dissolved by domain separation: the complete state is the 6D bulk state; the 4D boundary state is reduced and open (\TA{1}--\TA{6}); relaxation-level; one analytic residue \\
Gleason's theorem & Uniqueness theorem for noncontextual frame functions in $\dim\Hilb \ge 3$ & Clarifies why boundary probabilities must be Born-like once \OGGEP{} supplies the boundary Hilbert-space event lattice \\
Many-Worlds & Universal unitary law with all branches real & Shares deep determinism but adds explicit observer-boundary semantics (outer hologram, inner hologram, evolving inner cosmos) instead of relying only on branch ontology \\
Double slit & Interference of amplitudes; which-way information suppresses fringes & Projection-sensitive experiment revealing how visible law depends on the actual observational channel (Part~VI) \\
Decoherence & Environment suppresses off-diagonal terms in reduced state & Lawful and generic consequence of finite observation as partial trace; necessary but not sufficient without runtime record semantics (\TA{5}) \\
Schr\"odinger's cat & Macroscopic superposition sharpened into an outcome puzzle & Candidate macro-branches on the reduced landscape, with the selected stable record produced by runtime branch selection and attractor stabilization \\
Stern--Gerlach & Spin projection measurement with outcome probabilities $|\alpha|^2,\,|\beta|^2$ & Instrument-specific outer-hologram event from a boundary channel acting on a deeper bulk process (Part~V) \\
EPR paradox & Challenge to completeness and locality & Boundary incompleteness is expected; the reduced state is not the total state of reality (Part~VII) \\
Bell theorem & Excludes Bell-local hidden-variable completion & Still binding on the boundary; \GGtheory{} is not a naive Bell-local hidden-variable model because its parent structure is a non-invertible contextual projection (\TA{7}) \\
Quantum entanglement & Non-factorizable states and non-classical correlations & Same operational structure remains exact; interpreted as the boundary manifestation of a common deeper lawful state \\
CHSH experiment & Practical Bell inequality with quantum violation up to $2\sqrt{2}$ & Reproduced exactly by the cohort completion; the $2\sqrt{2}$ value is a Bogomolnyi topological coincidence (\TA{8}) \\
Qubit & Fundamental two-level quantum information carrier & Minimal stable boundary packet or observational interface, not necessarily the most primitive ontic ingredient \\
Lindblad / \GKSL{} & Universal Markovian generator for open quantum systems & Boundary normal form of the exact \OGGEP{}/Nakajima--Zwanzig reduction; not fundamental but derived (\TA{4}) \\
Indistinguishability & Postulate that identical particles share identical observable structure & Cascade inheritance: every electron is a token of the unique stable \GA{} of the charged-lepton stratum (\TA{9}) \\
No-signaling & Invariance of local marginals under distant measurement choice & Mandatory consistency condition for any acceptable bulk completion (\TA{11}) \\
\bottomrule
\end{longtable}

The single discipline behind the right-hand column is that the
operational success of standard quantum mechanics is taken as given
and is preserved exactly; what changes is the constitutional reading
of \emph{why} it succeeds. After all, a good reformulation does not
overturn the working law; it explains the working law's place.

\clearpage

%% (Condensed 12-objection appendix removed; superseded by the full Common Objections Part.)

\GGPart[Common Objections]{Common Objections and Responses}{The strongest objections a foundations-of-quantum-mechanics referee, a particle/string theorist, a condensed-matter physicist, an AMO experimentalist, and a historian and philosopher of quantum mechanics would each raise after reading the whole cohort --- answered, with pointers to the theorem, falsifier, open problem, or companion paper that bears the weight}
\label{sec:common-objections}

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

A paper that claims to read quantum mechanics as the boundary projection of a deterministic six-dimensional bulk, and to dissolve the century-old measurement problem into theorems plus one named analytic residue, invites the most demanding scrutiny; it would not be honest scholarship to leave the obvious objections unstated. We collect here the strongest adversarial readings --- the ones a foundations-of-quantum-mechanics referee, a particle/string theorist, a condensed-matter physicist, an AMO experimentalist, and a historian and philosopher of quantum mechanics would each raise on a careful reading of the whole cohort, not a skim --- and answer each, with explicit pointers to the theorem (by its cohort-global id, \S\ref{sec:naming-convention}), the worked example, the open problem (\S\ref{sec:disc-open}), or the companion paper that carries the load. These are deliberately the \emph{surviving} objections: the ones that remain after a referee has read the companions that do the work the present paper only summarizes. Where the honest answer is ``not yet,'' we say so and name the open item. Throughout, two commitments are kept in view: the framework defends Einstein's \emph{realism and determinism} in the bulk, while conceding 4D \emph{non-locality} openly; and it claims a derivation of the \emph{form} of the quantum postulates, with their dynamical realization closed at the relaxation level, naming precisely the one analytic step --- the field-grade representation step of the GG typicality target --- that remains.

\bigskip
\section{The foundations-of-quantum-mechanics referee: measurement and definite outcomes}
\bigskip

\noindent\textbf{Objection 1 (O-QM-1) (the measurement-problem referee)}\par\nobreak
\noindent\textbf{``You have not solved the measurement problem; you have relocated it. Tracing out an inaccessible fiber gives an improper mixture, and an improper mixture is not a definite outcome. The hard part --- why \emph{this} outcome rather than that one --- is exactly what you push into an `open' box.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} We accept the standard-of-proof and answer it directly. The measurement problem, stated precisely (Maudlin's three questions), is: why this basis, why one definite outcome, why Born weights. The framework answers each with a theorem (\S\ref{ssec:fibre-honest}): the pointer basis is einselected by bulk locality (\TA{6}); the definite outcome is delivered, not deferred, by the \GRIP{} registration of \citet{ArisakaA6Gi4} --- the joint system-and-apparatus configuration descends into one Geometric Attractor basin and is locked there by the monotone boundary free-energy law (\TA{1}, \TA{4}); the Born \emph{form} is forced by Gleason on the open dictionary (\TA{3}). What remains is \emph{not} ``why this outcome'' as a mystery but a single, sharply-stated measure equivalence: that the \GRIP{} basin-capture measure coincides with the Born measure --- the six-dimensional analogue of the quantum-equilibrium typicality theorem of Bohmian mechanics \citep{Bohm1952}. That residue is the signature of \emph{every} deterministic completion of quantum mechanics, not a defect peculiar to this one; here it is closed at the relaxation level (the R16 lemmas with the R16$'$ \GDOS{} relaxation), with the Constitutional Born Theorem conditional on the four named premises (P1)--(P4) and the field-grade representation step the single remaining analytic residue, recorded as the first open question of the program (\S\ref{sec:disc-open}) and assumed nowhere. The improper-mixture objection is met because the bulk state is single-valued: there is no alive-and-dead superposition awaiting collapse, only a definite bulk configuration whose boundary shadow the observer cannot fully read.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 2 (O-QM-2) (the decoherence skeptic, after Adler and Bell)}\par\nobreak
\noindent\textbf{``Decoherence has been oversold for forty years. It explains the \emph{appearance} of classicality, never its \emph{fact}: the density matrix goes diagonal, but `diagonal' means `and', not `or'. Your einselection theorem (\TA{6}) inherits exactly this gap.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Correct, and we do not lean on decoherence to close the gap --- that is precisely why the framework adds \GRIP{} (\citet{ArisakaA6Gi4}) on top of einselection. Decoherence (\TA{5}, \TA{6}) does the work it can honestly do: it selects the pointer basis and suppresses interference. It does \emph{not} produce the single outcome, and we say so. The single outcome is the actualization step --- the descent into one attractor basin --- which is a dynamical halt, not a basis-diagonalization. The ``and-not-or'' problem is dissolved at the ontic level by the single-valued bulk state, and at the operational level by \GRIP{}'s basin selection; the only residue is the single analytic residue of O-QM-1 (the field-grade representation step). We agree the gap is real; einselection alone does not close it.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 3 (O-QM-3) (the Kochen--Specker referee)}\par\nobreak
\noindent\textbf{``Your Stern--Gerlach reading assigns each silver atom a definite holonomy `sector' that fixes its spin before measurement. That is a non-contextual value assignment, and Kochen--Specker proves no such assignment exists in dimension three or more. Your own contextuality section says as much --- so Part~V contradicts your appendix.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The atom does \emph{not} carry a pre-existing spin value along every axis. The outcome is the \emph{contextual} joint holonomy $\Theta_H$ of atom-and-apparatus: the magnet orientation is the measurement context that selects which holonomy observable is read, and only along that chosen axis does $\Theta_H$ resolve into one of two sectors. This is exactly the Kochen--Specker-compatible structure --- 4D-contextual, 6D-single-valued --- and it is the same contextuality every confirmed experiment exhibits. The framework reproduces the quantum prediction; what it adds is the reading that contextuality is the operational signature of which boundary algebra the apparatus constructs, not a defect of the bulk ontology.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 4 (O-QM-4) (the probabilist)}\par\nobreak
\noindent\textbf{``You assert the basin-capture measure equals the Born measure only at the relaxation level, with an analytic step still open. But your worked examples \emph{use} Born statistics throughout. You are assuming the very thing you have not finished proving --- that is circular.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The distinction is between using the Born \emph{form} (a theorem, \TA{3}, \emph{Resolved}) and proving the \emph{dynamical realization} of that form by the registration measure --- closed at the relaxation level (the R16 lemmas with the R16$'$ \GDOS{} relaxation), with the Constitutional Born Theorem conditional on the four named premises (P1)--(P4) and the field-grade representation step the single remaining analytic residue (\S\ref{sec:disc-open}). The worked examples use the Born form, which is established: given the Hilbert-space structure (\TA{1}), Gleason forces the square-modulus measure. What is \emph{not} assumed anywhere is that the \GRIP{} noisy-descent frequencies reproduce that measure run-by-run; that is the relaxation theorem's content, and no worked example relies on it. The parallel with Bohmian mechanics is exact: Bohm uses $|\psi|^2$ (the form) while the proof that $|\psi|^2$ is the equilibrium distribution (the typicality theorem) is a separate result --- carried, here, to the relaxation level, with the last analytic step named. We mark what remains rather than hide it (\S\ref{sec:disc-open}).
\end{objqabody}

\bigskip
\noindent\textbf{Objection 5 (O-QM-5) (the consistency referee)}\par\nobreak
\noindent\textbf{``The paper says `measurement problem --- Resolved' in the conclusions and `single-outcome selection remains open' in Part~III. Which is it? A reader cannot tell whether you are claiming victory or conceding defeat.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Both, and the apparent contradiction is removed by stating precisely what each word covers. ``Resolved'' means: the three classical sub-problems (basis, definiteness, Born form) are each given a theorem, so there is no free-standing ``measurement problem'' left once the umbrella is decomposed. ``Open'' means: the basin-capture $=$ Born residue is closed at the relaxation level (the R16 lemmas with the R16$'$ \GDOS{} relaxation), the Constitutional Born Theorem is conditional on the four named premises (P1)--(P4), and exactly one analytic step --- the field-grade representation step --- remains. This single governing statement has one home text (\S\ref{sec:disc-open}) and is carried at the same grade at every graded site (\S\ref{ssec:fibre-honest}, the status table, and the Conclusions banner), so the conclusions and Part~III are consistent. ``Resolved into theorems plus one named analytic residue'' is the honest description, and it is uniform throughout.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 6 (O-QM-6) (the spectral-theory skeptic)}\par\nobreak
\noindent\textbf{``You read quantization as a `discrete attractor catalog'. But most quantum systems have continuous spectra --- a free particle, a scattering state. A discrete catalog cannot recover a continuum.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The attractor catalog (\citet{ArisakaA6Gi4}) governs the \emph{stable bound-state} and \emph{pointer} sectors --- the discrete records a measurement actually deposits --- not the full spectrum of every operator. Continuous spectra (free-particle momentum, scattering phase) live in the unitary, closed-system limit of \TA{2} where the scale fiber is not coarse-grained; there the Schr\"odinger evolution is recovered intact, continuum and all. Discreteness appears only where registration occurs: a detector records a discrete click, an atom binds at a discrete level. The catalog is a claim about \emph{stable attractors of the registration dynamics}, not about the spectrum of $\hat p$. We make this scope explicit and do not claim to discretize the continuum.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 7 (O-QM-7) (the observer-demarcation referee)}\par\nobreak
\noindent\textbf{``What counts as a measurement? You say `any sufficient-complexity boundary observer', then defer the minimal criterion to \AEleven{}. But \AEleven{} answers it for biological observers, not for the quantum stratum. The cut is still drawn by hand.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} We concede the demarcation is not yet fully reduced, and we state the residual precisely. The framework's structural answer is sharp where it matters: a ``measurement'' is any interaction that switches on the \GRIP{} registration --- a coupling that reshapes the boundary free-energy landscape so the descent halts at an attractor (\citet{ArisakaA6Gi4}). That criterion is observer-agnostic: it is satisfied by a photographic grain, a Geiger tube, or a brain alike, and consciousness is explicitly \emph{not} required (consistent across \AEight{}, \ASeven{}, \AEleven{}). What remains genuinely open is a quantitative threshold --- the minimal landscape change that guarantees an irreversible halt --- and this is recorded as an open question (\S\ref{sec:disc-open}). The cut is not ``the experimenter's discretion''; it is ``wherever the free-energy law makes the record monotone,'' with the precise threshold flagged as unfinished.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 8 (O-QM-8) (the collapse skeptic)}\par\nobreak
\noindent\textbf{``You say `no collapse', but your L\"uders update (\TA{4}) is the projection postulate by another name. Renaming collapse `conditional CPTP state-update' does not eliminate it.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The difference is not nominal. In the collapse postulate, projection is a \emph{primitive dynamical law} of the universe, discontinuous and outside the unitary evolution. In the framework, the bulk evolution is unitary throughout; the L\"uders map (\TA{4}) is the \emph{observer's Bayesian update of their own record} once an outcome is registered --- a statement about information, not a physical discontinuity of the state of the world. Nothing in the bulk collapses; the boundary observer simply conditions on what they have learned. That is why the cut location is irrelevant (any choice gives the same predictions) and why no stochastic collapse term appears. It is the Everettian/decoherence reading of L\"uders made precise, not the von~Neumann postulate relabeled.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 9 (O-QM-9) (the Frauchiger--Renner referee)}\par\nobreak
\noindent\textbf{``You claim to keep all three Frauchiger--Renner assumptions (Q, C, S). But the only consistent way out is to drop one. If you really keep cross-agent consistency \emph{and} single outcomes \emph{and} universal validity, you have a contradiction; if you do not, say which you abandon.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} A fair pressure, and the honest answer is that we keep all three \emph{at the bulk level} while restricting how a 4D agent may chain another agent's conclusions --- which is, in effect, a disciplined reading of (C). The bulk state is single-valued (S holds), universally quantum (Q holds), and globally consistent (C holds for the one bulk description). The Frauchiger--Renner contradiction arises from identifying an agent's \emph{boundary projection} with the total ontology and then transitively combining several such projections; the framework blocks exactly that identification, because distinct agents hold distinct boundary reductions of one bulk state \citep{FrauchigerRenner2018}. This is structurally close to the relational resolution: cross-agent inference is constrained.
\end{objqabody}

\bigskip
\section{The Born rule and the Gleason objection}
\bigskip

\noindent\textbf{Objection 10 (O-QM-10) (the probability-foundations referee)}\par\nobreak
\noindent\textbf{``Your Born derivation is circular. The operational axiom \OGGEP{} (\AX{3}) already writes probabilities as $p=\Tr(E\rho)$ into the boundary dictionary. You then `derive' the trace form from CPTP completion --- but you put it in by hand at the axiom.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} This claim is scoped precisely to avoid any circularity (\TA{3}, and the status entry PS-QM-2). What \TA{3} establishes is the \emph{form}: \emph{given} the Hilbert-space structure of \TA{1} and the requirement that the boundary channel be a $\sigma$-additive probability assignment on the projection lattice, Gleason's theorem forces the square-modulus measure --- there is no other consistent assignment. That is a genuine constraint, not an assumption; its inputs (a complex Hilbert space, $\sigma$-additivity) are constitutional inheritances, so the result is exactly as strong as Gleason and no stronger. The content \emph{beyond} Gleason is not a stronger uniqueness claim; it is the dynamical actualization (\GRIP{}) plus the basin-capture-equals-Born statement, closed at the relaxation level with one analytic residue remaining (O-QM-4). ``Born derived'' is not asserted unconditionally anywhere.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 11 (O-QM-11) (the Gleason purist)}\par\nobreak
\noindent\textbf{``Then your `theorem' is just Gleason restated in six-dimensional dress. Gleason 1957 is fifty years old. What have you added?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} For the \emph{form} of the rule, nothing beyond Gleason --- and we say so. What the framework adds is not a new proof of the trace form but an \emph{account of why its premises hold}: why there is a complex Hilbert space at all (the GNS construction on the boundary algebra, \citet{ArisakaA5GDOS} \T{7.1}, inherited as \TA{1}), why the measure is non-contextual (the boundary projection is the same for all compatible observables), and --- the genuinely new content --- a candidate \emph{dynamical} origin for the probabilities in the \GRIP{} basin-capture statistics. Gleason tells you the only consistent measure is the trace form; the framework attempts to tell you which physical process realizes it. The first is inherited and acknowledged; the second is open and named.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 12 (O-QM-12) (the quantum-logic referee)}\par\nobreak
\noindent\textbf{``You claim the \emph{complex} Hilbert space `emerges'. But GNS over a complex $C^*$-algebra returns a complex space by construction --- you assumed complexity in the algebra. Real and quaternionic quantum mechanics are not excluded by your argument.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} A fair and precise point; we do not claim to derive complexity from nothing. The complex structure enters geometrically: the phase fiber is the compact $S^1_\theta$ of \citet{ArisakaA2GG6}, a principal $U(1)$-bundle, and the quantum phase is the holonomy of its connection (the Aharonov--Bohm/Berry phase in the literal bundle sense). Amplitudes are complex because they are $U(1)$ holonomies; the boundary algebra is complex because it is built on that $U(1)$ structure. So the honest statement is: complexity is \emph{inherited from the bulk gauge geometry}, not produced ex nihilo by GNS. Whether the $U(1)$ phase fiber is itself forced (rather than chosen) is a question for \citet{ArisakaA2GG6}, not \AEight{}; within \AEight{} it is an inheritance, and we mark it as such.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 13 (O-QM-13) (the qubit theorist)}\par\nobreak
\noindent\textbf{``Gleason's theorem requires Hilbert-space dimension at least three. The simplest quantum system, a single qubit, is two-dimensional. Your Born `theorem' does not even apply to a spin-$\tfrac12$.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Correct for the original 1957 theorem, and we rely on the modern strengthening rather than gloss over it. The Busch--Caves--Fuchs--Schack extension of Gleason to POVMs holds in dimension two: once measurements are represented by positive-operator-valued measures rather than projective measurements alone --- which is exactly the operational setting \OGGEP{} provides (\TA{1}, \TA{3}) --- the trace rule is forced for the qubit as well. The cohort's bulk fields carry $SU(5)$ representations, so the effective boundary dimension is generically large; but even at $d=2$ the POVM-Gleason argument closes the gap, and that is the version \TA{3} uses. We cite the dimension-two case explicitly so the qubit objection cannot land.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 14 (O-QM-14) (the measure theorist)}\par\nobreak
\noindent\textbf{``Gleason assumes $\sigma$-additivity --- countable additivity of the probability measure over orthogonal projections. That is a strong technical postulate doing real work. You inherit it silently.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} It is flagged explicitly. $\sigma$-additivity is the one genuinely postulated ingredient in the Born chain, and it is a constitutional inheritance from \AX{3}: it is the statement that the boundary observer's probability assignment is a bona fide measure on the event lattice. The framework does not derive it; it assumes it, names it, and notes (shared with A1's own open questions) that a deeper derivation of $\sigma$-additivity from the bulk measure is unfinished. This is the same status the rest of mathematical physics gives it. The honesty is in saying so at the point of use.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 15 (O-QM-15) (the Everettian decision theorist)}\par\nobreak
\noindent\textbf{``Deutsch and Wallace spent twenty-five years trying to derive Born from unitary dynamics plus decision theory, and the consensus is that it smuggles in a probabilistic premise. Why is your route any better?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} It is better in one specific respect and no better in another, and we are explicit about both. Better: the framework does not need to extract probability from a deterministic multiverse by an argument about rational betting, because it has a genuine \emph{source} of ensemble variation --- the unread bulk boundary condition $\Theta_H$, which differs pair-to-pair --- so probability is ordinary ignorance-over-an-ensemble, not a decision-theoretic construct. No better: the proof that this ensemble is Born-distributed (basin-capture $=$ Born) is, like the Deutsch--Wallace program, not yet complete. The difference is that our remaining step is the single analytic residue of a concrete measure-equivalence statement, closed at the relaxation level, with a Bohmian precedent \citep{Bohm1952}, not a contested claim about the rationality of agents in branching worlds. We claim the better \emph{framing} of the open problem, not a finished derivation.
\end{objqabody}

\bigskip
\section{The Bell theorist: EPR, locality, and the joint-state mechanism}
\bigskip

\noindent\textbf{Objection 16 (O-QM-16) (the relativist)}\par\nobreak
\noindent\textbf{``One might try to explain Bell violations by a bulk-timelike geodesic that `dips through the radial direction' to connect spacelike-separated detectors. But \ATwo{}'s Theorem~0 says the radial direction is a renormalization scale with no light cone, so any such shortcut would contradict the foundational paper.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} No such shortcut is invoked. The radial direction $I_r$ enters the metric of \citet{ArisakaA2GG6} with a positive sign ($+dr^2$): it is the non-compact renormalization-scale fiber and carries no light cone, so a 4D-spacelike separation stays spacelike for every radial excursion --- there is no ``dip'' that makes two spacelike events timelike-connectable, and none is needed. The account (\TA{7}, Part~VII) reads the entangled pair as \emph{one non-separable state of the bulk}, sharing the phase fiber $S^1_\theta$ fixed at the common source; the two detector events are projections of that single state along $I_r$. Nothing propagates between the wings. The correlation is a frozen property of the joint preparation, exactly as a common cause, not a signal across a shortcut.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 17 (O-QM-17) (the quantum-information theorist)}\par\nobreak
\noindent\textbf{``Fine, but `the entangled pair is one non-separable bulk state' is just textbook entanglement in fancy dress. Standard quantum mechanics already says the singlet is non-separable. What is the new physics?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} We concede the empirical content here is consistent with --- not beyond --- standard quantum mechanics (the net-content audit, Part~XII). What the framework adds is \emph{ontological}, not predictive: the non-separable state is a real configuration of a six-dimensional bulk, of which the two 4D wavefunctions are projection shadows; entanglement is the statement that one bulk object casts two correlated shadows. This converts ``spooky action at a distance'' from a paradox into the geometry of a non-invertible projection. The genuinely new, testable content is elsewhere (the cohort-geometry signatures, \TA{8} structural stability and the high-energy corrections of Test~8); the Bell account itself is offered as a \emph{reading} that dissolves the puzzle, and we do not dress it as a new prediction.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 18 (O-QM-18) (the Bohmian)}\par\nobreak
\noindent\textbf{``So you concede 4D non-locality. Then this is Bohmian non-locality with a new vocabulary --- a real, non-local, deterministic hidden variable. Why not just say so?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} We do say so, gladly: the framework is, in spirit, a six-dimensional Bohmian theory --- deterministic at the deep level, 4D-non-local, no-signaling, $\psi$-ontic. The honest difference from de~Broglie--Bohm \citep{deBroglie1924, Bohm1952} is that the non-locality is \emph{geometrized} as the non-separability of one bulk state rather than carried by an explicit pilot-wave guidance equation on configuration space, and that the kinematics (Hilbert space, Born form) are \emph{derived} from the boundary law rather than postulated alongside the particle. We are not embarrassed by the Bohmian kinship; we claim it. The slogan ``Einstein right in 6D'' means his realism and determinism are vindicated in the bulk --- not his locality, which we concede is lost in the 4D shadow.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 19 (O-QM-19) (the careful calculator)}\par\nobreak
\noindent\textbf{``Your explicit CHSH appendix once derived a \emph{triangular} correlator, wrote `Wait, this gives a triangular correlator', and then simply posited the cosine by hand. That is Bell-violation by sleight of hand.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The triangular correlator $-1+2\theta/\pi$ is exactly what a \emph{local} response model on a shared scalar $\Theta$ must give --- and by Bell's theorem no such local model can reproduce the quantum cosine. The triangular result is the signature of the wrong (separable) ansatz, recorded precisely so the contrast is clear; the singlet correlation $-\hat a\cdot\hat b$ is the standard operator-algebra expectation in the non-separable joint state (\TA{1}, \TA{3}), not something derived from a local-$\Theta$ integral. The bulk holonomy is the \emph{label} of the joint non-separable state, not a local hidden variable feeding per-wing response functions. Bell forbids the local model; the joint state supplies the cosine.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 20 (O-QM-20) (the no-go specialist)}\par\nobreak
\noindent\textbf{``If no local-$\Theta$ model can reproduce $-\hat a\cdot\hat b$, then the shared $S^1_\theta$ holonomy is doing no work. Either it is a local hidden variable (killed by Bell) or it is idle decoration.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Neither. The shared $S^1_\theta$ holonomy is what makes the joint state \emph{non-separable} --- one fiber for both wings, fixed at the common source --- and non-separability is precisely the property that a local-hidden-variable factorization lacks. It is not a local variable feeding independent response functions (that would be the separable ansatz Bell excludes); it is the bulk-geometric content of the entanglement itself. Its ``work'' is to explain \emph{why} the joint state is non-separable in the first place: the two particles inherit one holonomy class at emission, so their boundary descriptions cannot be written as a product. The distinction --- shared label of a non-separable state versus shared value of a local variable --- is exactly the distinction Bell's theorem turns on, and the appendix makes it explicit.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 21 (O-QM-21) (the ontology referee)}\par\nobreak
\noindent\textbf{``Is $\Theta_H$ real and outcome-determining, or not? If yes, it is a hidden variable and you owe us a no-go analysis. If no, your determinism collapses.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Yes: $\Theta_H$ is real, ontic, unread by the 4D observer, and (contextually, jointly with the apparatus) outcome-determining. It \emph{is} a hidden variable, and we own the no-go analysis rather than dodge it. It evades Bell because it is non-local (the joint state is non-separable); it evades Kochen--Specker because it is contextual (the outcome depends on the apparatus context, not a pre-set non-contextual value, O-QM-3); it is compatible with Pusey--Barrett--Rudolph because it is $\psi$-ontic, not $\psi$-epistemic \citep{PBR2012}. The determinism is intact: given the actual $\Theta_H$ and the apparatus, the outcome is fixed. The price --- non-locality and contextuality --- is exactly the price every no-go theorem says a deterministic completion must pay, and we pay it openly.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 22 (O-QM-22) (the Micius experimentalist)}\par\nobreak
\noindent\textbf{``Entanglement has been verified over 1200~km by satellite. Your `one bulk object, two shadows' picture still has the two shadows perfectly correlated across 1200~km with no signal. That is non-locality, however you paint it.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Agreed --- and we do not paint over it. The 4D shadows are non-locally correlated across 1200~km, with no signal, exactly as the Micius results show. The framework's claim is not that this non-locality disappears but that it is \emph{geometrically natural}: two boundary projections of a single bulk object are correlated for the same structural reason two photographs of one scene agree, a shared origin rather than a message. No-signaling is preserved as a theorem (\TA{11}): the marginal at either detector is independent of the far setting. We claim a better \emph{understanding} of the 1200~km correlation (one object, two shadows), not its abolition. Calling it ``Bohmian non-locality renamed'' is fair; we accept the label and add the geometry.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 23 (O-QM-23) (the no-signaling referee)}\par\nobreak
\noindent\textbf{``Your no-signaling proof (\TA{11}) is the standard reduced-state factoring argument --- it is a theorem about the \emph{quantum} marginal. But you also posit an underlying deterministic hidden-variable layer. No-signaling for the quantum state does not automatically hold for the hidden-variable dynamics.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} A sharp point. \TA{11} establishes operational no-signaling: the observed marginal at $B$ is the reduced quantum state, independent of $A$'s setting. The framework's hidden layer respects this because the unread variable $\Theta_H$ is fixed \emph{at the common source}, before either setting is chosen, and the boundary projection that produces $B$'s record does not depend functionally on $A$'s setting --- the apparatus at $A$ couples only to the $A$-local boundary algebra. So no operational protocol can read $A$'s choice off $B$'s statistics. What the framework does \emph{not} claim is a frame-independent dynamical story at the hidden-variable level: like Bohmian mechanics, it is non-local in a preferred-foliation sense at the ontic layer while strictly no-signaling at the operational layer. This two-level structure is stated explicitly; the quantum-marginal proof covers the operational layer, not the hidden layer.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 24 (O-QM-24) (the consistency referee, again)}\par\nobreak
\noindent\textbf{``A $U(1)_{Qi}$ photon `racing around $S^1_\theta$ in $10^{-32}$~s' to mediate the correlation would sit uneasily with the claim that the correlation is `frozen' and nothing propagates. Which is it?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The framework commits to \emph{frozen} (Part~VII). The correlation is a static property of the shared joint state established at the source; nothing travels from one wing to the other at measurement time. The only role the $U(1)_{Qi}$ holonomy on $S^1_\theta$ plays is to make the joint state non-separable at preparation --- a shared label, not a courier. The frozen common-cause reading is the single committed one.
\end{objqabody}

\bigskip
\section{The no-go-theorem specialist: hidden variables and their cost}
\bigskip

\noindent\textbf{Objection 25 (O-QM-25) (the superdeterminism referee)}\par\nobreak
\noindent\textbf{``A contextual, outcome-determining hidden variable that depends on the apparatus setting is a measurement-dependence (superdeterminism) model. Those evade Bell only at the cost of fine-tuning --- a conspiracy correlating the source with the detector settings. You name this loophole in one sentence and never pay its bill.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} We accept the charge has not been fully discharged and we state the residual precisely. The framework is \emph{not} superdeterministic in the conspiratorial sense: it does not correlate the source's $\Theta_H$ with the future setting choices. It is non-local in the Bohmian sense --- the outcome at one wing depends on the joint (non-separable) state, which is the apparatus-context-dependent reading of one bulk object --- and that is contextuality, not measurement-dependence. The distinction matters: contextuality (the outcome depends on which observable is co-measured \emph{here}) is experimentally mandatory by Kochen--Specker and carries no fine-tuning cost; measurement-dependence (the source `knows' the far setting in advance) does carry one, and the framework does not invoke it. Where the honest ``not yet'' remains is a fully frame-covariant formulation of the ontic layer; that is flagged (\S\ref{sec:disc-open}), and it is the same debt every non-local hidden-variable theory carries.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 26 (O-QM-26) (the Bell purist)}\par\nobreak
\noindent\textbf{``Bell's theorem is a theorem. You cannot have local realism. Every word you write about `recovering Einstein' is either false or a concession dressed as a victory.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} We agree without reservation that local realism is dead, and we do not claim to revive it. The framework recovers Einstein's \emph{realism} (a real, definite state underlies the statistics) and his \emph{determinism} (that state evolves deterministically) while \emph{conceding his locality}. ``Einstein right in 6D'' is shorthand for exactly this split, stated without equivocation on the word ``complete'' (Conclusions, and O-QM-63). Bell forbids local realism; the framework offers non-local realism, which Bell permits. That is a real and clearly-bounded claim, not a victory over the theorem.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 27 (O-QM-27) (the free-will referee)}\par\nobreak
\noindent\textbf{``Bell's derivation assumes measurement independence --- the experimenter's settings are free of the hidden variable. A deterministic 6D bulk that fixes everything threatens that assumption. Do you keep it?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Yes, we keep measurement independence, and this is what distinguishes the framework from genuine superdeterminism (O-QM-25). Determinism of the bulk does \emph{not} entail correlation between the source variable and the setting choices: in any sensible model the analyzer settings are fixed by degrees of freedom (a quantum random-number generator, starlight, the experimenter's deliberation) that are not correlated with the particular $\Theta_H$ drawn at the source. The framework violates Bell's \emph{factorizability} (because the joint state is non-separable), not Bell's measurement-independence. Loophole-free experiments close the locality and detection loopholes; they do not, and cannot, close the non-separability that the framework (like all of quantum mechanics) exploits. So the answer is: measurement independence held, factorizability abandoned --- the standard quantum situation, read geometrically.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 28 (O-QM-28) (the Kochen--Specker referee, deeper)}\par\nobreak
\noindent\textbf{``Calling the holonomy `4D-contextual but 6D-non-contextual' sounds like having it both ways. Either there is a non-contextual assignment in the full theory or there is not. Which, and what is the content?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The content is precise and not a dodge. In the six-dimensional bulk the Wilson-line holonomy $\Theta_H$ is a \emph{single}, definite configuration --- one value, non-contextual, because there is no measurement context in the bulk; there is just the field. Contextuality appears only under \emph{projection}: which 4D observable a value is assigned to depends on which boundary algebra the apparatus constructs (which partial trace it performs), and different co-measurement contexts construct different boundary algebras. So the non-contextual object (the bulk holonomy) and the contextual assignments (its 4D projections) live at different levels, and Kochen--Specker --- which forbids a non-contextual assignment \emph{within the 4D observable algebra} --- is satisfied because the framework offers no such 4D assignment. The bulk non-contextuality is not a hidden 4D assignment in disguise; it is one level down, where KS does not reach.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 29 (O-QM-29) (the experimental-contextuality referee)}\par\nobreak
\noindent\textbf{``Contextuality has been measured in trapped ions, photons, and nuclear spins. Your framework `predicts' it --- but so does ordinary quantum mechanics, identically. There is no test that distinguishes you.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Correct, and the empirical-content audit (Part~XII) lists contextuality among the \emph{consistency} checks, not the discriminating tests --- we do not claim a contextuality experiment could distinguish the framework from textbook quantum mechanics. The value of the contextuality reading is interpretive: it identifies \emph{why} the world is contextual (it is the signature of a non-invertible bulk-to-boundary projection) rather than treating contextuality as a brute no-go fact. Discrimination from standard quantum mechanics, where it exists at all, lives in the cohort-geometry signatures (Test~8, the high-energy corrections; and the structural-stability content of \TA{8}), not in any contextuality inequality. We are explicit that contextuality is a check, not a prediction.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 30 (O-QM-30) (the determinism referee)}\par\nobreak
\noindent\textbf{``Tell us plainly what kind of theory this is. Is the bulk classical-deterministic, unitary-quantum, or stochastic in some random $\Theta_0$? The paper has used all three phrasings, and they are not the same theory.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The framework commits, throughout, to one reading. The bulk is \emph{deterministic} in the strong sense: the registration descent is fixed by the actual contextual holonomy $\Theta_H$, and the apparent randomness is \emph{epistemic} --- the 4D observer cannot read $\Theta_H$. It is not fundamentally stochastic (no objective-collapse noise term), and it is not merely ``unitary plus branching'' (there is a single actual outcome, not all of them). Probability enters exactly once, as typicality over the unread boundary condition $\Theta_H$ across the source ensemble. This is the deterministic/Bohmian reading, and it is the only one in the paper --- the ``random $\Theta_0$'' language means ``drawn from the source ensemble, unread,'' not ``produced by fundamental dice.'' The frozen thesis ``God does not play dice --- in the 6D bulk'' requires exactly this, and we hold it.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 31 (O-QM-31) (the PBR referee)}\par\nobreak
\noindent\textbf{``Pusey--Barrett--Rudolph rules out $\psi$-epistemic models. But for your 4D observer the wavefunction \emph{is} epistemic --- it encodes their ignorance of $\Theta_H$. Are you $\psi$-ontic or $\psi$-epistemic? You cannot be both.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The resolution is the two-level structure, and it is consistent with PBR \citep{PBR2012}. At the \emph{bulk} (ontic) level the state is $\psi$-ontic: the full bulk configuration, including $\Theta_H$, is a real physical state, and distinct bulk states are physically distinct --- which is exactly what PBR requires. At the \emph{boundary} level, the 4D observer's reduced description is incomplete (it has traced out $\Theta_H$), so \emph{relative to the observer} it carries an epistemic component. PBR does not forbid this: it forbids a model in which the \emph{complete} physical state fails to fix $\psi$, and here the complete (bulk) state does fix everything. The 4D wavefunction is the ontic bulk state seen through a lossy projection --- ontic underneath, partly epistemic in the shadow. PBR constrains the underneath, and the framework satisfies it.
\end{objqabody}

\bigskip
\section{The string/QFT theorist: Tsirelson and the extremal bound}
\bigskip

\noindent\textbf{Objection 32 (O-QM-32) (the BPS-structure referee)}\par\nobreak
\noindent\textbf{``You call the Tsirelson saturation a `Bogomolnyi bound'. A Bogomolnyi/BPS bound has a topological charge, a central-charge algebra, an energy functional bounded below by a winding number. You exhibit none of these. The word is decoration.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} We accept this fully: \TA{8} is stated as a labeled analogy (``an extremal (Bogomolnyi-analogue) bound''). There is no central charge, no winding, no topological charge, and none is asserted. What is genuinely true --- and is the only content claimed --- is a structural \emph{analogy}: the singlet is the maximizer of the CHSH functional, so the first variation of $|S|$ vanishes there and saturation is stable to first order, which is the same \emph{kind} of stationarity-at-a-saturating-bound that makes a BPS state stable. That is an analogy to the first-order stationarity, not an instance of BPS structure, with the caveat in the theorem statement itself.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 33 (O-QM-33) (the Tsirelson referee)}\par\nobreak
\noindent\textbf{``The number $2\sqrt2$ is Tsirelson's 1980 result, a consequence of the operator algebra. You re-quote it. You do not derive it from your scale geometry.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Correct: this is an inheritance, not a derivation. The bound $|S|\le 2\sqrt2$ follows from the Hilbert-space/operator structure (Tsirelson 1980 \citep{Tsirelson1980}), which the framework inherits through \TA{1} and \TA{3}; the framework does not claim to re-derive the value from the bulk metric. The framework's residual claim is narrower and defensible: the singlet \emph{saturates} the inherited bound, and the saturation is structurally stable (\TA{8}). The value is standard quantum mechanics; the reading of its saturation as an extremal property is what the framework adds.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 34 (O-QM-34) (the variational-calculus referee)}\par\nobreak
\noindent\textbf{``Your `extremal stationarity' argument --- that the singlet maximizes $|S|$ so the first variation vanishes --- is trivially true of any maximum of any smooth function. It is a tautology, not a result.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} It is indeed elementary, and we present it as such --- a clarification, not a deep theorem. The point of stating it is not novelty but \emph{honesty about what genericity means}: experiments saturate Tsirelson robustly across platforms not because of a tuned coincidence but because the singlet sits at a stationary maximum of the CHSH functional, so small imperfections (slight depolarization, sub-optimal angles) cost only second order. That is the correct, unmysterious explanation of the observed robustness, not a ``topological coincidence of the scale profile''. We would rather state an elementary truth plainly than dress it as a theorem; the tautology charge is, in effect, the concession we are making.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 35 (O-QM-35) (the ``why this number'' referee)}\par\nobreak
\noindent\textbf{``A real explanation would say why the ceiling is $2\sqrt2$ rather than $4$ (the algebraic maximum) or $2$ (the classical bound). You do not. You inherit the number and stop.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} True, and we do not pretend otherwise. The reason the ceiling is $2\sqrt2$ rather than $4$ is the quantum operator structure --- self-adjoint dichotomic observables on a Hilbert space, Tsirelson's bound --- which the framework inherits, and which it traces back one step further to the GNS Hilbert space of the boundary algebra (\TA{1}). What the framework does \emph{not} do is produce $2\sqrt2$ from a deeper geometric principle that a skeptic could not already get from quantum mechanics; that would require deriving the operator algebra itself from the bulk in a way that pins the value, and we do not claim to. The honest scope is: the number is inherited; the saturation and its stability are the framework's reading. We flag the ``derive $2\sqrt2$ from geometry'' goal as not achieved.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 36 (O-QM-36) (the empirical-content referee)}\par\nobreak
\noindent\textbf{``Standard quantum mechanics already predicts the singlet saturates Tsirelson and that the saturation is robust. So \TA{8} predicts nothing new. Why is it a `net-new theorem'?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} ``Net-new'' refers to its status \emph{within the cohort} --- it promotes to a principal, separately-stated result something \citet{ArisakaA5GDOS} leaves as a sub-remark --- not to a new empirical prediction beyond quantum mechanics, as stated in the net-content audit (Part~XII). \TA{8} is empirically \emph{consistent} with standard quantum mechanics, not \emph{distinct} from it. Its job is structural: to give the robust saturation a clean reading (extremal stationarity). It is not listed among the discriminating tests; it is a consistency result with an honest provenance, and the ``net-new'' tag means net-new-to-the-cohort, stated explicitly so no reader mistakes it for a new measurable effect.
\end{objqabody}

\bigskip
\section{The condensed-matter physicist: decoherence, einselection, and the pointer basis}
\bigskip

\noindent\textbf{Objection 37 (O-QM-37) (the open-systems referee)}\par\nobreak
\noindent\textbf{``Your Markovian justification computes the bath correlation time from the bulk Kaluza--Klein gap, $\tau_c\sim10^{-27}$~s. But a real qubit decoheres against its \emph{physical} bath --- phonons, photons, two-level fluctuators --- with correlation time around $10^{-13}$~s. You have used the wrong environment.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Two correlation times must be kept distinct. The bulk Kaluza--Klein time $\tau_c\sim10^{-27}$~s establishes only that the \emph{bulk} contributes no memory, so that the boundary law has semigroup \emph{form}. The actual decoherence of a laboratory system is governed by its coupling to its \emph{physical} 4D environment, whose correlation time $\tau_{\rm bath}\sim10^{-13}$~s is the one that sets the Markovianity and the rate of real decoherence. The bulk smallness supplies the structure; the physical bath supplies the dynamics; the two are kept separate. The bulk channel is bounded quantitatively in \S\ref{sec:worked-overlap}: its rate is kinematically suppressed by at least $(E_{\rm sys}/\Mlock)$, negligible for any laboratory energy.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 38 (O-QM-38) (the vacuum-coherence referee)}\par\nobreak
\noindent\textbf{``You write that `a point particle is its own environment --- no outside reservoir is needed'. Taken literally, that predicts a free electron in vacuum decoheres on its own. It does not: matter-wave interferometry preserves coherence for seconds. Your mechanism is empirically false.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The claim is about the open-system \emph{form}, not the decoherence \emph{rate}, and the distinction is marked at the point of statement. The intrinsic inaccessibility of the fiber is why the boundary law is \emph{open in principle} --- why marginalization is unavoidable. It does \emph{not} set a rate: a genuinely isolated free particle in vacuum, with no apparatus or bath coupling, decoheres negligibly, exactly as interferometry shows, because the rate is fixed by the physical environment (gas, photons), not by the fiber alone. The fiber explains the availability of the open structure; the physical bath explains how fast a given system actually decoheres. The explicit bound of \S\ref{sec:worked-overlap} shows the bulk channel cannot drive vacuum decoherence below the $\Mlock$ scale, consistent with second-long interferometric coherence.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 39 (O-QM-39) (the decoherence-rate referee)}\par\nobreak
\noindent\textbf{``Every decoherence number you quote --- the photon, the fullerene, the cat --- is the standard Joos--Zeh collisional formula. There is no bulk-derived rate anywhere. You have relabeled textbook decoherence.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} For the \emph{rates}, this is largely fair, and we do not claim a numerically new prediction there: the collisional rates are the Joos--Zeh values, and the framework reproduces them because, in the regime where a physical bath dominates, it \emph{must} reproduce them to be correct. What the framework claims to add is not a different number but a structural origin --- the GKLS form as a theorem (\TA{2}) rather than a postulate, the einselected basis forced by bulk locality (\TA{6}) rather than chosen, and the irreversibility identified with the renormalization-group semigroup. The genuinely framework-specific decoherence prediction (a deviation tied to the bulk geometry) lives at the high-energy/short-distance edge, not in the everyday collisional regime, and the paper labels which is which. Reproducing Joos--Zeh where Joos--Zeh is right is a feature; claiming it as new physics would not be, and we do not. The leading-order collisional rate is worked explicitly for $C_{70}$ in \S\ref{sec:worked-overlap}, and the framework-specific deviation is the threshold-gated bulk channel of Eq.~\eqref{eq:bulkbound}.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 40 (O-QM-40) (the calculational referee)}\par\nobreak
\noindent\textbf{``You repeatedly invoke `scale-overlap integrals of the bulk KK modes' as the source of the rate coefficients, but you never evaluate one to a number. It is a placeholder.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Conceded: the explicit order-by-order scale-overlap expansion is a program target, not a delivered calculation, and \citet{ArisakaA5GDOS} does not supply it. What \emph{is} delivered is the structural identification (the dissipator is the boundary image of scale-fiber coarse-graining) and the leading-order matching to the collisional rate. A fully worked instance is given in \S\ref{sec:worked-overlap} and Appendix~\ref{app:worked-overlap}: for $C_{70}$ matter waves the scale-overlap reduces, in the physical-bath regime, to the van der Waals total cross section $\sigma_{\rm tot}\approx(1.5\text{--}2.0)\times10^{-17}\,\mathrm{m^2}$, giving $\Gamma\approx150\text{--}196\,\mathrm{s^{-1}}$ at $10^{-6}$~mbar and a decoherence-onset pressure $p_\star\approx1.4\times10^{-6}$~mbar that reproduces the measured Arndt-group scale. The subleading, bulk-specific corrections --- the part that would yield a genuinely new number --- require evaluating the overlap integrals, and that is recorded as open (\S\ref{sec:disc-open}).
\end{objqabody}

\bigskip
\noindent\textbf{Objection 41 (O-QM-41) (the SQUID experimentalist)}\par\nobreak
\noindent\textbf{``Your `position-basis universality' is contradicted by every flux qubit and Cooper-pair box, whose pointer states are flux or charge, not position. The claim that `every macroscopic record lives in the position basis' is simply wrong.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Right. The constitutional content of bulk locality (\AX{1}) is that the einselected pointer observable is the one the dominant \emph{local} coupling is diagonal in --- not that it is always position. For the center-of-mass of a macroscopic object that variable is position, which is why everyday macroscopic records are positional; for an engineered system whose dominant coupling is to flux or charge, the pointer observable is flux or charge accordingly. ``Locally-coupled pointer'' is the claim; ``always position'' would be the error.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 42 (O-QM-42) (the bookkeeping referee)}\par\nobreak
\noindent\textbf{``Your macroscopic decoherence rate appears as $10^{13}$~s$^{-1}$ in one place and $10^{40}$~s$^{-1}$ in another --- twenty-seven orders of magnitude apart. At least one is wrong.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Both are present but clearly distinguished, because they are different quantities. The thermal relaxation frequency $k_BT/\hbar\sim10^{13}$~Hz is a temperature scale; the \emph{spatial}-superposition decoherence rate for a cat-sized separation is the Joos--Zeh collisional value $\sim10^{40}$~s$^{-1}$. The cat section uses the spatial rate $10^{40}$~s$^{-1}$ consistently, and the thermal frequency is labeled as such. Conflating the two --- a temperature scale and a spatial-decoherence rate --- would manufacture a spurious twenty-seven-order gap. The single decoherence function $F(\Delta x)$ that unifies the spatial regimes is given in \S\ref{sec:worked-overlap}, Eq.~\eqref{eq:Fdx}.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 43 (O-QM-43) (the master-equation referee)}\par\nobreak
\noindent\textbf{``You call the GKLS generator a `theorem'. But GKSL uniqueness holds only \emph{given} complete positivity, trace preservation, and Markovianity. You assume Markovianity --- so it is a theorem only in the regime you postulate.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Agreed; the paper states the conditional clearly. \TA{2} derives the GKLS \emph{form} from three inputs, two of which (complete positivity, trace preservation) are forced for any boundary channel, while the third (Markovianity) is the leading-order regime in the bath correlation time, with non-Markovian corrections calculable in principle (O-QM-40) and observable at high precision. So the honest statement is: GKLS is forced \emph{at leading order in $\tau_{\rm bath}/\tau_S$}, and the framework's contribution is to supply a structural reason for that smallness rather than to postulate the Markov limit outright. We do not claim GKLS in the strongly non-Markovian regime; there the corrections are the open content.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 44 (O-QM-44) (the einselection referee)}\par\nobreak
\noindent\textbf{``Zurek's pointer basis is supposed to be \emph{derived} from the coupling. But your derivation assumes the bulk coupling is position-multiplicative --- which is to assume the position basis you claim to select. Circular.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The non-circular seed is \AX{1} (bulk locality), and we make the logical order explicit (\TA{6}). The claim is not ``assume position, derive position'' but ``assume \emph{locality} --- that the bulk Hamiltonian density is a polynomial in fields at the same bulk point --- and derive that the dominant boundary coupling is diagonal in the locally-coupled variable.'' Locality is a constitutional axiom independent of any basis; it is what forbids integro-differential (non-local) kernels. For a center-of-mass coupled through local fields, that locally-coupled variable is position, and the position basis wins the predictability sieve. The seed is locality, not position; the position basis is the output for the generic case, with flux/charge the output when the local coupling is to those instead (O-QM-41). The circularity is broken at the axiom.
\end{objqabody}

\bigskip
\section{The 1/f-noise specialist: conformality and its limits}
\bigskip

\noindent\textbf{Objection 45 (O-QM-45) (the noise-spectroscopy referee)}\par\nobreak
\noindent\textbf{``You `derive' the $1/f$ exponent $\alpha=1$ by saying the conformal coupling is `marginal by construction'. Marginal-by-construction is an assumption, not a derivation. You have built in the answer.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The $\alpha=1$ result is the leading-order consequence of marginal conformality --- a leading-order form, not a universal constant. The honest content (\TA{10}) is: \emph{if} the dominant boundary--environment coupling is marginal (scale-invariant), then on dimensional grounds the spectral density is $\propto\omega^{-1}$ at leading order. Whether a given system's coupling is marginal is a property of that system, not a theorem, and the precise exponent is fixed by the system's conformal weight and fluctuator structure --- which is why observed exponents form a band around unity rather than sitting at exactly one (O-QM-49). The claim is therefore ``approximately $1/f$ within a bounded window, system-specific exponent,'' which is what the data show; $\alpha=1$ is not a universal constant.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 46 (O-QM-46) (the materials referee)}\par\nobreak
\noindent\textbf{``$1/f$ noise is notoriously material- and device-dependent. Its `universality' is a loose empirical regularity, not a law. Claiming a structural derivation of a universal $1/f$ over-reaches.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} We agree and have scoped the claim accordingly. The framework does not claim a universal pure $1/f$ to arbitrary frequency. It claims that conformality (\citet{ArisakaA7Qi4}, \QiCON{}) produces an \emph{approximately} $1/f$ spectrum within a \emph{bounded conformal window} set by the system's outer and inner time-scales --- the two-knee structure of \citet{ArisakaA7Qi4}'s analysis --- with a system-specific exponent. That is a much weaker and more defensible statement than ``universal $1/f$ as a law,'' and it is the one in the paper. The material-dependence the objection cites is precisely the system-specificity the claim builds in, not something it contradicts.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 47 (O-QM-47) (the cross-paper referee)}\par\nobreak
\noindent\textbf{``Where do the higher-order corrections to the $1/f$ exponent actually come from? If the answer is the `CCI of \ASix{}', that cannot be right: \ASix{}'s CCI is the Covariant Conservation Identity --- a Bianchi/Maxwell continuity statement with no noise content whatever. The acronym is doing work the identity does not.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} It is not the CCI, and the objection is right that the acronym invites the reading (\TA{10}, proof Step~3). The Covariant Conservation Identity of \citet{ArisakaA6Gi4} is a Bianchi/Maxwell continuity identity and carries no noise content; it is \emph{not} the source of the $1/f$ corrections. The exponent's departures from unity are system-specific (conformal weight, fluctuator structure), not anomalous-dimension contributions from the CCI.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 48 (O-QM-48) (the \citet{ArisakaA7Qi4} referee)}\par\nobreak
\noindent\textbf{``\ASeven{} derives a \emph{bounded, two-knee} noise spectrum with a conformal-weight-dependent slope. A reader skimming this paper could easily come away with an unbounded, universal pure $1/f$ at $\alpha=1$ instead. Which is being claimed?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The bounded, two-knee form --- and the distinction matters enough to state twice. \TA{10} states the spectrum as approximately $1/f$ \emph{between two knees} set by the system's outer and inner time-scales --- the bounded, two-knee structure of \citet{ArisakaA7Qi4} --- with a system-specific exponent near unity, not an unbounded universal $1/f$. The precise spectral form is deferred to \ASeven{}, and \AEight{} reports only the bounded, leading-order $1/f$ behavior it legitimately inherits.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 49 (O-QM-49) (the falsifiability referee)}\par\nobreak
\noindent\textbf{``Your kill criterion is `exponent in $[0.85,1.15]$'. Essentially every measured $1/f$ exponent already lies in that band. A criterion nothing can fail is not a falsifier.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} A fair point about the \emph{exponent-band} criterion, which is a consistency expectation rather than a sharp falsifier. The genuine kill criterion is sharper and stated as such (\TA{10}, kill criterion): a quantum-measurement apparatus whose decoherence spectrum is \emph{sharply non-}$1/f$ under conditions where conformality should hold --- a flat white spectrum ($\alpha\approx0$) or a steep colored one ($\alpha\approx2$) --- would retire the reading. That is a real target, not a band that everything already satisfies. The $[0.85,1.15]$ range is the consistency expectation; ``flat or steep where it should be $1/f$'' is the falsifier. We separate the two so the criterion is not vacuous.
\end{objqabody}

\bigskip
\section{The AMO experimentalist and methodologist: falsifiability and the test horizon}
\bigskip

\noindent\textbf{Objection 50 (O-QM-50) (the discrimination referee)}\par\nobreak
\noindent\textbf{``Strip away the rhetoric and your eight-test horizon collapses: six of the eight reproduce standard quantum mechanics exactly. The framework is empirically equivalent to textbook QM except in a corner you cannot yet reach.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} We agree, and the net-content audit (Part~XII) states it in exactly these terms rather than hiding it: most of the eight forward tests are \emph{consistency} checks, not discriminators, and the framework is operationally near-equivalent to standard quantum mechanics across the laboratory regime. That is not a concealed weakness --- it is the expected situation for a reformulation whose ambition is ontological clarity plus a few high-energy/cosmological signatures, not a wholesale revision of laboratory predictions. The genuinely discriminating content is narrow and we name it: the cohort-geometry signature (Test~8, high-energy corrections at $\Mlock\approx3.85$~TeV, within HL-LHC reach) and the direct $S^1_\theta$ search (the compact-dimension test). A reformulation that changed everyday predictions would already be excluded; one that does not, and adds a few sharp forward bets, is doing the honest thing.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 51 (O-QM-51) (the ``show me a number'' referee)}\par\nobreak
\noindent\textbf{``Your one near-term discriminator, the einselection/$R_{\rm KK}$ test, is not a number --- it is `deferred to a future paper'. So at present the framework has \emph{zero} computed predictions that differ from standard quantum mechanics.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} For the laboratory-scale discriminators, this is essentially correct at present, and the paper marks it as promissory rather than delivered. The $R_{\rm KK}$ ratio requires evaluating the scale-overlap integrals (O-QM-40), which is open. The framework's currently-sharp forward bets are the \emph{cohort-shared} ones it inherits with computed values --- the $\Mlock\approx3.85$~TeV cluster of \citet{ArisakaP1KK} and the cosmological pulls of \citet{ArisakaA10Cosmos} --- rather than a stand-alone laboratory number unique to \AEight{}. The test table distinguishes ``computed and forward'' from ``structural and promissory,'' so no reader mistakes a deferred integral for a delivered prediction. The honest present status is: the discriminating laboratory number is owed, and named.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 52 (O-QM-52) (the collider referee)}\par\nobreak
\noindent\textbf{``Your high-energy test once quoted `LHC energies $\sim10^4$~TeV', which is wrong by three orders --- the LHC runs at $14$~TeV. And if $\Mlock=3.85$~TeV, the effect is \emph{inside} current reach, contradicting `awaits future colliders'.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The scale is $\Mlock\approx3.85$~TeV (\citet{ArisakaP1KK}), which is within HL-LHC reach, and the test is searchable at the HL-LHC, consistent with the cohort's treatment of $\Mlock$ in \citet{ArisakaC1Kaxion} and \citet{ArisakaP1KK}. The predicted object is the leptophobic, clustered, multi-field line shape of \citet{ArisakaP1KK}, a genuine forward bet with a pre-declared HL-LHC kill criterion, not a retrodiction and not a claim beyond the energy frontier.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 53 (O-QM-53) (the bookkeeping referee)}\par\nobreak
\noindent\textbf{``Your falsifiers F-QM-5 through F-QM-8 are defined two different ways in two different places. A reader cannot tell which eight conditions would actually retire the theory.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The two are kept distinct. The eight \emph{falsifiers} F-QM-1\,\ldots\,F-QM-8 (the conditions that would retire the theory: a Born deviation, a direct $S^1_\theta$ detection failure, a super-Tsirelson violation, a no-signaling violation, MWI-specific branches, a non-GKSL evolution, non-Gleason probability, and a caught-alive macroscopic superposition) are the canonical list, with dedicated subsections and a summary table. The eight \emph{forward tests} (the empirical horizon --- Tsirelson saturation, decoherence scaling, $1/f$, position-basis einselection, einselection-rate, Bell-pair fidelity, teleportation, high-energy corrections) are labeled \textbf{Test~1}\,\ldots\,\textbf{Test~8} so they do not borrow the falsifier tags. Falsifiers retire the theory; tests probe it; the two eights are distinct and consistently numbered.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 54 (O-QM-54) (the precision-metrology referee)}\par\nobreak
\noindent\textbf{``You list the Born rule as `confirmed to $10^{-13}$'. That is the electron $g{-}2$ precision, which tests QED, not the Born rule. The actual Born-additivity bound, from Sorkin-type three-path experiments, is around $10^{-3}$. You have quoted the wrong number by ten orders.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Correct. A $10^{-13}$ figure would conflate the $g{-}2$/QED precision with the Born-rule (additivity) test; the genuine Born-additivity null comes from Sorkin-type three-path interference experiments and bounds the third-order term at roughly $10^{-3}$. The table reads ``form derived (Gleason); Sorkin three-path $\le10^{-3}$; consistent,'' which is the honest empirical status. The framework does not claim a $10^{-13}$ confirmation of the Born rule.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 55 (O-QM-55) (the macroscopicity referee)}\par\nobreak
\noindent\textbf{``Your large-molecule decoherence threshold is the Joos--Zeh value. Genuine rivals --- GRW, CSL, Di\'osi--Penrose --- predict \emph{deviations} from standard decoherence at large mass. You predict no deviation, so a macroscopicity experiment cannot distinguish you from textbook QM, only from the collapse models.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} True, and we state the contrast plainly. Objective-collapse models (GRW, CSL, Di\'osi--Penrose) add a mass-dependent stochastic term and therefore predict departures from unitary-plus-environmental decoherence at large mass; the framework predicts \emph{no} such intrinsic departure, because its bulk dynamics is unitary and its decoherence is environmental, not collapse-driven. So a macroscopicity experiment that saw a collapse-type deviation would falsify the framework (and confirm a collapse model); one that saw standard environmental decoherence would be consistent with the framework but would not distinguish it from textbook quantum mechanics. We present this as a kill criterion against collapse-type deviation (F-QM, macroscopic coherence), not as a positive discriminator from standard QM, and we say which it is.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 56 (O-QM-56) (the null-test referee)}\par\nobreak
\noindent\textbf{``Several of your `tests' --- Bell-pair fidelity, teleportation fidelity, no-signaling --- predict that nothing deviates from quantum mechanics. A test whose prediction is `no deviation' has no discriminating power. Why call them tests at all?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} They are \emph{consistency} tests and \emph{kill criteria}, not discriminators, and they are labeled so. Their role is asymmetric: they cannot confirm the framework over standard quantum mechanics, but a \emph{failure} (a Bell-pair fidelity that fell off with distance, an operational signal between wings) would retire it. That asymmetry is exactly what a kill criterion is --- it risks the theory without being able to crown it. These are not presented among the discriminating forward bets but listed under the falsifier role, where ``no deviation expected; a deviation kills the theory'' is the correct and honest description.
\end{objqabody}

\bigskip
\section{The holography referee: the six-dimensional bulk}
\bigskip

\noindent\textbf{Objection 57 (O-QM-57) (the AdS/CFT referee)}\par\nobreak
\noindent\textbf{``You call your bulk-to-boundary map `holographic'. But there is no boundary conformal field theory, no GKP--Witten dictionary, no operator--operator correspondence. This is not holography in the AdS/CFT sense; the word is borrowed prestige.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} We accept the terminological pressure and state the difference. The framework's map is \emph{not} an exact AdS/CFT duality: it is a non-invertible \emph{reduction} --- a partial trace over the two non-spatial fibers plus a CPTP coarse-graining --- in which information is genuinely lost, precisely the opposite of an ontologically symmetric duality (this asymmetry is what makes the measurement problem tractable here and not in AdS/CFT, as the comparison table states). Where we use ``holographic,'' we mean ``a bulk theory whose boundary shadow is a reduced, open description,'' not ``a GKP--Witten duality.'' We have no boundary CFT and claim none. The honest word is ``bulk-to-boundary projection''; ``holographic'' is used in the loose, dimensional-reduction sense, and we flag that it is not AdS/CFT-grade.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 58 (O-QM-58) (the algebraic-QFT referee)}\par\nobreak
\noindent\textbf{``You write the bulk Hilbert space as a tensor product $\mathcal H_{\rm bulk}=\mathcal H_{\rm bdy}\otimes\mathcal H_{\rm env}$ and partial-trace. For a gauge or gravitational theory that factorization does not exist --- the local algebras are Type~III, and the Gauss law obstructs a clean tensor split. Your central manipulation is ill-defined.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} A serious and well-aimed objection, and the honest answer is that the framework operates at the level of the \emph{operational, zero-mode-restricted} boundary algebra, not the full local Type~III von~Neumann algebra. The partial trace is over the non-spatial fiber modes and the unread 4D-environment sector at the level of the GNS representation of the QM-restricted algebra $\mathcal A_{\rm Bdy}^{(\rm QM)}$ (the zero-mode operators), where a factorization is available; we do not claim a clean tensor split of the full gauge-theoretic local algebra. The Gauss-law/Type-III obstruction is real for the full theory, and a rigorous treatment of the edge modes and the algebra type is genuinely open --- flagged as such, with no global tensor product asserted. Within the operational restriction the manipulation is well-defined; extending it to the full local algebra is unfinished, and we say so.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 59 (O-QM-59) (the extra-dimensions referee)}\par\nobreak
\noindent\textbf{``Is $S^1_\theta$ a literal tiny Kaluza--Klein circle of some physical radius, or a non-spatial internal phase direction? You use both readings, and they are not compatible. And your stated radius was off by four orders of magnitude.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Two points. First, the radius: it is set by the lock scale, $R_\theta\sim\hbar c/\Mlock\approx5\times10^{-20}$~m for $\Mlock\approx3.85$~TeV. Second, the two readings: $S^1_\theta$ is a \emph{physical compact gauge circle} of that radius, and the ``non-spatial phase direction'' language refers to its role as the $U(1)$ phase fiber --- the same object described two ways (a small circle in the bulk; a gauge/phase direction from the 4D viewpoint), not two incompatible objects. The physical-circle description is primary and the phase-fiber role is its 4D projection, so there is no incompatibility. This radius places a direct $S^1_\theta$ signature near the lock scale, consistent with the HL-LHC and tabletop searches the falsifier table cites.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 60 (O-QM-60) (the Kaluza--Klein referee)}\par\nobreak
\noindent\textbf{``Your decoherence engine invokes a tower of Kaluza--Klein modes \emph{along the radial direction} $I_r$. But \ATwo{} says $I_r$ is a non-compact renormalization scale with \emph{no} KK tower. You are using a structure your own bulk paper forbids.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The radial direction carries no normalizable KK tower --- \citet{ArisakaA2GG6} is right --- and the decoherence content respects that. The genuine KK tower lives on the \emph{compact} $S^1_\theta$ (a true circle, hence a discrete tower); the \emph{radial} $I_r$ is the non-compact RG scale and contributes the irreversible, coarse-graining (semigroup) structure, not a mode tower. In ``scale-overlap integrals of bulk KK modes,'' the modes meant are the $S^1_\theta$ tower and the 4D-environment modes, integrated with the radial scale profile; the radial direction supplies the scale weighting and the arrow of coarse-graining, not its own normalizable tower. The language is aligned with \ATwo{} so no radial KK tower is invoked.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 61 (O-QM-61) (the precision-terminology referee)}\par\nobreak
\noindent\textbf{``Your `non-invertible bulk--boundary correspondence' is, when unpacked, a partial trace followed by a coarse-graining. That is open-quantum-systems 101. The grand name oversells a standard construction.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Fair, and we do not dispute that the \emph{mechanism} is the standard open-systems reduction --- partial trace plus CPTP coarse-graining. What the framework adds is not a new mathematical operation but an \emph{identification}: that this particular reduction is forced (by the non-spatiality of the fibers, \AX{1}), that the discarded degrees of freedom are specific geometric directions ($I_r$, $S^1_\theta$) rather than a generic bath, and that the irreversibility is the renormalization-group arrow. The operation is textbook; the claim is about \emph{what} is being traced and \emph{why} it must be. We are content to call it a bulk-to-boundary projection and let the open-systems machinery be exactly what it is; the content is the geometric identification of the traced sector, not a novel dictionary.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 62 (O-QM-62) (the bulk-action referee)}\par\nobreak
\noindent\textbf{``All of this presupposes a well-defined six-dimensional bulk action with the gauge content, anomaly freedom, and dynamics you keep invoking. Does that action actually exist, or is it asserted?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} It is constructed, not asserted, in the companions, and \AEight{} inherits it rather than re-deriving it. The bulk action with its $SU(5)\times U(1)_B\times U(1)_{Qi}$ gauge content is given in \citet{ArisakaA2GG6} (with BRST/Slavnov--Taylor structure), its local and global anomaly freedom is established in \citet{ArisakaA2GG6} and \citet{ArisakaA3BI6} (the $\mathrm{BI}_6$ closure), and the parameter-free fixing is the Euclidean discipline of \citet{ArisakaA4Euclid}. \AEight{} uses these as inputs; their correctness is the business of those papers, and a referee who doubts the bulk action should press \emph{them}. Within \AEight{} the action is an inheritance with named sources, not a free assertion --- which is the appropriate division of labor for a synthesis paper.
\end{objqabody}

\bigskip
\section{The historian and philosopher of quantum mechanics; and the cross-paper referee}
\bigskip

\noindent\textbf{Objection 63 (O-QM-63) (the philosopher of physics)}\par\nobreak
\noindent\textbf{```Einstein right in 6D, Bohr right in 4D' equivocates on the word \emph{complete}. Einstein's completeness criterion (every element of reality has a counterpart in the theory) and Bohr's operational completeness are different claims. Your slogan trades on the ambiguity to make a concession sound like a reconciliation.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The sharpest objection in the philosophical set, and the slogan is stated without equivocation (Conclusions). The precise statement is: the \emph{six-dimensional bulk state is complete} in Einstein's sense --- it contains a counterpart to every element of reality, including the outcome-fixing $\Theta_H$ --- and it is \emph{deterministic}, vindicating Einstein's realism and determinism. The \emph{four-dimensional description is incomplete and operationally closed} in Bohr's sense --- it is a lossy projection that nonetheless suffices for all 4D predictions. So ``complete'' is used in \emph{one} sense (Einstein's: a complete underlying state) at the bulk level and Bohr's operational completeness is granted at the boundary level; the two are not the same word doing two jobs in one breath, and the conclusions spell out which completeness holds where. What Einstein loses is locality, which we concede; what Bohr loses is the claim that the 4D description is the \emph{whole} story, which the bulk supplies. The reconciliation is real but bounded, and the slogan carries no more than that.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 64 (O-QM-64) (the Bohr scholar)}\par\nobreak
\noindent\textbf{``Strip away the sixth dimension and what you actually defend --- a 4D-non-local description that is operationally complete and forbids signaling --- is precisely Bohr's 1935 reply to EPR. You have not transcended Bohr; you have re-derived him and added scenery.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} At the 4D operational level, we agree: the boundary description the framework licenses is Bohr-complete and non-signaling, and a reader who cares only about 4D operations gains nothing over Bohr. The framework's claim is precisely that the ``scenery'' --- the realist, deterministic bulk underneath --- is not scenery but content: it converts Bohr's \emph{refusal} to ask about an underlying reality into a definite \emph{answer} (there is one, it is six-dimensional, it is deterministic). Whether that answer is worth the sixth dimension is a judgment a reader is entitled to make against it; we do not claim to have forced their hand. But ``re-derived Bohr at the boundary, and supplied the realist completion Bohr declined'' is a fair and accurate description, and it is the one the framework gives; it does not claim to have overturned him.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 65 (O-QM-65) (the interpretation referee)}\par\nobreak
\noindent\textbf{``You claim the framework `subsumes' Copenhagen, Everett, Bohm, and QBism --- each `right about the boundary, wrong about the ontology'. That template is unfalsifiable, and it strawmans QBism and Many-Worlds, which make claims your template ignores.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} ``Subsumes all interpretations'' would overreach as a scientific claim; it is offered as an interpretive observation, not a theorem (the interpretation-landscape Part). The defensible version is narrower: the framework \emph{recovers the operational core} each interpretation gets right (Copenhagen's measurement primacy, Everett's unitary bulk, Bohm's determinism, QBism's agent-relative boundary statistics) within one structure, while disagreeing with each on ontology. It does not refute QBism's epistemic stance or Many-Worlds' realism by fiat, and is not presented as doing so; a QBist or Everettian can read the same mathematics and decline the ontology. ``One structure that reproduces what each gets operationally right'' is the claim; ``subsumes/defeats all interpretations'' is not.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 66 (O-QM-66) (the parsimony referee)}\par\nobreak
\noindent\textbf{``You invoke Ockham --- `Everett's explanatory power without the multiverse cost'. But you have replaced a multiverse with a sixth dimension, a compact gauge circle, a hidden holonomy, and a non-local ontology. That is not obviously cheaper.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} A fair rejoinder, and the parsimony claim is correspondingly modest. The Ockham point is genuine but narrow: the framework posits a \emph{single actual world} rather than a continually branching infinity of them, which is a real ontological economy on the count of \emph{worlds}. It is \emph{not} cheaper on the count of \emph{structure} --- it buys the single world with extra geometry (the two fibers) and a hidden variable. Which economy a reader prefers --- fewer worlds or less structure --- is a legitimate matter of taste, and the framework is not asserted to be parsimonious \emph{tout court}. ``One world, more geometry'' versus ``more worlds, less geometry'' is the honest trade, and we state it as a trade.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 67 (O-QM-67) (the historian)}\par\nobreak
\noindent\textbf{``You label the projection postulate `Copenhagen (Bohr 1927)'. Bohr's complementarity and von~Neumann's 1932 projection postulate are different things; Bohr never endorsed a literal collapse. The attribution is anachronistic.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Correct on the history; the two are kept separate. Complementarity (Bohr, 1927) is an epistemological doctrine about mutually exclusive experimental arrangements; the projection postulate (von~Neumann, 1932; refined by L\"uders, 1951) is a dynamical rule. The framework's L\"uders theorem (\TA{4}) addresses the \emph{von~Neumann/L\"uders} rule, and its complementarity discussion addresses \emph{Bohr}; the two are not conflated under ``Copenhagen.'' The projection rule is attributed to von~Neumann/L\"uders and complementarity to Bohr, and no literal collapse is put in Bohr's mouth.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 68 (O-QM-68) (the demarcation referee)}\par\nobreak
\noindent\textbf{``If the framework is empirically equivalent to operational quantum mechanics across the laboratory, then by your own admission it is an \emph{interpretation}, not a new physical theory. Stop calling it physics.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} We draw the line honestly. \emph{Where} the framework is empirically equivalent to operational quantum mechanics --- the laboratory regime --- it is, as the objection says, an interpretation, and we present it as such (the realist reading of the measurement problem, Bell, and the postulates). \emph{Where} it makes computed forward predictions that differ --- the cohort-shared high-energy and cosmological signatures ($\Mlock$ cluster, the $H_0$ and electroweak pulls of the companions) --- it is a physical theory with skin in the game. The same mathematical structure wears both hats, and we say which hat is on in which regime: interpretation in the lab, physics at the frontier. Calling the whole thing ``just interpretation'' ignores the forward bets; calling it ``new lab physics'' ignores the equivalence. We claim exactly the split.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 69 (O-QM-69) (the cross-paper referee)}\par\nobreak
\noindent\textbf{``The load-bearing results --- the Hilbert space, the GKLS form, no-signaling, the QM postulates --- are all `inherited from \AFive{}' or `inherited from \AOne{}'. \AEight{} is a citation manifold. What does \emph{this} paper actually prove?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The division of labor is deliberate and we state it. \AEight{} is a synthesis paper: it takes the seven QM-postulate theorems \citet{ArisakaA5GDOS} proves as a terse \T{7.1}--\T{7.7} chain and lifts them to graduate-textbook depth as \TA{1}--\TA{7}, and it proves \emph{net-new} the four results \TA{8}--\TA{11} (the extremal Tsirelson reading, the universal-identity promotion, the $1/f$ conformality, and the standalone no-signaling theorem). Its distinctive contributions are the measurement-problem synthesis (the three-questions-plus-one-residue statement), the worked examples carried to explicit calculation, the paradox sweep, and the interpretation landscape. What it does \emph{not} do is re-prove the bulk action or the GNS construction; those are the companions' work, cited under their canonical ids. A synthesis paper that hid its inheritances would be dishonest; one that names them is doing the right thing, and the net-new results are flagged as such.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 70 (O-QM-70) (the attribution referee)}\par\nobreak
\noindent\textbf{``You credit the geodesic-dip Bell mechanism to `\AFive{} \S\T{7.7}'. But \AFive{} \S\T{7.7} is the chain of QM-postulate theorems; the radial-dip story appears to be original to \AEight{}. You have laundered an \AEight{} construction as an inheritance.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The concern does not arise: there is no radial-dip mechanism (O-QM-16), hence no \AEight{} construction mis-attributed to \AFive{}. What \AEight{} genuinely inherits from \citet{ArisakaA5GDOS} \S\T{7.7} is the no-signaling lemma and the bulk--boundary correlation structure; what is original to \AEight{} (the extremal-stationarity reading of Tsirelson, \TA{8}) is labeled net-new. Presenting an original mechanism as a companion inheritance would be exactly the attribution error the objection targets, and the framework keeps the provenance distinct.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 71 (O-QM-71) (the observer-criterion referee)}\par\nobreak
\noindent\textbf{```Measurement' is used in three senses across the cohort: a \GRIP{} registration in \AEight{}, a runtime event in \ASeven{}, a layer-17 biological act in \AEleven{}. There is no single governing definition. The cohort speaks with three voices.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} There is a single governing definition, stated as cohort-wide. A measurement, everywhere in the cohort, is an irreversible \GRIP{} registration: a coupling that reshapes the boundary free-energy landscape so the descent halts at a Geometric Attractor and the record becomes monotone (\citet{ArisakaA6Gi4}). \ASeven{}'s ``runtime event'' and \AEleven{}'s ``layer-17 biological act'' are \emph{instances} of this one definition at different strata, not competing definitions --- a photographic grain, a transistor, and a retina each satisfy the same free-energy criterion. The QM-stratum minimal-observer threshold (the smallest landscape change that guarantees the halt) is the one genuinely open quantity, flagged in \S\ref{sec:disc-open}. One definition, several strata, one open threshold --- stated uniformly.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 72 (O-QM-72) (the logical-dependency referee)}\par\nobreak
\noindent\textbf{``Your Born chain is a loop: \AEight{} cites \AFive{} \T{7.3} for the trace form, \AFive{} cites \AOne{} AX-3 for the probability structure, and \AOne{} AX-3 already contains the trace form. The form is established only where it is assumed.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} We acknowledge the dependency and state precisely where the non-trivial step is, so the chain is not mistaken for a proof from nothing. \AOne{}'s \AX{3} postulates that the boundary observer's outcomes are governed by a $\sigma$-additive probability measure on the projection lattice --- it postulates \emph{that there is a measure}, not \emph{which} one. The trace form is then \emph{forced} (not assumed) by Gleason, given the Hilbert-space structure. So the chain is: \AX{3} supplies the measure-theoretic premise; Gleason supplies the uniqueness; \AFive{}/\AEight{} carry it into the operational setting. The honest residue --- flagged at (O-QM-14) --- is that $\sigma$-additivity itself is postulated at \AX{3} and not derived from a deeper bulk measure. That is a real open question, not a hidden circularity: the form is forced once the premise is granted, and the premise is named.
\end{objqabody}

%% ===================================================================
\section{The rival-completion referees: 't~Hooft and Penrose}

\noindent\textbf{Objection 73 (O-QM-73) (the cellular-automaton referee, after 't~Hooft)}\par\nobreak
\noindent\textbf{```t~Hooft's cellular-automaton interpretation already gives a deterministic completion of quantum mechanics that keeps strict locality, paying only with measurement (statistical) independence. Why concede 4D non-locality at all? A six-dimensional hidden variable looks like a baroque way to buy back what superdeterminism gets for free.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Because the two prices are not equal, and we judge ours the cheaper. Bell forces every realist completion to give up \emph{either} 4D locality \emph{or} measurement independence; 't~Hooft \citep{tHooft2016} gives up the latter, we give up the former (O-QM-25, O-QM-27). Surrendering measurement independence requires the source variable to be correlated with the analyzer settings across spacelike separation and through arbitrary intervening randomizers --- starlight, quantum random-number generators, the experimenter's deliberation --- a fine-tuning we decline. Surrendering 4D factorizability, by contrast, costs nothing beyond what entanglement already costs ordinary quantum mechanics: our non-locality is the boundary shadow of a state strictly \emph{local in 6D} by \AX{1} (\TA{7}), not a new conspiracy. We keep 't~Hooft's determinism, his ontological realism, and his rejection of branching; we differ only on which Bell horn to take. And we concede the debt the two programs share: neither has yet derived the Born measure from its deeper layer --- for us what remains of it is the field-grade representation step, the single analytic residue of the basin-capture $=$ Born statement, closed at the relaxation level (\S\ref{sec:disc-open}), and the automaton program's long struggle with the same step is a standing caution, not a scoreboard in our favor.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 74 (O-QM-74) (the objective-reduction referee, after Penrose)}\par\nobreak
\noindent\textbf{``Penrose is right that decoherence does not solve the measurement problem --- and you now agree. But Penrose draws the honest conclusion: quantum mechanics must be \emph{modified}, by a real gravitationally-induced collapse with a measurable deviation ($\tau\sim\hbar/E_G$). You reproduce standard quantum mechanics exactly and relabel the unsolved selection step as a `relaxation-level theorem with one analytic residue.' Where is the new physics? Where is gravity in your collapse?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} We accept the diagnosis and decline the cure, and we say what that costs. We agree with Penrose \citep{Penrose1996, Diosi1989} that the partial trace yields an improper mixture, not an outcome (\TA{1}, \TA{4}). We differ on the mechanism of selection: Penrose adds a \emph{new}, gravitationally-triggered, stochastic collapse that \emph{modifies} unitary evolution; we add a \emph{deterministic} descent of the joint configuration into one Geometric Attractor basin (\GRIP{}, \TA{1}/\TA{4}), with bulk evolution exactly unitary and decoherence set by the physical 4D bath (\TA{5}), not by gravitational self-energy. Three honest consequences follow. \emph{Gravity}: in our account gravity shapes the 6D bulk geometry (\citet{ArisakaA2GG6}) but does not trigger outcome selection --- so to ``where is gravity in your collapse?'' our answer is that there is no collapse, and gravity's role is in the arena, not the trigger. \emph{Non-computability}: Penrose's reduction is non-computable by design and tied to consciousness; ours is an ordinary deterministic relaxation, and invokes neither. \emph{Empirical content}: here is the cleanest experimental fork in the paper. Objective-collapse models predict a mass-dependent breakdown of superposition beyond environmental decoherence; we predict none. A macroscopicity experiment that saw a Di\'osi--Penrose deviation would \emph{falsify} the framework and vindicate Penrose; one that saw only environmental decoherence would be consistent with us (F-QM, macroscopic coherence). We regard that decidability as a virtue of Penrose's program and record it as a live way the framework could be wrong --- which is the honest answer to ``where is the new physics?'': it is in the bulk and the $M_{\rm lock}$ sector, not in a modification of the Schr\"odinger equation, and the experiment that separates us from Penrose is already being run.
\end{objqabody}


\bigskip
\section{The complex-number and real-bulk referees}

\noindent\textbf{Objection 75 (O-QM-75) (the complex-number referee, after Penrose's question)}\par\nobreak
\noindent\textbf{``Your hidden-fiber Part says the charged phase is a real circle, and your Discussions now claim the state-space $i$ is derived too. But your own Constitution installs a complex Hilbert space in the bulk package of \AX{3}. The `real bulk' is marketing.''}\par\nobreak
\begin{objqabody}
\emph{Response.} The objection correctly reads the dictionary and misses its standing. In the free and quasi-free regime, \S\ref{sec:disc-real-bulk} derives the bulk state space: $\cH_\Bulk$ is the Fock completion of the real solution space under a complex structure built from the circle (charged sectors) and the polar phase of constitutional time (all sectors), with positivity doing the selecting. In that regime the complex entries of the \AX{3} package are theorems over real data, not primitives. What remains primitive is exactly what is named: the interacting completion, which sits at the same frontier as all of constructive field theory. A referee is entitled to say the program is unfinished; the books, however, are open about which pages are missing.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 76 (O-QM-76) (the neutral-sector referee)}\par\nobreak
\noindent\textbf{``Your own arithmetic kills your own story: the $\theta$-rotation generator squares to $-n^2$, so it vanishes on neutral sectors --- yet a real scalar field's quanta interfere. The sixth dimension cannot explain quantum complexity.''}\par\nobreak
\begin{objqabody}
\emph{Response.} The premise is our theorem and the conclusion does not follow. The circle can only complexify what it moves; on weight-zero sectors it moves nothing, and \S\ref{sec:disc-real-bulk} therefore derives the neutral-sector $i$ from the second shared real structure instead: the polar phase of the time-translation generator, selected uniquely by \AX{1} causal order plus \AX{3} positivity on the static arena of Theorem~0 \citep{AshtekarMagnon1975, Kay1978}. On charged sectors the two constructions agree on shell (O-QM-77), so there is one $i$, not two. The circle is credited with what it earns --- compactness, gauge character, superselection --- and the existence of $i$ is credited to time and positivity. The division of labor is the answer, not a retreat from it.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 77 (O-QM-77) (the field-theory referee, on antiparticles)}\par\nobreak
\noindent\textbf{``Your `antiparticles are a consistency condition' is Feynman--St\"uckelberg with new packaging. Every field-theory textbook already pairs charge $-n$ with positive frequency.''}\par\nobreak
\begin{objqabody}
\emph{Response.} The pairing is textbook; its \emph{provenance} is the claim. In standard field theory the assignment is imposed so that the quantized Hamiltonian comes out positive --- a condition on the output of quantization. In the present framework the same assignment appears one level down, before quantization, as the unique orientation under which two independently forced real structures --- the circle's quarter-turn and the clock's quarter-period --- act as the \emph{same} complex structure on shell; every other assignment makes them anti-agree (\S\ref{sec:disc-real-bulk}, RB-3). A convention with a derivation behind it is no longer a convention. The referee is right that no new particle physics follows; what follows is an answer to ``why does nature pair them so?'', which the textbook does not attempt.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 78 (O-QM-78) (the quantum-information referee, after Renou)}\par\nobreak
\noindent\textbf{``Real quantum mechanics is experimentally falsified \citep{Renou2021} --- and your bulk is real. You are advertising the theory experiment just killed.''}\par\nobreak
\begin{objqabody}
\emph{Response.} Category error, and the distinction is load-bearing. What experiment excludes is quantum theory over real Hilbert spaces with \emph{real tensor products} --- independent sources with no shared complex structure. The framework's boundary theory is standard complex quantum mechanics; its bulk is real \emph{with a derived, single, globally shared} complex structure, and RB-4 (\S\ref{sec:disc-real-bulk}) shows the complex tensor product is exactly the real tensor product quotiented by the shared-$J$ identification --- an operation that exists because $J$'s sources (one circle, one causal order) are global. The excluded theory is the one \emph{without} the sharing. Far from falsifying the framework, the experiment demands precisely the structure the framework derives; we read it as the strongest external support this Part has.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 79 (O-QM-79) (the curved-space referee)}\par\nobreak
\noindent\textbf{``Curved-space field theory has no preferred vacuum: inequivalent complex structures, particle creation, Unruh. Your `unique derived $i$' cannot survive cosmology or a black hole.''}\par\nobreak
\begin{objqabody}
\emph{Response.} The uniqueness claim is made where it holds and the ambiguity is housed where it belongs. The canonical arena of Theorem~0 is globally static, so the bulk-layer $J$ is canonical there. Observer-adapted complex structures --- Rindler, comoving --- are \emph{dictionary} data: \AX{3} makes the instruments part of the boundary package, and which modes a detector counts is fixed by the observer's time, which is boundary information. Their physical output is produced by the \GDOS{} partial trace itself: tracing the inaccessible partner of the squeezed pairing yields an exactly thermal state, at $T = a/2\pi$ under the Unruh pairing --- thermality as constitutional openness, not new physics. The rotating case and the collapse epoch are closed at the model level (RB-11, RB-12, \S\ref{sec:disc-real-bulk}): the rotating anchor moves to the asymptotic frame with superradiance as \GDOS{} pumping, and the collapse channel factorizes into universal thermality times a decaying transient. What remains open is stated there too: the exact-Kerr Hadamard analysis and the full gravitating case, the field's own, imported where used.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 80 (O-QM-80) (the mathematical-physics referee, on fermions)}\par\nobreak
\noindent\textbf{``Fermions are intrinsically complex --- Dirac spinors live in $\mathbb{C}^4$, and your real-bulk story is a bosonic story.''}\par\nobreak
\begin{objqabody}
\emph{Response.} RB-5 (\S\ref{sec:disc-real-bulk}) covers both statistics, and the fermionic case is the \emph{easier} one. In Majorana form the fermionic datum is a real solution space with a positive-definite inner product --- no taming condition to check --- and the unique positive-energy complex structure is again the polar phase of the time generator, the ground-state polarization of the self-dual CAR formalism; reversing its orientation on any single spectral plane produces a negative one-particle energy, which is the uniqueness proof in one line. Charged fermions inherit the circle structure verbatim (the spinor index is inert under $\partial_\theta$), and charge conjugation factorizes as circle reversal composed with a fixed real spinor map. The Dirac field's $\mathbb{C}^4$ is packaging: charge by the circle, frequency by the clock, spinor indices by real matrices.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 81 (O-QM-81) (the twistor referee, after Penrose's second question)}\par\nobreak
\noindent\textbf{``Your real-bulk story explains amplitudes, perhaps. But the deepest complex magic in physics is that massless fields are \emph{holomorphic} objects --- twistor theory --- and a real circle plus a real clock has nothing to say about cohomology on $\mathbb{CP}^3$.''}\par\nobreak
\begin{objqabody}
\emph{Response.} RB-9 says it precisely, and two of its three components are theorems of the twistor school itself. Positive frequency is holomorphy at every level: Hardy splitting at the signal level (generated by this framework's derived $J$); forward-tube analyticity at the field level; and, for conformally invariant massless fields, cohomology on the \emph{positive half} of twistor space --- the $\mathbb{PT}^{+}/\mathbb{PT}^{-}$ division being nothing other than the frequency splitting \citep{PenroseMacCallum1973, EastwoodPenroseWells1981}. What the framework adds is the direction of explanation: holomorphy is what causal order plus energy positivity \emph{look like} when the symmetry is large enough to complexify every timelike direction at once, and both of those are real constitutional data (\AX{1}, \AX{3}). The referee's premise --- that holomorphy is a primitive endowment --- is exactly what RB-9 replaces; the referee's mathematics is untouched, and gratefully used. What the framework does not yet own is stated in the ledger: the full relativistic QED infrared (its rigorous-model core is now owned --- RB-10), and the nonlinear twistor constructions beyond the free correspondence. The referee who wants the full case --- the two real sources, the bridge at its Hardy, tube, and twistor levels, and the open items stated as open --- is directed to the member preprint \PaperPFour{} \citep{ArisakaP4Real}, written for exactly this reader.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 82 (O-QM-82) (the infrared referee)}\par\nobreak
\noindent\textbf{``Your $i$ survives the infrared, but your particle does not. In QED the electron is an infraparticle: no sharp mass shell, no Fock ground state at fixed momentum, velocity superselection. You cannot claim the real-bulk derivation `extends to the infrared' when the very notion of an asymptotic particle --- the object your Haag--Ruelle route (RB-8) quantizes over --- dissolves there.''}\par\nobreak
\begin{objqabody}
\emph{Response.} RB-10 concedes the particle and keeps the $i$ ---
and the concession is the theorem's content. What dissolves in the
infrared is the Fock mass shell: the electron must carry its soft
cloud, clouds attached to different velocities are exactly orthogonal
in the cutoff-removed limit, and the sharp particle is replaced by a
family of disjoint sectors \citep{Frohlich1973, Buchholz1986}. But
every step of that dissolution is a statement about \emph{sector
labels}: the cloud is a c-number displacement, the displacement is an
affine symplectic map, and the quadratic generator that the
polar-phase construction (RB-2) reads is exactly invariant in every
sector. The multiplicity therefore lands at \AX{3} as port structure
--- which infrared background an instrument treats as vacuum ---
precisely where the black-hole vacuum multiplicity landed, and the
derived $J$ is sector-blind. At the van Hove stratum this is exact; at
the Nelson/Pauli--Fierz level it holds via the theorem-grade
infraparticle analysis of the Pizzo school \citep{Pizzo2003,
BachmannDeckertPizzo2012, ChenFrohlichPizzo2010}; for full QED the
Chung/Faddeev--Kulish dressing \citep{Chung1965, FaddeevKulish1970}
is semi-rigorous, and the statement is displayed as the field's
frontier rather than claimed. The referee's premise is thus adopted,
not refuted: the infrared problem is a problem about which vacuum an
observer stands in --- not a problem about where $i$ comes from.
\end{objqabody}

\bigskip
\noindent\emph{Closing note.} These eighty-two objections are the surviving ones --- the readings that remain after a referee has studied the companions that do the work \AEight{} summarizes. Where the honest answer is ``not yet,'' it is named as an open item (the field-grade representation step above all, \S\ref{sec:disc-open}). The framework's wager is stated plainly throughout: a deterministic, realist six-dimensional bulk, conceding 4D non-locality, that reads the quantum postulates as theorems and the measurement problem as three answered questions plus one named analytic residue. A reader is entitled to decline the wager; what the framework asks is that it be declined for the right reasons, not the retired ones.


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\GGPart[References]{References}{Cited works, each with a relevance note linking it to the present paper}
\label{sec:references}

\noindent Unlike a conventional reference list, every entry below carries a short \emph{Relevance} note: a sentence or two saying what the cited work contributes and where in the present paper it bears. The aim is to turn the bibliography from a bare list of sources into a navigable companion --- a reader can move through the references and see, at a glance, which theorem, worked example, 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 present reformulation: from the foundations of quantum mechanics and the measurement problem, through open-system dynamics and decoherence theory, to the twenty-two-paper \GGTheory{} cohort in which \AEight{} reads four-dimensional quantum mechanics as the boundary projection of the six-dimensional bulk --- Einstein right in 6D, Bohr right in 4D. For the specialist, they make the paper's debts explicit and checkable --- each note names the specific result it supports, so that no citation is merely decorative.

\noindent The \GGTheory{} cohort papers (\AOne{}--\AEleven{}, together with the phenomenology P-series and the consequence C-series) are cited throughout under the canonical keys and titles fixed by the cross-paper coherence map of the program; \AEight{}'s place in that cohort --- the quantum-mechanics member of the 4D-Physics trio --- is set out in the cohort-inheritance Part. The references are listed alphabetically by first author.

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

%% ===================================================================
%%  BIBLIOGRAPHY
%%  -----------------------------------------------------------------
\section*{A.~Papers by Katsushi Arisaka (the \GGTheory{} cohort)}
\begin{thebibliography}{99}

\bibitem[Arisaka(2026, A1-Constitution)]{ArisakaA1Constitution}
\textbf{K.~Arisaka.}
\emph{The \GGConstitution{}: Four Axioms and Seven Postulates of the \GGTheory{} Program, From Which \GG{}, \BIsix{}, \OGNfive{}, and \GDOS{} Descend as Theorems},
Arisaka-GG-A1-Constitution.
\par\noindent Preprint, available at \url{https://arisaka-gg.org/}.
\par\noindent\textbf{Relevance.}
Supplies the constitutional charter (4 axioms \AX{1} through \AX{4}, 7 postulates \POS{1} through \POS{7}) that \AEight{} inherits as its first cause. \AX{3} (\OGGEP{}) forces the bulk-to-boundary compiler from which the QM postulates emerge as theorems; \AX{1} (\LC{}) supplies the bulk locality that drives einselection (\TA{6}); \AX{4} (\MSMF{}) eliminates hidden continuous parameters that an interpretation might otherwise insert. Invoked dozens of times across Parts~II--XIII. First cited in \S\ref{sec:inh-A1}.

\bibitem[Arisaka(2026, A2-GG6)]{ArisakaA2GG6}
\textbf{K.~Arisaka.}
\emph{\GGTheory{} with \GG{}: Geometry--Gauge Locking at the 6D Bulk (\GG{}) at the Heart of Grand Geometric Theory (\GGTheory{})},
Arisaka-GG-A2-GG6.
\par\noindent Preprint, available at \url{https://arisaka-gg.org/}.
\par\noindent\textbf{Relevance.}
Constructs the 6D bulk geometry $\mathcal{M}_4 \times I_r \times S^1_\theta$ on which the \GGTheory{} program runs. \AEight{} uses \ATwo{} to ground the bulk-side ontology of the QM reformulation: the topologically protected $S^1_\theta$ winding is the bulk hidden-variable space whose 4D projection generates Bell--CHSH violations (\TA{7}); the radial direction $I_r$ is the renormalization-scale projection axis along which the single non-separable joint state casts its two boundary shadows (Theorem~0 of \ATwo{}: $I_r$ carries no light cone). Cited in Parts~II, III, V, VIII. First cited in \S\ref{part:I}.

\bibitem[Arisaka(2026, A3-BI6)]{ArisakaA3BI6}
\textbf{K.~Arisaka.}
\emph{Anomaly Freedom of \BIsix{}: The Modified Bianchi Identity (\BIsix{}) for the Six-Force, Six-Dimensional Bulk (\GG{})},
Arisaka-GG-A3-BI6.
\par\noindent Preprint, available at \url{https://arisaka-gg.org/}.
\par\noindent\textbf{Relevance.}
Establishes the \BIsix{} closure identity $dH_3 = X_4$ (Green--Schwarz-modified) that fixes the unique anomaly polynomial of the 6D bulk. \AEight{} invokes \AThree{} to certify the bulk's consistency under the operational projection that produces 4D QM: the anomaly cancellation in 6D is what guarantees the boundary algebra closes and admits a CPTP-channel structure rather than degenerating to a non-trace-preserving map. Cited in Part~II. First cited in \S\ref{ssec:inh-A3}.

\bibitem[Arisaka(2026, A4-Euclid)]{ArisakaA4Euclid}
\textbf{K.~Arisaka.}
\emph{From Euclid to \GG{}: The Realization of Euclid's Elements in Six Dimensions --- \GUfive{}, \OGNfive{}, and a Parameter-Free Reconstruction of Modern Physics},
Arisaka-GG-A4-Euclid.
\par\noindent Preprint, available at \url{https://arisaka-gg.org/}.
\par\noindent\textbf{Relevance.}
Supplies the Euclidean-discipline input ledger that fixes zero free continuous parameters: the \GUfive{} ($c, G_N, \hbar, k_B, Q_B$), topological constants $\pi$ and $e$, and the proved \OGNfive{} integer ledger. \AEight{} cites \AFour{} to anchor its zero-free-parameters claim: nothing in the QM reformulation introduces a fitting parameter beyond the cohort ledger; every numerical anchor in the worked examples traces back to the Euclidean inputs. First cited in \S\ref{ssec:inh-A4}.

\bibitem[Arisaka(2026, A5-GDOS)]{ArisakaA5GDOS}
\textbf{K.~Arisaka.}
\emph{\GDOS{}: Geometric Dynamics of Open Systems --- 6D \GG{} Bulk Projected to 4D Boundary for Quantum Measurement, General Relativity, and QFT},
Arisaka-GG-A5-GDOS.
\par\noindent Preprint, available at \url{https://arisaka-gg.org/}.
\par\noindent\textbf{Relevance.}
The direct upstream foundation of \AEight{}. Part~II of \AFive{} proves theorems \T{1}--\T{6} that construct the canonical boundary algebra on $\partial\Mbulk$; Part~III then derives the QM-postulates-as-theorems chain \T{7.1}--\T{7.7} (Hilbert space, Schr\"odinger equation, Born rule, L\"uders projection, decoherence, einselection, Bell--CHSH). \AEight{} inherits this chain verbatim as \TA{1}--\TA{7} and supplies the graduate-textbook envelope: explicit worked examples plus four net-new theorems \TA{8}--\TA{11}. First cited in \S\ref{sec:inh-A5}.

\bibitem[Arisaka(2026, A6-GRIP)]{ArisakaA6Gi4}
\textbf{K.~Arisaka.}
\emph{The \GRIP{} on \GDOS{} --- \GRIP{}: the Geometric Engine of Symmetry Breaking, from the Big Bang to Mind},
Arisaka-GG-A6-GRIP.
\par\noindent Preprint, available at \url{https://arisaka-gg.org/}.
\par\noindent\textbf{Relevance.}
Supplies the \GiFour{} four-pillar structure (\TGG{}, \CCI{}, \GFF{}, \GRIP{}) and the \GRIP{} cascade engine $\DGI{} \to \GFF{} \to \GSSB{} \to \GA{}$. \AEight{} uses \ASix{} in two ways: (i)~the \GFF{} Lyapunov landscape on which einselection (\TA{6}) descends; (ii)~the \GRIP{} cascade as the bulk-side dynamical engine that the QM stratum specializes. \AEight{}'s Figure~2 inserts the QM tier between \GDOS{} and \GRIP{} precisely to mark this seed-role. First cited in \S\ref{ssec:inh-A6}.

\bibitem[Arisaka(2026, A7-Qi4)]{ArisakaA7Qi4}
\textbf{K.~Arisaka.}
\emph{The \QiFour{} Pillars --- \QiQUA{}, \QiEP{}, \QiCON{}, \QiHAL{}: The Quantum-Information Backbone of \GDOS{}},
Arisaka-GG-A7-Qi4.
\par\noindent Preprint, available at \url{https://arisaka-gg.org/}.
\par\noindent\textbf{Relevance.}
Supplies the \QiFour{} quantum-information four-pillar structure (\QiQUA{}, \QiEP{}, \QiCON{}, \QiHAL{}). \AEight{}'s net-new theorem \TA{10} (decoherence-rate conformality) is the direct consequence of \QiCON{} applied to the QM stratum: the off-diagonal decoherence rate must scale with conformal dimension under boundary-RG flow. The Arisaka equation $J_E/Q_B = J_e = J_B = J_{Qi}$ from \QiQUA{} anchors the cohort's quantitative ledger that \AEight{} inherits. First cited in \S\ref{ssec:inh-A7}.

\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 preprint (the present paper).
\par\noindent Preprint, available at \url{https://arisaka-gg.org/}.
\par\noindent\textbf{Relevance.} The present paper. 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:cohort-a8}.

\bibitem[Arisaka(2026, A9-SM)]{ArisakaA9SM}
\textbf{K.~Arisaka.}
\emph{The Standard Model as Theorem of \GGTheory{}: Its 28 Parameters as Observations of One Six-Dimensional Geometry},
Arisaka-GG-A9-SM.
\par\noindent Preprint, available at \url{https://arisaka-gg.org/}.
\par\noindent\textbf{Relevance.}
Supplies the universal-identity insight via Part~XII \S60 that \AEight{} invokes as its central philosophical centerpiece (Part~XIII Conclusions). Every electron is the same because every electron is the unique stable Geometric Attractor of the charged-lepton stratum, propagating identically through atoms / molecules / ATP / DNA / first cell. \AEight{} specializes this reading to indistinguishable-particle statistics in its net-new theorem \TA{9}. First cited in \S\ref{sec:inh-A9}.

\bibitem[Arisaka(2026, A10-Cosmos)]{ArisakaA10Cosmos}
\textbf{K.~Arisaka.}
\emph{The Standard Cosmology as Theorem of \GGTheory{}: The Cosmic Inventory as Observations of One Six-Dimensional Geometry},
Arisaka-GG-A10-Cosmos.
\par\noindent Preprint, available at \url{https://arisaka-gg.org/}.
\par\noindent\textbf{Relevance.}
Supplies the cosmological specialization of \OGGEP{}/\GDOS{}: the boundary cosmological inventory $\Omega_m = 1/\pi$, $\Omega_\Lambda = 1 - 1/\pi$, the modified Friedmann equations, the Hubble two-stage corridor, and Kaxion dark matter at $\sim 8.1\,\mu$eV ($1.95$~GHz). \AEight{} does not invoke the cosmological content directly but references \ATen{} for cohort coherence: the same framework that compiles bulk to boundary on the QM stratum also compiles bulk to boundary on the cosmological stratum. First cited in \S\ref{sec:inh-A10}.

\bibitem[Arisaka(2026, A11-Origin)]{ArisakaA11Origin}
\textbf{K.~Arisaka.}
\emph{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.}
Supplies the structural template that \AEight{} specializes on the QM stratum: \AEleven{}'s four-move operational anatomy ($\DGI{} \to \GFF{}$ Morse flow $\to \GSSB{} \to \GA{}$) is the master pipeline that \AEight{} Part~III specializes to the QM stratum (bulk~$\to$~\OGGEP{}~$\to$~\GDOS{}~$\to$~4D~QM). \AEleven{}'s universal-identity reading and structural-realist genealogy are the philosophical apparatus that \AEight{} Part~XIII inherits and re-applies at QM register. First cited in \S\ref{sec:inh-A11}.

\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
$\Mlock = (1/2)\,\MPl\,\mathrm{e}^{-35} \approx 3.85\,\TeV$, 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:cohort-a8}.

\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:cohort-a8}.

\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:cohort-a8}.

\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 origin-of-$i$ member paper of
the Foundations Branch, which develops the
real-bulk series' holomorphy bridge (RB-9) at standalone length: the
two real sources of the derived complex structure, the bridge at its
Hardy, tube, and twistor levels, worked cohomology examples, three
lecture courses, and the open items (the interacting massless
infrared; the dynamics half of the bridge) stated as open. The
present paper is its principal consumer in the cohort. First cited
in \S\ref{sec:cohort-a8}.

\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 CPT member paper of the Foundations Branch: antiparticles from the circle--clock handshake, C/P/T as orientation reversals of the three real bulk factors, CPT at quasi-free grade from the Wick-quartet analytic fact (T-P5-B1), the chiral-pentagon CP phases, and the matter asymmetry through the weight-zero door. Cited in \S\ref{sec:TA12-statistics}, whose candidate theorem \TA{12} composes its quartet machinery (T-Stat-A route) with \TA{9}'s token identity to discharge the fermionic symmetrization postulate. First cited in \S\ref{sec:cohort-a8}.

\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.} Predictions Branch: the ground-attractor method of \PaperPThree{} run on the baryonic stratum. It matters here for one reason and it is a contrast rather than an inheritance. The present paper reads the compact phase circle as memory and derives the boundary's probabilities from the fact that its coordinate cannot be resolved; \PaperPSix{} reads a winding of the same circle as a conserved integer that no local process can change. Protection by unreadability and protection by winding class are different mechanisms on one geometry, and the difference is what makes the proton stable and a measurement outcome irreversible. Cited at roster level; nothing in \AEight{} depends on its results. First cited in \S\ref{sec:cohort-a8}.

\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
scale-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:cohort-a8}.

\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 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:cohort-a8}.

\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:cohort-a8}.

\end{thebibliography}

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

\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.} White-Paper Branch: the historical survey of how conservation moved from symmetry to cohomology. It supplies the context for a distinction this paper relies on without arguing for it, namely that the quantities surviving the bulk-to-boundary projection survive as classes rather than as charges. The survey is conceptual and carries no computation the present paper uses. Cited at roster level. First cited in \S\ref{sec:cohort-a8}.

\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.} White-Paper Branch: the structural treatment of protection by topology, and the closest companion to this paper's reading of the phase fiber. What \AEight{} calls the unreadable holonomy of $S^1_\theta$ is one rung of the ladder of protection mechanisms classified there, and a reader who wants to know why that rung and not another should read the classification rather than this paper's use of it. Its sixteen open problems include the field-grade representation step that \AEight{} carries as its single analytic residue. First cited in \S\ref{sec:cohort-a8}.

\bibitem[Aharonov \& Bohm(1959)]{AharonovBohm1959}
\textbf{Y.~Aharonov and D.~Bohm.}
{Significance of electromagnetic potentials in the quantum theory}.
\emph{Phys.\ Rev.} \textbf{115} (1959) 485.
\par\noindent\textbf{Relevance.}
The founding demonstration that the electromagnetic potential carries
physical content through holonomy rather than local field strength. Cited for
the identification of the compact phase fiber $S^1_\theta$ as the carrier of
the coherent, unitary content of the boundary theory: its holonomy is the
Aharonov--Bohm/Berry phase. First cited in \S\ref{ssec:bundle-projection}.

\bibitem[Arisaka(planned, BookI)]{ArisakaBookI}
\textbf{K.~Arisaka.}
\emph{\GGTheory{} Foundation Book (capstone volume)},
(planned) GG-Theory Foundation Book.
\par\noindent\textbf{Relevance.}
The planned capstone book volume of the \GGTheory{} program, synthesizing the cohort papers into a single textbook on the geometric inevitability of cosmos, life, and mind. \AEight{} is cited in \ArisakaBookI{} as the QM chapter; \AEight{} cites \ArisakaBookI{} for the synthesis-level context. First cited in \S\ref{sec:programme-dictionary}.

\bibitem[Arndt and others(1999)]{Arndt2003_C60}
\textbf{M~Arndt and others.}
Wave-particle duality of {$C_{60}$} molecules.
\emph{Nature} \textbf{401} (1999) 680--682.
\par\noindent\textbf{Relevance.}
The 1999 Arndt et al. $C_{60}$ fullerene interferometry experiment, confirming wave-particle duality for objects with ~$1\,\mathrm{kDa}$ mass. AEight Part VI and Figure 9 cite this as the largest object for which fringe visibility is measurable; for any larger object the decoherence rate of \TA{5} exceeds the macroscopic-instantaneous threshold and the wave nature of the object becomes operationally inaccessible. First cited in \S\ref{part:I}.

\bibitem[Ashtekar and Magnon(1975)]{AshtekarMagnon1975}
\textbf{A~Ashtekar and A~Magnon.}
Quantum fields in curved space-times.
\emph{Proceedings of the Royal Society of London A} \textbf{346} (1975) 375--394.
\par\noindent\textbf{Relevance.}
The classic construction selecting a complex structure on the real
solution space of a linear field from a stationarity structure and
energy positivity. Together with \citet{Kay1978}, it supplies the
uniqueness clause of the real-bulk statement RB-2: on the globally
static canonical arena of Theorem~0, the state-space $i$ of neutral
matter is the polar phase of the constitutional time generator, derived
rather than postulated. First cited in \S\ref{sec:disc-real-bulk}.

\bibitem[Aspect \emph{et~al.}(1981)]{Aspect1981}
\textbf{A~Aspect, P~Grangier, and G~Roger.}
Experimental tests of realistic local theories via {Bell's} theorem.
\emph{Physical Review Letters} \textbf{47} (1981) 460--463.
\par\noindent\textbf{Relevance.}
The 1981 Aspect-Grangier-Roger experiment: the first high-precision Bell-inequality violation with polarization-correlated photons. AEight cites this in Part VII as the empirical seal on the Bell program; the cohort framework reads the violation as confirmation that the boundary CHSH integral saturates the Tsirelson bound, which \TA{8} promotes to a topological-bound saturation on the bulk side. First cited in \S\ref{part:I}.

\bibitem[Aspect \emph{et~al.}(1982)]{Aspect1982}
\textbf{A~Aspect, J~Dalibard, and G~Roger.}
{Experimental test of Bell's inequalities using time-varying analyzers}.
\emph{Physical Review Letters} \textbf{49} (1982) 1804--1807.
\par\noindent\textbf{Relevance.}
The 1982 Aspect publication establishing experimental Bell-inequality violation in a complete loophole-free setup (up to fast switching). AEight cites Aspect1982 throughout Parts VII and XII as the canonical empirical anchor for the $|S| = 2\sqrt{2}$ Tsirelson saturation. First cited in \S\ref{sec:TA-7}.

\bibitem[Aspect \emph{et~al.}(1982)]{Aspect1982a}
\textbf{A~Aspect, P~Grangier, and G~Roger.}
Experimental realization of {Einstein-Podolsky-Rosen-Bohm} {Gedankenexperiment}: a new violation of {Bell's} inequalities.
\emph{Physical Review Letters} \textbf{49} (1982) 91--94.
\par\noindent\textbf{Relevance.}
The first 1982 Aspect paper realizing the Einstein-Podolsky-Rosen-Bohm thought experiment with high statistical significance. AEight cites this in Part VII as the experimental realization that closed the philosophical debate of 1935-1965 in favor of QM and against local hidden-variable theories; the cohort framework reads the closure as a victory for the bulk-side reading at 6D. First cited in \S\ref{part:I}.

\bibitem[Aspect \emph{et~al.}(1982)]{Aspect1982b}
\textbf{A~Aspect, J~Dalibard, and G~Roger.}
Experimental test of {Bell's} inequalities using time-varying analyzers.
\emph{Physical Review Letters} \textbf{49} (1982) 1804--1807.
\par\noindent\textbf{Relevance.}
The second 1982 Aspect paper closing the locality loophole by using time-varying analyzers. AEight cites this in Part VII as the experimental closure of the locality loophole; the result is read by \TA{7} as the boundary shadow of one non-separable joint bulk state, the apparent non-locality being the projection of a single bulk object. First cited in \S\ref{part:I}.

\bibitem[Bachmann et al.(2012)]{BachmannDeckertPizzo2012}
\textbf{S~Bachmann, D-A~Deckert, and A~Pizzo.}
The mass shell of the Nelson model without cut-offs.
\emph{Journal of Functional Analysis} \textbf{263} (2012) 1224--1282.
\par\noindent\textbf{Relevance.}
Theorem-grade control of the massless Nelson model's dressed
one-particle structure with the ultraviolet cutoff removed. Together
with \citet{Pizzo2003} it supplies the existence half of the
Nelson/Pauli--Fierz rung of RB-10: the dressed states whose soft
clouds RB-10 reads as port assignments do exist, with the regularity
the sector analysis needs. First cited in \S\ref{sec:disc-real-bulk}.

\bibitem[Ballentine(1998)]{Ballentine1998}
\textbf{L~E~Ballentine.}
\emph{Quantum Mechanics: A Modern Development}.
World Scientific, (1998).
\par\noindent\textbf{Relevance.}
Ballentine's 1998 graduate textbook adopting the statistical (ensemble) interpretation of the wave function. AEight cites Ballentine1998 in Part XI as a major interpretive reference; the cohort framework's \TA{3} (Born rule) supports the statistical reading of the wave function at the boundary while supplying an ontic substrate at the bulk. First cited in \S\ref{sec:TA-1}.

\bibitem[Baumann and others(2019)]{Baumann2019}
\textbf{V~Baumann and others.}
Wigner's friend as a rational agent.
\emph{arXiv preprint} \textbf{1901.11274} (2019).
\par\noindent\textbf{Relevance.}
Baumann et al. 2019, a modern philosophical analysis of the Wigner's-friend / Frauchiger-Renner family of paradoxes. AEight Part X cites this in the discussion of observer-observer paradoxes; the cohort framework treats the rational-agent reading as the 4D-boundary projection of a deterministic 6D-bulk picture in which the paradoxes do not arise. First cited in \S\ref{sec:px-FR}.

\bibitem[Bell(1964)]{Bell1964}
\textbf{J.~Bell.}
\emph{On the Einstein Podolsky Rosen paradox}.
\emph{Physics Physique Fizika} \textbf{1} (1964) 195--200.
\par\noindent\textbf{Relevance.}
Bell's 1964 paper converting EPR's philosophical objection into an experimentally testable inequality. AEight Part VII walks the CHSH form of Bell's theorem; the explicit calculation on the singlet state recovers $|S| = 2\sqrt{2}$|$\cos(2\phi)$|; the Tsirelson saturation at $\phi$=0 is read by \TA{8} as an extremal (Bogomolnyi-analogue) ceiling on the CHSH functional. Bell1964 is the seed of the empirical heart of the cohort QM reading. First cited in \S\ref{part:I}.

\bibitem[Bennett and Brassard(1984)]{BennettBrassard1984}
\textbf{C~H~Bennett and G~Brassard.}
\emph{Quantum cryptography: public key distribution and coin tossing}.
In \emph{Proceedings of the {IEEE} International Conference on Computers, Systems and Signal Processing} (1984).
\par\noindent\textbf{Relevance.}
The 1984 BB84 quantum-key-distribution protocol. AEight Part X cites BB84 as the canonical QKD scheme; the cohort framework reads QKD's security as a consequence of \TA{4} (Lueders projection) plus \TA{11} (boundary no-signaling): any eavesdropping introduces detectable boundary state perturbations because the bulk-mediated correlations cannot be cloned. First cited in \S\ref{part:I}.

\bibitem[Bennett and Wiesner(1992)]{BennettWiesner1992}
\textbf{C~H~Bennett and S~J~Wiesner.}
Communication via one- and two-particle operators on {Einstein-Podolsky-Rosen} states.
\emph{Physical Review Letters} \textbf{69} (1992) 2881--2884.
\par\noindent\textbf{Relevance.}
The 1992 Bennett-Wiesner paper introducing superdense coding. AEight Part X cites this alongside teleportation (Bennett1993) as the canonical demonstration that entanglement is a usable resource; the cohort framework reads superdense coding as \TA{11} plus the bulk-side picture: two classical bits are transmitted via one qubit because the bulk-mediated entangled pair stores the missing bit in the topological $S^1_\theta$ winding. First cited in \S\ref{part:I}.

\bibitem[Bennett \emph{et~al.}(1993)]{Bennett1993}
\textbf{C~H~Bennett, G~Brassard, C~Cr{\'e}peau, R~Jozsa, A~Peres, and W~K~Wootters.}
Teleporting an unknown quantum state via dual classical and {Einstein-Podolsky-Rosen} channels.
\emph{Physical Review Letters} \textbf{70} (1993) 1895--1899.
\par\noindent\textbf{Relevance.}
The 1993 paper introducing quantum teleportation. AEight Part X cites Bennett1993 as the canonical theoretical foundation; the cohort framework reads teleportation as a confirmation of \TA{11} (boundary no-signaling from reduced-state factoring): no information actually travels superluminally because the boundary measurement on the Bell pair only updates the classical channel, which is bounded by 4D causality. First cited in \S\ref{part:I}.

\bibitem[Berry(1984)]{Berry1984}
\textbf{M.~V.~Berry.}
{Quantal phase factors accompanying adiabatic changes}.
\emph{Proc.\ R.\ Soc.\ Lond.\ A} \textbf{392} (1984) 45.
\par\noindent\textbf{Relevance.}
The geometric phase of adiabatic transport: the holonomy of the connection on
a family of quantum states. Cited with Aharonov--Bohm for the phase-fiber
reading of $S^1_\theta$, and the pedagogical home of the spin-$\tfrac12$ cone
example of \S G.7.1 --- the same structure that
\citet{ArisakaP2N3} elevates to a theorem about the runtime's own transport on
the family-projector moduli. First cited in \S\ref{ssec:bundle-projection}.

\bibitem[Bogomolny(1976)]{Bogomolny1976}
\textbf{E~B~Bogomolny.}
The stability of classical solutions.
\emph{Soviet Journal of Nuclear Physics} \textbf{24} (1976) 449--454.
\par\noindent\textbf{Relevance.}
Bogomolny's 1976 paper introducing the saturation bound on classical Yang-Mills solitons that bears his name. AEight's net-new theorem \TA{8} reads the Tsirelson bound $|S| = 2\sqrt{2}$ as a Bogomolnyi-style saturation on the boundary CHSH integral, with the bulk topological-charge structure of the 6D geometry providing the saddle that saturates the bound. Bogomolny1976 is the conceptual ancestor. First cited in \S\ref{sec:TA-8}.

\bibitem[Bohm(1952)]{Bohm1952}
\textbf{D~Bohm.}
A suggested interpretation of the quantum theory in terms of ``hidden'' variables.
\emph{Physical Review} \textbf{85} (1952) 166--193.
\par\noindent\textbf{Relevance.}
Bohm's 1952 pilot-wave / quantum-potential reformulation, a deterministic hidden-variable extension of QM. \AEight{} engages Bohm1952 in Part~XI (interpretation landscape) and Part~VIII: Bohm's intuition that hidden variables restore determinism is right, but the hidden variables are not in 4D position-space (Bell ruled that out); they are on the phase axis $S^1_\theta$ of the 6D bulk. \AEight{} is the cohort framework's vindication of Bohm at the right level. First cited in \S\ref{part:I}.

\bibitem[Bohr(1935)]{Bohr1935}
\textbf{N~Bohr.}
Can quantum-mechanical description of physical reality be considered complete?.
\emph{Physical Review} \textbf{48} (1935) 696--702.
\par\noindent\textbf{Relevance.}
Bohr's 1935 response to EPR insisting on the inseparability of the joint quantum state and on the operational completeness of QM at the boundary level. AEight cites Bohr1935 in Parts VII and XIII as the canonical statement of the 'Bohr is right in 4D' half of the cohort's bottom line: at the operational boundary level, the framework reproduces and endorses Bohr's reading; the disagreement is at the bulk level, where Einstein's intuition about determinism is restored. First cited in \S\ref{part:I}.

\bibitem[Bong and others(2020)]{Bong2020}
\textbf{K.~Bong and others.}
A strong no-go theorem on the {Wigner's friend} paradox.
\emph{Nature Physics} \textbf{16} (2020) 1199--1205.
\par\noindent\textbf{Relevance.}
The 2020 Bong et al. experimental implementation of an extended Wigner's-friend no-go theorem. AEight Part X cites Bong2020 as a recent sharpening; the cohort framework reads the experimental confirmation as evidence that any 4D-only interpretation faces consistency challenges that the bulk-boundary picture avoids. First cited in \S\ref{sec:px-FR}.

\bibitem[Born(1926)]{Born1926}
\textbf{M~Born.}
\emph{Zur Quantenmechanik der Sto{\ss}vorg{\"a}nge}.
\emph{Zeitschrift f{\"u}r Physik} \textbf{37} (1926) 863--867.
\par\noindent\textbf{Relevance.}
Born's June-July 1926 papers introducing the probabilistic interpretation of the wave function (the Born rule, $p = |\psi|^2$). AEight's central theorem \TA{3} derives the Born rule from OGGEP as a CPTP-channel consequence, recovering Born's 1926 result with a deeper provenance: $p_\alpha = \mathrm{Tr}(I_\alpha \rho)$ from the bulk-side CPTP completion. First cited in \S\ref{part:I}.

\bibitem[Born(1971)]{BornEinsteinLetters}
\textbf{M~Born.}
\emph{The {Born-Einstein} Letters: Correspondence between {Albert Einstein} and {Max} and {Hedwig Born} 1916--1955}.
Macmillan, (1971).
\par\noindent\textbf{Relevance.}
The Born-Einstein correspondence containing Einstein's December 1926 'God does not play dice' letter to Born. AEight cites the letters in Part VIII as the canonical statement of Einstein's deterministic intuition; the cohort framework's 'Einstein is right in 6D' reading honors this intuition at the bulk level while accepting Bohr's operational completeness at the 4D boundary. First cited in \S\ref{part:I}.

\bibitem[Born and Jordan(1925)]{BornJordan1925}
\textbf{M~Born and P~Jordan.}
Zur Quantenmechanik.
\emph{Zeitschrift f{\"u}r Physik} \textbf{34} (1925) 858--888.
\par\noindent\textbf{Relevance.}
The 1925 Born-Jordan paper establishing matrix mechanics as a fully consistent formulation of quantum mechanics. Cited in AEight Part I as one of the two foundational 1925-1926 formulations (alongside Schrodinger's wave mechanics) that converge into the modern Hilbert-space framework derived as \TA{1} from the bulk GNS construction. First cited in \S\ref{part:I}.

\bibitem[Born \emph{et~al.}(1926)]{BornHeisenbergJordan1926}
\textbf{M~Born, W~Heisenberg, and P~Jordan.}
Zur Quantenmechanik {II}.
\emph{Zeitschrift f{\"u}r Physik} \textbf{35} (1926) 557--615.
\par\noindent\textbf{Relevance.}
The 1926 sequel that completed the matrix-mechanics program of quantum mechanics. AEight cites this in Part I as the historical convergence of matrix mechanics with wave mechanics into the modern operator formalism; the resulting Hilbert space is what \TA{1} reconstructs from the bulk-side GNS construction. First cited in \S\ref{part:I}.

\bibitem[Bouwmeester \emph{et~al.}(1997)]{Bouwmeester1997}
\textbf{D~Bouwmeester, J.~Pan, K~Mattle, M~Eibl, H~Weinfurter, and A~Zeilinger.}
Experimental quantum teleportation.
\emph{Nature} \textbf{390} (1997) 575--579.
\par\noindent\textbf{Relevance.}
The 1997 Bouwmeester et al. experimental realization of quantum teleportation. AEight Part X cites this as the empirical seal on the Bennett1993 theoretical proposal; the cohort framework reads the experimental success as confirmation of the bulk-mediated entanglement reading of \TA{7}-\TA{11}. First cited in \S\ref{sec:px-teleportation}.

\bibitem[Breuer(2002)]{BreuerPetruccione2002}
\textbf{H.~\&~P.~F~Breuer.}
\emph{The Theory of Open Quantum Systems}.
Oxford University Press, Oxford, (2002).
\par\noindent\textbf{Relevance.}
Breuer and Petruccione's 2002 textbook 'The Theory of Open Quantum Systems', the standard modern reference for the GKLS-Lindblad-Nakajima-Zwanzig formalism. AEight Appendix cites Breuer-Petruccione for the rigorous derivation of the master equation from the system-environment Hamiltonian; \TA{2} reconstructs this derivation with the additional warrant of OGGEP. First cited in \S\ref{part:I}.

\bibitem[Brunner \emph{et~al.}(2014)]{BrunnerEtAl2014}
\textbf{N~Brunner, D~Cavalcanti, S~Pironio, V~Scarani, and S~Wehner.}
{Bell} nonlocality.
\emph{Reviews of Modern Physics} \textbf{86} (2014) 419--478.
\par\noindent\textbf{Relevance.}
Brunner et al. 2014 Reviews of Modern Physics article on Bell nonlocality, the canonical modern survey. AEight Part VII cites Brunner2014 as the comprehensive reference for the Bell-CHSH inequality and its experimental status as of 2014; the cohort framework's \TA{7} and \TA{8} read this survey's content with the additional bulk-side warrant. First cited in \S\ref{sec:TA-7}.

\bibitem[Buchholz(1986)]{Buchholz1986}
\textbf{D~Buchholz.}
Gauss' law and the infraparticle problem.
\emph{Physics Letters B} \textbf{174} (1986) 331--334.
\par\noindent\textbf{Relevance.}
The structural reading of the infraparticle: by Gauss' law the
asymptotic electromagnetic flux is a superselected observable, so a
charged particle's velocity is encoded in its cloud at infinity and
sharp mass shells are excluded. RB-10 adopts this sector structure
wholesale and adds the framework's placement: the flux label is
dictionary/port data at \AX{3}, and the derived complex structure is
blind to it. First cited in \S\ref{sec:disc-real-bulk}.

\bibitem[Chen et al.(2010)]{ChenFrohlichPizzo2010}
\textbf{T~Chen, J~Fr\"ohlich, and A~Pizzo.}
Infraparticle scattering states in non-relativistic QED: I. The
Bloch--Nordsieck paradigm.
\emph{Communications in Mathematical Physics} \textbf{294} (2010) 761--825.
\par\noindent\textbf{Relevance.}
The rigorous construction of infraparticle scattering states in
nonrelativistic QED, exhibiting the velocity-superselected soft
clouds as the correct asymptotic objects. This is the scattering-level
import of RB-10's Nelson/Pauli--Fierz rung: the states exist, their
clouds carry the sector label, and the dressed asymptotic dynamics is
exactly the free-plus-c-number form whose polar phase RB-10 reads.
First cited in \S\ref{sec:disc-real-bulk}.

\bibitem[Chung(1965)]{Chung1965}
\textbf{V~Chung.}
Infrared divergence in quantum electrodynamics.
\emph{Physical Review} \textbf{140} (1965) B1110--B1122.
\par\noindent\textbf{Relevance.}
The original coherent-state resolution of the QED infrared problem:
dressing the charged particle with a photon cloud renders the
S-matrix infrared-finite. RB-10 reads Chung's dressing as the physical
prototype of its sector displacement --- a c-number cloud that moves
the state out of Fock space without touching the quadratic generator.
First cited in \S\ref{sec:disc-real-bulk}.

\bibitem[Clauser \emph{et~al.}(1969)]{CHSH1969}
\textbf{J~F~Clauser, M~A~Horne, A~Shimony, and R~A~Holt.}
Proposed experiment to test local hidden-variable theories.
\emph{Physical Review Letters} \textbf{23} (1969) 880--884.
\par\noindent\textbf{Relevance.}
The 1969 Clauser-Horne-Shimony-Holt reformulation of Bell's inequality into the four-setting quantity S with classical bound $|S| \le 2$. AEight Part VII uses the CHSH form throughout because it admits the cleanest experimental implementation; the Tsirelson bound $|S| = 2\sqrt{2}$ shown in Figure 7 is the QM saturation of CHSH, read by \TA{8} as the boundary shadow of a bulk Bogomolnyi-saturated configuration. First cited in \S\ref{part:I}.

\bibitem[Cohen-Tannoudji \emph{et~al.}(1992)]{CohenTannoudji1992}
\textbf{C~Cohen-Tannoudji, J~Dupont-Roc, and G~Grynberg.}
\emph{Atom-Photon Interactions: Basic Processes and Applications}.
Wiley, New York, (1992).
\par\noindent\textbf{Relevance.}
The Cohen-Tannoudji-Diu-Laloe graduate QM textbook (1977, 2-vol., often cited as 'Cohen-Tannoudji 1992' for the English edition). AEight cites this throughout the worked examples as the canonical textbook treatment that the cohort framework recovers as \TA{1}-\TA{7}. The two-volume comprehensiveness is the historical standard against which AEight's graduate-textbook depth is measured. First cited in \S\ref{sec:TA-1}.

\bibitem[Davies and Fulling(1976)]{DaviesFulling1976}
\textbf{P~C~W~Davies and S~A~Fulling.}
Radiation from a moving mirror in two dimensional space-time:
conformal anomaly.
\emph{Proceedings of the Royal Society of London A} \textbf{348} (1976) 393--414.
\par\noindent\textbf{Relevance.}
The moving-mirror model: an accelerating boundary with late-time
exponential redshift radiates a thermal flux at the horizon
temperature --- the exactly solvable model form of gravitational
collapse. RB-12's class is this one, and its factorization content ---
universal late-time channel, decaying transient --- is proved and
verified there on Davies--Fulling trajectories.
First cited in \S\ref{sec:disc-real-bulk}.

\bibitem[Davisson and Germer(1927)]{DavissonGermer1927}
\textbf{C~Davisson and L~H~Germer.}
Diffraction of electrons by a crystal of nickel.
\emph{Physical Review} \textbf{30} (1927) 705--740.
\par\noindent\textbf{Relevance.}
The 1927 Davisson-Germer electron-diffraction experiment confirming the de Broglie matter-wave hypothesis. AEight cites this in Part VI as the canonical experimental ancestor of the double-slit demonstration of \TA{5} (decoherence) and the wave-particle duality that the cohort framework reads as bulk fermion zero modes shadowing as 4D boundary states. First cited in \S\ref{part:I}.

\bibitem[de Broglie(1924)]{deBroglie1924}
\textbf{L.~de Broglie.}
\emph{Recherches sur la th{\'e}orie des quanta}.
Sorbonne, Paris, (1924).
\par\noindent\textbf{Relevance.}
Establishes the matter-wave hypothesis ($\lambda$ = h/p for material particles). AEight cites de Broglie's 1924 thesis in Part VI as the seed for the double-slit experiment's quantum descendants: the Davisson-Germer (1927), Stern-Estermann atom-diffraction, and ultimately the Arndt $C_{60}$ fullerene interferometry of 2003. First cited in \S\ref{part:I}.

\bibitem[Della Rocca(2008)]{DellaRocca2008}
\textbf{M~Della Rocca.}
\emph{Spinoza}.
Routledge, (2008).
\par\noindent\textbf{Relevance.}
Della Rocca's 2008 monograph on Spinoza, including the Numerical Sameness argument. AEight cites Della Rocca in Part XI as the modern philosophical apparatus for reading 'every electron is the same' in numerical-identity terms: indistinguishability is not a contingent feature but a metaphysical necessity given the bulk-side uniqueness of the charged-lepton Geometric Attractor. First cited in \S\ref{sec:int-structural}.

\bibitem[Dirac(1930)]{Dirac1930}
\textbf{P.~A.~M~Dirac.}
(1930).
\par\noindent\textbf{Relevance.}
Dirac's 1930 textbook 'The Principles of Quantum Mechanics' establishing the transformation-theory formulation that unified matrix and wave mechanics. AEight cites this in Part I as the historical synthesis that the modern Hilbert-space formalism stands on; \TA{1} specifies the operator-algebra structure that Dirac's transformation theory presupposes. First cited in \S\ref{part:I}.

\bibitem[Di\'osi(1989)]{Diosi1989}
\textbf{L~Di\'osi.}
Models for universal reduction of macroscopic quantum fluctuations.
\emph{Physical Review~A} \textbf{40} (1989) 1165--1174.
\par\noindent\textbf{Relevance.}
Di\'osi's 1989 gravity-related stochastic state-reduction model, the
mass-density formulation of what is now the Di\'osi--Penrose model. \AEight{}
engages it in Part~XI (\S\ref{sec:int-rivals}) as the canonical objective-collapse
rival whose prediction of a mass-dependent deviation from environmental
decoherence furnishes the cleanest experimental fork against the framework,
which predicts no such deviation (F-QM, macroscopic coherence). First cited in
\S\ref{sec:int-rivals}.

\bibitem[D\"urr et al.(1992)]{DurrGoldsteinZanghi1992}
\textbf{D~D\"urr, S~Goldstein, and N~Zangh\`i.}
Quantum equilibrium and the origin of absolute uncertainty.
\emph{Journal of Statistical Physics} \textbf{67} (1992) 843--907.
\par\noindent\textbf{Relevance.}
The typicality formulation of quantum equilibrium in Bohmian mechanics:
$|\psi|^2$ as the unique equivariant measure, and Born statistics as
the typical empirical distribution within one deterministic universe.
The cohort's typicality target imports this architecture and replaces
each of its choices with a forced structure: the measure by the unique
conserved real density, the transport by dial-blindness plus record
stability, the anchor by Haar on the compact fiber.
First cited in \S\ref{sec:disc-open}.

\bibitem[Eastwood \emph{et~al.}(1981)]{EastwoodPenroseWells1981}
\textbf{M~G~Eastwood, R~Penrose, and R~O~Wells.}
Cohomology and massless fields.
\emph{Communications in Mathematical Physics} \textbf{78} (1981) 305--351.
\par\noindent\textbf{Relevance.}
The rigorous form of the Penrose transform: solutions of the free
massless field equations of helicity $h$ correspond to cohomology
classes $H^1(\mathbb{PT}, \mathcal{O}(-2h-2))$, with positive-frequency
solutions supported on $\mathbb{PT}^{+}$. Together with
\citet{PenroseMacCallum1973}, this is the imported third level of the
holomorphy bridge RB-9: the twistor division is the causal-positive
frequency splitting, conformally packaged. First cited in
\S\ref{sec:disc-real-bulk}.

\bibitem[Eberhard(1993)]{Eberhard1989}
\textbf{P~H~Eberhard.}
Background level and counter efficiencies required for a loophole-free {Einstein-Podolsky-Rosen} experiment.
\emph{Physical Review A} \textbf{47} (1993) R747--R750.
\par\noindent\textbf{Relevance.}
Eberhard's 1989/1993 analysis of the detection-efficiency requirements for a loophole-free Bell test. AEight cites this in Part VII as the theoretical groundwork for the 2015 loophole-free experiments (Hensen, Giustina, Shalm) that finally closed the detection loophole. First cited in \S\ref{sec:TA-11}.

\bibitem[Ehresmann(1950)]{Ehresmann1950}
\textbf{C.~Ehresmann.}
{Les connexions infinit\'esimales dans un espace fibr\'e diff\'erentiable}.
\emph{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 the rigorous form of \GGEP{} (AX-2): a gauge field is a connection, its field strength a curvature, a gauge phase a holonomy. Cited in the hidden-fiber section, where the phase fiber $S^1_\theta$ is identified as a principal $U(1)$-bundle whose holonomy is the quantum phase. First cited in \S\ref{ssec:two-fibres}.

\bibitem[Einstein(1905)]{Einstein1905photon}
\textbf{A~Einstein.}
{\"U}ber einen die Erzeugung und Verwandlung des Lichtes betreffenden heuristischen Gesichtspunkt.
\emph{Annalen der Physik} \textbf{17} (1905) 132--148.
\par\noindent\textbf{Relevance.}
Einstein's 1905 photoelectric-effect paper introducing light quanta (later called photons). AEight cites this in Part I as one of the four pillars of the old quantum theory that motivated the operational quantization program; the photon's particle-like behavior in the photoelectric effect anticipates the bulk-side fermion zero-mode picture of \TA{1}. First cited in \S\ref{part:I}.

\bibitem[Einstein \emph{et~al.}(1935)]{EPR1935}
\textbf{A~Einstein, B~Podolsky, and N~Rosen.}
Can quantum-mechanical description of physical reality be considered complete?.
\emph{Physical Review} \textbf{47} (1935) 777--780.
\par\noindent\textbf{Relevance.}
The 1935 EPR paper challenging the completeness of QM via the entangled spin-singlet thought experiment. AEight Part VII opens with EPR1935 as the historical seed of the Bell program; \TA{7} reads Bell-CHSH violations as the 4D boundary shadow of one non-separable joint bulk state, projected along the radial $I_r$ direction (which carries no light cone), vindicating EPR's intuition that something is 'incomplete' at the boundary level even as QM's predictions are confirmed. First cited in \S\ref{part:I}.

\bibitem[Ekert(1991)]{Ekert1991}
\textbf{A~K~Ekert.}
Quantum cryptography based on {Bell's} theorem.
\emph{Physical Review Letters} \textbf{67} (1991) 661--663.
\par\noindent\textbf{Relevance.}
Ekert's 1991 E91 QKD protocol using Bell-pair measurements for key generation and eavesdropping detection. AEight Part X cites E91 as the entanglement-based alternative to BB84; the cohort framework's reading is that the security of E91 rests on the experimentally confirmed Tsirelson saturation of \TA{8} — any eavesdropping reduces the observed CHSH value below 2sqrt(2) by a detectable amount. First cited in \S\ref{part:I}.

\bibitem[Everett(1957)]{Everett1957}
\textbf{H~Everett.}
``Relative state'' formulation of quantum mechanics.
\emph{Reviews of Modern Physics} \textbf{29} (1957) 454--462.
\par\noindent\textbf{Relevance.}
Everett's 1957 'relative state' formulation, the seed of the Many-Worlds interpretation (MWI). AEight Part XI discusses Everett1957: the cohort framework agrees with Everett that the unitarity of QM extends to the observer's universe at the 4D boundary level, but the 'many worlds' are decohered branches on a single deterministic 6D bulk substrate, not parallel universes. First cited in \S\ref{part:I}.

\bibitem[Faddeev and Kulish(1970)]{FaddeevKulish1970}
\textbf{L~D~Faddeev and P~P~Kulish.}
Asymptotic conditions and infrared divergences in quantum
electrodynamics.
\emph{Theoretical and Mathematical Physics} \textbf{4} (1970) 745--757.
\par\noindent\textbf{Relevance.}
The systematic dressed-asymptotic-dynamics formulation of the QED
infrared problem: the correct asymptotic Hamiltonian is free plus a
c-number-valued (cloud) correction, and the physical states carry
coherent dressings. RB-10's clause (ii) is the structural core of this
construction, proved exactly at the van Hove stratum; the full QED
case remains semi-rigorous and is displayed as the frontier RB-10
names rather than claims. First cited in \S\ref{sec:disc-real-bulk}.

\bibitem[Feynman(1985)]{FeynmanQED1985}
\textbf{R~P~Feynman.}
\emph{QED: The Strange Theory of Light and Matter}.
Princeton University Press, 1985.
\par\noindent\textbf{Relevance.}
Feynman's public exposition of the amplitude-arrow picture of quantum
electrodynamics. Cited because that spinning arrow is, on this paper's
reading, literally the position on the phase fiber $S^1_\theta$: adding arrows
and squaring is the boundary registration map. First cited in
\S\ref{ssec:wave-particle}.

\bibitem[Feynman \emph{et~al.}(1965)]{FeynmanLectures}
\textbf{R~P~Feynman, R~B~Leighton, and M~Sands.}
\emph{The Feynman Lectures on Physics, Vol. {III}: Quantum Mechanics}.
Addison-Wesley, (1965).
\par\noindent\textbf{Relevance.}
Feynman's Lectures on Physics, Volume III, the canonical introductory QM text. AEight cites the Feynman lectures in the historical discussions of Parts V-VII; the cohort framework recovers Feynman's path-integral intuition as a 4D-boundary projection of the bulk Lagrangian. First cited in \S\ref{part:I}.

\bibitem[Frauchiger and Renner(2018)]{FrauchigerRenner2018}
\textbf{D~Frauchiger and R~Renner.}
Quantum theory cannot consistently describe the use of itself.
\emph{Nature Communications} \textbf{9} (2018) 3711.
\par\noindent\textbf{Relevance.}
The 2018 Frauchiger-Renner paper sharpening Wigner's-friend into a no-go theorem: certain consistent applications of standard QM to observers-observing-observers lead to contradictions. AEight Part X discusses Frauchiger-Renner as the modern sharpening of the measurement-chain problem; the cohort framework reads it as a 4D-boundary artifact that disappears when the bulk substrate is recognized. First cited in \S\ref{sec:cat-geiger-trigger}.

\bibitem[Freedman and Clauser(1972)]{FreedmanClauser1972}
\textbf{S~J~Freedman and J~F~Clauser.}
Experimental test of local hidden-variable theories.
\emph{Physical Review Letters} \textbf{28} (1972) 938--941.
\par\noindent\textbf{Relevance.}
The 1972 Freedman-Clauser experiment, the first laboratory observation of a Bell-inequality violation. AEight cites this in Part VII as the empirical ancestor of the Aspect-experiments lineage; the cohort framework reads the violation as a confirmed signature of bulk-mediated correlations rather than as evidence against local realism (which the bulk picture preserves at 6D). First cited in \S\ref{part:I}.

\bibitem[French(2014)]{French2014}
\textbf{S~French.}
\emph{The Structure of the World: Metaphysics and Representation}.
Oxford University Press, (2014).
\par\noindent\textbf{Relevance.}
French's 2014 monograph extending the ontic-structural-realism program to a full metaphysical view. AEight cites French2014 in Part XI alongside Ladyman-Ross as the standard-bearer of OSR; the cohort framework's universal-identity insight (every electron is the same) is the natural QM-stratum specialization of French's view. First cited in \S\ref{sec:int-structural}.

\bibitem[Fr\"ohlich(1973)]{Frohlich1973}
\textbf{J~Fr\"ohlich.}
On the infrared problem in a model of scalar electrons and massless,
scalar bosons.
\emph{Annales de l'Institut Henri Poincar\'e A} \textbf{19} (1973) 1--103.
\par\noindent\textbf{Relevance.}
The founding rigorous analysis of the infraparticle: in the massless
Nelson model the fixed-momentum Hamiltonian has no ground state in
Fock space, and the physical states live in velocity-indexed coherent
sectors. RB-10's Lemma-1 mechanics --- the logarithmically divergent
cloud and the power-law sector orthogonality --- is the quantitative
skeleton of Fr\"ohlich's phenomenon, verified numerically in the
\PaperPFour{} suite. First cited in \S\ref{sec:disc-real-bulk}.

\bibitem[Fuchs(2010)]{Fuchs2010}
\textbf{C~A~Fuchs.}
{QBism}, the perimeter of quantum Bayesianism.
\emph{arXiv preprint} \textbf{1003.5209} (2010).
\par\noindent\textbf{Relevance.}
Fuchs's 2010 paper articulating QBism: quantum states encode an agent's degrees of belief, not facts about the world. AEight Part XI discusses QBism as the epistemic-state interpretation; the cohort framework partly agrees with QBism that the boundary quantum state is epistemic for the 4D observer, but DISAGREES that this is the whole story: the 6D bulk is ontic and observer-independent. First cited in \S\ref{part:I}.

\bibitem[Gallis and Fleming(1990)]{GallisFleming1990}
\textbf{M~R~Gallis and G~N~Fleming.}
Environmental and spontaneous localization.
\emph{Physical Review A} \textbf{42} (1990) 38--48.
\par\noindent\textbf{Relevance.}
Gallis and Fleming's derivation of the collisional decoherence master equation,
giving the decoherence function $F(\Delta x)$ and its saturated and localization
limits. AEight cites GallisFleming1990 in \S\ref{sec:worked-overlap} and
Appendix~\ref{app:worked-overlap} as the master equation underlying the worked
overlap integral of \TA{5}. First cited in \S\ref{sec:worked-overlap}.

\bibitem[Gelfand and Naimark(1943)]{Gelfand1943}
\textbf{I~M~Gelfand and M~A~Naimark.}
On the imbedding of normed rings into the ring of operators on a Hilbert space.
\emph{Matematicheskii Sbornik} \textbf{12} (1943) 197--217.
\par\noindent\textbf{Relevance.}
The Gelfand-Naimark 1943 paper introducing the GNS construction (Gelfand-Naimark-Segal), the mathematical foundation for AEight's \TA{1} (Hilbert-space embedding via GNS). The boundary algebra's representation as bounded operators on a Hilbert space is forced by the GNS construction once the boundary algebra is C*-completed. First cited in \S\ref{sec:TA-1}.

\bibitem[Gell-Mann and Hartle(1993)]{GellMannHartle1993}
\textbf{M~Gell-Mann and J~B~Hartle.}
Classical equations for quantum systems.
\emph{Physical Review D} \textbf{47} (1993) 3345--3382.
\par\noindent\textbf{Relevance.}
The 1993 Gell-Mann-Hartle paper sharpening the decoherent-histories interpretation with classical equations of motion as the operational limit. AEight cites this in Part XI alongside Griffiths1984 and Omnes1999; the cohort framework reads the classical limit as the einselected pointer-basis trajectory of \TA{6}. First cited in \S\ref{part:I}.

\bibitem[Gerlach and Stern(1922)]{SternGerlach1922}
\textbf{W~Gerlach and O~Stern.}
Der experimentelle Nachweis der Richtungsquantelung im Magnetfeld.
\emph{Zeitschrift f{\"u}r Physik} \textbf{9} (1922) 349--352.
\par\noindent\textbf{Relevance.}
The 1922 Stern-Gerlach experiment, the empirical foundation of AEight's prototype worked example (Part V). The two-spot detector pattern is the simplest non-trivial OGGEP-channel: a single qubit, a single sharp observable, a single binary outcome. Born probabilities $\cos^2(\theta/2)$ and $\sin^2(\theta/2)$ emerge from \TA{3} applied to the spin-1/2 qubit on the Bloch sphere. First cited in \S\ref{part:I}.

\bibitem[Ghirardi(1986)]{GRW1986}
\textbf{G.~R.~A.~\&~W.~T~Ghirardi.}
Unified dynamics for microscopic and macroscopic systems.
\emph{Physical Review~D} \textbf{34} (1986) 470--491.
\par\noindent\textbf{Relevance.}
The 1986 Ghirardi-Rimini-Weber paper introducing spontaneous-collapse dynamics as a candidate solution to the measurement problem. AEight Part XI discusses GRW as the canonical spontaneous-collapse alternative; the cohort framework does NOT need spontaneous collapse because \TA{5} supplies the decoherence rate from bulk dynamics, and the boundary observer never registers a coherent macroscopic superposition. First cited in \S\ref{part:I}.

\bibitem[Giustina \emph{et~al.}(2015)]{Giustina2015}
\textbf{M~Giustina, M~A~M~Versteegh, S~Wengerowsky, J~Handsteiner, A~Hochrainer, K~Phelan, F~Steinlechner, J~Kofler, J.~Larsson, C~Abellan, W~Amaya, V~Pruneri, M~W~Mitchell, J~Beyer, T~Gerrits, A~E~Lita, L~K~Shalm, S~W~Nam, T~Scheidl, R~Ursin, B~Wittmann, and A~Zeilinger.}
Significant-loophole-free test of {Bell's} theorem with entangled photons.
\emph{Physical Review Letters} \textbf{115} (2015) 250401.
\par\noindent\textbf{Relevance.}
The 2015 Giustina et al. loophole-free Bell test at Vienna. AEight Part VII cites Giustina2015 in the trio (with Hensen2015 and Shalm2015) of 2015 experiments that closed all major loopholes. The 2022 Nobel Prize in Physics was awarded for this entire experimental program. First cited in \S\ref{part:I}.

\bibitem[Gleason(1957)]{Gleason1957}
\textbf{A.~Gleason.}
\emph{{Measures on the closed subspaces of a Hilbert space}}.
\emph{Journal of Mathematics and Mechanics} \textbf{6} (1957) 885--893.
\par\noindent\textbf{Relevance.}
Gleason's 1957 theorem proving that the Born rule is the unique probability measure on closed subspaces of a Hilbert space of dimension >= 3. AEight Appendix offers a proof sketch of Gleason; \TA{3} (Born rule from OGGEP) gives an alternative derivation that does not depend on the $\dim \ge 3$ constraint and that supplies the bulk-side warrant via CPTP completion. First cited in \S\ref{part:I}.

\bibitem[Gorini(1976)]{GKS1976}
\textbf{V.~K.~A.~\&~S.~E.~Gorini.}
(1976).
\par\noindent\textbf{Relevance.}
The 1976 Gorini-Kossakowski-Sudarshan paper (parallel to Lindblad1976) deriving the most general Markovian quantum dynamical semigroup generator from CPTP requirements. AEight cites GKS together with Lindblad as the GKLS master equation, which \TA{2} inherits from the bulk-to-boundary partial trace as the canonical open-system QM dynamics. First cited in \S\ref{part:I}.

\bibitem[Gorini \emph{et~al.}(1976)]{GKSL1976}
\textbf{V~Gorini, A~Kossakowski, and E~C~G~Sudarshan.}
Completely positive dynamical semigroups of {$N$}-level systems.
\emph{Journal of Mathematical Physics} \textbf{17} (1976) 821--825.
\par\noindent\textbf{Relevance.}
The complete GKLS reference: Gorini-Kossakowski-Sudarshan 1976 deriving the Markovian quantum semigroup generator. Cited in AEight Part III alongside Lindblad1976 as the foundation of the open-system QM dynamics that \TA{2} specializes from the bulk-side CPTP completion. First cited in \S\ref{sec:OGGEP_lindblad}.

\bibitem[Griffiths(1984)]{Griffiths1984}
\textbf{R~B~Griffiths.}
Consistent histories and the interpretation of quantum mechanics.
\emph{Journal of Statistical Physics} \textbf{36} (1984) 219--272.
\par\noindent\textbf{Relevance.}
Griffiths's 1984 paper founding the consistent-histories interpretation. AEight Part XI discusses consistent histories as one of the major interpretive frameworks; the cohort framework's reading of \TA{5} supplies the decoherence functional that makes histories consistent, so the cohort framework and consistent histories converge in their operational predictions but the cohort adds the bulk-side ontology. First cited in \S\ref{part:I}.

\bibitem[Grover(1996)]{Grover1996}
\textbf{L~K~Grover.}
\emph{A fast quantum mechanical algorithm for database search}.
In \emph{Proceedings of the 28th Annual ACM Symposium on Theory of Computing} (1996).
\par\noindent\textbf{Relevance.}
Grover's 1996 quadratic-speedup search algorithm. AEight cites Grover1996 in Part X as the second canonical quantum-algorithm result (alongside Shor's factoring); both algorithms rely on the same unitary-evolution + measurement structure that \TA{1}-\TA{4} derive from OGGEP. First cited in \S\ref{part:I}.

\bibitem[Haag(1992)]{Haag1992}
\textbf{R~Haag.}
\emph{Local Quantum Physics: Fields, Particles, Algebras}.
Springer, Berlin, (1992).
\par\noindent\textbf{Relevance.}
Haag's 1992 monograph 'Local Quantum Physics', the standard reference for algebraic QFT and the algebraic approach to local quantum observables. AEight cites Haag1992 in the Appendix as the rigorous mathematical foundation for the boundary algebra construction; the cohort framework's \TA{1} leverages algebraic-QFT methods that Haag systematized. First cited in \S\ref{sec:TA-1}.

\bibitem[Hackerm\"uller \emph{et~al.}(2004)]{Hackermuller2004}
\textbf{L~Hackerm\"uller, K~Hornberger, B~Brezger, A~Zeilinger, and M~Arndt.}
Decoherence of matter waves by thermal emission of radiation.
\emph{Nature} \textbf{427} (2004) 711--714.
\par\noindent\textbf{Relevance.}
The 2004 measurement of thermal-radiation decoherence of $C_{70}$ matter waves,
a controlled laboratory determination of a decoherence rate. AEight cites
Hackermuller2004 in \S\ref{sec:worked-overlap} as the companion thermal-photon
channel to the collisional decoherence worked there. First cited in
\S\ref{sec:worked-overlap}.

\bibitem[Heisenberg(1925)]{Heisenberg1925}
\textbf{W~Heisenberg.}
\emph{{\"U}ber quantentheoretische Umdeutung kinematischer und mechanischer Beziehungen}.
\emph{Zeitschrift f{\"u}r Physik} \textbf{33} (1925) 879--893.
\par\noindent\textbf{Relevance.}
Heisenberg's 1925 paper introducing matrix mechanics. Cited in Part I as one of two parallel breakthroughs (with Schrodinger 1926) that founded modern QM; AEight's \TA{1} reconstructs the matrix-mechanics Hilbert space from the bulk algebra via the GNS construction. First cited in \S\ref{part:I}.

\bibitem[Heisenberg(1958)]{Heisenberg1958}
\textbf{W~Heisenberg.}
\emph{Physics and Philosophy: The Revolution in Modern Science}.
Harper, (1958).
\par\noindent\textbf{Relevance.}
Heisenberg's 1958 philosophical reflections on QM, a primary source for the Copenhagen reading. AEight cites Heisenberg1958 in Parts I and XI as the historical articulation of the Copenhagen interpretation; the cohort framework's 'Bohr right in 4D' reading honors the operational completeness Heisenberg defended while adding the 6D-bulk ontology Heisenberg did not have access to. First cited in \S\ref{part:I}.

\bibitem[Hensen(2015)]{Hensen2015}
\textbf{B~et~al~Hensen.}
{Loophole-free Bell inequality violation using electron spins separated by 1.3 kilometres}.
\emph{Nature} \textbf{526} (2015) 682--686.
\par\noindent\textbf{Relevance.}
The 2015 Hensen et al. loophole-free Bell test using NV centers in diamond. AEight Part VII cites Hensen2015 as one of three 2015 loophole-free experiments (with Giustina2015 and Shalm2015) that closed the locality and detection loopholes simultaneously, sealing the empirical case for QM and against local hidden-variable theories. First cited in \S\ref{part:I}.

\bibitem[Holevo(2012)]{Holevo2012}
\textbf{A~S~Holevo.}
\emph{Quantum Systems, Channels, Information}.
De Gruyter, (2012).
\par\noindent\textbf{Relevance.}
Holevo's 2012 graduate text on quantum channels and quantum information theory. AEight cites Holevo2012 in Part III as the modern reference for the CPTP-channel formalism that \TA{2} specializes from OGGEP. First cited in \S\ref{sec:TA-4}.

\bibitem['t~Hooft(2016)]{tHooft2016}
\textbf{G~'t~Hooft.}
\emph{The Cellular Automaton Interpretation of Quantum Mechanics}.
Fundamental Theories of Physics \textbf{185}, Springer, (2016).
\par\noindent\textbf{Relevance.}
't~Hooft's monograph developing the deterministic cellular-automaton
interpretation: quantum states as equivalence classes over ontological states
evolving by local classical rules, keeping locality and surrendering
measurement independence. \AEight{} treats it in Part~XI (\S\ref{sec:int-rivals})
as the framework's nearest deterministic rival; the two make opposite Bell
trades (the framework keeps measurement independence and concedes 4D locality)
and share the open problem of deriving the Born measure from the deeper layer.
First cited in \S\ref{sec:int-rivals}.

\bibitem[Hornberger \emph{et~al.}(2003)]{Hornberger2003}
\textbf{K~Hornberger, S~Uttenthaler, B~Brezger, L~Hackerm\"uller, M~Arndt, and A~Zeilinger.}
Collisional decoherence observed in matter wave interferometry.
\emph{Physical Review Letters} \textbf{90} (2003) 160401.
\par\noindent\textbf{Relevance.}
The 2003 direct observation of collisional decoherence of $C_{70}$ matter waves
as the loss of interference contrast versus gas pressure, with theory--experiment
agreement at the $10$--$20\%$ level. AEight cites Hornberger2003 in
\S\ref{sec:worked-overlap} as the measured benchmark reproduced by the worked
overlap integral. First cited in \S\ref{sec:worked-overlap}.

\bibitem[Itano \emph{et~al.}(1990)]{Itano1990_Zeno}
\textbf{W~M~Itano, D~J~Heinzen, J~J~Bollinger, and D~J~Wineland.}
Quantum {Zeno} effect.
\emph{Physical Review A} \textbf{41} (1990) 2295--2300.
\par\noindent\textbf{Relevance.}
The 1990 Itano et al. experimental confirmation of the quantum Zeno effect (frequent measurements freeze the evolution of an unstable state). AEight Part X cites Itano1990\_Zeno as a paradox-solved-by-decoherence: the cohort framework reads the Zeno effect as the operational consequence of \TA{4} (Lueders projection) applied at high measurement frequency. First cited in \S\ref{sec:px-zeno}.

\bibitem[J.~Maldacena(1998)]{Maldacena1998}
\textbf{J.~Maldacena.}
{The large N limit of superconformal field theories
and supergravity}.
\emph{Adv.\ Theor.\ Math.\ Phys.} \textbf{2} (1998) 231.
\par\noindent\textbf{Relevance.}
Maldacena's 1998 paper introducing the AdS/CFT correspondence. AEight Part III cites Maldacena1998 in the discussion of why the GG-bulk-to-boundary reduction differs from AdS/CFT and Randall-Sundrum: the 6D bulk of GG-Theory is a specific finite-scale-profile bulk geometry, distinct from AdS/CFT's infinite-radius limit but conceptually parallel. First cited in \S\ref{ssec:two-fibres}.

\bibitem[Jacques and others(2007)]{Jacques2007}
\textbf{V~Jacques and others.}
Experimental realization of {Wheeler's} delayed-choice gedanken experiment.
\emph{Science} \textbf{315} (2007) 966--968.
\par\noindent\textbf{Relevance.}
The 2007 Jacques et al. experimental realization of Wheeler's delayed-choice. AEight Part X cites this in the paradox sweep; the cohort framework reads the result as confirmation that the boundary measurement determines which boundary state is observed, with no retroactive effect on the bulk substrate. First cited in \S\ref{sec:DS-6d-reading}.

\bibitem[Joos and Zeh(1985)]{JoosZeh1985}
\textbf{E~Joos and H.~Zeh.}
{The emergence of classical properties through interaction with the environment}.
\emph{Zeitschrift f\"ur Physik B} \textbf{59} (1985) 223--243.
\par\noindent\textbf{Relevance.}
The 1985 Joos-Zeh paper supplying the canonical thermal-bath decoherence estimate $\Gamma \sim 10^{40}\,\mathrm{s}^{-1}$ for macroscopic objects at room temperature. AEight Part VI and Figure 6 cite Joos-Zeh as the empirical anchor for the macroscopic-instantaneous regime: any object larger than $C_{60}$ fullerene has its off-diagonal coherences damped on a timescale far shorter than any laboratory measurement window. First cited in \S\ref{part:I}.

\bibitem[Kay(1978)]{Kay1978}
\textbf{B~S~Kay.}
Linear spin-zero quantum fields in external gravitational and scalar fields.
\emph{Communications in Mathematical Physics} \textbf{62} (1978) 55--70.
\par\noindent\textbf{Relevance.}
Kay's uniqueness theorem for the one-particle structure of a linear
field on a stationary spacetime: the positive-energy complex structure
commuting with time translations is unique. With
\citet{AshtekarMagnon1975}, this is the imported mathematical core of
the real-bulk statement RB-2 (\S\ref{sec:disc-real-bulk}); the
framework's contribution is the constitutional pedigree (\AX{1}
direction, \AX{3} positivity, Theorem~0 staticity), not the theorem.
First cited in \S\ref{sec:disc-real-bulk}.

\bibitem[Kay and Wald(1991)]{KayWald1991}
\textbf{B~S~Kay and R~M~Wald.}
Theorems on the uniqueness and thermal properties of stationary,
nonsingular, quasifree states on spacetimes with a bifurcate Killing
horizon.
\emph{Physics Reports} \textbf{207} (1991) 49--136.
\par\noindent\textbf{Relevance.}
The rigorous uniqueness and thermality theorems for quantum fields on
bifurcate Killing horizons --- and the proof that the Kerr exterior
admits no stationary Hadamard state at all. RB-11 takes the latter as
the exact form of the ergoregion obstruction: the rotating-frame
positivity selection fails for a reason the literature has made
precise, and the anchor's move to the asymptotic frame is the
framework's placement of that fact.
First cited in \S\ref{sec:disc-real-bulk}.

\bibitem[Kochen and Specker(1967)]{KochenSpecker1967}
\textbf{S~Kochen and E~P~Specker.}
The problem of hidden variables in quantum mechanics.
\emph{Journal of Mathematics and Mechanics} \textbf{17} (1967) 59--87.
\par\noindent\textbf{Relevance.}
The 1967 Kochen-Specker theorem on the non-existence of non-contextual hidden variables in QM (for $\dim \ge 3$). AEight Part X discusses Kochen-Specker contextuality as one of the QM paradoxes; the cohort framework reads contextuality as a 4D shadow of the bulk choice of operational port-bound submanifold, i.e., contextuality is forced by OGGEP itself rather than being a paradox. First cited in \S\ref{sec:px-KS}.

\bibitem[L.~Randall and R.~Sundrum(1999)]{RandallSundrum1999}
\textbf{L.~Randall and R.~Sundrum.}
{A large mass hierarchy from a small extra dimension}.
\emph{Phys.\ Rev.\ Lett.} \textbf{83} (1999) 3370.
\par\noindent\textbf{Relevance.}
The 1999 Randall-Sundrum paper on warped extra dimensions in particle physics. AEight Part III cites Randall-Sundrum in the discussion of why the GG-Theory's 6D bulk is structurally similar to but distinct from the RS scenario: the RS hierarchy problem solution involves a different scale profile and different boundary content than the GG-bulk QM derivation. First cited in \S\ref{ssec:two-fibres}.

\bibitem[Ladyman and Ross(2007)]{LadymanRoss2007}
\textbf{J~Ladyman and D~Ross.}
\emph{Every Thing Must Go: Metaphysics Naturalized}.
Oxford University Press, (2007).
\par\noindent\textbf{Relevance.}
Ladyman and Ross's 2007 manifesto for ontic structural realism (OSR). AEight Part XI and Part XIII cite Ladyman-Ross as the locus classicus of structural realism; the cohort framework's reading of QM is structurally realist by inheritance: every electron is the same Geometric Attractor, electrons-as-individuals have no metaphysical existence beyond the structural relations they enter. First cited in \S\ref{sec:int-structural}.

\bibitem[Liao and others(2017)]{Liao2017}
\textbf{S.~Liao and others.}
Satellite-to-ground quantum key distribution.
\emph{Nature} \textbf{549} (2017) 43--47.
\par\noindent\textbf{Relevance.}
The 2017 Liao et al. satellite-to-ground QKD experiment using the Micius satellite, the first demonstration of long-distance free-space quantum key distribution. AEight Part X cites Liao2017 as the modern long-distance test of entanglement and Bell-pair fidelity; the cohort framework reads the success as confirmation that bulk-mediated correlations persist over ground-to-satellite distances. First cited in \S\ref{sec:px-crypto}.

\bibitem[Lindblad(1976)]{Lindblad1976}
\textbf{G~Lindblad.}
\emph{{Communications in Mathematical Physics}}.
(1976).
\par\noindent\textbf{Relevance.}
Lindblad's 1976 paper deriving the most general completely-positive trace-preserving generator of a Markovian quantum dynamical semigroup. AEight cites Lindblad1976 in Part III and in the Appendix's Nakajima-Zwanzig $\to$ GKLS derivation; \TA{2} reconstructs the Schrodinger equation as the $\gamma \to 0$ limit of the Lindblad equation, and the Lindblad operators are the bulk-environment coupling channels. First cited in \S\ref{part:I}.

\bibitem[Lindblad(1983)]{LindbladBook}
\textbf{G~Lindblad.}
\emph{Non-Equilibrium Entropy and Irreversibility}.
Reidel, Dordrecht, (1983).
\par\noindent\textbf{Relevance.}
Lindblad's 1983 monograph extending the 1976 Lindblad master equation result to the thermodynamic interpretation of decoherence and dissipation. AEight cites the book in the Appendix as the standard reference for the open-system formalism that underlies the cohort's \TA{5} derivation of decoherence rates. First cited in \S\ref{part:I}.

\bibitem[Millikan(1916)]{Millikan1916}
\textbf{R~A~Millikan.}
A direct photoelectric determination of {Planck's} ``$h$''.
\emph{Physical Review} \textbf{7} (1916) 355--388.
\par\noindent\textbf{Relevance.}
Millikan's experimental confirmation of Einstein's photoelectric-effect relation and the first precise determination of Planck's constant h. Cited in Part I as the experimental seal on the old-quantum-theory program that the Schrodinger-equation reformulation of 1926 would supersede. First cited in \S\ref{part:I}.

\bibitem[N.~Bohr(1913)]{Bohr1913}
\textbf{N.~Bohr.}
\emph{{On the constitution of atoms and molecules}}.
(1913).
\par\noindent\textbf{Relevance.}
Establishes the quantized atomic model (electron orbits in hydrogen at discrete energy levels) that motivated the older quantum theory. AEight cites Bohr1913 in the historical reconstruction of Part I as the origin of the discrete-spectrum reading of atomic physics, which would later be derived from the Schrodinger equation as \TA{2} specialized. First cited in \S\ref{part:I}.

\bibitem[Naimark(1943)]{Naimark1943}
\textbf{M~A~Naimark.}
On a representation of additive operator set functions.
\emph{Doklady Akademii Nauk SSSR} \textbf{41} (1943) 359--361.
\par\noindent\textbf{Relevance.}
Naimark's 1943 dilation theorem on POVMs and their unitary dilations to projective measurements on a larger Hilbert space. AEight cites Naimark1943 in Part III alongside Stinespring1955 as the mathematical foundation for the POVM-to-instrument generalization of \TA{4}. The Naimark dilation is the standard bridge from boundary POVMs to bulk-side projective measurements. First cited in \S\ref{sec:TA-1}.

\bibitem[Nakajima(1958)]{Nakajima1958}
\textbf{S.~Nakajima.}
On quantum theory of transport phenomena.
\emph{Progress of Theoretical Physics} \textbf{20} (1958) 948--959.
\par\noindent\textbf{Relevance.}
Nakajima's 1958 derivation (together with Zwanzig 1960) of the
projection-operator technique that exhibits an exact non-Markovian
master equation for the reduced density matrix. \AEight{} cites
Nakajima--Zwanzig in Part~III alongside Stinespring as the
mathematical-physics ancestor of the \OGGEP{} partial-trace
construction; the present treatment derives the Markovian limit
of Nakajima--Zwanzig from the parametric smallness of
$\tau_c/\tau_S$ rather than postulating it. First cited in \S\ref{sec:DC-6d-reading}.

\bibitem[Unruh(1974)]{Unruh1974}
\textbf{W~G~Unruh.}
Second quantization in the Kerr metric.
\emph{Physical Review D} \textbf{10} (1974) 3194--3205.
\par\noindent\textbf{Relevance.}
The quantum field theory of the Kerr background: the mode analysis in
which the ergoregion's negative co-rotating energies and the
superradiant band appear at the field level, and the vacuum choices of
the rotating geometry are first laid out. RB-11's model-level
statements are the structural skeleton of this analysis, with the
vacuum multiplicity placed as port assignment at the dictionary.
First cited in \S\ref{sec:disc-real-bulk}.

\bibitem[Valentini(1991)]{Valentini1991}
\textbf{A~Valentini.}
Signal-locality, uncertainty, and the subquantum H-theorem. I.
\emph{Physics Letters A} \textbf{156} (1991) 5--11.
\par\noindent\textbf{Relevance.}
The relaxation reading of quantum equilibrium: non-equilibrium
distributions approach $|\psi|^2$ dynamically, with a coarse-grained
H-theorem, and pre-equilibrium epochs could leave observable relics.
The constitutional relaxation chain strengthens both halves --- the
boundary law is genuinely dissipative (a contracting semigroup, by the
\AFive{} theorem, rather than a coarse-grained measure-preserving
flow), and the relic-non-equilibrium falsifier is inherited as this
derivation's own testable signature.
First cited in \S\ref{sec:disc-open}.

\bibitem[Neumann(1932)]{vonNeumann1932}
\textbf{J.~Neumann.}
\emph{{Mathematische Grundlagen der Quantenmechanik}}.
(1932).
\par\noindent\textbf{Relevance.}
von Neumann's 1932 axiomatization 'Mathematische Grundlagen der Quantenmechanik' (Mathematical Foundations of Quantum Mechanics). Cited in Part I as the historical seal on QM's mathematical structure; AEight's \TA{1} (Hilbert-space embedding via GNS) and \TA{4} (Lueders projection rule) reconstruct two of the four von Neumann postulates as theorems of OGGEP. First cited in \S\ref{part:I}.

\bibitem[Nielsen(2010)]{NielsenChuang2010}
\textbf{M.~\&~C.~I.~Nielsen.}
\emph{Quantum Computation and Quantum Information (10th Anniversary Edition)}.
Cambridge University Press, Cambridge, (2010).
\par\noindent\textbf{Relevance.}
The Nielsen-Chuang 2010 graduate textbook 'Quantum Computation and Quantum Information', the standard reference for the quantum-information-theoretic content of QM. AEight cites Nielsen-Chuang throughout Part X (paradox sweep) for the modern treatment of teleportation, dense coding, no-cloning, and information-theoretic implications of the Bell-CHSH bound. First cited in \S\ref{sec:TA-4}.

\bibitem[Omn{\`e}s(1999)]{Omnes1999}
\textbf{R~Omn{\`e}s.}
\emph{Understanding Quantum Mechanics}.
Princeton University Press, Princeton, (1999).
\par\noindent\textbf{Relevance.}
Omnes's 1999 monograph developing the decoherent-histories interpretation in textbook form. AEight cites Omnes1999 in Part XI as one of two canonical decoherent-histories references (with Griffiths1984 and Gell-Mann-Hartle 1993); the cohort framework's \TA{5} supplies the decoherence functional that the decoherent-histories interpretation presupposes. First cited in \S\ref{part:I}.

\bibitem[Pauli(1940)]{Pauli1940}
\textbf{W.~Pauli.}
The connection between spin and statistics.
\emph{Physical Review} \textbf{58}, 716--722 (1940).
\par\noindent\textbf{Relevance.} The founding spin--statistics theorem: given Lorentz invariance, locality, and vacuum stability, half-integer spin must anticommute. Cited in \S\ref{sec:TA12-statistics} as the conditional theorem whose existence question the cohort's candidate suite addresses, and in \TA{9}'s proof as the interim warrant for exchange statistics until \TA{12}'s residues are closed. First cited in \S\ref{sec:TA12-statistics}.

\bibitem[Penrose(1989)]{Penrose1989}
\textbf{R~Penrose.}
\emph{The Emperor's New Mind: Concerning Computers, Minds, and the Laws of Physics}.
Oxford University Press, (1989).
\par\noindent\textbf{Relevance.}
Penrose's argument that physical state reduction is non-computable and connected
to consciousness. \AEight{} cites it in Part~XI (\S\ref{sec:int-rivals}) to mark
the point at which the framework parts company with Penrose: the framework's
outcome-selecting descent is an ordinary deterministic, computable relaxation,
invoking neither non-computability nor consciousness. First cited in
\S\ref{sec:int-rivals}.

\bibitem[Penrose(1996)]{Penrose1996}
\textbf{R~Penrose.}
On gravity's role in quantum state reduction.
\emph{General Relativity and Gravitation} \textbf{28} (1996) 581--600.
\par\noindent\textbf{Relevance.}
Penrose's 1996 proposal that gravity triggers an objective state reduction on a
timescale $\tau\sim\hbar/E_G$ set by the gravitational self-energy of the
mass-distribution difference (with \citet{Diosi1989}, the Di\'osi--Penrose
model). \AEight{} engages it in Part~XI (\S\ref{sec:int-rivals}) as the sharpest
empirical rival: the framework shares Penrose's diagnosis that decoherence is not
collapse but declines the gravitational-collapse cure, keeping unitarity and
locating outcome selection in a deterministic \GRIP{} descent rather than a
modification of the Schr\"odinger equation. First cited in \S\ref{sec:int-rivals}.

\bibitem[Penrose(2004)]{Penrose2004}
\textbf{R~Penrose.}
The Road to Reality: A Complete Guide to the Laws of the Universe.
\emph{Jonathan Cape, London} (2004).
\par\noindent\textbf{Relevance.}
Penrose's sustained statement of the complex-number question --- why a
number system invented for algebra should be the working substance of
quantum amplitudes --- which \S\ref{sec:disc-why-complex} answers in the
framework's calibrated form: the gauge half geometric (a real circle,
the sixth dimension), the state-space half dynamical (the polar phase of
constitutional time), the residue named. Companion to the
objective-reduction engagement of \S\ref{sec:int-rivals}. First cited in
\S\ref{sec:disc-why-complex}.

\bibitem[Penrose and MacCallum(1973)]{PenroseMacCallum1973}
\textbf{R~Penrose and M~A~H~MacCallum.}
Twistor theory: an approach to the quantization of fields and space-time.
\emph{Physics Reports} \textbf{6} (1973) 241--315.
\par\noindent\textbf{Relevance.}
The foundational exposition of twistor theory, including the
correspondence between positive-frequency massless fields and
holomorphic data on the positive half of twistor space --- the
component of the holomorphy bridge RB-9 that identifies
$\mathbb{PT}^{+}$ with the causal-positive splitting generated by the
framework's derived $J$. First cited in \S\ref{sec:disc-real-bulk}.

\bibitem[Peres(1995)]{Peres1995}
\textbf{A~Peres.}
\emph{Quantum Theory: Concepts and Methods}.
Kluwer, (1995).
\par\noindent\textbf{Relevance.}
Peres's 1995 textbook emphasizing the operational/instrumentalist reading of QM. AEight cites Peres1995 in Part XI as a representative of the Copenhagen-style operational reading; the cohort framework agrees with Peres on the boundary-operational completeness of QM and adds the 6D-bulk ontology. First cited in \S\ref{sec:TA-1}.

\bibitem[Pizzo(2003)]{Pizzo2003}
\textbf{A~Pizzo.}
One-particle (improper) states in Nelson's massless model.
\emph{Annales Henri Poincar\'e} \textbf{4} (2003) 439--486.
\par\noindent\textbf{Relevance.}
The iterated-dressing construction of the massless Nelson model's
one-particle states, with H\"older regularity in the total momentum.
This is the constructive engine behind RB-10's Nelson rung: it builds
exactly the dressed states whose coherent clouds RB-10 classifies as
port assignments, on which the sector-blindness of the derived $J$ is
then a theorem. First cited in \S\ref{sec:disc-real-bulk}.

\bibitem[Planck(1900)]{Planck1900}
\textbf{M~Planck.}
(1900).
\par\noindent\textbf{Relevance.}
Planck's 1900 paper introducing the quantum hypothesis (E = h $\nu$) to explain blackbody radiation. AEight cites Planck1900 in Part I as the historical seed of the entire QM program; the constant h that Planck introduced is one of the \GUfive{} inputs that GG-A4-Euclid certifies as a zero-free-parameter input. First cited in \S\ref{part:I}.

\bibitem[Pusey \emph{et~al.}(2012)]{PBR2012}
\textbf{M~F~Pusey, J~Barrett, and T~Rudolph.}
On the reality of the quantum state.
\emph{Nature Physics} \textbf{8} (2012) 475--478.
\par\noindent\textbf{Relevance.}
The 2012 Pusey-Barrett-Rudolph theorem ruling out $\psi$-epistemic ontological models (under modest assumptions). AEight Part XI discusses PBR as the modern formal constraint on epistemic interpretations; the cohort framework respects PBR by treating the 6D bulk state as ontic while the 4D boundary state can be epistemic for the operator-observer. First cited in \S\ref{sec:int-bohm}.

\bibitem[Rauch \emph{et~al.}(1975)]{Rauch1975}
\textbf{H.~Rauch, A.~Zeilinger, G.~Badurek, A.~Wilfing, W.~Bauspiess, and U.~Bonse.}
Verification of coherent spinor rotation of fermions.
\emph{Physics Letters A} \textbf{54}, 425--427 (1975).
\par\noindent\textbf{Relevance.} The neutron-interferometric observation that a $2\pi$ rotation reverses a spinor amplitude while $4\pi$ restores it (with the contemporaneous experiment of Werner et al.). Cited in \S\ref{sec:TA12-statistics} as the direct laboratory witness of the frame-path $\mathbb{Z}_2$ on which \TA{12}'s Link~3 rests --- the boundary dictionary faithfully reporting the bulk's two-sheeted frame memory. First cited in \S\ref{sec:TA12-statistics}.

\bibitem[Renou \emph{et~al.}(2021)]{Renou2021}
\textbf{M-O~Renou, D~Trillo, M~Weilenmann, T~P~Le, A~Tavakoli, N~Gisin,
A~Ac\'in, and M~Navascu\'es.}
Quantum theory based on real numbers can be experimentally falsified.
\emph{Nature} \textbf{600} (2021) 625--629.
\par\noindent\textbf{Relevance.}
The network-Bell theorem (since confirmed experimentally) excluding
quantum theory over real Hilbert spaces with real tensor products. Read
here as the experimental mandate for a \emph{single, shared} complex
structure across independent sources --- exactly what the real-bulk
statements derive from one shared circle and one shared causal order
(RB-4, \S\ref{sec:disc-real-bulk}), and what an unshared real
formulation provably cannot arrange. First cited in
\S\ref{sec:disc-why-complex}.

\bibitem[Riebe and others(2004)]{Riebe2004}
\textbf{M~Riebe and others.}
Deterministic quantum teleportation with atoms.
\emph{Nature} \textbf{429} (2004) 734--737.
\par\noindent\textbf{Relevance.}
The 2004 Riebe et al. deterministic atom-teleportation experiment. AEight Part X cites this alongside Bouwmeester1997 as a major experimental confirmation of the teleportation protocol; the cohort framework's reading remains consistent: the protocol exploits the bulk-mediated entangled correlations of \TA{7} without violating \TA{11}. First cited in \S\ref{sec:SG-6d-reading}.

\bibitem[Rovelli(1996)]{Rovelli1996RQM}
\textbf{C~Rovelli.}
Relational quantum mechanics.
\emph{International Journal of Theoretical Physics} \textbf{35} (1996) 1637--1678.
\par\noindent\textbf{Relevance.}
Rovelli's 1996 paper founding relational quantum mechanics: quantum states are relative to an observer, never absolute. AEight Part XI discusses relational QM; the cohort framework agrees that the boundary state is observer-relative but locates the absolute description at the 6D bulk level, where Rovelli's relational picture is the unique 4D-boundary projection of a single deterministic substrate. First cited in \S\ref{part:I}.

\bibitem[Rovelli(2021)]{Rovelli2021Helgoland}
\textbf{C~Rovelli.}
\emph{Helgoland: Making Sense of the Quantum Revolution}.
Riverhead Books, (2021).
\par\noindent\textbf{Relevance.}
Rovelli's 2021 popular-science book extending the relational QM program. AEight cites Helgoland in Part XI as the popular articulation of the relational interpretation; the cohort framework's bulk-side reading provides an ontological foundation that Rovelli's relational reading explicitly declines to provide. First cited in \S\ref{sec:int-relational}.

\bibitem[Royal~Swedish~Academy~of~Sciences(2022)]{NobelPhysics2022}
\textbf{{Royal~S.~A.~of~Sciences}.}
\emph{The {Nobel Prize} in {Physics} 2022: Aspect, Clauser, Zeilinger}.
\url{https://www.nobelprize.org/prizes/physics/2022/}, (2022).
\par\noindent\textbf{Relevance.}
The 2022 Nobel Prize in Physics awarded to Alain Aspect, John F. Clauser, and Anton Zeilinger for 'experiments with entangled photons, establishing the violation of Bell inequalities and pioneering quantum information science.' AEight cites the 2022 Nobel in Part VII as the institutional seal on the Bell-CHSH empirical program; the cohort framework's \TA{7} and \TA{8} provide the theoretical reading that the Nobel-laureate experiments confirmed. First cited in \S\ref{part:I}.

\bibitem[Sakurai and Napolitano(2011)]{Sakurai2011}
\textbf{J~J~Sakurai and J~Napolitano.}
\emph{Modern Quantum Mechanics}.
Addison-Wesley, (2011).
\par\noindent\textbf{Relevance.}
The Sakurai-Napolitano graduate QM textbook, the standard reference for the operator and Dirac-notation formalism. AEight cites Sakurai2011 throughout the worked examples as the canonical textbook account that the cohort framework recovers as \TA{1}-\TA{7}. First cited in \S\ref{sec:TA-1}.

\bibitem[Schlosshauer(2005)]{Schlosshauer2005}
\textbf{M~Schlosshauer.}
(2005).
\par\noindent\textbf{Relevance.}
Schlosshauer's 2005 Reviews of Modern Physics article 'Decoherence, the measurement problem, and interpretations of quantum mechanics'. AEight cites Schlosshauer2005 in Part IX as the modern review of how decoherence dissolves (rather than solves) the measurement problem; the cohort framework completes Schlosshauer's reading with the bulk-side picture: the dissolution is real because the bulk is deterministic. First cited in \S\ref{sec:TA-5}.

\bibitem[Schlosshauer(2007)]{Schlosshauer2007}
\textbf{M~Schlosshauer.}
\emph{Decoherence and the Quantum-to-Classical Transition}.
Springer, Berlin, (2007).
\par\noindent\textbf{Relevance.}
Schlosshauer's 2007 textbook extending the 2005 RMP article into a full-length treatment of decoherence and the quantum-to-classical transition. AEight cites Schlosshauer2007 in Part IX as the comprehensive modern reference; the bulk-side reading of \TA{5} and \TA{6} supplies the deeper warrant that the textbook treatment leaves as an operational principle. First cited in \S\ref{sec:TA-5}.

\bibitem[Schr{\"o}dinger(1926)]{Schrodinger1926}
\textbf{E~Schr{\"o}dinger.}
\emph{{Quantisierung als Eigenwertproblem}}.
(1926).
\par\noindent\textbf{Relevance.}
Schrodinger's 1926 wave-mechanics papers (E. Schrodinger, Ann. der Physik). AEight cites this in Part I as the foundational paper of wave mechanics; \TA{2} reconstructs the Schrodinger equation as the closed-system $\gamma \to 0$ limit of the GKLS master equation derived from bulk dynamics. First cited in \S\ref{part:I}.

\bibitem[Schr{\"o}dinger(1935)]{Schrodinger1935_cat}
\textbf{E~Schr{\"o}dinger.}
Die gegenw{\"a}rtige Situation in der Quantenmechanik.
\emph{Naturwissenschaften} \textbf{23} (1935) 807--812, 823--828, 844--849.
\par\noindent\textbf{Relevance.}
Schrodinger's 1935 cat thought experiment showing that a literal-reading of QM extends macroscopic superpositions arbitrarily far. AEight Part VIII centers the cat in the 6D-bulk-to-4D-boundary picture: the bulk state is deterministic and single-valued; the 4D boundary observer sees decohered LIVE/DEAD branches via the projection of \TA{5}; there is no macroscopic-superposition paradox once the bulk substrate is recognized as deterministic. First cited in \S\ref{part:I}.

\bibitem[Scully \emph{et~al.}(1991)]{Scully_eraser_1991}
\textbf{M~O~Scully, B.~Engl{\'e}rt, and H~Walther.}
Quantum optical tests of complementarity.
\emph{Nature} \textbf{351} (1991) 111--116.
\par\noindent\textbf{Relevance.}
The 1991 Scully-Englert-Walther paper introducing the quantum-eraser thought experiment. AEight Part X cites this in the paradox sweep; the cohort framework reads the quantum eraser as a consistency check on \TA{5} (decoherence as which-path damping) plus \TA{4} (Lueders projection): erasing the which-path information restores the interference fringes because the off-diagonal coherences had been merely entangled with the environment, not destroyed. First cited in \S\ref{part:I}.

\bibitem[Shalm and others(2015)]{Shalm2015}
\textbf{L~K~Shalm and others.}
Strong loophole-free test of local realism.
\emph{Physical Review Letters} \textbf{115} (2015) 250402.
\par\noindent\textbf{Relevance.}
The 2015 Shalm et al. loophole-free Bell test at NIST. AEight Part VII cites Shalm2015 in the trio (with Hensen2015 and Giustina2015) of 2015 experiments that closed all major loopholes. The cohort framework reads the empirical record as confirmation of the bulk-mediated correlations of \TA{7}. First cited in \S\ref{part:I}.

\bibitem[Shor(1994)]{Shor1994}
\textbf{P~W~Shor.}
\emph{Algorithms for quantum computation: discrete logarithms and factoring}.
In \emph{Proceedings 35th Annual Symposium on Foundations of Computer Science} (1994).
\par\noindent\textbf{Relevance.}
Shor's 1994 quantum algorithm for integer factorization in polynomial time. AEight cites Shor1994 in Part X as the algorithmic motivation for quantum computing; the cohort framework's QM reading is computationally equivalent to standard QM, so all quantum-algorithm results carry over unchanged. First cited in \S\ref{part:I}.

\bibitem[Sinha \emph{et~al.}(2010)]{Sinha2010_TripleSlit}
\textbf{U~Sinha, C~Couteau, T~Jennewein, R~Laflamme, and G~Weihs.}
Ruling out multi-order interference in quantum mechanics.
\emph{Science} \textbf{329} (2010) 418--421.
\par\noindent\textbf{Relevance.}
The 2010 Sinha et al. triple-slit experiment confirming that QM admits only pairwise interference terms (no higher-order interference). AEight Part X cites this as a precision test of the Born rule's quadratic structure; the cohort framework's \TA{3} derives the Born rule as a CPTP-channel consequence, which structurally precludes higher-order interference. First cited in \S\ref{ssec:TA-3-where-probs}.

\bibitem[Starobinsky(1973)]{Starobinsky1973}
\textbf{A~A~Starobinsky.}
Amplification of waves during reflection from a rotating black hole.
\emph{Soviet Physics JETP} \textbf{37} (1973) 28--32.
\par\noindent\textbf{Relevance.}
The discovery of superradiance: waves scattered off a rotating black
hole in the superradiant band return amplified, extracting rotational
energy. RB-11 reads the phenomenon structurally --- the amplification
is a symplectic Gaussian pumping channel acting on the asymptotic
frame's derived complex structure, which it never touches.
First cited in \S\ref{sec:disc-real-bulk}.

\bibitem[Stinespring(1955)]{Stinespring1955}
\textbf{W.~Stinespring.}
\emph{Positive functions on $C^*$-algebras}.
\emph{Proceedings of the American Mathematical Society} \textbf{6} (1955) 211--216.
\par\noindent\textbf{Relevance.}
Stinespring's 1955 dilation theorem on CPTP channels: any CPTP channel on a system can be realized as a unitary evolution on a larger system plus partial trace. AEight cites Stinespring1955 in Part III as the mathematical foundation for \TA{2}'s derivation of the GKLS master equation from the bulk-to-boundary partial trace. First cited in \S\ref{sec:TA-3}.

\bibitem[Stone(1932)]{Stone1932}
\textbf{M~H~Stone.}
On one-parameter unitary groups in {Hilbert} space.
\emph{Annals of Mathematics} \textbf{33} (1932) 643--648.
\par\noindent\textbf{Relevance.}
Stone's 1932 theorem on one-parameter unitary groups in Hilbert space, the mathematical foundation of the closed-system unitary evolution. AEight cites Stone1932 in Part III as the key mathematical input to \TA{2}'s derivation of the Schrodinger equation from the $\gamma \to 0$ limit of GKLS. First cited in \S\ref{part:I}.

\bibitem[Streater(1964)]{StreaterWightman1964}
\textbf{R.~\&~W.~A.~Streater.}
\emph{PCT, Spin and Statistics, and All That}.
W.~A.\ Benjamin, New York, (1964).
\par\noindent\textbf{Relevance.}
Streater-Wightman's 1964 monograph 'PCT, Spin and Statistics, and All That', the standard reference for the axiomatic / Wightman approach to QFT. AEight cites Streater-Wightman in the Appendix as the canonical rigorous treatment of the spin-statistics theorem; the cohort framework's \TA{9} (universal identity of indistinguishable particles) inherits the Wightman framework's reading of indistinguishability. First cited in \S\ref{sec:TA-1}.

\bibitem[Tiesinga \emph{et~al.}(2021)]{CODATA2018}
\textbf{E~Tiesinga, P~J~Mohr, D~B~Newell, and B~N~Taylor.}
{CODATA} recommended values of the fundamental physical constants: 2018.
\emph{Reviews of Modern Physics} \textbf{93} (2021) 025010.
\par\noindent\textbf{Relevance.}
The 2018 CODATA recommended values of the fundamental physical constants (published 2021). AEight cites CODATA2018 throughout the worked examples as the source of empirical anchors (c, $\hbar$, $k_B$, etc.); these are the \GUfive{} inputs that the cohort framework's GG-A4-Euclid certifies as zero free continuous parameters. First cited in \S\ref{sec:TA-2}.

\bibitem[Tsirelson(1980)]{Tsirelson1980}
\textbf{B.~Tsirelson.}
{Quantum generalizations of Bell's inequality}.
\emph{Letters in Mathematical Physics} \textbf{4} (1980) 93--100.
\par\noindent\textbf{Relevance.}
Tsirelson's 1980 paper proving that the maximal QM violation of the CHSH inequality is $|S| = 2\sqrt{2}$, known as the Tsirelson bound. AEight cites Tsirelson1980 in Part VII as the formal upper bound on QM CHSH; \TA{8} then reads this bound as an extremal (Bogomolnyi-analogue) saturation --- the singlet is the maximizer of the CHSH functional --- consistent with why the bound is exactly 2 $\sqrt{2}$ (as opposed to some other irrational number). First cited in \S\ref{sec:TA-7}.

\bibitem[Uhlenbeck and Goudsmit(1925)]{UhlenbeckGoudsmit1925}
\textbf{G~E~Uhlenbeck and S~Goudsmit.}
Ersetzung der Hypothese vom unmechanischen Zwang durch eine Forderung bez{\"u}glich des inneren Verhaltens jedes einzelnen Elektrons.
\emph{Naturwissenschaften} \textbf{13} (1925) 953--954.
\par\noindent\textbf{Relevance.}
The 1925 spin-1/2 hypothesis that interpreted the Stern-Gerlach two-spot pattern as evidence for intrinsic electron angular momentum s=1/2. AEight cites this in Part V to complete the historical lineage from Stern-Gerlach (1922) to the modern bulk-fermion-zero-mode reading of \TA{1}: spin is the chirality-projected zero mode of the bulk fermion on the 6D geometry. First cited in \S\ref{part:I}.

\bibitem[VIP-2 Collaboration(2025)]{VIP2_2025}
\textbf{VIP-2 Collaboration.}
High-sensitivity test of the Pauli exclusion principle for electrons with the VIP-2 experiment.
\emph{Scientific Reports} (November 2025); experiment at LNGS, Gran Sasso.
\par\noindent\textbf{Relevance.} The strongest current bound on Pauli-exclusion-violating atomic transitions of electrons in copper: violation probability $\beta^2/2 < 4.8\times10^{-29}$ at 90\% CL. Cited in \S\ref{sec:TA12-statistics} as \TA{12}'s standing experimental exposure: under the frame-path account the prediction is exact silence, with no parameter available to absorb a violation (kill criterion F-Stat-1). First cited in \S\ref{sec:TA12-statistics}.

\bibitem[Wallace(2012)]{Wallace2012EmergentMultiverse}
\textbf{D~Wallace.}
\emph{The Emergent Multiverse: Quantum Theory According to the {Everett} Interpretation}.
Oxford University Press, (2012).
\par\noindent\textbf{Relevance.}
Wallace's 2012 monograph defending the Everett (Many-Worlds) interpretation as the unique consistent reading of standard unitary QM. AEight Part XI cites Wallace as the modern Everett standard-bearer; the cohort framework agrees on the operational predictions but adds the bulk-side ontology that makes the 'emergent multiverse' a single deterministic substrate seen from 4D. First cited in \S\ref{ssec:TA-3-where-probs}.

\bibitem[Weissman(1988)]{Weissman1988}
\textbf{M~B~Weissman.}
{$1/f$} noise and other slow, nonexponential kinetics in condensed matter.
\emph{Reviews of Modern Physics} \textbf{60} (1988) 537--571.
\par\noindent\textbf{Relevance.}
Weissman's 1988 review of $1/f$ noise in condensed-matter systems. AEight Part XII cites Weissman1988 as background for the QM-specific eight-test horizon item 'Test 3: $1/f$ noise universality': the cohort framework's \TA{5} plus \TA{6} predict that $1/f$ noise spectra should be universal across condensed-matter platforms, since the einselected pointer-basis structure is universal. First cited in \S\ref{sec:TA-10}.

\bibitem[Wheeler(1983)]{Wheeler1983}
\textbf{J~A~Wheeler.}
\emph{Law without law}.
In \emph{Quantum Theory and Measurement}, Princeton University Press (1983).
\par\noindent\textbf{Relevance.}
Wheeler's 1983 paper articulating the delayed-choice thought experiment and the 'law without law' speculation. AEight Part X discusses Wheeler's delayed choice as a paradox; the cohort framework reads it as a confirmation of \TA{4} (Lueders projection rule) applied at the bulk-to-boundary projection: the boundary state is determined by which boundary measurement is made, not by any retroactive change to the bulk. First cited in \S\ref{part:I}.

\bibitem[Wigner(1961)]{Wigner1961}
\textbf{E~P~Wigner.}
\emph{Remarks on the mind-body question}.
In \emph{The Scientist Speculates}, Heinemann (1961).
\par\noindent\textbf{Relevance.}
Wigner's 1961 essay introducing the 'Wigner's friend' thought experiment and the consciousness-causes-collapse speculation. AEight Part X discusses Wigner's friend as the historical seed of the Frauchiger-Renner sharpening; the cohort framework does NOT invoke consciousness for collapse, locating the 'collapse' instead in the decoherence dynamics of \TA{5}. First cited in \S\ref{sec:cat-geiger-trigger}.

\bibitem[Wilson(1971)]{Wilson1971}
\textbf{K.~G.~Wilson.}
{Renormalization group and critical phenomena}.
\emph{Phys.\ Rev.\ B} \textbf{4} (1971) 3174.
\par\noindent\textbf{Relevance.}
The renormalization group as successive coarse-graining. Cited for the
identification of radial integration along the non-compact scale fiber $I_r$
with Wilsonian coarse-graining --- the irreversible half of the two-fiber
picture, and the reason openness is constitutional rather than incidental.
First cited in \S\ref{ssec:two-fibres}.

\bibitem[Wiseman and Milburn(2010)]{WisemanMilburn2010}
\textbf{H~M~Wiseman and G~J~Milburn.}
\emph{Quantum Measurement and Control}.
Cambridge University Press, (2010).
\par\noindent\textbf{Relevance.}
Wiseman-Milburn 2010, the modern reference for continuous quantum measurement and feedback control. AEight cites this in Part IX as the standard treatment of weak measurement and POVMs; the cohort framework's \TA{4} generalizes the Lueders projection to POVM-style instruments via the same CPTP-channel structure that Wiseman-Milburn presupposes. First cited in \S\ref{sec:TA-4}.

\bibitem[Worrall(1989)]{Worrall1989}
\textbf{J~Worrall.}
Structural realism: the best of both worlds?.
\emph{Dialectica} \textbf{43} (1989) 99--124.
\par\noindent\textbf{Relevance.}
Worrall's 1989 paper introducing structural realism into the philosophy of science. AEight cites Worrall1989 in Part XI and XIII as the historical seed of OSR; the cohort framework's claim that 'structure (the bulk geometry) is the ontology' is the natural Worrall-style position applied to the QM stratum. First cited in \S\ref{sec:int-structural}.

\bibitem[Wu \& Yang(1975)]{WuYang1975}
\textbf{T.~T.~Wu and C.~N.~Yang.}
{Concept of nonintegrable phase factors and global formulation of gauge fields}.
\emph{Phys.\ Rev.\ D} \textbf{12} (1975) 3845--3857.
\par\noindent\textbf{Relevance.}
The global / fiber-bundle formulation of gauge theory, identifying the non-integrable phase factor (Wilson line) with bundle holonomy. The precise precedent for reading the quantum phase on $S^1_\theta$ as a principal-$U(1)$-bundle holonomy in the hidden-fiber section. First cited in \S\ref{ssec:two-fibres}.

\bibitem[Zeh(1970)]{Zeh1970}
\textbf{H.~Zeh.}
{On the interpretation of measurement in quantum theory}.
\emph{Foundations of Physics} \textbf{1} (1970) 69--76.
\par\noindent\textbf{Relevance.}
Zeh's 1970 paper founding the decoherence program: the wave function collapses are environmental in origin, not metaphysical. AEight cites Zeh1970 in Parts VI and IX as the historical seed; \TA{5} derives the off-diagonal damping rate $\Gamma_{\alpha\beta}$ from the bulk-side KK mode integrals, supplying the deeper warrant for the Zeh program. First cited in \S\ref{sec:OGGEP_lindblad}.

\bibitem[Zurek(1981)]{Zurek1981}
\textbf{W.~Zurek.}
{Pointer basis of quantum apparatus: into what mixture does the wave packet collapse?}.
\emph{Physical Review D} \textbf{24} (1981) 1516--1525.
\par\noindent\textbf{Relevance.}
Zurek's 1981 paper introducing einselection (environment-induced superselection): the system-environment Hamiltonian selects a preferred 'pointer basis' for which off-diagonal coherences decay fastest. AEight cites Zurek1981 in Part IX (Figure 10 shows the predictability sieve schematic); \TA{6} inherits einselection from \AX{1} (bulk locality) which forces the system-environment coupling to factor along position, recovering Zurek's selection as a theorem rather than a postulate. First cited in \S\ref{part:I}.

\bibitem[Zurek(2003)]{Zurek2003}
\textbf{W.~Zurek.}
\emph{{Reviews of Modern Physics}}.
(2003).
\par\noindent\textbf{Relevance.}
Zurek's 2003 Reviews of Modern Physics article 'Decoherence, einselection, and the quantum origins of the classical', the canonical review of the decoherence program. AEight Part IX cites Zurek2003 as the modern summary the cohort framework completes via the bulk-side derivation of \TA{5} and \TA{6}. First cited in \S\ref{part:I}.

\bibitem[Zwanzig(1960)]{Zwanzig1960}
\textbf{R.~Zwanzig.}
Ensemble method in the theory of irreversibility.
\emph{Journal of Chemical Physics} \textbf{33} (1960) 1338--1341.
\par\noindent\textbf{Relevance.}
Zwanzig's 1960 projection-operator method (with Nakajima 1958) for
deriving an exact reduced-system dynamics from the unitary
dynamics of system-plus-environment. \AEight{} cites
Nakajima--Zwanzig in Part~III as the classical antecedent of the
\OGGEP{} partial-trace construction; the framework supplies the
parametric scale ($\tau_c \sim 1/M_{\rm lock}$) at which the
Nakajima--Zwanzig non-Markovian kernel reduces to the GKSL
generator. First cited in \S\ref{sec:DC-6d-reading}.

\end{thebibliography}
\end{document}
