%% v20.2 (August 22, 2026): THE TWENTY-TWO-PAPER DISCLOSURE.
%%   Every reference to a companion paper now carries that companion's own
%%   permanent page -- e.g. https://preprints.arisaka-gg.org/a3-bi6/ -- in place
%%   of the single program address every reference used to share.  Papers that
%%   are not cohort members keep their own locators: the 1985 Kamiokande paper,
%%   the three PsyArXiv works, and the CKM paper still in preparation are
%%   untouched.
%%   Sixty-four companion references carried no locator at all and now carry
%%   one, in this paper's own format.
%%   A11-Origin is posted with the rest, so the twenty-one papers that called it
%%   "in preparation" no longer do, and fifteen printed statements about its
%%   status -- footnotes, table rows, captions and the staged-submission
%%   paragraph -- were corrected to match.  The author's own reservation about
%%   its upper layers is kept verbatim; only its posting status changed.
%%   Two malformed sentences were repaired: A3's "UCLA preprint,." and P2's
%%   title running into the identifier.
%%   No claim, value, figure, table or grade differs from the previous version.
%% ===================================================================
%% ===================================================================
%% v15.9 (August 18, 2026) -- BT-1: THE BIBLIOGRAPHY TITLE PASS.
%%
%% EVERY COMPANION IS NOW CITED BY THE TITLE PRINTED ON ITS OWN COVER.
%%
%% 20 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.
%% ====================================================================
%% ===================================================================
%% v15.8 (August 18, 2026) -- FP-5: THE DISCLOSURE-PREP PASS.
%%
%% The cohort is posted as TWENTY-ONE papers with A11-Origin held back.
%%
%% (1) THE A11 BIBLIOGRAPHY ENTRY HAD NO AVAILABILITY LINE AT ALL.  A
%%     reader was told only "Arisaka-GG-A11-Origin (2026)", with nothing said about where
%%     the paper can be found.  It now reads "In preparation.  To be
%%     posted ... on completion; no result of the present paper depends
%%     on it."
%% (2) AND IT CITED A11 UNDER A TITLE A11 DOES NOT HAVE.  This paper
%%     called it "Origin: Why Are We Here?".
%%     A11 v15.14's own \GGTitlePage reads "The Geometric Inevitability
%%     of Cosmos, Life, and Mind --- The Self-Referential Closure of the
%%     GG-Theory Program".  That was a working title from before the paper was renamed, and
%%     it mattered little while the entry pointed at a downloadable
%%     preprint.  It matters now: the entry says IN PREPARATION, and a
%%     title is the only handle a reader has on a paper they cannot yet
%%     fetch.  Corrected.
%% (3) catalogue -> catalog in printed text (Report 443's pair-list
%%     hole).  16 printed sites; \label / \ref identifiers and changelog
%%     comments protected.
%%     THE FIRST RUN OF THIS SWEEP WAS WRONG AND THE GATE PASSED IT.  The
%%     suffix alternation read (e|ed|es|ing), and Python's | is
%%     first-match-wins, so "catalogued" matched the bare "e" and left the
%%     "d" stranded: the sweep wrote "catalogd".  No spelling audit has
%%     that string in its pair list, so nothing reported it; it was found
%%     by reading a rendered page.  Alternation reordered longest-first,
%%     the suffix map asserted exhaustive, and the loosened check re-run
%%     against text carrying the known defect to confirm it still fires.
%%
%% NO OTHER CONTENT CHANGED.
%% ===================================================================
%% ===================================================================
% v15.5 (August 17, 2026) -- MIGRATED TO STYLE FILE v11.5.
%%
%% FOUR MACROS THAT LIVED IN THE PAPERS COME HOME, AND NOT ONE PAGE
%% MOVES.  This paper's part in that:
%%
%% * \GGShowcaseHead removed; this paper prints the sans form with \bigskip -- the style file default,
%%   and needs no setup line.
%% * \GGtocbreak removed.  It is one of THREE payloads of the
%%   same defect -- content written into the .toc for the full
%%   Contents is replayed by etoc into the Overview.  The
%%   style file's \GGOverviewOfContents clears all three.
%% * the private \GGOverviewOfContents is gone.  It was the
%%   style file's own body with two numbers changed, and P5
%%   and P6 carried it BYTE-IDENTICALLY.  Three
%%   \renewcommands now say the two numbers and the head
%%   skip, and the page is the page it was.
%%
%% The style file is v11.5 (2026/08/17).  Every one of the twenty-four
%% MANIFEST.sha256 files covers it, so changing it meant rebuilding and
%% repackaging all twenty-four -- which is why this was deferred five
%% times and is why it was done in one pass.
%%
%% ASSERTED, NOT ASSUMED: this paper's printed text is line for line
% what v15.4 printed, except for the lines named above.
%% ===================================================================
%% ===================================================================
%% v15.4 (August 17, 2026) -- THE -ise FAMILY.
%%
%% centraliser -> centralizer (3); polariser -> polarizer (3).  Charter 1.1's pair list names these forms; spelling_audit.py's
%% PAIRS list did not carry them, so this paper has reported MIXED
%% PAIRS 0 and ALWAYS-BRITISH 0 at every gate while printing them.  The
%% cohort sweep of August 17, 2026 found sixty-one across twelve of the
%% twenty-four live papers; Report 406 section 6 has the inventory and
%% spelling_audit.py now carries the family as a RULE rather than a list.
%%
%% Substitution is on PRINTED TEXT ONLY.  Changelog comments are the
%% historical record, \label/\ref/\cite arguments are keys, and published
%% titles are quoted rather than written -- all three are masked, and every
%% frozen title is asserted present and unchanged after the edit.
%% NOTHING ELSE IN THIS PAPER CHANGES: no claim, no number, no float, no
%% label, and no page.
%% ===================================================================
%% ===================================================================
%% v15.3 (August 17, 2026) -- THE OVERVIEW'S FORCED BREAK, REMOVED.
%%
%% Page 8 of v15.2 opened with "Part XV. Common Objections and Responses
%% 309" and "Part XVI. References 328" above its List of Figures heading
%% -- two orphaned lines under the wrong running head.  I read that page
%% during the v15.2 ship and did not see it; it was found by scanning the
%% whole cohort after the identical defect turned up on P6-Proton.
%%
%% The cause: \addtocontents{toc}{\protect\newpage} before Part XV, which
%% keeps the FULL Contents from orphaning "Part XVI. References" -- and
%% which etoc replays into the Overview of Contents, because the Overview
%% is built from the same .toc file.
%%
%% The raw \newpage becomes \GGtocbreak, conditional on a flag the
%% Overview clears: the Contents keeps its break, the Overview loses it.
%% A local \GGOverviewOfContents is added at the same time, tightened for
%% sixteen Parts, exactly as on P6-Proton v15.3.
%%
%% NO PROSE CHANGED.  No word, no figure, no table, no equation, no
%% number.  This is a page-break repair and nothing else.
%% ===================================================================
%% ------------------------------------------------------------
%% v15.1 -> v15.2, August 16, 2026.  THE EXPANSION, AND THE
%% SHOWCASE PAGE.  Commissioned by the author: P5 absorbs what
%% W2-Holonomy and S2-Leptogenesis developed out of it, W2 being a
%% white paper and S2 a supplementary study, while P5 carries all
%% the theorems and the new concepts itself.
%%   FOUR PLATES IMPORTED: W2's anatomy (the circle remembers, the
%%   interval forgets); S2's door (only a neutral particle can be
%%   its own opposite), RECOLOURED to the house palette; S2's two
%%   unitarity triangles at true relative area; S2's DeltaB in 3Z
%%   selection rule.  Three unchanged, one recoloured, all four
%%   attributed to the paper that OWNS the result rather than the
%%   one that drew it first.
%%   FIVE NEW SECTIONS: the anatomy of a theorem (why CPT is exact
%%   and C, P, T are not); the door drawn, inside the existing
%%   self-conjugate section; why the quark sector cannot pay; the
%%   selection rule that does both jobs, with a prediction that can
%%   only fail; and the three horizons of sameness.
%%   TWO DEPARTURES FROM THE APPROVED PLAN, BOTH FOUND IN THE DOING.
%%   (1) fig:chain is NOT recoloured.  Its three colours already
%%   carry a load-bearing meaning -- orange marks the block resting
%%   on conjecture C-P5-C -- and recolouring to the fibre convention
%%   would erase an honesty device to gain a reading.  The caption
%%   is extended with the fibre reading instead, so both survive.
%%   (2) S2's flavour-sum bar chart is NOT imported.  It carries
%%   three flavoured epsilon values computed at S2's own inputs;
%%   this paper does not derive them and S2 is an internal study
%%   with no bibitem here and none on the arXiv path, so they would
%%   print under no citation.  The concept is carried in prose and
%%   the chart is not, and the omission is stated where it falls.
%%   THE SHOWCASE PAGE: the anatomy above, the door below -- the
%%   machinery, then what it decides.
%% NEW PROSE IN THIS SHIP: five sections, four figure captions, one
%% showcase headline and caption, and one caption extension on
%% fig:chain.  NO EXISTING SENTENCE OF THE PAPER IS REWRITTEN, no
%% float is deleted or renumbered, no numerical value is changed,
%% and no grade is promoted.
%% ------------------------------------------------------------
%% ===================================================================
%%  P5-CPT v3.52 (August 12, 2026) -- the consistency sweep (Report 309,
%%  author-approved).  (1) "eighteen companion treatments" -> twenty-one
%%  at the three roster-count sites.  (2) Option A: the heavy triplet's
%%  equilibrium yield displayed (1.25/g*), eta_B central 7.34e-10 (+20%),
%%  headline sites quoted at the display standard ~7e-10 with the x3
%%  band (2.4-22)e-10; audit table gains the yield row.  (3) "lands on
%%  the ledger value" retired at both sites for the bracket statement.
%%  (4) C4 prose conditioned to grade; C4 gains the external-source gap
%%  sentence; C6/kill route (iii)/the window numbers/roadmap row 9 carry
%%  the parent's measured E = 7.1 (68% [4.4, 11.4]).  (5) Omega_b bridge
%%  recomputed on the new central (0.0268; +16% at the corridor h,
%%  through zero near the ladder branch) with the owner decomposition;
%%  "0.3%" -> 0.7% at three sites.  (6) J_PMNS layer annotated.  (7) QD
%%  step: size ledger-quantized, direction/necessity asserted per site
%%  (A9's frozen completion class displays no sign rule).  (8) The
%%  factor-13 convention spread collapsed to the single-convention ~12.
%%  LHCb gamma (62.8 +/- 2.6, Nov 2025) verified current -- unchanged.
%% ===================================================================
%% ===================================================================
%%  P5-CPT v3.51  (August 6, 2026)  -- the cohort-wide figure-size sweep.
%%  17 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.
%%  One exception: fig:parts-roadmap-p5 is capped at 0.83 rather than
%%  its computed 0.936.  That plate is 172 x 231mm, so at the house
%%  target it stands 200mm tall and the clearpage-isolated minipage
%%  that carries it no longer fits its page with the caption -- it
%%  overflowed by 35pt and left the preceding page blank.  0.765 is
%%  the tallest that fits, measured from that overflow; its type lands
%%  at 6.4pt rather than 7.79.  Trimming the ten-line caption would
%%  buy the rest, as it did for GG-P6-Proton's sixteen-Part roadmap,
%%  and is left for the author.
%%  Per-figure table and recipe: [v13-support]/allsweep.py and
%%  [v13-support]/figure_census.json.
%% ===================================================================
%% ===================================================================
%%  P5-CPT v3.50  (August 5, 2026)  -- the cohort funnel at the house size.
%%  The central-claim funnel was drawn with 164mm boxes and then
%%  left at natural size, so its \footnotesize landed 16% larger than the roadmap
%%  and cohort-map plates beside it.  The plate target is now
%%  0.951\textwidth, computed rather than chosen: the picture measures
%%  175.0mm wide compiled standalone, and the house type size is set by
%%  GG-P3-Electron Figure 3 (0.93\textwidth on 171.1mm, \textwidth =
%%  159.2mm, so type at 0.865 of natural size), which for any picture
%%  of width W means f = W / 184.0.
%%  Nothing inside the picture changes: no font command, no box
%%  dimension, no text, no relative position.
%%  Recipe and the measured table for all fourteen papers:
%%  [v13-support]/funnel_sweep.py.
%% ===================================================================
%% ===================================================================
%%  P5-CPT v3.49  (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.
%% ===================================================================
%% ===================================================================
%%  P5-CPT v3.46  (July 30, 2026)  -- Phase-2 references completion.
%%  146 of 179 bibitems carried no first-citation pointer -- the whole B
%%  (external literature) section.  Each now closes with "First cited in
%%  S<label>", the label resolved from the entry's first body citation to
%%  its nearest labelled ancestor section.  Charter sec.2 reference standard.
%%  No entry text, key, or citation target changed.
%% ===================================================================
%% ===================================================================
%% v3.45 (July 25, 2026): P3 PART-V POINTER (one sentence)
%%  The positron-as-conjugate-readout episode now closes with a
%%  pointer to the electron paper's dedicated treatment: its Part V,
%%  "The Origin of Positrons" (its sixth question, Q6), which imports
%%  the origin of the twin at theorem grade from the present paper
%%  and reads the positron exactly as developed here. No other
%%  change anywhere.
%% ===================================================================
%% ===================================================================
%% v3.44 (July 24, 2026): COMPANION SWEEP -- P3 DEBT-REGISTRY RENAME
%%  The electron paper's open-problem registry is renamed from the O-
%%  prefix to D- ("registered debt"), the cohort reserving O-Suffix-N
%%  for numbered objections. The six sites here that quote the
%%  conjugate-readout debt (including the named-statement deck row)
%%  are renamed to match. No other change anywhere.
%% ===================================================================
%% ===== v3.43 (2026-07-20): H-PHIGEO-1 CANDIDATE MECHANISM =====
%% Ship, author-approved; spec: internal report HPhigeo1-Transport
%% 2026-07-20 + v2 extension 2026-07-21; harmonized with A10-Cosmos
%% v13.6 Appendix I.3, the derivation home.  The flavour-measurement
%% regulator installed at candidate grade across the paper's
%% H-phigeo-1 register: statement, closed form, physics intuition,
%% pointer -- the full kinetic derivation stays in A10 I.3.
%% (1) sec:resonance-faced, NEW block 'The mechanism, identified
%% (candidate grade)': Gamma_eff = Gamma_tau = c_tau y_tau^2 T
%% (2-4 MeV vs hypothesized ~3 MeV); zero-dial closed form
%% DeltaM/Gamma_eff = 1/(35 c_tau y_tau^2) = (2.7-5.3)e4 vs
%% hypothesized 3e4; landings stated honestly (single-T 0.28-0.55
%% of observed; T-integrated 0.05-0.30, favourable corner ~0.6,
%% always from below; the deficit named as condition C6);
%% four-regulator landscape spans sixteen orders, only the
%% flavour-measurement rate lands; conditions C1-C6 listed with
%% statuses; two clean kills + the soft C6 kill registered;
%% H-phigeo-1 not retired, sharpened.  (2) The g^2T rescue sketch
%% SUPERSEDED (computed to overshoot ~3e4, flavour-blind besides):
%% re-pointed to c_tau y_tau^2 T at the regime-map green arrow +
%% caption, the candidate-resolutions passage, F.3, and F.6 Step 3.
%% (3) Part VIII step 5 (sanctioned amendment): mechanism clause
%% added, every grade word kept.  (4) Mention-site sweep: minimal
%% mechanism clause at the audit caption, Part-VIII status ledger,
%% open-problem 5, symbol table, F.3/F.6/F.6.1 target-value sites,
%% closure-economy summary, O-P5-40, O-P5-41; bare naming sites
%% left untouched.  (5) Promotion-roadmap row 9: mechanism recorded
%% at candidate grade (mirrors A10's row-9 clause).  eps_res value,
%% eta_B, all falsifier tables, Part IX, abstract, exec summary,
%% frozen map, Part-I close: byte-identical (abstract at 1898 chars
%% against the 1920 cap: no room for the mechanism clause; noted).
%% v3.42 -> v3.43.  Script: p5_v3.43_mechanism.py.
%% ===================================================================
%% Arisaka-GG-P5-CPT -- version history
%% v3.42 (July 20, 2026): STAGE 5A HOUSEKEEPING SHIP (Wave-2
%%   housekeeping batch, ship 1).  (1) Cover-page date sync: the
%%   \GGTitlePage third-line date literal 'July 18, 2026' -> 'July
%%   20, 2026', matching \PaperDate.  (2) Apparatus count: executive
%%   summary 'twenty-two tables' -> 'thirty-two tables' (verified
%%   against the rendered PDF: Tables 1-32; 28 table + 4 longtable
%%   environments); 'twenty-one figures' verified exact (Figures
%%   1-21) and unchanged.  Physics, theorem keys, numbers, values,
%%   grades: unchanged.  Script: p5_v3.42_stage5a.py.
%% v3.41 (July 19, 2026): FOUNDATIONS-BRANCH MEMBERSHIP (Wave 1 of the
%%   19-paper campaign; edit script p5_v3.41_wave1.py). The paper's
%%   self-description converted from satellite standing to full
%%   membership: member of the nineteen-paper cohort, Foundations
%%   Branch, the A9-SM--A10-Cosmos interface. (1) Title block:
%%   \PaperStatus{Foundations Branch}; pdfsubject and arXiv comment to
%%   member language (nineteen-paper). (2) All reader-facing "satellite"
%%   self-descriptions converted (abstract, exec summary, Parts
%%   II/IV/IX/XI-XVI sites, source ledger, suffix roster, objections
%%   O-P5-27/43/53/62, references intro); "collider satellite" (P1-KK)
%%   -> "collider companion" (4 sites). (3) Cohort-sense counts:
%%   seventeen -> nineteen/eighteen (7 sites); the suffix roster adopts
%%   the canonical nineteen-paper form (Real (P4); CPT (P5, the present
%%   paper)). (4) P4 designator: \PaperPFour -> GG-P4-Real; bib key
%%   ArisakaP4Twistor -> ArisakaP4Real (11 sites); the P4 bibitem takes
%%   the P4-Real v3.2 cover title; \ASix renewed to GG-A6-GRIP (7
%%   render sites, matching the bibitem that already reads A6-GRIP).
%%   (5) Part I close standardized to the house architecture: NEW
%%   closing section "The cohort inheritance and the roadmap of the
%%   sixteen Parts" replacing the former "The two parents..." and
%%   "Roadmap" sections (labels sec:inheritance / sec:roadmap kept):
%%   member-language cohort prose + "The Foundations Branch" paragraph
%%   from this paper's seat, closing with the frozen v1.4 nineteen-paper
%%   cohort map (verbatim master; internal node names MP4/MP5), and a
%%   new sixteen-Part roadmap walkthrough closing with the NEW roadmap
%%   figure (fig:parts-roadmap-p5; Parts IV-VIII highlighted as the
%%   five-station chain per fig:p5-central-claim). Part-end summary
%%   kept verbatim. (6) Exec-summary figure count: sixteen ->
%%   twenty-one (the two new figures; also closes a pre-existing
%%   drift, actual count had reached nineteen). Physics, numbers,
%%   grades, falsifiers, H-phigeo-1 sites, the Part-VIII step-5 block,
%%   and Part IX content: unchanged (protect-asserted).
%%   Script: p5_v3.41_wave1.py.
%% v3.40 (July 19, 2026): PHI_GEO INSTALL (ship, author-approved; spec:
%%   internal report PhiGeo-Projection 2026-07-18). The flavoured
%%   resonant-leptogenesis recomputation of the phi_geo projection
%%   installed. (1) Chain step 5 regraded asserted -> derived-conditional
%%   on the NEW named hypothesis H-phigeo-1 (finite-temperature partial
%%   resonance, Delta M/Gamma_eff ~= 3e4 delivering throughput e^-K/2);
%%   provenance stated: U_PMNS cancels in the unflavoured invariant, the
%%   dial-free real-R door gives unflavoured eps = 0, at the TeV scale the
%%   flavoured asymmetries eps_{i,alpha} are the operative object; driver
%%   alpha_31 = pi (sets the sign of matter; J_PMNS-blind at the
%%   few-percent level); nonzero at m_1 = 0 (not riding on the QD layer);
%%   honest bracket printed (peak overshoot ~3e4, vacuum undershoot
%%   ~4e6); eps_res ~= 2.5e-6 kept as the H-phigeo-1 TARGET value, not a
%%   derived output, at every site (step 5, audit row 5, flowchart node,
%%   symbol table, F.3/F.3.3/F.6.1, transport-receipt chapter).
%%   (2) Audit row 5: asserted -> derived-cond. (H-phigeo-1); caption
%%   updated (the m_1/m_3-order-factor language withdrawn as a
%%   reconstruction artefact). (3) Promotion-roadmap row 9 rewritten per
%%   the recomputation (open task: finite-T flavoured transport testing
%%   H-phigeo-1; acceptance band +-O(1) in eps_eff). (4) Seam-Yukawa/
%%   washout convention reconciled at the Gamma_N and K_wash sites (steps
%%   3/6, F.2.3, transport chapter): factor ~13 in Y_nu^2 between the
%%   seam m-tilde ~= 8.4 meV and the washout m-tilde = m_3; declared and
%%   priced as O(10) inside the order-of-magnitude register.
%%   (5) sec:resonance-faced: H-phigeo-1 stated beside C-P5-C (the sharp
%%   form of the condition is partial resonance; the exact-peak form
%%   overshoots by ~3e4); regime-map caption and success criterion
%%   restated on the partial-resonance target; closure-economy summary,
%%   open-problem 5, Part-VIII status ledger, T-P5-5 conditional clause,
%%   and front-cover grade phrase aligned. (6) O-P5-40/41 responses
%%   updated to the new state (flavoured computation displayed; open item
%%   H-phigeo-1). eps_res value, eta_B, and all falsifiers unchanged.
%%   Script: p5_v3.40_phigeo_install.py.
%% v3.39 (July 18, 2026): REGISTER ALIGNMENT (ship).
%%   Hostile-review action plan, section 4: the headline language
%%   (abstract, executive summary, Fig. 2 audit bar, Step-language,
%%   cross-paper grade language) aligned with the paper's own
%%   grades.  No claim weakened below its grade; none priced above
%%   it.  (1) Abstract: "exact geometric values" -> "exact lattice
%%   values ... the first reached by one frozen, pre-registered
%%   completion step"; the antiparticle "derived at its stated
%%   grade"; closing register sentence added (every claim carries
%%   its grade -- derived, conditional, candidate, or asserted --
%%   and the paper's tables say which).  (2) Exec summary: the
%%   eighty-two objections re-described honestly (adversarial
%%   questions constructed in-house across eleven referee
%%   perspectives, answered with pointers into the paper); "every
%%   input forced" -> "every input inherited at its stated grade",
%%   with the asserted projection step named beside the named
%%   conjecture; the parity projection priced inline (asserted, per
%%   Part VI's ledger); the QD completion priced at its site ("the
%%   codebook's one asserted quantum-defect step"); "The forced
%%   values" -> "The values, each at its grade".  (3) Fig. 2 (the
%%   27-row audit bar): "SM group from one hand" and "anomaly
%%   freedom" tagged (c) in the derived legend; caption sentence
%%   added -- rows tagged (c) are conditional on named parent-paper
%%   items, with the parents listed in the inheritance and
%%   promotion-roadmap appendices.  (4) Part VII Step 1: "inherited;
%%   derived-conditional upstream" -> the parent codebook's own
%%   grade for the cascade inputs (Asserted, chosen post-hoc)
%%   carried here without upgrade, stated at the site.  (5) Part VI
%%   first claims: the projection matrix, the Z2's existence, and
%%   the chirality choice each stated "asserted, per Table 8"
%%   (tab:parity-ledger) where first claimed.  (6) Part III: one
%%   register sentence at theorem visibility -- Theorem 0 is, in
%%   its home paper's own words (Remark 0.5), a constitutional
%%   forcing argument rather than a theorem of constructive quantum
%%   field theory, and every "forced" downstream inherits that
%%   register.  (7) NEW Part VII subsection + table
%%   (tab:qd-disclosure): every QD completion used here or
%%   inherited, with size, unit, and direction vs data (incl. the
%%   away-from-data Cabibbo modulus); the standing defence (bounded,
%%   quantized, frozen class, incapable of further adjustment) and
%%   the audit counterweight (heterogeneous class, asserted
%%   necessity, data-favouring direction record) printed together.
%%   (8) Part IX: the shore's naming claim scoped -- the analytic
%%   structure itself is classical (Paley-Wiener, KMS, tube
%%   analyticity); what is claimed is the naming; endowment-two
%%   scope note added (constitutional arena vs the expanding
%%   universe of the cosmology companion).  (9) Priority claims
%%   audited: "the first framework" and "for the first time in
%%   seventy years" now carry "to the author's knowledge"; the
%%   law-level-arrow strength claim kept as the standing point,
%%   already costed in Part IX.  (10) Part XV preamble: the
%%   objections named as self-authored stress tests in referee
%%   personas, published so real referees can start where they end,
%%   now including the findings of an adversarial multi-reviewer
%%   audit of this version's numerics.  No numbers, tables of
%%   values, theorem statements, or falsifiers changed.
%% v3.38 (July 18, 2026): NUMERICS + FALSIFIER INTEGRITY (ship).
%%   Hostile-review action plan, sections 2-3, all items. NUMERICS:
%%   (1) J_CKM printed as ONE stated-layer evaluation: PDG-class
%%   moduli (s12=0.2250, s23=0.0418, s13=0.0037; prefactor 0.9735)
%%   with the QD phase -> 3.1e-5; the 2.7-3.2e-5 band restated as the
%%   spread across the framework's evaluation layers (exec summary,
%%   WE II, Part VII Step 6, both confrontation tables, symbol
%%   table, E.6.3). (2) eps_res regraded derived-cond. -> asserted at
%%   chain step 5 and audit row 5; the undisplayed m1/m3-order
%%   suppression in the J_PMNS -> eps_res projection named; new
%%   promotion-roadmap row 9 (the phi_geo projection, venue A10/this
%%   paper). kappa printed as 1/K_wash = 2.0e-2 with the O(1)
%%   adoption kappa = 1.0e-2 owned in the audit caption, the
%%   throughput text, the transport chapter, and F.6.1 (the
%%   displayed identity uses that freedom). (3) The +1.17 raw pull
%%   disclosed at every site as carrying the one-lattice-step theory
%%   width pi/35 in quadrature (raw +4.2 sigma on experimental error
%%   alone); E.6.1 gains the missing step (Step 4). (4) sin 2beta
%%   recomputed at the stated layer (lattice phase 66.857 deg,
%%   PDG-class moduli s12=0.2250, s23=0.0418, s13=0.0037): 0.708,
%%   pull -0.1 sigma, still watched, layer stated for auditability;
%%   all -1.3 sigma sites updated; the tense watched number is now
%%   the direct gamma (lattice triangle tracks the QD phase, ~66.9
%%   deg, vs LHCb 62.8+-2.6 -> +1.6 sigma, printed per the
%%   no-cosmetic-delay rule). (5) WE IV asymmetry corrected to
%%   A_CP ~ +8% at the lattice point (0.28 x 0.283); sign statement
%%   fixed (neutrinos ENHANCED for sin delta < 0, per H.3's
%%   formula); matter-effect caveat -> problem H.3.2; the two other
%%   28% sites annotated with the lattice-point value. (6)
%%   KamLAND-Zen band 36-156 -> 28-122 meV at both table sites and
%%   H.4; H.5.2 rescaled to the matched pair: T(1.72 meV) ~
%%   1.0e29-1.9e30 yr, one to two orders beyond the ton scale
%%   (reach sentences and the KamLAND-Zen reference note updated).
%%   (7) m_bb evaluation layers stated: first-order 1.72 meV
%%   (registered), QD layer 1.79 meV (H.4.2 arithmetic corrected);
%%   layer-robust-null sentence at the registered-null row and the
%%   Part VII / Part X sites. (8) J_PMNS normalized to -9.4e-3 at
%%   all sites; H.6.2 notes the prefactor spread (-9.4 to -9.5);
%%   H.6.1 surd decimal -0.28284 -> -0.28278 (and -0.282 -> -0.283);
%%   "quantum-descent" -> "quantum-defect"; arXiv comment ~300 pp;
%%   H.6 splittings pulls printed honestly (solar -0.6 sigma vs
%%   NuFit 7.49+-0.19, -1.0 sigma vs JUNO 7.50+-0.12; atmospheric
%%   +0.7 to +1.9 sigma vs the 2.51-2.53e-3 class). FALSIFIER
%%   ARCHITECTURE: (a) new derived-splittings kill row appended to
%%   tab:falsifiers (row 9) and the Appendix N catalogue (JUNO,
%%   sub-percent on both splittings, ~2031); statement-deck
%%   inventory 8 -> 9 falsifier rows. (b) The subdominant 3pi/2
%%   assignment branch pre-registered as dead, bindingly, at the
%%   Part VII degeneracy site and in the falsification catalogue.
%%   (c) gamma printed now and watched: present-frontier paragraph,
%%   conf-quark, the refresh section, the falsification narratives,
%%   and the watched rows of the catalogue. (d) Conditioning
%%   sentence on sensitivity dates added to the falsifier-table and
%%   kill-test-timeline captions. (e) NuFit 6.0 variant pinned
%%   (IC19 without SK atmospherics; the with-SK variant flips the
%%   theta_23 octant, moves delta to 212 deg, lattice point at -0.4
%%   sigma). (f) The joint (w, Sum m_nu) DESI exposure registered
%%   as one correlated test, not two independent outs. No forced
%%   value moved; every numerical change states its evaluation
%%   layer.
%% v3.37 (July 18, 2026): WHOLE-DOCUMENT READ-THROUGH (ship).
%%   Self-consistency, Part-to-Part flow, and house-voice pass
%%   before hostile review.  (a) Part I Roadmap: the missing
%%   Part IX (What is time?) entry restored between the Part
%%   VIII and Part X entries.  (b) Part VIII Part-end summary:
%%   stale handoff to "the roads" replaced by a handoff to Part
%%   IX (time).  (c) Part X Part-end summary: handoff now points
%%   to Part XI (the instruments) instead of skipping to the
%%   Discussions housekeeping.  (d) Part X opening: the account-
%%   complete list now includes Part IX's four endowments.
%%   (e) Part XIII Part-end summary expanded to do both jobs
%%   (summary + handoff).  (f) Naming-convention appendix:
%%   objection count "fifty" corrected to eighty-two.  (g) Voice,
%%   light touch: bridge sentences opening Part IV (from Part
%%   III's closing question) and Part VI (from Part V's promise);
%%   one physical picture opening the transport preliminaries.
%%   No physics claims, grades, numbers (beyond (f)), theorem
%%   statements, tables, abstract, executive summary, or
%%   appendix solutions touched.
%% v3.36 (July 18, 2026): FRONT-MATTER REFINEMENT (author-directed
%%   ship). (a) Abstract REWRITTEN plain-English on page 1: the
%%   question, the orientation answer, CPT as the undetectable
%%   total reversal, the antiparticle as the other reading of a
%%   real circle (spin-1/2 and exclusion located, candidate
%%   grade), time's four endowments (Part IX), the six-per-ten-
%%   billion receipt with the exact CP phases (66.9/196.4 deg),
%%   and the dated mortality clause; under the 1920-char limit.
%%   (b) Executive Summary refactored: every paragraph short; the
%%   theorems paragraph split four ways (handshake+Wick quartet /
%%   mirrors T-P5-1/2 / flavours T-P5-3/4 / matter chain T-P5-5);
%%   light de-jargonizing; spin-statistics and the time Part named
%%   in the apparatus (sixteen Parts); NEW closing philosophy
%%   paragraph ending on the falsifiability creed. (c) Overview of
%%   Contents trimmed to one line per Part (local \GGPart override
%%   writes short toc part entries) and every caption given a
%%   <= 60-char short LoF argument: Overview + List of Figures on
%%   one page. Body Parts untouched.
%% v3.35 (July 18, 2026): TIME DEFINITION WOVEN DOWNSTREAM
%%   (author instruction): Part X gains a closing section "The
%%   sixth comparison: what each road says time is"; Part XII's
%%   summary registers the four-endowment definition; Part XIII's
%%   established-arc and last word carry the time answer.
%% v3.34 (July 18, 2026): PART IX EXPANDED (author commission):
%%   NEW section on the problem of time in quantum gravity and its
%%   dissolution-by-architecture (honest register: the cost,
%%   Theorem 0's staticity, stated); worked KMS/quartet arithmetic
%%   (single-mode drill: correlator computed, KMS periodicity
%%   derived, the four faces read off); two figures: the four
%%   endowments diagram and the shore half-plane sketch.  Part IX
%%   now ~10 pages.  GGPart subtitle + Part-end summary extended.
%% v3.33 (July 18, 2026): NEW PART IX, "WHAT IS TIME?" (author
%%   commission).  The four-endowment definition consolidated from
%%   Part IV's third-face physics (order/cone, repetition/Killing,
%%   arrow/OGGEP, shore/positivity); the standard accounts of time
%%   reviewed and honoured (classical+relativity, thermodynamics,
%%   QM, QFT+SM, string/QG); the OGGEP/GDOS/GRIP arrow stated as
%%   the first law-level irreversibility with custody kept; energy
%%   and entropy as the two currencies of one direction (GFF = GEP
%%   + CEP, A6); one-table comparison; cousins engaged
%%   (Connes-Rovelli thermal time, Page-Wootters, Boyle-Turok);
%%   plain-language philosophical close.  Former Parts IX-XV are
%%   now X-XVI (64 tilde-forms, 2 ranges, 4 table cells shifted;
%%   preface day-plan extended).  Seven new bibitems with
%%   paragraph relevance: Augustine, Boltzmann, Connes-Rovelli,
%%   DeWitt, Eddington, McTaggart, Page-Wootters.
%% v3.32 (July 18, 2026): PART XV PHASE-2 (author instruction:
%%   every Relevance note a solid paragraph, as in the parent
%%   papers).  All 148 sub-paragraph notes expanded to full
%%   paragraphs (3-5 sentences: what the work established, its
%%   significance, its precise consumption point in this paper);
%%   every pre-existing pointer preserved; the six "See X" stubs
%%   (Kay, Pati-Salam, Pauli, Salam, Yanagida, Yoshimura) given
%%   their own content with companion pointers retained.
%%   COLLATERAL: four objection-number references left stale by the
%%   v3.31 renumbering corrected (Objection 20->31, 44->69, 21->34;
%%   O-P5-20 -> O-P5-31).
%% v3.31 (July 18, 2026): PART XIV PHASE-2 (author instruction:
%%   update the Q&A, add >= 30 more with profound physics).  +32
%%   objections across the ten referee sections plus a NEW eleventh
%%   section, "The quantum-foundations referee" (letter Z; alphabet
%%   exactly filled).  New exposures registered: the cosmological
%%   mass-sum (DESI < 64 meV vs derived 59.2 meV --- a new
%%   independent kill/confirmation channel), light steriles as a
%%   T-P5-4 uniqueness kill, AMS antihelium as a Part-VIII kill,
%%   antimatter gravity (ALPHA-g) as a registered retrodiction.
%%   Structural cousins engaged without polemic: Boyle-Turok
%%   CPT-symmetric universe, Nelson-Barr, modular flavour symmetry,
%%   swampland folklore.  Stale O-38 (courses/C.3) re-aimed at
%%   Appendix F.3.  All \PObj renumbered 1..82; count sites updated
%%   (fifty -> eighty-two; ten -> eleven referee communities).
%%   New bibitems: BoyleFinnTurok2018, ALPHAg2023.
%% v3.30 (July 18, 2026): BLOCK III REGISTRIES (the ledger-and-
%%   registry appendices, A10's O-T analogues): I internal source
%%   ledger (9 papers, role + relevance); J cross-paper inheritance
%%   table (compact; 11 rows, keyed to P5 Parts); K consolidated
%%   benchmark audit (10 problems, mechanism + kill switch); L
%%   named-theorem-and-conjecture deck (every formal named
%%   statement with label and one-line summary; inventory verified
%%   programmatically); M named-conjecture promotion roadmap (8
%%   targets with venue and status); N falsification-test catalogue
%%   (8 tests with precision and venue, keyed to the falsifier
%%   table and the kill-test timeline).  Reading guide becomes O;
%%   Part XIV letters shift (no letter-keyed inbound refs exist).
%%   Front cover / GGPart Block III wording updated.
%% v3.29 (July 18, 2026): BLOCK I AND THE PART-XIII FRONT COVER TO
%%   THE A9/A10 STANDARD (author instruction).  Front cover gains
%%   the two stylistic commitments and the Prerequisites +
%%   recommended-textbooks shelf (A10 model, adapted to the five
%%   chapters).  Appendix A rebuilt as the grouped abbreviations
%%   longtable (cohort groups 1-6 shared with A9/A10; satellite
%%   terms; standard physics) + the canonical acronym-hierarchy
%%   figure (fig:abbrev-flow, verbatim from A1/A10).  Appendix B
%%   rebuilt as the grouped key-symbol longtable (bulk/ledger,
%%   scale chain, handshake, flavour, neutrino, leptogenesis,
%%   laboratory).  Appendix C rebuilt in the A10 format with the
%%   prefix itemization, the seventeen+two suffix roster, and the
%%   satellite's letter/number usage stated.  Collateral: Appendix
%%   D shelf line updated to "complete solutions included" (stale
%%   since v3.18).
%% v3.28 (July 18, 2026): APPENDIX H -- THE NEUTRINO'S CENTURY,
%%   FROM SCRATCH (Part XIII campaign, installment 5 -- the last
%%   course dissolved; Block II now five Step-graded chapters, zero
%%   courses).  The course's lecture bodies retained and
%%   restructured to the locked architecture: H.1 oscillations from
%%   two-state QM; H.2 the PMNS matrix and the leptonic Jarlskog;
%%   H.3 the appearance asymmetry; H.4 Dirac/Majorana/seesaw and
%%   double beta; H.5 Step 2 (what the campaign cannot price); H.6
%%   Step 3 (the lattice point end to end).  All 18 problems
%%   upgraded to complete worked solutions + intuition.  TWO LOOSE
%%   COURSE ANSWERS CORRECTED: JUNO solar first-maximum at 3.1 MeV
%%   (was "~2 MeV"); J_PMNS^max uncertainty ~2% dominated by
%%   theta13/theta12 with theta23 suppressed by near-maximality
%%   (was "~5% dominated by theta23").  Five bib relevance notes
%%   re-aimed; front matter five chapters; ninety problems, all
%%   with complete solutions; "Five shelves" -> "Six shelves".
%% v3.27 (July 18, 2026): APPENDIX G -- THE ANTIPARTICLE, FROM
%%   SCRATCH (Part XIII campaign, installment 4; locked standard).
%%   G.1 the Dirac sea, derived then dismantled as theorems; G.2
%%   Feynman-Stueckelberg and the four contours; G.3 Weinberg's
%%   causality theorem with its three undischarged axioms; G.4 the
%%   bulk spinor algebra as textbook math (mod-8 reality clock,
%%   B Bbar = -1 in (1,5), anomaly flip); G.5 Step 2 (what the
%%   three orthodox answers cannot say, graded on the three-depths
%%   ruler); G.6 Step 3 (the handshake T-P5-A/B/C with the mode
%%   computation, the forced conjugate content, and the readouts:
%%   positron, Wheeler, Hawking partner, custody).  Antiparticle
%%   course DISSOLVED (18 problems absorbed/upgraded to complete
%%   solutions); four inbound Lecture-D pointers re-aimed; front
%%   matter now four chapters + one course; ninety problems exact
%%   (18*4 chapters + 18 neutrino course = 90).
%% v3.26 (July 18, 2026): APPENDIX F -- THE ORIGIN OF THE MATTER
%%   EXCESS, FROM SCRATCH (Part XIII campaign, installment 3; locked
%%   standard as Appendices D and E).  F.1 Sakharov's three
%%   conditions derived (equilibrium theorem, C-vs-CP toy, the SM's
%%   double failure); F.2 the seesaw (Weinberg operator, block
%%   diagonalisation, Casas-Ibarra); F.3 the CP asymmetry (tree-loop
%%   interference, Covi-Roulet, Davidson-Ibarra bound derived,
%%   resonant regime); F.4 transport (Boltzmann pair, m_star,
%%   washout, sphalerons with 28/79 derived, entropy); F.5 Step 2
%%   (six things the SM+seesaw cannot price); F.6 Step 3 (the
%%   ledger's receipt: the eight-step chain, the throughput
%%   identity, eta_B = 5.87e-10 at -4.1%, D* grade, C-P5-C carried
%%   honestly).  Leptogenesis course DISSOLVED (18 problems
%%   absorbed/upgraded to complete solutions); six inbound Lecture-C
%%   pointers re-aimed; front matter now three chapters + two
%%   courses; ninety problems exact (18*3 chapters + 18*2 courses).
%% v3.25 (July 18, 2026): APPENDIX E -- THE FLAVOUR SECTOR, FROM
%%   SCRATCH (Part XIII campaign; built to the locked standard of
%%   the memory addendum in ONE ship: Step-1/2/3, displayed
%%   derivations, 18 problems w/ complete solutions + intuition,
%%   even paragraphs via the v3.22 algorithm run inline, three-level
%%   citations, chapter shelf).  E.1 Yukawas -> CKM diagonalised;
%%   E.2 rephasing invariants + Jarlskog + unitarity triangle;
%%   E.3 kaons: epsilon and epsilon-prime derived, KTeV/NA48 double
%%   ratio; E.4 B factories: sin2beta interferometry derived;
%%   E.5 Step 2 (what the SM cannot price); E.6 Step 3 (the
%%   pentagon, with the confrontation rows).  Flavour-holonomy
%%   course DISSOLVED; main-text "Lecture B.6" pointers re-aimed;
%%   front page/curriculum/exec updated (two chapters + three
%%   courses; ninety problems remains exact 18+18+54).  New
%%   bibitems: Cabibbo1963, Wolfenstein1983.
%% v3.24 (July 18, 2026): STEP NUMBERING SIMPLIFIED (author
%%   instruction: 1a/1b/1c/1d sub-letters redundant given D.1-D.4).
%%   Subsection titles now "Step 1 ---" for D.1-D.4; Step 2 / Step 3
%%   unchanged; all in-body and bibliography "Step 1x" references
%%   converted to subsection cross-references.
%% v3.23 (July 18, 2026): APPENDIX D INLINE CITATIONS (author
%%   instruction): three levels threaded through the chapter ---
%%   internal P5 cross-references (Parts II-X, falsifier and
%%   timeline tables, T-P5 names), cohort citations (A2 sec.20,
%%   A5-GDOS, A8-QM; separate citealp per charter), and external
%%   references (Sakurai, Bigi-Sanda, Sozzi, Branco-Lavoura-Silva,
%%   Peskin-Schroeder, Wu 1957, Luders 1954 + Pauli 1955 at the
%%   Lagrangian road, NuFit, PDG).  Miss-tolerant seam insertion;
%%   all keys verified in the bibliography.
%% v3.22 (July 18, 2026): APPENDIX D EVEN-PARAGRAPH PASS (author
%%   instruction: v3.21's hand-picked splits left uneven lengths).
%%   Systematic algorithm: every prose paragraph > ~700 chars in
%%   app:discrete-qft split at sentence boundaries into ~400-550
%%   char blocks (4-6 printed lines); prose stretches between
%%   displayed equations processed likewise; tabular blocks
%%   untouched.  Whitespace only; no wording changes.
%% v3.21 (July 18, 2026): APPENDIX D READABILITY PASS (author
%%   instruction: undergraduate-textbook rhythm; paragraphs were
%%   too long).  Whitespace-only within app:discrete-qft: worked
%%   solutions split at Step 2/3/4 boundaries; Physics-intuition
%%   passages given their own paragraphs; ~28 long expository
%%   paragraphs split at natural sentence seams.  No wording
%%   changes.
%% v3.20 (July 18, 2026): APPENDIX D EQUATIONS THREADED (author
%%   instruction: v3.19 passages too prose-heavy; A9/A10 carry
%%   derivations in displayed math).  Step-by-step displays added:
%%   no-T-eigenvalue proof; Panofsky parity balance + statistics
%%   constraint; Goldhaber angular-momentum ledger; Wu rate and
%%   asymmetry formulae; CPLEAR Kabir asymmetry with epsilon
%%   relation; CPT comparison observables (kaon delta_CPT, 1S-2S,
%%   q/m); the three-reversal maps displayed in Step 3.
%% v3.19 (July 18, 2026): APPENDIX D LECTURES EXPANDED (author
%%   instruction: bodies were short and dry versus A9/A10; the
%%   exposition, not only the solutions, must teach).  All six
%%   subsection bodies expanded to parent-paper depth for the
%%   active experimenter: operational meaning of symmetry at the
%%   bench; why T has no quantum number; Panofsky's pion-parity
%%   argument; Goldhaber's neutrino-helicity experiment told
%%   quantitatively; Furry's theorem; Wu's cryostat in full;
%%   T-odd triple products (D and R coefficients); final-state-
%%   interaction mimicry; CPLEAR's Kabir asymmetry; the modern
%%   CPT programme (antihydrogen 1S-2S, Penning-trap q/m,
%%   muonium) and Greenberg's CPT-Lorentz theorem; the six
%%   limitations each given its bench consequence; the Step-3
%%   account expanded with the experimenter's translation table.
%%   Problems reformatted per author (bold statements, no
%%   indent, \medskip before each; solutions non-indented).
%%   New bibitem:
%%   Goldhaber1958.
%% v3.18 (July 18, 2026): APPENDIX D WORKED SOLUTIONS (author
%%   instruction: no problems without solutions).  All 18 problems
%%   of app:discrete-qft keep their statements; the hint/answer
%%   format is replaced by complete step-by-step solutions with
%%   physics intuition, no steps skipped.  PHYSICS FIX folded in:
%%   old D.6.1 answer said "T and CT antiunitary, all else
%%   unitary" -- corrected to the right statement (antiunitary =
%%   the four clock-touching operations T, CT, PT, CPT).  Wording
%%   in GGPart subtitle / curriculum / exec updated: the opening
%%   chapter's problems carry complete worked solutions; course
%%   problems retain hints-and-answers pending their dissolution.
%% v3.17 (July 18, 2026): PART XIII FROM-SCRATCH REBUILD, INST. 1
%%   (author-approved blueprint after the v3.16 critique: length,
%%   \appendix lettering, Block architecture, and the Step-1/2/3
%%   principle of A9/A10 now enforced).  \appendix issued; GGPart
%%   front page rewritten to the three-Block architecture; pocket
%%   overview dissolved into curriculum prose; NEW Appendix C
%%   (cross-cohort naming); NEW Appendix D "Discrete symmetries in
%%   QM and QFT, from scratch" -- full Step-1/2/3 chapter (Wigner
%%   proved; P and C derived on the Dirac theory with the bilinear
%%   table; T, Kramers, Wu, and the kaon matrix; CPT by both the
%%   Luders-Pauli and the Wightman routes with consequences
%%   derived; the six limitations; the GG account) with chapter
%%   shelf and 18 problems -- Course A dissolved into it and
%%   deleted.  Remaining courses retitled without course letters
%%   (dissolution in installments 2-4).  Apparatus/exec/roadmap/
%%   pdfsubject updated (90 problems remains exact: 72+18).
%%   New bibitems: PeskinSchroeder1995, Sakurai2017.
%% v3.16 (July 18, 2026): PART XIII CAMPAIGN, INSTALLMENT 1 (author
%%   commission: appendices toward the A9/A10 standard, for incoming
%%   graduate students in neutrino experiments and phenomenology;
%%   complete derivations, textbooks, no page limit).  NEW Course E,
%%   "The neutrino's century --- from oscillations to the lattice
%%   point": six lectures at displayed-equation depth in the A9
%%   Step-1/2/3 register, eighteen problems with hints and answers,
%%   opened by the annotated textbook shelf.  Five textbook bibitems
%%   added (Giunti-Kim, Zuber, Mohapatra-Pal, Bigi-Sanda, Sozzi).
%%   COUNT RECONCILIATION (pre-existing staleness fixed): overview
%%   "Three courses, eighteen lectures" and Part-end "Eighteen
%%   lectures" had lagged Course D; all sites now five courses /
%%   thirty lectures / ninety problems; pdfsubject "four" -> "five"
%%   graduate courses; roadmap line updated.  Installments 2+
%%   (deepening Courses A-D to the same depth; registries) queued
%%   pending sign-off.
%% v3.15 (July 18, 2026): PART XII (CONCLUSIONS) REBUILT to the
%%   standard of C1-Kaxion Part XII and P1-KK Part XI (both read in
%%   full first), per author instruction: plain English, physics
%%   intuition, broad audience; profound philosophical close going
%%   into A11-Origin's topic.  Preserved verbatim (extracted, not
%%   retyped): the plain-English arc, the limitations, the personal
%%   fifty-years section, mirrors-to-mind, why-are-we-here, the
%%   last word.  New: opener "The oldest question in the mirror";
%%   "The paper in ten numbers" table (tab:p5-ten-numbers, C1
%%   pattern: value/status/proved/killed-by); "An answer we can
%%   check --- soon"; the vision expanded (eighty-year measurement,
%%   spectroscopy of symmetries); "The trailhead, not the summit";
%%   four-movement philosophical close "What it would mean" with
%%   movements on the self-referential closure (A11) at honest
%%   grades; \GGAcknowledgments added at Part end (cohort standard,
%%   present in both models, previously absent here).  All
%%   pre-existing Part-XII labels preserved.  Apparatus:
%%   twenty-one -> twenty-two tables.
%% v3.14 (July 18, 2026): PART XI (DISCUSSIONS) REBUILT to the
%%   standard of the cohort's model Discussion Parts (C1-Kaxion
%%   v13.14 Part XI, P1-KK v13.4 Part X; both read in full first).
%%   Same structure, density, coverage: deep Significance (seventy-
%%   year arc, zero-dial epistemics, the over-constraint, calendar-
%%   as-result, quiet-CPV inversion, per-community significance);
%%   PS-P5-1..8 classic-puzzle registry (resolved/dissolved, each
%%   with verdict + conditionality); I-P5-1..10 innovation registry
%%   (before/after form); nobility clause (four checkable
%%   commitments); five-roads summary + NEW discrimination matrix
%%   tab:p5-discrimination (12 observables x 5 rivals); claims
%%   ledger extended (second-child candidate line); falsification
%%   section: three narratives (watched tension / ordering wager /
%%   quiet signature), the eight predictions walked individually
%%   (what/derived-where/measured-how/killed-by), the falsification
%%   lattice; tab:falsifiers and fig:timeline preserved verbatim;
%%   open questions restructured (why open / tried / closing
%%   condition / impact) + item 10 (spin-statistics promotion
%%   path); NEW "completed cohort and future prospects" section
%%   (imports/exports/operational request); Summary with strengths
%%   AND weaknesses.  All pre-existing Part-XI labels preserved.
%%   Apparatus: twenty -> twenty-one tables.
%% v3.13 (July 18, 2026): SPIN-STATISTICS COMPLETION IN P5 (the
%%   quartet's home; author-commissioned closure of the campaign
%%   installed in A2-GG6 v13.1 sec.20 and A8-QM v13.1 T-A8.12, both
%%   signed off).  (1) New Part-IV section sec:quartet-second-child
%%   after "One i for the whole world": T-Stat-A stated as the Wick
%%   quartet's second child (candidate; home A2 sec.20; residues
%%   exactly T-P5-2's), with the analytic mechanism in P5 vocabulary
%%   and the THIRD job of the intertwiner B (obstruction /
%%   conjugation / exclusion; L-Stat-2) extending C2's two-job
%%   poetry.  (2) Part IX QFT-road verdict: second-lock addendum
%%   (two sentences).  (3) Part VIII claims section: FD dependency
%%   of the transport acknowledged (one sentence).  (4) Pauli1940
%%   bibitem added.  P5's claims/falsifier ledgers intentionally
%%   unchanged: the second child is a cohort-external candidate
%%   (F-Stat-1 lives in A2).
%% v3.12 (July 18, 2026): DEVIATION-PLOT SPACING + TABLE-18 DEV/PULL
%%   (author requests).  (1) Figures 13/14 regenerated with more
%%   vertical room per entry (p5_dev_plots_gen.py: row height 0.72 ->
%%   1.45 in/entry, FO/QD pair separation widened; data and captions
%%   unchanged).  (2) tab:exp-predicted-measured reformatted with
%%   explicit Dev and Pull columns (A9 exec-table convention; Pull
%%   uses sigma_obs (+) sigma_th in quadrature, which reproduces the
%%   previously printed +1.17/+0.70, -1.3, +1.0 exactly; all values
%%   verified programmatically).  Part VII's Table 9 unchanged.
%% v3.11 (July 18, 2026): TABLE-9 DUPLICATE + DEVIATION PLOTS
%%   (author correction of the v3.10 commission). "A9-SM Figures 1
%%   and 2" = the executive-summary deviation forest plots (A9 pp.
%%   5-6), not the Part-I chain figures.  Installed at the head of
%%   sec:exp-confrontation: tab:exp-predicted-measured (verbatim
%%   duplicate of Part VII's tab:predicted-measured, kept for the
%%   Part's self-containedness) + fig:p5-dev-linear +
%%   fig:p5-dev-symlog (PNGs from p5_dev_plots_gen.py, cloned from
%%   [v11-Support]/a9_dev_plots_gen.py; P5 rows: sin th12 CKM,
%%   delta_CKM, sin 2beta, J_CKM, delta_PMNS, eta_B; the four
%%   unmeasured/bound rows excluded from the deviation axis, same
%%   rule as A9), with main-text explanations.  The v3.10 funnel and
%%   audit-bar figures are kept (author sign-off).  Apparatus:
%%   fourteen -> sixteen figures; nineteen -> twenty tables.
%% v3.10 (July 18, 2026): TWO GRAPHICAL PRESENTATIONS (commissioned).
%%   (1) Following A9-SM Figures 1 and 2: fig:p5-central-claim (the
%%   five claims drawn as one deductive funnel, A9 Fig.-1 style) and
%%   fig:p5-status (the 27-row honesty audit at a glance, A9 Fig.-2
%%   stacked-bar style; counts computed from tab:parity-ledger,
%%   tab:part4-ledger, tab:part6-ledger: 12 derived / 7 conditional /
%%   3 asserted / 2 named conjectures / 3 other), both installed at
%%   the end of sec:claim with main-text explanations.
%%   (2) Following C1-Kaxion Figure 10: fig:p5-killtest-timeline
%%   (falsification / kill-test timeline, cohort convention: violet
%%   standing record, orange watch, blue accelerator fronts, green
%%   ton-scale edge, red kill dates; axis 2026-2041), installed in
%%   sec:exp-timeline after tab:exp-timeline with explanation.
%%   Apparatus: eleven -> fourteen figures; exec-summary figure list
%%   extended (funnel, audit bar, kill-test timeline).
%% v3.9 (July 18, 2026): PART X COMPLETED (author-approved outline).
%%   NEW section "The confrontation, forced value by forced value":
%%   eight subsections, each with the value at its grade, the
%%   instruments, the quantitative reach, and a pre-declared kill --
%%   (1) the lepton phase (196.4 deg within 1.0 sigma of the NO
%%   global fit; the QUIET-CPV SIGNATURE made explicit: sin(delta)
%%   = -0.28 means Hyper-K/DUNE should NOT find 5-sigma CPV if the
%%   lattice is right -- the near-null is the prediction; 270 deg
%%   would be a ~5-sigma kill at +-15 deg; 196-vs-180 separation
%%   needs the late-2030s +-7-8 deg combined reach, ~2-2.5 sigma);
%%   (2) the ordering (NO required by the derived spectrum; JUNO ~3
%%   sigma ~2031, DUNE 5 sigma ~2033; the joint-fit's mild IO lean
%%   logged as the watch item; kill = IO at 5 sigma); (3) the
%%   Majorana mass (1.72 meV one-sided: ton-scale discovery at
%%   inverted-band values kills; null is the prediction; the value
%%   itself is a kiloton-era target; linked kill via ordering);
%%   (4) the strong-CP window (n2EDM at 1e-27 e cm probes theta-bar
%%   1e-11; any confirmed EDM in the eight-order window kills the
%%   two-fate architecture with the SM untouched; the lattice also
%%   forbids new continuous phases, so eEDM nulls are predictions);
%%   (5) the proton (silence at every depth; Hyper-K to 1e35;
%%   contrast with unification-with-decay windows; n-nbar included);
%%   (6) the right-handed absence (F-P5-1 at HL-LHC and forever);
%%   (7) the pentagon's quark numbers watched (sin2beta -1.3 sigma;
%%   gamma confrontation queued at Belle II/LHCb ~1 deg; NA62/KOTO
%%   as loop-level texture probes); (8) the triad joined (P1-KK
%%   lock cluster + C1 Kaxion + P5 battery; AND-logic; any clean
%%   kill ends the programme). NEW section "The world numbers,
%%   refreshed" (the freshness duty executed): Part VII's
%%   confrontation table updated in place -- sin2beta 0.699+-0.017
%%   -> 0.709+-0.011 (pull -0.2 -> -1.3, watched), delta_PMNS
%%   measured column -> NuFit 6.0 NO/IO values with +1.0 sigma
%%   status; Part VII Step-3 prose updated (0.2 sigma/202 deg ->
%%   1 sigma of NO fit + 3 sigma IO disfavour); WE III and Course B
%%   sin2beta updated; the v3.8 interim bookkeeping note in the
%%   present-frontier section replaced by the executed-refresh
%%   sentence. NEW section "Simulating the verdicts: 2028, 2031,
%%   2035, 2040" (P1-KK pattern): per-milestone GG-true vs GG-false
%%   ledgers; the quiet-universe-with-one-loud-cluster global
%%   signature; the combined-likelihood trajectory. NEW section
%%   "The falsifier timeline, dated" with the schedule-centric
%%   table (kill/support/watch verdict types; cross-ref to the
%%   claims-level falsifier table of Part XI). NEW closing section
%%   "What is claimed, and what is inherited" (all sensitivities
%%   the collaborations'; claimed: dial-lessness of the
%%   confrontation, the one-sided silence predictions, the
%%   quiet-CPV signature, the triad AND-logic). Interim part-end
%%   summary replaced by the final one. Apparatus count updated
%%   (nineteen tables); roadmap Part-X sentence extended. No new
%%   references.
%% v3.8 (July 18, 2026): NEW PART X "EXPERIMENTS --- PAST, NOW, AND
%%   FUTURE", FIRST INSTALLMENT (author-commissioned: the paper's
%%   experimental Part, following the P1-KK / C1-Kaxion pattern ---
%%   P5 as the cohort's third falsification channel). Built on three
%%   web-verified deep-research dossiers (July 2026): CP-violation
%%   history and frontier; neutrino-experiment history and frontier;
%%   0vbb + proton decay. Shipped in this installment: the Part
%%   opening ("The third channel": the cohort's three falsification
%%   channels, the freshness clause, the two hats); "A century of
%%   broken mirrors, measured" (kaon era with the epsilon'/epsilon
%%   saga at corrected final values -- KTeV final 19.2 +- 2.1e-4,
%%   NA48 14.7 +- 2.2e-4, world 16.6 +- 2.3e-4; CPLEAR; the CPT
%%   mass bound; B era from KM through ARGUS, the 2001 sin2beta
%%   observations, BaBar's 2012 direct T violation, LHCb's 2019
%%   charm and 2025 BARYON CP-violation discoveries, the Nov 2025
%%   gamma combination 62.8 +- 2.6 deg; the EDM era) with the
%%   past-CPV milestones table; "The neutrino century" (Pauli to
%%   the 2015 Nobel, with the author's Kamiokande years placed)
%%   with the past-neutrino milestones table; "The present frontier
%%   (mid-2026)" (NuFit 6.0 digits; the T2K+NOvA joint fit, Nature
%%   646 (2025); JUNO first results Nov 2025; NA62 combined
%%   9.6e-11; KOTO bound; sin2beta updated world average 0.709 +-
%%   0.011 with the recomputed pull -1.3 sigma stated honestly;
%%   KamLAND-Zen 3.8e26 yr; LEGEND-200 first result; Super-K
%%   proton-decay ledger incl. the 2024 eta and 2026 pi0pi0
%%   channels; KATRIN 0.45 eV; DESI Sum m_nu < 0.064 eV with the
%%   ordering tension noted; DSNB at threshold); "The instruments
%%   of the decisive decade" (Hyper-K 2028, DUNE beam ~2031, JUNO
%%   running, IceCube Upgrade, ESSnuSB; ton-scale 0vbb:
%%   LEGEND-1000, nEXO, CUPID, AMoRE-II, SNO+; proton decay at
%%   Hyper-K/DUNE/JUNO; n2EDM and JILA III; supernova/SNEWS 2.0;
%%   KATRIN/Project 8) with the future-instruments table whose
%%   last column names the GG target each instrument touches.
%%   Interim Part-end summary (the confrontation sections follow
%%   in the second installment upon author approval). LATER PARTS
%%   RENUMBERED: hardcoded Part X->XI ... XIV->XV swept (11 sites);
%%   roadmap gains the Part-X sentence; apparatus counts updated
%%   (eighteen tables); arXiv comment ~180 pp. NEW REFERENCES (29,
%%   annotated): ReinesCowan1956, Danby1962, DONUT2001, OPERA2018,
%%   ARGUS1987, MoeElliottHahn1987, GeorgiGlashow1974, GERDA2020,
%%   KamLANDZen2024, CUORE2024, CUPID2025, LEGEND2025, LEGEND1000,
%%   nEXO2022, T2KNOvA2025, JUNO2025, HyperK2025, DUNE2020,
%%   KATRIN2025, DESI2025, NA622025, KOTO2025, LHCbCharm2019,
%%   LHCbBaryon2025, SuperKPdecay2020, SuperKnnbar2021,
%%   Borexino2020, SNEWS2021, Roussy2023.
%% v3.7 (July 17, 2026): PART IX (THE FIVE ROADS) TRIPLED
%%   (author-directed: dramatic revision; for each of QFT, the
%%   Standard Model, string theory, and Lambda-CDM show what it
%%   achieved, where it ends, and how the geometry completes the
%%   story; rigorous, systematic, analytical, quantitative; deep
%%   philosophical close). Each road rebuilt in the three-movement
%%   architecture with the existing passages preserved inside:
%%   QFT road -- NEW "What quantum field theory achieved" honour
%%   roll (the framework of the century; g-2 at the part-in-10^12
%%   level; renormalization and the Wilsonian reading; spin-
%%   statistics; the CPT theorem; the vacuum as medium), existing
%%   twins/mirrors/matter subsections kept (Weinberg theorem env
%%   and all ssec labels preserved), NEW closing subsection "Where
%%   the road ends, and how the geometry continues it" (the three
%%   undischarged axioms collected; T-P5-B1's Wick-license added
%%   to the completion map; itemized QFT-question -> GG-answer
%%   ledger with grades). SM road -- NEW achievements subsection
%%   (most precise theory in history: LEP lineshape, exactly three
%%   light neutrinos, third generation predicted by CP, W/Z masses
%%   predicted, Higgs at prescribed couplings, one-generation
%%   anomaly cancellation), existing five-silences/two-failures
%%   text kept as "where it ends" with the dial count added,
%%   NEW completion subsection (silences and failures answered
%%   item by item with Part numbers and grades; dials to lattice
%%   points). String road -- rebuilt in three movements with all
%%   concessions repeated (GS-inflow lineage, CHSW chirality,
%%   conjugate excitations, orientation-reversal C, worldsheet
%%   CPT, CP-as-gauge-symmetry), "where it ends" (vacuum
%%   selection; no numbers; moduli problem; the spatial-
%%   compactification burdens vs Theorem 0's non-spatial fibre,
%%   structural contrast only), completion (same three geometric
%%   moves at forced valuation; locked vs landscape-valued).
%%   Lambda-CDM road -- three movements (six numbers that fit the
%%   sky; the concordance certificate; all six as inputs; dials
%%   re-derived as outputs at their grades; the bookkeeper
%%   certifies the tripod's receipt; the 7.5% seam republished).
%%   NEW section "The inventory: who predicts what" with the
%%   cross-framework observable table (SM fit / string unvalued /
%%   Lambda-CDM input / GG forced value, pulls cross-ref'd to the
%%   forced-vs-measured table). Depth scoreboard kept. NEW closing
%%   section "The five roads, seen from above": the four verbs
%%   (constrain, codify, enumerate, measure) and the fifth verb;
%%   the hand-off circle of the twentieth century; Einstein's
%%   question given each road's answer; the jigsaw humility clause
%%   verbatim-grade; determinacy-with-mortality as the scientific
%%   posture; the two-hats close. Part subtitle and roadmap
%%   extended; apparatus count updated to fifteen tables (verified
%%   programmatically). SECOND PASS additions: QFT subsection
%%   "The road's own summit" (Wilson's RG as the filial ancestor of
%%   the scale dimension); SM subsection "Reading the dials" (the
%%   28-parameter anatomy 3+2+6+4+3+3+4+2+1 and the four rhymes,
%%   each mapped to its Part); NEW section "Reading the pulls, and
%%   the falsification symmetry" (pull spread +0.01..+1.17 with the
%%   flags counted as data; road-by-road vindication list -- W_R,
%%   270 deg, proton decay, fourth generation, EDM window, moduli
%%   relics); century-arc paragraph added to the close.
%% v3.6 (July 17, 2026): THE GEOMETRIC ORIGIN OF SAKHAROV'S
%%   CONDITIONS (author-commissioned: "everyone takes them for
%%   granted without questioning their deeper origin -- we have
%%   finally found the answer"). Section sec:sakharov-custody
%%   expanded with the derivation of the keys: three origin walks
%%   -- (i) the two books and the still-point door (Gauss-law
%%   absoluteness of the gauged bulk ledger; the boundary B-L
%%   budget; the weight-zero door as the one gap in the winding
%%   bookkeeping; sphalerons as boundary machinery; OGGEP owns the
%%   ledger), (ii) the oriented descent and the chiral lattice
%%   (C and P broken by the projection's absence, CP by the
%%   pentagon's angle; delta = 0 lawful in the SM, CP symmetry
%%   UNWRITABLE here; GRIP owns the angle), (iii) the always-on
%%   projection and the fall (equilibrium given its geometric face
%%   via the thermal shore of T-P5-B1 -- a KMS state on the clock's
%%   half-plane, CPT-balanced; the constitution denies the boundary
%%   that rest; the Morse descent as a falling system by
%%   construction; GDOS owns the observability) -- followed by the
%%   custody/Sakharov alignment displayed, NEW proposition P-P5-E
%%   (the Sakharov tripod): the three conditions as boundary
%%   shadows of three constitutional structures, each carried by a
%%   named result, with the corollary that a GG universe cannot
%%   fail Sakharov -- some asymmetry is structural, only the
%%   magnitude hangs on C-P5-C; proof by assembly with clause
%%   grades; and the closing intuition "Sakharov, read backward"
%%   (necessity 1967 vs structural satisfaction here; the
%%   conditions as the parts-list of an oriented descent; falls
%%   that keep books leave receipts). Status-ledger table gains the
%%   P-P5-E row; Part VIII subtitle, roadmap, and Executive Summary
%%   updated. No new floats or references.
%% v3.5 (July 17, 2026): NEW THEOREM T-P5-B1 (THE WICK QUARTET)
%%   installed in Part IV as the new section "The clock's third
%%   face: imaginary time", directly following the handshake
%%   section (author-commissioned, from the author's question on
%%   Minkowski's it, Wick's rotation, and the Schrodinger i as one
%%   structure). Content: the polar-decomposition lemma (A = J0|A|,
%%   the clock as rotor times speed); theorem T-P5-B1 stating the
%%   canonical bijection of four data -- the clock J, the
%%   harmonic-conjugate (Hardy) splitting, the Euclidean contraction
%%   semigroup with reflection positivity, and the holomorphic
%%   time-tube -- with orientations tracked and the one-analytic-
%%   fact clause (one-sided spectrum relative to causal
%%   orientation); proof in seven explicit steps at the quasi-free
%%   grade (mode reduction; damping construction with the CR pair
%%   displayed; Paley-Wiener converse; Euclidean semigroup and
%%   reflection positivity with Osterwalder-Schrader cited; the
%%   holomorphic tube; the orientation bijection; uniqueness cycle);
%%   physics intuition on the spot (motor = rotor x speed; the
%%   direction in which oscillation relaxes; why Wick rotation is
%%   licensed and by which axiom clause; boosts/rotations merging as
%%   the germ of Jost's CPT, cross-linked to Part V's modular
%%   discussion); Corollary B1-a (the boundary's three shorthands:
%%   i d/dt, t -> it, Wick); Corollary B1-b (custody: imaginary time
%%   belongs to the clock, on all sectors, feeding the one-J-one-i
%%   doctrine; why there is no imaginary space -- two-sided momentum
%%   spectrum -- and why the fibre could not supply the shore --
%%   compact two-sided windings); conditional remarks (complex
%%   Lorentz group/Jost; KMS periodicity as boundary temperature,
%%   C3 cross-ref); status-and-inheritance paragraph
%%   (Ashtekar-Magnon/Kay, Paley-Wiener, Osterwalder-Schrader,
%%   Wightman tube conceded; the packaging-with-provenance, the
%%   orientation bookkeeping, and the custody corollary claimed;
%%   grade derived (quasi-free), full-Lorentz clause conditional);
%%   closing physics-intuition essay "What, then, is time?" linking
%%   the theorem to AX-1 (locality-causality: the cone -- time's
%%   order), AX-2 (geometry-gauge equivalence, via Theorem 0: the
%%   static Killing flow -- time's repetition) and AX-3 (the
%%   operational dictionary: the arrow and the shore -- time's
%%   openness), with the two-sidedness contrast (space and fibre)
%%   and the Minkowski-1908 lineage. NEW annotated reference:
%%   Osterwalder-Schrader I/II (1973/1975) with relevance note.
%%   Part IV subtitle, roadmap, and Executive Summary extended;
%%   arXiv comment theorem count updated to eleven.
%% v3.4 (July 17, 2026): PART VIII (THE ORIGIN OF MATTER) DOUBLED
%%   (author-directed: the heart of P5-CPT; warm register, rigorous
%%   derivation, the longer the better). NEW opening section "The
%%   number that survived the fire" (the annihilation epoch told
%%   warmly -- a furnace of pairs burning to light, the one-in-a-
%%   billion remainder, "who wrote the one?"; the Part as the
%%   destination of the whole paper). Observational facts expanded:
%%   each certificate walked (how the acoustic peaks read eta_B; the
%%   BBN concordance as a two-epoch, five-orders-in-temperature
%%   agreement certifying the asymmetry predates t ~ 1 s; the
%%   domain-exclusion and antinuclei certificates; the symmetric-
%%   universe counterfactual's freeze-out logic walked to 10^-18).
%%   Sakharov section expanded: the three conditions DERIVED (why
%%   each is necessary from symmetric initial data, including the
%%   equilibrium/CPT argument), then the custody map, then the
%%   three-locks intuition (the architecture had already paid each
%%   condition for other reasons). Why-leptogenesis section
%%   expanded: the exclusion walk (GUT route structurally closed
%%   and why that is a virtue; EWBG closed by the inherited
%%   crossover AND the SM's 10^-20 CP shortfall, cross-ref'd; the
%%   lepton route as the only Sakharov-complete channel left), NEW
%%   Davidson-Ibarra paragraph (the 10^9 GeV bound stated, cited,
%%   and evaded by the resonant regime -- the reason TeV-scale
%%   leptogenesis is testable at all). Chain deepened in place:
%%   step 6's m_star given its three-line meaning (the equilibrium
%%   neutrino mass, measured constants only); step 7's exponent
%%   given the alphabet-echo intuition (the third appearance of
%%   K = 35). Zero dilution: minted-coin intuition added. Resonance
%%   section: NEW "what success and failure look like" paragraph
%%   (the finite-T criterion; the clean kill; why this is an open
%%   problem with a computation attached, not a mystery).
%%   Falsifiers: NEW testability-inversion paragraph (hierarchical
%%   leptogenesis explains and cannot be tested; this chain is
%%   testable four ways within a decade; more predictive and more
%%   protective simultaneously). NEW closing section "What is
%%   claimed, and what is inherited" (Sakharov, Fukugita-Yanagida,
%%   Pilaftsis(-Underwood), Casas-Ibarra, Kuzmin-Rubakov-
%%   Shaposhnikov, Harvey-Turner conceded -- "not one new formula";
%%   dial-lessness, the e^{-K/2} reconstruction, zero dilution,
%%   proton-safe baryogenesis, and the testability inversion
%%   claimed, ranked, at the ledger's grades). Part-end summary
%%   extended with the remainder arc. Subtitle and roadmap
%%   extended. No new floats (verified programmatically).
%% v3.3 (July 17, 2026): PART VII (THE CHIRAL PENTAGON) EXPANDED IN
%%   EXTREME DETAIL (author-directed: the most profound Part;
%%   rigorous, systematic, the longer the better). NEW opening
%%   section "The question no dial can answer" (1964's shock, 1973's
%%   accommodation-not-explanation, delta = 0 as lawful as 1.147 in
%%   the SM, the author's own decades inside epsilon'/epsilon and
%%   sin 2beta, and the Part's plan). T-P5-3 now UNPACKED in two new
%%   walk subsections: "The calculus, walked" (the dial group
%%   displayed, edge transformation, telescoping on loops, spanning-
%%   tree gauge fixing, the (N-1)^2 count at N = 3, plaquettes as
%%   smallest invariants, J as the one area at N = 3) and "The
%%   criterion and the lattice, walked" (CP negates every edge angle
%%   -- Part V's C walk cashed in; the {0, pi} criterion; then the
%%   valuation: what could float a holonomy off the lattice and why
%%   the arena forbids each escape -- no second torus, no moduli;
%%   the rigidity residue stated honestly). Quark chain rebuilt as
%%   six numbered steps, each graded, with NEW material: the
%%   pentagon's trigonometry from scratch (phi from the pentagram,
%%   cos 36 = phi/2, cos 72 = (sqrt5-1)/4, three lines); the branch
%%   phase recognized as the pentagon's INTERIOR angle (108 deg) and
%%   the physical phase as its CENTRAL angle (72 deg); the Cabibbo
%%   interference formula derived in public (|a - b e^{i phi}|^2
%%   expansion, the golden coefficient emerging from cos(3pi/5),
%%   numerical walk to 0.22502); the rephasing step and QD slip with
%%   flags verbatim. Lepton chain rebuilt as five numbered steps
%%   (the odd counter J = 7, arcsin(1/7) at the reactor block, the
%%   pi shift explained, the {pi, 3pi/2} degeneracy given its
%%   sentence, Majorana pair as parity data, m_bb, J_PMNS, dates).
%%   T-P5-4: NEW dial-count paragraph (what a generic seesaw offers
%%   -- masses, textures, phases -- vs what remains here: nothing;
%%   Part IV still-point seed cross-linked). Strong CP: the problem
%%   given its full statement (theta-bar defined for the
%%   non-specialist, the 10^-10 absurdity); the export argument
%%   walked (EFT logic vs exported Lagrangian); each factor of the
%%   leakage bound given its origin sentence (pentagon sine, throat
%%   exponential, chirality power). NEW closing section "What is
%%   claimed, and what is inherited" (Jarlskog calculus, string CP
%%   as gauge symmetry (Strominger-Witten, Dine-Leigh-MacIntire,
%%   Choi-Kaplan-Nelson), discrete-flavour model building, PQ, and
%%   Vafa-Witten conceded; the lattice valuation with forced
%%   falsifiable numbers, the two-fate architecture, the geometrized
%%   criterion, and the dial-free door claimed, ranked, at the
%%   ledger's grades). Part-end summary extended. Subtitle and
%%   roadmap extended. No new floats: figure and table counts
%%   unchanged (verified programmatically).
%% v3.2 (July 17, 2026): PART VI (PARITY) DOUBLED (author-directed:
%%   warm register, rigorous derivation, the longer the better). The
%%   five superlatives now carry their bench faces (W/Z mass vs the
%%   massless photon; Wu; every muon decay and the shining Sun; the
%%   10^-18 m range; the kaon/B home of CP). Basin/gate architecture:
%%   the landscape formalized (critical points, gradient flow), the
%%   executor's five properties derived one clause at a time, and the
%%   canal-lock intuition passage added. Projection section: the
%%   centraliser computed in public (block form of the SU(5) adjoint
%%   under conjugation by P, displayed; hypercharge as the surviving
%%   trace direction, displayed), NEW block-grid figure
%%   fig:parity-blocks (which blocks commute, which are projected),
%%   the fermion halving displayed (internal parity tied to 4D
%%   chirality, one hand per representation), and the fifteen-state
%%   count check (5bar+10 = one left-handed SM generation, listed).
%%   Orientation section: bare-mass prohibition derived (doublet x
%%   singlet), every mass through the projection's own debris, the
%%   toll-at-the-gate intuition. Two-partial-answers section:
%%   Vafa-Witten and survival argument each given a sketch of WHY
%%   they hold. Gate-Operator Lemma: clause-by-clause discussion
%%   (what each clause asserts, the orientation-pairing proof
%%   strategy for (ii), what a proof would deliver, what would kill
%%   the lemma). Handedness invariant: the Ozma measurement walk
%%   (Wu sign + descent direction + Jarlskog sign = one datum, not
%%   three). NEW closing section "What is claimed, and what is
%%   inherited" (Kawamura/Hall-Nomura, DHVW, CHSW, Vafa-Witten,
%%   't Hooft conceded in full; basin/gate as the only framework
%%   addressing "why always the weak force", the forcing-vs-
%%   selection distinction, P-P5-D, and F-P5-1 claimed, ranked,
%%   at the ledger's grades). Part-end summary extended. Subtitle
%%   and roadmap extended; apparatus figure count updated to eleven
%%   (verified programmatically).
%% v3.1 (July 17, 2026): PART V TRANSFORMED (author-directed: the
%%   mirror-definition Part, previously the paper's shortest, brought
%%   to the depth of its source material). NEW opening section "The
%%   operations, given bodies" (the standard formalism's anatomical
%%   silence -- per-field phases, equivalence classes, antiunitarity
%%   as selection -- stated from its own texts; the Part's agenda).
%%   NEW section "The three reversals, walked one at a time": C walked
%%   on Part IV's modes (green cell to green cell, turning sense
%%   preserved), P as the exchange of the Weyl modules (chiral vs
%%   vector-like criterion stated as the bridge to Part VI), T walked
%%   into the clash quadrant (the image lies where the handshake
%%   forbade J-linear physics -- antiunitarity as the only exit),
%%   each with laboratory faces. NEW section "First dividends of the
%%   table": Kramers' theorem as a corollary (T^2 = -1 from the fixed
%%   real spinor map, no phase conventions; Course A drill
%%   cross-linked) with its laboratory face; the no-T-quantum-number
%%   theorem (antilinear operators carry no invariant eigenvalue) and
%%   why T is tested by asymmetries and EDMs -- 1957 vs 1998/2012
%%   explained. T-P5-2 section: NEW paragraph on the converse
%%   discipline (Greenberg: CPT violation implies Lorentz violation
%%   -- the arena itself is what would have to go) and the theorem's
%%   laboratory record (equal masses/widths/moments; cross-ref to
%%   Part I's six-comparison table). NEW section "The five fates of
%%   the mirrors" (absorbing and expanding the old factors-break
%%   paragraph): C and P broken by an absence (the projected-out
%%   mirror sector -- why the first fall was maximal), CP surviving
%%   the projection and broken only by the pentagon angle (why the
%%   second fall was small and eight years later), T falling with CP
%%   through the unbreakable product, CPT blind to every real
%%   texture; NEW five-fates table. NEW closing section "What is
%%   claimed, and what is inherited" (concede-then-claim: Wigner/
%%   Jost/BW/Greenberg and all quantum-number bookkeeping conceded;
%%   the derived table, the anatomy trade, the five-fates pattern,
%%   and the T/arrow custody split claimed at their stated grades;
%%   grades cross-ref'd to the mirror-Parts ledger). Part-end summary
%%   rewritten. Part subtitle, roadmap, and Executive Summary
%%   extended; apparatus count updated (fourteen tables, verified
%%   programmatically). "Proved next" for T-P5-3 corrected to
%%   "proved in Part VII".
%% v3.0 (July 17, 2026): PART IV DOUBLED (author-directed; the
%%   antiparticle Part brought to the depth of its sources, with the
%%   real-number framing deepened throughout). NEW section "A world
%%   written in real numbers" (the stage: below quantisation, on real
%%   solution spaces; the no-i pledge; the two working rules; why the
%%   claim must be proved at this level; scope stated). Handshake:
%%   T-P5-B's proof expanded to four explicit steps (skewness, polar
%%   phase, sign from AX-3 positivity, uniqueness + real frequency
%%   splitting, Hilbert-transform kernel displayed); NEW subsection
%%   "The handshake executed: one charged mode, in rotation blocks"
%%   (the whole computation in 2x2 real matrices, boundary packaging
%%   only at the last line, closing on one-J-upstairs-one-i-downstairs);
%%   NEW quadrant figure fig:quadrants (winding x frequency, kept
%%   diagonal vs clash cells, Feynman-Stueckelberg rereading arrows).
%%   NEW section "One i for the whole world": T-P5-C1 (composites and
%%   Fock extend the shared J with no new complex input) with the
%%   entanglement intuition and the Renou network-Bell instrument
%%   paragraph; T-P5-C2 (fermionic conjugation = fibre reversal
%%   composed with the fixed real intertwiner B) with the same-matrix-
%%   both-jobs intuition and the Dirac-i inventory. NEW section "The
%%   self-conjugate exceptions": self-conjugacy derived from weight
%%   zero (photon C-parity -1, pi0 C-even with the 2-gamma/3-gamma
%%   check, metric), the neutrino as the unique weight-zero matter
%%   field seeding T-P5-4 and the door (m_bb pointer), carousel
%%   still-point intuition. Bulk encoding: the three-step Diophantine
%%   walk with displayed equations (core block, d2 chirality flip,
%%   reservoir pattern, 32-Omega singlet sea with n_F/n_SD = 8).
%%   Descent: new physics-intuition passage (customs office; audit in
%%   reverse). Boundary readout: three episodes promoted to
%%   subsections and deepened (positron with Depth-3 mass-equality
%%   argument; Wheeler with the 1940 telephone story and the
%%   wrong-custodian resolution; Hawking partner with the trace
%%   derivation and readout-is-not-resident close). Ledger: Depth-3
%%   sentence on the Part-II ruler. Part opener re-pointed at the new
%%   flow; Part subtitle, roadmap, and Executive Summary extended
%%   (T-P5-C1/C2 sentence added). STALE CROSS-REFERENCE REPAIRS
%%   (v1.1-numbering survivors, verified against the v2.9 roadmap):
%%   Exec Summary "Part V argues"->VI and "Part VI assembles"->VIII;
%%   Part IV opener "(Part VIII rehearses)"->IX; descent opener
%%   "Question 1"->"Question 2"; worked-examples intro "owns Part IV"
%%   ->"owns the mirror Parts"; References relevance notes Hall-
%%   Nomura V->VI, Kuzmin VI->VIII, Pilaftsis VI->VIII, Sakharov
%%   VI->VIII. COUNT REPAIRS (verified programmatically): apparatus
%%   "nine figures"->ten, "twelve tables"->thirteen; pdfsubject and
%%   arXiv comment "three graduate courses"->four; arXiv comment
%%   page count and theorem count refreshed. RENDERING FIX
%%   (pre-existing): "\GG{}-6" printed "GG-6-6" at two Part IV
%%   sites; now "\GG{}". Post-preview sweep of every Part~N
%%   mention added two more stale repairs: basin/gate "CP violation
%%   joins through Part IV"->VII; Kawamura note "Part V's
%%   mechanism"->VI.
%% v2.9 (July 16, 2026): PART III REBUILT (author-directed) on the
%%   companion satellite's arena template, which presents the
%%   framework far more completely. ADDED: axioms rendered "in
%%   full" with the forcing chain made explicit (locality+
%%   interaction => a scale; locality+charge => a phase; "AX-1
%%   hands exactly two structures to the next axiom"); AX-2's two
%%   clauses (locked equivalence = Einstein's elevator upgraded;
%%   invariant physics), complex-free, "AX-2 builds the fibre";
%%   AX-3's dictionary TUPLE displayed as an equation (Kraus
%%   instruments, POVM, CPTP maps, three compatibility clauses,
%%   operational positivity = clause (iii)), "AX-3 collapses the
%%   fibre that AX-2 built"; AX-4 for completeness; the RELAY
%%   physics-intuition (knob/dial/reader). Theorem 0 now displays
%%   THE METRIC, the three properties (non-spatiality by
%%   generators + isometry algebra; global staticity => the
%%   constitutional clock as a Killing field; reality), the
%%   six-step forcing chain, and the "do not picture six axes"
%%   intuition; the non-spatiality dissolutions relocated here.
%%   Ledger now displays the closure identity and per-integer
%%   structural meanings. NEW sections: the fibre dictionary
%%   (harmonic-pair equation + the full dictionary list + "the
%%   photon carries no dial"); the boundary law (OGGEP/GDOS: the
%%   projection forced, always-on, its derived form, and the two
%%   fibres carrying the two halves of quantum mechanics); the
%%   actualization layer (GRIP: records are real angle classes,
%%   and the flavour textures are GRIP outputs -- why this paper
%%   may say "lattice-locked"); the arc (real bulk, complex
%%   middleware, real records); the two shared structures named
%%   (G1/G2) with a P5 inheritance table; a plain-terms Part
%%   close. P5-specific material retained: the warm "why axioms
%%   at all" opening, the Weyl/Minkowski lineage, the AX-3
%%   measurement calibration, POS-3, the two-sources subsections,
%%   the observer subsection, the custody map and table.
%% v2.8 (July 16, 2026): NEW subsection 29.5 "Why the real bulk
%%   matters: the observer built to read it" (author-directed).
%%   Elaborates why real-variable-only in the bulk is essential,
%%   via the companion neuroscience: the fourth Abelian symmetry
%%   U(1)_Qi and its informational current; DGI as the unbroken
%%   freely-propagating substrate; the Inner Hologram --- the
%%   cortex's allocentric world-model reconstructed and carried in
%%   neural PHASE (theta/gamma), the same real-angle medium the
%%   bulk uses for charge; the parallel "the world writes reality
%%   in real phase; the brain reads it in real phase; i is notation
%%   in both." Calibrated strictly to the companion papers'
%%   Asserted grade; every concept (U(1)_Qi, DGI, allocentric,
%%   Inner Hologram, phase coding) defined from scratch, no assumed
%%   prior knowledge, no internal-affairs language. Refs added:
%%   OKeefe--Nadel, Hafting (grid cells), Buzsaki (rhythms),
%%   Friston (free energy).
%% v2.7 (July 16, 2026): THE J/i DISTINCTION SHARPENED (author's
%%   question: "I thought we had got rid of all i from the 6D
%%   bulk"). Correct: the bulk contains NO i --- it contains J, a
%%   REAL quarter-turn operator (a complex structure, never a
%%   complex number); i is the 4D boundary's shorthand name for J,
%%   arising only when boundary QFT packages real pairs (a,b) into
%%   amplitudes a+ib. Doctrine line installed: "one J upstairs is
%%   why there is one i downstairs." Section 29.4 retitled ("One J
%%   in the bulk, one i on the boundary") and rewritten at the two
%%   levels; handshake preview, T-P5-C intuition, the
%%   price-of-one-i closing, and the abstract's source clause all
%%   tightened to keep J (bulk, real) and i (boundary, notation)
%%   in their own custody.
%% v2.6 (July 16, 2026): READER-FACING HYGIENE SWEEP (author-flagged,
%%   standing rules): (1) all internal-history references removed
%%   from body text ("the earlier version", "new in this version",
%%   "the five reports behind this paper", "The author asked",
%%   "an earlier presentation" x2); (2) no-prior-knowledge rule
%%   enforced --- "The two sources, at depth" and the Part-IV
%%   handshake opening no longer lead with companion-paper names or
%%   labels; concepts introduced from scratch, citations attached
%%   after the physics.
%% v2.5 (July 16, 2026): DOCTRINAL CORRECTION (author-flagged,
%%   serious): the fibre circle and scale direction are NON-SPATIAL
%%   dimensions --- the circle subsection of Part III had imported
%%   the Kaluza--Klein spatialization ("too small to walk in") that
%%   A2-GG6's Theorem 0 exists to refuse. Rewritten from A2's
%%   canonical doctrine: dimensions classified by their GENERATORS
%%   (translation vs dilatation vs compact charge); the isometry
%%   algebra iso(1,3)+R_D+u(1)_Q, not iso(1,5); coordinates
%%   dimensionless by construction (a log of an energy ratio; an
%%   angle); motion in theta a gauge transformation, not a
%%   displacement; no light cone, no radius modulus, no KK tower,
%%   no landscape --- non-spatiality as the derived conclusion that
%%   dissolves the problems of every spatial-extra-dimension
%%   programme. AX-2 paragraph re-lineaged through Weyl 1918/1929
%%   (both insights), with Kaluza--Klein honestly placed as the
%%   spatializing wrong turn and Minkowski's time as the first
%%   non-spatial geometric dimension. KK-adjacent passages audited
%%   and corrected throughout (Part II conjugate-field line, Part
%%   IV ledger clause, Part IX string road, Objection 42; Kaluza/
%%   Klein relevance notes rewritten). New refs: Weyl 1918, Weyl
%%   1929, Minkowski 1908.
%% v2.4 (July 16, 2026): Part III rebuilt and DOUBLED (author-directed
%%   round 4, in the Part-II voice): warm Part opening ("why axioms
%%   at all" --- laws vs bylaws); each axiom given the Feynman-style
%%   treatment (AX-1 as "news and records"; AX-2 as "nature keeps
%%   one set of books" --- the scale and the phase must be geometry,
%%   with Kaluza 1921 / Klein 1926 cited as the lineage; AX-3 as
%%   the dictionary and the projection, calibrated; POS-3 as "the
%%   ledger must balance"). Theorem 0 + ledger warmed (the five
%%   integers, one line each). The two-sources section GROWN into
%%   "The two sources, at depth" with subsections: the circle (a
%%   dimension too small to walk in, big enough to wind; KK lineage
%%   honestly placed), the clock (the quarter-period advance; Gabor
%%   1946 cited at the analytic signal), the handshake preview, and
%%   the one shared i (Renou). Three new references with notes.
%% v2.3 (July 16, 2026): Part II rebuilt and DOUBLED (author-directed
%%   round 3): accessible, warm register throughout; every section
%%   opens with plain-language motivation before formality; inline
%%   citations added to the standard textbooks (Branco--Lavoura--
%%   Silva, Kolb--Turner, Lawson--Michelsohn, Weinberg QTF) and the
%%   original papers ('t Hooft 1976, Klinkhamer--Manton 1984,
%%   Alvarez-Gaume--Witten 1984, Mohapatra--Senjanovic 1980) at
%%   their points of use. The flavour-graph figure REBUILT to cohort
%%   standard (styled vertices, shaded plaquette, legend). New
%%   physics-intuition passages: the phasor pair, the gear rule for
%%   (anti)linearity, dials-and-wires, the area meaning of J, the
%%   seesaw balance, sphalerons as topological unwinding, the Bott
%%   clock for spinor reality, the depths run through on one
%%   example. Seven new references with notes.
%% v2.2 (July 16, 2026): Part I rebuilt and DOUBLED (author-directed
%%   round 2). New opening "The oldest asymmetry, asked precisely"
%%   (the A9/A10-interface framing; the two hats declared as method).
%%   NEW sections: "Before the fall" (Kant's incongruent
%%   counterparts, Noether, Wigner 1927, Laporte; Purcell--Ramsey
%%   1950 as the lone early doubt); "The repaired mirror and the
%%   Ozma problem" (Landau's CP; Gardner's Ozma problem resolved by
%%   1964, promoted to P-P5-D); "One phase, three families" (KM
%%   counting theorem; Upsilon 1977; top 1995; Jarlskog);
%%   "What experiment still cannot tell us" (eight open measurements
%%   with parent-paper stakes) and "What theory still cannot say"
%%   (seven philosophical ledger entries, incl. Kant modernized and
%%   the measurement shadow, calibrated). History deepened in place:
%%   Pauli's letter + Feynman's bet; Davis/Homestake, SNO, KamLAND,
%%   the proton-decay origin of the water detectors; the EDM
%%   programme (Purcell--Ramsey to nEDM 2020 + ACME 2018); Dirac's
%%   1933 antimatter stars + Alfven--Klein cosmology and its
%%   gamma-ray death. Five-questions section dissolved per author
%%   instruction; "The two parents" section rewritten (A9-SM and
%%   A10-Cosmos as P5's parents, cited as such). Thirteen new
%%   references with notes. Question-number cross-refs neutralized.
%% v2.1 (July 16, 2026): front-matter refinement (author-directed,
%%   round 1 of the Part-by-Part deepening). (a) Title page moved to
%%   the canonical A9/A10 L9 form: two-line plain-English subtitle
%%   ("Why Every Particle Has a Twin..."), the satellite blurb
%%   paragraph REMOVED, date in the canonical slot, abstract fitted
%%   on page 1 (trimmed ~20%). (b) Executive Summary opening
%%   REWRITTEN: the five-questions frame removed per author
%%   instruction; opens instead with the field's remaining big
%%   questions and the paper's position as the interface between
%%   A9-SM and A10-Cosmos --- the asymmetry of matter as the seam
%%   between the derived Standard Model and the derived cosmos,
%%   hence our origin. The real-bulk spirit (P4) named in the ES.
%%   (c) Preface pointer updated. Body Parts untouched this round.
%% v2.0 (July 16, 2026): THE MAJOR SURGERY (author-commissioned, on
%%   the approved P5-CPT-v2.0-Surgery-Proposal). Root repair: the
%%   direction of explanation inverted to GG-FIRST throughout ---
%%   the five reports' question->GG-answer->comparison architecture
%%   restored faithfully, zero-loss coverage matrix honoured.
%%   FOURTEEN Parts. NEW Part I opening: the author's five questions
%%   verbatim, each with its GG answer up front; history re-housed
%%   after. Part II extended: (1,5) spinor algebra + no-Majorana
%%   lemma, weight arithmetic, anomaly-flip, the three-depths
%%   instrument. Part III extended: the axiom-custody table restored
%%   in full; the two sources at depth. NEW Part IV "The Origin of
%%   Antiparticles" (Reports I-II GG content in full): the handshake
%%   theorems with mode-level proof, one shared i, the bulk encoding
%%   (no-Majorana applied, core-block conjugates, I8 chirality flip,
%%   d2 -> D2-, the 32*Omega singlet sea, two-kinds distinction),
%%   descent as repackaging, boundary readouts (positron/Wheeler/
%%   Hawking/custody), negative results, "the price of one i",
%%   Part-end originality ledger. Parts V-VII deepened: strong-CP
%%   leakage formula + Kaxion double protection displayed; Crooks
%%   product; PMNS degeneracy; parity derived-vs-asserted table +
%%   survival-argument subsection; m_star derivation; per-Part
%%   concede-then-claim ledgers. NEW Part VIII "The Five Roads, at
%%   Full Depth": QFT (Dirac sea risen and dismantled; Feynman-
%%   Stueckelberg; WEINBERG'S CAUSALITY THEOREM stated; Wigner;
%%   Jost's anatomy-silence; Bisognano-Wichmann; Greenberg; direct
%%   T violation; Vafa-Witten + survival argument; the baryogenesis
%%   toolbox incl. ARS + Affleck-Dine), the SM (five silences,
%%   10^-20, crossover), string theory (worldsheet CPT, Chan-Paton
%%   + KK orientation-reversal precedent conceded, CP as gauge
%%   symmetry, Kawamura, moduli problem), LambdaCDM (the
%%   bookkeeper), and the Depth-1/2/3 synthesis with consolidated
%%   tables. Conclusions: NEW "From mirrors to mind" --- the
%%   oriented-emergence story (particles, life, mind) at A11 grade.
%%   NEW Course D "The antiparticle, four ways" (6 lectures, 18
%%   problems). Objections 41-50 added (Weinberg-in-costume, KK
%%   precedent, Ashtekar-Magnon, BW scope, free-field scope, Dirac-
%%   sea nostalgia, readout wordplay, Hawking partner, conflation,
%%   sixth-dimension dependence). ~20 new references with notes
%%   (Dirac trilogy, Anderson, Stueckelberg 1941/1960, Feynman
%%   1949, Weinberg QTF, Sonoda, Polchinski, Berkooz-Douglas-Leigh,
%%   Strominger-Witten, CPLEAR, BaBar 2012, Kostelecky, ARS,
%%   Affleck-Dine...). Abstract/ES/Preface rewritten answer-first,
%%   in the rigorous-yet-warm register; front-matter counts
%%   refreshed.
%% v1.1 (July 16, 2026): the growth pass (author-commissioned:
%%   "expand all recommended areas; add more graphs and tables").
%%   Part I history deepened: C-violation double-fall paragraph;
%%   kaon formalism displayed (eps, eps'/eps, the double ratio,
%%   superweak + Wolfenstein1964 ref); NEW chronology table
%%   (18 rows, cited actors) and NEW CPT-tests table (6 flagship
%%   comparisons). EIGHT new figures: flavour graph + plaquette
%%   (Part II); orientation reversals, pentagon, two-fate Wilson
%%   lines (Part IV); basin/gate landscape (Part V); leptogenesis
%%   flowchart, resonance regime map (Part VI); falsifier timeline
%%   (Part VII). NEW tables: basin/gate (Part V), numeric audit
%%   (Part VI), predicted-vs-measured (10 rows, Part IV). NEW
%%   Worked Examples section (five, end to end with numbers: Wu,
%%   CKM plaquette, sin2beta, nu-asymmetry at 196 deg, the door in
%%   matrices). Problem sets expanded to three per lecture for all
%%   eighteen lectures (36 further problems with hints + answers,
%%   consolidated per course). Front-matter apparatus counts
%%   refreshed.
%% v1.0 (July 16, 2026): first complete draft, on the author-approved
%%   TOC (P5-CPT-TOC-Proposal-2026-07-16). Title (a). Full scope:
%%   discrete symmetries + parity + baryogenesis. Eleven Parts on the
%%   P4-Twistor chassis: history; preliminaries; arena; the discrete
%%   symmetries derived (T-P5-1..4); parity as the oriented descent
%%   (incl. the new handedness-invariant proposition P-P5-D and the
%%   registered no-W_R falsifier); the origin of matter (T-P5-5 at D*
%%   under Conj. VI, with the resonance gap faced openly); Discussions
%%   (five roads: QFT, SM, strings, LambdaCDM, GG); Conclusions (incl.
%%   the extended personal perspective "Fifty years in the mirrors");
%%   Technical Appendices (three courses, eighteen lectures);
%%   forty objections in nine categories; A/B references with the
%%   inherited-mechanism citations paid in full (Fukugita-Yanagida,
%%   Pilaftsis, Davidson-Ibarra, Kawamura, Dine-Leigh-MacIntire...).
%%   Radial projection stated as canonical per author decision.
%% ===================================================================
%% \pdfoutput=1 added v15.0: section 12 item 10 requires it in every
%% arXiv package.  P4-Real reached a built package without it and the
%% full nine-tool gate passed clean, because the item is on the v20
%% checklist and not in the battery.  Added at Stage 1 here instead.
\pdfoutput=1
%% ---- vv15.10 (August 19, 2026) ----
%% The front-matter showcase pages given breathing room.
%% A showcase page's free height is now distributed by weighted stretch
%% glue -- a sixth at the top, a third above the plate, a third below it,
%% a sixth at the foot -- instead of pooling above the headline and below
%% the caption, which left a broad band of white at each end of the page
%% and crowded the three elements into the middle third.
%% Three source forms were found in the cohort and each is handled.  A
%% minipage frame becomes a minipage of EXPLICIT height with the [s]
%% inner alignment, because a minipage is a vbox of natural height and
%% glue inside one does not stretch; the height is \textheight minus
%% \topskip.  A \begingroup frame keeps its page-level glue and changes
%% only the weights.  An unframed page -- there were eight, five of them
%% showcase tables carrying a longtable, which cannot live inside a
%% minipage -- gains page-level glue for the first time, which is why it
%% had been printing hard against the top of the text block.
%% 1 showcase page in this paper.  Layout only: not one word,
%% number, figure or table row is changed by this ship.
\documentclass[11pt,letterpaper]{article}

\usepackage{ucla-arisaka-gg-11}
\usepackage{indentfirst}
\renewcommand{\LaTeX}{LaTeX} % cohort-standard shim

%% ---- Per-paper identity -------------------------------------------
\PaperSeries{GGP}
\PaperStatus{Foundations Branch}% P-series designator retained; branch named per the 19-paper map
\PaperNumber{Arisaka-GG-P5-CPT}
%% ------------------------------------------------------------
%% v3.52 -> v15.0, August 15, 2026.  STAGE 1: MECHANICAL.
%%   The nine-stage v15 pipeline, seventh paper conversion.
%%   a4paper -> letterpaper.  Letter is the SHORTER sheet, so the
%%   conversion ADDS pages; the widening is +3.7 %% and pushes
%%   plates toward 8.08 pt of type, which section 5 leaves alone
%%   while it is inside the house band.
%%   880 PRINTED BRITISH SPELLINGS -> 0, over 125 distinct words:
%%   THE LARGEST IN THE COHORT, against P4-Real's previous 307.
%%   flavour 182, programme 183, fibre 113, flavoured 36,
%%   quantisation 34, relabelling 22, honoured 21, normalisation
%%   20.  This paper's subject is the flavor sector, so the word it
%%   uses most was the one spelled the British way throughout.
%%   2 IN THE PDF METADATA -- pdfsubject and pdfkeywords both
%%   carried `flavour'.  No printed-text sweep sees them and any
%%   properties panel shows them.
%%   THE MAP WAS GENERATED BY RULE AND PRINTED WHOLE, not
%%   hand-written: 125 entries is where a typo becomes a shipped
%%   defect.  Every generated replacement is asserted NOT to be
%%   matched by the sweep rule itself -- the P4 `fiberwise' failure
%%   made a hard check instead of a discovery.
%%   `Teller' WOULD HAVE BEEN SWEPT.  The doubled-l branch that
%%   reaches traveller and labeller reaches teller, and all four
%%   sites here are the SURNAME, in `Gamow--Teller'.  Protected in
%%   gg_spelling, beside telling/selling/swelling/retelling.
%%   \pdfoutput=1 added -- section 12 item 10, absent here as it
%%   was in P4-Real.
%%   P5 IS NOT A STALE ARTIFACT: deployed 319 pages, rebuild 319,
%%   and pagecmp reports 0 of 319 pages differing.
%%   NO PROSE WAS WRITTEN IN THIS STAGE.
%% ------------------------------------------------------------
%% ------------------------------------------------------------
%% Stage 1B, same day.  `storey' -> `story', and the tool fixed.
%%   P5 wrote the building metaphor `one more storey' once while
%%   writing `story' 44 times elsewhere in the same paper.  P4-Real
%%   carried the identical sentence and the identical defect.
%%   THE CAUSE WAS THE TOOL, and it is now fixed rather than
%%   worked around: gg_spelling had storey removed from the sweep
%%   RULE *and* placed in the cohort-global PROTECT set, because
%%   A11-Origin uses `storey-port' and `a storey count S' as defined
%%   terms at 142 sites.  A global exemption for one paper's defined
%%   term let the ordinary word through everywhere else.
%%   storey[s]? is back in the rule, and the exemption moved to a new
%%   PAPER_PROTECT map keyed by filename, which A11's own migration
%%   will pass as extra_protect.  Re-run against a file with the known
%%   defect (P4-Real v3.8) it fires; against the fixed v15.0 it does
%%   not.
%%   MEASURED COHORT-WIDE: A6-GRIP 1, A7-Qi4 2, C2-Hubble 3, C3-BH 2
%%   printed sites still live, plus A2, A5 and W2 at v13 -- while
%%   A10-Cosmos had already converted its own 23.  A rider follows.
%% ------------------------------------------------------------
%% ------------------------------------------------------------
%% Stage 2A, same day.  Four bare grids into captioned floats.
%%   float_audit on the baseline: 4 uncaptioned bare tabulars and 1
%%   unbreakable tabular over six rows.  They are the same objects;
%%   a float cannot break across a page, so promoting each answers
%%   both counts at once.
%%   THE GRIDS ARE COPIED VERBATIM.  Not a rule, a column spec, a
%%   row or a value is touched.  What is added is a caption above
%%   (section 8), a label, a written short form, and \small on the
%%   one grid that did not carry it.
%%   CAPTIONS DESCRIBE, THEY DO NOT CLAIM.  Every statement in each
%%   caption was read off the grid it sits on; where the
%%   surrounding prose makes a stronger claim than the table shows,
%%   the caption stays with the table.
%%   Located by build_mask, not by text anchor: this paper
%%   hard-wraps and one grid sits inside a paragraph whose
%%   flattened form does not exist in the source.
%%   NEW PROSE: four captions and four short forms.
%% ------------------------------------------------------------
%% ------------------------------------------------------------
%% Stage 2B, same day.  Table captions moved above the grid.
%%   Section 8's float standard ends "caption above" and section
%%   11 lists caption position among the Stage 1 mechanical items.
%%   P5 set 22 of 32 table captions BELOW and 10 above: it was
%%   inconsistent with itself before it was inconsistent with the
%%   cohort.  MEASURED, not assumed -- every shipped v15 paper
%%   checked is 100 %% above (A5-GDOS 32/0, A2-GG6 21/0, P4 5/0).
%%   Caption and label move together to just after \centering,
%%   where this paper's own ten correct tables put them.  Caption
%%   text, short form, label and grid are copied byte for byte;
%%   only their order inside the float changes.
%%   FIGURES ARE NOT TOUCHED.  Every figure in the cohort captions
%%   below the picture, P4-Real included, and float_audit does not
%%   test position; moving them would be a house change, not a
%%   repair, and section 14.2 owns house changes.
%%   NO PROSE WAS WRITTEN IN THIS STAGE.
%% ------------------------------------------------------------
%% ------------------------------------------------------------
%% Stage 2C, same day.  Nineteen thin captions extended.
%%   Nineteen is the most of any paper in the cohort.  EVERY
%%   EXISTING SENTENCE IS KEPT VERBATIM and appended to; section
%%   14.2 reserves rewriting a caption's claim and none is
%%   rewritten.
%%   WHAT IS ADDED WAS READ OFF THE GRID, NOT OFF THE PROSE.  Each
%%   extension names the table's own columns, its own row count,
%%   and what the columns are for.  Where the surrounding prose
%%   makes a stronger statement than the grid shows, the caption
%%   stays with the grid.
%%   ONE NUMBER DELIBERATELY NOT REPEATED: tab:part6-ledger's last
%%   row carries a percentage for the two-route baryon-density gap
%%   which is NOT section 8's 16.4 %% two-road gap.  They are
%%   different quantities and a caption carrying either invites
%%   the reader to conflate them, so the caption says the gap's
%%   owner is unidentified -- what the row says -- and no figure.
%%   All nineteen short forms already existed and none is touched:
%%   P5 is the first paper in the cohort to arrive with one
%%   written on every thin caption.
%%   NEW PROSE: nineteen caption extensions.
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%% Stage 2D, same day.  List of Tables; 13 Plain-English Parts.
%%   FRONT MATTER: \GGListOfTables was absent against thirty-six
%%   captioned tables, the most of any paper in the cohort.
%%   THIRTEEN PLAIN-ENGLISH PART INTRODUCTIONS, the most in the
%%   cohort -- P4-Real and P3-Electron took eight, P2-N3 five.  P5
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%%   Part and page in GG-Report-367 for review or rewriting.  Each
%%   states the orthodoxy at full strength before contradicting it
%%   and names the grade in the same breath as the result.  Part
%%   V's calls the relation to Jost's theorem COMPLEMENTARITY and
%%   not replacement, because the paper explicitly disclaims the
%%   stronger reading and an earlier draft of mine did not.  Part
%%   VIII's names the baryon-bridge gap in the same paragraph as
%%   the zero dial count.
%% ------------------------------------------------------------
%% ------------------------------------------------------------
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%% ------------------------------------------------------------
%% Stage 5, same day.  Six missing cohort companions.
%%   P5 carried 15 of 21.  Missing: A4-Euclid, P2-N3, C2-Hubble,
%%   P6-Proton, W1-Symmetry, W2-Holonomy -- the classic three
%%   again, the sixth paper running, plus three more.  The widest
%%   gap since P2-N3's nine.
%%   ADDED WITH THE PROSE THAT CITES THEM: one new paragraph, `The
%%   rest of the map', closing sec:inheritance and placed before
%%   the sentence that hands off to the cohort figure.
%%   RELEVANCE NOTES WRITTEN FOR THIS PAPER, not copied.  Each says
%%   what the companion does for the mirrors, the orientations and
%%   the matter excess.
%%   TWO EARN MORE THAN A MENTION.  P2-N3 is a PRECONDITION and not
%%   a neighbor: a nonzero Jarlskog invariant needs three families,
%%   so without that paper's count this one has no CP phase to
%%   explain.  W2-Holonomy's exact statement -- a holonomy requires
%%   a closed loop and the interval fiber has none -- is the same
%%   asymmetry Part IX derives the four endowments of time from.
%%   \citealp used singly throughout: section 7 forbids two Arisaka
%%   keys in one \citep, and reading that as a false-positive class
%%   cost A10-Cosmos and C2-Hubble printed defects.
%%   NEW PROSE: one paragraph and six Relevance notes.
%% ------------------------------------------------------------
%% ------------------------------------------------------------
%% Stage 6A, same day.  Front-matter list overflow.
%%   The new List of Tables pushed one entry onto a page of its own
%%   and the Contents pushed `Part XVI. References' onto another --
%%   two near-empty pages at 0.55 %% and 0.59 %% ink.  Section 7's
%%   remedy for front-matter overflow is \captionsetup{list=no} on a
%%   COHERENT CLASS of floats, named in the report.
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%%   ALREADY ITS OWN NUMBERED APPENDIX SECTION.  Eight of them --
%%   Summary of Abbreviations, Glossary of Key Symbols, Internal
%%   source ledger, Cross-paper inheritance table, Consolidated
%%   benchmark audit, Named-theorem-and-conjecture deck,
%%   Named-conjecture promotion roadmap, Falsification-test
%%   catalogue.  Each has a \section heading carrying its own name,
%%   so every one is still reachable from the Contents by that name;
%%   being listed a second time in the List of Tables was the
%%   duplication, not the omission.  They are also the paper's
%%   reference apparatus rather than its results.
%%   THE FIRST ATTEMPT USED A WORSE CLASS -- `the four longtables' --
%%   which is a typesetting property and not a property a reader can
%%   see.  It also saved only four lines and left the Contents
%%   orphan standing.  The rule that a class must be one the reader
%%   could state is not decoration.
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%%   the List of Tables and nothing else changes.
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% ------------------------------------------------------------
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% ------------------------------------------------------------
\PaperVersion{v20.2}
\PaperDate{August 22, 2026}

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\hypersetup{
  pdftitle = {The Mirrors of Matter: Discrete Symmetries as Orientation, CP Violation as a Chiral Pentagon, and the Origin of the Cosmic Matter Excess},
  pdfauthor = {Katsushi Arisaka},
  pdfsubject = {Foundations Branch member paper of the twenty-two-paper GG-Theory cohort. Derives the C/P/T table as orientation bookkeeping of three real factors, the CPT theorem as total orientation reversal, CP violation as chirality of pentagonal-lattice flavor holonomies with forced values of the CKM and PMNS phases, parity violation as the orientation of the family descent, and the cosmic baryon asymmetry via TeV-scale resonant leptogenesis on an inherited ledger. Theorem-proof form with a technical-appendix machine room of Step-graded chapters, worked examples, and five-way structural comparisons.},
  pdfkeywords = {CPT theorem, discrete symmetries, charge conjugation, parity violation, time reversal, antiunitarity, CP violation, Jarlskog invariant, pentagonal lattice, flavor holonomy, CKM phase, PMNS phase, Majorana phases, strong CP problem, Kaxion, seesaw, leptogenesis, resonant leptogenesis, sphalerons, baryon asymmetry, baryogenesis, Sakharov conditions, matter-antimatter asymmetry, GG-Theory, sixth dimension, phase circle, OGGEP, GDOS, GRIP, falsifiability}
}
%% ---- arXiv submission metadata -------------------------------------
%% Primary: hep-ph. Cross-lists: hep-th, hep-ex, astro-ph.CO.
%% Comments field: ~300 pp; figures + tables; eleven named theorems, one
%%   new proposition, three named conjectures carried at honest grade;
%%   five Step-graded appendix chapters with ninety fully worked
%%   problems; eighty-two answered objections; annotated references; Foundations Branch member of the
%%   nineteen-paper GG-Theory cohort (arisaka-gg.org).
%% Abstract: within the 1920-character arXiv limit (verify at ship).
%% ---------------------------------------------------------------------


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\begin{document}
% ---- v3.48 (August 2, 2026) ----
% Part IX 'What Is Time?' dramatically expanded in the two-fibre
% language (grounding note GG-Note-What-Is-Time). New sections:
% 'The two fibres: where the endowments live' (anatomy table; tick
% and arrow on different fibres; why mechanics and thermodynamics
% grew up apart; clock vs hourglass; uniqueness strengthened);
% 'Clocks: every clock is a winding counter' (Compton/caesium/
% Kaxion 1.95 GHz/Hubble; e^140 range); 'The chronology,
% quantified' (d ln t/dr = 2 exact; eight-epoch table in both
% r-conventions; age ledger 140.24 = 36.1+91.6+12.5; ln(1+z_sw) =
% J; flagged 4K candidate at numerology-discipline grade; the
% chronology figure); 'Temperature, the shore, and the scale rate'
% (KMS married to Unruh-Tolman; CMB as oldest thermometer); 'The
% significance: the new definition against every previous one'
% (Newton/Leibniz/Einstein/Boltzmann/QM/Wheeler-DeWitt/
% Page-Wootters/Connes-Rovelli honoured and located; the two
% claims no predecessor makes). 'How much time is there?' added to
% the answer-for-everyone; Part subtitle and Part-end summary
% refreshed to match. All numbers verified programmatically.

% ---- v3.47 (August 2, 2026) ----
% New Part VIII section 'The alphabet audit: every dial of the
% birth, valued' (sec:alphabet-audit), after the resonance section:
% every CP quantity of baryogenesis as a letter of the pentagonal
% alphabet (delta_CKM 2pi/5 chiral/bystander; delta_PMNS anchor pi
% + 1/J deviation, spectator; alpha_21 = 0 inert; alpha_31 = pi THE
% driver, half-angle i); the paradox of the achiral drivers stated
% and resolved (half-angle / saturation / pentagon-chirality); the
% existence-sign-magnitude trichotomy; native-TikZ alphabet figure;
% the determinism claim with its honest boundary; falsifier
% pointers. One pointer sentence appended to PS-P5-7. Numbers
% re-verified programmatically. First v13-project P5 ship.


%% ===================================================================
%%  COVER PAGE
%% ===================================================================
\thispagestyle{empty}
\GGTitlePage%
  {The Mirrors of Matter}%
  {Why Every Particle Has a Twin, Why Nature's Mirrors Break,\\
   and Why the Universe Is Made of Matter}%
  {August 12, 2026}
\thispagestyle{empty}

\vspace{6pt}
\begin{abstract}
\noindent
Why is there something rather than nothing? A universe born
even-handed between matter and antimatter should have annihilated
to light. And why do the three mirrors of nature --- charge (C),
space (P), and time (T) --- each fail to reflect the world
faithfully, while their combination CPT never fails? This paper,
a member of the \GGTheory{} cohort's Foundations Branch, answers
both questions with one idea: the mirrors are not laws but orientations. The
world is built from real, oriented geometry --- a compact circle,
space, and a one-way clock --- and C, P, and T are the reversals
of those three orientations. Each can break because the geometry
has texture; the total reversal CPT survives because it is the
one reflection no real world can detect. The antiparticle is
derived at its stated grade, not postulated: it is the same
object read the other way around a real circle, and the reading
forced there also locates
the origin of spin-$\tfrac12$ and of the exclusion principle (at
candidate grade). Time itself is defined by four endowments ---
it orders, it repeats, it opens one way, and it shelters the
mathematics of the quantum world --- the subject of a new Part.
The receipt is quantitative: on a fully inherited ledger the
quark and lepton CP phases take exact lattice values,
$66.9^{\circ}$ and $196.4^{\circ}$ --- the first reached by one
frozen, pre-registered completion step --- the neutrino masses
are derived, and the universe retains about six extra particles of
matter per ten billion photons --- the measured excess from which
chemistry, life, and mind followed. Every claim carries its
grade --- derived, conditional, candidate, or asserted --- and
the paper's tables say which. The claim is mortal: DUNE,
Hyper-Kamiokande, JUNO, the ton-scale double-beta searches, and
the high-luminosity LHC can each kill it within this decade, and
the paper says where.
\end{abstract}

\clearpage
\begin{ExecutiveSummary}
\noindent
\textbf{The questions that remain.} Two companion papers of the
\GGTheory{} program derive, from one six-dimensional geometry,
the twenty-eight parameters of the Standard Model
\citep{ArisakaA9SM} and the inventory of the standard cosmology
\citep{ArisakaA10Cosmos}. Between those two derivations lies a
seam, and on that seam live the oldest unanswered questions in
fundamental physics.

Where do antiparticles come from --- what forces every charged
particle to have a twin of equal mass and opposite charge? What
\emph{are} the discrete symmetries C, P, and T --- charge, mirror,
and time reversal --- and why is exactly one of them antiunitary?
Why did parity fall maximally, and in exactly one force? Why does
CP violation exist, at exactly the measured strength --- and why
does the strong interaction, alone, conserve it to one part in
ten billion? And the question that contains the others: why did
the universe keep six baryons per ten billion photons instead of
annihilating to light --- why is there matter, and therefore
chemistry, and therefore us?

The Standard Model stores each of these facts as an input; this
paper, standing at the interface of its two companions, derives
them as consequences of one structure. It is, in the program's
own terms, the paper about the asymmetry of matter --- which is
to say, about our origin.

\smallskip\noindent
\textbf{The one answer.} The world of the \GG{} bulk --- the real
six-dimensional arena beneath the boundary we inhabit --- is
built from \emph{real, oriented} geometry: the
thesis of its Foundations-Branch partner \citep{ArisakaP4Real}. Nature
requires no complex primitive, and every imaginary unit in
physics is the derived packaging of real orientations. Taken
seriously, that single idea makes this paper's subject
transparent.

Antiparticles are the unique alignment of the bulk's two real
rotation structures --- the quarter-turn $J$ the boundary will
write as $i$ --- forced wherever charge lives. The discrete
symmetries are the reversals of the bulk's three orientations;
their table, including the antiunitarity of T, is derived, not
assigned.

Their violations are not accidents but the texture of the
descent: a chiral pentagon where the flavors turn, an oriented
cascade where the families fall --- and, at the end of the chain,
a universe that kept six parts in ten billion of itself, from
which chemistry, life, and mind became possible. The mirrors
broke because the world has a direction; we are among the
consequences.

\smallskip\noindent
\textbf{The theorems that carry it.} Before the mirrors, the
twins: Part~IV derives the antiparticle itself, one level below
quantization. T-P5-A: on each charged sector the circle's
quarter-turn $\Jq = \partial_\theta/Q$ is the unique
circle-compatible complex structure, and conjugation is
orientation reversal of the fiber. T-P5-B: the clock's polar
phase $\Ja$ --- the Hilbert transform, a real operator --- is the
unique positive-energy complex structure. T-P5-C: the two agree
on shell precisely under the Feynman--St\"uckelberg pairing ---
\emph{antiparticles are the consistency condition between the
sixth dimension and causal time}. T-P5-C1/C2 extend the one
shared $J$ to composites and Fock space with no new complex
input; T-P5-B1, the Wick quartet, shows that Minkowski's $it$,
Wick's rotation, and the Schr\"odinger $i$ are one datum --- the
polar decomposition of the time generator, whose lower half-plane
is the only lawful home of ``imaginary time.''

Then the mirrors. T-P5-1: C, P, and T are the orientation
reversals of the three real factors of the \GG{} bulk --- the
compact fiber circle, the spatial slice, and the causal clock ---
and a discrete symmetry is antiunitary if and only if it contains
T, because of the three orientations only the clock's is what the
imaginary unit is made of: the textbook unitary/antiunitary table
is derived, not assigned. T-P5-2: the total reversal CPT
preserves each Feynman--St\"uckelberg-paired mode set and is a
symmetry for every coupling texture built from real geometric
data --- the CPT theorem as orientation bookkeeping. The
dividends follow free of charge: Kramers' degeneracy with no
phase conventions anywhere, and the five fates of the mirrors ---
which broke, in what order, by how much --- as consequences of
one projection (itself asserted, per Part~VI's ledger) and one
lattice.

Then the flavors. T-P5-3: a Yukawa texture is a $\mathrm{U}(1)$
connection on the flavor graph; its physical content is its loop
holonomies; CP is a symmetry iff every loop angle lies in
$\{0,\pi\}$; and on the single circle of the \GG{} bulk, with the
block integer $L=5$, first-order holonomies are locked to the
pentagonal lattice $(\pi/5)\bZ$ --- so CP violation exists
because $3\pi/5$ is not a multiple of $\pi$: the flavor pentagon
is chiral. The values, each at its grade: the quark phase
$\dCKM = 2\pi/5$ at first order, completed --- by the codebook's
one asserted quantum-defect step --- to $13\pi/35 \simeq
66.9^{\circ}$; the
lepton phase $\dPMNS = \pi + 2\arcsin(1/7) \simeq 196.4^{\circ}$;
Majorana phases $(0,\pi)$; the Jarlskog invariant $\JCKM \simeq
3.1\times10^{-5}$ at the stated evaluation layer (layer band
$2.7$--$3.2\times10^{-5}$). T-P5-4: a Majorana self-pairing exists iff the
fiber weight vanishes --- lepton number is winding number, and
the weight-zero neutrino is the unique dial-free door.

Then the matter. Part~VI argues --- as a structural claim whose
missing lemma is named --- that parity violation is the
orientation of the family descent: the unique broken force is the
unique gate force, and a parity-even coupling cannot carry the
direction of a one-way cascade. Part~VIII assembles the origin of
matter: Sakharov's conditions are derived as the boundary shadows
of the constitutional tripod (proposition P-P5-E: the ledger and
its door, the oriented descent, the always-on projection), and
T-P5-5 assembles the excess via TeV-scale resonant leptogenesis
on the inherited neutrino and CP ledger, every input inherited
at its stated grade ---
$\Mlock = \tfrac12 M_{\mathrm{Pl}}e^{-35} \simeq 3.85$~TeV, the
dial-free door, the seesaw-matched Yukawa, the holonomy CP phase
--- yielding $\etaB \approx 7\times10^{-10}$ (central $7.34$; band $\times3^{\pm1}$) against the observed $(6.12\pm0.04)\times10^{-10}$, at order-of-magnitude grade under one named conjecture and one flavored projection
step carried derived-conditional on the named partial-resonance
hypothesis H-$\phi_{\mathrm{geo}}$-1, with zero dilution derived
and the resonance question quantified rather than hidden.

\smallskip\noindent
\textbf{The apparatus.} The paper is built to be checked rather
than believed, across its sixteen Parts: a seventy-year
experimental history with its eighteen-row chronology and the
six-comparison CPT-test record; a preliminaries Part defining the
real presentation, the flavor graph, and the transport equations
from scratch; the antiparticle Part, which also locates the
spin--statistics connection and the exclusion principle as the
Wick quartet's second child (carried at candidate grade); and a
Part asking what time \emph{is}, answering with the four
endowments --- order, repetition, the one-way arrow, and the
analytic shore that shelters the quantum world's mathematics.

The machine room: a lettered technical appendix in the cohort
standard --- the reference dictionaries and cross-cohort naming;
five from-scratch, Step-graded chapters, on the discrete
symmetries in quantum mechanics and quantum field theory, on the
flavor sector (with $\varepsilon$, $\varepsilon'$, and
$\sin 2\beta$ derived from the textbooks and then priced by the
pentagon), on the origin of the matter excess (Sakharov's
conditions to the assembled $\etaB$, priced by the ledger), on
the antiparticle (the sea, the contour, and the causality
theorem, then the handshake as the fourth way), and on the
neutrino's century (two-state quantum mechanics to the lattice
point) --- ninety problems across the block, all with complete
worked solutions.

The instruments: five worked examples executed with numbers, from
Wu's cryostat to the neutrino asymmetry at $196^{\circ}$;
twenty-one figures --- among them the twenty-two-paper cohort map,
the roadmap of the sixteen Parts, the deductive funnel that draws
the central claim on one page, the custody map, the pentagon, the
basin/gate landscape, the leptogenesis flowchart, the resonance
regime map that draws the paper's sharpest open item, the 27-row
honesty audit at a glance, the twin deviation panels, and the
dated kill-test timeline --- and
thirty-two tables, including the axiom-custody table, the ten-row
forced-versus-measured confrontation, the three-depths
scoreboard, and the dated falsifier table with its timeline.

The comparisons: a five-way structural comparison at full depth
--- quantum field theory, the Standard Model, string theory,
$\Lambda$CDM, and \GGTheory{}, each given its own chapter and its
own concessions; eighty-two adversarial questions, constructed
in-house across eleven referee perspectives --- the kaon and
neutrino experimentalists first among them --- and answered with
pointers into the paper; and references each carrying a
relevance note, with
the inherited mechanisms --- Fukugita--Yanagida, Pilaftsis,
Kawamura, Dine--Leigh--MacIntire --- cited at their source.

\smallskip\noindent
\textbf{The register.} Derived where provable, conditional where
conditional, open where open. The discrete-symmetry table and the
holonomy calculus are theorems at the quasi-free grade; the
gate-operator lemma (parity), the determinant-phase cancellation
(strong CP), and the sphaleron-transport normalization (the $K/2$
exponent) are named conjectures, stated as such; the resonance
condition of the leptogenesis chain is its one named input, and
the twelve-order gap between the geometric splittings and the
width is quantified in the text.

Two small additions are made to the cohort's ledger: a
convention-free handedness invariant (the surviving chirality,
the descent orientation, and the sign of the CP holonomy as one
alignment), and a registered falsifier --- \emph{no right-handed
weak sector at any scale}: the mirror modes were projected out,
not made heavy, so a $W_R$ observed at any energy kills the
geometric construction.

\smallskip\noindent
\textbf{Who should read it, and how.} \emph{A kaon or $B$-physics
experimentalist} meets the phases of their own measurements ---
$\varepsilon$, $\varepsilon'/\varepsilon$, $\sin 2\beta$ ---
re-read as interferometry of a single geometric angle, with the
lattice prediction laid against the world averages. \emph{A
neutrino physicist} meets a forced leptonic phase within
$0.2\sigma$ of the current global fit, a definite double-beta
amplitude $\mbb$, and the statement that their next decade of
data adjudicates this paper. \emph{A cosmologist} meets a
baryogenesis chain with no dials and an honest error budget.

\emph{A graduate student} meets a self-contained course sequence
connecting Wigner's theorem, the CPT theorem, flavor physics,
and the Boltzmann transport of leptogenesis to one geometric
source. \emph{A general physicist} meets a plain thesis: the
universe is oriented --- circle, clock, and cascade --- and the
broken mirrors, the pentagon of flavor, and the matter in your
hand are three measurements of that one orientation.

\smallskip\noindent
\textbf{The philosophy, last.} The mirrors broke on the descent's
orientation; the clock stands on time's four endowments; and the
deepest claim of the paper is that orientation, not substance, is
what distinguishes matter from antimatter and past from future. A
twin is the same object read the other way around a real circle;
an arrow is a reading the world cannot unread. And it is held to
the one standard that makes it science: the paper can be wrong by
2035, and it says where.
\end{ExecutiveSummary}

%% ---- v15.2: the showcase headline and the four imported plates ---
%% \Large is the house default as of Report 391.
%% 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.
%% Each box holds the UNSCALED tikzpicture; the \resizebox stays at
%% every use site, so the body copy and the showcase copy are the same
%% drawing at different scales and never diverge.  Filled at
%% \setstretch{1}, which is what P5's figure environments run --
%% probed INSIDE the environment, per Report 392.
%% THE DOOR PLATE IS RECOLOURED to the house palette of section 8:
%% S2's Warn/Good/Accent become red!60!black / green!45!black /
%% blue!55!black.  Verified against the source: identical extracted
%% text and identical glyph bounding boxes, 97 boxes, colour only.
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\begin{scope}
\node[align=center] at (0,2.8) {\textbf{(a) The circle $S^1_\theta$ (6th dimension)}\\[-1pt]
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\draw[->,line width=0.95pt] (0.2,0.4) -- (6.6,0.4);
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\def\PW{4.9}

\foreach \i/\who/\laps in {0/{electron}/{1 lap}, 1/{quark}/{a third of a
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  \begin{scope}[shift={(\i*\PW,0)}]
    \draw[frame] (-1.9,-3.7) rectangle (1.9,2.5);
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\begin{scope}[shift={(0,0)}]
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\begin{scope}[shift={(\PW,0)}]
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  \draw[leg, draw=black!70] (2.05,0) -- (2.222,0) -- (2.136,0.149) -- cycle;
  \fill[black!70, opacity=0.5] (2.05,0) -- (2.222,0) -- (2.136,0.149) -- cycle;
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  \path (current bounding box.south) coordinate (s2drawctr);
  \node[lab, align=left, anchor=north, text width=295.0pt]
       at ([yshift=-7pt]s2drawctr)
    {Both areas are real and both are measured. The unitarity triangles are
     drawn here to \emph{true relative area}: the lepton area is
     \textbf{three hundred times} the quark area. \S\ref{sec:road-sm} prices
     the disparity that actually matters for baryogenesis: the effective
     $CP$ violation of the quark route computes to $\sim\!10^{-20}$, ten
     orders short.};
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    {Every canonical proton-decay mode has $\Delta B = 1$; neutron--
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     $\Delta B \in 3\mathbb{Z}$ and is unaffected. \textbf{A single event
     in either forbidden class falsifies the framework} --- and Hyper-
     Kamiokande will improve the bound on the first class by about an
     order of magnitude.};
\end{tikzpicture}}

\GGKeywords{CPT theorem; discrete symmetries; charge conjugation;
parity violation; time reversal; antiunitarity; Wigner's theorem;
CP violation; Jarlskog invariant; rephasing invariance; flavor
holonomy; pentagonal lattice; CKM matrix; PMNS matrix; Majorana
phases; neutrinoless double beta decay; strong CP problem; Kaxion;
seesaw mechanism; dial-free door; leptogenesis; resonant
leptogenesis; sphalerons; baryon asymmetry; Sakharov conditions;
matter--antimatter asymmetry; basin and gate forces; oriented
descent; orbifold projection; handedness invariant; right-handed
$W$; \GGTheory{}; sixth dimension; phase circle; integer ledger;
OGGEP; GDOS; GRIP; falsifiability}

%% ---- v15.2 showcase page: THE TWO FIBERS, AND THE ONE DOOR ----
%% P5 had no showcase page.  The two plates are the paper's deepest
%% picture and its most arresting one: the anatomy that makes CPT
%% exact, and the criterion that leaves exactly one species able to
%% be its own antiparticle.  The machinery, then what it decides --
%% the pairing P3-Electron's page uses.
%% Unnumbered (\captionof*), so every numbered float keeps its number.
%% ---------------------------------------------------------------
\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{The Circle Remembers What the Interval Forgets ---\\[3pt]
and Only One Particle Can Be Its Own Opposite}
\par\vspace*{\stretch{2}}
{\setlength{\abovecaptionskip}{5pt}%
\small\captionof*{figure}{\textbf{Why one mirror survives, and which door it leaves open.} \emph{Above:} the arena has two fibers and they hold different kinds of thing. On the compact phase circle every path closes, so everything a connection does there is holonomy --- quantized, protected, and unable to be deformed by anything the boundary does. Along the scale interval no path closes, nothing is protected, and the descent visits each scale exactly once. $CPT$ is exact because it is a statement about classes on the circle; $C$, $P$ and $T$ break individually because their breaking is an address written along the interval, once and in one direction. \emph{Below:} the same circle decides which particle can be its own antiparticle. A particle and its conjugate wind in opposite senses, and winding is what electric charge is --- so the electron's one lap and the quark's third of a lap weld both doors shut, and exactly one species of known matter is left. That is why the origin of matter runs through neutrino physics, and the reason is structural rather than a matter of degree. In full: \S\ref{sec:anatomy-of-a-theorem}, p.~\pageref{sec:anatomy-of-a-theorem}, and Figure~\ref{fig:door-p5}, p.~\pageref{fig:door-p5}.}}
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\par\vspace{9pt}
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\par\vspace*{\stretch{1}}
\end{minipage}
\clearpage
%% ---- end showcase page ----------------------------------------


\GGfrontmatteronepagetrue
\GGOverviewOfContents
\GGListOfFigures
%% \GGListOfTables added v15.0: P5 carried the Overview, the List of
%% Figures and the Contents but no List of Tables, against thirty-six
%% captioned tables -- the most of any paper in the cohort, and four of
%% them promoted out of bare grids at Stage 2A.
\GGListOfTables
\GGTableOfContents

\GGPreface

This paper is written to be readable, end to end, by an experimental
particle physicist --- the reader who has sat shifts on a kaon beam,
tuned a $B$-factory trigger, or watched phototubes in a mine, and who
has measured the numbers this paper claims to explain. It was
structured so that such a reader loses nothing by knowing no
\GGTheory{} in advance: everything imported is restated with its
hypotheses, every definition is made in Part~II, and every theorem is
proved or proof-sketched on the page, with the physics said in words
immediately after the mathematics.

Here is the honest pitch for spending a week with it. You already own
the phenomena: parity violation from your first course, the
$\varepsilon$ and $\varepsilon'$ of the kaon system, the unitarity
triangle, $\theta_{13}$ and the hunt for $\delta_{CP}$, the six
baryons per ten billion photons of the CMB. What you likely do not
own is a single statement connecting them: that C, P, and T are the
orientation reversals of three real factors of one geometry; that the
antiunitarity of T --- the odd entry in every textbook table --- is a
one-line consequence; that the CP phase you measured is a closed-loop
angle on a real circle, quantized in pentagonal steps; that the same
orientation that breaks parity orders the mass hierarchy and biases
the heavy-neutrino decays; and that the matter excess is the last
entry in that chain of one fact. Those identifications are theorems
or honestly graded conditionals here, each about a page, and each
checkable with tools you already have.

A suggested week. Day one: Part~I for the questions and the story
you lived part of, and Part~II for the definitions --- do
not skip the flavor graph or the spinor algebra; the whole CP
story is bookkeeping on the first and the whole antiparticle
story stands on the second. Day two: Part~IV --- the handshake,
slowly, with the mode-level proof done by hand; it is the
deepest half-page in the paper and everything downstream uses
it. Day three: Part~V's two theorems with Appendix~\ref{app:discrete-qft} beside them,
then Part~VI (parity) --- the shortest and most structural.
Day four: Part~VII (the pentagon) in the morning with the worked
examples, and Part~VIII (the origin of matter) in the afternoon
with the audit table in hand. Day five: Part~IX --- \emph{what is time?} --- then Part~X, the five roads
--- where the comparisons you have been silently making are made
out loud, with every concession paid --- and
Appendix~\ref{app:flavour}, where $\varepsilon$, $\varepsilon'/\varepsilon$,
and $\sin 2\beta$ are recomputed as holonomy interferometry.
The weekend: the falsifier table with its dates, the Conclusions
--- including the story that runs from the mirrors to the reader
--- and as many of the eighty-two objections as you have appetite
for; if you have spent time at a kaon beam or a neutrino
detector, your objection is in there under your own job title.

Three further notes. The register of the paper --- what is proved,
what is conditional, what is open, what is conceded --- is stated at
every claim, and the reader is invited to audit it; the named gaps
(a lemma for parity, a normalization for the baryon number, a
resonance mechanism faced in its own section) are the paper's work
orders, not its fine print. The problem sets carry hints and answers
because they are meant to be done, not admired. And the personal
perspective at the end is an experimentalist's, not a theorist's:
the author spent 1980--1985 designing and constructing Kamiokande,
a decade inside CP violation at KTeV and BaBar, and the fifty years
since his first encounter with the broken mirrors wondering what
they were trying to say. This paper is the answer he wishes he could
have mailed to himself in 1976 --- offered now with the calibration
the intervening decades of experimental discipline demand.

\GGPart[The question and its history]{The Question and Its History}{The oldest asymmetry, asked precisely; a century of falling mirrors, worn with two hats --- the experimenter's and the philosopher's; and the problems that remain}
\label{part:history}

\section[Part~I in plain English]{Part~I in plain English: the mirrors, and seventy years of watching them break}
For most of the history of physics the mirrors were not a subject. That a law should look the same reflected in a mirror, or with matter exchanged for antimatter, or with time run backward, was not a hypothesis anyone thought to test. It was what a law was. Then in the space of eight years two of the three fell, and the second fell by about two parts in a thousand --- small enough to have been missed, and large enough that the universe is made of something.

This Part tells that history at full strength, because the strength is the point. Lee and Yang's audit found that parity had never actually been tested in the weak interaction. Wu tested it and it failed maximally. Cronin and Fitch found CP failing in a kaon. Through all of it CPT did not fail once, and a symmetry that survives everything its parts do not is a fact that wants an explanation rather than a certificate.

Nothing is derived in this Part. It states the questions, lays seventy years of measurement out as a single table, says plainly what experiment still cannot tell us, and locates the paper in its cohort. The claim comes at the end and not before: the mirrors are not laws but orientations, and a reflection can fail because the geometry has texture, while the total reversal cannot.

\section{The oldest asymmetry, asked precisely}
\label{sec:oldest-asymmetry}

Two companion papers of the \GGTheory{} program stand on either
side of this one. The particle companion derives the Standard
Model --- its gauge structure, its twenty-eight parameters, its
flavor geometry --- as the boundary theorem of one
six-dimensional bulk \citep{ArisakaA9SM}. The cosmology companion
derives the standard cosmology --- its six dials, its epochs, its
inventory --- from the same geometry \citep{ArisakaA10Cosmos}.
Between them lies a seam, and the seam is not a technicality: it
is the single place where the laboratory's smallest measured
asymmetries and the sky's largest one turn out to be one subject.
For the deepest number the cosmology paper must inherit from the
particle paper is not a mass or a coupling; it is a
\emph{bias} --- the microscopic preference for matter over
antimatter without which the derived universe would contain
neither stars nor readers. This preprint is the paper of that
seam. In the program's own terms, it is the paper about the
asymmetry of matter --- which is to say, about our origin.

Asked precisely, the seam's question splits into a ladder of
questions, and it will pay to state them at the outset in their
natural order, because the entire paper climbs this ladder.
\emph{Why does every charged particle have a twin} --- what
enforces, to twelve digits, the existence of a partner of equal
mass and opposite charge? \emph{What are the operations} C, P,
and T that exchange the twins and reverse the mirrors --- are
they conventions of our bookkeeping, or structures of the world?
\emph{Why did the mirrors break selectively} --- parity
maximally, and only in one force; CP subtly, and by exactly the
measured amount; the combined reversal CPT never, to one part in
$10^{18}$? \emph{And why did matter win} --- by what mechanism
did a universe that should have annihilated to light keep six
baryons per ten billion photons, the residue from which
everything else is made? The Standard Model, magnificent as
bookkeeping, stores every one of these facts as an input
\citep{ArisakaA9SM}; the standard cosmology, exact as
accounting, holds the last one as an initial condition
\citep{ArisakaA10Cosmos}. Between the two, the questions simply
wait.

One further word about method, because it governs the whole
paper. This subject, uniquely in fundamental physics, has always
been worn with two hats. It is the most \emph{experimental} of
deep subjects --- every one of its turning points was forced by
an instrument: a cryostat, a spark chamber, a kaon beam, a tank
of water in a mine --- and it is simultaneously the most
\emph{philosophical}: the questions ``is left--right a
convention?'' and ``why is there something rather than
nothing?'' were asked by philosophers centuries before physics
could touch them. This paper keeps both hats on throughout ---
every claim will carry both an instrument and an argument ---
and this Part tells the history the same way, because the
history is itself the best introduction the subject has: it is
the record of philosophy's oldest symmetry questions becoming,
one by one, measurements.

\section{Before the fall: mirror symmetry as philosophy and as
law}
\label{sec:before-the-fall}

The belief that nature cannot tell left from right is far older
than the physics that tested it. Kant, in 1768, built an entire
argument about the nature of space on the puzzle of
\emph{incongruent counterparts} --- the left hand and the right
hand, alike in every internal relation yet not superposable ---
and concluded that handedness must reside not in the objects but
in space itself \citep{Kant1768}. The philosophical instinct
that followed --- that a mirror-reversed world would be not
merely similar to ours but \emph{indistinguishable} from it ---
hardened, over two centuries, from instinct into unexamined
law. When physics acquired the language to say it precisely, it
did: Wigner, in 1927, made parity a quantum number
\citep{Wigner1927}, building on Laporte's empirical selection
rule in atomic spectra \citep{Laporte1924}, and within a decade
parity conservation was load-bearing structure --- used daily,
cited automatically, confirmed silently by every
parity-labeled nuclear level. Noether's theorem
\citep{Noether1918} had meanwhile taught physics its deepest
lesson about symmetry --- that continuous invariances are
conservation laws --- and the discrete inversions P, C, and T,
though outside her theorem's hypotheses, borrowed its prestige:
symmetries were what the world \emph{had}, and the mirror
symmetries were the simplest of all.

Almost no one thought to check. The honorable exception
deserves its place in every retelling: in 1950 --- six years
before Lee and Yang --- Purcell and Ramsey argued that the
question of a neutron electric dipole moment, forbidden by
parity, was \emph{experimental}, not a matter for assumption:
``the question of the possible existence of an electric dipole
moment of a nucleus or of an elementary particle\ldots becomes
a purely experimental matter'' \citep{PurcellRamsey1950}. Their
proposal --- test the mirror by hunting a dipole --- founded
what is now the seventy-year EDM program
(\S\ref{sec:history-cpt}), and their methodological point ---
that no symmetry, however beautiful, is above measurement ---
is the experimental hat of this paper in a single sentence from
1950. The philosophical hat has its own pre-1956 landmark:
Gardner would later crystallize it as the \emph{Ozma problem}
--- can one communicate the meaning of ``left'' to a distant
civilization by radio, using physics alone
\citep{Gardner1964}? Before 1956 the answer was believed to be
no, and the belief was wrong twice over, as the next sections
record.

\section{The mirror that broke: 1956--1957}
\label{sec:history-parity}

Until 1956, parity conservation was not a hypothesis of physics; it
was part of its common sense. That the laws of nature should not
distinguish left from right had the standing of an accounting
identity --- used daily in atomic spectroscopy through Laporte's
selection rule \citep{Laporte1924}, confirmed silently by every
parity-labeled nuclear level, and so entrenched that no one had
thought to test it where it had never been checked. The
$\theta$--$\tau$ puzzle forced the issue: two strange mesons,
identical in mass and lifetime, decaying to final states of opposite
parity. The community's instinct was to duplicate the particle. Lee
and Yang's instinct \citep{LeeYang1956} was to audit the books ---
and the audit revealed that parity conservation, exhaustively
verified in the strong and electromagnetic interactions, had
\emph{no experimental support whatsoever} in the weak interaction.
They proposed the decisive tests. Within months, Wu and her
collaborators \citep{Wu1957} had polarized $^{60}$Co nuclei in a
cryostat and watched the decay electrons emerge preferentially
opposite to the nuclear spin --- a pseudoscalar expectation value,
$\langle \vec{\sigma}\cdot\vec{p}\,\rangle \neq 0$, impossible in a
parity-symmetric world. Garwin, Lederman, and Weinrich
\citep{Garwin1957} confirmed it in pion--muon decay within days.

The community's disbelief is itself part of the historical
record, and worth preserving for the generation that has only
known the aftermath. Pauli, days before the news reached him,
wrote that he was ``ready to bet a very large sum'' that the
experiments would come out symmetric --- ``I do not believe that
the Lord is a weak left-hander'' --- and had the grace, days
after, to marvel at his own certainty about a question he had
never examined. Feynman had wagered fifty dollars that parity
would survive, and paid. The lesson both men drew is the one this
Part exists to transmit: the mirror had never been \emph{tested}
where it broke; an assumption had circulated as a law for thirty
years on the strength of its beauty alone. Purcell and Ramsey's
1950 warning (\S
ef{sec:before-the-fall}) had been exactly
right, and almost exactly unread.

Two features of the fall deserve emphasis, because this paper will
return to both. First, the violation was not small. The weak
interaction does not slightly prefer one hand; it is \emph{maximally}
chiral, coupling to left-handed fermion components and right-handed
antifermion components and to nothing else --- codified within two
years as the two-component neutrino \citep{LeeYang1957, Landau1957,
Salam1957} and the $V{-}A$ current \citep{SudarshanMarshak1958,
FeynmanGellMann1958}. An asymmetry that is total is not a
perturbation to be parameterized; it is a structure to be explained.
Second, the fall was a \emph{double} fall, a point often compressed
in retellings: charge conjugation broke in the same experiments. A
C-symmetric world with broken P would pair every left-handed
neutrino with a left-handed antineutrino; the observed correlations
(the $\mu^{+}$ from $\pi^{+}$ decay is left-polarized where the
$\mu^{-}$ from $\pi^{-}$ is right-polarized) showed C violated as
maximally as P, with the product CP surviving --- which is why
Landau's repair (Section~\ref{sec:history-cp}) was available at
all. Third, the violation was \emph{selective}. The strong and
electromagnetic interactions conserve parity exactly --- a fact later
promoted to a theorem for vector-like theories by Vafa and Witten
\citep{VafaWitten1984} --- so whatever breaks the mirror does so in
exactly one of the four forces. Seventy years later, the Standard
Model holds both facts as inputs. Part~VI of this paper holds them as
one output.

\section{The repaired mirror, and the Ozma problem}
\label{sec:ozma}

Parity's fall was absorbed with remarkable speed, because a
repaired mirror lay ready at hand. The 1957 experiments broke P
and C separately but preserved their product: the mirror image
of a left-handed neutrino is a right-handed antineutrino, and
that particle exists. Landau proposed CP as the true mirror of
nature \citep{Landau1957}, and for seven years the repair held
--- with a philosophical consequence that deserves to be made
explicit, because it is the cleanest illustration in physics of
what a discrete symmetry \emph{means} operationally. Consider
Gardner's Ozma problem \citep{Gardner1964}: by radio alone ---
no shared artifacts, no parcels --- teach a distant civilization
which hand is ``left.'' Before 1957 it cannot be done: every
law then known is mirror-blind, so every instruction (``wind
the wire this way\ldots'') presupposes the very convention it
is meant to convey. After Wu, it seems possible --- ``build the
cobalt experiment; the electrons emerge opposite the spin;
call that direction down and the rest follows'' --- \emph{until}
one notices the loophole Landau's repair creates: a civilization
made of antimatter, performing the same instructions with
anti-cobalt, would derive the opposite hand and be equally
convinced. Under exact CP, handedness and matter-ness are
jointly conventional: one can be defined only relative to the
other. The Ozma problem's full resolution therefore had to wait
for 1964: once CP itself fails, the kaon system's small
asymmetries single out matter from antimatter \emph{absolutely}
--- the long-lived kaon decays slightly more often to the
positron than to the electron --- and with matter-ness fixed by
physics, handedness follows. That a parlor puzzle about radio
messages is settled by a $2	imes10^{-3}$ asymmetry in a
strange meson is the two-hats character of this subject in
miniature; and the \GGTheory{} reading of the same fact ---
that the world carries an absolute orientation, of which the
kaon asymmetry is one laboratory shadow --- is Proposition
P-P5-D of Part~VI, the Ozma problem promoted to an invariant.

\section{The second fall: CP, 1964}
\label{sec:history-cp}

Parity's fall was absorbed quickly, because a repaired mirror was at
hand: the combined operation CP, spatial reflection composed with
charge conjugation, survived the 1957 experiments intact, and Landau
proposed it as the true symmetry --- the mirror image of a
left-handed neutrino is a right-handed \emph{anti}neutrino, and that
particle exists. For seven years the repair held. Then Christenson,
Cronin, Fitch, and Turlay, studying the neutral kaon beam at
Brookhaven \citep{Cronin1964}, observed the long-lived kaon --- the
CP-odd combination, forbidden to decay to two pions --- decaying to
two pions at the level of two per thousand.

The neutral kaon system deserves its place of honor in this history,
because it is the first and still the most exquisite
\emph{interferometer of the discrete symmetries}: a two-state quantum
system whose mass eigenstates are coherent superpositions of a
particle and its antiparticle, in which a fractional asymmetry of
order $10^{-3}$ imprints itself on macroscopic decay distributions.
Two parameters organize the system, and their definitions repay
display because the courses use them. The mass eigenstates are
\begin{equation}
|K_{S,L}\rangle \;\propto\;
(1+\varepsilon)\,|K^0\rangle \;\pm\; (1-\varepsilon)\,|\bar K^0\rangle,
\label{eq:kaon-states}
\end{equation}
so $\varepsilon \neq 0$ --- \emph{indirect} CP violation --- means
the long-lived state is not the CP eigenstate but carries a small
admixture of the wrong parity, $|\varepsilon| = (2.228 \pm
0.011)\times10^{-3}$; while
\begin{equation}
\mathrm{Re}\!\left(\frac{\varepsilon'}{\varepsilon}\right)
\;\neq\; 0
\label{eq:epsprime}
\end{equation}
--- \emph{direct} CP violation --- means the decay amplitudes
themselves distinguish matter from antimatter, isospin channel by
isospin channel. The distinction carried forty years of physics: a
``superweak'' interaction acting only in the mixing
\citep{Wolfenstein1964} could fake $\varepsilon$ but predicts
$\varepsilon' = 0$ identically, so the double ratio became the
discriminator between a new feeble force and CP violation resident
in the weak amplitudes --- the two parameters were the field's
precision targets for the next four decades. CP
violation was different in kind from parity violation: not maximal
but small, not structural but seemingly accidental --- a phase that
had no reason to be there and no reason to be $2\times10^{-3}$. Sakharov
understood within three years what the phase was \emph{for}
\citep{Sakharov1967}; Section~\ref{sec:history-stake} returns to
that. Kobayashi and Maskawa understood within a decade where it could
\emph{live}: with three quark generations, the quark mixing matrix
possesses exactly one physical phase \citep{KobayashiMaskawa1973} ---
a counting theorem that predicted a third family from a kaon
asymmetry. What no one has explained in the sixty years since is why
the phase takes the value it does. That is the assignment Part~VII
accepts.

\section{One phase, three families: 1973, and the twenty-year
confirmation}
\label{sec:one-phase}

Kobayashi and Maskawa's 1973 observation
\citep{KobayashiMaskawa1973} deserves a section of its own,
because it is simultaneously one of the great counting theorems
of physics and the precise statement of what remained
unexplained. With two quark families --- all that were then
known, and the charm quark itself still a year from discovery
--- the quark mixing matrix can always be made real: no CP
violation. With three families, exactly one physical phase
survives the rephasing freedom (the counting this paper
geometrizes as Eq.~\eqref{eq:phase-count}). The kaon asymmetry
of 1964 therefore \emph{predicted a third generation} --- an
extraordinary inference from a $10^{-3}$ effect in one meson
--- and the prediction was honored in instalments across
twenty-two years: the $b$ quark at Fermilab in 1977
\citep{Herb1977}, the top quark, completing the family, in
1995. Jarlskog then distilled the phase's entire physical
content into one basis-independent number
\citep{Jarlskog1985}, the invariant $J$ that Part~VII's
pentagon must and does price. The ledger of 1973--1995 is thus:
the \emph{existence} of the phase explained (three families
admit one), its \emph{location} explained (the weak
charged current), and its \emph{value} explained by nothing ---
$\delta$ could sit anywhere on the circle, and the Standard
Model would shrug. Holding that open door in mind is the right
way to read everything that follows; it is the door
Part~VII claims the pentagon closes, and the door the parent
particle paper first opened geometrically \citep{ArisakaA9SM}.

\section{The precision era: KTeV, NA48, and the \texorpdfstring{$B$}{B} factories}
\label{sec:history-precision}

The program that followed was one of experimental physics' long
marches. Direct CP violation required measuring the double ratio
\begin{equation}
R \;=\;
\frac{\Gamma(K_L \to \pi^{+}\pi^{-})\,/\,\Gamma(K_S \to \pi^{+}\pi^{-})}
     {\Gamma(K_L \to \pi^{0}\pi^{0})\,/\,\Gamma(K_S \to \pi^{0}\pi^{0})}
\;\simeq\; 1 + 6\,\mathrm{Re}\!\left(\frac{\varepsilon'}{\varepsilon}\right)
\label{eq:double-ratio}
\end{equation}
--- four rates, two of them into the notoriously difficult neutral
mode, with every systematic (energy scale, acceptance, accidental
activity, regeneration) entering coherently --- to a precision of
parts in $10^{4}$. The experimental design problem this posed
(simultaneous beams, regenerator versus vacuum, a calorimeter good
enough to make $\pi^0\pi^0$ a precision channel) consumed two
generations of instrument-building on two continents. After a generation of
effort, KTeV at Fermilab \citep{KTeV1999} and NA48 at CERN
\citep{NA48_2002} established
$\mathrm{Re}(\varepsilon'/\varepsilon) = (16.6\pm2.3)\times10^{-4}$:
the asymmetry lives in the amplitudes, as the CKM paradigm requires,
and superweak alternatives died. The $B$ factories then subjected the
paradigm to its decisive test in a second meson system: BaBar at SLAC
and Belle at KEK measured the time-dependent CP asymmetry in
$B \to J/\psi K_S$ \citep{BaBar2001, Belle2001}, reading
$\sin 2\beta$ --- an angle of the unitarity triangle --- from the
interference of mixing and decay. The triangle closed. Every
measured CP asymmetry, in kaons and in $B$ mesons, traced back to the
single Kobayashi--Maskawa phase, with the Jarlskog invariant
\citep{Jarlskog1985} --- the basis-independent measure of all CP
violation in the quark sector --- pinned at
$\JCKM = (3.12^{+0.13}_{-0.12})\times10^{-5}$.

It is worth pausing on what this triumph did and did not establish.
It established that the Standard Model's bookkeeping is correct: one
phase, entering every observable through the same rephasing-invariant
combinations. It did not establish anything about the phase itself.
$\delta \simeq 1.147\pm0.026$ radians is a measured number with no
derivation, in a model where it could have been anything, including
zero. The precision era answered ``how'' completely and ``why'' not
at all --- and, as Section~\ref{sec:history-stake} records, the
measured value is quantitatively \emph{incapable} of producing the
matter excess it was once hoped to explain. The author spent a decade
of his scientific life inside these measurements, at KTeV and then at
BaBar; Part~XIII records what that decade taught, and what question
it left standing.

\section{The third arena: neutrinos}
\label{sec:history-neutrinos}

The discrete-symmetry program's third arena opened underground. The
Kamioka detector --- designed and constructed in 1980--1985, with the
author among its designers --- was built to watch for proton decay
and found, instead, the neutrino sky: supernova SN1987A's burst, the
solar neutrino deficit confirmed in real time, and, in its successor
Super-Kamiokande, the 1998 discovery that atmospheric neutrinos
oscillate \citep{SuperK1998} --- neutrinos have mass, and the lepton
sector has its own mixing matrix. The arena had in fact been opened twice before, and the full
lineage bears telling. Davis's chlorine tank in the Homestake
mine had counted solar neutrinos since 1968 and found a third of
the predicted rate \citep{Davis1968} --- a deficit that stood as
an anomaly for thirty years, too strange to accept and too solid
to dismiss. The water detectors --- built, it must be
remembered, to watch for \emph{proton decay}, the signature
prediction of the grand unified theories --- found the protons
silent and the sky loud: Kamiokande confirmed the solar deficit
in real time, caught the neutrino burst of SN1987A, and turned
its successor toward the atmospheric flux, where the 1998
oscillation discovery was made. SNO then closed the solar case
with heavy water --- the neutrinos were not missing but
\emph{changed} \citep{SNO2002} --- and KamLAND closed it from
the reactor side \citep{KamLAND2003}: neutrinos have mass, the
lepton sector mixes, and a second flavor matrix exists with
its own potential CP phase. The reactor experiments completed
the matrix's angles with $\theta_{13}$ \citep{DayaBay2012}, and with
all three angles sizeable, the lepton sector's CP phase $\dPMNS$
became measurable in principle --- and became, over the following
decade, the flagship quest of experimental neutrino physics. T2K's
early running already disfavors CP conservation in the lepton
sector \citep{T2K2020}; the global fits center near
$\dPMNS \sim 200^{\circ}$ \citep{NuFit2024}; and the next-generation
long-baseline program --- DUNE and Hyper-Kamiokande --- will
measure the phase to $\sim\pm15^{\circ}$ in the early 2030s.

The stakes of that measurement are higher than a matrix element. If
neutrinos are Majorana --- their own antiparticles, the unique
possibility opened by their electrical neutrality --- then the lepton
sector harbors two further CP phases invisible to oscillations, a
mass mechanism (the seesaw \citep{Minkowski1977, YanagidaSeesaw,
GellMannRamondSlansky}) reaching to scales no collider will visit,
and, through Fukugita and Yanagida's leptogenesis
\citep{FukugitaYanagida1986}, a candidate answer to Sakharov's
question: the matter excess made first as a lepton excess, in the
CP-biased decays of the seesaw's heavy partners, then reprocessed
into baryons by the electroweak sphalerons. The experimental keys are
exactly the quantities the current program pursues: $\dPMNS$, the
neutrino mass ordering, and the neutrinoless double-beta-decay rate
that would certify the Majorana nature. The detectors that opened
this arena were built to answer ``does the proton die?''; the arena
they opened may instead answer ``why was the proton born?''

\section{The theorem that survived: CPT}
\label{sec:history-cpt}

Against this history of falling mirrors stands one unbroken theorem.
The combined reversal CPT --- all three operations at once --- is a
symmetry of any local, Lorentz-invariant quantum field theory with a
stable vacuum: proved in increasing generality by L\"uders, Pauli,
and Schwinger \citep{Luders1954, Pauli1955}, and given its definitive
axiomatic form by Jost \citep{Jost1957, StreaterWightman}, whose
proof runs through the analytic continuation of the Wightman
functions to the complexified Lorentz group. Its modern algebraic
descendant is sharper still: Bisognano and Wichmann
\citep{BisognanoWichmann1976} identified the CPT operation, combined
with a rotation, with the Tomita--Takesaki modular conjugation of the
wedge algebra --- CPT as a structural feature of the local net
itself. And Greenberg proved the converse discipline
\citep{Greenberg2002}: CPT violation cannot be had cheaply, for it
entails Lorentz violation.

Experiment has treated the theorem with corresponding respect. The
neutral kaon system again leads: the $K^0$--$\bar K^0$ mass
difference bounds fractional CPT violation near $10^{-18}$
\citep{PDG2024}. Antihydrogen spectroscopy compares matter and
antimatter atom by atom at the $10^{-12}$ level \citep{ALPHA2020}.
Every particle's mass, lifetime, and magnetic moment agrees with its
antiparticle's to the precision measured;
Table~\ref{tab:cpt-tests} collects the flagship comparisons.

\begin{table}[t]
\centering
\caption[The experimental record of the unbroken theorem.]{The experimental record of the unbroken theorem
\citep{PDG2024, ALPHA2020}. Every entry is a matter--antimatter
identity that T-P5-2 forces exactly and any real texture preserves;
a confirmed violation at any order would kill T-P5-2 and, via
Greenberg's theorem \citep{Greenberg2002}, the Lorentzian arena
itself.}
\label{tab:cpt-tests}
\small
\renewcommand{\arraystretch}{1.2}
\begin{tabular}{p{4.2cm} p{4.2cm} p{3.1cm} p{2.6cm}}
\toprule
\textbf{CPT comparison} & \textbf{Observable} &
\textbf{Precision} & \textbf{System} \\
\midrule
$K^0$ vs $\bar K^0$ mass & $|m_{K^0} - m_{\bar K^0}|/m_K$ &
$\lesssim 10^{-18}$ & neutral kaons \\
$e^{+}$ vs $e^{-}$ moment & $|g_{+} - g_{-}|/g$ &
$\sim 2\times10^{-12}$ & Penning traps \\
$p$ vs $\bar p$ charge/mass & $|(q/m)_p - (q/m)_{\bar p}|$ &
$\sim 10^{-12}$ & Penning traps \\
$\bar p$ magnetic moment & $|\mu_p - \mu_{\bar p}|/\mu_p$ &
$\sim 10^{-9}$ & BASE \\
H vs $\bar{\mathrm{H}}$ 1S--2S & fractional frequency &
$\sim 2\times10^{-12}$ & ALPHA \\
$\mu^{+}$ vs $\mu^{-}$ lifetime & $|\tau_{+} - \tau_{-}|/\tau$ &
$\sim 10^{-4}$ & storage rings \\
\bottomrule
\end{tabular}
\end{table}

One experimental program belongs to this section although it
tests the factors rather than the product, because it is the
oldest continuous thread in the whole subject: the electric
dipole moments. A permanent EDM of a fundamental particle
violates both P and T --- hence, by CPT, CP --- and the hunt
Purcell and Ramsey proposed in 1950
\citep{PurcellRamsey1950} has now run for seventy years and
twelve orders of magnitude: the neutron bound stands at
$|d_n| < 1.8\times10^{-26}\,e\,$cm \citep{nEDM2020}, the
electron bound --- via molecular amplifiers --- at
$|d_e| < 4.1\times10^{-30}\,e\,$cm \citep{ACME2018}. These
nulls are load-bearing twice in this paper: they enforce
$|\bar\theta| \lesssim 10^{-10}$, the strong-CP smallness whose
\GG{} resolution is the two-fate architecture of Part~VII with
its own falsifiable leakage floor \eqref{eq:thetabar-leakage};
and they discipline every proposed new CP phase, this paper's
included. The EDM program is the experimental hat at its
purest: a 1950 philosophical challenge to an assumption, run as
precision metrology ever since.

The survival of CPT is as
deep a fact as the fall of its factors --- and the standard theorem,
for all its rigor, is anatomically silent: it proves that the
antiunitary operator $\Theta$ exists and commutes with the dynamics,
but does not say what C, P, and T individually \emph{are}, why T is
the antiunitary one, or why nature violates the factors while
preserving the product. This paper's Part~V offers the anatomy: the
factors are orientation reversals of three real geometric structures;
the product is total reversal, which no real texture can detect; and
the factors break precisely because the texture of our world --- the
flavor pentagon, the oriented descent --- is not orientation-blind.

\section{The cosmological stake: Sakharov, 1967}
\label{sec:history-stake}

One number makes the discrete symmetries everyone's business:
\begin{equation}
\etaB \;\equiv\; \frac{n_B - n_{\bar B}}{n_\gamma}
\;=\; (6.12 \pm 0.04)\times10^{-10},
\label{eq:etaB-obs}
\end{equation}
the baryon-to-photon ratio, read at the $0.3\%$ level from the
acoustic peaks of the cosmic microwave background \citep{Planck2018}
and confirmed, independently and five orders of magnitude earlier in
temperature, by the primordial deuterium abundance
\citep{Cooke2018}. Two epochs, one number: the asymmetry was in place
before the first three minutes and has been conserved since. The
cosmos, meanwhile, contains no antimatter to find: no annihilation
glow at domain boundaries \citep{CohenDeRujulaGlashow1998}, no
confirmed antinuclei in the cosmic rays \citep{AMS02}. And a
universe born symmetric would have kept essentially nothing:
nucleon--antinucleon annihilation at freeze-out leaves a relic
abundance near $10^{-18}$ --- nine orders too small --- a cosmos of
pure radiation, with no stars, no chemistry, and no one to be
puzzled. For every billion antiprotons in the early universe there
were a billion and one protons, and everything that exists is the
one.

Before Sakharov's answer, the alternatives deserve their
paragraph, because both were reasonable and both are dead ---
and their deaths are measurements, not fashions. Dirac himself,
in his 1933 Nobel lecture, suggested that half the stars might
be made of antimatter --- ``it is quite possible that\ldots
these stars being built up mainly of positrons and negative
protons'' \citep{Dirac1933} --- a symmetric universe with
segregated domains; and Alfv\'en and Klein built a full
cosmology on that premise \citep{Alfven1965}. The gamma-ray
sky killed it: domain boundaries would annihilate, and the
diffuse background contains no such glow at any scale out to
the visible horizon \citep{CohenDeRujulaGlashow1998}. The
second alternative --- that the excess is simply an initial
condition, written into the universe at birth --- survives
logically but has been priced out structurally: inflation,
required by the cosmology on independent grounds
\citep{ArisakaA10Cosmos}, dilutes any primordial asymmetry to
irrelevance, so the excess must have been \emph{made},
dynamically, after the beginning. What remains is Sakharov's
road, and it runs directly through this paper's subject.

Sakharov saw in 1967 \citep{Sakharov1967} --- three years after the
kaon result, when its significance was visible to almost no one ---
that a dynamical origin of the excess requires exactly three
ingredients: baryon-number violation; C \emph{and} CP violation
(the decays must know matter from antimatter); and a departure from
thermal equilibrium (in equilibrium, CPT itself forces the books to
balance). The three conditions are this paper's closing scoreboard.
The Standard Model, remarkably, fails them quantitatively while
satisfying them qualitatively: its sphalerons violate baryon number,
its CKM phase violates CP, its electroweak transition could have
provided the non-equilibrium --- but the effective CP violation
computes to $\sim10^{-20}$ \citep{GavelaEtAl1994}, ten orders short,
and the transition is, for the measured Higgs mass, a smooth
crossover \citep{Kajantie1996}: no bubble walls, no departure, no
baryons. Every gram of matter is thereby an experimental proof of
physics beyond the Standard Model --- arguably the oldest and most
secure such proof we possess. Part~VIII presents the \GGTheory{}
candidate for that physics, with its inputs inherited rather than
chosen and its failure modes listed with dates.

\section{Seventy years in one table, and the detail behind it}
\label{sec:history-table}

Table~\ref{tab:chronology} lays out the experimental chronology as
one continuous program; Table~\ref{tab:seventy-years} then
compresses the standing of each question it raised.

\begin{table}[t]
\centering
\caption[Seventy years as one experiment.]{Seventy years as one experiment. The final row is the
reason this paper can be wrong on a date --- the property
Table~\ref{tab:falsifiers} itemizes. Nineteen rows, from Laporte's selection rule to the measurements of the 2030s, each with the actor, the episode, and what that episode settled. Read as one column, the table is the case for treating seventy years of mirror experiments as a single experiment that is still running, and whose last row has not been performed yet.}
\label{tab:chronology}
\small
\renewcommand{\arraystretch}{1.18}
\begin{tabular}{p{1.2cm} p{4.1cm} p{4.6cm} p{4.4cm}}
\toprule
\textbf{Year} & \textbf{Actor} & \textbf{Episode} &
\textbf{What it established} \\
\midrule
1924 & \citet{Laporte1924} & parity selection rule & P as atomic
bookkeeping \\
1956 & \citet{LeeYang1956} & the audit & P untested in weak
interactions \\
1957 & \citet{Wu1957}; \citet{Garwin1957} & $^{60}$Co;
$\pi$--$\mu$--$e$ & P and C fall, maximally \\
1957--58 & \citet{LeeYang1957}; \citet{SudarshanMarshak1958};
\citet{FeynmanGellMann1958} & two-component $\nu$; $V{-}A$ &
maximal violation codified \\
1954--57 & \citet{Luders1954}; \citet{Pauli1955};
\citet{Jost1957} & the CPT theorem & the product protected \\
1964 & \citet{Cronin1964} & $K_L \to \pi\pi$ & CP falls, at
$2\times10^{-3}$ \\
1967 & \citet{Sakharov1967} & the three conditions & CP violation
is why matter exists \\
1973 & \citet{KobayashiMaskawa1973} & one phase at $N = 3$ & a
third family predicted from a kaon \\
1985 & \citet{KuzminRubakovShaposhnikov1985} & sphalerons &
$B{+}L$ erased; $B{-}L$ is the currency \\
1985 & \citet{Jarlskog1985} & the invariant & CP violation
basis-free: one number \\
1986 & \citet{FukugitaYanagida1986} & leptogenesis & the lepton
route to the excess \\
1998 & \citet{SuperK1998} & atmospheric oscillations & neutrino
mass; the third arena opens \\
1999--2002 & \citet{KTeV1999}; \citet{NA48_2002} &
$\varepsilon'/\varepsilon$ & direct CP violation; superweak dies \\
2001 & \citet{BaBar2001}; \citet{Belle2001} & $\sin 2\beta$ & the
triangle closes on one phase \\
2004 & \citet{PilaftsisUnderwood2004} & resonant leptogenesis &
the TeV window opens \\
2012 & \citet{DayaBay2012} & $\theta_{13}$ & $\dPMNS$ becomes
measurable \\
2020 & \citet{T2K2020} & CP conservation disfavored & the
leptonic phase within reach \\
2030s & DUNE, Hyper-K, tonne-scale $0\nu\beta\beta$ &
$\dPMNS$ to $\pm15^{\circ}$; Majorana nature & this paper's
adjudication \\
\bottomrule
\end{tabular}
\end{table} The columns deserve one sentence each. The
\emph{QFT} column records what quantum field theory as a framework
provides: definitions and one great theorem, but no principle
selecting which discrete symmetries hold. The \emph{Standard Model}
column records the paradigm's honest ledger: violation patterns
written into the field content, one fitted phase, two unexplained
zeros (the strong-CP angle and its own baryon asymmetry). The
\emph{string} column records the landscape's position: mechanisms in
abundance, valuations nowhere. The \GGTheory{} column carries the
claims of this paper, each tagged with its register --- and the
register column is part of the physics, not a disclaimer.

\begin{table}[t]
\centering
\caption[Seventy years of questions, and where each stands.]{Seventy years of discrete-symmetry questions and where each
framework stands. Register tags in the final column: \emph{derived}
(theorem at stated grade), \emph{conditional} (named lemma or
conjecture outstanding), D$^{*}$ (derived-conditional,
order-of-magnitude). The table is defended, row by row, in
Parts~IV--X.}
\label{tab:seventy-years}
\small
\renewcommand{\arraystretch}{1.25}
\begin{tabular}{p{2.5cm} p{2.6cm} p{2.7cm} p{2.6cm} p{3.6cm}}
\toprule
\textbf{Question} & \textbf{QFT} & \textbf{Standard Model} &
\textbf{String theory} & \textbf{\GGTheory{} (this paper)} \\
\midrule
What are C, P, T? & operators, phases free (Wigner) & same &
same; C $=$ orientation reversal of the string & orientation
reversals of fiber, space, clock (T-P5-1; derived) \\
Why is T antiunitary? & input & input & input & derived: only the
clock's orientation is what $i$ is made of \\
Why does CPT hold? & Jost's theorem (interacting; anatomy silent) &
inherited & worldsheet CPT & total orientation reversal
(T-P5-2; quasi-free grade) \\
Why is P violated, in one force? & no principle & input (V$-$A by
hand) & vacuum choice & the oriented descent: broken $=$ gate $=$
chiral (Part~VI; conditional, L11) \\
Why does CP break, by this much? & no principle & fitted
($\delta = 1.147\pm0.026$) & spontaneous, moduli-valued, unpredicted
& chiral pentagon: $\dCKM = 2\pi/5 \to 13\pi/35$ (T-P5-3;
conditional) \\
Why is $\bar\theta < 10^{-10}$? & unexplained & unexplained (axion
added by hand) & axions generic, values not fixed & no knob exported
$+$ Kaxion relaxation (conditional, L9) \\
Why did matter win? & mechanism toolbox, dials free & fails by ten
orders & ingredients $+$ moduli problem & TeV resonant leptogenesis
on the inherited ledger: $\etaB \simeq 5.9\times10^{-10}$
(Part~VIII; D$^{*}$, Conj.~VI) \\
\bottomrule
\end{tabular}
\end{table}

\section{What experiment still cannot tell us}
\label{sec:remaining-experimental}

Seventy years of instruments have converted most of this
subject's questions into numbers; this section records, one by
one, the measurements that remain open --- the experimental
ledger this paper's claims will be judged against, with the
parent papers' stakes noted where they ride on the same
instruments. \emph{(1) The leptonic CP phase.} $\dPMNS$ is the
last mixing parameter of either flavor sector still
unmeasured at definitive precision; the global fits center near
$200^{\circ}$ \citep{NuFit2024}, T2K disfavors conservation
\citep{T2K2020}, and DUNE and Hyper-Kamiokande will reach
$\pm15^{\circ}$ in the early 2030s --- the single most
consequential measurement pending in this subject, for the
Standard Model (is the lepton sector CP-violating at all?), for
this paper (the pentagon says $196^{\circ}$), and for the
cosmology parent (the door's bias rides on it,
\citealp{ArisakaA10Cosmos}). \emph{(2) The neutrino mass
ordering.} Normal or inverted --- JUNO and the atmospheric fits
will settle it; the parent particle paper's derived spectrum
requires normal \citep{ArisakaA9SM}. \emph{(3) The Majorana
question.} Is the neutrino its own antiparticle? Neutrinoless
double beta decay is the only known direct probe; the
tonne-scale generation reaches the inverted band, and the
meV-class normal-ordering target ($\mbb \simeq 1.7$~meV here)
awaits the generation after. \emph{(4) The EDM frontier.}
Every order of magnitude of improvement on $d_n$ and $d_e$
either finds new CP violation or tightens the strong-CP
mystery; this paper's two-fate architecture places a floor at
$\sim10^{-18}$ in $\bar\theta$ terms --- a discovery band, not
just a null. \emph{(5) The right-handed question.} Is parity
restored at high energy? Left--right symmetric models say yes,
at some scale; the geometric projection of Part~VI says
never; $W_R$ searches adjudicate, permanently. \emph{(6) The
TeV frontier.} The lock cluster at $3.85$~TeV --- the heavy
sector of the matter chain --- sits within HL-LHC reach with a
pre-declared criterion. \emph{(7) The antimatter sky.}
AMS-02's antihelium candidates remain unconfirmed; a single
confirmed antinucleus of primary origin would reopen the
domain question the gamma-ray sky closed. \emph{(8) Proton
decay.} Still the sharpest single discriminator in fundamental
physics: the grand-unified expectation is a rate, the \GG{}
ledger's is exact stability; Hyper-Kamiokande watches both
claims at once.

\section{What theory still cannot say}
\label{sec:remaining-philosophical}

The philosopher's ledger is shorter to state and has stood
open longer. \emph{(1) What are the operations?} The standard
formalism defines C, P, and T as operators with adjustable
phases and offers no account of what they are reversals
\emph{of}; the question ``is the antiunitarity of T a fact
about nature or about our bookkeeping?'' has no orthodox
answer. \emph{(2) Kant's question, modernized.} Is orientation
itself conventional or physical? The Ozma resolution
(\S\ref{sec:ozma}) shows the world can define handedness
absolutely --- but the standard framework locates the
capability in two measured numbers (the kaon charge asymmetry,
the CKM phase) without saying why the world possesses it.
\emph{(3) The value problem.} Even granting one CP phase by
counting, nothing in any standard framework prices it:
$\delta = 1.147$ and $\delta = 0$ are equally lawful, and on
the second there is no matter and no one to notice. The
anthropic shrug explains at most why we do not observe zero,
never why we observe this. \emph{(4) The coincidence
problem.} Five superlatives --- broken, chiral,
family-changing, short-range, CP-bearing --- on one gauge
factor, unconnected in the formalism that hosts them.
\emph{(5) The strong-CP problem.} A phase that could be
anything is zero to ten digits, beside a phase that is
measured to be large; the standard cure installs a new field
by hand. \emph{(6) Why matter won.} The Standard Model's own
answer fails by ten orders of magnitude
(\S\ref{sec:road-sm}); every working mechanism buys the
excess with fitted parameters; and the deeper question ---
whether the excess is an accident of initial conditions or a
theorem of structure --- is not even posable in a framework
that stores its symmetry violations as inputs. \emph{(7) The
measurement shadow.} Behind all of these stands the oldest
open problem of quantum mechanics itself --- what an
observation \emph{is} --- which the parent program addresses
with the bulk-to-boundary projection (\OGGEP{}), and whose
arrow (records accumulate, the past is fixed) is the same
orientation this paper traces through the mirrors
(\citealp{ArisakaA5GDOS}; \citealp{ArisakaA10Cosmos}). This paper does not
argue that case here; it inherits the framework in which, for
the first time, questions (1) through (6) become derivations
--- and it flags, with the calibration the register demands,
that a framework able to do so would constitute a shift in
how physical law is read. Whether it \emph{has} done so is
exactly what the falsifier tables of Part~XII put before the
instruments.

\section{The claim of this preprint}
\label{sec:claim}

This preprint makes five claims, stated here with their registers so
that no reader mistakes the paper's center of gravity.

\emph{Claim 1 (derived).} C, P, and T are the orientation reversals
of the three real factors of the \GG{} bulk --- the compact fiber
circle, the spatial slice, and the causal clock --- and the
unitary/antiunitary table is a consequence: a discrete symmetry is
antiunitary iff it contains T (T-P5-1).

\emph{Claim 2 (derived, quasi-free grade).} CPT is total orientation
reversal, and is a symmetry of the construction for every coupling
texture built from real geometric data; its interacting extension
rides on the standard scattering hypotheses (T-P5-2).

\emph{Claim 3 (derived through clause (iii); clause (iv) forced by
the ledger).} CP violation is the chirality of flavor-texture loop
holonomies; on the single circle with $L=5$, first-order holonomies
are pentagonal-lattice points, giving forced central values for the
quark and lepton CP phases and the Majorana pair $(0,\pi)$
(T-P5-3, with the completion step's status stated where it is used).

\emph{Claim 4 (structural, one named lemma outstanding).} Parity
violation is the orientation of the family descent: the unique broken
force is the unique gate force, and only a chiral coupling can carry
a one-way flow. The mechanism (the projection) is in place; the
uniqueness argument awaits the gate-operator lemma (Part~VI).

\emph{Claim 5 (derived-conditional, order of magnitude).} The baryon
asymmetry follows from TeV-scale resonant leptogenesis on the
inherited ledger, $\etaB \approx 7\times10^{-10}$ ($\times3^{\pm1}$), under one named
normalization conjecture and with the resonance condition as the
chain's declared input (Part~VIII).

What this paper does \emph{not} claim: that the standard CPT theorem
is superseded (Jost's scope exceeds ours; the relation is
complementarity, argued in Part~X); that the flavor-holonomy
calculus's rephasing machinery is new (it is Jarlskog's, geometrized;
what is new is the lattice valuation); that the projection mechanism
of Part~VI is new (it is orbifold-GUT technology, cited to its
authors); or that the leptogenesis transport is new (it is
Fukugita--Yanagida in the Pilaftsis regime; what is new is that
every dial is inherited). The paper's originality lies in the
\emph{anatomy} (what the mirrors are), the \emph{valuation} (what the
phases must be), and the \emph{unification} (that one orientation
answers five questions) --- and its honesty lies in the named list of
what remains open.

Five claims, five registers --- but one chain. Before the chain is
walked slowly (which is the business of the next hundred and fifty
pages), it is worth seeing whole, because its shape is the paper's
real argument: not five separate assertions but one deduction passing
through five stations, each station forced by the one above it.
Figure~\ref{fig:p5-central-claim} draws it.

\begin{figure}[H]
\centering
\resizebox{0.951\textwidth}{!}{%
\begin{tikzpicture}[node distance=21mm,
 every node/.style={font=\sffamily\footnotesize, align=center},
 box/.style={draw=black!55, rounded corners, minimum width=118mm,
   minimum height=12mm, inner sep=5pt},
 arrow/.style={-{Stealth[length=3mm]}, line width=1.10pt},
 elab/.style={font=\sffamily\scriptsize\itshape, fill=white,
   inner sep=1.5pt}]
\node[box, fill=violet!10] (bulk)
 {\textbf{The bulk is real, oriented geometry}\\
  one compact fiber circle $+$ oriented space $+$ one causal clock
  --- and no $i$ anywhere upstairs};
\node[box, fill=blue!8, below of=bulk] (hand)
 {\textbf{Two quarter-turns must agree on every charged mode}\\
  the circle's phase turn and the clock's positive-frequency turn
  (AX-1 $+$ AX-3; Theorem~0)};
\node[box, fill=yellow!13, below of=hand] (twin)
 {\textbf{Antiparticles: the same ripple, read against both
  orientations}\\
  C, P, T $=$ the three orientation reversals; the table is derived;
  CPT is its bookkeeping (T-P5-1/2)};
\node[box, fill=orange!14, below of=twin] (mirr)
 {\textbf{The mirrors break selectively}\\
  P and C fall by an \emph{absence} (one hand kept; no $W_R$);
  CP falls by an \emph{angle} (the pentagon:
  $2\pi/5 \to 13\pi/35$; $\dPMNS = 196.4^{\circ}$)};
\node[box, fill=green!12, below of=mirr] (door)
 {\textbf{Biased traffic through one door at a one-way boundary}\\
  Sakharov's three conditions $=$ one geometric tripod (P-P5-E);
  every dial inherited};
\node[box, fill=orange!16, below of=door] (rcpt)
 {\textbf{The receipt: $\etaB \approx 5.9\times10^{-10}$
  (grade D$^{*}$)}\\
  with the cash-out $\dCKM = 13\pi/35$;
  $(\alpha_{21},\alpha_{31}) = (0,\pi)$; $\mbb = 1.72$~meV;
  $\bar\theta \lesssim 5\times10^{-18}$; normal ordering; no $W_R$};
\draw[arrow] (bulk) -- node[elab, right]
 {the handshake (T-P5-A/B/C): consistency forces a partner} (hand);
\draw[arrow] (hand) -- node[elab, right]
 {Feynman--St\"uckelberg pairing, forced not chosen} (twin);
\draw[arrow] (twin) -- node[elab, right]
 {the descent to 4D is oriented} (mirr);
\draw[arrow] (mirr) -- node[elab, right]
 {the still point opens the one weight-zero door (T-P5-4)} (door);
\draw[arrow] (door) -- node[elab, right]
 {zero fitted parameters; one grade-D$^{*}$ transport} (rcpt);
\end{tikzpicture}}
\caption[The paper's central claim as one deductive funnel.]{%
\textbf{The paper's central claim as one deductive funnel.} The five
claims of \S\ref{sec:claim} drawn as the single chain they form, from
the real oriented bulk at the top to the six-per-ten-billion receipt
at the bottom. Each arrow names the theorem or mechanism that forces
the step; each box carries its register in the text. The chain is
walked slowly in Parts~IV--VIII; the funnel is the paper on one
page.}
\label{fig:p5-central-claim}
\end{figure}

Read the funnel from the top. The first box is inherited from the
parent papers: a real, oriented six-dimensional geometry with no $i$
in it. The first arrow is this paper's opening act --- the handshake
theorem, which says that the circle's quarter-turn and the clock's
quarter-turn must agree on every charged ripple, and that the only
way they can agree is if every ripple has a partner. That single
consistency condition delivers the third box: antiparticles, and with
them the three reversal operations and the CPT bookkeeping. The
fourth box is where the story turns from symmetry to asymmetry ---
the oriented descent keeps one hand and tilts the pentagon --- and
the fifth is where asymmetry becomes traffic: Sakharov's three
famous conditions, reassembled as one tripod whose every leg is
geometric. The bottom box is the number. Notice what the funnel does
\emph{not} contain: a dial. Nothing in any box was fitted to make
the bottom line come out; every input was fixed upstream, for other
reasons, in other papers. That is the sense in which the funnel is
one claim rather than five --- and the sense in which a failure
anywhere in it is a failure of the whole.

One more picture completes the accounting. A funnel shows the logic;
it does not show the \emph{grades} --- and this paper's method is
that every step carries one. The three Part ledgers
(Tables~\ref{tab:parity-ledger}, \ref{tab:part4-ledger},
and~\ref{tab:part6-ledger}) audit the chain row by row;
Figure~\ref{fig:p5-status} compresses their twenty-seven rows into a
single bar, in the manner of the parent paper's parameter audit.

\begin{figure}[H]
\centering
\resizebox{\textwidth}{!}{%
\begin{tikzpicture}[
 every node/.style={font=\fontsize{13}{15}\selectfont, align=center}
]
\definecolor{derivedG}{RGB}{46, 139, 87}
\definecolor{condY}{RGB}{218, 165, 32}
\definecolor{assertedR}{RGB}{178, 34, 34}
\definecolor{conjP}{RGB}{106, 76, 147}
\definecolor{otherN}{RGB}{110, 110, 110}
% 27 units: 12 derived | 7 cond | 3 asserted | 2 conjecture | 3 other
\fill[derivedG] (0,0) rectangle (12,1.1);
\fill[condY] (12,0) rectangle (19,1.1);
\fill[assertedR] (19,0) rectangle (22,1.1);
\fill[conjP] (22,0) rectangle (24,1.1);
\fill[otherN] (24,0) rectangle (27,1.1);
\draw (0,0) rectangle (27,1.1);
\node[white] at (6,0.55) {\textbf{12 DERIVED}};
\node[white] at (15.5,0.55) {\textbf{7 COND.}};
\node[white, font=\fontsize{11}{13}\selectfont]
 at (20.5,0.55) {\textbf{3}};
\node[white, font=\fontsize{11}{13}\selectfont]
 at (23,0.55) {\textbf{2}};
\node[white, font=\fontsize{11}{13}\selectfont]
 at (25.5,0.55) {\textbf{3}};
\foreach \x in {0,...,27}{
 \draw (\x,1.1) -- (\x,1.25);
}
\foreach \x/\lab in {0/0, 12/12, 19/19, 22/22, 24/24, 27/27}{
 \node[font=\fontsize{9}{11}\selectfont] at (\x,1.5) {\lab};
}
\node[font=\fontsize{9}{11}\selectfont] at (13.5,1.95)
 {ledger row index};
\node[anchor=west, derivedG,
 font=\fontsize{16}{19}\selectfont\bfseries] at (0.0,-1.2)
 {\textbullet\ Derived (12):};
\node[anchor=west, align=left, text width=190mm,
 font=\fontsize{12}{14}\selectfont] at (5.6,-1.2)
 {C/P/T table; CPT (quasi-free); holonomy calculus; Majorana door
  T-P5-4; SM group from one hand (c); chirality; anomaly freedom
  (c); Higgs
  from projection; Sakharov tripod; lock chain; portal; matched
  Yukawa.};
\node[anchor=west, condY,
 font=\fontsize{16}{19}\selectfont\bfseries] at (0.0,-2.8)
 {\textbullet\ Derived (cond., 7):};
\node[anchor=west, font=\fontsize{12}{14}\selectfont] at (7.1,-2.8)
 {pentagonal lattice; $\dCKM$; the PMNS set; $\bar\theta$;
  $\epsilon^{\mathrm{res}}$; washout $\kappa$; zero dilution.};
\node[anchor=west, assertedR,
 font=\fontsize{16}{19}\selectfont\bfseries] at (0.0,-3.8)
 {\textbullet\ Asserted (3):};
\node[anchor=west, font=\fontsize{12}{14}\selectfont] at (5.2,-3.8)
 {the projection matrix $P$; the $\bZ_2$ itself; which hand
  survives.};
\node[anchor=west, conjP,
 font=\fontsize{16}{19}\selectfont\bfseries] at (0.0,-4.8)
 {\textbullet\ Named conjectures (2):};
\node[anchor=west, font=\fontsize{12}{14}\selectfont] at (8.1,-4.8)
 {C-P5-A (one gate force); C-P5-C (resonance $+$ exponent).};
\node[anchor=west, otherN,
 font=\fontsize{16}{19}\selectfont\bfseries] at (0.0,-5.8)
 {\textbullet\ Other (3):};
\node[anchor=west, font=\fontsize{12}{14}\selectfont] at (4.4,-5.8)
 {the throughput reconstruction; the D$^{*}$ transport band; the
  open $\Omega_b$ bridge.};
\end{tikzpicture}}
\caption[The 27-row honesty audit at a glance.]{%
\textbf{The 27-row honesty audit at a glance:} the union of the
paper's three status ledgers --- the parity mechanism's
(Table~\ref{tab:parity-ledger}), the mirror Parts'
(Table~\ref{tab:part4-ledger}), and Part~VIII's
(Table~\ref{tab:part6-ledger}) --- compiled into one bar. Twelve
rows derived at their stated grade
(\textcolor{green!50!black}{\textbf{green}}), seven derived modulo
one named step each (\textcolor{olive!80!black}{\textbf{gold}}),
three asserted with the assertion printed
(\textcolor{red!80!black}{\textbf{red}}), two carried by named
conjectures (\textcolor{violet!80!black}{\textbf{purple}}), and
three in other registers --- one reconstruction, one D$^{*}$
band, one open seam (\textcolor{black!60}{\textbf{gray}}). The bar
is the paper's honesty policy made visible: what is proved is
separated from what is owed, and the owed list is short enough to
print. Rows tagged (c) are booked derived here \emph{conditional
on named parent-paper items} --- the constitutional hand choice
and the anomaly-closure program carried open in the
constitution papers; the parents are listed in the inheritance
and promotion-roadmap appendices
(Appendices~\ref{app:inheritance-table}
and~\ref{app:promotion-roadmap}).}
\label{fig:p5-status}
\end{figure}

The bar rewards a moment's stare. Nearly half the audit is green ---
and the green rows are not the easy ones; they include the C/P/T
table, the CPT theorem at its stated grade, the Majorana door, and
the Sakharov tripod, which is to say the paper's spine. The gold
band is the honest middle: seven chains each explicit end to end
with exactly one named step outstanding, so that a reader who
distrusts any gold row knows precisely which lemma to attack. The
red band is deliberately small and deliberately loud --- three
assertions, all living in the parity projection, all flagged in
Part~VI before any referee could flag them. And the purple and gray
tails are the frontier: two conjectures with names and one seam with
an unidentified owner. Roughly speaking, the bar says: the anatomy
and the valuation are theorems, the mechanism's necessity is a
program, and the transport is physics at quantified grade. The
rest of the paper is the row-by-row defense of exactly that
sentence.

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

\noindent
Before the vocabulary is built (Part~II) and the arena is
assembled (Part~III), 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:inheritance}, Figure~\ref{fig:cohort-map-p5}), and a
panorama of the sixteen Parts of the present paper
(\S\ref{sec:roadmap}, Figure~\ref{fig:parts-roadmap-p5}). 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:inheritance}

\GGTheory{} is a twenty-two-paper research program
\citep{ArisakaA1Constitution} --- eleven A-papers, six P-papers, three C-papers and two white papers --- whose structural claim is that the
observed laws are the boundary operational description of a real
six-dimensional bulk --- four macroscopic spacetime dimensions
plus one scale direction and one compact phase circle \Stheta{}
--- governed by four axioms and an integer ledger
$(M,N,L;J,K) = (1,3,5;7,35)$ \citep{ArisakaA3BI6}. Within that
program this paper is a member of the Foundations Branch, and
it has two parents: everything in it descends from one or the
other. The particle parent \citep{ArisakaA9SM} supplies the
flavor geometry: the discrete-%
symmetry and flavor-holonomy theorems this paper consolidates
(its T-SM-46 pair), the Majorana-door theorem (T-SM-44.1), the
basin/gate architecture of the weak interaction, the strong-CP
two-part vanishing, and the derived neutrino ledger --- masses,
$\dPMNS$, the Majorana parity pair --- on which everything
downstream rides. The cosmology parent \citep{ArisakaA10Cosmos}
supplies the transport: the Sakharov custody, the ten-step
resonant-leptogenesis chain, the zero-dilution clause, and the
honesty objections this paper adopts verbatim. \PaperPFive{} is
their interface --- the place where the particle parent's
smallest derived asymmetries are handed to the cosmology parent's
largest one --- and the reader will find both cited on nearly
every page that follows, because nearly every claim is a
consolidation, sharpening, or audit of one of theirs. Like every
member paper of the cohort, this one restates what it imports and
owns what it proves, so that a reader who never opens a companion
loses the provenance but not the argument.

\paragraph{The Foundations Branch.} On the map's left flank
stands the branch this paper belongs to: the two member papers
whose shared subject is the boundary's complex structure and its
mirrors, a single column beside the spine. The partner paper,
\PaperPFour{} \citep{ArisakaP4Real}, is the origin-of-$i$ member:
it owns the question ``why complex numbers?'' and answers it with
two real sources --- the compact phase circle and the causal
clock --- whose forced agreement on charged sectors is the
antiparticle pairing (its theorems T-P4-1 through T-P4-4,
restated in Part~III below as this paper's inputs), under the
division of labor both papers state in the same words: \emph{the
dictionary owns the sectors; the geometry owns the $i$}. On the
map that inheritance is the solid shared-$i$ edge descending from
\PaperPFour{} into this paper's node. \PaperPFive{} owns the
questions those sources make answerable next: what the discrete
symmetries are (reversals of the sources' orientations), why CP
breaks (the texture on the circle is chiral), why P breaks in one
force (the descent between basins is oriented), and why matter
won (the oriented, CP-biased decays of the heavy neutrino sector)
--- and its own position on the map is the \ANine{}--\ATen{}
interface: fed by the ledger phases derived in \ANine{} --- the
PMNS Dirac phase and the Majorana parity $\alpha_{31} = \pi$ ---
and feeding the $\etaB$ chain of \ATen{} across the width of the
map, at the grade Part~VIII records: derived-conditional on the
named finite-temperature hypothesis
H-$\phi_{\mathrm{geo}}$-1. From the rest of the cohort it
consolidates: the discrete-symmetry and flavor-holonomy theorems
of the Standard-Model treatment \citep{ArisakaA9SM}; the
Majorana-door theorem of the same paper; the baryogenesis chain
of the cosmology treatment \citep{ArisakaA10Cosmos}; the
Wilson-line computations of the gauge-runtime treatment
\citep{ArisakaA6Gi4}; and the strong-CP resolution shared with
the Kaxion companion \citep{ArisakaC1Kaxion}. Each import is
restated at its point of use, with its grade. Both branch members
carry P-series designators; their place at the spine's flank, not
the phenomenology rows, marks their foundational role.

\paragraph{The rest of the map.} Six companions are cited in this paper
without being parents to it, and each earns its place here for a
different reason. \citealp{ArisakaP2N3} is a precondition rather than a
neighbor: a nonzero Jarlskog invariant needs three families, so without
that paper's count there is no CP phase in this one to explain, and the
chiral pentagon of Part~\ref{part:pentagon} has nothing to be drawn on.
\citealp{ArisakaP6Proton} stands at the other end of the same number:
Part~\ref{part:matter} makes the baryon asymmetry and that paper shows
the resulting protons are kept, baryon number being carried as a winding
of the compact circle rather than as a bookkeeping label.
\citealp{ArisakaC2Hubble} carries the expansion history the matter
excess is measured against, and its corridor reading is where the
inventory this paper's chain must land on is read off.
\citealp{ArisakaA4Euclid} traces the geometrization lineage from Euclid
forward, of which the present claim --- that the mirrors are orientations
of real geometry rather than laws --- is one more step, and it is the
companion that says how large a step it is not. The two white papers
close the roster. \citealp{ArisakaW1Symmetry} follows a century of
symmetry and conservation in the order the results were found and watches
the conviction that symmetry causes conservation loosen step by step;
the mirrors of this paper are the sharpest case in that history, being
symmetries that were assumed for three centuries and then measured to
fail. \citealp{ArisakaW2Holonomy} turns on a single exact statement ---
a holonomy requires a closed loop, and the interval fiber has none ---
which is precisely the asymmetry Part~\ref{part:time} derives the four
endowments of time from, put to work there on sixteen open questions
instead.

The twenty-two-paper map, identical in every paper of the cohort,
is Figure~\ref{fig:cohort-map-p5}, 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).
%% ===================================================================
\clearpage
\begin{figure}[p]
\centering
\resizebox{\textwidth}{!}{%
%% v22 type scaling, local to this group: the picture sets node text
%% in \footnotesize (the root in \small) and its edge and branch labels
%% in \scriptsize, so redefining the three commands here enlarges box
%% type by 15%% and everything outside a box by 20%% without touching a
%% single node.
\renewcommand{\footnotesize}{\fontsize{10.35}{12.4}\selectfont}%%
\renewcommand{\small}{\fontsize{11.5}{13.8}\selectfont}%%
\renewcommand{\scriptsize}{\fontsize{9.6}{11.5}\selectfont}%%
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%% --- Row 0: axiomatic root ---
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%% --- Row 1: GG6 Trio ---
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 {\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 ---
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 {\textbf{\AFive{}} \\[-1pt] \footnotesize \GDOS{} boundary law};

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

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

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

%% --- Row 6: Predictions Branch (P-series) ---
%% v22: four predictions on the row at 50mm pitch.
\node[pred] (P1) at (-75mm, -138mm)
 {\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,
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\node[branchframe, draw=teal!60!black, fill=teal!4,
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\node[branchlabel, text=cyan!40!black, anchor=east]
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\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 ---
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\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) ---
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 node[edgelabel,pos=0.55] {Thm~0} (MP4.north);
%% MP4 -> A8 (solid; the origin of i). Departs the east side (MP4.south
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%% 6mm left of the A6->A8 arrowhead, staying left of A6 throughout.
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 node[edgelabel,pos=0.45] {origin of $i$} ([xshift=-6mm]A8.north);
%% MP4 -> A2 (dashed deployment: complex-structure machinery), nudged
%% 3mm up the east side so the two east-side departures stay distinct.
\draw[arrowdash] ([yshift=3mm]MP4.east) .. controls (-70mm,-52mm) .. (A2.south west);
%% MP4 -> MP5 (v1.3: SOLID, per the reading conventions -- forced
%% descent, labelled). A short straight vertical between the pair;
%% the "shared i" label sits on the right side of the edge, the
%% Foundations Branch label on the left.
\draw[arrow] (MP4.south) --
 node[edgelabel, pos=0.5, anchor=west, xshift=1mm] {shared $i$}
 (MP5.north);
%% A9 -> MP5 (solid; ledger phases). Leaves A9's west side, dips
%% through the corridor below the A8 row and enters MP5.east
%% horizontally at the A11 row height. Exactly one clean crossing
%% (the A8->A11 spine edge, near x = -40mm); stays above the dashed
%% A9->PFRAME.north-west edge throughout.
\draw[arrow] (A9.west) .. controls (-40mm,-97mm) and (-58mm,-113mm) ..
 node[edgelabel,pos=0.6] {$\delta,\ \alpha_{31}$} (MP5.east);
%% MP5 -> A10 (solid; eta_B). The one long edge: runs left-to-right
%% through the clear corridor BELOW the A11 node (above the
%% Predictions frame), then climbs into A10's bottom border right of
%% the A10->A11 arrowtail; it leaves MP5.south vertically. Controls
%% lifted 5mm with the raised MP5 so the dip still clears A11's
%% bottom border. One clean crossing with each of the two dashed
%% A9->Predictions-frame edges.
\draw[arrow] (MP5.south) .. controls (-96mm,-128mm) and (55mm,-130mm) ..
 node[edgelabel,pos=0.5] {$\eta_B$} ([xshift=8mm]A10.south);
%% MP5 -> A8 (dashed; Wick-quartet/statistics feed for T-A8.12): with
%% MP5 level with A11 this is now a short straight upward edge from
%% the column's top-right corner into A8's bottom border, 8mm left
%% of the A8->A11 arrowtail. No crossings.
\draw[arrowdash] (MP5.north east) -- ([xshift=-8mm]A8.south);
%% --- White-Paper Branch edges (v22) ---
%% A3 -> W1: \AThree{} is \PaperWOne{}'s heaviest spine source and its
%% thematic terminus -- the \BIsix{} anomaly proof is where its
%% Green--Schwarz / Hopkins--Singer ladder ends.
\draw[arrowdash] ([xshift=8mm]A3.south) .. controls (58mm,-43mm) ..
 node[edgelabel,pos=0.55] {\BIsix{}} (W1.west);
%% A10 -> W2: \ATen{} is \PaperWTwo{}'s heaviest source of any kind.
%% At +15%% those two boxes overlap in x (never in y), so this is drawn
%% off A10's south-east corner, clear of the W1->W2 arrowhead.
\draw[arrowdash] (A10.south east) .. controls (84mm,-101mm) ..
 ([xshift=-6mm]W2.north);
%% W1 -> W2, solid and labelled: the mirror of the shared-i edge on the
%% left flank.  \PaperWTwo{} states the dependency itself -- it takes
%% "the structure that the arc arrives at, protection by topology
%% rather than by symmetry".
\draw[arrow] (W1.south) --
 node[edgelabel, pos=0.80, anchor=west, xshift=1mm]
 {symmetry $\to$ topology} (W2.north);

%% Centring pad, retired (v22): it cancelled a Bezier phantom that the
%% polyline above no longer creates; the bbox is now the ink.
\end{tikzpicture}}
\caption[The twenty-two-paper cohort map.]{The
twenty-two-paper cohort map of \GGTheory{}. The constitutional root
\AOne{} (violet, top) supplies the four axioms and seven postulates
from which the rest of the cohort descends: the GG6 Trio (\ATwo{},
\AThree{}, \AFour{}, red) realizes the bulk geometry; \AFive{}
(yellow) the boundary law; the \GDOS{} Trio pillars (\ASix{},
\ASeven{}, green) the geometric and quantum-information runtime; the
4D-Physics Trio (\AEight{}, \ANine{}, \ATen{}, blue) the textbook
readouts; and the closing capstone \AEleven{} (gold) the
quarks-to-consciousness identity cascade. Two framed branches carry
the phenomenology --- the \textbf{Predictions Branch} (cyan), whose
four papers hand their results along the row by the internal arrows
(\PaperPTwo{}'s $N{=}3$ into \PaperPThree{}, and that paper's
ground-attractor method, \textsf{GGA}, on into \PaperPSix{}), and the
\textbf{Cosmology \& Astrophysics Branch} (teal). The spine is
flanked by two columns of two, which do opposite jobs. On the left the
\textbf{Foundations Branch} (violet) feeds it: \PaperPFour{}, fed by
\ATwo{}'s Theorem~0 arena, manufactures the $i$ that \AEight{}
consumes, and \PaperPFive{} is the \ANine{}--\ATen{} interface, fed by
the ledger phases and feeding the $\eta_B$ chain across the width of
the map, with the shared-$i$ edge descending solid between them. On
the right the \textbf{White-Paper Branch} (brown) reads it:
\PaperWOne{} walks the century from Noether to Hopkins--Singer, and
\PaperWTwo{} dissects the structure that arc arrives at ---
protection by topology rather than by symmetry --- across sixteen open
problems. The white papers derive nothing new, carrying every result
at its parent's declared grade; the single dashed feed drawn into each
names its heaviest source, not its only one. \emph{Reading
conventions:} solid arrows mark axiom-forced descent and carry an edge
label naming the driving axiom or postulate where most illustrative;
dashed arrows mark the deployment of derived results, or historical
links inherited without being constitutionally forced.}
\label{fig:cohort-map-p5}
\end{figure}
\clearpage

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

The rest of the paper is organized into fifteen further Parts,
and the walk through them is plotted in
Figure~\ref{fig:parts-roadmap-p5}. The present Part --- the
entrance --- is nearly done: it has asked the oldest asymmetry
precisely, worn the two hats through a century of falling
mirrors, put the five claims on the table with their registers
attached, and laid out the two orientation maps of this closing
section. Two Parts then put the machine on the table. Part~II
defines everything the argument stands on: the real presentation
of charged fields, orientation reversals and (anti)linearity, the
flavor graph and its holonomies, the seesaw in real blocks, the
transport equations of leptogenesis, the $(1,5)$ spinor algebra
with its no-Majorana lemma, the anomaly flip, and the
three-depths ruler. Part~III assembles the \GGTheory{} arena: the
constitutional clauses, the ledger, the four theorems inherited
from the Foundations-Branch partner, the axiom-custody table that
assigns every symmetry ingredient its axiom, and the custody map
--- \emph{the bulk owns the symmetries; the descent owns their
breaking; the boundary owns their observability}.

The five Parts that follow are the paper's center of gravity, and
they are highlighted in the roadmap because together they are the
deductive funnel of Figure~\ref{fig:p5-central-claim} walked
slowly: one deduction passing through five stations, each station
forced by the one above it. Part~IV derives the twin at full
depth --- the handshake theorems T-P5-A/B/C with the mode-level
proof, the clock's third face (T-P5-B1, the Wick quartet, where
Minkowski's $it$, Wick's rotation, and the Schr\"odinger $i$ are
proved to be one analytic fact), the extension theorems that
carry one shared $J$ into composites, Fock space, and the Dirac
fermion, the self-conjugate exceptions the circle cannot move,
and the boundary that reads the twins. Part~V gives the mirrors
their bodies: the three reversals walked one at a time on the
modes of Part~IV, the table theorem (T-P5-1) with Kramers'
degeneracy as its first dividend, and CPT as total reversal
(T-P5-2), with its converse discipline and its laboratory record.
Part~VI answers the parity question --- why exactly one force is
chiral: the basin/gate architecture with each clause of the
gate's job description derived, the projection computed in
blocks, the handedness invariant with its Ozma measurement, and
the no-$W_R$ falsifier. Part~VII answers the CP question: the
flavor-holonomy theorem (T-P5-3) with its calculus, the
pentagon's own trigonometry from scratch, the quark chain and the
lepton chain step by graded step, the dial-free door (T-P5-4),
and the strong-CP two-fate architecture. Part~VIII answers the
matter question at the depth the heart of the paper deserves: the
annihilation epoch and the surviving number certified three ways,
Sakharov's three conditions derived and traced to their geometric
origin, the exclusion walk that leaves leptogenesis as the only
open channel, the ten-step inherited chain with its audit table,
zero dilution, the resonance question faced in public, and the
dated kill list.

Part~IX pauses the march to ask the question underneath every
page so far --- what \emph{is} time? --- and answers with the
four endowments: order from the cone, repetition from the
equivalence, the arrow and the analytic shore from the
dictionary; the standard accounts honored as partial readings;
the quantum-gravity problem of time dissolved by architecture;
and the first law-level arrow.

Three Parts then confront the derived mirrors with their rivals
and their instruments. Part~X gives the four rival roads their
full say in three movements each --- what the road achieved,
where it ends, and how the geometry completes the story --- for
QFT, the Standard Model, string theory, and $\Lambda$CDM, then
scores the five on the three-depths ruler and closes with the
roads seen from above. Part~XI turns from theorems to
instruments: the century of measured mirrors and the neutrino
century with their milestone tables, the present frontier as of
mid-2026, the instruments of the decisive decade --- each with
its schedule, its sensitivity, and the forced value it touches
--- the confrontation with its pre-declared kills, and the
pre-registered verdicts of 2028--2040. Part~XII discusses:
significance, the innovations registered, the claims ledger, the
falsifier table, and the open questions stated as open.

Part~XIII concludes in plain English --- the arc, the ten
numbers, the limitations, the author's fifty years in the
mirrors, and the oriented emergence of particles, life, and mind.
The back matter is the workshop. Part~XIV carries the lettered
appendix machine room --- five Step-graded chapters, ninety
problems, all with complete worked solutions --- together with
the reference dictionaries and the ledger-and-registry
appendices; Part~XV answers eighty-two objections, each under its
reader's own job title; and Part~XVI annotates every reference
with its relevance note. Read in order, the sixteen Parts compose
a single self-contained argument; 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 /
%%  A10-Cosmos v13.3 pattern -- vertical Part column, right phase
%%  sidebar, highlighted Parts IV--VIII carrying the central claim,
%%  per fig:p5-central-claim), clearpage-isolated minipage +
%%  \captionof.
%% ===================================================================
\clearpage
\noindent
\begin{minipage}{\textwidth}
\centering
\resizebox{0.7377\textwidth}{!}{%
\begin{tikzpicture}[
 font=\sffamily\footnotesize,
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%% --- Part I: entrance ---
\node[found] (I) at (0, 0)
 {\textbf{Part~I}\quad\textbf{The Question and Its History} \\[-1pt]
 \footnotesize The oldest asymmetry, asked precisely; a century of falling mirrors; the five claims, registers attached};

%% --- Parts II--III: the toolbox and the arena ---
\node[appar] (II) at (0, -13mm)
 {\textbf{Part~II}\quad\textbf{Mathematical Preliminaries} \\[-1pt]
 \footnotesize The real presentation, orientation reversals, flavor graphs, seesaw, transport --- everything defined};
\node[appar] (III) at (0, -26mm)
 {\textbf{Part~III}\quad\textbf{The \GGTheory{} Arena} \\[-1pt]
 \footnotesize Axioms, forced geometry, the ledger, the four inherited theorems, and the custody map};

%% --- Parts IV--VIII: the five-station chain (highlighted) ---
\node[innov] (IV) at (0, -42mm)
 {\textbf{Part~IV}\quad\textbf{The Origin of Antiparticles} \\[1pt]
 \footnotesize The handshake T-P5-A/B/C and the Wick quartet (T-P5-B1): \\[-1pt]
 \footnotesize the twin derived; one shared $J$ for the whole tower};
\node[innov] (V) at (0, -59mm)
 {\textbf{Part~V}\quad\textbf{The Mirrors Defined: C, P, T and the CPT Theorem} \\[1pt]
 \footnotesize The three reversals given bodies; the table theorem (T-P5-1), Kramers for free; \\[-1pt]
 \footnotesize CPT as total reversal (T-P5-2), blind to every real texture};
\node[innov] (VI) at (0, -76mm)
 {\textbf{Part~VI}\quad\textbf{Parity: The Oriented Descent} \\[1pt]
 \footnotesize Why exactly one force is chiral: basin/gate architecture, the projection \\[-1pt]
 \footnotesize computed in blocks, the handedness invariant, a registered falsifier};
\node[innov] (VII) at (0, -93mm)
 {\textbf{Part~VII}\quad\textbf{CP Violation: The Chiral Pentagon} \\[1pt]
 \footnotesize Loop holonomies valued on the pentagonal lattice: $\dCKM = 66.9^{\circ}$, \\[-1pt]
 \footnotesize $\dPMNS = 196.4^{\circ}$, $(0,\pi)$; the dial-free door; the strong-CP two fates};
\node[innov] (VIII) at (0, -110mm)
 {\textbf{Part~VIII}\quad\textbf{The Origin of Matter} \\[1pt]
 \footnotesize Sakharov derived; the one open channel; the ten-step inherited chain, audited; \\[-1pt]
 \footnotesize $\etaB \approx 7\times10^{-10}$ --- seven per ten billion, with its sign};

%% --- Parts IX--XI: the pause, the rivals, the instruments ---
\node[princ] (IX) at (0, -125mm)
 {\textbf{Part~IX}\quad\textbf{What Is Time?} \\[-1pt]
 \footnotesize The four endowments: order, repetition, the arrow, and the analytic shore};
\node[princ] (X) at (0, -138mm)
 {\textbf{Part~X}\quad\textbf{The Five Roads, at Full Depth} \\[-1pt]
 \footnotesize QFT, the Standard Model, string theory, $\Lambda$CDM --- and the three-depths scoreboard};
\node[princ] (XI) at (0, -151mm)
 {\textbf{Part~XI}\quad\textbf{Experiments --- Past, Now, and Future} \\[-1pt]
 \footnotesize The measured mirrors, the frontier of mid-2026, and the decisive decade's instruments};

%% --- Parts XII--XIII: weighing and summit ---
\node[results] (XII) at (0, -164mm)
 {\textbf{Part~XII}\quad\textbf{Discussions} \\[-1pt]
 \footnotesize Significance \,\textbar\, innovations \,\textbar\, claims ledger \,\textbar\, falsifiers \,\textbar\, open questions \,\textbar\, the cohort};
\node[summit] (XIII) at (0, -177mm)
 {\textbf{Part~XIII}\quad\textbf{Conclusions --- Why Matter Won} \\[-1pt]
 \footnotesize The arc in plain English and ten numbers; the vision: symmetries read as geometry};

%% --- Parts XIV--XVI: back matter ---
\node[back] (XIV) at (0, -190.5mm)
 {\textbf{Part~XIV}\quad\textbf{Technical Appendices} \\[-1pt]
 \footnotesize The machine room: five Step-graded chapters, ninety worked problems, the ledgers};
\node[back] (XV) at (0, -204mm)
 {\textbf{Part~XV}\quad\textbf{Common Objections and Responses} \\[-1pt]
 \footnotesize Eighty-two adversarial readings, answered with pointers};
\node[back] (XVI) at (0, -217.5mm)
 {\textbf{Part~XVI}\quad\textbf{References} \\[-1pt]
 \footnotesize Cited works, each with a relevance note};

%% --- Vertical flow arrows (the highlighted chain in innov-orange) ---
\draw[arrow] (I.south) -- (II.north);
\draw[arrow] (II.south) -- (III.north);
\draw[arrowinnov] (III.south) -- (IV.north);
\draw[arrowinnov] (IV.south) -- (V.north);
\draw[arrowinnov] (V.south) -- (VI.north);
\draw[arrowinnov] (VI.south) -- (VII.north);
\draw[arrowinnov] (VII.south) -- (VIII.north);
\draw[arrowinnov] (VIII.south) -- (IX.north);
\draw[arrow] (IX.south) -- (X.north);
\draw[arrow] (X.south) -- (XI.north);
\draw[arrow] (XI.south) -- (XII.north);
\draw[arrow] (XII.south) -- (XIII.north);
\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; the innov boxes are wider (half-width 70mm),
%% their labels sit at x = 75mm.
\node[phaselabel] at (67mm, 0mm) {\emph{Entrance}};
\node[phaselabel] at (67mm, -19.5mm) {\emph{The toolbox} \\ \emph{\& the arena}};
\node[phaselabel, text=orange!70!black] at (75mm, -76mm)
 {\emph{\textbf{The five-station}} \\ \emph{\textbf{chain}}};
\node[phaselabel] at (67mm, -125mm) {\emph{The pause:} \\ \emph{what is time?}};
\node[phaselabel] at (67mm, -144.5mm) {\emph{Rivals \&} \\ \emph{instruments}};
\node[phaselabel] at (67mm, -170.5mm) {\emph{Weighing \&} \\ \emph{summit}};
\node[phaselabel] at (67mm, -204mm) {\emph{Back matter}};

\end{tikzpicture}%
}
\par\bigskip
\captionof{figure}[Roadmap of the sixteen Parts of GG-P5-CPT.]{Roadmap of the sixteen Parts of \textsf{GG-P5-CPT}. The argument develops in seven phases. \emph{Entrance} (Part~I): the oldest asymmetry asked precisely, the century of falling mirrors, and the five claims with their registers. \emph{The toolbox and the arena} (Parts~II--III): every object the argument uses, defined from scratch, and the \GGTheory{} stage with its ledger and custody map. \emph{The five-station chain} (Parts~IV--VIII, highlighted): the twin derived from the handshake of the circle and the clock, the mirrors given bodies and the CPT theorem, the oriented descent of parity, the pentagon's CP phases, and the origin of matter --- the deductive funnel of Figure~\ref{fig:p5-central-claim} walked slowly. \emph{The pause} (Part~IX): what time is --- the four endowments. \emph{Confrontation} (Parts~X--XI): the four rival roads at full depth and the instruments of the decisive decade with their pre-registered verdicts. \emph{Weighing and summit} (Parts~XII--XIII): the discussions and the conclusions --- why matter won. The three back-matter Parts collect the technical appendices with their ninety worked problems (Part~XIV), the eighty-two common objections and responses (Part~XV), and the annotated references (Part~XVI). Parts~IV--VIII are highlighted because together they carry the paper's central claim --- the single deduction of \S\ref{sec:claim}, from the real oriented bulk to the six-per-ten-billion receipt --- on which every later Part, and the cohort downstream of \PaperPFive{}, stands.}
\label{fig:parts-roadmap-p5}
\end{minipage}
\clearpage

\medskip
\noindent\emph{Part-end summary.} Seventy years of experiment broke
three mirrors, spared the fourth, measured one phase to three
digits, and counted six baryons per ten billion photons --- while
the theory that organizes all of it explains none of it. The claims
of this paper are on the table with their registers attached. The
next Part builds the vocabulary in which they can be stated
precisely.

\GGPart[Mathematical preliminaries]{Mathematical Preliminaries}{The real presentation, orientation reversals, flavor graphs, and transport --- everything the argument stands on, defined}
\label{part:prelim}

\section[Part~II in plain English]{Part~II in plain English: the toolbox, opened before it is used}
Nothing is proved in this Part. It defines every object the argument uses --- the real presentation of a charged field, orientation reversals and antilinearity, the flavor graph, loop holonomies and the Jarlskog construction, the seesaw in real blocks, leptogenesis transport, spinors in $(1,5)$, and the weight arithmetic --- with proofs where they are short and precise citations where they are classical.

Two of the definitions do more work than the rest and are worth slowing down for. The first is the real presentation: a complex field written as a pair of real components with a rule between them, so that the $i$ becomes a rotation one can point at. The second is the flavor graph, on which a CP phase becomes the holonomy of a loop. Most of what happens later in this paper is one of those two objects being read carefully.

The Part also fixes the three depths, which is the instrument used in Part~\ref{part:fiveroads} to compare this account with the four it is set beside. A reader who wants to know how that comparison is scored should read the depths before reading the comparison, because an instrument introduced after the measurement is not an instrument.

\section{The real presentation of a charged field}
\label{sec:real-presentation}

Every physicist has met the trick at the heart of this Part,
probably in the second year of undergraduate study, wearing
humbler clothes. An electrical engineer writes an oscillating
voltage as the real part of $V_0 e^{i\omega t}$ --- not because
the circuit contains imaginary voltages, but because a rotating
point in a plane is the tidiest way to carry an amplitude and a
phase together. The complex number is a \emph{picture of a
rotation}. The thesis this paper inherits from its
Foundations-Branch partner \citep{ArisakaP4Real} is that in fundamental
physics the engineer's situation is the true one: nothing in
nature is complex; wherever an $i$ appears, something real is
rotating, and the job of foundations is to find the rotor. In
the \GG{} bulk the rotors are physical --- a compact fiber
circle and the causal clock --- and this Part builds the
dictionary for working with them directly, in what the parent
program calls the \emph{real presentation}
\citep{ArisakaA8QM}. Nothing in the dictionary is difficult;
the reader who can rotate a vector in a plane can follow every
step of this paper. What the dictionary buys is not rigor for
its own sake but \emph{visibility}: once fields are real pairs
and phases are real angles, the discrete symmetries become
things one can \emph{see} --- reversals of drawn arrows --- and
theorems about them become bookkeeping a student can check. A field of charge $n$ under a
compact $\mathrm{U}(1)$ with angle coordinate
$\theta \in [0, 2\pi)$ is presented not as a complex function
$\psi \sim e^{in\theta}$ but as a \emph{real pair} $(a, b)$ on which
the circle acts by rotation:
\begin{equation}
\begin{pmatrix} a \\ b \end{pmatrix}
\;\longmapsto\;
R(n\alpha)\begin{pmatrix} a \\ b \end{pmatrix},
\qquad
R(\varphi) = \begin{pmatrix}
\cos\varphi & -\sin\varphi \\ \sin\varphi & \cos\varphi
\end{pmatrix},
\label{eq:real-pair}
\end{equation}
under the circle translation $\theta \mapsto \theta + \alpha$.
Concretely: a ``charge-2 field'' is a real pair that turns
twice while the fiber turns once, exactly as the engineer's
second harmonic turns twice per cycle --- winding number is all
that charge is, which is why charge is quantized (windings are
integers) and why neutral fields do not rotate at all. A
coupling phase is a rigid fiber rotation --- a real $R(\varphi)$
block, not a complex number --- and complex conjugation is
\emph{orientation reversal of the circle}, $\theta \mapsto -\theta$,
which sends the weight-$n$ pair to the weight-$(-n)$ pair: the
``conjugate field'' is the same pair, read with the fiber's
orientation reversed --- an angle run backward, not a path
retraced. On each
charged sector the charge-normalized circle generator
\begin{equation}
\Jq \;=\; \frac{\partial_\theta}{Q}
\label{eq:JQ}
\end{equation}
squares to $-\mathbf{1}$ and is the sector's complex structure: the
``$i$'' of a charged field is a quarter-turn of a real fiber. On the
neutral sector $\Jq$ acts as zero --- the circle can only complexify
what it moves --- and the state-space complex structure comes instead
from the clock (Section~\ref{sec:inherited-theorems}). Nothing in the
paper's arguments requires the symbol $i$; where complex notation
appears it is shorthand for real pairs and rotation blocks; the
flagship computations of Appendix~\ref{app:flavour} use that
shorthand, every $i$ redeemable by the same rule.

\emph{Physics intuition.} The real presentation is not a notational
austerity exercise; it is what makes the discrete symmetries
\emph{geometric}. In complex notation, ``charge conjugation''
conjugates symbols; in the real presentation it reverses a circle.
An operation that reverses a circle has an anatomy --- it either
preserves or reverses each of the other orientations in the theory
--- and anatomies can be theorems. That is the entire method of
Parts~IV and~V.

\section{Orientation reversals and (anti)linearity}
\label{sec:orientation-reversals}

Let $V$ be a real vector space carrying a complex structure $J$
(a real operator with $J^2 = -\mathbf{1}$). A real-linear operator
$S$ on $V$ is \emph{$J$-linear} if $SJ = JS$ and
\emph{$J$-antilinear} if $SJ = -JS$. Under the identification of
$(V, J)$ with a complex space, $J$-linear operators are complex
linear and $J$-antilinear operators are conjugate-linear; if they
preserve the relevant inner product they are, respectively, unitary
and antiunitary. Two composition rules do all the work of
Part~V: the product of two $J$-linear or two $J$-antilinear
operators is $J$-linear, and a mixed product is $J$-antilinear ---
(anti)linearity multiplies like a sign.

There is a homely way to hold the two rules just stated, and it
will carry the reader through every proof in Part~V. Think of
$J$ --- the operator ``rotate a quarter turn'' --- as a gear
meshed into the machinery of the theory. An operation that
leaves the gear turning the same way ($SJ = JS$) is
\emph{unitary}: it may rearrange the machinery, but clockwise
stays clockwise. An operation that makes the gear turn the
other way ($SJ = -JS$) is \emph{antiunitary}: every phase runs
backward through it, which is exactly the familiar
complex-conjugation of textbook antiunitarity. And because
reversing a rotation twice restores it, the composition rule
--- mixed products are antiunitary, matched products are
unitary --- is nothing but ``two reversals make a forward.''
The formal statements live in the standard texts
\citep{StreaterWightman, WeinbergQTF}; the geometric reading is
this paper's method.

The standard route to the discrete symmetries --- Wigner's theorem
\citep{Wigner1931} --- states that any symmetry of quantum-mechanical
transition probabilities is implemented either unitarily or
antiunitarily, and leaves the choice to consistency arguments made
case by case: time reversal is \emph{assigned} its antiunitary
character because a unitary $T$ would flip the sign of the
Hamiltonian and wreck the spectrum. This paper's departure, stated
now so the reader can watch for it: when the complex structure $J$
itself has geometric \emph{sources} --- when $i$ is made of specific
orientations --- then whether a given reversal commutes or
anticommutes with $J$ is not a choice but a computation, and the
unitary/antiunitary table becomes a theorem about which reversals
touch the sources (T-P5-1).

\section{The flavor graph}
\label{sec:flavour-graph}

The CP-violation half of the paper runs on one combinatorial
object, and the object is friendlier than its name. Picture a
patch panel: six sockets (the quark fields), nine wires between
them (the Yukawa couplings), and on each socket a small dial
that anyone may turn freely (the phase convention of that
field). Each wire carries a twist --- a real angle, a rigid
rotation of the fiber. The physics question ``which phases are
real and which are convention?'' becomes the electrician's
question ``which twists can be dialed away?'' --- and the
electrician answers it instantly: turning dials can untwist any
\emph{tree} of wires, but the total twist around a closed
\emph{loop} cannot be touched, because every dial feeds a loop
twice with opposite signs. That is the whole rephasing-invariant
theory of CP violation, in wiring language, and the standard
monograph develops the same content in matrix language
\citep{BrancoLavouraSilva}. Formally: consider $N$ generations
of up-type and down-type fields (the lepton case is identical
with its own labels). The \emph{flavor graph} is
the bipartite graph with $2N$ vertices --- one per field --- and
$N^2$ edges --- one per Yukawa coupling. In the real presentation
each coupling is a block $|y_{ij}|\,R(\varphi_{ij})$: a magnitude
and a fiber twist. A Yukawa \emph{texture} is thus a
$\mathrm{U}(1)$-valued connection on the flavor graph: each edge
carries an angle.

\begin{figure}[t]
\centering
\resizebox{0.609\textwidth}{!}{%
\begin{tikzpicture}[font=\sffamily\small, >=Stealth,
 uvert/.style={draw=blue!60!black, line width=0.80pt, fill=blue!8,
   rounded corners=2pt, minimum width=13mm, minimum height=7.5mm},
 dvert/.style={draw=orange!75!black, line width=0.80pt, fill=orange!12,
   rounded corners=2pt, minimum width=13mm, minimum height=7.5mm}]
 \fill[red!10] (0.65,3.6) -- (5.35,3.6) -- (5.35,1.8) -- (0.65,1.8)
   -- cycle;
 \node[uvert] (u1) at (0,3.6) {$u$};
 \node[uvert] (u2) at (0,1.8) {$c$};
 \node[uvert] (u3) at (0,0.0) {$t$};
 \node[dvert] (d1) at (6,3.6) {$d$};
 \node[dvert] (d2) at (6,1.8) {$s$};
 \node[dvert] (d3) at (6,0.0) {$b$};
 \foreach \i in {1,2,3} \foreach \j in {1,2,3} {
   \draw[gray!45, line width=0.60pt] (u\i) -- (d\j);
 }
 \draw[line width=1.10pt, red!75!black] (u1) --
   node[above=0.5mm, pos=0.22, font=\footnotesize]{$\varphi_{ud}$} (d1);
 \draw[line width=1.10pt, red!75!black] (u1) --
   node[below=0.4mm, pos=0.16, font=\footnotesize]{$\varphi_{us}$} (d2);
 \draw[line width=1.10pt, red!75!black] (u2) --
   node[above=0.4mm, pos=0.16, font=\footnotesize]{$\varphi_{cd}$} (d1);
 \draw[line width=1.10pt, red!75!black] (u2) --
   node[below=0.5mm, pos=0.22, font=\footnotesize]{$\varphi_{cs}$} (d2);
 \foreach \v in {u1,u2,u3,d1,d2,d3} {
   \draw[line width=0.80pt] ([xshift=0mm,yshift=5.5mm]\v.center) circle (1.6mm);
   \draw[line width=0.80pt] ([xshift=0mm,yshift=5.5mm]\v.center) --
     ([xshift=1.0mm,yshift=6.6mm]\v.center);
 }
 \node[draw=black!55, rounded corners=2pt, fill=white, align=left,
   font=\footnotesize, anchor=north west] at (7.4,3.8)
  {\textbf{vertices}: fields, each with a\\ free phase dial
   (rephasing)\\[2pt]
   \textbf{edges}: Yukawa couplings\\ $|y_{ij}|\,R(\varphi_{ij})$
   --- a magnitude\\ and a fiber twist\\[2pt]
   \textbf{shaded loop}: the plaquette\\
   $\Phi_{\square} = \varphi_{ud} - \varphi_{us}
     + \varphi_{cs} - \varphi_{cd}$\\[2pt]
   dial turns cancel around any\\ closed loop:
   $\Phi_{\square}$ is physical};
\end{tikzpicture}}
\caption[The flavor graph for three generations.]{The flavor graph for three generations, drawn to the
cohort's wiring-diagram standard. Up-type and down-type fields
are vertices, each carrying a freely re-settable phase dial; the
nine Yukawa couplings are edges, each carrying a fiber twist
$\varphi_{ij}$. Turning any dial shifts every edge at that vertex
equally, so the shifts telescope to zero around a closed loop:
the shaded four-edge plaquette holonomy $\Phi_{\square}$ is
invariant --- a flavor-space Aharonov--Bohm phase
\citep{AharonovBohm1959}. For $N = 3$, unitarity ties all such
plaquettes to a single physical angle, the Jarlskog phase
(\S\ref{sec:jarlskog}).}
\label{fig:flavour-graph}
\end{figure}

Not all the angles are physical. Each vertex carries a ``dial'' ---
the freedom to re-zero the phase convention of that field --- and a
\emph{rephasing} acts as a gauge transformation on the graph,
\begin{equation}
\varphi_{ij} \;\longmapsto\; \varphi_{ij} + \alpha_i - \beta_j ,
\label{eq:rephasing}
\end{equation}
with $\alpha_i$ the up-side and $\beta_j$ the down-side dial turns.
The gauge-invariant content of a connection on a graph is its
\emph{holonomies}: the sums of edge angles around closed loops, on
which the action \eqref{eq:rephasing} telescopes to zero. Choosing a
spanning tree of the graph and re-zeroing every tree edge (always
possible, since tree edges can be nulled one by one), the residual
physical data are exactly one angle per independent loop:
\begin{equation}
\#\{\text{physical phases}\}
= N^2 - (2N - 1) = (N-1)^2 ,
\label{eq:phase-count}
\end{equation}
the familiar counting --- zero for one generation, one for two
(where it can be removed by a further discrete choice), one
surviving CP phase for $N = 3$.

\section{Loop holonomies and the Jarlskog construction}
\label{sec:jarlskog}

The connection between loop holonomies and observable CP violation
is the standard rephasing-invariant calculus, due in its
basis-independent form to Jarlskog \citep{Jarlskog1985}; this
section states it in graph language and draws the boundary between
what is inherited and what this paper adds. For a mixing matrix $V$,
the \emph{quartet invariants}
$\mathrm{Im}(V_{ij}V_{kl}V_{il}^{*}V_{kj}^{*})$ are the imaginary
parts of the smallest rephasing-invariant products --- in graph
terms, the sines of the plaquette holonomies, the loop angles of the
four-edge cycles. For $N = 3$, unitarity ties all nine quartets to a
single number, the Jarlskog invariant $J$; every CP-violating
observable in the quark sector is proportional to it; and CP is
conserved iff every quartet vanishes --- iff every loop holonomy
lies in $\{0, \pi\}$. Two properties deserve emphasis. First, the
loop angles are exactly as gauge-invariant as an Aharonov--Bohm
phase \citep{AharonovBohm1959}, and the analogy is not decorative:
in the real presentation both are holonomies of a
$\mathrm{U}(1)$ connection, one on spacetime, one on the flavor
graph. Second, everything in this section is inherited mathematics
--- the graph packaging is convenient, not novel. What the parent
program adds, and what T-P5-3 proves, is a \emph{valuation}: a
reason the loop angles take particular values, namely that on a
single compact circle with block integer $L = 5$ the first-order
holonomies are confined to the pentagonal lattice $(\pi/5)\bZ$. The
standard calculus says where CP violation \emph{lives}; the lattice
says what it \emph{costs}.

\emph{Physics intuition --- why an area?} It is worth pausing on
the geometric meaning of $J$, because it recurs. The six
unitarity relations of the CKM matrix are triangles in the
complex plane, drawn from products of matrix elements --- and
all six triangles, however different their shapes, have the
\emph{same area}, $J/2$. Why? Because CP violation is a
holonomy, and a holonomy is a flux: the loop angle is the
``magnetic flux'' through the plaquette of the flavor graph,
and areas are what fluxes are. When the $B$ factories measured
the triangle's angles and the kaon experiments its sides, they
were surveying one plaquette from two directions --- which is
why the global fit's consistency is a one-number statement, and
why Part~VII can price all of quark CP violation by valuing a
single angle. An experimentalist who has drawn the unitarity
triangle has been doing flavor-space magnetostatics all
along.

\section{The seesaw and Casas--Ibarra in real blocks}
\label{sec:seesaw}

Part~VIII requires the seesaw mechanism, and its physical idea
deserves its playground name. Put a child and an adult on a
seesaw: the harder the adult sits down, the higher the child
rides. The neutrino is the child. If a very heavy neutral
partner exists --- mass $M_N$ --- and mixes with the light
neutrino through an ordinary electroweak coupling, then
diagonalizing the pair sends one mass \emph{up} to $M_N$ and
the other \emph{down} to $m_\nu \sim (\text{ordinary
mass})^2/M_N$: the observed sub-eV lightness is not a mystery
to be tuned but the shadow of a heaviness not yet seen. The
mechanism was found independently several times within three
years \citep{Minkowski1977, YanagidaSeesaw,
GellMannRamondSlansky, MohapatraSenjanovic1980} --- a sign, as
with most good ideas, that it was ready. In the real
presentation it acquires one further virtue: the question
``\emph{may} the heavy partner have a Majorana mass at all?''
--- in standard model-building a choice --- becomes weight
arithmetic with a unique answer (T-P5-4). A
Majorana mass term is a \emph{self}-pairing of a field --- in real
terms, a symmetric bilinear on a single weight sector --- and
Part~VII's T-P5-4 proves such a pairing is invariant under the circle
action iff the weight is zero. Given heavy Majorana states $N_i$
with mass matrix $M_N$ and Yukawa couplings $Y_\nu$ to the lepton
doublets, the light-neutrino mass matrix is
\begin{equation}
m_\nu \;=\; \frac{v^2}{2}\, Y_\nu^{T} M_N^{-1} Y_\nu ,
\label{eq:seesaw}
\end{equation}
suppressed by the heavy scale --- the seesaw. Inverting the relation
to express the Yukawa in terms of the observed light masses and
mixings is the Casas--Ibarra parameterization
\citep{CasasIbarra2001},
$Y_\nu = M_N^{1/2} R\, m_\nu^{1/2} U_{\mathrm{PMNS}}^{\dagger}/v$
with $R$ a complex orthogonal matrix; in the inherited chain of
Part~VIII the scale of $M_N$, the light masses, and the phases of
$U_{\mathrm{PMNS}}$ are all fixed upstream, so the parameterization
carries no freedom beyond the $R$-matrix structure it makes
explicit.

\section{Leptogenesis transport in one section}
\label{sec:transport}

Part~VIII's chain calls the following standard results, stated here
once with their sources. The picture to hold throughout is a
leaky bathtub: heavy decays pour asymmetry in, inverse decays
and scatterings drain it away, and what survives is the level
in the tub at the moment the drain freezes. \emph{Decay asymmetry:} a heavy Majorana
state $N_i$ decaying to lepton--Higgs pairs generates a CP asymmetry
$\epsilon_i = [\Gamma(N_i \to \ell H) - \Gamma(N_i \to \bar\ell
\bar H)]/[\text{sum}]$, arising at one loop from the interference of
tree and self-energy/vertex graphs \citep{FukugitaYanagida1986,
CoviRoulet1996}. \emph{Resonant regime:} when two heavy states are
quasi-degenerate, the self-energy contribution is resonantly
enhanced; at $|M_i - M_j| \simeq \Gamma_j/2$ the asymmetry saturates
near $\epsilon \sim \tfrac12\sin(2\phi)$ with $\phi$ the relevant
CP phase --- the Yukawa suppression cancels between numerator and
denominator \citep{Pilaftsis1997, PilaftsisUnderwood2004}. This is
the mechanism that permits TeV-scale leptogenesis, evading the
Davidson--Ibarra bound $M_N \gtrsim 10^9$~GeV that governs the
hierarchical regime \citep{DavidsonIbarra2002}. \emph{Washout:} the
same Yukawas that produce the asymmetry can erase it via inverse
decays and scatterings; the strength is measured by the effective
mass $\widetilde m$ against the equilibrium scale
$m_\star \simeq 1.1\times10^{-3}$~eV, and in the strong-washout
regime $K_{\mathrm{wash}} = \widetilde m/m_\star \gg 1$ the
surviving efficiency is $\kappa \simeq 1/K_{\mathrm{wash}}$ up to
$O(1)$ factors \citep{BuchmullerDiBariPlumacher}. \emph{Sphaleron
conversion:} the strangest and most beautiful entry in the
toolbox, worth three sentences of intuition. 't~Hooft showed
that baryon and lepton number are not exactly conserved in the
Standard Model itself --- the electroweak vacuum has a winding
structure, and tunneling between windings changes $B{+}L$
\citep{tHooft1976} --- but at zero temperature the rate is
absurdly small. Klinkhamer and Manton found the mountain pass
between the windings --- the \emph{sphaleron}, a static
unstable solution sitting at the top of the barrier
\citep{KlinkhamerManton1984} --- and Kuzmin, Rubakov, and
Shaposhnikov realized that the hot early universe simply walks
over the pass: above the electroweak crossover, $B{+}L$ is
erased and re-made freely, while $B{-}L$ is untouchable
\citep{KuzminRubakovShaposhnikov1985}. A lepton asymmetry
(negative $B{-}L$) is therefore reprocessed into baryons with a
coefficient fixed by chemical-potential bookkeeping over the
plasma, $B = \tfrac{28}{79}(B{-}L)$ for one Higgs doublet
\citep{HarveyTurner1990} --- the number every baryogenesis
mechanism in existence must multiply by, derived in
Appendix~\ref{app:lepto-transport}.
\emph{Entropy bookkeeping:} the late-time photon bath dilutes the
asymmetry by the standard factor, $\etaB \simeq 7.04\, Y_B$, with
$\gstar = 106.75$ relativistic degrees of freedom at production.
The standard reference for all of this machinery is Kolb and
Turner \citep{KolbTurner}, and the reader meeting transport for
the first time should feel no intimidation: every equation in
the kit answers a question a household understands ---
\emph{how fast does it decay} (the width), \emph{does the
crowd undo it} (washout), \emph{who converts the currency}
(sphalerons), and \emph{how much did the room grow} (entropy).
Every symbol above reappears in Part~VIII with a number
attached, and nothing harder than these four questions ever
enters.

\section{Spinors in \texorpdfstring{$(1,5)$}{(1,5)}: the algebra
that forbids particle $=$ antiparticle}
\label{sec:spinor-algebra}

Part~IV's bulk-encoding argument rests on the Clifford algebra of
six-dimensional Lorentzian spacetime, so it is built here from
scratch. The Dirac spinor representation of $\mathrm{Spin}(1,5)$
has complex dimension $2^{6/2} = 8$. The chirality operator
$\Gamma^7 = \Gamma^0\Gamma^1\cdots\Gamma^5$ squares to $+\mathbf 1$
in $(1,5)$ signature, so the Dirac representation splits into two
irreducible Weyl representations of complex dimension four,
\begin{equation}
\mathbf{8}_{\mathrm{Dirac}} \;=\; \mathbf{4} \oplus \bar{\mathbf{4}},
\label{eq:weyl-split}
\end{equation}
exchanged by complex conjugation. The decisive fact is the
\emph{reality classification}: in $d = 6$ Lorentzian signature the
chiral spinor representation is \textbf{complex} --- not real, not
pseudoreal --- and consequently:

\begin{lemma}[no Majorana condition in $(1,5)$]
\label{lem:no-majorana}
There is no Majorana condition compatible with the Clifford reality
structure of $(1,5)$ signature: no charge-conjugation matrix $B$
with $B\bar B = +\mathbf 1$ exists in the chiral representation, so
the condition $\psi = B\psi^{*}$ has no solutions. A Weyl fermion
and its conjugate are genuinely independent objects.
(Symplectic-Majorana--Weyl conditions, which pair spinors using an
internal doublet index, exist but identify nothing with its own
conjugate; they are not used in the \GG{} construction.)
\end{lemma}

\begin{proof}[Proof sketch]
The classification of spinor reality types by signature $(t, s)$
is periodic in $(t{-}s) \bmod 8$; for $(1,5)$, $t - s \equiv 4$,
where the intertwiner $B$ satisfies $B\bar B = -\mathbf 1$: the
would-be Majorana projection squares to $-1$ and is empty. The
standard reference treatment is Lawson and Michelsohn
\citep{LawsonMichelsohn}; the \GG{} arena paper carries the
applied statement \citep{ArisakaA2GG6}.
\end{proof}

\emph{Physics intuition --- the eight-hour clock.} The mod-8
periodicity is not numerology; it is the deepest clock in
mathematics (Bott's), and one can feel why dimension matters
without any machinery. A spinor is a ``square root of a
vector,'' and taking square roots interacts with reality the
way it always has --- sometimes the root of a real thing is
real, sometimes it comes in conjugate pairs. As the dimension
ticks upward, the Clifford algebra cycles through the real
numbers, the complex numbers, and the quaternions with period
eight, and the available reality conditions cycle with it.
Four-dimensional spacetime happens to sit at an hour where a
spinor can be its own conjugate (Majorana is \emph{possible}
there --- an option, famously still undecided for the
neutrino); six-dimensional spacetime sits four hours on, where
it cannot. The \GG{} bulk did not choose to keep particles and
antiparticles distinct; it was built at the hour of the clock
where identifying them is not on the menu. This is what it
means for a conclusion to come from geometry rather than
model-building: the option was closed before any physics was
put in.

\emph{Why this matters --- said now, used in Part~IV.} In four
dimensions a neutral fermion may be its own antiparticle
(Majorana), and whether nature uses that option is an experimental
question. In the \GG{} bulk the option \emph{does not exist} for
any charged chiral fermion: the algebra itself forbids
identifying a bulk Weyl fermion with its conjugate. The conjugate
sector --- the antiparticle content --- is therefore necessarily a
\emph{separate, genuinely present} module of the bulk matter, not
a notational reflection. When Part~IV says the antiparticle is
``really there'' before any descent, this lemma is what the words
mean.

\section{Weight arithmetic and the anomaly flip}
\label{sec:weight-anomaly}

Two more tools, stated once. \emph{Weight arithmetic:} under the
circle action, the symmetric square of a weight-$n$ real pair
decomposes into the invariant $a^2 + b^2$ (the Dirac pairing,
winding-conserving) plus a doublet rotating with weight $2n$,
invariant iff $n = 0$ --- the three-line computation
(Worked example~\ref{ssec:we5}) that will carry the Majorana door.
\emph{The anomaly flip:} each 6D Weyl species of chirality
$\sigma_s = \pm1$ in representation $\rho_s$ contributes to the
anomaly eight-form proportionally to its chirality,
\begin{equation}
I_8\!\left[D^{-}\right] \;=\; -\,I_8\!\left[D^{+}\right],
\label{eq:anomaly-flip}
\end{equation}
so a chirality-flipped Weyl fermion in the same representation is
a \emph{physically distinct species} contributing with opposite
sign. The eight-form itself descends from the index theorem, and
its structure in $4k{+}2$ dimensions --- where purely
gravitational anomalies exist and chirality is a signed currency
--- was mapped in the classic analysis of Alvarez-Gaum\'e and
Witten \citep{AlvarezGaumeWitten1984}; the \GG{} ledger paper
executes the full arithmetic for \BIsix{}
\citep{ArisakaA3BI6}. The flip \eqref{eq:anomaly-flip} is the
algebraic hinge on which those books balance, and (Part~IV
\S\ref{sec:bulk-encoding}) the mechanism by which the anomaly
arithmetic \emph{forces} opposite-chirality matter into the
bulk spectrum. Said warmly: in six dimensions the bookkeeper
cannot be bribed --- every chiral species signs its
contribution, the total must vanish, and a spectrum written
all in one sign is simply not a theory. Nature's ledgers,
here as everywhere in the program, do the model-building
themselves.

\section{The three depths: an instrument for comparing
explanations}
\label{sec:three-depths}

The five-roads comparison of Part~X needs a calibrated ruler,
defined here so it cannot be bent later. A question like ``where
do antiparticles come from?'' is answered at three possible
depths. \textbf{Depth 1 (spectral):} given a quantum theory, why
does its spectrum contain, for every charged particle, a partner
of equal mass and opposite charge? \textbf{Depth 2 (structural):}
why must any theory of the relevant type contain such partners ---
what consistency condition forces them? \textbf{Depth 3
(ontological):} what \emph{are} the paired degrees of freedom, and
what is the pairing itself made of --- in particular, what is the
complex structure whose conjugation defines it? The same ruler
applies to each discrete symmetry (what is the operation made of?)
and to the matter excess (are the inputs of the mechanism
themselves explained?). To feel the ruler's teeth, run it once on a familiar fact:
``the positron has the electron's mass.'' Depth~1: it is in the
spectrum --- measured, tabulated, no explanation offered.
Depth~2: it \emph{must} be, by the CPT theorem --- any local,
Lorentz-invariant theory pairs the masses; a magnificent
constraint that works equally well whatever the electron and
positron turn out to be. Depth~3: the two are \emph{one
object read in two orientations} --- the mass equality is not
enforced between two things but trivial for one thing
(\S\ref{sec:boundary-readout}) --- an answer that tells you
what the twins are, at the price of committing to an ontology
that can be wrong. The three depths are not ranks of virtue;
they are different trades of scope against insight, and a
mature comparison states which trade each framework has made.
The honest headline of the whole comparison, defended in
Part~X: QFT answers Depths 1 and 2 rigorously and stops;
string theory answers Depth 1 by inheritance and gestures at
Depth 3 without valuation; $\Lambda$CDM measures the answers
and explains none; the \GG{} claim --- with its registers and
its named residues --- lives at Depth 3, with Depths 1--2
recovered as corollaries.

\section{Notation table}
\label{sec:notation}

$\Jq = \partial_\theta/Q$: circle complex structure on a charged
sector. $\Ja$: clock complex structure (polar phase of the time
generator; the Hilbert transform). $R(\varphi)$: rotation block.
$\varphi_{ij}$: texture edge angle; loop angles are their cycle
sums. $J$ (unadorned, in flavor contexts): Jarlskog invariant.
$(M,N,L;J,K) = (1,3,5;7,35)$: the integer ledger (the flavor-count
$N$ and ledger $J$ are always disambiguated by context).
$\Mlock = \tfrac12 M_{\mathrm{Pl}}e^{-K} \simeq 3.85$~TeV: the lock
scale. $\etaB$: baryon-to-photon ratio. Grades: \emph{derived} /
\emph{conditional (named lemma)} / \emph{D$^{*}$} (derived
conditional, order of magnitude) / \emph{open}.

\medskip
\noindent\emph{Part-end summary.} The vocabulary is in place: fields
as real pairs, discrete operations as orientation reversals whose
(anti)linearity multiplies like a sign, CP phases as loop holonomies
on the flavor graph, and the transport toolkit of leptogenesis with
its standard coefficients. The next Part assembles the stage on
which the theorems act.

\GGPart[The GG arena]{The \GGTheory{} Arena}{The axioms, the forced geometry, the ledger, the four inherited theorems, and the custody map}
\label{part:arena}

\section[Part~III in plain English]{Part~III in plain English: the stage, inherited and restated}
This paper does not build the framework it uses. The axioms, the theorem that forces the six-dimensional arena, the integer ledger, the boundary law and the runtime are all inherited from companion papers, and this Part restates each at the precision the later proofs need, naming the paper that owns it in every case.

What comes out is small, and it is the whole basis of the argument: a four-dimensional spacetime, a non-spatial radial interval and a non-spatial compact circle, with a globally static metric --- and, attached to every system in the universe, two real structures. The circle can carry a holonomy. The interval cannot, because a holonomy needs a closed loop and the interval has none. That asymmetry is what the rest of the paper spends itself on.

Restating an inherited result is not the same as proving it. The inheritance table records which result draws on which companion, and which classical mathematics is imported to reach it, so that a reader can go to the owning paper and check that it says what is claimed here. Where a clause is load-bearing later, that is the check worth making.

\section{Why axioms at all --- and which ones this paper needs}
\label{sec:clauses}

Most physics papers do not begin with axioms, and most should
not. But a paper that claims to \emph{derive} things usually
taken as given --- antiparticles, the discrete symmetries, the
value of a CP phase --- owes the reader an exact statement of
what it is deriving them \emph{from}, or the claim is theater.
The framework this paper works in organizes its assumptions as a
short constitution \citep{ArisakaA1Constitution}: four axioms,
seven postulates, and one set of integers, written once and then
never renegotiated. The spirit is worth internalizing before the
letter. A constitution is not a summary of laws; it is the small
set of commitments from which laws are \emph{argued}, and its
value lies precisely in how little it says --- every clause is a
hostage, offered in advance, to every future measurement. This
paper uses three of the four axioms and one postulate. Each is
given first in one line, then in full, because the reader should
know at every later point which commitment is doing the work.

\emph{In one line each.} \textbf{AX-1} (locality and causality):
physics is registered on a Lorentzian causal manifold through
retarded, microcausal protocols --- it supplies a global time
orientation, and, applied to interacting charged matter, it
\emph{forces} the two structures the next axiom must geometrize.
\textbf{AX-2} (geometry--gauge equivalence): every structure the
dynamics cannot avoid must be realized as geometry --- one
composite connection, with all physical content in curvature and
holonomy invariants. \textbf{AX-3} (the operational dictionary):
a bulk theory is not physically defined until one says how
boundary observers register outcomes --- instruments, channels,
and, decisively here, an operational positivity clause.
\textbf{POS-3} (anomaly freedom): chiral sectors must cancel all
anomalies, by one inflow mechanism and no other.

\emph{AX-1 in full.} Locality--causality is not one requirement
but a bundle: interactions built from products of fields at
coincident points; retarded support --- no response before its
cause; and microcausality --- spacelike-separated observables
commute. Two consequences are then \emph{forced}, not assumed,
and both are the reason this paper exists. First: an interacting
local theory is singular at coincident points and therefore
cannot be specified without a resolution scale and the flow
connecting resolutions --- the renormalization group, made
compulsory. Second: local, causal interaction of \emph{charged}
matter cannot proceed by action at a distance; a phase frame
must be carried at every point, with a connection compensating
its arbitrariness --- the gauge principle, read as a consequence
rather than a postulate. A scale and a phase: \emph{AX-1 hands
exactly two structures to the next axiom.} The same axiom also
hands this paper two of its three orientations --- the clock's
(causal order gives time a direction, the orientation T
reverses) and space's (through the Lorentzian arena it demands,
the orientation P reverses) --- and, at the operational level,
the forward-composing semigroup ($t \ge 0$: records accumulate,
a record once made is not un-makeable) that Part~VIII will
recognize as Sakharov's third condition before cosmology is ever
mentioned.

\emph{AX-2 in full.} The geometry--gauge equivalence principle
has two clauses. \emph{Locked equivalence}: along any timelike
curve there is a combined choice of coordinates, frame, and
gauge in which the composite connection is flattened to first
order --- Einstein's elevator, upgraded so that geometry and
gauge are one locked structure rather than two. \emph{Invariant
physics}: all physical content resides in the curvature and
holonomy invariants of that one connection; ``forces'' are
artifacts of unlocked descriptions. Both clauses are
\emph{complex-free} --- a real connection, real curvature, real
holonomy angles --- and the operative reading is a single rule:
whatever the dynamics cannot avoid must be realized as geometry.
Here is the axiom that answers a question every student has
asked and been told not to: \emph{where does the gauge phase
live?} It lives in a dimension. Given AX-1's two burdens, AX-2
makes the scale a real dimension $\Ir$ and the phase a real
circle $\Stheta$, whose windings are charge and whose reversal
is C. In the framework's own words: \emph{AX-2 builds the fiber.}
One word of that sentence carries the program's most original
content, and it is the word that is \emph{absent}: neither new
dimension is \emph{spatial}. The lineage makes the point
sharpest. Geometric dimensions that are not dimensions of space
begin with Minkowski, whose time is exactly that
\citep{Minkowski1908}; Weyl then had both of the great gauge
insights --- local \emph{scale} as geometry in 1918
\citep{Weyl1918}, local \emph{phase} as geometry in 1929, the
birth of gauge theory itself \citep{Weyl1929}. Kaluza and Klein
geometrized the phase first \citep{Kaluza1921, Klein1926} --- and
made it a circle \emph{of space}, the move that burdened every
extra-dimension program since with a radius to stabilize, a
charged tower to hide, and ultimately a landscape to wander.
AX-2 keeps Weyl's two insights and refuses the spatialization:
the scale and the phase enter as what they \emph{are} --- a
resolution and an angle, dimensionless by construction --- and
Theorem~0 below \emph{derives}, rather than assumes, that they
are non-spatial \citep{ArisakaA2GG6}.

\emph{AX-3 in full.} The operational axiom declares that a bulk
theory is not physically defined without a fixed dictionary from
bulk configurations to registered boundary physics,
\begin{equation}
\mathcal{D}(B) \;=\;
\bigl(\, \cH,\ \rho,\ \{\mathcal{I}_\alpha\},\
\{\Phi_{t,s}\},\ \ldots \bigr),
\label{eq:ax3-tuple}
\end{equation}
with $\rho$ a boundary state, $\{\mathcal{I}_\alpha\}$
measurement \emph{instruments} in Kraus form with
positive-operator-valued semantics, and $\Phi_{t,s}$ completely
positive, trace-preserving evolution maps --- the whole subject
to three compatibility clauses: retardedness (from AX-1),
gauge--geometry covariance (from AX-2), and \emph{operational
positivity}, clause (iii), the one this paper leans on when the
clock's sign is selected. Measurement is thereby part of the
theory's syntax rather than a metaphysical addendum, and the
instruments' membership in the dictionary is why observer-adapted
structures --- which modes a detector counts, which sector it
calls ``the particle'' --- are boundary data throughout this
paper. \emph{AX-3 collapses the fiber that AX-2 built.} One
calibration, stated plainly and not re-argued here: in the wider
framework this clause is also where quantum measurement finds a
home --- the collapse, the Born weights, the arrow of records
read as properties of the projection rather than of the bulk
(\citealp{ArisakaA5GDOS}; \citealp{ArisakaA8QM}). That case is
made in those treatments; this paper borrows only the positivity
clause and the custody rule that boundary \emph{labels} are the
dictionary's property. Readers who withhold judgment on the
measurement claim lose nothing downstream.

\emph{AX-4, for completeness.} Maximum symmetry, minimum
freedom: no continuous structure is admissible unless a prior
clause compels it. Its role here is quiet but real --- it closes
Theorem~0's count from above (no seventh dimension) and forbids
the unforced extras that would otherwise let a phase float free
of the one circle, which is exactly what makes the flavor
angles of Part~VII lattice-locked rather than free.

\emph{POS-3, the one postulate.} Chiral gauge sectors must
cancel all anomalies --- gauge, mixed, and global --- and may do
so only through a single Green--Schwarz-type inflow implemented
as a modified Bianchi identity \citep{ArisakaA3BI6}. Warmly: in
six dimensions every chiral species signs its ledger entry
(Eq.~\eqref{eq:anomaly-flip}), the books must close exactly, and
no theory gets to exist whose books do not. It polices every
chirality assignment in Part~VI --- and, as Part~IV will show, it
is one of the two independent reasons the bulk cannot be written
without conjugate matter.

\paragraph{Physics intuition.} The axioms are a relay, and the
handoffs are the whole story. AX-1 says: to be local, causal, and
charged, you must carry a zoom knob and a dial. AX-2 says: if you
must carry them, then they are not bookkeeping --- they are
places in the geometry, real and non-spatial. AX-3 says: and here
is what a four-dimensional being can actually read of them ---
states, channels, instruments, with positivity. The first hands
two burdens to the second; the second makes them real; the third
makes them readable. Every complex number in this paper is born
in the gap between the second and the third: \emph{real geometry,
read by a bounded reader}.

\section{Theorem 0: the forced arena}
\label{sec:theorem0}

The two axioms together do not merely permit a six-dimensional
bulk; they compel one, and they compel its shape. The arena
theorem of the framework's geometry paper \citep{ArisakaA2GG6}
establishes
\begin{equation}
\cM_6 \;=\; \cM_4 \times \Ir \times \Stheta,
\qquad
ds_6^2 \;=\; e^{-2r}\,\eta_{\mu\nu}\,dx^\mu dx^\nu
\;+\; dr^2 \;+\; d\theta^2 ,
\label{eq:theorem0-metric}
\end{equation}
with $\Ir$ a non-compact \emph{scale} direction --- the
geometrized renormalization group --- and $\Stheta$ a compact
\emph{phase} circle --- the geometrized gauge fiber. Three
properties of \eqref{eq:theorem0-metric} carry everything this
paper does.

\emph{Non-spatiality.} The two extra directions are classified
by their \emph{generators}, and this is the decisive move: a
spatial or temporal direction is generated by a translation ---
that is what makes it a place one could in principle go --- while
$\Ir$ is generated by a dilatation and $\Stheta$ by a compact
charge rotation. No signal propagates along them; no boundary
protocol resolves a position on them; the bulk isometry algebra
is
\begin{equation}
\mathfrak{iso}(1,3) \oplus \bR_D \oplus \mathfrak{u}(1)_Q ,
\qquad \text{not} \quad \mathfrak{iso}(1,5).
\label{eq:isometry}
\end{equation}
Both coordinates are moreover dimensionless by construction --- a
logarithm of an energy ratio and an angle, pure numbers carrying
no unit of length --- which is the sharpest possible statement
that they are not spatial.

\emph{Global staticity.} In \eqref{eq:theorem0-metric} the
four-dimensional metric is Minkowski and the warp depends only on
$r$; hence $\partial_t$ is a global, hypersurface-orthogonal,
timelike Killing field --- the \emph{constitutional clock}. This
is not a decoration: it is precisely the hypothesis under which
the positive-energy complex structure of
\S\ref{ssec:source-clock} is \emph{unique}, so the arena's
staticity is what makes the clock's quarter-turn canonical rather
than conventional.

\emph{Reality.} Every datum in \eqref{eq:theorem0-metric} ---
metric, interval, angle, connection --- is real. No complex
number appears anywhere in the arena, and none will be added.

The proof is a six-step forcing chain, worth possessing in
outline because each step is load-bearing later.
\emph{Step 1}: AX-1 is defined relative to a light cone, so the
registered arena is Lorentzian. \emph{Step 2}: locality plus
interaction forces the renormalization scale (AX-1's first
burden). \emph{Step 3}: AX-2 promotes it to a coordinate --- the
scale direction $\Ir$, along which the connection component is
the running coupling and the $\beta$-function is literally a
covariant derivative. \emph{Step 4}: locality plus charge forces
the periodic gauge phase (AX-1's second burden). \emph{Step 5}:
AX-2 promotes it to the compact direction $\Stheta$, whose
Wilson-line holonomy is the gauge-invariant datum.
\emph{Step 6}: the count closes two-sidedly --- from below,
because deleting either direction breaks AX-1 itself (no scale:
the theory is ill-defined at short distance; no phase: charged
interaction becomes action at a distance), and from above,
because AX-4 admits no third structure. Two properties proved en
route are used constantly here: the added directions carry
\emph{no light cone}, and motion along $\theta$ is a gauge
transformation --- a change of description, not a physical
displacement.

\emph{Register, at theorem visibility.} The arena theorem is, in
its home paper's own words (Remark~0.5 of \citealp{ArisakaA2GG6}),
a constitutional forcing argument rather than a theorem of
constructive quantum field theory, and every ``forced''
downstream of it in this paper inherits that register.

\paragraph{Physics intuition.} Do not picture six axes. Picture
the stage --- spacetime, where things happen --- with a zoom knob
attached to it and a dial at every point of it. Theorem~0 says:
charged, causal physics cannot be written without the knob and
the dial, and the constitution obliges both to be geometry. The
knob is not a corridor; the dial is not a room. That is the whole
of ``non-spatial,'' and it is why the dial's angle can be
physical only through \emph{differences} --- holonomies --- which
is where every complex exponential in physics will come from.

\emph{What this dissolves.} Feel the force of the classification
before asking what it builds, because it removes at a stroke the
questions that have shadowed every spatial extra-dimension
program since 1921. ``Why don't we see the extra dimensions?''
--- the question does not arise: there is nothing to see because
there is no place to look; the phase dimension is not hiding, it
is \emph{present in every interference experiment ever
performed}, as the phase of every charged field. ``What
stabilizes the radius?'' --- there is no radius; an angle has
period $2\pi$ by what an angle is, and a resolution has no size.
``Where is the tower of charged Kaluza--Klein copies?'' --- there
is no tower, because the circle's characters are not momentum
modes of a spatial loop but the \emph{charges themselves}: the
integer labeling a winding sector is not the mass level of a
heavy replica, it is the electric charge of the field you already
know. The problems that killed the spatial fifth dimension are
not solved here; they are unposable --- and that is the sense in
which the two extra dimensions being \emph{derived} (Theorem~0,
from AX-1 and AX-2 alone), and derived to be non-spatial,
departs from every predecessor: the two things they geometrize,
a resolution and an angle, never were places.

\section{The integer ledger}
\label{sec:ledger}

One more inheritance is load-bearing and deserves its own
section: on this arena the anomaly-closure architecture fixes a
five-integer ledger --- forced, not fitted
\citep{ArisakaA3BI6}:
\begin{equation}
(M,\, N,\, L;\, J,\, K) \;=\; (1,\, 3,\, 5;\, 7,\, 35),
\qquad
K \;=\; L \cdot J \;=\; M^2 + N^2 + L^2 \;=\; 35 .
\label{eq:ledger}
\end{equation}
Each integer carries a structural meaning, and each deserves one
warm line, because these five numbers are this paper's entire
``parameter set.'' $M = 1$ is the loneliest: the tensor channel
is primitive and unique --- one indivisible winding unit, which
is why charge comes in one quantum and why the circle's
characters are integers at all. $N = 3$ is the crowd: the
baryonic trace ratio isolates a triplicity inside the unified
block, whose later echoes are color and the three families ---
and three is the smallest number of families whose mixing can
hide a phase. $L = 5$ is the artist: the unified simple block is
the five-dimensional one, and its regular polygon --- the
pentagon --- will turn out to draw both the Cabibbo angle and the
CP phase. $J = 2N{+}1 = 7$ is the counter: the centered odd
closure built from the triplicity, whose oddness the leptonic
phase inherits. And $K = LJ = 35$ is the deep one: the throat
depth, the exponent that turns the Planck scale into the TeV lock
and the ledger into the baryon excess.

Four of the five do direct work here, each at a stated point.
$L = 5$ is the \emph{angle alphabet}: with one circle and one
five-block --- and, decisively, no second flavor torus on which
an independent angle could float --- every first-order physical
phase is assembled from steps of $\pi/5$, which is why the
CP-violating loop holonomies of Part~VII are lattice-locked and
why $2\pi/5$ appears where a free parameter would otherwise sit.
$N = 3$ is why exactly one quark CP phase survives
(Eq.~\eqref{eq:phase-count}). $J = 7$ sets the leptonic phase
$\dPMNS = \pi + 2\arcsin(1/J)$. And $K = 35$ sets both the lock
scale $\Mlock = \tfrac12 M_{\mathrm{Pl}}e^{-K} \simeq 3.85$~TeV
and the throughput exponent $e^{-K/2}$ of the matter chain
(Part~VIII). The ledger is the reason this paper may say
``quantized'' and ``locked'' where other frameworks must say
``measured'' --- and it was fixed, published with its own
falsifiers, before any of the computations below were assembled.

\paragraph{Physics intuition.} A dial that can be re-zeroed
freely carries no numbers of its own. What the anomaly ledger
adds is a tick-mark structure: the bookkeeping closes only for
one integer packet, and those integers become the dial's
engravings --- how many windings make a charge ($M$), how many
steps make the flavor pentagon ($L$). The circle supplies the
geometry of the phase; the ledger supplies its arithmetic.

\section{The fiber dictionary: charged fields as real harmonic
pairs}
\label{sec:fibre-dictionary}

With the arena fixed, the translation between bulk geometry and
the familiar language of charged fields can be written down once
and used everywhere. Single-valuedness on the compact circle
gives every bulk matter field a real harmonic decomposition; the
charge-$n$ sector is the pair of real weight-$n$ harmonics
\begin{equation}
\Psi_n(x;\theta) \;=\; a_n(x)\cos n\theta \;+\; b_n(x)\sin n\theta,
\qquad
R_\alpha : (a_n, b_n) \;\mapsto\; R(n\alpha)\,(a_n, b_n),
\label{eq:fibre-dictionary}
\end{equation}
and the textbook complex charged field is the boundary's
packaging $\psi_n = a_n + i b_n$ of that real pair
(\S\ref{sec:real-presentation}). Under this dictionary every
piece of gauge folklore becomes a statement about a real circle:
charge is the winding number $n$; charge quantization is
compactness ($n \in \bZ$); a gauge transformation is a fiber
shift $\theta \mapsto \theta + \alpha(x)$; the covariant
derivative is the horizontal lift, acting on the $n$-harmonic as
$\partial_\mu - i n A_\mu$; the Aharonov--Bohm phase is a real
holonomy angle of a real connection \citep{AharonovBohm1959};
and complex conjugation is orientation reversal
$\theta \mapsto -\theta$ --- the operation this paper will call
C. One consequence is worth its own sentence, because it settles
a question students ask and textbooks answer only by convention:
the photon is the connection \emph{itself} --- not a section of
any associated bundle --- and therefore correctly carries no
dial. It has no phase to reverse, no charge to conjugate, and is
its own antiparticle for a reason one can see rather than
memorize.

\section{The boundary law: how a four-dimensional reader reads a
six-dimensional book}
\label{sec:boundary-law}

The dictionary AX-3 demands is not left abstract in the wider
framework; a dedicated treatment constructs it
\citep{ArisakaA5GDOS}, and four of its structural results frame
everything this paper does with observers, records, and
asymmetries.

\emph{First, the projection is forced.} A boundary protocol reads
records sorted by a light cone; Theorem~0 gave the fiber
directions no light cone; therefore no operation resolves where a
system sits along $\Ir$ or around $\Stheta$, and \emph{every} act
of registration marginalizes the fiber. Openness is not an
accident of imperfect apparatus --- it is constitutional; unitary
closure is the unreachable idealization in which the fiber is
never coarse-grained.

\emph{Second, the projection is always on.} It happens at every
spacetime point, observer or no observer. ``Measurement'' is its
macroscopic, record-forming special case; decoherence is its
microscopic one. Nothing waits for a physicist.

\emph{Third, the boundary law has a derived form.} Exact
projection gives a memory-kernel equation; complete positivity is
inherited from unitary conjugation plus partial trace; and in the
Markov regime the generator takes the standard
Gorini--Kossakowski--Sudarshan--Lindblad form --- from which the
textbook quantum postulates follow as theorems rather than
axioms.

\emph{Fourth --- and this is the one to carry into every later
Part --- the two fibers carry the two halves of quantum
mechanics.} The compact phase circle carries the coherent,
unitary, interference-generating half: transport along it is pure
phase. The non-compact scale direction carries the open,
irreversible half: its forward renormalization-group semigroup is
the dissipative generator, and decoherence is the loss of
$\theta$-coherence induced by coarse-graining along $r$. Two
dimensions, two faces of the quantum world --- and this paper
lives on both. The mirrors, the twins, and the pentagon are the
first fiber's business; the arrow of time, the readout of which
twin is ``matter,'' and the transport that fixes the baryon
number are the second's. When \S\ref{sec:custody} says that the
bulk owns the symmetries while the boundary owns their
observability, this is the structure it is naming.

\paragraph{Physics intuition.} The bulk is a six-dimensional
book; the boundary observer reads a four-dimensional shadow of
it, always, whether or not anyone calls the reading an
experiment. What the shadow cannot show --- the dial angle, the
zoom setting --- comes out as probability and irreversibility;
what it can show --- differences, rates, records --- comes out
real. Quantum mechanics, on this reading, is the grammar of
reading a book through its shadow, and the one ``imaginary'' rule
in that grammar is the shadow's compression of the dial and the
clock into a single symbol.

\section{The actualization layer: why records are real, and why
the textures are locked}
\label{sec:actualization}

One more story bears directly on this paper, because it is where
outcomes --- the \emph{real} end of the chain --- are fixed. The
runtime treatment \citep{ArisakaA6Gi4} builds the descent in four
pillars: the total geometric--gauge object (the one connection as
a dynamical whole); the covariant conservation identity (the
boundary Ward currents --- electric, baryonic, informational,
metric --- that any lawful runtime must preserve); a geometric
free-energy functional (a strict Morse landscape ranking
admissible boundary configurations); and the pipeline itself, in
which descent proceeds as a finite, ordered cascade of Morse
steps, each transition costing a quantized action, halting at the
ground attractor. Two consequences matter here, and both are
load-bearing.

\emph{Every registered output of physics is a real quantity} ---
an angle class, a count, a rate. The registered datum of a
measurement is a contextual joint holonomy: an element of the
circle, a real angle class. So the complex layer of the formalism
begins \emph{after} preparation and ends \emph{before}
registration. It is genuinely middleware.

\emph{And the coupling textures are not free functions.} The
Yukawa textures whose loop holonomies measure CP chirality
(Part~VII) are outputs of this descent --- stabilized attractor
data, not inputs a model-builder may dial. This is the precise
reason this paper may call the flavor angles
\emph{lattice-locked} where every other framework must call them
measured: they are not parameters at all but the settled
positions of a cascade that had nowhere else to stop.

\paragraph{Physics intuition.} If the boundary law says how the
shadow is read, the actualization layer says how the ink dries: a
measurement completes when the joint configuration of system and
apparatus rolls into a basin it cannot leave, and what is written
is an angle --- a dial reading, as real as any pointer. Between
the real bulk that supplies the story and the real record that
ends it, the complex amplitude lives only in the telling.

\section{The arc, assembled: real bulk, complex middleware, real
records}
\label{sec:arc}

Part~III can now be summarized as a single pipeline, and the
summary is the frame the rest of the paper fills in. The
\emph{inputs} are real: an arena (Theorem~0), one connection
(AX-2), an integer ledger \eqref{eq:ledger}, and matter as real
harmonic pairs on the circle \eqref{eq:fibre-dictionary}. The
\emph{outputs} are real: probabilities and rates, records,
holonomy classes --- and, in this paper, laboratory asymmetries
and one number counted in the sky. In between sits the layer
every physicist was trained in: amplitudes, phases, state spaces
--- the compressed notation a fiber-blind reader is forced into.
Nothing complex enters the pipeline's ends. That is the sense in
which the discrete symmetries can have an \emph{anatomy} at all:
C, P, and T are operations on the real ends, and the boundary's
$i$ --- which they appear to act on --- is only the middle's
shorthand for a real quarter-turn.

\section{The two shared structures, named}
\label{sec:two-structures}

Everything this paper derives comes from exactly two real
structures, both supplied by the arena and both attached to
\emph{every} system in the universe:
\begin{align}
\textbf{G1:}&\quad \text{the phase circle } \Stheta
\quad (\text{generator } \partial_\theta),
\label{eq:G1}\\
\textbf{G2:}&\quad \text{the constitutional clock } \partial_t
\quad \text{with the positivity clause of AX-3}.
\label{eq:G2}
\end{align}
G1 supplies the gauge phase and everything quantized about it; G2
supplies the state-space rotation and everything oriented about
it; their consistency on charged sectors will be the
particle--antiparticle pairing (Part~IV); their three
orientations --- G1's, G2's, and space's --- will be C, P, and T
(Part~V); the texture written on G1 during the descent will be
parity violation and the pentagon (Parts~VI and~VII); and the
door that texture biases will be the matter of the universe
(Part~VIII). Table~\ref{tab:inheritance} records what each result
uses.

\begin{table}[htbp]
\centering
\caption[What each result uses from the arena.]{What each result of this paper uses from the arena, and
which classical mathematics it imports. Nothing in the left
column is postulated by this paper; nothing in the right column
is claimed by it. One row per numbered result. The middle column names the axiom, theorem or ledger entry the result stands on; the right column names the classical mathematics imported to reach it, with an em dash where none is needed. The table is here so that a reader can see at a glance which parts of the argument are new to this paper and which are inherited.}
\label{tab:inheritance}
\small
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{@{}p{4.6cm} p{3.6cm} p{6.4cm}@{}}
\toprule
\textbf{Result (this paper)} & \textbf{Arena source} &
\textbf{Imported classical mathematics} \\
\midrule
T-P5-A (the circle) & AX-2; Theorem 0 (G1) & harmonic analysis on
$S^1$ \\
T-P5-B (the clock) & AX-1; AX-3(iii) (G2) &
\citet{AshtekarMagnon1975}; \citet{Kay1978} \\
T-P5-C (antiparticles) & G1 $+$ G2 jointly & --- \\
one $J$, one $i$ & G1, G2 global & \citet{RenouEtAl2021} (as
constraint) \\
T-P5-1 (the C/P/T table) & orientations of G1, G2, space & ---
\\
T-P5-2 (CPT) & total reversal of the three & (contrast:
\citealp{Jost1957}) \\
T-P5-3 (holonomy calculus) & G1; ledger $L = 5$ &
\citet{Jarlskog1985} \\
T-P5-4 (the door) & G1 weight arithmetic & seesaw
(\citealp{Minkowski1977}) \\
Parity: the projection & POS-3; chirality of the bulk & orbifold
GUT (\citealp{Kawamura2000}) \\
T-P5-5 ($\etaB$) & ledger $K = 35$; the door & transport
(\citealp{KolbTurner}) \\
Lattice-locked textures & the descent
(\citealp{ArisakaA6Gi4}) & Morse theory \\
Real records; readouts & the boundary law
(\citealp{ArisakaA5GDOS}) & GKSL form \\
The anatomy (\S\ref{sec:anatomy-of-a-theorem}) & G1 compact, G2
directed & the two-fiber formulation of
(\citealp{ArisakaW2Holonomy}) \\
$\Delta B \in 3\mathbb{Z}$ (\S\ref{sec:deltab-three}) & T-SM-60.1,
hypotheses H1--H3 & (\citealp{ArisakaA9SM}; \citealp{ArisakaP6Proton}) \\
\bottomrule
\end{tabular}
\end{table}

\paragraph{Physics intuition.} The whole construction can be said
in one breath: nature carries one dial and one clock, the same
pair everywhere. The dial cannot be read, so only its differences
--- holonomies --- are physics; the clock cannot be run backward,
so only one frequency orientation is physics. Complex numbers are
what a four-dimensional observer writes down when forced to log
differences of an invisible dial and orientations of an
irreversible clock in a single symbol. And the subject of this
paper --- the twins, the mirrors, the pentagon, the matter ---
is what happens when you ask that dial and that clock to agree.

\section{The two sources, at depth}
\label{sec:inherited-theorems}

Everything this paper proves grows from four facts about the
arena's two sources of complex structure. Each is introduced
here from scratch, at the blackboard, assuming no prior
acquaintance --- they are the paper's foundation stones, and
the reader deserves to stand on each one before building
begins. Each also carries a citation to the companion treatment
in which it was established at full length
\citep{ArisakaP4Real}; none is re-proved in this section ---
Part~IV re-proves the ones this paper's claims ride on.

\subsection{The circle: a dimension of phase, not of place}
\label{ssec:source-circle}

The first source is not a place, and that sentence is the most
important in this Part. Every prior higher-dimensional program
--- Kaluza--Klein, Randall--Sundrum, string theory --- added
dimensions \emph{of space} and then had to hide them: curl them
small, warp them away, explain why no experiment walks into
them, stabilize their radii, and live with the moduli, the
charged towers, and eventually the landscape that spatial extra
dimensions bring \citep{ArisakaA2GG6}. The \GG{} bulk's two
extra dimensions are a different \emph{kind of thing}, and the
arena paper gives the classification its precise form: a
dimension is known by its \emph{generator}. A spatial or
temporal direction is generated by a translation --- that is
what makes it a place one could in principle go. The scale
dimension $I_r$ is generated by a \emph{dilatation}: a value of
$r$ is a resolution, a magnification at which the same world is
read, and one does not travel along it any more than one
travels through the zoom of a microscope. The phase dimension
\Stheta{} is generated by a \emph{compact charge rotation}: a
value of $\theta$ is a gauge angle, and local motion along it
is a gauge transformation --- a change of description, not a
displacement; no signal propagates there, no light cone
extends there, and the bulk's isometry algebra is
$\mathfrak{iso}(1,3) \oplus \mathbb{R}_D \oplus
\mathfrak{u}(1)_Q$, emphatically not $\mathfrak{iso}(1,5)$.
Both coordinates are dimensionless by construction --- a
logarithm of an energy ratio and an angle, pure numbers
carrying no unit of length --- which is, in the arena paper's
words, the sharpest possible statement that they are not
spatial.

Feel what this dissolves before asking what it builds. ``Why
don't we see the extra dimensions?'' --- the question that has
shadowed every spatial program since 1921 --- simply does not
arise: there is nothing to see because there is no place to
look; the phase dimension is not hiding, it is \emph{present in
every interference experiment ever performed}, as the phase of
every charged field. ``What stabilizes the radius?'' --- there
is no radius; an angle has period $2\pi$ by what an angle is,
and a resolution has no size. ``Where is the tower of charged
Kaluza--Klein copies?'' --- there is no tower, because the
circle's characters are not momentum modes of a spatial loop
but the \emph{charges themselves}: the integer that labels a
winding sector is not the mass level of a heavy replica, it is
the electric charge of the field you already know. The
problems that killed the spatial fifth dimension are not solved
here; they are unposable --- and this is the sense, stated
once and carried silently hereafter, in which the author's
construction departs from every predecessor: the two extra
dimensions are \emph{derived} (Theorem~0, from AX-1 and AX-2
alone), and derived to be non-spatial, because the two things
they geometrize --- a resolution and an angle --- never were
places.

With the category corrected, the physics of the fiber is
exactly as Part~II drew it, and three old mysteries remain
dissolved. Why is charge quantized? --- because the phase
dimension is compact: windings of an angle are integers. Why do
charged wavefunctions carry phases? --- because the field's
``complex phase'' \emph{is} its fiber angle: the quantum phase
is the holonomy of a real, non-spatial circle, which is why the
Aharonov--Bohm effect measures it while no local field does
\citep{AharonovBohm1959}. Why can conjugation be geometric? ---
because reversing an angle's orientation is as real an
operation as reversing a spatial axis, without the angle being
an axis: C is $\theta \mapsto -\theta$, the reversal of a
phase, not a mirror of a place. What the parent program adds
as theorem (T-P4-1) is uniqueness: on each charged sector the
circle's quarter-turn $\Jq = \partial_\theta/Q$ is the unique
circle-compatible complex structure --- up to one sign the
circle itself cannot fix --- together with the two honest
limitations that make a second source necessary: the circle
complexifies only what it moves (neutral fields get nothing),
and it does not know its own orientation. Hold both
limitations; they are load-bearing.

\subsection{The clock: the quarter-period advance}
\label{ssec:source-clock}

The second source is not a place but a direction: causal time,
with a bound below on energy. Its gift (T-P4-2) is the operator
this paper has already met in engineer's clothing --- the
Hilbert transform, the ``advance every oscillation by a quarter
period'' operation of the analytic-signal construction
\citep{Gabor1946}. On the real solution space of any free bulk
field, that operator is a complex structure; and the theorem
that elevates it from convenience to physics is
\emph{uniqueness}: among all complex structures commuting with
the dynamics, exactly one gives every mode positive energy,
and on the constitutional arena it is this one
\citep{AshtekarMagnon1975, Kay1978}. The constitutional
pedigree is worth reciting like a verse, because each clause
comes from a different axiom: \emph{AX-1 supplies the
direction} (causal order says which way time points),
\emph{the positivity clause of AX-3 the selection} (which
orientation of the splitting is ``positive''), \emph{Theorem~0
the uniqueness} (the arena is globally static, so the choice
is canonical). Where the circle knew charge but not direction,
the clock knows direction but not charge: it acts identically
on every field, neutral ones included --- which is exactly
what the circle could not do. Two half-blind sources; each
supplies what the other lacks.

\subsection{The handshake, previewed}
\label{ssec:source-handshake}

On every charged sector, both sources are present and each
supplies a quarter-turn that claims to be \emph{the} rotation
of that sector. They agree --- Part~IV
proves, mode by mode --- under exactly one matching of charge
to frequency, and that matching is the particle--antiparticle
pairing. The preview is stated here only so the reader carries
the right expectation into Part~IV: the antiparticle will not
be introduced, postulated, or discovered there. It will be
\emph{forced}, by the requirement that a bulk with two real
sources of rotation share one $J$ --- and hence that the
boundary description have one $i$ --- with the sign the circle
could not fix fixed in the same stroke.

\subsection{One $J$ in the bulk, one $i$ on the boundary}
\label{ssec:source-shared}

Before stating the fourth fact, a distinction must be drawn so
cleanly that it can never again blur, because everything in this
paper turns on it. \emph{The bulk contains no imaginary unit.}
What the bulk contains is $J$ --- a \emph{real} operator, the
quarter-turn, exactly as real as the rotation matrix
$R(90^{\circ})$ it is. That $J^2 = -\mathbf{1}$ makes $J$ a
complex \emph{structure}; it does not make $J$ a complex
\emph{number}, any more than a screw is a helix because it
traces one. The symbol $i$ belongs to the other side of the
projection: when the four-dimensional boundary description
mimics the real bulk --- and mimic it must, for the boundary
theory is quantum field theory, the downstream description of
bulk physics --- it packages each real pair $(a, b)$ into a
single amplitude $a + ib$, and that packaging is lawful only
along $J$. The boundary's $i$ is its shorthand \emph{name} for
the bulk's $J$: a notational contraction of ``apply the
quarter-turn.'' In the custody language this paper uses
throughout: the geometry owns $J$; the dictionary owns the
symbol.

With the levels separated, the fourth fact can be stated
exactly (T-P4-4). \emph{Upstairs, in the bulk:} the two sources
supply one shared $J$ --- composites and the Fock construction
extend it canonically, with no second structure ever appearing
--- a statement about real geometry, in which $i$ does not
occur. \emph{Downstairs, on the boundary:} because there is
exactly one $J$ to name, there is exactly one $i$ in quantum
mechanics --- one global phase convention that entangled,
composite, many-particle states all share. One $J$ upstairs is
why there is one $i$ downstairs. And the downstairs statement
is tested: the network Bell experiments that exclude
real-Hilbert-space quantum mechanics \citep{RenouEtAl2021}
constrain precisely the many-unshared-structures alternative,
and are satisfied by construction in a boundary description fed
by a bulk with one $J$. Note carefully what the experiments do
and do not test: they test the boundary formalism (it needs a
single shared imaginary unit) and they cannot see the bulk; the
bulk is what \emph{explains} the formalism's need. What looked
like a threat to any ``real quantum mechanics'' is, for this
architecture, the sharpest available support --- because the
architecture never claimed the boundary description could do
without $i$; it claims the \emph{world} does, and supplies the
one real structure the notation was naming all along.

The logical dependence of what follows is asymmetric, and
stating it plainly is part of the register: Part~V's table and
CPT theorem need all four inputs; Part~VII's holonomy calculus
needs only the fiber; Part~VII's door theorem needs only the
circle action on weight sectors. A reader who doubts the clock
derivation still faces the pentagon; a reader who doubts
everything still faces the falsifier table.

\subsection{Why the real bulk matters: the observer built to read
it}
\label{ssec:real-bulk-mind}

It would be fair for a reader to ask, at this point, why so much
weight is placed on a bookkeeping preference. The boundary
description works perfectly well with its $i$; the antiparticle
pairing could be stated in either language; what, beyond
tidiness, is bought by insisting that underneath the notation the
world is real? The honest answer reaches further than the rest
of this paper, into territory the program develops in dedicated
companion treatments and grades as its most tentative --- and it
is offered here, at that same tentative grade, because it is the
reason the real-bulk view is not a stylistic choice but a claim
about what kind of universe this is. The short form: \emph{the
world writes reality in real phase, and the observer is built to
read it in real phase.} If that parallel holds, the reality of
the bulk is not a convenience of the physicist; it is the common
substrate of the thing observed and the thing observing.

Three ideas are needed, and each is introduced from scratch.

\emph{First, a fourth charge --- of information.} Alongside the
familiar conserved quantities (electric charge, and the gauged
baryon number this paper's proton stability rests on), the
program admits one further Abelian symmetry, written
$\Uone{Qi}$, whose conserved current is not a flow of energy or
matter but a flow of \emph{information}
\citep{ArisakaA7Qi4}. The reader need not accept the full
apparatus to follow the point; what matters is its carrier. A
$\Uone{Qi}$ charge is stored and transported the way any
$\mathrm{U}(1)$ quantity is stored on the fiber of Part~II ---
as a \emph{phase}, a real angle --- and its only physical,
convention-independent content is the holonomy of that phase
around a closed loop, exactly the Aharonov--Bohm-type invariant
of \S\ref{ssec:source-circle}. Information, in this reading, is
carried the same way charge is: as a real angle on a real
circle, whose $i$ (should one write it) is again only the name of
a quarter-turn.

\emph{Second, the freely running substrate --- ``DGI.''} Before
any symmetry breaks, the bulk carries this information current on
an unbroken gauge-equivalence background --- the program's term
is \emph{Dynamical Gauge Invariance} --- on which the current
propagates freely: not yet a particle, not yet a measured value,
not yet localized \citep{ArisakaA6Gi4}. It is the rawest layer of
the descent, the informational analogue of the unprojected,
still-real bulk field this whole Part has been describing. One
holds it in mind as the state of ``information before it has been
read'': present, lawful, and --- crucially for the parallel about
to be drawn --- entirely real.

\emph{Third, the reader in the loop --- the Inner Hologram.} Now
the striking part, and the reason the author asks that this view
be taken seriously rather than as decoration. Consider what a
brain must do. Its senses deliver a lossy, self-centered,
low-dimensional shadow of the world --- a two-dimensional retinal
image does not determine the three-dimensional scene that cast
it, and a stream of such shadows, tagged ``I turned left, I
stepped forward,'' accumulates error the way dead reckoning
always has. To survive, an animal cannot store the shadows; it
must \emph{reconstruct}, from them, a stable model of the world
as the world is --- a \emph{world-centered}, or \emph{allocentric},
frame in which objects hold their places while the self moves
through them \citep{OKeefeNadel1978}. The companion treatments
call this reconstructed model the \emph{Inner Hologram}: the
brain's running, self-updating, allocentric picture of what lies
beyond the sensory boundary (\citealp{ArisakaA7Qi4}; \citealp{ArisakaA11Origin}).
It is not a metaphor drawn loosely; the neural hardware is
identified --- the place cells, grid cells, and head-direction
cells of the mammalian hippocampal system are exactly the
machinery a world-centered frame requires
(\citealp{OKeefeNadel1978}; \citealp{HaftingEtAl2005}) --- and the
reconstruction runs, behaviorally, as a transform whose
magnitude is paid for in \emph{time}: recognizing a rotated or
displaced object takes longer in proportion to the transform
needed to align it with memory, in steps clocked by the brain's
rhythms.

And here is the payoff, the sentence this subsection exists to
earn. The medium in which the Inner Hologram is written is
neural \emph{phase}. The pose of a reconstructed object is
carried by the phase of traveling cortical waves --- the
theta rhythm ($\sim$5~Hz) that paces the integration of the
world-model against fresh evidence, nested with the gamma
rhythm ($\sim$40~Hz) that carries the sensory content
\citep{Buzsaki2006} --- and a memory, in this account, is a
$\Uone{Qi}$ holonomy: a gauge-invariant phase difference, the
same real-angle-around-a-loop invariant that the fiber uses for
charge. The brain, on this reading, does not merely happen to
use oscillations; it must, because the quantity it traffics in
--- reconstructed, relational, world-centered information --- is a
$\Uone{Qi}$ charge, and a $\Uone{Qi}$ charge lives on a phase.
The improvement of the model over time --- learning --- is then
a theorem of the same kind as the rest of the program: any
memory-bearing system carrying this information current
necessarily refines its allocentric reconstruction, driving the
mismatch between the world and its inner picture monotonically
down (\citealp{ArisakaA11Origin}; \citealp{Friston2010}).

Now stand back and see the parallel whole, because it is the
answer to the reader's opening question. The physics half of
this paper says: charge is a real winding on a real circle;
antiparticles are the real handshake of two real rotation
structures; the imaginary unit is the boundary's name for a real
quarter-turn --- \emph{the world is real, and $i$ is how the
four-dimensional shadow writes it down.} The neuroscience half,
developed in the companion treatments and invoked here only to
illuminate the physics, says: the brain reconstructs the world
in a real, world-centered model, carries that model in real
neural phase, and traffics in information as a real
$\Uone{Qi}$ holonomy --- \emph{the mind is real, and any $i$ in
its equations is again how a description writes down a rotation
of real angles.} The reality of the bulk and the phase-coded
reality of the cortex are, on this view, the same fact seen at
opposite ends of the cascade that runs from the fiber circle to
the reader: reality is written in real phase, and the reader is
an instrument built, by the same descent, to read real phase.
The universal $i$ that experiment demands
(\S\ref{ssec:source-shared}) is not evidence that nature is
complex; it is evidence that nature keeps \emph{one} real
rotation structure --- and that the observer, being part of
nature, inherited it.

Two cautions close the subsection, and they are not
afterthoughts. First, register: the neuroscience developed in
the companion treatments is graded there as the program's most
tentative reach --- an offered structural identification awaiting
the derivations that dedicated later work must supply, not an
established result --- and nothing in this paper's physics
depends on it. The mirrors break, the pentagon turns, and the
matter of the universe is counted whether or not the mind story
holds. Second, proportion: this subsection is a widening of view,
placed here because the author regards the real-bulk doctrine as
inseparable from the question of the observer, and because a
reader who has followed the ``one $J$, one $i$'' argument
deserves to know how far its author believes it reaches. The
paper now returns to what it can defend with instruments and
theorems.

\section{The custody principle, extended}
\label{sec:custody}

One principle governs every use of the two levels this Part has
separated, and it is worth stating as a rule before it is used as
one. Wherever a physical structure and a descriptive label meet
--- a charge sector and its name, a rotation and its symbol ---
the two belong to different owners: \emph{the dictionary owns the
sectors; the geometry owns the} $i$. The rule was drawn from
three independent encounters with the same pattern (charge
superselection, horizon vacua, and the infrared clouds of
soft radiation), each of which dissolves once one asks which
owner a disputed object belongs to
\citep{ArisakaP4Real}. This paper requires its extension to
the discrete symmetries, and states that extension now as the
organizing map (Figure~\ref{fig:custody}):

\begin{center}
\emph{The bulk owns the symmetries; the descent owns their
breaking; the boundary owns their observability.}
\end{center}

\begin{figure}[t]
\centering
\resizebox{0.395\textwidth}{!}{%
\begin{tikzpicture}[
 font=\sffamily\small, >=Stealth, node distance=6mm,
 box/.style={draw=black!55, rounded corners=2pt, align=center,
   inner sep=5pt, minimum width=44mm, minimum height=11mm}]
\node[box, fill=blue!8] (bulk)
  {\textbf{BULK} (AX-1 $+$ AX-2, sign clause AX-3)\\
   fiber $\cdot$ space $\cdot$ clock\\
   C, P, T defined; CPT a theorem};
\node[box, fill=orange!12, below=of bulk] (descent)
  {\textbf{DESCENT} (\GRIP{})\\
   oriented cascade; chiral texture\\
   P and CP broken here};
\node[box, fill=green!10, below=of descent] (boundary)
  {\textbf{BOUNDARY} (\GDOS{}/\OGGEP{})\\
   CPTP projection; arrow of time\\
   asymmetries observed; $\etaB$ recorded};
\draw[->, line width=0.80pt] (bulk) -- node[right=1mm]{orientation written into texture} (descent);
\draw[->, line width=0.80pt] (descent) -- node[right=1mm]{transport, irreversibly read out} (boundary);
\end{tikzpicture}}
\caption[The custody map: bulk, descent, boundary.]{The custody map. The bulk owns the symmetries (the three
orientations and the CPT theorem); the descent owns their breaking
(the chiral texture: parity and CP); the boundary owns their
observability (the arrow of time and every recorded asymmetry, the
baryon excess included).}
\label{fig:custody}
\end{figure}

\emph{The bulk owns the symmetries.} The three orientations that C,
P, and T reverse are bulk structures (fiber, space, clock: AX-2,
AX-1, AX-1 with AX-3's sign clause), and the 6D action is invariant
under each reversal separately --- in particular, the bulk is
exactly time-reversal symmetric \citep{ArisakaA5GDOS}. The
symmetries and the CPT theorem are therefore Part-IV property,
proved upstairs.

\emph{The descent owns their breaking.} What breaks P and CP is not
the bulk but the \emph{texture} the descent writes on it: the
orbifold projection that keeps one hand (Part~VI), and the flavor
holonomies --- \GRIP{} outputs \citep{ArisakaA6Gi4} --- whose
pentagonal chirality is CP violation (Part~VII). In the cosmology
companion's formula: \emph{\GRIP{} owns the angle}
\citep{ArisakaA10Cosmos}.

\emph{The boundary owns their observability.} The arrow of time is
not T-violation but the non-invertibility of the OGGEP projection
--- an AX-3 property, with the bulk T-symmetric --- and every
observable consequence of the broken mirrors, from a kaon asymmetry
to the baryon excess, is read at the boundary through \GDOS{}
transport: \emph{\GDOS{} owns the observability, \OGGEP{} owns the
ledger}.

This three-line map is the paper's answer to the custody
question posed in Part~I: are C, P, T and their theorem defined in the 6D bulk under
AX-1 and AX-2, or do they belong to AX-3 and the GDOS/GRIP
machinery? The answer is layered, and
Table~\ref{tab:axiom-custody} states it row by row, in the
papers' own words.

\begin{table}[t]
\centering
\caption[The axiom-custody table.]{The axiom-custody table: the custody question answered
row by row. The operations and the CPT theorem are bulk-level
(AX-1 $+$ AX-2, with one clause of AX-3 selecting the sign of
$J$); the violations are descent-level (\GRIP{} holonomies); the
observations are boundary-level (\GDOS{}/\OGGEP{}).}
\label{tab:axiom-custody}
\small
\renewcommand{\arraystretch}{1.28}
\begin{tabular}{p{5.1cm} p{2.6cm} p{6.0cm}}
\toprule
\textbf{Ingredient} & \textbf{Custodian} & \textbf{The cohort's
own words} \\
\midrule
The clock's time-orientation (what T reverses) & AX-1 (bulk) &
``AX-1 supplies the direction'' \\
The fiber circle (what C reverses) & AX-2 / Theorem 0 (bulk) &
``AX-2 \emph{builds} the fiber'' \\
Space orientation (what P reverses) & AX-1's arena (bulk) &
time-oriented Lorentzian manifold \\
The sign of the state-space $J$ & AX-3 positivity clause &
``the positivity clause of AX-3 the selection'' \\
Bulk T-symmetry of the 6D action & AX-1 $+$ AX-2 & ``bulk
time-reversal symmetry\ldots an AX-1$+$AX-2 property'' \\
Arrow of time / irreversibility & AX-3 / \OGGEP{} (boundary) &
``the asymmetric projection of the \OGGEP{} CPTP channel'' \\
The CP-violating angle & \GRIP{} (descent) & ``\GRIP{} owns the
angle'' \\
Its observability & \GDOS{} (boundary) & ``\GDOS{} owns its
observability'' \\
The conserved ledger it acts on & \OGGEP{} & ``\OGGEP{} owns the
ledger'' \\
The one relaxed CP angle ($\bar\theta$) & Kaxion (companion) &
dynamical relaxation $+$ static suppression \\
\bottomrule
\end{tabular}
\end{table}

One entry of the table deserves its theorem-level footnote,
because it protects the whole architecture from a classic
confusion. The bulk action is \emph{exactly} time-reversal
symmetric, while every boundary record is irreversible --- and
these two statements are not in tension but in \emph{product}:
the cohort derives the Crooks fluctuation theorem as precisely
the combination of ``bulk time-reversal symmetry of the 6D
action (an AX-1$+$AX-2 property)'' with ``the asymmetric
projection of the \OGGEP{} CPTP channel (an AX-3 property)''
\citep{ArisakaA5GDOS}. The arrow of time is the operational
shadow of the projection, not a microscopic property of the
bulk; T-violation (a symmetry statement about dynamics) and
irreversibility (an operational statement about records) live in
different rows of the custody table, and every later use of
``the descent is oriented'' respects the separation.

\bigskip
\noindent\textbf{Close of Part~III, in plain terms.} The stage is
set, and it is entirely real: an arena forced by locality and
equivalence --- a stage, a zoom knob, and a dial at every point,
neither knob nor dial a place; an integer ledger engraving the
dial; a boundary law that reads the bulk through an always-on
projection, the phase fiber carrying quantum mechanics' coherent
half and the scale fiber its irreversible half; and an
actualization layer that dries the ink into real records and locks
the flavor textures the later Parts will price. Two structures on
this stage are shared by everything in the universe --- the dial
G1 and the clock G2 --- and three orientations live on it: the
dial's, the clock's, and space's. The custody map tells the rest
of the paper where each question belongs: a bulk whose action is
symmetric under each orientation reversal separately (Part~V
proves the table and the CPT theorem there); a descent that is
\emph{oriented} --- a downhill flow through the family landscape,
writing a chiral texture (Part~VI) with pentagonal holonomies
(Part~VII) --- so that parity and CP break together as two faces
of one one-wayness; and an open boundary that records the
consequences irreversibly, from laboratory asymmetries to the
six-per-ten-billion of the sky (Part~VIII). The mirrors are bulk
symmetries; the broken mirrors are descent facts; the matter
excess is the boundary's permanent record of the breakage. The
next Part asks the first question in the natural order, and the
one on which all the others rest: what happens when the dial and
the clock are asked to agree?

\GGPart[The origin of antiparticles]{The Origin of Antiparticles}{The handshake of the circle and the clock; the clock's third face --- imaginary time as the polar decomposition's shore; one shared $J$ for the whole tower; the exceptions on the circle's still point; the bulk that cannot hide its conjugates; the descent that only relabels; and the boundary that reads the twins}
\label{part:antiparticles}

\section[Part~IV in plain English]{Part~IV in plain English: why everything has a twin}
The twins are the first thing the two structures buy. The circle gives charged matter its phase; the clock gives the state space its $i$; and the two agree only if every charged field arrives with a partner of opposite charge. Antiparticles are not an extra postulate in this account. They are the consistency condition between two derivations that were carried out separately and could have disagreed.

The Part proves that in three theorems and then follows it. The clock's third face is imaginary time. There is one $i$ and not one per system, which is what a composite system requires and what a 2021 experiment tested. Spin and statistics appear as the same structure's second child, at candidate grade and labeled as candidate. And the self-conjugate exceptions are the particles the circle cannot move, which is why a Majorana neutrino would be its own antiparticle for a reason one can see rather than memorize.

The negative results are recorded in this Part rather than in a footnote. What the descent does not do is stated in its own section, and the boundary's job in reading the twins is kept separate from the geometry's job in making them, because the two run together easily and the argument needs them apart.

\section{The question, sharpened three ways}
\label{sec:anti-question}

Part~III closed with a question: what happens when the dial
and the clock are asked to agree? This Part answers it, and the
answer is the antiparticle.
``Where do antiparticles come from?'' is three questions wearing
one coat, and the coat has hidden them from each other for ninety
years. \emph{(Q1) Why do antiparticles exist at all?} --- what
consistency of nature forces, for every charged particle, a twin
of equal mass and opposite charge? \emph{(Q2) Where in the
architecture do they first exist?} --- in the 6D bulk, or only
after the descent to the Standard Model, or only in the
quantized theory? \emph{(Q3) Why, then, is the universe made of
one twin and not the other?} The standard account answers Q1 at
the structural level (Part~X rehearses how, and how well),
declines Q2 as meaningless, and outsources Q3 to speculative
mechanisms. The \GG{} account answers all three, in order, and
the answers are the sections of this Part: the stage is set in
real numbers (\S\ref{sec:real-stage}); the twins are forced by
the agreement of two real geometric structures
(\S\ref{sec:handshake}); the agreement extends, with no new
input, to one shared $J$ for everything a boundary can build
(\S\ref{sec:one-i}), with the fields the circle cannot move as
the lawful exceptions (\S\ref{sec:self-conjugate}); the twins
exist in the bulk, before quantization and before any descent,
both as pairing and as content (\S\ref{sec:bulk-encoding}); the descent only relabels
them (\S\ref{sec:descent-relabels}); the boundary reads them
(\S\ref{sec:boundary-readout}); and the answer to Q3 is a
theorem-guarded door and an oriented, CP-biased cascade,
delivered quantitatively in Part~VIII.

\section{A world written in real numbers}
\label{sec:real-stage}

Let us be completely clear about the stage before the first
theorem walks onto it, because the stage is half the point.
Part~II introduced the real presentation as a dictionary ---
fields as real pairs, phases as real angles, conjugation as a
reversed circle. Part~III derived the doctrine behind the
dictionary: the bulk contains no imaginary unit, only $J$, a real
quarter-turn operator, and the symbol $i$ is the boundary's
shorthand name for $J$ (\S\ref{ssec:source-shared}). This Part
now goes one step further than adopting a notation. It makes a
pledge. \emph{From here to the end of the Part, no argument will
use the symbol $i$} --- not in a definition, not in a proof, not
in a displayed equation --- except where the boundary's shorthand
is itself the thing under discussion, and is announced as such.
Everything on the stage is real. Fields are the real pairs
$(a,b)$ of Eq.~\eqref{eq:real-pair}, and the circle acts on them
by rotation blocks $R(n\alpha)$. Coupling phases are rigid
rotations. The dynamics is generated by a real operator on a real
solution space. And the two candidates for ``the imaginary unit
of physics'' are two real operators --- a quarter-turn of the
fiber and a quarter-period advance of the clock --- whose
credentials the next section examines one at a time, in public.

Why insist on the pledge, rather than merely permit it? Because
of the shape of what this Part claims to prove. The claim is that
the particle--antiparticle pairing is forced \emph{one level
below quantization} --- below the complex Hilbert space, below
the creation and annihilation operators, below every structure
that already carries an $i$ inside it. A derivation conducted in
complex notation could not establish that claim even if every one
of its steps were sound, for the same reason a witness cannot
testify to events before his arrival: complex notation enters the
theory at quantization, carrying the pairing's fingerprints
already upon it. In the standard development the
Feynman--St\"uckelberg assignment is imposed \emph{after}
quantization, as the condition that the Hamiltonian come out
positive --- a correct selection, made instrumentally, on the
output. To ask whether the assignment was forced \emph{before}
the machinery existed, one must stand where the machinery does
not yet exist: on the real solution space of the classical bulk
field, where the only structures in sight are the symplectic
form, the circle action, and the causal clock. That is where the
three theorems of the next section stand. If they succeed, the
pairing will have been derived from real geometry alone, and the
complex Hilbert space --- with its one $i$ --- will sit
downstream of the result rather than upstream of it.

Two working rules, then, both already earned upstream and
restated here as the Part's grammar. \emph{Rule one: the circle
owns charge.} A charged field is a pair that turns; its charge is
how many times it turns per turn of the fiber; conjugation is the
fiber run backward. \emph{Rule two: the clock owns direction.}
Causal time is oriented (AX-1), energy is bounded below (the
positivity clause of AX-3), and together --- the next section
proves --- these select a unique way of turning ``real
oscillation'' into ``rotation with a direction.'' Each rule alone
will turn out to supply half an imaginary unit. The whole Part is
the story of what happens where the two halves meet. Roughly
speaking, we are about to watch two gears, machined in different
shops to different drawings, get lowered onto one shaft --- and
discover that they mesh in exactly one relative orientation, and
that the orientation has borne a famous name since 1949.

One more word, on scope, because the register requires it. The
solution spaces on which the theorems are proved are those of
free and quasi-free bulk fields; interacting matter enters
through the asymptotic frame, conditionally, as
\S\ref{sec:anti-ledger} prices. Nothing in the pledge is weakened
by this: the level claim (below quantization) and the scope claim
(quasi-free, interacting conditional) are independent statements,
and both are attached wherever they bind.

\section{The handshake: three theorems}
\label{sec:handshake}

The derivation lives at a level most treatments never visit:
below quantization, in the real geometry of the bulk's solution
spaces. Three theorems carry it. Each is stated and proved here
in full, assuming nothing beyond Parts~II and~III; readers who
want the same results developed at monograph length, with their
interacting extensions, will find them in the companion
treatments of quantum mechanics and of the origin of complex
numbers (\citealp{ArisakaA8QM}; \citealp{ArisakaP4Real}).

\begin{theorem}[\textbf{T-P5-A}: the circle --- the gauge $i$]
\label{thm:p5-A}
On each charge-$n$ sector of a real bulk matter field, the
charge-normalized circle generator $\Jq = \partial_\theta/Q$ is a
complex structure: it squares to $-\mathbf 1$ and identifies the
real weight-$n$ harmonic pair with the standard complex charged
field, with complex conjugation realized as orientation reversal
of the circle, $\theta \mapsto -\theta$. It is the unique such
structure commuting with the circle action --- \emph{up to sign}.
On the neutral sector the same generator acts as zero: the circle
can only complexify what it moves.
\end{theorem}

\begin{proof}
On the weight-$n$ pair \eqref{eq:real-pair} the generator acts as
$\partial_\theta = n\,\varepsilon$ with $\varepsilon$ the
$90^{\circ}$ rotation; normalizing by the signed charge $Q = n$
gives $\Jq = \varepsilon$, and $\varepsilon^2 = -\mathbf 1$. Any
complex structure commuting with all the rotations $R(n\alpha)$
must be a polynomial in $\varepsilon$ on each sector, hence
$\pm\varepsilon$: uniqueness up to sign. On weight zero,
$\partial_\theta = 0$ and no structure is supplied.
\end{proof}

\emph{Physics intuition, at length.} Everything familiar about
charge is repackaged in this small theorem, and it is worth
walking through slowly. Charge quantization is the compactness of
the circle: winding numbers are integers. The Aharonov--Bohm
phase is the circle's holonomy. The ``complex phase'' of a
charged wavefunction is a real angle of position on a real fiber;
multiplying by $i$ is a quarter-turn. And --- the clause that
carries this Part --- \emph{complex conjugation is going around
the circle the other way}. A ``negatively charged'' mode is not a
new substance; it is the same real pair winding oppositely. But
the theorem also confesses the circle's two blindnesses, and both
are load-bearing. The circle cannot complexify the neutral
sector: a neutral field needs an $i$ exactly as much as a
charged one (its wave equation, its interference, its quantum
amplitudes), and the circle has none to give --- so a second
source must exist. And the circle does not know its own
orientation: nothing in the fiber geometry distinguishes
$\varepsilon$ from $-\varepsilon$, winding-up from winding-down
--- the circle knows the plane of rotation but not the
direction. Two blindnesses, one cure, and the cure is the clock.

\begin{theorem}[\textbf{T-P5-B}: the clock --- the state-space
$i$]
\label{thm:p5-B}
On the real solution space of any free or quasi-free bulk field,
the polar phase of the time-translation generator $A$,
\begin{equation}
\Ja \;=\; \pm\, A\,(-A^2)^{-1/2},
\label{eq:polar-phase}
\end{equation}
is a complex structure --- concretely, the Hilbert transform in
constitutional time --- and it is the \emph{unique}
positive-energy complex structure commuting with the dynamics on
the globally static arena, with the sign fixed by the positivity
clause of AX-3 \citep{AshtekarMagnon1975, Kay1978}. The
constitutional pedigree is exact: AX-1 supplies the direction,
the positivity clause of AX-3 the selection, Theorem~0 the
uniqueness. In the boundary's shorthand, its $(-i)$-eigenspace is
the positive-frequency subspace: the frequency splitting is
derived, not assumed.
\end{theorem}

\begin{proof}
We walk the construction in four steps, at the level of rigor the
quasi-free setting permits, and flag the one step whose
functional-analytic care is carried by the cited theorems.

\emph{Step 1: the generator is skew.} The real solution space $V$
of a free bulk field carries the symplectic form $\omega(u,v)$
built from the field and its conjugate momentum on a Cauchy
slice, and time translation preserves it:
$\omega(e^{tA}u, e^{tA}v) = \omega(u,v)$ for all $t$.
Differentiating at $t = 0$ gives
$\omega(Au, v) + \omega(u, Av) = 0$: the generator $A$ is skew
with respect to $\omega$. Skewness is what a rotation generator
looks like before anyone has said ``rotation'': it forbids $A$
from stretching the planes it acts in.

\emph{Step 2: the polar phase exists and squares to
$-\mathbf 1$.} On the globally static arena (Theorem~0) the
physical modes oscillate --- $A$ has no kernel on them --- so
$-A^2$ is a positive operator and its inverse square root exists
on the completion; the domains and closures here are precisely
the care the cited authors supply
\citep{AshtekarMagnon1975, Kay1978}. Then, since $A$ commutes
with every function of $A^2$,
\begin{equation}
\Ja^2 \;=\; A\,(-A^2)^{-1/2}\,A\,(-A^2)^{-1/2}
\;=\; A^2\,(-A^2)^{-1} \;=\; -\mathbf 1 .
\label{eq:polar-square}
\end{equation}
The polar phase is the generator with its speed divided out:
what remains is pure turning.

\emph{Step 3: the sign.} Both $+\Ja$ and $-\Ja$ are complex
structures commuting with the dynamics; they are distinguished by
the energy form $\mu(u,v) = \omega(\Ja u, v)$, which is positive
definite for exactly one choice of the sign. The positivity
clause of AX-3 selects that choice. This is the precise point at
which ``energy is bounded below'' enters the geometry --- not as
a condition on a quantized Hamiltonian, but as the orientation of
a classical quadratic form.

\emph{Step 4: uniqueness, and the frequency splitting, in real
terms.} Ashtekar--Magnon and Kay prove that on a stationary
region there is exactly one complex structure commuting with the
dynamics that renders the energy positive; Theorem~0's global
staticity makes the region the whole arena and the selection
canonical. The frequency splitting is then a statement about
oriented planes, with no complex number required: every mode of
frequency $E > 0$ spans a two-dimensional real plane on which $A$
acts as $E$ times a quarter-turn, and $\Ja$ acts as that same
quarter-turn with the speed removed --- one fixed rotation sense
per plane, the same for every $E$. ``Positive frequency'' names
the orientation $\Ja$ induces on each plane, and it is derived
because $\Ja$ is. (In the boundary's complex shorthand, the same
statement reads: the $(-i)$-eigenspace of $\Ja$ is the
positive-frequency subspace.) Concretely, in the time domain
$\Ja$ is the Hilbert transform,
\begin{equation}
(\Ja u)(t) \;=\; \frac{1}{\pi}\,
\mathrm{p.v.}\!\int \frac{u(s)}{t - s}\, ds ,
\label{eq:hilbert-kernel}
\end{equation}
the real integral kernel that advances every oscillation by a
quarter period --- the analytic-signal operator of the engineer,
promoted to constitutional office.
\end{proof}

\emph{Physics intuition, at length.} The clock's $i$ answers the
question the circle could not: where does the $i$ of
$i\hbar\,\partial_t$ come from --- the one a strictly neutral
field needs as much as a charged one? Answer: from time itself,
run through causality and positivity. ``Multiply by $i$'' means
``advance the phase by a quarter period,'' and \emph{that} is a
real operation --- the Hilbert transform, an integral kernel any
engineer has met as the analytic-signal construction. What
elevates it from convenience to structure is the uniqueness: on
the constitutional arena there is exactly one way to split real
oscillations into ``positive frequency'' and ``negative
frequency'' consistent with causal order and positive energy,
and the clock's polar phase is it. Note what each source knows
that the other does not. The clock knows the \emph{direction} ---
causality orients it, positivity picks its sign --- but it
belongs to every field equally and knows nothing of charge. The
circle knows \emph{charge} --- it is charge --- but not
direction. Two half-blind sources of the same $i$; the theory
works only if they agree wherever they overlap. That agreement
has a name.

\begin{theorem}[\textbf{T-P5-C}: the handshake --- antiparticles
as the consistency condition]
\label{thm:p5-C}
On the on-shell charge-$n$ sectors the two structures agree,
$\Ja = \Jq$, precisely under the Feynman--St\"uckelberg pairing
--- charge $+n$ with positive frequency \emph{and} charge $-n$
with positive frequency --- and anti-agree on the opposite
pairing. The pairing is therefore not a convention but the unique
orientation making two independently forced real structures act
as the same complex structure on shell. It fixes the sign left
open in T-P5-A, and yields one global $J$: \emph{the existence of
antiparticles is the consistency condition between the sixth
dimension and causal time.}
\end{theorem}

\begin{proof}
The proof is a computation every reader should perform once. An
on-shell mode of charge $n$ and energy $E > 0$ is the real
oscillation with total dial angle $n\theta - Et$:
\begin{equation}
u_{+}(\theta, t) \;=\; \cos(n\theta - Et).
\label{eq:mode-plus}
\end{equation}
Apply the two structures. The circle turns the fiber:
$\Jq\!: \cos(n\theta - Et) \mapsto -\sin(n\theta - Et)$. The
clock advances the phase a quarter period:
$\Ja\!: \cos(n\theta - Et) \mapsto -\sin(n\theta - Et)$. They
agree. Now take the \emph{wrong} pairing --- charge $n$ with
negative frequency, the mode $\cos(n\theta + Et)$: the circle
still gives $-\sin(n\theta + Et)$, but the clock, acting on a
negative-frequency oscillation, gives $+\sin(n\theta + Et)$.
They anti-agree: on this mode the two forced structures differ by
a sign, and no global $J$ exists. The unique repair is the
relabeling: read the negative-frequency charge-$n$ modes as
\emph{positive}-frequency charge-$(-n)$ modes --- under which
$\Jq = \partial_\theta/Q$ flips its normalization with $Q$ and
agreement is restored on every physical mode. That relabeling is
the Feynman--St\"uckelberg assignment; the relabeled modes are
the antiparticles; and the assignment simultaneously fixes the
circle's sign ambiguity, since only one orientation of
$\varepsilon$ matches the clock on the physical pairing.
\end{proof}

\emph{Physics intuition, at length --- the gear alignment.} The
circle knows the plane of rotation but not the direction; the
clock knows the direction. Wherever charge lives, the two must
mesh --- like two gears that can engage in exactly one relative
orientation. The meshing condition \emph{is} the particle--%
antiparticle pairing: positive charge running forward in phase,
negative charge running forward in phase, and nothing running
backward. Antiparticles are not an extra ingredient of
relativistic quantum theory; they are the gear alignment between
its two real sources. Three further remarks complete the picture.
\emph{First, the level.} In standard field theory the same
assignment is imposed \emph{after} quantization, so that the
Hamiltonian comes out positive --- a condition on the output. The
handshake operates one level down, before quantization, as the
compatibility of two classical-geometric structures; in the
parent program's words, \emph{a convention with a derivation
behind it is no longer a convention}. \emph{Second, the reach.}
The extension is canonical: composite systems and the Fock
construction extend the shared $J$ with no new complex input, so
the boundary description carries one $i$, globally shared ---
exactly the configuration the network Bell experiments demand
\citep{RenouEtAl2021}; and for fermions, charge conjugation
factorizes as circle-orientation reversal composed with a fixed
real spinor map, so the Dirac field's celebrated complexity is
the $J$-packaging of real data --- charge by the circle,
frequency by the clock, spin by real matrices. No Dirac sea is
invoked at any point, here or anywhere in the cohort.
\emph{Third, the unification.} The literature that asks ``why is
quantum mechanics complex?'' and the literature that asks ``why
do antiparticles exist?'' are disjoint literatures. On this
account they are one question with one answer: the reason
quantum mechanics is complex and the reason antiparticles exist
are the same fact, read in two directions. That identification
--- not the pairing itself, which is textbook --- is the
novelty, and \S\ref{sec:anti-ledger} prices it honestly.


\subsection{The handshake executed: one charged mode, in rotation
blocks}
\label{ssec:handshake-blocks}

Theorems deserve rehearsals. Let us now run the entire handshake,
end to end, on a single charged mode, with every object a real
number or a real $2\times2$ matrix --- the reader is invited to
follow with a pencil, because the whole computation fits on one
sheet and there is no better way to own it. Take winding $n = +1$
and energy $E > 0$, and call the mode an electron by a harmless
abuse (the electron's conventional winding is $-1$; reversing
every winding below changes nothing).

\emph{The stage.} The candidate mode plane is the real span of
\begin{equation}
u_1 = \cos(\theta - Et), \qquad u_2 = \sin(\theta - Et),
\label{eq:mode-basis}
\end{equation}
and every operator below is a $2\times2$ real matrix in the basis
$(u_1, u_2)$.

\emph{The circle's quarter-turn.} Differentiating along the
fiber, $\partial_\theta u_1 = -u_2$ and
$\partial_\theta u_2 = +u_1$; normalizing by the signed charge
$Q = +1$,
\begin{equation}
\Jq \;=\; \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix},
\qquad \Jq^2 = -\mathbf 1 .
\label{eq:JQ-matrix}
\end{equation}

\emph{The clock's quarter-turn.} As functions of time the two
basis modes oscillate with phase $Et - \theta$, and on every
positive-frequency oscillation the clock's operator acts by the
quarter-period rule $\cos \mapsto \sin$, $\sin \mapsto -\cos$
(Eq.~\eqref{eq:hilbert-kernel}). Applied to
$u_1 = \cos(Et - \theta)$ and $u_2 = -\sin(Et - \theta)$ this
gives $\Ja u_1 = \sin(Et - \theta) = -u_2$ and
$\Ja u_2 = +\cos(Et - \theta) = u_1$. In the same basis,
\begin{equation}
\Ja \;=\; \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix} .
\label{eq:JA-matrix}
\end{equation}
Compare \eqref{eq:JQ-matrix} and \eqref{eq:JA-matrix}: the two
structures are not merely compatible --- they are \emph{equal,
entry by entry}. On this pairing of winding with frequency, the
fiber's geometry and the clock's causality have manufactured the
same quarter-turn twice.

\emph{The wrong pairing.} Now run the same computation on the
opposite matching --- winding $+1$ against the mode whose clock
phase runs the other way:
\begin{equation}
v_1 = \cos(\theta + Et), \qquad v_2 = \sin(\theta + Et).
\label{eq:wrong-basis}
\end{equation}
The circle does not care --- $\partial_\theta v_1 = -v_2$,
$\partial_\theta v_2 = +v_1$, so $\Jq$ is the matrix of
\eqref{eq:JQ-matrix} again. But the clock reads the phase as
$Et + \theta$ and advances it: $\Ja v_1 = \sin(Et+\theta) = +v_2$
and $\Ja v_2 = -v_1$. In the basis $(v_1, v_2)$,
\begin{equation}
\Jq = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix},
\qquad
\Ja = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}
= -\Jq .
\label{eq:clash-matrix}
\end{equation}
The two forced structures now differ by a sign on every vector of
the plane. No global $J$ exists; the theory, as paired, is
inconsistent with its own geometry.

\emph{The repair, watched closely.} Here is the beautiful step.
The clash plane is not new territory --- because cosine is even,
\begin{equation}
\cos(\theta + Et) \;=\; \cos\!\big((-1)\theta - Et\big),
\qquad
\sin(\theta + Et) \;=\; -\sin\!\big((-1)\theta - Et\big):
\label{eq:reread}
\end{equation}
the same two-dimensional plane, re-read as the
\emph{positive}-frequency plane of winding $-1$. Run the
operators once more in the re-read basis
$w_1 = \cos(-\theta - Et)$, $w_2 = \sin(-\theta - Et)$. The
circle gives $\partial_\theta w_1 = +w_2$, and --- this is where
the signed charge earns its place in the definition
$\Jq = \partial_\theta / Q$ --- the normalization is now
$Q = -1$, so $\Jq w_1 = -w_2$. The clock reads phase $Et+\theta$
as before and gives $\Ja w_1 = \sin(Et + \theta) = -w_2$. They
agree again, entry by entry. The relabeling that repairs the
clash is exactly the Feynman--St\"uckelberg assignment, and the
repaired object is the antiparticle: same plane, opposite
winding, clock running forward.

\emph{The packaging, last of all.} Only now, with the real
computation complete, let the boundary introduce its shorthand:
write $\psi = a + ib$ for the pair $(a,b)$, so that the shared
quarter-turn becomes multiplication by $i$. The kept plane of
\eqref{eq:mode-basis} packages as $e^{\,i(\theta - Et)} =
e^{\,i\theta} e^{-iEt}$ --- charge $+1$, positive frequency: the
particle. The re-read plane packages as $e^{\,i(-\theta - Et)} =
e^{-i\theta} e^{-iEt}$ --- charge $-1$, positive frequency: the
antiparticle. Both amplitudes rotate forward in time; their fiber
windings are opposite; and the $i$ in both is one and the same
symbol because the $J$ it names is one and the same matrix. At no
step before this paragraph did the symbol $i$ occur --- and when
it arrived, it arrived as a name. That is the whole doctrine of
this paper in one computation: one $J$ upstairs is why there is
one $i$ downstairs.

\begin{figure}[t]
\centering
\resizebox{0.649\textwidth}{!}{%
\begin{tikzpicture}[font=\sffamily\small, >=Stealth,
  cell/.style={draw=black!55, rounded corners=2pt, align=center,
    inner sep=5pt, minimum width=54mm, minimum height=15mm}]
 \node at (0,1.45) {\textbf{winding $+n$}};
 \node at (6.6,1.45) {\textbf{winding $-n$}};
 \node[rotate=90, anchor=center] at (-3.75,0)
   {clock: forward};
 \node[rotate=90, anchor=center] at (-3.75,-2.7)
   {clock: backward};
 \node[cell, fill=green!10] (pp) at (0,0)
   {$\cos(n\theta - Et)$\\[1pt] $\Jq = \Ja$: \textbf{kept} --- the particle};
 \node[cell, fill=green!10] (mp) at (6.6,0)
   {$\cos(-n\theta - Et)$\\[1pt] $\Jq = \Ja$: \textbf{kept} --- the antiparticle};
 \node[cell, fill=red!8] (pm) at (0,-2.7)
   {$\cos(n\theta + Et)$\\[1pt] $\Jq = -\Ja$: \textbf{clash}};
 \node[cell, fill=red!8] (mm) at (6.6,-2.7)
   {$\cos(-n\theta + Et)$\\[1pt] $\Jq = -\Ja$: \textbf{clash}};
 \draw[->, line width=1.10pt, blue, dashed] (pm.east)
   to[bend right=14]
   node[midway, below, sloped]{re-read (F--S)} (mp.south);
 \draw[->, line width=1.10pt, blue, dashed] (mm.west)
   to[bend left=14] (pp.south);
\end{tikzpicture}}
\caption[The handshake: four pairings of winding and clock.]{The four candidate pairings of winding and clock sense,
and the handshake's verdict. On the two kept cells the circle's
quarter-turn and the clock's quarter-period advance are the same
real matrix, Eqs.~\eqref{eq:JQ-matrix}--\eqref{eq:JA-matrix}; on
the clash cells they differ by a sign and no shared $J$ exists,
Eq.~\eqref{eq:clash-matrix}. The dashed arrows are the
Feynman--St\"uckelberg re-reading of Eq.~\eqref{eq:reread}:
because cosine is even, each clash plane \emph{is} a kept plane
read with the opposite winding. The two kept cells together carry
one complex world with one $i$.}
\label{fig:quadrants}
\end{figure}

\section{The clock's third face: imaginary time}
\label{sec:wick-quartet}

Every physicist carries three devices that involve the letter $i$
and the letter $t$ together, and has been taught to regard them as
unrelated conveniences. Minkowski's $it$ (1908): write the time
coordinate as imaginary and the Lorentz metric looks Euclidean,
boosts look like rotations ``by an imaginary angle''
\citep{Minkowski1908}. Schr\"odinger's $i\,d/dt$ (1926): the
quantum evolution law places the imaginary unit against the time
derivative. And Wick's rotation (mid-century, named for
G.~C.~Wick): continue $t \to -i\tau$, turn oscillation into
damping, quantum theory into statistical mechanics, and compute
--- the daily bread of lattice field theory. Three tricks, three
eras, no stated connection. This section proves that with T-P5-B
on the table the three are \emph{one datum}: the polar
decomposition of the constitutional time generator. The theorem
is lettered into the clock's family --- T-P5-B1 --- because it is
the clock's own structure, unpacked; nothing in it touches the
fiber, which is precisely one of its lessons.

One word on register before the mathematics, because this Part
made a pledge (\S\ref{sec:real-stage}) and this section is where
the pledge earns its keep. To speak of ``holomorphy'' we shall
complexify --- announced now, as the boundary's shorthand, used
knowingly. But the reader should watch for the striking fact
that the theorem's second clause has a \emph{fully real}
formulation: holomorphy in imaginary time is the statement that
a profile and its quarter-turned partner form a
harmonic-conjugate pair --- the Cauchy--Riemann equations, which
are real equations about real functions, and which turn out to
be nothing but the clock's quarter-turn written as a field
equation. Even imaginary time, it will emerge, is real
underneath.

\emph{The setting, recalled, and the polar lemma.} Let
$(V, \omega, A)$ be as in T-P5-B: the real symplectic solution
space of a free or quasi-free bulk field on the globally static
arena, $A$ the constitutional time-translation generator, skew
with respect to $\omega$, with no kernel on the physical modes.
Define the clock's \emph{modulus}
\begin{equation}
|A| \;:=\; (-A^2)^{1/2},
\label{eq:clock-modulus}
\end{equation}
positive and dynamics-commuting by the spectral calculus of
T-P5-B, Step~2. Then $A$ factorizes:
\begin{equation}
A \;=\; J_0\,|A| \;=\; |A|\,J_0,
\qquad J_0 := A\,|A|^{-1},
\qquad J_0^2 = -\mathbf 1,
\label{eq:polar-decomposition}
\end{equation}
and the pair $(J_0, |A|)$ is unique with these properties: a
skew generator splits, exactly once, into a \emph{rotor} (a
complex structure --- pure turning, no speed) and a \emph{speed}
(a positive operator --- pure magnitude, no turning). T-P5-B's
clock structure is this rotor with its orientation fixed,
$\Ja = \pm J_0$, the sign selected by the positivity clause of
AX-3. \emph{Physics intuition --- the motor.} Every motor is a
rotor times a speed, and \eqref{eq:polar-decomposition} says the
constitutional clock is a motor: $A$ turns each mode plane
($J_0$) at a rate ($|A|$). Real time evolution $e^{tA}$
exponentiates the whole motor --- rotation at frequency $E$ on
each plane. The theorem below says: imaginary time is what one
gets by exponentiating \emph{the speed alone}, $e^{-\sigma|A|}$
--- switching off the rotor and letting the magnitude act as
pure damping --- and that this switch, the modulus of the polar
decomposition, is simultaneously Minkowski's $it$, Wick's
rotation, and the analytic face of Schr\"odinger's $i$.

\begin{theorem}[\textbf{T-P5-B1}: the Wick quartet --- one
analytic fact]
\label{thm:p5-B1}
On $(V, \omega, A)$ as above, the following four data are in
canonical bijection --- each determines the others uniquely,
orientations included:

(1) \emph{The clock.} A complex structure $J$ commuting with the
dynamics $e^{tA}$ whose energy form $\mu(u,v) = \omega(Ju, v)$ is
positive definite.

(2) \emph{The harmonic-conjugate (Hardy) splitting.} An
assignment, to each solution $u$, of a partner $v$ such that the
pair extends from the time axis to a bounded harmonic-conjugate
pair $(U, V)$ on a half-plane $\{(t,\sigma): \sigma > 0\}$ ---
equivalently, in the announced shorthand, a conjugate pair of
dynamics-invariant subspaces $V_{\mathbb C} = H^{+} \oplus H^{-}$
whose profiles extend holomorphically and boundedly to the two
half-planes of complexified time.

(3) \emph{The Wick rotation.} A strongly continuous contraction
semigroup $S(\sigma) = e^{-\sigma|A|}$, $\sigma \geq 0$, arising
as the analytic continuation of the evolution to imaginary time,
with the energy form's positivity holding on the imaginary-time
section (reflection positivity)
\citep{OsterwalderSchrader}.

(4) \emph{The Minkowski section.} A bounded holomorphic extension
of the time-translation representation to the complexified
translation semigroup $\{t + i\sigma : \sigma \geq 0\}$ --- the
time-direction tube on which ``rotation by an imaginary angle''
is a lawful operation rather than a notation.

Moreover the orientations correspond: the sign of $J$
$\leftrightarrow$ which half-plane carries the bounded extension
$\leftrightarrow$ which direction of imaginary time damps rather
than explodes $\leftrightarrow$ the orientation of the tube ---
and all four are fixed at once by one constitutional choice,
AX-1's causal orientation composed with the positivity clause of
AX-3. Beneath all four lies a single analytic fact: \emph{the
spectrum of the generator is one-sided relative to the causal
orientation.} Clauses (1)--(4) are its four notations.
\end{theorem}

\begin{proof}
Seven steps, none skipped; every functional-analytic closure is
carried by the cited literature, and every computation is
displayed on the mode planes where the quasi-free setting lives.

\emph{Step 1 (mode reduction).} By T-P5-B, Steps 1--2, $V$
decomposes into two-dimensional $A$-invariant oriented planes; on
the plane of frequency $E > 0$ with basis
$e_1 = \cos(Et)$, $e_2 = \sin(Et)$, the generator acts as $E$
times a quarter-turn, the modulus as $|A| = E\,\mathbf 1$, and
the rotor $J_0$ as the quarter-turn itself. All four clauses are
statements about dynamics-commuting structures, so all four may
be verified plane by plane and assembled by the spectral
calculus.

\emph{Step 2 ((1) $\Rightarrow$ (2): the damping construction,
and the Cauchy--Riemann pair).} Given the clock, define for each
solution $u$ and each $\sigma \geq 0$ the damped profile
\begin{equation}
U(t, \sigma) \;:=\; \bigl(e^{-\sigma|A|} u\bigr)(t),
\qquad
V(t, \sigma) \;:=\; \bigl(e^{-\sigma|A|} \Ja u\bigr)(t):
\label{eq:damped-profiles}
\end{equation}
the solution and its quarter-turned partner, both run through
the pure-speed semigroup. On a mode, with
$u = a\cos(Et) + b\sin(Et)$ and the clock acting by the
quarter-period rule of T-P5-B ($\cos \mapsto \sin$,
$\sin \mapsto -\cos$),
\begin{equation}
U = e^{-\sigma E}\,[\,a\cos Et + b\sin Et\,],
\qquad
V = e^{-\sigma E}\,[\,a\sin Et - b\cos Et\,].
\label{eq:mode-damped}
\end{equation}
Three properties, each a one-line check on
\eqref{eq:mode-damped}. \emph{Boundedness:} $|U|, |V| \leq
(|a|+|b|)\,e^{-\sigma E}$, bounded on $\sigma > 0$ precisely
because $E > 0$. \emph{Harmonicity:}
$(\partial_t^2 + \partial_\sigma^2)\,U = (-E^2 + E^2)\,U = 0$,
and likewise for $V$. \emph{Conjugacy:} differentiating
\eqref{eq:mode-damped},
\begin{equation}
\partial_\sigma U = -\,\partial_t V,
\qquad
\partial_\sigma V = +\,\partial_t U
\label{eq:CR-pair}
\end{equation}
--- the Cauchy--Riemann equations. So $(U, V)$ is a bounded
harmonic-conjugate pair on the half-plane with boundary values
$(u, \Ja u)$: clause (2), in its real form. In the announced
shorthand, $F := U + iV$ is holomorphic in $z = t + i\sigma$ on
the upper half-plane --- on the mode,
$F = (a - ib)\,e^{iEz}$, with $|F| = |a-ib|\,e^{-E\sigma}$ ---
and the $H^{\pm}$ splitting is the eigenspace decomposition of
$\Ja$. Pause on \eqref{eq:CR-pair}, because it is the section's
conceptual hinge: \emph{the Cauchy--Riemann equations are the
clock's quarter-turn written as a field equation.} Translating
into the second coordinate is time translation composed with the
quarter-turn --- $\sigma$ is not a place, not a hidden axis; it
is the direction the rotor defines, opened into a half-plane by
the damping the modulus supplies.

\emph{Step 3 ((2) $\Rightarrow$ (1): the Paley--Wiener
converse).} Conversely, suppose the bounded conjugate extension
exists on the upper half-plane. Expand the boundary profile in
the mode planes; a component of frequency $-E < 0$ would extend
as $e^{-iEz}$, of modulus $e^{+E\sigma}$ --- unbounded. Bounded
holomorphic extension to a half-plane therefore forces
\emph{one-sided} frequency content (this is the Paley--Wiener
theorem, in its simplest clothing), and the splitting
$H^{\pm}$ is recovered; $J$ is then multiplication by $\mp i$ on
the two summands --- equivalently, on the real space, the
quarter-turn whose partner assignment $u \mapsto v$ the
extension realized. The half-plane choice recovers the sign of
$J$: extension upward pairs with one orientation of the
quarter-turn, downward with the other.

\emph{Step 4 ((1) $\Leftrightarrow$ (3): the Euclidean
semigroup, and where positivity re-enters).} Given the clock,
$S(\sigma) := e^{-\sigma|A|}$ is a strongly continuous semigroup
by the spectral calculus, and a \emph{contraction} ---
$\|S(\sigma)\| \leq 1$ --- precisely because $|A| \geq 0$: the
one-sidedness of the spectrum is what damps rather than
explodes. It is the analytic continuation of the evolution in
the exact sense of Step 2: the matrix elements
$\mu(u, e^{tA} v)$, which on a mode are trigonometric in $t$,
extend holomorphically in $t$ to the half-plane, and their
restriction to the imaginary axis $z = i\sigma$ is the
Euclidean two-point kernel $\mu(u, S(\sigma) v)$, symmetric in
its arguments. Reflection positivity is the statement that this
kernel is positive-definite with respect to reflection in
$\sigma$ --- on the modes, that Gram matrices of decaying
exponentials $e^{-(\sigma_j + \sigma_k)E}$ are positive
semi-definite, which they are, being Gram matrices of the
vectors $e^{-\sigma_j |A|}v$ in the $\mu$-inner product ---
\emph{and the $\mu$ whose positivity this is, is AX-3's}: the
same clause that fixed the clock's sign now certifies the
Euclidean section. Conversely, from a reflection-positive
contraction semigroup one reconstructs the Hilbert space, the
positive generator $|A|$, and the unitary evolution --- this is
the Osterwalder--Schrader reconstruction
\citep{OsterwalderSchrader} --- and the polar recombination of
$|A|$ with the reconstructed $A$ returns $J$, with its sign
fixed by which reflection direction was positive. \emph{Physics
intuition --- why Wick rotation works.} Every practitioner has
rotated a contour or run a Euclidean lattice, trusting that the
answers continue back. The trust has a precise source, and it
is not cleverness of technique: it is clause (3)'s equivalence
with clause (1). The continuation exists because the spectrum
is one-sided; the spectrum is one-sided because of the
positivity clause; so \emph{the license for every Wick rotation
ever performed is the same axiom clause that selects the
Schr\"odinger $i$}. A theory with energy of both signs would
have no consistent Euclidean section --- some mode would
explode in every direction of imaginary time --- and no lattice
simulation of it could converge. The daily bread is baked from
AX-3.

\emph{Step 5 ((1) $\Leftrightarrow$ (4): the holomorphic
tube).} Clause (4) is clause (2) read group-theoretically. On
$H^{+}$, define $T(z) := $ translation by the complexified time
$z = t + i\sigma$; on a mode, $T(z) = e^{iEz}$, of modulus
$e^{-E\sigma}$: a family holomorphic in $z$, unitary on the real
axis ($\sigma = 0$), contractive on the tube ($\sigma > 0$),
and a semigroup in $z$. Its existence as a \emph{bounded}
holomorphic extension is, again by Paley--Wiener, equivalent to
the one-sidedness of the spectrum, hence to clause (1). This is
the honest content of Minkowski's device: writing $it$ is
lawful exactly as the coordinate on this tube --- the domain
where the operations of rotation and damping commute into one
holomorphic family --- and unlawful as a statement that the
arena possesses an imaginary axis. The tube is attached to the
time direction by the dynamics; it is nowhere in the metric.

\emph{Step 6 (the orientation bijection).} Track the one binary
choice through the four clauses. The clock's sign ($J = +J_0$
versus $-J_0$) selects which eigenspace is called $H^{+}$
(Step 2); $H^{+}$ selects the half-plane of bounded extension
(Step 2's damping computation); the half-plane selects the
direction of imaginary time in which $S(\sigma)$ contracts
rather than explodes (Step 4); and that direction orients the
tube (Step 5). Each arrow is invertible (Steps 3--5), so the
four orientation data are one bit --- and the bit is not free:
AX-1 orients real time, and the positivity clause of AX-3
selects the sign of $J$ (T-P5-B, Step 3), fixing the whole
quartet at once. There is nothing left to choose anywhere in
clauses (1)--(4).

\emph{Step 7 (uniqueness, and the single fact).} The bijections
compose into a cycle: $J \mapsto (H^{+}, H^{-}) \mapsto
S(\sigma) \mapsto |A| \mapsto J$ (polar recombination), and
each map is uniquely determined by its input (Steps 2--5), so
each datum determines the other three with no residual freedom.
Finally, each clause was shown equivalent to the same
statement: the spectrum of $A$, read against the causal
orientation, lies on one side. One-sidedness \emph{is} the
analytic fact; the quartet is its notation.
\end{proof}

\begin{corollary}[\textbf{B1-a}: the boundary's three
shorthands]
\label{cor:B1a}
(a) ``$i\,d/dt$'' is clause (1): the Schr\"odinger $i$ is the
clock's rotor $\Ja$, the boundary's name for a real
quarter-turn. (b) ``$t \to it$'' is clauses (2) and (4):
Minkowski's imaginary time is the coordinate on the continuation
half-plane that energy positivity opens --- the name of an
analytic direction, not of an axis of the arena. (c) ``Wick
rotation'' is clause (3): passing from the motor to its speed,
$e^{tA}$ to $e^{-\sigma|A|}$ --- available exactly where, and
exactly because, clause (1)'s clock exists.
\end{corollary}

\begin{corollary}[\textbf{B1-b}: custody --- imaginary time
belongs to the clock]
\label{cor:B1b}
The quartet is constructed from $(V, \omega, A)$ alone: no
fiber datum enters. Hence imaginary time exists on every
sector, neutral included, sourced by AX-1 and AX-3 --- it is
the \emph{clock's} third face, and the sixth dimension neither
supplies it nor could: the fiber's generator has two-sided
integer spectrum (windings $\pm n$), so its ``modulus'' damps
no direction, and the compact circle admits no half-plane.
Likewise there is no ``imaginary space'': spatial translations
are skew generators too, but their spectra are two-sided (left-
and right-movers are equally lawful), no positivity clause
orients them, and their polar rotors are pure convention.
Among all the arena's directions, time alone carries a
one-sided spectrum --- which is why time alone has a shore. The
bulk, meanwhile, remains real throughout: ``$it$'' is the
boundary's shorthand for the half-plane exactly as ``$i$'' is
its shorthand for $J$ (\S\ref{ssec:source-shared}), and the
one shared $J$ of \S\ref{sec:one-i} now carries one shared
shore with it.
\end{corollary}

\emph{Two conditional remarks, graded.} \emph{First --- the
Lorentz completion and the germ of CPT.} Clause (4) is proved
here in the time direction. Its covariant completion ---
extension to the complexified Lorentz group, where boosts and
rotations genuinely merge and ``rotation by an imaginary
angle'' becomes group theory --- holds at the interacting,
axiomatic grade in Wightman theory \citep{StreaterWightman},
and it is precisely the continuation through which Jost proved
the CPT theorem. The resonance deserves a sentence: \emph{the
same half-plane that licenses Wick's rotation is the germ of
the continuation that makes CPT provable} --- the quartet of
this section and the total reversal of Part~V
(\S\ref{sec:cpt}) are two harvests of one analytic structure,
and the open modular route named there (Bisognano--Wichmann)
is the natural bridge for both. Grade: inherited at axiomatic
scope; conditional in this framework, alongside T-P5-2's
interacting extension. \emph{Second --- the thermal shore.}
Periodicity in imaginary time is the boundary's reading of
temperature (the KMS condition); the black-hole companion's
derivation of Hawking thermality \citep{ArisakaC3BH} is its
worked instance, with the periodicity supplied by the
projection's reading of the horizon. Grade: companion result,
conditional as there stated.

\emph{Status, and what is inherited.} Concessions first, in
full. Every clause of T-P5-B1 is classical in isolation: the
positive-energy complex structure is Ashtekar--Magnon and
Kay's \citep{AshtekarMagnon1975, Kay1978}; the half-plane
characterization of one-sided spectra is Paley and Wiener's;
the Euclidean reconstruction is Osterwalder and Schrader's
\citep{OsterwalderSchrader}; the tube analyticity is the
Wightman school's \citep{StreaterWightman}. What this section
claims is the \emph{packaging with provenance}: the four data
assembled into a canonical bijection with the orientations
tracked and anchored --- one bit, fixed by AX-1 composed with
AX-3's positivity clause --- so that three devices physics
folklore treats as separate tricks are exhibited as one datum,
the polar decomposition of the constitutional clock; and the
custody corollary, which extends the one-$J$-one-$i$ doctrine
to imaginary time and explains, from spectra, why time has a
shore and space and the fiber do not. Grade: derived
(quasi-free), matching T-P5-B and T-P5-2's register; the
Lorentz completion conditional as flagged. The claim survives
a hostile reading at exactly this calibration: nothing here is
new analysis; what is new is knowing, clause by clause, whose
structure it always was.

\emph{Physics intuition, at length --- what, then, is time?}
Let us stand back, because this small theorem quietly answers a
large question, and the answer is geometric in exactly the
manner this program keeps finding. Ask what the constitution
gives to the direction we call time, and read the gifts off the
axioms one at a time. \emph{AX-1 --- locality and causality ---
gives time its order.} The light cone at every event sorts the
world into before, after, and elsewhere; influence runs one way
along it. This is the endowment Minkowski geometrized in 1908,
the first non-spatial structure ever drawn as geometry --- time
as an ordered direction, not a flowing substance. \emph{AX-2
--- geometry--gauge equivalence --- gives time its repetition.}
Through Theorem 0, AX-1 and AX-2 together force the globally
static arena: the constitutional clock is a Killing flow, a
direction along which the geometry repeats itself exactly ---
and only because time is such a symmetry can frequency, energy,
and the mode planes of this Part exist at all. A time that did
not repeat could not tick. \emph{AX-3 --- the operational
dictionary --- gives time its openness, twice.} Once
irreversibly: the \OGGEP{} projection is one-way, records
accumulate, and the arrow of time is the dictionary's
non-invertibility over a bulk that is itself exactly
T-symmetric (\S\ref{sec:custody}). And once analytically ---
this section's addition: the positivity clause makes the
clock's spectrum one-sided, and a one-sided spectrum opens a
\emph{shore} --- the half-plane of lawful continuation, the
sheltered side where oscillation relaxes into damping, where
the quantum theory keeps its statistical shadow, and where, at
the covariant completion, the CPT theorem lives. Order, from
the cone; repetition, from the equivalence; arrow and shore,
from the dictionary. That is what time \emph{is}, in this
geometry: not a container and not a river, but the unique
direction of the arena that carries all four endowments at
once.

And the uniqueness is the profound part, so let it be said
sharply. Space has translations, but their spectra are
two-sided --- a left-mover is as lawful as a right-mover --- so
space has no orientation the dynamics must respect, no polar
rotor worth naming, no shore: there is no imaginary $x$ because
nothing damps there. The fiber has rotation, but its windings
are two-sided integers on a compact circle --- it knows charge,
not direction (\S\ref{sec:handshake}) --- so the fiber has no
shore either: there is no imaginary $\theta$. Time alone is the
direction along which the world is \emph{stable} --- bounded
below in energy, by the dictionary's own consistency --- and
stability is the whole secret. One-sidedness of the spectrum is
the mathematical name of ``nothing falls forever''; the polar
decomposition turns that stability into a rotor and a speed;
the rotor is the $i$ of every Schr\"odinger equation, the
speed's semigroup is every Euclidean computation, and the
half-plane between them is the only thing Minkowski's $it$ ever
lawfully named. The three mirrors of this paper broke on the
descent's orientation; the three faces of the clock stand on
time's. Indeed, the custody line of this Part extends by one
clause and then closes: the geometry owns the cone, the dial,
and the shore; the dictionary owns the words $it$, $i$, and
Wick. After all, the deepest habit this program unlearns is
the mistaking of names for places --- and time's imaginary
direction, like the sixth dimension itself, turns out to be not
a place one could go but a structure the world cannot do
without: the shore on which a stable universe writes its
quantum mechanics on one side and reads its thermodynamics on
the other.

\section{One \texorpdfstring{$i$}{i} for the whole world}
\label{sec:one-i}

The handshake of the last section settled one mode at a time: on
each charged sector, the circle's quarter-turn and the clock's
agree under exactly one pairing. But a world is not one mode. A
world is composites --- atoms built of charged parts, mesons
built of conjugate pairs; it is many-particle states and the Fock
tower above every field; it is fermions, whose spinor components
entangle the story with spin; and it is, above all,
\emph{entangled} states, whose interference is the signature
phenomenon of quantum mechanics. If each new floor of this tower
demanded its own complex structure, the handshake would be a
curiosity of the ground floor. The two theorems of this section
show the opposite: the shared $J$ climbs the whole tower with no
new input, and spin contributes a real matrix, not a new
rotation. The tower is complex because the ground floor is ---
and for no other reason.

\begin{theorem}[\textbf{T-P5-C1}: one shared $J$ --- composites
and Fock space]
\label{thm:p5-C1}
Let real solution-space sectors carry the shared complex
structure $J$ of T-P5-B/C. Then $J$ extends to direct sums, to
$J$-tensor products, and to the symmetric and antisymmetric Fock
spaces over the one-particle space, and in each case the
extension is the unique one restricting to $J$ on the generating
sectors: the constructions introduce no new complex structure.
The boundary description of composite and many-particle systems
therefore carries a single global complex structure --- in its
own shorthand, one $i$.
\end{theorem}

\begin{proof}
\emph{Sums.} On $V_1 \oplus V_2$ the block operator $J \oplus J$
squares to $-\mathbf 1$, and any complex structure commuting with
the two projections and restricting to $J$ on each summand is
$J \oplus J$: no freedom.

\emph{Products.} On the real tensor product
$V_1 \otimes_{\bR} V_2$ there are two candidates,
$J \otimes \mathbf 1$ and $\mathbf 1 \otimes J$, and a choice
between them would be a genuine new input. But the composite
state space of two systems that share a phase plane is not the
real tensor product; it is its quotient by exactly that
difference --- the $J$-tensor product, in which $Ju \otimes v$
and $u \otimes Jv$ are identified. Physically the identification
is the additivity of phase: turning the joint amplitude a quarter
period \emph{is} turning either factor a quarter period, because
the dial angles of subsystems add. On the quotient the two
candidates are one operator, and it restricts correctly: again no
freedom.

\emph{Fock.} The $m$-particle spaces are the symmetrized or
antisymmetrized $m$-fold $J$-tensor powers, each inheriting its
unique $J$ by the previous two steps, and the Fock sum inherits
$J$ summand by summand by the first step. At no floor of the
tower was a choice available; the full statement, with the
operator domains attended to, is established in the companion
treatment of quantum mechanics \citep{ArisakaA8QM}.
\end{proof}

\emph{Physics intuition --- why entanglement needs the theorem.}
The reader may feel that extending $J$ to composites is
bookkeeping. Entanglement is why it is physics. Take two charged
modes far apart and superpose ``both excited'' with ``neither
excited'' --- a two-particle amplitude interfering with the
vacuum. Interference means the branches' phases turn \emph{in the
same plane}: there must be one quarter-turn that advances the
joint amplitude, not one per subsystem with no relation between
them. If each factor carried its own unrelated structure, the
superposition would have no lawful plane to turn in, and the
interference term --- the entire observable content of
entanglement --- would be undefined. T-P5-C1 says the bulk
supplies exactly the shared plane the phenomenon requires, for
free, from the sector-level handshake. And the requirement has
been put on the bench. The network Bell experiments
\citep{RenouEtAl2021} test precisely whether quantum mechanics
can run on real amplitudes with \emph{unshared} rotation
structures distributed across a network, and the data exclude
it: the boundary formalism demands its single, globally shared
$i$. Note once more, with the custody discipline of Part~III,
what the experiment does and does not see. It tests the boundary
formalism, and the formalism passes only in the one-shared-$i$
configuration; the bulk is what \emph{explains} why that
configuration, and no other, was ever on offer. One experiment,
two hats: the instrument confirms the need; the geometry supplies
the provenance.

\begin{theorem}[\textbf{T-P5-C2}: fermionic conjugation,
factorized]
\label{thm:p5-C2}
On the fermionic sectors of the bulk, charge conjugation
factorizes as
\begin{equation}
C \;=\; (\theta \mapsto -\theta) \,\circ\, B,
\label{eq:C-factorised}
\end{equation}
where the first factor is orientation reversal of the fiber,
acting on weight sectors exactly as for scalars, and $B$ is a
fixed real intertwiner on the spinor indices, independent of the
state and determined by the Clifford algebra. Consequently the
Dirac field's complex structure is the $J$-packaging of real data
--- charge carried by the circle, frequency by the clock, spin by
real matrices --- and spin introduces no new complex input.
\end{theorem}

\begin{proof}
Conjugation must reverse every fiber weight --- that is what
carries a charged field to its conjugate --- and on the spinor
factor it must exchange the two Weyl modules of the split
$\mathbf 8_{\mathrm{Dirac}} = \mathbf 4 \oplus \bar{\mathbf 4}$
of Eq.~\eqref{eq:weyl-split}. The exchange is implemented by the
intertwiner $B$ of the Clifford reality structure --- the same
matrix whose obstruction $B\bar B = -\mathbf 1$ closed the
Majorana option in Lemma~\ref{lem:no-majorana} --- and $B$ is
fixed by the algebra up to conventions, with no dependence on the
state or the dynamics. The weight reversal is the scalar story of
T-P5-A; composing the two gives \eqref{eq:C-factorised}. Since
$B$ is a real matrix, the composite is $J$-linear for the shared
$J$ --- the content of ``no new complex input'' --- and the full
statement at Fock level is again the companion treatment's
\citep{ArisakaA8QM}.
\end{proof}

\emph{Physics intuition --- the same matrix, both jobs.} There is
a small piece of poetry in this proof worth pausing over. The
matrix $B$ appears twice in this paper, in opposite roles. In
Part~II it was the obstruction: $B\bar B = -\mathbf 1$ is the
algebraic fact that \emph{forbids} a $(1,5)$ fermion from being
its own conjugate --- too weak, as it were, to close the loop and
identify a spinor with its reflection. Here it is the
implementation: the same matrix is exactly strong enough to
\emph{carry} each Weyl module onto the other. The map that cannot
make a fermion self-conjugate is precisely the map that exchanges
it with its twin. And with $B$ in hand, the celebrated mystery of
the Dirac field's complexity dissolves by inventory. Eight real
components; a circle that turns the charged pairs (charge); a
clock that advances every oscillation (frequency); one fixed real
matrix (the spin structure); and one shared $J$ wearing the
boundary name $i$. The famous $i$ of the Dirac equation --- the
one in $i\gamma^\mu\partial_\mu\psi = m\psi$ --- is not a fifth
ingredient. It is the same quarter-turn, written where the
boundary's shorthand puts it.

\section{The quartet's second child: spin--statistics, and the
third job of \texorpdfstring{$B$}{B}}
\label{sec:quartet-second-child}

Two theorems of this Part now stand side by side: T-P5-B1, which
isolated the single analytic fact --- the one-sided spectrum, the
clock's quarter-turn --- behind Minkowski's $it$, Wick's rotation,
and the Schr\"odinger $i$; and T-P5-2, which derived CPT from that
fact at quasi-free grade. This section records that the quartet
has a \emph{second} child, and that this paper --- the quartet's
home --- is where the kinship belongs on the page.

The axiomatic literature has known the kinship in its own
language for sixty years. In the Wightman framework
\citep{StreaterWightman}, the same forward-tube analyticity that
the spectrum condition grants is what proves \emph{both} great
structure theorems: CPT, read off at the Jost points
\citep{Jost1957}, and spin--statistics \citep{Pauli1940}, where
the exchange of two spacelike-separated insertions is implemented
by an analytically continued $2\pi$ rotation contributing exactly
the sign $(-1)^{2s}$. The modular refinement says it more
sharply still: the Bisognano--Wichmann structure
\citep{BisognanoWichmann1976} yields the CPT conjugation and the
spin--statistics connection from one wedge. What the literature
lacks --- and what this paper's Part~IV supplies --- is a
geometric parent \emph{for the analyticity itself}: here the
one-sided spectrum is not an axiom but the operational face of
the causal clock (AX-1 positivity, unique by Theorem~0). The
kinship, restated in this paper's vocabulary:

\begin{theorem}[\textbf{T-Stat-A}, restated: spin--statistics as
the quartet's second child (\emph{candidate}; home:
\citealp{ArisakaA2GG6}, \S20)]
\label{thm:quartet-second-child}
For the real quasi-free textures of this Part --- fields built on
the circle $\oplus$ clock $\oplus$ space data of T-P5-A/B/C ---
the spin--statistics connection is forced by the same single
analytic fact as CPT. The clock's one-sided spectrum supplies the
forward-tube analyticity (T-P5-B1); the real orientation
reversals supply the Jost-point geometry (as in the proof of
T-P5-2); and the analytically continued $2\pi$ rotation that
implements exchange contributes $(-1)^{2s}$, the sign of the
bulk's two-valued frame-path register. Consequently CPT and
Fermi--Dirac statistics are the two children of one geometric
parent: \emph{the clock's quarter-turn is why the mirrors
multiply to the identity, and why matter excludes itself.}
Grade: candidate, target derived (quasi-free), with residues
\emph{identical} to T-P5-2's --- the interacting case on
scattering hypotheses, and the modular route open.
\end{theorem}

The statement is printed at candidate grade and stays out of this
paper's claims ledger deliberately: its bulk-side premises ---
the frame-path memory of \GDOS{} states (L-Spin-1), the
genericity clause that populates the spinorial register
(T-Spin-A), and the exchange-equals-rotation lemma for tokens of
one attractor (L-Stat-1) --- are installed, with their residues
named, in the companion section of the geometry paper
(\citealp{ArisakaA2GG6}, \S20), and the QM-register composition
--- the discharge of the textbook symmetrization postulate as
boundary inheritance, theorem T-A8.12 --- in the quantum
mechanics paper (\citealp{ArisakaA8QM}). What belongs
irreducibly to \emph{this} paper is the parent and the first
child: the quartet, and CPT. The candidate falsifier F-Stat-1
(any confirmed violation of Fermi--Dirac statistics or the
spin--statistics connection, at any magnitude) is registered in
the companion section; the standing experimental record --- the
$4\pi$ periodicity of neutron interferometry, the
$\beta^2/2 < 4.8\times10^{-29}$ exclusion bound --- is collected
there and in T-A8.12's anchors.

\emph{The third job of the matrix.} The proof of T-P5-C2 paused
over a small piece of poetry: the intertwiner $B$ does two jobs,
first as obstruction ($B\bar B = -\mathbf 1$ forbids a $(1,5)$
fermion from being its own conjugate --- Lemma~\ref{lem:no-majorana}),
then as implementation (the same matrix carries each Weyl module
onto the other in $C = (\theta \mapsto -\theta) \circ B$). The
poetry has a third stanza. $B\bar B = -\mathbf 1$ is precisely
the statement that the Weyl module is \emph{quaternionic}
(pseudo-real); on a quaternionic module the invariant bilinear
form is antisymmetric --- symplectic --- and the symmetry of the
invariant bilinear is what fixes the quantization sign of the
sector: an antisymmetric invariant forces anticommutators, on
pain of a vanishing or indefinite two-point structure (the
Clifford bookkeeping is lemma L-Stat-2 of
\citealp{ArisakaA2GG6}). The Kramers degeneracy of Part~IV
($T^2 = -\mathbf 1$ on half-integer states) is the boundary
shadow of the same quaternionic structure. So the one matrix
does three jobs: \emph{it forbids the fermion its own mirror, it
carries the fermion to its twin, and it makes the fermion's
tokens exclude one another} --- obstruction, conjugation,
exclusion, all read off one reality structure that the $(1,5)$
signature of the real bulk fixed before any field was written
down. The signature fact that made the descent chiral is the
signature fact that makes matter solid.

\emph{Why this section is short.} Everything load-bearing in it
lives elsewhere by design: the machinery in T-P5-B1 and T-P5-2
above, the bulk lemmas in the geometry paper, the boundary
composition in the quantum mechanics paper. Its business here is
custody: the quartet is this paper's theorem, and a reader of
this paper should learn from this paper that the quartet has two
children --- one derived (CPT, T-P5-2), one at candidate
grade with a named promotion path --- and that both fall to the
same falsifiers, because both are faces of the same one-sided
clock. \emph{The twins were forced by the handshake; the mirrors
multiply to the identity by the quartet; and matter takes up
room by the quartet's second child. One geometry; two theorems
proved here; one candidate named here and owed.}


\section{The self-conjugate exceptions: what the circle cannot
move}
\label{sec:self-conjugate}

Every theorem so far has concerned what the circle moves. It is
time to ask about what it does not --- because the exceptions are
not loose ends; they are the theory probing its own rule. The
photon has no antiparticle distinct from itself. Neither does the
neutral pion, nor the metric's ripples. The textbook records this
as a definition --- ``a neutral particle may be its own
antiparticle'' --- a label attached by hand, species by species.
Here it is a theorem with a mechanism, and the mechanism is the
last clause of T-P5-A: \emph{the circle can only complexify what
it moves}. A weight-zero field does not rotate under the fiber;
on it $\Jq$ acts as zero, its complex structure comes from the
clock alone, and fiber reversal --- which is all that charge
conjugation is --- does not move it:
\begin{equation}
n = 0: \qquad (\theta \mapsto -\theta)\, u \;=\; u .
\label{eq:self-conjugate}
\end{equation}
Such a field is not \emph{permitted} to be its own conjugate; it
is \emph{unable not to be}. And the handshake, which forced a
twin for every charged mode, forces nothing here: with no winding
to match against frequency, there is no gear to align and no twin
to relabel. The existence of antiparticles and the self-conjugacy
of neutral fields are the two values of one criterion --- does
the circle move you? --- where the standard account carries them
as separate conventions.

The inventory checks, with both hats on. The photon is the
boundary field of the circle's own connection --- in the fiber
dictionary of Part~III, the photon carries no dial; it is the
dial's messenger --- and its self-conjugacy comes with a
signature the laboratory confirmed long ago: because the winding
current it couples to reverses under fiber reversal, the field
itself must reverse sign. The photon is its own antiparticle
\emph{with intrinsic C-parity $-1$}. The neutral pion, a bound
pair of conjugate windings, is C-even, and the bookkeeping shows
in its decays: $\pi^0 \to 2\gamma$ dominates, with
$(-1)^2 = +1$ checking, while $\pi^0 \to 3\gamma$ --- allowed by
every conservation law except this one --- has never been seen.
The metric field carries no winding at all, and its ripples are
real for the plainest reason in the paper: they are geometry. In
each case self-conjugacy is not an assignment; it is the absence
of a dial to reverse.

Then there is the exception among the exceptions, and the paper
turns on it. Among the \emph{matter} fields --- the fermions ---
exactly one sits at weight zero: the neutrino. For a fermion,
self-conjugacy is not automatic even at weight zero, because the
spinor structure has its own say --- Part~II's eight-hour clock
permits a Majorana option in four boundary dimensions while
forbidding it in the six-dimensional bulk --- so the neutrino's
situation is subtler and richer: it is the one matter field for
which the question ``are you your own antiparticle?'' is even
grammatical. The weight arithmetic of Part~II
(\S\ref{sec:weight-anomaly}) already contains the seed: the
symmetric square of a weight-$n$ pair contains the
winding-conserving Dirac pairing for every $n$, and a second
invariant --- the self-pairing --- exactly at $n = 0$. Part~VII
sharpens the seed into the theorem this paper calls the dial-free
door (T-P5-4): a Majorana self-pairing is lawful iff the weight
vanishes; the neutrino is the unique such matter field; and the
seesaw structure follows with no dial to turn. Part~VIII then
walks the matter of the universe through that door. The
laboratory hat is on as well: Majorana nature plus the derived
masses and phases give an effective neutrinoless-double-beta mass
$\mbb \simeq 1.7$~meV \citep{ArisakaA9SM} --- below current
sensitivity, within the ambitions of the next-generation
program, and one of the dated falsifiers of Part~XII.

\emph{Physics intuition --- the still point.} Picture the fiber
as a carousel and every charged field as a rider on its platform.
Conjugation spins the carousel backward: every rider sees the
world reverse --- except whoever stands on the axis, for whom
backward and forward are the same motion. The axis is not a
defect of the carousel; it is a consequence of there being a
carousel at all. A rotation must have a still point, and whatever
lives there is its own reflection. The remarkable fact is not the
existence of the still point but what nature chose to put on it:
exactly one matter field --- the lightest, the most elusive, the
only one able to pair with itself --- and, as Part~VIII argues
quantitatively, the field through whose self-pairing the entire
matter content of the universe was minted. After all, if one
wished to design a universe in which the origin of matter hangs
on a single geometric exception, one could do no better than to
place one neutral fermion at the center of the turning circle and
open the one door only it can use.


\begin{figure}[htbp]
\centering
\resizebox{0.83\textwidth}{!}{%
\usebox{\GGplateDoor}}
\caption[Only a neutral particle can be its own opposite.]{The criterion of
Eq.~\eqref{eq:self-conjugate}, applied to the three species a reader will
reach for. A particle and its antiparticle wind around
the phase circle in opposite senses. If the winding is anything other than
zero the two windings differ, and no relabeling can make the particle
identical to its own conjugate --- the electron winds once, every quark
winds a third of a turn, and both doors are welded shut. Only a field the
circle does not move escapes the argument, and winding around this circle is
precisely what electric charge is. Exactly one particle in known matter
survives the test. This is why the origin of matter runs through neutrino
physics and not through the quark sector, and the reason is structural
rather than a matter of degree.}
\label{fig:door-p5}
\end{figure}

\paragraph{The same criterion, drawn.} Figure~\ref{fig:door-p5} is
Eq.~\eqref{eq:self-conjugate} with three species substituted into it, and it
is worth a page because the conclusion is stronger than the textbook's and
is usually met as a weaker one. The textbook says a neutral particle
\emph{may} be its own antiparticle: a permission, granted species by
species. The criterion here grants nothing and forbids a great deal. A
charged field is not \emph{disallowed} from being self-conjugate by some
further rule; it is \emph{unable}, because the circle moves it and moving it
is what having charge means. The Majorana option is not one modeling choice
among several. It is the only door the geometry leaves open, and it opens
onto exactly one species of known matter.

\section{The bulk that cannot hide its conjugates}
\label{sec:bulk-encoding}

The handshake forces the \emph{pairing}. Independently --- and
this bears emphasis, because the two arguments could each fail
without the other --- the bulk's representation theory forces the
conjugate \emph{content}, twice over.

\emph{First forcing: the spinor algebra.}
Lemma~\ref{lem:no-majorana} did the work in advance: in $(1,5)$
signature the Dirac spinor splits into the conjugate Weyl pair
$\mathbf 4 \oplus \bar{\mathbf 4}$ of Eq.~\eqref{eq:weyl-split},
the chiral representation is complex, and no Majorana condition
exists, so a bulk Weyl fermion and its
conjugate are genuinely independent objects --- there is no
reality condition available that could identify a charged fermion
with its own reflection. The matter of the \GG{} bulk is
packaged as left-handed Weyl spinors (the $\mathbf 4$ of
$\mathrm{Spin}(1,5)$) tensored with
$\bar{\mathbf 5} \oplus \mathbf{10}$ of $\mathrm{SU}(5)$ per
generation, \emph{with an analogous right-handed structure for
the conjugate sector} \citep{ArisakaA2GG6}: the antiparticle
content is a separate, present module of the bulk, by algebra.

\emph{Second forcing: the anomaly ledger.} The anomaly flip
\eqref{eq:anomaly-flip} makes chirality a signed currency, and
the \BIsix{} books cannot balance in one sign alone
\citep{ArisakaA3BI6}. It is worth walking the arithmetic in three
steps, because ``the books force conjugate content'' is the kind
of claim a reader should watch happen rather than take on faith.

\emph{Step 1: the brick already contains conjugates.} The
elementary building block of the matter spectrum --- the core
block --- is
\begin{equation}
\mathcal{C}^{\sigma} \;=\;
\sigma\left[\,\mathbf{10} \oplus \mathbf{5} \oplus
2\cdot\bar{\mathbf{5}} \oplus 11\cdot\mathbf{1}\,\right],
\label{eq:core-block}
\end{equation}
one chirality $\sigma$ carrying a fixed $\mathrm{SU}(5)$ content
that mixes unbarred and barred representations: the two
$\bar{\mathbf 5}$'s sit inside the brick, next to the
$\mathbf{10}$ and the $\mathbf{5}$. Conjugate representations are
not an addition to the elementary unit; they are part of its
definition.

\emph{Step 2: the solution flips a chirality.} Balancing the full
anomaly polynomial over the orbit reservoir is a Diophantine
problem --- integer multiplicities, fixed coefficients, no dials
--- and its solution returns one negative coefficient,
$d_2 < 0$. A negative number of fields is not physics; the unique
physical reading is supplied by the flip
\eqref{eq:anomaly-flip}:
\begin{equation}
d_2\, D_2^{+} \;=\; |d_2|\, D_2^{-} \, ,
\label{eq:d2-flip}
\end{equation}
a negative multiplicity of positive-chirality fields \emph{is} a
positive multiplicity of the opposite-chirality species. The
physical reservoir that results,
\begin{equation}
(8,\, 4,\, 48,\, 12,\, 2)\cdot\frac{\Omega}{25}
\qquad \text{(fourth orbit of negative chirality)},
\label{eq:reservoir}
\end{equation}
has no negative multiplicity anywhere; what it has instead is a
\emph{definite mixed-chirality pattern}. The ledger's arithmetic
cannot be satisfied by one chirality, and the solution says
exactly which orbit must carry the other sign.

\emph{Step 3: gravity forces a conjugate sea.} Closing the
gravitational sector of the same ledger forces, in addition,
\begin{equation}
n_F \;=\; 32\,\Omega
\label{eq:singlet-sea}
\end{equation}
gauge-singlet Weyl fermions of chirality \emph{opposite} to the
visible reservoir --- eight for every self-dual tensor the
closure employs, $n_F/n_{SD} = 8$ --- whose members admit Dirac
pairings (vector-like, liftable). None of this consulted an
observation, and no field was added by hand: the
opposite-chirality content is demanded by the arithmetic of
anomaly freedom, executed on integers.

\emph{A distinction that must never be blurred.} The two
forcings supply two logically different kinds of ``antiparticle-%
like'' content, and a hostile reader will conflate them if the
paper does not forbid it here. The representation conjugates
(the $\bar{\mathbf 4}$ partners, the barred reps) are what
descend to the Standard Model's antiparticles --- every charged
species' twin. The anomaly-forced opposite-chirality
\emph{species} ($D_2^{-}$, the singlet sea) are independent
chiral fields whose opposite anomaly sign lets the books
balance; they are not ``the antiparticles of'' the visible
matter. The bulk contains conjugate content in both senses; only
the first sense is the antiparticle of Q1; and the cohort's
papers keep the two ledgers separate throughout.

\emph{Physics intuition.} Put the two forcings beside the
handshake and a satisfying closure appears. The handshake says:
\emph{if} charged matter exists, its modes must pair as particle
and antiparticle --- a statement about orientations. The algebra
says: the charged matter that exists cannot be its own conjugate
--- a statement about representations. The ledger says: the
spectrum as a whole must carry both chirality signs --- a
statement about integers. Three independent structures, none
adjustable, all pointing the same way: a \GG{} bulk without
antiparticle content is not merely un-observed, it is
\emph{unwritable}. That is what ``naturally and inevitably''
means in question (Q1) of \S\ref{sec:anti-question}, and no weaker claim would deserve the
words.

\section{What the descent does, and does not do}
\label{sec:descent-relabels}

Question 2 asked whether antiparticles emerge after \GG{} is
decomposed to the Standard Model. The answer, now provable in
one paragraph: \textbf{the decomposition creates nothing; it
relabels.} In the boundary description one Standard-Model family
is written entirely in the left-handed Weyl convention, with the
right-handed particles appearing as left-handed conjugates ---
$u_R^c$ in $(\bar{\mathbf 3}, \mathbf 1, -\tfrac23)$, $d_R^c$,
$e_R^c$ --- and charge conjugation realized as the same fiber
reversal $\theta \mapsto -\theta$ it always was
\citep{ArisakaA9SM}. The mode-level pair is the $\pm n$ winding
pair on the fiber, exactly the pair of
Eq.~\eqref{eq:mode-plus}'s proof. What \emph{is} new at or after
the descent is threefold, and each novelty is operational, not
ontological. (i) After electroweak breaking, the Yukawas pair
each left field with its conjugate partner into massive Dirac
fermions --- the familiar sense in which particle and
antiparticle content ``merge'' into one massive field; $d_R$ and
$d^c$ are two descriptions of the same degree of freedom.
(ii) The one exception is the field the circle cannot
distinguish from its conjugate: the weight-zero neutrino,
uniquely permitted a Majorana self-pairing (T-P5-4), the door of
Part~VIII. (iii) The particle-versus-antiparticle \emph{label}
--- which twin an observer calls matter --- is a superselection-%
sector assignment, and sectors are boundary dictionary data.

\emph{Physics intuition --- why ``relabeling'' is the deep
answer, not the deflationary one.} It may seem that ``the descent
only relabels'' demotes the question. The opposite is true.
Consider what the alternative would require. If antiparticles
were created at the descent, the descent would need creation
machinery --- some step at which a spectrum without conjugates
acquires them --- and that machinery would face the full force of
the question this Part opened with: why a twin for every charged
species, with exactly equal mass, and not some other arrangement?
No such machinery exists anywhere in the construction, and none
is needed, because the bulk spectrum never lacked the conjugates:
the algebra could not write them out
(\S\ref{sec:bulk-encoding}), and the handshake had already
paired them (\S\ref{sec:handshake}). What the descent does is
what a customs office does at a border: it inspects, sorts, and
relabels the goods; it manufactures nothing. The left-handed
convention, the $u_R^c$ notation, the Dirac pairing at
electroweak breaking --- each is a repackaging of content the
bulk supplied, very much like a chord renamed when the clef
changes: the notes do not move. Indeed, the reader can run the
audit in reverse. Every property of the boundary twins that is
\emph{structural} --- existence, pairing, equal masses, conjugate
charges --- traces upstairs; every property that is
\emph{conventional} --- which twin is called matter --- traces to
the dictionary. Which brings us to the readout.

\section{The boundary that reads the twins}
\label{sec:boundary-readout}

Three episodes from the cohort's boundary physics show the
operational layer at work. Each is a classic puzzle; each
dissolves the same way; and the reader who watches all three
dissolve will come to trust the custody principle as a tool
rather than a slogan.

\subsection{The positron as conjugate readout}
\label{ssec:positron-readout}

The electron paper of the program derives the electron as a
unique attractor basin --- one stable resting place for charged
matter at that level of the landscape, its properties fixed by
the geometry \citep{ArisakaP3Electron}. Uniqueness claims invite
their own falsification, and here the candidate falsifier walks
into every detector on Earth: the positron. If the framework
permits exactly one basin, what is this second particle --- equal
mass to extraordinary precision, opposite charge, predicted on
paper in 1931 and found in a cloud chamber the following year? The
resolution is the handshake, applied: \emph{the positron is the
conjugate readout of the same basin --- same level (mass),
opposite winding eigenvalue (charge) --- exactly as CPT requires;
what the framework forbids is a second basin, not a second
orientation}. One object, two orientations of reading it.

Three consequences repay attention. First, the mass equality of
electron and positron is not \emph{enforced} between two objects;
it is \emph{trivial} for one object --- where there are not two
things, nothing can differ. This is Depth~3 answering a question
Depth~2 can only constrain: the CPT theorem bounds the mass
difference of the pair; the readout picture abolishes the
difference by abolishing the pair --- the same economy by which a
string's conjugate excitations are automatically degenerate, and
for the same structural reason. Second, the charge flip is the
\emph{definition} of the second readout, not a property that
might have been otherwise: the conjugate readout is the reversed
winding, by T-P5-C2's factorization. Third, the register must
record a debt: the promotion of this picture to a proved
corollary of the bulk spinor structure --- ``the positron, as a
theorem'' --- is a registered obligation of the program (its
label there: D-P3-4), carried here honestly as owed rather than
claimed. What is claimed is narrower and firm: nothing in the
positron's measured properties requires a second basin, and the
conjugate-readout reading uses only structures that are theorems
at their stated grade. The reading now runs in both directions:
the electron paper devotes a dedicated Part --- its Part~V, the
origin of positrons, the sixth of its questions --- to importing
the origin of the twin at theorem grade from the present paper
and closing that question as the conjugate readout developed
here \citep{ArisakaP3Electron}.

\subsection{Wheeler's one-electron universe, rejected and
preserved}
\label{ssec:wheeler}

One day in 1940, John Wheeler telephoned his graduate student
with an idea too beautiful to keep: perhaps there is only
\emph{one} electron in the universe, threading forward and
backward through time --- its forward strands the world's
electrons, its backward strands the positrons --- which would
explain, in a single stroke, why every electron has exactly the
same mass and charge. The student was Richard Feynman, and half
of the idea stayed with him: the reinterpretation of positrons as
electrons running backward became, nine years later, the
propagator picture of his positron theory \citep{Feynman1949}.
The cosmology did not survive the telephone call. As Feynman
objected at once, a single worldline must return as often as it
departs: the picture predicts as many positrons as electrons, and
the universe conspicuously declines to supply them. (Wheeler had
an answer --- perhaps they hide in the protons --- to which
history has not been kind.) Nor has the one-electron universe any
account of net lepton number.

The present reading preserves what was right in the idea in the
only form that survives the objection. What is unique is not the
particle's \emph{worldline} but the \emph{attractor it reads
out}: all electrons are alike, not because one particle visits us
many times, but because every electron is a readout of one basin
--- and the positron is the same basin read with the fiber
reversed. Sameness is explained with no bookkeeping of strands,
and no symmetric-universe prediction follows, because readouts
are not worldline segments: how many of each orientation the
universe contains is a question about its \emph{history} ---
Part~VIII's question --- not about the basin. Wheeler, one might
say, filed the sameness of electrons under the wrong custodian:
he assigned it to the geometry of one worldline when it belongs
to the uniqueness of one attractor. Move the file, and the absurd
prediction evaporates while the explanation survives.

\subsection{The Hawking partner, dissolved}
\label{ssec:hawking-partner}

The popular picture of Hawking radiation is a pair-creation
story: a particle--antiparticle pair fluctuates into being at the
horizon; one twin falls in while the other escapes; the escaping
flux is the radiation, and the infalling twin books the negative
energy. The picture is vivid, useful for rough arithmetic --- and
notoriously treacherous when reified: treat the interior partner
as a real, independent quantum, permanently entangled with its
exterior twin, and one is soon assembling the firewall-flavored
paradoxes of the modern literature. The black-hole companion
paper of the program derives the thermality of the radiation
with no pair story at all: the horizon's boundary description is
the partial trace of a two-mode-squeezed \emph{bulk} state ---
the standard squeezed correlation between exterior and interior
modes, read through the boundary's one-way projection --- and
tracing the interior mode yields the exact thermal spectrum
\citep{ArisakaC3BH}. In that derivation the ``interior partner
particle'' never appears as an independent antiquantum: it is
part of the traced-over 6D bulk --- structure, not population.
The paradoxes that come from reifying the partner do not arise,
and for the same reason no paradox arises from the two readouts
of the electron basin: a readout is not a resident.

The pattern of the three episodes is now visible, and it is the
custody principle of \S\ref{sec:custody} doing its work three
times over: \emph{the dictionary owns the sectors; the geometry
owns the} $i$. The pairing --- the antiparticle as structure ---
is bulk geometry, proved in \S\ref{sec:handshake} and never in
danger in any episode. The label --- the antiparticle as
something an observer registers, counts, or reifies --- is a
boundary port assignment. And each of the three puzzles was
manufactured by promoting a label to a structure: a second basin
for the positron, a worldline for the sameness of electrons, an
independent quantum for the interior partner. Keep each item with
its owner, and there is nothing left to be puzzled by.

\section{The negative results, recorded}
\label{sec:anti-negatives}

Three silences matter as much as the theorems, and an audit that
omitted them would flatter the case. \emph{First}, the boundary-%
law and gauge-runtime papers of the cohort contain no
antiparticle mechanism at all --- verified by exhaustive search
--- and this is as it should be: on the custody principle the
boundary owns readout, not origin. \emph{Second}, the bulk's
quaternionic structures are spent elsewhere: the three
commuting complex structures of the $\mathbb{H}^2$ spinor
geometry buy the \emph{three families}, not particle--%
antiparticle doubling \citep{ArisakaA6Gi4}. The complex-%
structure budget is spent twice on two different questions ---
circle-and-clock on the single $i$ and the pairing;
the quaternionic triple on $N = 3$ --- and no cohort paper
conflates the two purchases. \emph{Third}, nowhere in the
twenty-two-paper cohort does a Dirac sea, a filled vacuum, or a
zitterbewegung picture appear: the account is Feynman--%
St\"uckelberg from top to bottom, with the relabeling now
derived rather than imposed.

\section{The answer, in one paragraph}
\label{sec:price-of-one-i}

Antiparticles do not emerge from the decomposition of \GG{} to
the Standard Model --- they are older than the decomposition and
older than quantization. The theory has exactly two sources of
complex structure, both real and both constitutionally forced:
the compact sixth-dimension circle, which complexifies charged
sectors but does not know its own orientation, and causal time
with energy positivity, which supplies the unique positive-%
frequency complex structure --- the Hilbert transform --- and
does know its direction. These two structures must act as the
same imaginary unit on every charged mode, and there is exactly
one way to arrange that: pair charge $+n$ with positive
frequency and charge $-n$ with positive frequency --- the
Feynman--St\"uckelberg assignment. That forced pairing \emph{is}
the antiparticle. Charge conjugation is nothing but orientation
reversal of the circle; the 6D spinor algebra guarantees the
conjugate content is genuinely there in the bulk, since no
Majorana condition exists in $(1,5)$ to identify a charged
fermion with its conjugate; and the anomaly ledger of \BIsix{}
independently forces opposite-chirality species into the
spectrum. What the descent to the Standard Model adds is only
the operational layer: the boundary dictionary labels which
sector an observer calls ``particle,'' the positron is the
conjugate readout of the same electron basin, and the one field
the circle cannot distinguish from its conjugate --- the
weight-zero neutrino --- is the one allowed to be its own
antiparticle, which is precisely the door through which the
matter--antimatter asymmetry of the cosmos is later generated.
\emph{Antiparticles, in this framework, are the price of having
one $i$ instead of two --- where ``$i$,'' here as everywhere, is
the boundary's name for the bulk's one real quarter-turn.}

\section{Originality ledger for this Part}
\label{sec:anti-ledger}

Concessions first, in full; claims after, ranked --- the
register's order. \emph{Not new here:} the Feynman--%
St\"uckelberg pairing itself (1941/1949 --- ``the pairing is
textbook; its provenance is the claim''); the slogan
``antiparticles as a consistency condition,'' which Weinberg's
causality argument already embodies at the structural level
(stated in full in Part~X); conjugation as orientation
reversal of a compact circle, whose \emph{image} is old
Kaluza--Klein lore made vivid by string theory's oriented
strings (conceded in Part~X with its sources) --- though the
\GG{} circle is categorically different, a non-spatial phase
dimension with no radius, no tower, and no landscape
\citep{ArisakaA2GG6}; the positive-energy complex
structure, which is Ashtekar--Magnon and Kay's theorem,
promoted here to constitutional status; and the real-quantum-%
mechanics lineage from St\"uckelberg's 1960 program through the
modern exclusion experiments. \emph{New here or in the parent
theorems this Part consolidates, ranked by confidence:}
(1)~the two-source compatibility theorem itself --- no prior
framework possesses \emph{two independently derived} real
structures whose forced agreement is the pairing, so no prior
framework could derive the Feynman--St\"uckelberg assignment
before quantization; KK has the circle but borrows its
state-space $i$, Ashtekar--Magnon has the clock but no circle to
shake hands with; (2)~the unification --- ``why is quantum
mechanics complex?'' and ``why do antiparticles exist?'' made
one question with one answer; (3)~the custody division of the
antiparticle concept into bulk pairing and boundary label,
with its three dissolved puzzles; (4)~the double bulk-forcing
of conjugate content (algebra plus ledger), which upgrades
``natural'' to ``unwritable otherwise.'' \emph{Scope, stated
plainly:} the handshake is proved at the free/quasi-free grade;
interacting matter rides on the asymptotic frame,
conditionally; and every sentence of this Part inherits that
qualifier. \emph{Depth, stated on the Part~II ruler
(\S\ref{sec:three-depths}):} the claims of this Part live at
Depth~3 --- what the twins \emph{are}, and what the pairing is
made of --- with Depth~1 (the spectrum pairs) and Depth~2 (it
must pair) recovered as corollaries of the handshake rather than
assumed; no rival framework, Part~X will argue, competes at that
depth, and the register above is the price of admission. The comparison of these claims against QFT, the
Standard Model, string theory, and $\Lambda$CDM --- including
Weinberg's theorem, the Dirac sea's rise and fall, and the
string-theoretic precedents --- is the business of Part~X,
where each rival is given its full say.

\medskip
\noindent\emph{Part-end summary.} Two half-blind sources of one
quarter-turn; one forced alignment, extended without new input to
composites, Fock space, and the Dirac fermion; lawful exceptions
exactly where the circle stands still; a bulk that cannot be
written without its conjugates, twice over; a descent that
relabels; a boundary that reads. The twins exist because the
world must agree with itself. And because they exist --- and
because, as the coming Parts show, the mirrors that exchange them
broke selectively, by computable amounts, in a universe whose
ledger kept six parts in ten billion --- there are atoms rather
than annihilation light, chemistry rather than silence, and
readers rather than none. Now that the twins are on stage, the
operations that exchange them can be defined and derived. That is
the next Part.

\GGPart[The mirrors defined]{The Mirrors Defined: C, P, T and the CPT Theorem}{The three reversals given bodies and walked one at a time; the table from orientation; Kramers for free; CPT as total reversal, blind to every real texture; and the five fates of the mirrors, derived}
\label{part:theorems}

\section[Part~V in plain English]{Part~V in plain English: the three mirrors, given bodies}
C, P and T are normally introduced as operations on fields: rules with signs, justified by what they produce. This Part gives them bodies instead. Each is the reversal of the orientation of one real factor of the arena --- the circle, space, and the clock --- and the operations follow from the reversals rather than the other way about.

Two results come out. The discrete-symmetry table is derived rather than assembled, including the fork that makes T antiunitary while C and P are unitary, which in the textbook account is a choice justified after the fact. And CPT is total orientation reversal: the one reflection a real texture cannot see, which is why it survives while each of its parts fails.

The Part ends by separating what is claimed from what is inherited, as every mirror Part in this paper does, and the separation matters most here. The relation to Jost's theorem is complementarity, not replacement: Jost's scope exceeds the geometric account's, which is quasi-free with the interacting case conditional on scattering hypotheses. The Part says that in those words rather than leaving a reader to discover it.

\section{The operations, given bodies}
\label{sec:mirrors-question}

Part~I told the story of three symbols; this Part finally says
what they name. It is worth pausing on how strange the standard
situation is, because familiarity has hidden it. For seventy
years, C, P, and T have been used daily by every particle
physicist alive --- assigned as quantum numbers, tested in decay
after decay, broken in headlines --- and yet the formalism that
uses them declines to say what they \emph{are}. In the standard
development each is an operator defined by its action on creation
and annihilation operators, \emph{up to intrinsic phases chosen
field by field} \citep{StreaterWightman}: ``the'' parity operator
is really an equivalence class of operators, and the tables of
intrinsic parities and C-parities record conventions as much as
facts. The antiunitarity of T is not derived but selected ---
Wigner's theorem permits either character, and antiunitarity is
the choice that spares the spectrum
(\S\ref{sec:orientation-reversals}). None of this is a defect of
rigor; the formalism is exact, and its predictions are the
best-tested in science. It is a silence of \emph{anatomy}.
Nothing in the definitions says what the operations are made of
--- and therefore nothing in them could have warned, before 1956,
that two of the three would fail, or explained, after 1964, why
the failures came in that order and at those sizes.

This Part removes the silence, with the tools Parts~II--IV built.
The three operations will be three specific real maps ---
reversals of the three orientations the bulk possesses --- walked
one at a time on the very modes Part~IV constructed
(\S\ref{sec:three-reversals}). The unitary/antiunitary table will
then be a one-line theorem rather than a case analysis (T-P5-1),
and it will immediately pay dividends the standard formalism
buys separately: Kramers' degeneracy with no phase conventions,
and the reason T has no quantum number of its own
(\S\ref{sec:table-dividends}). The CPT theorem will be
bookkeeping over the anatomy (T-P5-2). And the anatomy will then
answer the question the standard account cannot pose: \emph{why
did the mirrors fall selectively --- these ones and not those, in
that order, by those amounts?} The five fates of the mirrors
(\S\ref{sec:five-fates}) are the Part's harvest, and the Part
closes by pricing exactly what in all this is new
(\S\ref{sec:mirrors-ledger}).

\section{The three reversals, walked one at a time}
\label{sec:three-reversals}

Before any theorem, let us simply watch each operation act. The
stage is Part~IV's: real mode planes, the quadrant map of
Figure~\ref{fig:quadrants}, and the two kept (green) cells ---
the particle plane spanned by $\cos(n\theta - Et)$,
$\sin(n\theta - Et)$ and the antiparticle plane spanned by
$\cos(-n\theta - Et)$, $\sin(-n\theta - Et)$.

\subsection{C: the fiber run backward}
\label{ssec:walk-C}

Charge conjugation is the reversal $\theta \mapsto -\theta$ ---
for fermions composed with the fixed real spinor map $B$ of
T-P5-C2, which we may set aside here since it is
state-independent. Apply it to the particle plane's basis:
\begin{equation}
C:\quad
\cos(n\theta - Et) \,\mapsto\, \cos(-n\theta - Et),
\qquad
\sin(n\theta - Et) \,\mapsto\, \sin(-n\theta - Et).
\label{eq:C-walk}
\end{equation}
Both basis vectors land squarely on the antiparticle plane's
basis, in order, with no signs: C carries one green cell of
Figure~\ref{fig:quadrants} onto the other, \emph{preserving the
turning sense} --- the particle's forward rotation maps to the
antiparticle's forward rotation. An operation that preserves
every plane's turning sense commutes with the quarter-turn: this
is the observation T-P5-1 will turn into ``C is unitary.'' On the
couplings, the same reversal negates every fiber angle,
$\varphi \mapsto -\varphi$ --- the fact that will decide CP's
fate in Part~VII. And on the weight-zero sector C acts as the
identity (\S\ref{sec:self-conjugate}): the still point of the
carousel, where the C-parities $\pm1$ live as honest quantum
numbers --- the photon's $-1$, the neutral pion's $+1$, each
derived there rather than assigned.

\subsection{P: the hands exchanged}
\label{ssec:walk-P}

Parity is the reversal of the spatial orientation,
$\vec{x} \mapsto -\vec{x}$. On the fiber--clock dial angle
$n\theta - Et$ it does nothing at all --- the walk is, on these
coordinates, a standstill --- and that quiet is precisely its
content: P touches neither source of the quarter-turn. Where P
acts is on the suppressed spatial and spin data: it reverses
momenta, and it \emph{exchanges the two Weyl modules} --- in
four-dimensional language, the left-handed $(\tfrac12, 0)$ and
right-handed $(0, \tfrac12)$ representations
(\S\ref{sec:spinor-algebra}). This single structural fact carries
all of parity physics. A theory treats P as a symmetry if and
only if its two hands carry the same charges --- a
\emph{vector-like} theory; it violates P if and only if the two
hands are charged differently --- a \emph{chiral} theory. The
intrinsic parities of the tables (scalar even, pseudoscalar odd,
vector odd, axial even) are the bookkeeping of how each field's
real components contract with the spatial orientation. Nothing
here is new physics; what is new is what Part~VI will do with it:
in the \GG{} descent, one hand of the spectrum is not
differently charged but \emph{absent}, and P will fail for the
strongest possible reason.

\subsection{T: the clock run backward}
\label{ssec:walk-T}

Time reversal is the reversal $t \mapsto -t$ --- the operation
that plays every film backward. Apply it to the particle plane:
\begin{equation}
T:\quad
\cos(n\theta - Et) \,\mapsto\, \cos(n\theta + Et).
\label{eq:T-walk}
\end{equation}
Look at where the image lies on Figure~\ref{fig:quadrants}: in
the \emph{clash quadrant} --- exactly the planes on which
Part~IV proved the circle's quarter-turn and the clock's
disagree, the planes the handshake outlawed for any $J$-linear
physics. This is the geometric heart of time reversal's
strangeness, and it deserves a slow sentence. The clock reversal
cannot be read as a $J$-linear map at all: its image escapes the
kept world unless the turning sense is conjugated along the way
--- $T \Ja T^{-1} = -\Ja$ --- and an operation that conjugates
the quarter-turn is, by the gear rule of Part~II, antiunitary.
Antiunitarity is not a property assigned to T for the spectrum's
sake; it is \emph{the only exit from the clash quadrant}. The
laboratory has long known T's strangeness operationally: unlike
C and P, time reversal furnishes no quantum number to tabulate
(\S\ref{sec:table-dividends} derives why), so it is tested not by
selection rules but by asymmetries and by static moments ---
matters this paper's Parts~I and~IX take up with the
instruments. Hold, for now, the single geometric fact: of the
three reversals, only the clock's collides with the handshake.

The three walks are done, and the reader has seen everything the
table theorem needs: C hops between the green cells with the
turning sense preserved; P stands still on the dial and exchanges
the hands elsewhere; T walks into the forbidden quadrant and must
conjugate the turning to survive. The theorem now only records
this.

\section{T-P5-1: the discrete-symmetry table}
\label{sec:table}

\begin{theorem}[\textbf{T-P5-1}: the table from orientation]
\label{thm:p5-1}
On the quasi-free real bulk, let $C$, $P$, and $T$ act as orientation
reversals of the fiber circle, the spatial slice, and the causal
clock respectively. Then: $C$ is $J$-linear --- it reverses
$\partial_\theta$ and the signed charge together, leaving
$\Jq = \partial_\theta/Q$ invariant; $T$ is $J$-antilinear --- it
reverses the polar phase of the time generator, $T \Ja T^{-1} =
-\Ja$; $P$ is $J$-linear --- it touches neither fiber nor clock.
Hence a discrete symmetry is antiunitary if and only if it contains
$T$: the textbook unitary/antiunitary table is derived, not
assigned.
\end{theorem}

\begin{proof}
Under $\theta \mapsto -\theta$ the weight-$n$ sector maps to the
weight-$(-n)$ sector and $\partial_\theta \mapsto -\partial_\theta$;
since $\Jq$ is normalized by the \emph{signed} charge,
$\Jq \mapsto (-\partial_\theta)/(-Q) = \Jq$: $C$ commutes with the
sector complex structure, and by the handshake (T-P4-3) with the one
global $J$. Under $t \mapsto -t$ the frequency orientation flips:
the polar phase of the generator (the Hilbert transform in
constitutional time) is odd under clock reversal, so
$T \Ja T^{-1} = -\Ja$. $P$ reverses the spatial orientation, which
enters neither $\Jq$ (a fiber object) nor $\Ja$ (a time-domain
object): $P$ commutes with $J$. (Anti)linearity multiplies under
composition like a sign (Section~\ref{sec:orientation-reversals}),
so a product of reversals is $J$-antilinear iff it contains the one
$J$-anticommuting factor --- iff it contains $T$.
\end{proof}

\begin{figure}[t]
\centering
\resizebox{0.604\textwidth}{!}{%
\begin{tikzpicture}[font=\sffamily\small, >=Stealth]
 % C: circle
 \begin{scope}[shift={(0,0)}]
  \draw[line width=0.80pt] (0,0) circle (0.75);
  \draw[->, line width=1.10pt, blue] (0.75,0) arc (0:120:0.75);
  \draw[->, line width=1.10pt, red, dashed] (0.75,0) arc (0:-120:0.75);
  \node at (0,-1.35) {\textbf{C}: fiber reversal};
  \node[font=\footnotesize] at (0,-1.8) {$\theta \mapsto -\theta$;\ \ $\Jq \mapsto \Jq$};
  \node[font=\footnotesize, blue] at (0,-2.25) {$J$-linear (unitary)};
 \end{scope}
 % P: space
 \begin{scope}[shift={(4.6,0)}]
  \draw[->, line width=1.10pt, blue] (-0.9,0) -- (0.9,0) node[right]{$\vec{x}$};
  \draw[->, line width=1.10pt, red, dashed] (0.55,-0.3) -- (-0.9,-0.3) node[left]{$-\vec{x}$};
  \draw[line width=0.80pt] (0,0.5) -- (0,-0.6);
  \node at (0,-1.35) {\textbf{P}: space reversal};
  \node[font=\footnotesize] at (0,-1.8) {touches neither source};
  \node[font=\footnotesize, blue] at (0,-2.25) {$J$-linear (unitary)};
 \end{scope}
 % T: clock
 \begin{scope}[shift={(9.2,0)}]
  \draw[line width=0.80pt] (0,0) circle (0.75);
  \draw[line width=0.80pt] (0,0) -- (0,0.55);
  \draw[->, line width=1.10pt, blue] (0,0) -- (25:0.6);
  \draw[->, line width=1.10pt, red, dashed] (0,0) -- (155:0.6);
  \node at (0,-1.35) {\textbf{T}: clock reversal};
  \node[font=\footnotesize] at (0,-1.8) {$t \mapsto -t$;\ \ $\Ja \mapsto -\Ja$};
  \node[font=\footnotesize, red] at (0,-2.25) {$J$-antilinear (antiunitary)};
 \end{scope}
\end{tikzpicture}}
\caption[The three orientation reversals of T-P5-1.]{The three orientation reversals of T-P5-1. The fiber
reversal flips the rotation generator and the signed charge
together, leaving $\Jq = \partial_\theta/Q$ invariant; the space
reversal touches neither source of $i$; the clock reversal is the
one flip the imaginary unit is made of. The unitary/antiunitary
table is the map of which reversals touch the sources.}
\label{fig:reversals}
\end{figure}

\emph{Physics intuition.} Why is time reversal the odd one out ---
antiunitary, where C and P are unitary? Because of the three
orientations a discrete symmetry can flip, \emph{only the clock's is
what the $i$ is made of}. Flip space, and nothing happens to $i$.
Flip the fiber, and the charge label flips with the rotation
generator, so $i$ survives the double negative. Flip the clock, and
``a quarter period ahead'' becomes ``a quarter period behind'' ---
$i$ becomes $-i$, which is antilinearity. Wigner's theorem
\citep{Wigner1931} offers the fork (unitary or antiunitary) and
standard quantum mechanics chooses by consistency; here the choice
is a computation, because $i$ has sources and the table is the map
of which reversals touch them. This is the paper's smallest theorem
and, in the author's view, its purest: a fact memorized by every
graduate student for ninety years, turned into one line of
orientation bookkeeping.

The theorem also fixes the operations' \emph{anatomy} in a way the
standard formalism cannot. In QFT the discrete operators carry
per-field phase conventions, adjustable within wide limits
\citep{StreaterWightman}; here C \emph{is} $\theta \mapsto -\theta$
composed (for fermions) with a fixed real spinor map
\citep{ArisakaA8QM}, P \emph{is} the spatial reversal, T \emph{is}
the clock reversal --- three specific real maps, no convention
anywhere. When Part~VI speaks of ``which hand survives the
projection'' and Part~VIII of ``CP-biased decays,'' the operations
being violated are these three maps, not representatives of
equivalence classes.

\section{First dividends of the table}
\label{sec:table-dividends}

A good theorem pays immediately. Here are two payments the
standard formalism makes separately, with conventions, and the
table makes free of charge.

\begin{corollary}[Kramers' degeneracy, with no phase choice]
\label{cor:kramers}
On fermionic sectors, $T$ composes the clock reversal with the
fixed real spinor map of the bulk construction, and the
composite squares to $-\mathbf 1$ on half-integer spin. In any
$T$-symmetric dynamics, every energy level of a half-integer-spin
system is therefore at least doubly degenerate: no state is its
own $T$-image.
\end{corollary}

\begin{proof}
The clock factor squares to $+\mathbf 1$ (two reversals make a
forward), and the fiber is untouched; the spinor factor is the
fixed real intertwiner whose square is $-\mathbf 1$ on
half-integer spin --- the same algebra, used at the same office,
as the map $B$ of T-P5-C2 \citep{ArisakaA8QM}. Hence
$T^2 = -\mathbf 1$. Now suppose $T\psi = \lambda\psi$ for some
state $\psi$ and scalar $\lambda$. Applying $T$ again and using
antilinearity, $T^2\psi = \bar\lambda\lambda\,\psi =
|\lambda|^2\psi \neq -\psi$: contradiction. So $\psi$ and
$T\psi$ are distinct states of the same energy, and the level is
paired. No intrinsic phase was chosen at any step ---
Wigner's two classes $T^2 = \pm 1$ \citep{Wigner1931} are here a
computation on the real spinor algebra (problem D.1.3 of
Appendix~\ref{app:d-wigner} drills it).
\end{proof}

\emph{Physics intuition, and the laboratory face.} Kramers'
theorem is the quiet workhorse of atomic and solid-state physics:
every level of an odd-electron atom, ion, or molecule comes in
protected pairs that no electric field, however violent, can
split --- only a magnetic field, which breaks T itself, lifts
them. In the standard development this protection descends from a
phase-convention-laden analysis; here it is three squares
multiplied together. The reader should feel the economy: the same
real spinor map that could not make a fermion self-conjugate
(Lemma~\ref{lem:no-majorana}), and that implements its
conjugation (T-P5-C2), now signs its time-reversal square. One
matrix, three offices, no conventions.

\emph{The second dividend: why T has no quantum number.} C and P
are $J$-linear with square one, so their eigenvalues $\pm1$ are
honest, basis-independent labels --- hence the tables of parities
and C-parities, and the selection rules that carried half a
century of spectroscopy. T admits no such label, and the table
says why with unusual crispness: an antilinear operator has no
invariant eigenvalue at all. If $T\psi = \lambda\psi$, then
rephasing the state rotates $\lambda$ --- $T(c\psi) =
\bar c\,\lambda\psi$ --- so ``the T-parity of a state'' is not a
property of the state. This is not a technicality; it dictated
seventy years of experimental design. Parity violation, once
suspected, was confirmed within months by a selection-rule-scale
effect (Part~I); direct time-reversal violation, requiring a
genuine rate asymmetry between a process and its motion-reversed
partner, waited until 1998 in the kaons and 2012 in the $B$
mesons (\S\ref{sec:road-qft}), and the static searches --- the
electric dipole moments, T-odd by the same bookkeeping --- remain
the deepest null results in physics
(\S\ref{sec:history-cpt}). The instruments were not slow; the
anatomy is genuinely different. A mirror with a quantum number
breaks loudly; a mirror without one must be caught moving.

\section{T-P5-2: CPT as orientation bookkeeping}
\label{sec:cpt}

\begin{theorem}[\textbf{T-P5-2}: CPT as total reversal]
\label{thm:p5-2}
The total orientation reversal $CPT$ maps each
Feynman--St\"uckelberg-paired mode set to itself, and is a symmetry
of the construction for every coupling texture built from real
geometric data. Consequently every particle and its antiparticle
share mass and total width, and all CPT-odd observables vanish, at
the quasi-free grade; the extension to interacting matter holds
conditionally on the standard scattering hypotheses (the asymptotic
frame of T-P4-8).
\end{theorem}

\begin{proof}
On the paired real modes $\cos(n\theta - Et)$ and
$\cos(-n\theta - Et)$ the total reversal acts as the identity map
\emph{of the pair}: fiber reversal exchanges the members, clock
reversal exchanges them back, and space reversal relabels
propagation directions within each. It remains to check the
couplings. A texture built from real geometric data consists of
magnitudes (coupling moduli, metric and symplectic contractions,
winding sums) and fiber angles. The magnitudes are even under total
reversal. The angles reverse sign under the fiber factor --- but
reversing the sign of every texture angle while conjugating every
loop is a \emph{relabeling} of the same real texture (each loop
holonomy $\varphi \mapsto -\varphi$ composed with the orientation
reversal of the loop itself, leaving every holonomy functional
invariant). Hence the action, and every correlation function built
from it, is invariant. Equal masses and widths for
Feynman--St\"uckelberg partners follow because the pair is a single
orbit of the symmetry; CPT-odd observables are odd functions on a
set the symmetry maps to itself.
\end{proof}

\emph{Relation to the standard theorem, stated exactly.} Jost's CPT
theorem \citep{Jost1957, StreaterWightman} is stronger in scope: it
covers interacting Wightman field theories, with the proof running
through analytic continuation to the complexified Lorentz group. It
is, however, anatomically silent --- it constructs the antiunitary
$\Theta$ and proves it a symmetry without saying what C, P, or T
individually are, or why $\Theta$'s antiunitarity traces to the time
factor. T-P5-2 inverts the trade: narrower scope (quasi-free, with
the interacting case conditional), full anatomy (the three factors
named as reversals of constitutional structures, antiunitarity
derived in T-P5-1). The two theorems are complementary, and the
natural bridge between them is named in the open questions: the
Bisognano--Wichmann identification of the CPT operation with modular
conjugation \citep{BisognanoWichmann1976} suggests that the clock
structure --- hence $\Ja$, hence the table --- may be derivable from
modular flow, tying the geometric account to the algebraic one at
interacting strength. That derivation is open, and it is listed as
such (Part~XII).

\emph{The converse discipline, and the laboratory record.} Two
further facts frame the theorem from outside. First, the
converse: CPT violation cannot be had cheaply. Greenberg's
theorem \citep{Greenberg2002} shows that an interacting theory
which violates CPT necessarily violates Lorentz invariance ---
so what a CPT-violating signal would overturn is not a detail of
the construction but the arena itself, the time-oriented
Lorentzian stage that AX-1 supplies and everything in this paper
stands on. The falsifier is correspondingly total, and
Part~I's register of the tests is worth re-reading with this in
mind: the neutral-kaon mass comparison at the $10^{-18}$ level,
antihydrogen spectroscopy at $10^{-12}$, the six-comparison
table of \S\ref{sec:history-cpt} --- every entry is an
equal-mass, equal-width, opposite-moment corollary of the total
reversal, and every entry has held. Second, the register: those
corollaries are exactly what T-P5-2 delivers for
Feynman--St\"uckelberg partners --- the pair is a single orbit
of the symmetry, so mass and total width cannot differ between
its members --- and the quasi-free grade plus the conditional
interacting extension is the honest scope within which this
paper claims them.


\section{The anatomy of a theorem: why \texorpdfstring{$CPT$}{CPT} is exact
and \texorpdfstring{$C$}{C}, \texorpdfstring{$P$}{P},
\texorpdfstring{$T$}{T} are not}
\label{sec:anatomy-of-a-theorem}

The two results just proved sit oddly beside each other until one asks where
each of them lives. T-P5-1 says the three reversals are orientation flips of
three constitutional structures. T-P5-2 says their product is a symmetry of
every texture built from real geometric data. Parts~VI and~VII will then
show that the individual mirrors break --- comprehensively, and by
construction. A reader is entitled to ask why the product should be safe
when every factor is not.

The answer is not in the algebra. It is in the arena, and the arena has
exactly two kinds of place to put a thing.

\begin{figure}[htbp]
\centering
\resizebox{0.851\textwidth}{!}{%
\usebox{\GGplateAnatomy}}
\caption[The circle remembers; the interval forgets.]{The two fibers of the
bulk, and the different things they can hold, adapted from the holonomy
companion \citep{ArisakaW2Holonomy}. \emph{(a)} The compact phase circle
$\Stheta$: every path closes, so everything a connection does there is
holonomy --- quantized, gauge-protected, transported by a group. Classes on
a circle do not age: no boundary operation can continuously deform one into
another. \emph{(b)} The scale interval $\Ir$: no path closes, so nothing is
protected; transport composes only forward, a semigroup, and the descent
visits each scale exactly once. Every deposit is final. \emph{The circle
remembers; the interval forgets} --- and the two sentences of this section
are that figure, applied to the three reversals.}
\label{fig:anatomy-p5}
\end{figure}

\paragraph{Protection has an address.} Whenever physics presents a quantity
that is exact --- a charge that never leaks, an anomaly coefficient that
does not run, a family count that neither drifts nor renormalizes --- the
anatomy says where it lives: on the circle, deposited in a holonomy class no
boundary operation can continuously deform. And whenever physics presents a
quantity that is \emph{placed} --- a mass, a scale, a rate, an epoch --- the
anatomy says that lives along the interval, as an address on the fiber that
forgets. The formulation is the holonomy companion's
\citep{ArisakaW2Holonomy}; the two structures are this paper's G1 and G2 of
\S\ref{sec:two-structures}, seen from the side.

\paragraph{Which side each reversal is read from.} Now the asymmetry
between the theorem and its factors is not a puzzle but a location.

\emph{$CPT$ is exact because it is a statement about the circle alone.} The
proof of T-P5-2 turns on one step: reversing the sign of every texture angle
while conjugating every loop is a \emph{relabeling} of the same real
texture, because each loop holonomy $\varphi \mapsto -\varphi$ composed with
the orientation reversal of the loop itself leaves every holonomy functional
invariant. That is a statement about holonomy classes, and holonomy classes
are circle-side. Nothing the descent does along the interval can reach them,
because reaching them would mean continuously deforming a class --- which is
what a compact fiber forbids. The exactness of $CPT$ is not a coincidence
that survived; it is a class label that had nowhere to leak to.

\emph{$C$, $P$ and $T$ individually are not exact, because their breaking is
an address.} The chiral texture of Parts~VI and~VII is written \emph{onto}
the fiber \emph{during} the descent --- at a scale, once, in one direction.
Parity violation and the pentagon are not properties of the circle; they are
what the interval deposited on it at a particular place along a one-way
road. An address can be anything, and a once-only pass cannot be undone.
That is why the mirrors break and stay broken, and why they break at
specific scales rather than everywhere at once.

\paragraph{Two corollaries the reader should collect.} First, the
five fates of \S\ref{sec:five-fates} are not five independent facts but one
sorting rule applied five times: a mirror survives exactly when its
statement is about a class, and fails exactly when its statement is about an
address. Second, the strength of the two claims should differ, and it does:
the exactness of $CPT$ is held at the grade of the arena's own topology,
while every statement about \emph{where} a mirror broke inherits the grade
of the descent that broke it. The status ledgers of
\S\ref{sec:part4-ledger} and \S\ref{sec:part6-ledger} record the difference
row by row; this section says why the difference exists.

\paragraph{And the honest limit.} The anatomy explains the shape of the
result. It does not extend its scope. T-P5-2 remains a quasi-free theorem
with the interacting case conditional on the asymptotic frame of T-P4-8, and
no amount of pointing at a compact fiber changes that --- Jost's theorem is
still the one with the wider reach, and this paper's contribution to that
comparison is anatomy, not coverage. The circle explains why the exactness
is not an accident. It does not prove the exactness on a domain the proof
does not cover.

\section{The five fates of the mirrors}
\label{sec:five-fates}

T-P5-2 contains the seed of everything that follows in this
paper, and this section grows it into the Part's harvest. CPT is
blind to any real texture --- no texture, however contrived, can
break it. But the \emph{factors} are not blind, and with the
anatomy in hand we can now do what the standard account cannot:
derive, rather than record, the fate of each mirror --- which
broke, why, in what order, and by how much. The reader should
hold Part~I's history in mind throughout; this section is that
history, explained.

\emph{The fates of C and P: broken by an absence.} The descent of
Part~VI keeps one hand of the bulk spectrum: left-handed
particles, and --- since conjugation reverses the fiber without
touching space --- their right-handed antiparticle readouts. Now
apply the walks of \S\ref{sec:three-reversals}. P maps a
surviving left-handed particle to its right-handed mirror ---
\emph{a state the projection removed}. C maps it to a left-handed
antiparticle --- removed likewise. Neither operation fails
because some coupling was mistuned; each fails because its image
is not in the world. This is why the first fall was
\emph{maximal}: an asymmetry produced by an absent sector is an
all-or-nothing asymmetry --- Wu's electrons could not be partly
missing --- and it is why C and P fell \emph{together}, in the
same experiments of the same winter: one projection, two
reversals, one absence seen twice.

\emph{The fate of CP: survives the absence, falls to an angle.}
Compose the two failures and something remarkable happens: CP
maps a left-handed particle to a right-handed antiparticle ---
\emph{present}. The combined mirror hops between states the
projection kept; the absence that killed its factors cannot touch
it. This is the structural content of the 1957 repair (Part~I):
the community's instinct that the true mirror pairs matter with
reflected antimatter was, in this language, the observation that
CP survives the projection. What the repaired mirror could still
fall to is subtler: a fiber reversal negates every texture angle
without conjugating the loops, so CP survives iff every
closed-loop angle equals its own negative --- iff the texture is
\emph{achiral on the circle}, every loop angle in $\{0, \pi\}$
(T-P5-3's criterion, proved in Part~VII). And the \GG{} texture
is not achiral: the pentagon locks the Cabibbo loop at an angle
off the real axis. Hence the second fall's character, so
different from the first's: not an absence but an \emph{angle};
not all-or-nothing but small (the kaon asymmetry at parts per
thousand); not immediate but eight years subtle, hiding in a
regeneration-plagued decay. The standard account holds ``P
maximally violated'' and ``CP barely violated'' as two unrelated
inputs. Here they are one architecture read twice: the projection
is total, the lattice angle is small, and each mirror broke
exactly as hard as its breaker.

\emph{The fate of T: the product's alibi.} T is broken wherever
CP is --- not by an independent mechanism but by the unbreakable
product: if CPT holds and CP fails, T must fail in exact
compensation. The direct observations (\S\ref{sec:road-qft})
confirm the bookkeeping: the kaon and $B$-meson T-asymmetries
are the CP asymmetries read through the total reversal. And the
custody discipline of Part~III must be kept in force here, one
last time: this dynamical T-violation --- an angle in a texture
--- has nothing to do with the arrow of time, which is the
non-invertibility of the boundary projection over a bulk whose
action is exactly T-symmetric (\S\ref{sec:custody}). One is a
property of couplings, measured in asymmetries; the other is a
property of readout, present even where every coupling is
T-even. The film runs forward because the projector is one-way,
not because the reel is twisted.

\emph{The fate of CPT: no fate at all.} The total reversal is
blind to every real texture (T-P5-2): there is nothing in a world
built from real geometric data for it to see, hence nothing that
can break it. The five fates are collected in
Table~\ref{tab:five-fates}.

\begin{table}[t]
\centering
\caption[The five fates of the mirrors.]{The five fates of the mirrors, derived from one
projection and one lattice. The pattern the twentieth century
measured --- P and C falling together and totally, CP falling
later and slightly, T tracking CP, CPT never falling --- is
here the fingerprint of a one-handed descent writing a chiral
texture on an orientation-blind bulk.}
\label{tab:five-fates}
\small
\renewcommand{\arraystretch}{1.28}
\begin{tabular}{p{1.2cm} p{2.9cm} p{4.6cm} p{4.6cm}}
\toprule
\textbf{Mirror} & \textbf{Reverses} & \textbf{Fate, derived} &
\textbf{Breaker, and its size} \\
\midrule
P & space & broken maximally & an absence: the mirror hand is
projected out (Part~VI) \\
C & fiber & broken maximally, with P & the same absence, read
through the fiber \\
CP & fiber $+$ space & survives the projection; broken small &
an angle: the chiral pentagon locks a loop off $\{0,\pi\}$
(Part~VII) \\
T & clock & broken exactly with CP & the product's bookkeeping:
CPT holds, so T pays CP's bill \\
CPT & all three & unbroken, for every real texture & none
exists: real data are blind to total reversal (T-P5-2) \\
\bottomrule
\end{tabular}
\end{table}

\emph{Physics intuition --- the pattern was the message.} Step
back from the table and read the century again. Five mirrors;
four fates; and the fates line up with the breakers' kinds ---
absences break totally, angles break slightly, products pay
bills, and blindness never breaks. Nothing about that pattern is
visible in a formalism where each mirror's fate is an independent
input; everything about it is forced in a geometry where one
projection removes a hand and one lattice twists a loop. Indeed,
had history run in the other order --- the small fall first, the
maximal one second --- this architecture could not have
accommodated it. The mirrors did not merely break; they broke
\emph{informatively}, and the pattern of the breakage was the
geometry, speaking.

\section{What is claimed, and what is inherited}
\label{sec:mirrors-ledger}

Concessions first, in full --- the register's order. The
representation-theoretic framework of the discrete symmetries is
Wigner's \citep{Wigner1931}, and every quantum-number fact
rehearsed above --- intrinsic parities, C-parities, Kramers'
degeneracy, the two classes $T^2 = \pm1$ --- is standard physics,
re-derived here from the anatomy but discovered decades ago
without it. The CPT theorem in its full interacting, axiomatic
scope is Jost's \citep{Jost1957, StreaterWightman}, with the
modular grounding Bisognano--Wichmann's
\citep{BisognanoWichmann1976}; T-P5-2 is narrower in scope and
must always be presented as the complementary theorem it is ---
anatomy for scope, with the modular route the open bridge. The
converse discipline is Greenberg's \citep{Greenberg2002}. What
this Part claims as its own, at the grades of
Table~\ref{tab:part4-ledger}: \emph{first}, the derived table ---
antiunitary if and only if the operation contains T, obtained as
a computation on the sources of the quarter-turn rather than a
Wigner-fork selection; a result, it is worth noting, that
survives even for a reader who rejects everything else in the
program, since it needs only the premise that the imaginary
unit has physical sources with orientations. \emph{Second}, the
anatomy itself: C, P, T as three specific real maps with no
per-field conventions, so that ``P is violated'' becomes a
statement about a specific reversal of a specific orientation.
\emph{Third}, the five-fates architecture: the selective pattern
of the mirrors' historical falls --- which, together, in what
order, at what sizes --- derived from one projection and one
lattice, where the standard account carries the pattern as so
many unrelated inputs. \emph{Fourth}, the theorem-level custody
split between dynamical T-violation and the arrow of time. The
first claim is graded derived; the second is definitional and
carries the first; the third inherits Part~VI's and Part~VII's
grades and is exactly as strong as they are; the fourth is
carried by the constitutional Crooks decomposition cited in
Part~III. Nothing above depends on any claim graded weaker than
its row in the ledger.

\medskip
\noindent\emph{Part-end summary.} The operations now have
bodies: three reversals of three constitutional orientations,
walked one at a time on the modes of Part~IV. Where a convention
stood, a theorem stands --- antiunitary iff it contains T ---
and the theorem pays on the spot: Kramers for free, the missing
T quantum number explained, CPT as bookkeeping, and the century's
pattern of broken mirrors --- total, then slight, then the bill,
then never --- derived from one projection and one lattice. The
factors can break only where the world's texture carries a
direction, and it carries two. The first is the orientation of
the descent itself: parity, next.

\GGPart[Parity: the oriented descent]{Parity: The Oriented Descent}{Why exactly one force is chiral --- the five superlatives collapsed to one; the basin/gate architecture with its clauses derived; the projection computed in blocks; the handedness invariant; and a registered falsifier}
\label{part:parity}

\section[Part~VI in plain English]{Part~VI in plain English: why exactly one force is chiral}
The Standard Model can describe parity violation to any precision an experiment requires, and it cannot ask why the violation happens in one force and in no other. This Part asks it. Gravity, the strong force and electromagnetism are basins; the weak force is the gate; and the weak column differs from the other three in every property one can name, which in the Standard Model is five unconnected accidents of the input charges.

The mechanism offered is the projection that makes the descent oriented: chirality is the orientation of the descent, and the gate force is the one that carries it. The Part states the mechanism, derives what follows from it, and then puts the honesty ledger on the page --- four statements derived, three asserted, one conjectured. The three asserted are the projection matrix itself, the existence of the projection, and which chirality survives, and they are the three to press hardest on.

A registered falsifier closes the Part: there is no right-handed weak sector, at any scale. That is not a prediction with a window around it. Any $W_R$, ever, ends this account, and the Part says so before a reader has to ask whether anything here could be wrong.

\section{The question the Standard Model cannot ask}
\label{sec:parity-question}

Part~V closed on a promise: the factors can break only where
the world's texture carries a direction, and the first of the
two directions is the orientation of the descent itself. To see
what that promise must explain, restate the 1957 discovery as
the Standard Model holds it today: the
left-handed fermions are $\SUtwoL$ doublets, the right-handed
fermions are singlets, and parity violation is thereby written into
the field content --- an input codifying $V{-}A$, not an output of
anything. What the model then cannot ask is the question its own
structure poses. The weak interaction is
simultaneously five different kinds of exceptional, and each
superlative has a bench face the reader has met. It is the only
\emph{broken} gauge force: its messengers weigh nearly a hundred
protons each while the photon and the gluon weigh nothing --- the
single most conspicuous asymmetry in the force table. It is the
only \emph{chiral} force: Wu's cryostat measured its parity
violation at full strength, and no experiment has ever caught the
strong or electromagnetic force telling left from right. It is
the only \emph{family-changing} force: every muon that has ever
decayed did so by this force and no other, and the Sun shines
because only this force can turn a proton's quark into its
sibling. It is the only \emph{short-range} force among the
fermion couplings --- $10^{-18}$~m, a thousandth of a proton,
which is why it took until 1896 to notice it at all. And it is
the unique home of \emph{CP violation}: of the four forces, only
this one has ever been caught distinguishing matter from
antimatter, in the kaons and the $B$ mesons where the author
spent his experimental decades. Five superlatives, one gauge
factor. In the Standard
Model these are five separate accidents of the input charges; no two
of them are connected by anything in the formalism. Quantum field
theory at large adds two genuine structural results --- parity
cannot even break spontaneously in vector-like theories
\citep{VafaWitten1984}, and only chiral fermions are protected from
acquiring superheavy masses \citep{tHooft1979}, which explains why
the \emph{light} spectrum is chiral --- but neither says why
chirality sits in $\SUtwoL$, nor why the broken force and the
chiral force coincide. String theory geometrizes the possibility
(chirality from compactification topology
\citep{CandelasHorowitzStromingerWitten1985, DixonHarveyVafaWitten1985})
without selecting it: the landscape holds mirror-symmetric vacua in
abundance. The claim of this Part is that the five superlatives are
five faces of one geometric fact --- \emph{the descent between
family basins is oriented} --- and that the fact has a mechanism in
place, one named lemma outstanding, and a falsifier all its own.

\section{The basin/gate architecture}
\label{sec:basin-gate}

The parent Standard-Model treatment \citep{ArisakaA9SM} divides the
four forces by one structural criterion. A force whose gauge
symmetry is \emph{unbroken} holds its charge fixed: it can move a
state only within a fixed-charge sheet --- within a family basin.
Gravity, the strong force, and electromagnetism are such
\emph{basin forces}: exactly conserved, family-diagonal,
parity-even, long-range where massless. An exact symmetry forbids
precisely the family-changing transition a cascade needs. The
electroweak factor $\SUtwoL \times \UoneY$ is the one broken gauge
structure --- and the claim is that this is not a coincidence
list but an identity: \emph{being the broken force and being the
gate force are the same statement}. Let us set the stage with more care, because the argument
deserves to be walked rather than gestured at. The family
landscape of the parent treatment is a potential over the space
of family configurations: its critical points are the three
family levels, its basins their neighborhoods, and the descent
--- the Morse gradient flow --- runs strictly downhill, heavier
to lighter, which is why the mass hierarchy has an order rather
than a shuffle. Now ask what kind of coupling can \emph{execute}
that descent, and derive the gate's job description one clause at
a time.

\emph{First: the executor is family-off-diagonal.} A transition
from basin~3 to basin~2 connects distinct critical points; a
coupling that conserves the family label can shuffle states
within a basin forever and move nothing between them. The
executor must be the force that does not commute with family ---
and among the observed forces exactly one changes flavor.

\emph{Second: the executor is massive, hence short-range, hence
broken.} A gate is a localized crossing --- the transition
happens \emph{at} the saddle between basins, not everywhere at
once --- and localization requires a length scale. A massless
mediator has none: its influence is scale-free and long-range,
the signature of a basin force. The executor must carry a mass;
a gauge force carries a mass only if its symmetry is
spontaneously broken; so the gate force and the broken force are
one.

\emph{Third: the executor is dissipative.} The descent is a
decay --- basin~3 states \emph{fall} to basin~2 and the energy
difference leaves, irreversibly, through the boundary's one-way
law (the open-system clause of Part~III). The executor is the
force whose characteristic processes are decays: the force of
finite lifetimes.

\emph{Fourth --- the parity clause --- the executor carries a
direction:}

\begin{quote}
A Morse gradient flow has a direction: strictly downhill, heavier
$\to$ lighter. A direction is an orientation, and an orientation can
be carried only by a parity-violating coupling. The weak force is
maximally chiral, and that maximal parity violation is the
microscopic image of the cascade's one-wayness. A parity-even force
cannot encode ``the flow goes this way and not the other.''
\end{quote}

\emph{Physics intuition --- the canal lock.} Picture the family
landscape as a river system stepping down through a set of
canal locks. The open water of each pool is the domain of the
basin forces: currents may swirl a boat anywhere within its
pool, tirelessly and reversibly, and they never lower it. The
locks are different machinery altogether. A lock is
\emph{local} --- it works only at the step; it is \emph{built},
not free --- there is structure at the crossing that the open
water does not have; it is \emph{dissipative} --- every descent
spills water downstream and cannot be undone by the lock
itself; and above all a lock is \emph{one-way machinery in its
operation}: gravity gives the traffic through it a direction,
and the lock's mechanism is shaped by that direction --- the
gates open downstream. The weak interaction, on the reading this
Part defends, is the lock machinery of the family landscape ---
and once one sees it so, the five superlatives stop being a
list. Of course the lock force is the broken one: locks are
built structure at the crossings. Of course it changes families:
that is what a lock is \emph{for}. Of course it is short-range:
locks live at the steps. Of course its processes are decays:
locks lower boats. And of course it is chiral --- the clause the
next block states --- because one-way machinery must be able to
tell which way is downstream. After all, no one is surprised
that the machinery of a canal lock looks nothing like open
water; the surprise would be a landscape with steps and no
locks.

\begin{table}[t]
\centering
\caption[The basin/gate table.]{The basin/gate table. Five superlatives, one column: in
the Standard Model five unconnected accidents of the input
charges; in the oriented-descent reading, five faces of ``the
weak interaction is the gate force.'' Six properties across four forces. The weak column is set in bold because it is the one that differs in every row: broken where the others are unbroken, maximally parity-violating where the others are even, off-diagonal in family space where the others are diagonal, short-ranged, and the only home of CP violation.}
\label{tab:basin-gate}
\small
\renewcommand{\arraystretch}{1.25}
\begin{tabular}{p{2.4cm} p{2.2cm} p{2.2cm} p{2.2cm} p{3.4cm}}
\toprule
\textbf{Property} & \textbf{Gravity} & \textbf{Strong} &
\textbf{EM} & \textbf{Weak} \\
\midrule
Gauge symmetry & unbroken (universal) & unbroken & unbroken &
\textbf{spontaneously broken} \\
Parity & even & even (Vafa--Witten protected) & even &
\textbf{maximally violating} \\
Family action & diagonal & diagonal & diagonal &
\textbf{off-diagonal (unique)} \\
Range & long & confined, P-even & long &
\textbf{short (massive $W$, $Z$)} \\
CP violation & none & none ($\bar\theta \to 0$) & none &
\textbf{unique home} \\
Role & basin & basin & basin & \textbf{gate} \\
\bottomrule
\end{tabular}
\end{table}

\begin{figure}[t]
\centering
\resizebox{0.641\textwidth}{!}{%
\begin{tikzpicture}[font=\sffamily\small, >=Stealth, xscale=1.15]
 % landscape curve
 \draw[line width=1.10pt]
  (0,3.0) .. controls (0.8,3.1) and (1.2,1.9) .. (1.8,1.9)
  .. controls (2.5,1.9) and (2.7,2.6) .. (3.3,2.0)
  .. controls (3.8,1.5) and (4.2,0.9) .. (4.9,0.9)
  .. controls (5.7,0.9) and (5.9,1.5) .. (6.5,1.0)
  .. controls (7.0,0.6) and (7.4,0.15) .. (8.2,0.12);
 % basins
 \node[font=\footnotesize] at (1.8,1.55) {basin 3 (heavy)};
 \node[font=\footnotesize] at (4.9,0.55) {basin 2};
 \node[font=\footnotesize] at (8.0,-0.25) {basin 1 (light)};
 % basin forces circles
 \draw[->, blue, line width=0.80pt] (1.5,2.15) arc (150:390:0.32);
 \draw[->, blue, line width=0.80pt] (4.6,1.15) arc (150:390:0.32);
 \node[font=\footnotesize, blue, align=center] at (0.9,3.55)
   {basin forces\\ $\gamma$, $g$, $G$ --- P-even};
 \draw[blue!60] (1.35,3.15) -- (1.6,2.5);
 % gates
 \draw[->, red, line width=1.10pt] (3.05,2.35) -- (4.0,1.35);
 \draw[->, red, line width=1.10pt] (6.2,1.35) -- (7.15,0.5);
 \node[font=\footnotesize, red, align=center] at (6.3,3.55)
   {gate force: $W^{\pm}$ --- broken, chiral, short-range\\
    transport holonomy $=$ CP phase};
 \draw[red!60] (5.0,3.15) -- (3.6,1.95);
 \draw[red!60] (6.8,3.15) -- (6.7,1.05);
 % orientation arrow
 \draw[->, line width=1.10pt, gray] (0.4,-0.6) -- (7.8,-0.6)
   node[midway, below]{the descent is oriented: heavier
   $\to$ lighter; an orientation only a chiral coupling can carry};
\end{tikzpicture}}
\caption[The family landscape: basins and gates.]{The family landscape. Unbroken forces hold their charges
fixed and act within basins (blue); the unique broken force
executes the gates (red), carries the downhill orientation ---
parity violation --- and acquires the transport holonomy --- the
CP phase --- along the way. ``The descent is oriented, seen
twice.''}
\label{fig:basin-gate}
\end{figure}

On this reading the five superlatives collapse to one: the weak
interaction is the force of the landscape's \emph{gates}, and every
gate property --- broken, chiral, off-diagonal, short-range,
CP-bearing --- is the same orientation seen through a different
observable (Table~\ref{tab:basin-gate};
Figure~\ref{fig:basin-gate}). CP violation joins through Part~VII: the gate crossings
are where the family projector is transported, the transport's
holonomy is the CP phase, and \emph{chirality and CP violation are
the single statement ``the descent is oriented,'' seen twice}.

\emph{Register.} The parent paper grades this synthesis honestly ---
``a structural argument, not yet a theorem'' --- and names the
missing piece, quoted in full in Section~\ref{sec:gate-lemma}. This
Part carries the same grade and adds two small pieces the argument
was missing: a convention-free invariant
(Section~\ref{sec:handedness}) and a falsifier
(Section~\ref{sec:wr-falsifier}).

\section{The mechanism: the projection}
\label{sec:projection}

Beneath the architecture sits a concrete mechanism, and its pedigree
should be stated before its role. The bulk matter of the \GG{}
program is genuinely chiral six-dimensional Weyl matter: in
$(1,5)$ signature the chiral spinor representation is complex, no
Majorana condition exists, and the left- and right-handed Weyl
modules are independent \citep{ArisakaA2GG6} --- there is a hand to
pick. The reduction to the four-dimensional boundary projects the
compact directions with a $\bZ_2$ parity whose matrix on the
$\mathrm{SU}(5)$ fundamental is
\begin{equation}
P \;=\; \mathrm{diag}(+1, +1, +1, -1, -1),
\label{eq:parity-matrix}
\end{equation}
acting on the \emph{radial} direction of the bulk (the canonical
assignment in the parent program; the boundary sits at one end of
the scale direction). Register, first and plainly: the $\bZ_2$
itself and the matrix \eqref{eq:parity-matrix} are
\emph{asserted}, per Table~\ref{tab:parity-ledger} below ---
part of the construction, not derived from the axioms; their
derivation from the admissibility filters is the ledger's
registered open item. Two consequences follow at once. Gauge fields
survive as zero modes only along generators commuting with $P$, so
the unbroken group is the centralizer of $P$ in $\mathrm{SU}(5)$:
\begin{equation}
\SUthreeC \times \SUtwoL \times \UoneY
\label{eq:centraliser}
\end{equation}
--- exactly the Standard-Model group \citep{ArisakaA5GDOS}. It is
worth doing this computation in public, because it takes four
lines and the reader who does it owns the Standard-Model group as
a \emph{result}. Write a general traceless hermitian generator of
$\mathrm{SU}(5)$ in block form adapted to the signs of $P$ ---
three ``$+$'' slots, two ``$-$'' slots:
\begin{equation}
M \;=\;
\begin{pmatrix}
A_{3\times3} & B_{3\times2} \\
B^{\dagger}_{2\times3} & D_{2\times2}
\end{pmatrix},
\qquad
P M P^{-1} \;=\;
\begin{pmatrix}
A & -B \\
-B^{\dagger} & D
\end{pmatrix}.
\label{eq:block-conjugation}
\end{equation}
Conjugation by $P$ leaves the diagonal blocks alone and flips the
sign of the off-diagonal ones. The generators commuting with $P$
are therefore exactly the block-diagonal ones: the traceless
$A$-block ($\mathrm{SU}(3)$, the gluons), the traceless $D$-block
($\mathrm{SU}(2)$, the weak bosons), and the one remaining trace
direction that keeps $M$ traceless overall,
\begin{equation}
Y \;\propto\; \mathrm{diag}\!\left(
-\tfrac13, -\tfrac13, -\tfrac13, +\tfrac12, +\tfrac12\right),
\label{eq:hypercharge}
\end{equation}
which is precisely the hypercharge assignment of the fields the
$\mathbf 5$ contains. Count: $8 + 3 + 1 = 12$ surviving gauge
directions out of $24$ --- the Standard-Model group, obtained by
striking out two blocks (Figure~\ref{fig:parity-blocks}). The
twelve projected directions --- the off-diagonal $B$ blocks, the
$X$ and $Y$ bosons of grand unification --- are removed from the
four-dimensional gauge sector, which is also why this
construction, unlike four-dimensional grand unification, puts no
proton-decay mediator in the spectrum. The broken-direction
components of the \emph{internal} gauge field survive as boundary
scalars: the Higgs sector is the projection's own debris, and the
doublet--triplet economics --- the celebrated headache of
four-dimensional unification --- is handled by the same signs
\citep{Kawamura2000}, so the agent that breaks the symmetry and
the agent that delivers the masses descend from one geometric
object.

\begin{figure}[t]
\centering
\resizebox{0.312\textwidth}{!}{%
\begin{tikzpicture}[font=\sffamily\footnotesize, scale=1.05]
 % fills first, then grid
 \fill[green!12] (0,2) rectangle (3,5);
 \fill[green!12] (3,0) rectangle (5,2);
 \fill[red!10] (3,2) rectangle (5,5);
 \fill[red!10] (0,0) rectangle (3,2);
 \draw[line width=0.80pt] (0,0) rectangle (5,5);
 \draw[line width=0.80pt] (3,0) -- (3,5); \draw[line width=0.80pt] (0,2) -- (5,2);
 % labels
 \node[align=center] at (1.5,3.5)
   {$A$: $\mathrm{SU}(3)_C$\\ \textbf{kept}};
 \node[align=center] at (4.0,1.0)
   {$D$: $\mathrm{SU}(2)_L$, $Y$\\ \textbf{kept}};
 \node[align=center] at (4.0,3.5)
   {$B$: $X, Y$\\ projected};
 \node[align=center] at (1.5,1.0)
   {$B^{\dagger}$: $X, Y$\\ projected};
 % sign headers
 \node at (0.5,5.3) {$+$}; \node at (1.5,5.3) {$+$};
 \node at (2.5,5.3) {$+$}; \node at (3.5,5.3) {$-$};
 \node at (4.5,5.3) {$-$};
 \node at (-0.35,4.5) {$+$}; \node at (-0.35,3.5) {$+$};
 \node at (-0.35,2.5) {$+$}; \node at (-0.35,1.5) {$-$};
 \node at (-0.35,0.5) {$-$};
\end{tikzpicture}}
\caption[The projection at a glance.]{The projection at a glance: the $\mathrm{SU}(5)$
generators in the block form of Eq.~\eqref{eq:block-conjugation},
with the signs of $P = \mathrm{diag}(+,+,+,-,-)$ on the margins.
An entry survives in the four-dimensional gauge sector iff its
row and column signs agree. The green blocks are the
Standard-Model group, Eq.~\eqref{eq:centraliser}; the red blocks
--- the $X, Y$ directions of grand unification --- are projected
out of the gauge sector, their internal components surviving as
the scalar debris the parent computation identifies with the
Higgs sector.}
\label{fig:parity-blocks}
\end{figure}

The fermions halve by the same signs, tied now to four-dimensional
chirality. The projection condition couples the internal parity to
the boundary chirality operator --- schematically, on each
$\mathrm{SU}(5)$ component of a bulk Weyl multiplet,
\begin{equation}
\Psi \;\longmapsto\; (P \otimes \gamma_5)\,\Psi
\quad\text{under the reflection of the projected direction},
\label{eq:fermion-halving}
\end{equation}
so a component's zero mode survives only if its $P$-sign and its
four-dimensional chirality agree: of the two hands inside each
six-dimensional Weyl field, \emph{exactly one} keeps a zero mode
per representation, and the assignments select the left-handed
set --- a choice that is itself asserted, per
Table~\ref{tab:parity-ledger} below. The count is worth
performing once, because it lands
exactly: the $\bar{\mathbf 5}$ supplies five left-handed Weyl
fields ($d^c$ in three colors, and the lepton doublet $L$), the
$\mathbf{10}$ supplies ten ($Q$ in six components, $u^c$ in
three, and $e^c$) --- fifteen left-handed Weyl fermions: one
complete Standard-Model generation, nothing missing, nothing
extra, and every mirror partner projected out rather than
hidden. In one line from the parent paper: \emph{the ``L'' in
$\SUtwoL$ is geometric, not phenomenological}.

\emph{Pedigree, conceded in full.} This mechanism is orbifold-GUT
technology: the projection matrix \eqref{eq:parity-matrix}, the
centralizer breaking, the chirality selection, and the
doublet--triplet economics are the construction of Kawamura and of
Hall and Nomura \citep{Kawamura2000, HallNomura2001}, built in turn
on the string orbifolds of Dixon, Harvey, Vafa, and Witten
\citep{DixonHarveyVafaWitten1985}. Nothing in this section is
original to the \GG{} program, and the comparison of Part~X
credits it without reservation. What the program claims as its
own is the \emph{forcing}: in the orbifold-GUT literature the
projection is a model choice among many; here the output --- the
Standard-Model group with three chiral generations --- is claimed
forced by the constitutional ledger, with the projection's own
derivation from the admissibility filters (POS-1 smoothness,
POS-3 anomaly freedom) registered as an open item rather than
claimed. The distinction between \emph{mechanism} (inherited) and
\emph{selection} (claimed, partially open) is the entire
originality claim of this section, and it is deliberately modest.

\subsection{What is derived and what is asserted --- the ledger of
the mechanism}
\label{ssec:parity-ledger-table}

\begin{table}[t]
\centering
\caption[The parity mechanism's honesty ledger.]{The parity mechanism's honesty ledger. The outputs are
verified; the projection's own necessity is the story's open
frontier --- stated here before any referee states it. Eight statements, each with its grade and the basis or residue behind it. Four are derived, three are marked asserted in bold, and one is a named conjecture. The three asserted rows are the projection matrix itself, the existence of the projection, and which chirality survives --- and they are the three a reader should press on hardest.}
\label{tab:parity-ledger}
\small
\renewcommand{\arraystretch}{1.25}
\begin{tabular}{p{6.2cm} p{2.6cm} p{5.0cm}}
\toprule
\textbf{Statement} & \textbf{Grade} & \textbf{Basis / residue} \\
\midrule
Centralizer of $P$ in $\mathrm{SU}(5)$ is exactly
$\SUthreeC \times \SUtwoL \times \UoneY$ & derived & explicit
block computation \citep{ArisakaA5GDOS} \\
6D matter is genuinely chiral (a hand exists to pick) & derived &
Lemma~\ref{lem:no-majorana}; \citep{ArisakaA2GG6} \\
Surviving spectrum is anomaly-free & derived & \BIsix{} inflow
\citep{ArisakaA3BI6} \\
Higgs sector $=$ projection's broken-direction scalars & derived &
zero-mode computation \\
The projection matrix $P$ itself & \textbf{asserted} & ``a
standard parity assignment''; derivation from POS-1/POS-3 is open
question (7) \\
The $\bZ_2$ projection exists & \textbf{asserted} & part of the
construction, not an axiom \\
\emph{Which} chirality survives & \textbf{asserted} & the
canonical Ramond-sector assignment; anchored physically only via
P-P5-D \\
Gate force $=$ broken force $=$ chiral force (uniqueness) &
conjecture & C-P5-A (L11) \\
\bottomrule
\end{tabular}
\end{table}

\section{Chirality is the orientation of the descent}
\label{sec:orientation-argument}

The architecture and the mechanism meet in the reading this Part
exists to defend. The projection says \emph{how} one hand
survives; the oriented descent says \emph{why it must}: a cascade
with a direction requires a carrier that can encode direction,
and the carrier's chirality is that encoding.

Mass belongs to the same web, and the link is a two-line
derivation worth displaying. A Dirac mass term pairs the two
hands, $m\,\bar\psi_L \psi_R$. But the projection left the two
hands \emph{differently charged}: $\psi_L$ sits in an $\SUtwoL$
doublet and $\psi_R$ (as its conjugate's partner) in a singlet,
so the pairing
\begin{equation}
m\,\bar\psi_L\psi_R
\qquad (\mathbf{2} \otimes \mathbf{1} \neq \text{invariant})
\label{eq:bare-mass-forbidden}
\end{equation}
is gauge-forbidden outright: \emph{no fermion of the projected
world may simply have a mass}. The only lawful route is through a
doublet scalar --- $y\,\bar\psi_L H \psi_R$ --- and the
projection has already supplied the very object: its own scalar
debris. Every fermion mass in the world is therefore a Yukawa
coupling times the condensate of the projection's leftovers, and
the hierarchy of those couplings is read in the parent treatment
as the cohomology of the doubled left/right stratum, ordered by
the same Morse flow whose orientation is the parity violation.

\emph{Physics intuition --- the toll at the gates.} There is a
warm way to hold this. In a vector-like world, mass is a
birthright: any fermion may carry any mass, up to the deepest
scale in the theory --- which is why, by the survival argument
below, such fermions are gone from the light world. In the
projected world, mass is a \emph{toll collected at the gates}:
a fermion acquires inertia only by coupling to the gate
structure's own scalar field, and how much it acquires is set by
where it sits along the descent. That is why the mass hierarchy
is ordered by the family cascade rather than scattered at
random --- the toll schedule and the road map are the same
geometry --- and why the particle that delivers all masses (the
Higgs) and the force that changes all families (the weak
interaction) were destined to sit in the same corner of the
Standard Model. Parity violation, electroweak breaking, the mass
hierarchy's direction, CP violation, and the weak monopoly on
family change: one oriented structure, five projections onto
experiment.

\section{The two partial answers the orthodoxy does own}
\label{sec:orthodox-partials}

Honesty requires recording that quantum field theory, though it
cannot ask ``why only the weak force,'' does own two genuine
structural results about parity, and the \GG{} architecture nests
them rather than replacing them. \emph{Vafa--Witten:} in
vector-like theories --- QCD above all --- parity cannot break
even spontaneously \citep{VafaWitten1984}; the strong
interaction's mirror is theorem-protected. The reason is worth a
sentence, because it is beautiful: in a vector-like theory the
fermionic path-integral measure is positive, so the free energy
of any parity-breaking configuration is provably no lower than
that of its mirror-symmetric competitor --- the vacuum cannot
\emph{afford} to break parity. In the basin/gate language this is
the interior guarantee of the clause ``unbroken $\Rightarrow$
parity-even'': not only does an unbroken force carry no
orientation, its vacuum cannot manufacture one.
\emph{The survival argument:} vector-like fermions admit
gauge-invariant bare masses and are generically superheavy; only
chiral fermions are \emph{forced} to be light, their masses
forbidden until the symmetry breaks \citep{tHooft1979}. This is
QFT's deepest ``why is the light world chiral'' --- a
quasi-anthropic structural selection: everything parity-even in
the fermion sector has long since been paid its heavy mass and
left the stage. The \GG{} account absorbs it gratefully --- the
descent's light spectrum is chiral for exactly this reason among
others --- and then answers the two questions the survival
argument leaves open: why the chirality sits in $\SUtwoL$
specifically, and why the broken force and the chiral force
coincide. Those are the basin/gate clauses, and no statement
internal to QFT reaches them.

\section{The Gate-Operator Lemma: statement and status}
\label{sec:gate-lemma}

The missing theorem is named in the parent treatment, and this paper
adopts it as its principal conjecture:

\begin{conjecture}[\textbf{C-P5-A} (the Gate-Operator Lemma, L11)]
\label{conj:gate}
Any connection that parallel-transports the family projector between
adjacent critical points of the family landscape is necessarily:
(i) family-off-diagonal; (ii) \emph{chiral} --- it carries the
orientation of the downhill gradient flow, and no parity-even
connection can; (iii) massive, with its mass set by the
symmetry-breaking vacuum at the gate. Within the boundary gauge
group, the unique factor admitting all three properties is
$\SUtwoL$.
\end{conjecture}

Read the three clauses slowly, because each carries a different
weight. Clause~(i) is the lightest: transport between distinct
critical points is family-off-diagonal by construction; its
content is uniqueness --- that \emph{only} the gate coupling does
this --- which is an observed fact awaiting its structural proof.
Clause~(iii) is the localization argument of
\S\ref{sec:basin-gate} sharpened: the transporting connection
lives at the saddle, the saddle has a symmetry-breaking vacuum,
and the connection's mass is that vacuum's scale --- the $W$'s
mass, on this reading, is the width of the gate. Clause~(ii) is
the crown, for it is precisely the statement ``parity violation
is a theorem of \GGTheory{}.'' Its natural proof strategy is an
orientation-pairing argument, and the shape is easy to hold:
suppose a parity-even connection could execute the transport.
Parity-evenness means the connection commutes with the reversal
that inverts the flow's orientation; but that reversal exchanges
the downhill and uphill readings of the same gate, so the
transport it commutes with cannot encode which end of the gate
is the lower --- and a transport that cannot tell downhill from
uphill cannot \emph{execute a descent}, only a shuttle. What
remains open is the formalization of ``encode'': the precise
statement of how the flow's orientation is written into the
connection's chirality, which is exactly the lemma. The
completion is queued in the parent program; this paper does
not claim it. Note also what would \emph{kill} the conjecture:
a demonstrably parity-even coupling observed to mediate
family-changing decay --- or, within the formalism, a
construction of an even connection transporting the family
projector down a Morse gate --- would sever ``chiral'' from
``gate'' and collapse the architecture to a coincidence list.
Until the lemma lands, the honest summary of this Part is:
mechanism in place (inherited), architecture original and
compelling, theorem pending.

\section{The handedness invariant}
\label{sec:handedness}

One objection to any chirality claim must be met head-on: ``left''
versus ``right'' is a naming convention, so what physical content
does ``the left hand survives'' possess? The answer is that a lone
chirality label is convention, but a \emph{correlation of
orientations} is not. This paper registers the correlation as a new
proposition of the program:

\begin{proposition}[\textbf{P-P5-D}: the handedness invariant]
\label{prop:handedness}
The following three orientation data are not independent
conventions but one alignment, invariant under all relabelings:
(i) the internal chirality of the surviving fermion projection;
(ii) the orientation of the family descent (the sign of the Morse
flow, heavier $\to$ lighter); (iii) the sign of the CP-violating
loop holonomy (the chirality of the flavor pentagon,
$+2\pi/5$ versus $-2\pi/5$). Reversing any one of the three while
fixing the other two yields a physically distinct world; reversing
all three is a relabeling.
\end{proposition}

\emph{Status and content.} The proposition is stated here at
proposition grade with its proof strategy, not its proof: (i) and
(ii) are linked by C-P5-A(ii) (the carrier's chirality encodes the
flow direction); (ii) and (iii) are linked by T-P5-3 plus the gate
reading (the holonomy is acquired \emph{along} the descent, so its
sign is measured against the descent's direction); the invariance
of the triple under total relabeling is T-P5-2's total-reversal
blindness. Its experimental content is real: nature could have
presented the opposite \emph{relative} sign --- the measured signs
of $\JCKM$ and the mass-ordering direction test the alignment ---
and the alignment is exactly what Wu's experiment, the unitarity
triangle's area sign, and the mass hierarchy jointly measure. The
proposition sharpens ``the descent is oriented, seen twice'' into
``seen three times, with one sign.''

\emph{The Ozma measurement, walked.} Part~I posed the old riddle
--- how would one communicate ``left'' to a distant civilization
by physics alone? --- and told its classic resolution: Wu's
asymmetry defines a handedness, and the kaons' charge asymmetry
resolves the residual matter--antimatter ambiguity in the
instructions. P-P5-D upgrades the riddle's answer from a protocol
to an invariant, and it is worth walking the three measurements
that pin it. \emph{Measure (i)}, the surviving chirality: Wu's
correlation $\langle \vec\sigma \cdot \vec p\rangle < 0$ for
the electrons of beta decay --- the sign that says which hand the
weak force couples. \emph{Measure (ii)}, the descent's
orientation: the direction of every family decay --- muons become
electrons and never the reverse --- the arrow heavier $\to$
lighter. \emph{Measure (iii)}, the pentagon's chirality: the sign
of the Jarlskog invariant, read as the orientation of the
unitarity triangle's area in the standard convention --- which
way the flavor pentagon turns. The proposition asserts that
these three signs are \emph{one datum}: any world with the same
laws realizes either this triple or its total reversal, never a
mixed alignment. A civilization that measures all three and finds
the mixed alignment has falsified the construction; a
civilization that finds our alignment can now be told ``left''
in one sentence rather than three protocols. The Ozma problem,
promoted from a puzzle about communication to a statement about
the rigidity of nature's orientations.

\section{The falsifier, registered: no right-handed weak sector}
\label{sec:wr-falsifier}

Left--right symmetric extensions of the Standard Model
\citep{PatiSalam1974, MohapatraPati1975} restore parity at high
energy: the mirror fermions and a $W_R$ exist, made heavy by a
second breaking scale, and parity violation is an accident of our
low vantage. The \GG{} construction predicts the opposite, and this
paper registers the prediction as a falsifier:

\begin{quote}
\textbf{F-P5-1 (no $W_R$, at any scale).} The mirror modes are
projected out of the spectrum, not made heavy: there is no
right-handed weak gauge sector at any energy, and parity is never
restored. The observation of a $W_R$, of mirror fermions completing
the right-handed doublets, or of parity restoration at any scale
falsifies the geometric-projection construction of
Section~\ref{sec:projection} outright.
\end{quote}

This converts what Report-level review identified as a gap (the
cohort offered no parity falsifier) into an asset: the construction
is \emph{more} falsifiable than left--right symmetry, since every
future collider limit on $W_R$ \citep{PDG2024} bears on it, and a
single discovery kills it with no parameter to hide behind.
(Registration in the cohort's canonical falsifier ledger, with its
F-number assigned against the master list, is queued with the
author's approval of this paper.)

\section{Note on the projection's location}
\label{sec:projection-location}

For the cohort-literate reader, one internal note. The fermion
chirality projection acts on the \emph{radial} (scale) direction of
the bulk --- the canonical assignment, stated as such in the
Standard-Model treatment. The compact phase circle \Stheta{}
carries no $\bZ_2$ identification --- the smoothness clause of
POS-1 forbids it, and the Kaxion's Wilson line requires the full
smooth circle \citep{ArisakaC1Kaxion}. Earlier presentations of the
zero-mode computation phrased the projection on the angular
coordinate; the radial phrasing is the canonical one, and the
harmonization of the older text is a housekeeping item of the
parent program, noted here so that no reader takes the variation
in phrasing for a variation in physics.

\section{What is claimed, and what is inherited}
\label{sec:parity-claims}

Concessions first, in full --- the register's order, and for this
Part the concessions are unusually large. The mechanism of
\S\ref{sec:projection} is inherited item for item: the projection
matrix, the centralizer breaking, the chirality selection, and
the doublet--triplet economics are the orbifold GUT of Kawamura
and Hall--Nomura \citep{Kawamura2000, HallNomura2001}, built on
the string orbifolds of Dixon--Harvey--Vafa--Witten
\citep{DixonHarveyVafaWitten1985}; chirality from the geometry of
extra dimensions is string theory's oldest four-dimensional
success \citep{CandelasHorowitzStromingerWitten1985}; the
theorem-protection of QCD's mirror is Vafa--Witten's
\citep{VafaWitten1984}; and the deepest orthodox ``why is the
light world chiral'' is the survival argument
\citep{tHooft1979}. None of the machinery on these pages is
original, and the comparison of Part~X pays each source in its
own section. What this Part claims, ranked by confidence, at the
grades of Table~\ref{tab:parity-ledger}: \emph{first}, the
basin/gate architecture as an answer to the question no other
framework poses --- why the broken, chiral, family-changing,
short-range, CP-bearing superlatives all land on one gauge
factor; the Standard Model holds them as five accidents, string
theory as five vacuum properties, and no formalism known to this
program connects any two of them, while the oriented descent
derives all five from one fact (grade: structural argument,
C-P5-A pending). \emph{Second}, the forcing-versus-selection
distinction: in the orbifold-GUT literature the projection is a
choice from a catalog; here the output is claimed forced by the
constitutional ledger, with the projection's own derivation
honestly open --- the difference between a mechanism and an
explanation, kept explicit. \emph{Third}, the handedness
invariant P-P5-D --- the Ozma problem promoted to a convention-free
alignment with measurable content. \emph{Fourth}, the falsifier
F-P5-1 --- the conversion of ``no parity falsifier registered''
into the sharpest one available: projected out, not made heavy;
never restored, at any scale. Nothing in the four claims leans on
a grade stronger than the ledger prints.

\medskip
\noindent\emph{Part-end summary.} One projection breaks the
unified group to the Standard-Model group --- a four-line block
computation the reader has now performed --- and keeps one hand
of fifteen states, a complete generation with its mirrors
removed rather than hidden. One architecture says why the
hand-keeping force must be the broken, family-changing,
short-range, dissipative one: the canal locks of a landscape
whose descent has a direction. One conjecture (C-P5-A) stands
between architecture and theorem; one proposition (P-P5-D) makes
the handedness a measurable alignment rather than a convention;
one falsifier (F-P5-1) makes the whole story mortal at every
future collider. The same orientation is now to be seen a second
time --- as the chiral holonomy the descent writes on the
flavor texture: the pentagon, and the price of CP.

\GGPart[CP violation: the chiral pentagon]{CP Violation: The Chiral Pentagon}{Where CP phases live (loop holonomies), what values them (the pentagonal lattice), and what follows: the forced quark and lepton phases, step by step; the dial-free door; the strong-CP complementary fate; five worked examples}
\label{part:pentagon}

\section[Part~VII in plain English]{Part~VII in plain English: where the CP phases live, and what values them}
A phase that can be dialed to any value explains nothing; it records what was measured. This Part argues that in this geometry the CP phases cannot be dialed. They are holonomies of loops in the flavor graph, and the lattice those loops live on is pentagonal, so the admissible values form a list rather than a continuum.

Two theorems carry it. The flavor holonomy theorem says where the phases live and how they are computed; the dial-free door says which channel is open and forces the Majorana case where it does. Between them they turn the question ``what is the CP phase?'' into the question ``which lattice point?'' --- and a list can be wrong in a way that a dial cannot, which is the whole reason for preferring one.

Strong CP appears here as the complementary fate: the one angle relaxed dynamically rather than locked to a lattice point. The Part closes with worked examples, the claimed-and-inherited separation, and a status ledger for the mirror Parts, so the standing of everything proved since Part~\ref{part:theorems} can be read down a single column.

\section{The question no dial can answer}
\label{sec:pentagon-question}

Of all the questions this paper asks, the one this Part answers
is the author's oldest debt. Part~I told the history: in 1964 the
long-lived kaon decayed into two pions, and the repaired mirror
--- the one symmetry the community had agreed nature surely kept
--- broke, at two parts in a thousand
(\S\ref{sec:history-cp}). In 1973 Kobayashi and Maskawa found
where such a breakage could \emph{live}: with three generations,
one complex phase survives every field redefinition, and CP
violation can be written into the quark matrix
\citep{KobayashiMaskawa1973}. It was a discovery of the first
rank --- it predicted the third generation --- and yet the reader
should see with complete clarity what it did and did not do. It
located the dial. It did not turn it. In the Standard Model the
phase $\delta$ is a free parameter: $\delta = 0$ --- no CP
violation, no baryogenesis, no us --- is exactly as lawful as the
measured $1.147$, and nothing in the formalism prefers one to the
other. The author spent two experimental decades inside this
dial's consequences --- $\varepsilon'/\varepsilon$ at KTeV,
$\sin 2\beta$ at BaBar --- measuring, to three digits, a number
that the reigning theory declared unexplainable in principle.
Why does the mirror between matter and antimatter break at all?
And why by \emph{exactly this much}? The Standard Model's honest
answer to both questions is: no reason.

This Part derives the dial's setting, and the derivation has a
shape worth announcing in advance, because each stage is a
different kind of result. \emph{First}, a calculus (T-P5-3,
clauses i--iii): where CP phases live --- not in couplings but in
\emph{closed loops} of couplings, as rephasing-invariant
holonomies on the flavor graph; this stage is inherited
mathematics, geometrized, and the Part says so. \emph{Second}, a
valuation (clause iv): what sets the holonomies' values --- on
the \GG{} bulk they are not continuous dials but points of a
\emph{pentagonal lattice}, because the one compact circle and the
block integer $L = 5$ are the only alphabet available; this stage
is the new physics, and everything falsifiable descends from it.
\emph{Third}, the outputs, chain by chain and step by step: the
quark phase (with the Cabibbo angle's golden-ratio closed form as
a by-product), the lepton phase, the Majorana pair, and the two
Jarlskog invariants --- each a number with a pull, not a fit.
\emph{Fourth}, the unique door the phases will bias (T-P5-4).
And \emph{fifth}, the complementary fate: why the same geometry
that forbids the weak CP angle to vanish forces the strong one
to. Roughly speaking, the Part replaces ``one dial, unexplained''
with ``one alphabet, no dials'' --- and the alphabet has five
letters.

\section{T-P5-3: the flavor holonomy theorem}
\label{sec:holonomy}

\begin{theorem}[\textbf{T-P5-3}: flavor holonomy and the chirality
criterion]
\label{thm:p5-3}
Regard a Yukawa texture as a $\mathrm{U}(1)$-valued connection on
the flavor graph (Section~\ref{sec:flavour-graph}). Then:
(i) the rephasing group acts as the gauge group of the graph, and
in a spanning-tree gauge the residual physical data are exactly the
loop holonomies, numbering $(N-1)^2$ for $N$ generations;
(ii) the quartet invariants
$\mathrm{Im}(V_{ij}V_{kl}V_{il}^{*}V_{kj}^{*})$ are the plaquette
sines, and for $N = 3$ unitarity ties all of them to the single
Jarlskog invariant;
(iii) $CP$ is a symmetry if and only if every loop holonomy lies in
$\{0, \pi\}$ --- CP violation is chirality of the texture on the
circle;
(iv) on the \GG{} bulk --- one compact circle, no second flavor
torus, block integer $L = 5$ --- the first-order loop holonomies are
valued in the pentagonal lattice $(\pi/5)\bZ$.
\end{theorem}

\begin{proof}
(i) The rephasing \eqref{eq:rephasing} is the gauge action of
$\mathrm{U}(1)^{2N}$ on edge angles; on any closed loop the dial
turns telescope to zero, so loop holonomies are invariant; nulling a
spanning tree's $2N - 1$ edges (one dial per vertex, one overall
turn acting trivially) leaves one angle per independent cycle:
$N^2 - (2N-1) = (N-1)^2$. (ii) A quartet is the smallest
rephasing-invariant product with an imaginary part; its phase is the
four-edge plaquette holonomy; the standard unitarity identities
\citep{Jarlskog1985} equate the nine plaquettes up to sign for
$N = 3$. (iii) By T-P5-1, CP reverses the fiber and negates every
edge angle; the transformed texture is a rephasing of the original
iff every loop angle satisfies $\varphi \equiv -\varphi \bmod
2\pi$, i.e. $\varphi \in \{0, \pi\}$. (iv) A loop holonomy is a
closed transport on the single circle \Stheta{}; the fiber carries
no continuous modulus (Theorem~0's arena is rigid, and the holonomy
moduli of the flavor connection are set by the block structure),
so first-order holonomies are built from the block-geometric
angles available, all of which are multiples of $\pi/L = \pi/5$:
the Higgs doublet retains the fraction $2/L$ of the unified block,
forcing the branch phase $\pi(1 - 2/L) = 3\pi/5$, and per-family
re-zeroings move angles by the lattice step $\pi/5$
\citep{ArisakaA9SM}. Absent a second torus there is no continuous
direction in which a holonomy could float off the lattice.
\end{proof}

\emph{Physics intuition.} An experimentalist has met clause (iii)
before, in different clothes: it is the statement that CP violation
requires a phase that cannot be rephased away, promoted from
bookkeeping to geometry --- the unremovable phase is a closed-loop
angle, as gauge-invariant as the Aharonov--Bohm phase and for the
same reason. Clause (iv) is the new physics. In the Standard Model
the surviving loop angle is a continuous parameter: nothing values
it, and $\delta = 0$ (no CP violation, no us) would have been as
lawful as the measured $1.147$. On the \GG{} bulk the angle is a
\emph{lattice point}: the pentagon of $L = 5$ is the only alphabet
available, and CP violation exists because the forced branch phase
$3\pi/5$ is not a multiple of $\pi$ --- \emph{the flavor pentagon
is chiral}. The same pentagon whose trigonometry (the golden ratio)
sets the Cabibbo angle's magnitude sets the existence and the size
of CP violation. One geometric object, two of flavor physics'
deepest numbers.

\subsection{The calculus, walked: dials, trees, loops, and one
area}
\label{ssec:calculus-walked}

The proof above is complete but compressed, and this Part's
subject deserves the uncompressed version. Let us walk clauses
(i) and (ii) slowly, with every object displayed.

\emph{The dial group.} The flavor graph of
\S\ref{sec:flavour-graph} has $2N$ vertices (the up-type and
down-type family fields) and $N^2$ edges (the Yukawa couplings),
each edge carrying a real twist angle $\varphi_{ij}$. A
rephasing is a turn of the dials: one free angle per vertex,
\begin{equation}
\varphi_{ij} \;\longmapsto\;
\varphi_{ij} + \alpha_i - \beta_j ,
\label{eq:rephasing-action}
\end{equation}
the gauge action of $\mathrm{U}(1)^{2N}$ on the edge data. Now
transport around a closed loop of edges, alternating direction
as the bipartite structure demands, and add the angles with
signs. Every dial in the loop is entered once and left once, so
its two contributions cancel --- the sum \emph{telescopes} ---
and the loop angle is untouched by every rephasing: a gauge
invariant, for exactly the reason the Aharonov--Bohm phase is
one.

\emph{The count.} How many independent invariants survive?
Fix a spanning tree of the graph --- $2N - 1$ edges touching
every vertex with no loop --- and use the dials to null its
angles one edge at a time, working outward from a root; $2N - 1$
of the $2N$ dials are spent (the overall simultaneous turn acts
on no edge), and no freedom remains. What is left is one angle
per independent cycle:
\begin{equation}
N^2 - (2N - 1) \;=\; (N-1)^2
\qquad\xrightarrow{\;N=3\;}\qquad 4
\label{eq:holonomy-count}
\end{equation}
physical angles for three generations --- the gauge-fixed
content of the texture, obtained by an electrician's procedure
rather than a matrix decomposition.

\emph{The smallest invariant, and the one area.} The smallest
closed loop in a bipartite graph has four edges: up-family $i$
to down-family $j$ to up-family $k$ to down-family $l$ and back
--- a \emph{plaquette}. Its holonomy is the phase of the quartet
$V_{ij}V_{kl}V_{il}^{*}V_{kj}^{*}$, the smallest
rephasing-invariant product with an imaginary part, and its sine
is the quartet's imaginary part in unit-modulus normalization.
For $N = 3$, unitarity of the mixing matrix ties the nine
plaquettes together: all their sines are equal up to sign, and
the common magnitude is the Jarlskog invariant
\citep{Jarlskog1985},
\begin{equation}
\bigl|\mathrm{Im}\,(V_{ij}V_{kl}V_{il}^{*}V_{kj}^{*})\bigr|
\;=\; J \quad\text{for every plaquette}:
\label{eq:one-area}
\end{equation}
one area for the whole graph --- geometrically, twice the area
of the unitarity triangle, whichever of the six triangles one
draws. This is why the global fit's consistency is a one-number
statement, why all of quark CP violation is priced by a single
angle, and why the four angles of \eqref{eq:holonomy-count}
collapse, for observables, to one invariant phase and three
magnitudes.

\subsection{The criterion and the lattice, walked}
\label{ssec:lattice-walked}

\emph{The criterion (clause iii), cashed from Part~V.} Part~V's
walk established that fiber reversal negates every texture angle
(\S\ref{ssec:walk-C}); composing with the space reversal, which
touches no fiber angle, CP sends $\varphi_{ij} \mapsto
-\varphi_{ij}$ on every edge. The transformed texture describes
the same physics iff it is a rephasing of the original --- iff
every \emph{loop} angle, which no rephasing can touch, equals
its own negative:
\begin{equation}
\varphi_{\mathrm{loop}} \;\equiv\; -\varphi_{\mathrm{loop}}
\pmod{2\pi}
\quad\Longleftrightarrow\quad
\varphi_{\mathrm{loop}} \in \{0, \pi\}.
\label{eq:achiral-criterion}
\end{equation}
A texture all of whose loops sit at $0$ or $\pi$ is
\emph{achiral on the circle} --- its twists are indistinguishable
from their mirror --- and CP holds. A single loop off the real
axis makes the texture chiral, and CP breaks. The criterion is
exact, convention-free, and now a theorem of the anatomy rather
than a rule of thumb.

\emph{The valuation (clause iv), and what it forbids.} Here the
Standard Model and the \GG{} bulk part company, and the reader
should see precisely where. Both frameworks agree on
\eqref{eq:achiral-criterion}; the question is what \emph{sets}
$\varphi_{\mathrm{loop}}$. In the Standard Model the loop angle
is a continuous parameter of the Yukawa matrices --- any value
is lawful, nothing prefers any value, and the measured $1.147$
is an initial condition of the universe. On the \GG{} bulk, run
the inventory of what could possibly value an angle.
\emph{Could it float on a continuous modulus?} No: Theorem~0's
arena is rigid --- the fiber has no radius to vary, no shape
parameter, no landscape of vacua (\S\ref{ssec:source-circle});
the moduli that make string-theoretic CP phases
continuous simply do not exist here. \emph{Could it live on a
second flavor torus, with its own alphabet?} No: the arena has
exactly one compact direction --- one circle, hence one alphabet
of angles. \emph{What angles does that alphabet contain?} The
block geometry's own: the unified block carries the integer
$L = 5$, the Higgs doublet retains the fraction $2/L$ of it,
and the branch phase forced by that retention is $\pi(1 - 2/L)
= 3\pi/5$, with per-family re-zeroings moving angles by the
step $\pi/L = \pi/5$ \citep{ArisakaA9SM}. Every first-order
holonomy is therefore a multiple of $\pi/5$: the pentagonal
lattice. The honest residue is stated in the mirror-Parts
ledger: that \emph{no} further continuous deformation of the
holonomy moduli exists is graded derived-conditional
(``rigidity''), and it is the single hinge on which clause (iv)
turns. Granted the rigidity, the conclusion is remarkable
enough to state twice: \emph{CP violation exists because
$3\pi/5$ is not a multiple of $\pi$} --- the flavor pentagon
is chiral --- and no choice, anywhere, was available to make it
otherwise.

\subsection{The quark sector: from the pentagon to
\texorpdfstring{$\dCKM$}{delta-CKM}}
\label{ssec:ckm}

\begin{figure}[t]
\centering
\resizebox{0.373\textwidth}{!}{%
\begin{tikzpicture}[font=\sffamily\small, scale=1.5, >=Stealth]
 % pentagon
 \foreach \k in {0,1,2,3,4} {
   \coordinate (P\k) at ({90+72*\k}:1.25);
 }
 \draw[line width=0.80pt] (P0)--(P1)--(P2)--(P3)--(P4)--cycle;
 \foreach \k in {0,1,2,3,4} \fill (P\k) circle (1.1pt);
 % lattice angles from center
 \draw[->, gray] (0,0) -- (P0);
 \draw[->, line width=1.10pt, blue] (0,0) -- (P1);
 \draw[->, line width=1.10pt, red] (0,0) -- ({90+72*1.5}:1.25);
 \node[font=\footnotesize] at (0.02,1.5) {$0$};
 \node[font=\footnotesize, blue] at ({90+72}:1.55) {$2\pi/5$ \ ($=\dCKM^{(1)}$)};
 \node[font=\footnotesize, red] at ({90+108}:1.62) {$3\pi/5$ \ ($=\varphi_{12}$)};
 % annotations
 \node[align=center, font=\footnotesize] at (0,-1.85)
  {lattice step $\pi/5$;\quad chiral because $3\pi/5 \notin \pi\bZ$\\
   golden ratio in $\cos(3\pi/5)$ $\Rightarrow$ the Cabibbo closed form};
\end{tikzpicture}}
\caption[The flavor pentagon.]{The flavor pentagon. With one compact circle and block
integer $L = 5$, first-order holonomies live on the lattice
$(\pi/5)\bZ$: the forced branch phase $3\pi/5$ (red) is not a
multiple of $\pi$, so CP violation must exist; one unphysical
lattice step leaves the physical phase $2\pi/5$ (blue); and the
pentagon's trigonometry --- the golden ratio --- sets the Cabibbo
magnitude \eqref{eq:cabibbo}. One object, two of flavor physics'
deepest numbers.}
\label{fig:pentagon}
\end{figure}

Before the chain, the pentagon itself --- because the whole
sector runs on the trigonometry of the number five, and the
reader deserves it from scratch. In a regular pentagon the
diagonal and the side are locked in the most famous ratio in
mathematics: drawing two diagonals from one vertex cuts the
pentagon into triangles similar to those of the pentagram, and
the similarity forces $d/s = 1 + s/d$, whose positive root is
the golden ratio $d/s = \phi = (1 + \sqrt5)/2$. From it, two
lines of trigonometry deliver everything this Part uses:
\begin{equation}
\cos 36^{\circ} = \frac{\phi}{2} = \frac{1 + \sqrt5}{4},
\qquad
\cos 72^{\circ} = 2\cos^2 36^{\circ} - 1
= \frac{\sqrt5 - 1}{4}.
\label{eq:pentagon-trig}
\end{equation}
And two of the pentagon's canonical angles will now appear in
the physics with their geometric identities intact: the
\emph{interior} angle $108^{\circ} = 3\pi/5$ --- the angle at
each corner --- will be the forced branch phase, and the
\emph{central} angle $72^{\circ} = 2\pi/5$ --- the angle each
side subtends at the center --- will be the physical CP phase.
The reader may reasonably suspect numerology at this point; the
chain below is the antidote, six steps, each with its
provenance and its grade.

\emph{Step 1 (inherited; parent grade Asserted, chosen post-hoc;
carried without upgrade): the mixing magnitudes.} The up- and down-sector nearest-neighbor
mixing angles are fixed upstream by the mass-ratio geometry of
the parent treatment, $\theta_{12}^{(u)} = \sqrt{y_u/y_c} =
e^{-\pi}$ and $\theta_{12}^{(d)} = \sqrt{y_d/y_s} = e^{-\pi/2}$
\citep{ArisakaA9SM}: the square-root-of-mass-ratio form is the
standard seesaw-texture relation, and the specific exponentials
are the parent codebook's cascade mass ratios --- whose grade in
that codebook's own provenance table is Asserted (chosen
post-hoc), a grade this paper carries verbatim, without upgrade.
Nothing in this Part adjusts them.

\emph{Step 2 (derived-conditional): the branch phase.} The
relative branch phase between the two sectors is the forced
pentagonal angle $\varphi_{12} = \pi(1 - 2/L) = 3\pi/5$ ---
the Higgs doublet retains two of the block's five directions,
and the branch phase is $\pi$ diminished by the retained
fraction (\S\ref{ssec:lattice-walked}). Note the geometry
keeping its word: $3\pi/5$ is the pentagon's interior angle.

\emph{Step 3 (derived, given Steps 1--2): the Cabibbo
interference.} The observable $1$--$2$ mixing is the
\emph{phasor difference} of the two sectors' rotations, carried
out at the relative angle $\varphi_{12}$:
\begin{equation}
\sin\theta_{12} \;=\;
\bigl| e^{-\pi/2} - e^{\,i\,3\pi/5} e^{-\pi} \bigr|
\;=\; 0.22502,
\qquad
\sin^2\theta_{12} = e^{-\pi} + e^{-2\pi}
+ \tfrac{\sqrt5 - 1}{2}\, e^{-3\pi/2},
\label{eq:cabibbo}
\end{equation}
and the closed form is worth deriving in public. For any two
magnitudes $a, b$ at relative angle $\varphi$,
$|a - b e^{i\varphi}|^2 = a^2 + b^2 - 2ab\cos\varphi$; here
$a = e^{-\pi/2}$, $b = e^{-\pi}$, and
$\cos(3\pi/5) = -\cos 72^{\circ} = -(\sqrt5 - 1)/4$ by
\eqref{eq:pentagon-trig}, so the cross term is
$+2\,e^{-3\pi/2}(\sqrt5-1)/4 = \tfrac{\sqrt5-1}{2}\,
e^{-3\pi/2}$: the golden-ratio coefficient is not decoration
but the cosine of the pentagon's central angle doing its work.
Numerically: $e^{-\pi} = 0.043214$, $e^{-2\pi} = 0.001867$,
$\tfrac{\sqrt5-1}{2}e^{-3\pi/2} = 0.618034 \times 0.008983 =
0.005552$; the sum is $0.050633$, whose square root is
$0.22502$ --- against the measured $0.22501 \pm 0.00068$. The
pentagon's literal signature, at four decimal places.

\emph{Step 4 (derived-conditional): from branch phase to
physical phase.} The branch phase is not yet the CP phase: one
common family rephasing is unphysical, and on the lattice the
shortest step such a rephasing can take is $\pi/L = \pi/5$.
Removing it,
\begin{equation}
\dCKM^{(1)} \;=\; \frac{3\pi}{5} - \frac{\pi}{5}
\;=\; \frac{2\pi}{5} \;=\; 72^{\circ}
\label{eq:deltackm-first}
\end{equation}
at first order --- the raw mismatch is three pentagonal steps,
one is basis choice, two are physical. Interior angle in,
central angle out: the two canonical angles of the pentagon,
divided by one lattice step of convention.

\emph{Step 5 (asserted; flagged): the quantum-defect
completion.} The QD completion applies the shortest allowed
throat slip, $-\pi/K$ with $K = 35$:
\begin{equation}
\dCKM^{\mathrm{QD}} \;=\; \frac{2\pi}{5} - \frac{\pi}{35}
\;=\; \frac{13\pi}{35} \;=\; 1.16688~\mathrm{rad}
\;\simeq\; 66.9^{\circ},
\label{eq:deltackm-qd}
\end{equation}
against the measured $1.147 \pm 0.026$~rad: a pull of $+0.70$.
\emph{Status, stated plainly:} the first-order value
\eqref{eq:deltackm-first} is the forced result (pull $+1.17$
with a one-lattice-step theory width $\pi/35$ added in
quadrature; against the experimental error alone the raw value
sits at $+4.2\sigma$);
the QD step is a bounded lattice correction whose size is ledger-quantized (one lattice unit, drawn from the parent's frozen completion class) and whose direction and necessity are asserted per site ---
the parent paper's own codebook flags it ``post-hoc,'' and this
paper carries the flag verbatim rather than absorbing it.

\emph{Step 6 (derived, given the above): the area.} The
Jarlskog invariant follows with no further input beyond the
measured moduli, and the evaluation layer is stated so the
number can be audited --- PDG-class moduli ($s_{12} = 0.2250$,
$s_{23} = 0.0418$, $s_{13} = 0.0037$; prefactor
$c_{12}c_{23}c_{13}^{2} = 0.9735$) with the QD phase:
\begin{equation}
\JCKM = c_{12} c_{23} c_{13}^2\, s_{12} s_{23} s_{13}
\sin\dCKM^{\mathrm{QD}} \;=\; 3.1\times10^{-5},
\label{eq:jarlskog-pred}
\end{equation}
squarely on the measured $(3.12^{+0.13}_{-0.12})\times10^{-5}$.
Across the framework's evaluation layers --- first-order versus
QD phase, all-first-order moduli versus measured moduli --- the
same area evaluates between $2.7$ and $3.2\times10^{-5}$; that
layer spread, not an error bar, is the band the tables quote. Every downstream quark-sector CP observable is this one
area read through different interferometers:
Appendix~\ref{app:flavour} recomputes
$\varepsilon$, $\varepsilon'/\varepsilon$, and $\sin 2\beta$
from this single angle as holonomy interferometry, and
\S\ref{app:flavour-kaons} is the version for the reader who has
sat the shifts.

\emph{Physics intuition --- what kind of answer this is.} Stand
back and ask what has actually been claimed. The question ``why
is CP violation exactly this strong?'' has received an answer of
an unfamiliar kind: \emph{because a pentagon's angles are what
they are.} The branch phase is an interior angle; the physical
phase is a central angle; the Cabibbo magnitude carries the
golden ratio, which is the pentagon's aspect; and the lattice
step that separates convention from physics is the pentagon's
own fifth of a half-turn. In the Standard Model these four
numbers are one fitted dial and three fitted moduli; here they
are four readings of one geometric object, and the object was
not chosen --- $L = 5$ is the ledger's, forced upstream of this
paper, in use across twenty-one companion treatments for
everything from the Cabibbo angle to the lock scale. Indeed,
that is the real force of the pentagon: it was not built for CP
violation, and CP violation falls out of it.

\subsection{The lepton sector: \texorpdfstring{$\dPMNS$}{delta-PMNS}
and the Majorana pair}
\label{ssec:pmns}

The lepton chain runs on a different ledger integer, and the
difference is itself structural: where the quark alphabet is the
block integer $L = 5$, the lepton alphabet is the odd chirality
counter $J = 2N + 1 = 7$ --- the same integer that counts the
generations' doubled hands plus the axis, and the same $7$ whose
power suppresses the strong-CP leakage below. Five steps, each
graded.

\emph{Step 1 (inherited): the reactor-block phase.} The minimal
neutrino mixing operator of the parent treatment carries the
reactor-block angle $\phi_\nu = \arcsin\!\bigl(1/(2N{+}1)\bigr)
= \arcsin(1/7)$ \citep{ArisakaA9SM}: the smallest angle the
odd-integer alphabet writes, appearing where the smallest lepton
mixing (the reactor angle's sector) demands it.

\emph{Step 2 (derived-conditional, on the leptonic assignment
lemma): the reference shift.} The physical Dirac phase is read
against the parameterization's real reference channel, and that
channel enters the structural form with a negative sign: the
measured phase is the block phase shifted by $\pi$ ---
\begin{equation}
\dPMNS^{(1)} \;=\; \pi + 2\arcsin\!\frac{1}{7}
\;=\; 196.43^{\circ},
\qquad
\dPMNS^{\mathrm{QD}} \;=\; 196.06^{\circ},
\label{eq:deltapmns}
\end{equation}
with the QD completion applying the same bounded throat slip as
the quark chain. A subdominant assignment degeneracy
$\{\pi, 3\pi/2\}$ exists at the lemma level and is resolved in
favor of the structural form; the resolution is part of the
assignment lemma's residue, carried in the ledger rather than
hidden. The subdominant $3\pi/2$ assignment branch is hereby
pre-registered as dead: a measurement near
$270$--$286^{\circ}$ may \emph{not} be reinterpreted under the
alternative branch; the assignment lemma is frozen in the form
stated here, and its failure is the chain's failure.

\emph{Step 3 (the confrontation).} The prediction sits within
$1.0\sigma$ of the current normal-ordering global fit
($177^{\circ}{}^{+19}_{-20}$ \citep{NuFit2024}) --- and is
already disfavored at $\approx 3\sigma$ under inverted
ordering, which the derived spectrum excludes in any case ---
and, more importantly, it sits squarely inside the
$\pm15^{\circ}$-class resolution DUNE and Hyper-Kamiokande
will deliver in the 2030s. This
is the falsifier with the nearest date in the whole Part: a
measured $\dPMNS$ at, say, $270^{\circ}$ kills the chain with
no parameter to hide behind.

\emph{Step 4 (derived, given T-P5-4): the Majorana pair.} The
Majorana phases are not fitted either. The constitutionally
forced rank-two Majorana block (T-P5-4 below) fixes one state of
each intrinsic parity --- the two surviving heavy states are a
parity pair, not a phase continuum --- so
\begin{equation}
(\alpha_{21}, \alpha_{31}) \;=\; (0, \pi)
\label{eq:majorana-phases}
\end{equation}
--- parity data, not dials. With the parent treatment's derived
light masses, the effective neutrinoless double-beta mass is
$\mbb \simeq 1.72$~meV (the first-order evaluation, which is
the registered value; the QD-layer evaluation is $1.79$~meV):
below current reach, inside the
ambitions of the tonne-scale program, and the cleanest
laboratory shadow the Majorana pair casts.

\emph{Step 5 (derived, given the above): the leptonic area.}
The leptonic Jarlskog is $\JPMNS \simeq -9.4\times10^{-3}$ (the NuFit-angle layer; the parent's first-order-angle evaluation gives $-9.2\times10^{-3}$) ---
three hundred times the quark area, which is why the
long-baseline program can see it this decade (Worked
Example~IV works the rate asymmetry). Every number in this
subsection is a falsifier with a date attached (Part~XII).

\subsection{The completion class, disclosed in one table}
\label{ssec:qd-disclosure}

The quantum-defect completion deserves one table rather than
scattered flags, and here it is: every QD completion used in this
paper or inherited from its parent codebook, with its size, its
unit, and its direction relative to the data. The standing defense
is stated fairly: each step is a bounded, quantized element of a
frozen completion class --- a lattice or throat step the ledger
already owns, incapable of further adjustment once applied, and
falsifiable as such. The audit's counterweight is printed beside
it, because both facts belong to the record: the class is
heterogeneous in size and unit (an angle here, a ratio there), its
necessity at each site is asserted rather than derived, and
wherever the step was exercised its direction has favored the
data --- with the single exception of the Cabibbo modulus, where
the completion points away from the data and the registered tables
quote the first-order value.

\begin{table}[H]
\centering
\caption[The QD completions, disclosed.]{The quantum-defect (QD)
completions used in this paper or inherited from its parent
codebook, each with size, unit, and direction relative to the
data. The parent's own grade for the class is Asserted (chosen
post-hoc), carried here without upgrade; the standing defense and
the audit's counterweight are stated together in the text above.}
\label{tab:qd-disclosure}
\small
\renewcommand{\arraystretch}{1.25}
\begin{tabular}{p{2.5cm} p{3.0cm} p{2.9cm} p{2.2cm} p{2.8cm}}
\toprule
\textbf{Observable} & \textbf{First order} & \textbf{Completion} &
\textbf{Size (unit)} & \textbf{Direction vs data} \\
\midrule
$\dCKM$ & $2\pi/5 = 72^{\circ}$ & one throat slip $-\pi/K$ &
$-\pi/35$ (angle) & toward \\
$\dPMNS$ & $\pi + 2\arcsin(1/7)$ & the same slip, lepton alphabet &
$-\pi/490$ (angle) & toward \\
$s_{13}$ (CKM) & cascade product & multiplicative defect &
$+12\%$ (ratio) & toward \\
$\sin\theta_{12}$ (CKM) & $0.22502$, Eq.~\eqref{eq:cabibbo} &
modulus defect & $-0.02\%$ (ratio) & \textbf{away}; the registered
tables quote first order \\
fermion masses & parent cascade exponentials & per-particle
defects & per-particle (various) & toward, by construction \\
\bottomrule
\end{tabular}
\end{table}

\section{T-P5-4: the dial-free door}
\label{sec:door}

\begin{theorem}[\textbf{T-P5-4}: Majorana iff weight zero]
\label{thm:p5-4}
Let a fermion multiplet carry weight $n$ under the compact phase
circle \Stheta{}. A Lorentz-invariant, gauge-invariant
Majorana-type self-pairing --- equivalently a $\Delta L = 2$
bilinear --- exists iff $n = 0$. On every $n \neq 0$ sector the
unique invariant symmetric self-pairing is the charge-conserving
Dirac pairing; and the trilinear $LHN$ is the unique fiber-neutral
portal touching the weight-zero sector linearly.
\end{theorem}

\begin{proof}
In the real presentation the symmetric square of a weight-$n$ pair
decomposes into the rotation-invariant combination $a^2 + b^2$ ---
the Dirac pairing, which conserves winding --- and a doublet
rotating with weight $2n$, which is invariant iff $2n\alpha \equiv 0$
for all $\alpha$, i.e. iff $n = 0$. For $n = 0$ the reality
condition of the sector \emph{is} the Majorana condition: the field
is its own conjugate because the circle does not move it. The portal
statement follows by weight counting on the trilinear: only the
weight-zero singlet can couple linearly to a fiber-neutral
combination of doublet and Higgs \citep{ArisakaA9SM}.
\end{proof}

\emph{Physics intuition.} Lepton number is winding number. A charged
field cannot be its own antiparticle because conjugation reverses a
circle that moves it; the neutrino, alone in the matter sector, has
nowhere to wind --- and so it, alone, may pair with itself. The
theorem converts the seesaw from a model-builder's choice into a
constitutional consequence: if neutrinos have mass at all, the
unique lawful $\Delta L = 2$ structure is the weight-zero
self-pairing, the heavy completion enters through the unique portal,
and every dial that a seesaw model normally offers is absent ---
which is why the parent program calls it the \emph{dial-free
door}. Part~VIII walks through it: the door's CP-biased traffic,
transported by the boundary's one-way law, is the matter of the
universe. And the door carries its own laboratory test: Majorana
nature plus the parity pair \eqref{eq:majorana-phases} plus the
derived masses give $\mbb \simeq 1.7$~meV --- below current reach,
squarely in the sights of the next-generation program.

\emph{The dial count, performed.} To feel what ``dial-free''
buys, count what a generic seesaw offers the model builder: three
heavy Majorana masses, free; a $3\times3$ complex Yukawa texture
coupling the heavy states to the doublets, free up to rephasing;
and the high-scale phases the Casas--Ibarra parameterization
makes explicit --- a continuum of them
\citep{CasasIbarra2001}. Every one is a place where the answer
can be adjusted after the question is asked. Now count here. The
portal is unique (the trilinear of the theorem); the weight-zero
sector is unique (Part~IV's still point, \S\ref{sec:self-conjugate},
where this door was seeded); the heavy masses are set by the lock
scale of the ledger, not chosen (Part~VIII); the Yukawa is fixed
by seesaw matching to the derived light masses; and the phases
are the lattice and parity data of this Part. The count of
adjustable quantities is zero --- which is precisely why every
row of Table~\ref{tab:predicted-measured} is a confrontation
rather than a fit, and why Part~VIII can hand the cosmology a
chain with nothing left to turn.

\section{Strong CP: the complementary fate}
\label{sec:strongcp}

The quark and lepton phases of Section~\ref{sec:holonomy} are
\emph{locked}: lattice points, unable to relax. This section
concerns the one CP angle with the opposite experimental
character, and the reader deserves its statement from scratch.
The strong interaction's Lagrangian admits one more CP-odd term
than the ones this Part has priced: the topological density,
whose coefficient combines with the quark mass matrix's overall
phase into the invariant
$\bar\theta = \theta_{\mathrm{top}} + \arg\det(M_u M_d)$. A
nonzero $\bar\theta$ would give the neutron an electric dipole
moment --- a T-odd, P-odd static moment of exactly the kind
Part~V's dividends section said the instruments hunt --- and the
hunt has come back empty at extraordinary depth:
$|\bar\theta| \lesssim 10^{-10}$ \citep{PDG2024, nEDM2020}. Here
is the Standard Model's second CP embarrassment, the mirror
image of the first: a phase that may take any value takes,
inexplicably, the most special value available --- while sitting
in the same Lagrangian as a weak phase of order one. Two dials,
one spun to $66^{\circ}$, one parked at zero to ten digits, no
connection between them. On the \GG{} bulk the pair is not a
coincidence but a complementarity: the same alphabet that
\emph{locks} the weak angle on the lattice gives the strong
angle a fate of the opposite kind --- it is the one angle the
geometry lets \emph{relax}. Three statements, each with its
grade.

\emph{No knob is exported (derived).} The first statement turns
on a difference in logical direction that deserves one slow
sentence. Effective-field-theory practice writes down \emph{every}
gauge-invariant term the symmetries permit and lets data set the
coefficients --- on that logic $\theta_{\mathrm{top}}$ exists
because nothing forbids it, and its smallness is a fine-tuning.
The visible four-dimensional Lagrangian of this program is not
assembled that way: it is \emph{exported} from the bulk --- every
term arrives with a coefficient the descent computes --- and the
topological density $G\widetilde G$ appears only if the descent
generates its coefficient. The bulk construction generates none
(\citealp{ArisakaA2GG6}; \citealp{ArisakaA3BI6}):
$\theta_{\mathrm{top}}^{(1)} = 0$ --- the strong-CP problem is
solved first ``by the absence of a knob.'' Not a symmetry
imposed, not a relaxation arranged: a term that was never
written, because nothing upstairs writes it.

\emph{The determinant phase cancels at first order (conditional,
lemma L9).} In the first-order quark codebook the singular values
are positive and the physical phases live only in the misalignment
matrices, so $\arg\det(M_u M_d) = 0$: a relative family phase can
generate CKM CP violation without generating strong CP --- the
precise distinction the Standard Model makes by hand, here
geometric. The end-to-end proof is the named lemma L9,
outstanding; the residual leakage carries a closed-form bound
worth displaying, because every factor is a ledger integer:
\begin{equation}
\bigl|\bar\theta\bigr|
\;\lesssim\;
\sin\!\Bigl(\frac{2\pi}{L}\Bigr)\,\frac{e^{-K}}{2^{J}}
\;=\;
\sin\!\Bigl(\frac{2\pi}{5}\Bigr)\,\frac{e^{-35}}{2^{7}}
\;\simeq\; 4.7\times10^{-18},
\label{eq:thetabar-leakage}
\end{equation}
eight orders of magnitude below the neutron-EDM bound
$|\bar\theta| \lesssim 10^{-10}$. Read the three factors, because
each is a ledger integer doing its characteristic work:
$\sin(2\pi/5)$ is the pentagon's own sine --- the largest CP
content the lattice can leak, the same $L = 5$ that locks the
weak phase; $e^{-K} = e^{-35}$ is the throat's exponential ---
the same $K = 35$ that sets the lock scale and the baryon
throughput of Part~VIII; and $2^{-J} = 2^{-7}$ is the chirality
counter's power of two --- the same $J = 7$ that wrote the
lepton phase one subsection ago. No new number was needed to
bound the leakage; the alphabet priced its own imperfection. A
measured EDM between $10^{-18}$ and $10^{-10}$ would break this
architecture while leaving the Standard Model untouched: the
bound is a falsifier, not a comfort
(Table~\ref{tab:falsifiers}).

\emph{The Kaxion relaxes what remains (companion result).} The
compact-angle pseudoscalar of the Kaxion companion
\citep{ArisakaC1Kaxion} couples through the gluonic portal, and QCD
itself supplies the potential whose minimum the Vafa--Witten
theorem \citep{VafaWitten1984} certifies at
$\bar\theta_{\mathrm{eff}} = 0$: Peccei--Quinn relaxation
\citep{PecceiQuinn1977}, executed by a field the geometry forced
--- a gauge-topological Wilson-line angle, not an installed scalar
--- with the laboratory line at $8.1~\mu$eV. The strong-CP closure
is doubly protected: dynamically by relaxation, statically by the
scale profile.

\begin{figure}[t]
\centering
\resizebox{0.607\textwidth}{!}{%
\begin{tikzpicture}[font=\sffamily\small, >=Stealth]
 % locked side
 \begin{scope}[shift={(0,0)}]
  \draw[line width=0.80pt] (0,0) circle (0.85);
  \foreach \k in {0,1,2,3,4,5,6,7,8,9} {\fill ({36*\k}:0.85) circle (0.9pt);}
  \draw[->, line width=1.10pt, blue] (0,0) -- (90-72:0.85);
  \node[font=\footnotesize, blue] at (0.1,-1.2) {$\alpha_B = 2\pi/5$ (lattice point)};
  \node at (0,1.25) {\textbf{$\Uone{B}$ Wilson line: LOCKED}};
  \node[font=\footnotesize] at (0,-1.65) {weak CP phase cannot relax:};
  \node[font=\footnotesize] at (0,-2.0) {CP violation \emph{must} exist};
 \end{scope}
 % relaxed side
 \begin{scope}[shift={(7.6,0)}]
  \draw[line width=0.80pt] (0,0) circle (0.85);
  \draw[->, line width=1.10pt, red] (0,0) -- (40:0.85);
  \draw[->, red!60, dashed] (40:0.6) arc (40:5:0.6);
  \node[font=\footnotesize, red] at (0,-1.2) {$a_K/f_K$: dynamical, rolls to $0$};
  \node at (0,1.25) {\textbf{$\Uone{Qi}$ Wilson line: RELAXED}};
  \node[font=\footnotesize] at (0,-1.65) {Vafa--Witten minimum:};
  \node[font=\footnotesize] at (0,-2.0) {$\bar\theta$ \emph{must} vanish};
 \end{scope}
 \draw[<->, line width=0.80pt, gray] (2.1,0) -- (5.5,0)
   node[midway, above, align=center, font=\footnotesize]
   {one compact geometry,\\ two fates};
\end{tikzpicture}}
\caption[The two-fate architecture of the two CP phases.]{The two-fate architecture (\S\ref{sec:strongcp}; Wilson
lines computed side by side in \citealp{ArisakaA6Gi4}). The
$\Uone{B}$ holonomy is a pentagonal lattice point --- the weak CP
phase is locked on, and CP violation cannot relax away. The
$\Uone{Qi}$ holonomy is the dynamical Kaxion angle --- the strong
CP phase rolls to the Vafa--Witten zero. The Standard Model holds
these two numbers as unrelated embarrassments; here they are the
two fates one alphabet allows.}
\label{fig:two-fate}
\end{figure}

\emph{The two-fate architecture, seen whole.} The same compact
geometry gives the weak-sector CP angle a locked fate (a pentagonal
lattice point --- CP violation \emph{must} exist and cannot relax
away) and the strong-sector angle a relaxed fate (a dynamical
Wilson line --- $\bar\theta$ \emph{must} vanish). In the Standard
Model the coexistence of a large CKM phase with a vanishing
$\bar\theta$ is a standing embarrassment --- two unrelated
parameters, one inexplicably maximal-ish, one inexplicably zero.
Here they are the two faces of one alphabet: the angles the lattice
holds, and the angle the geometry lets go. The gauge-runtime
companion computes both Wilson lines side by side --- the
$\Uone{B}$ holonomy locked at $2\pi/L = 2\pi/5$, the $\Uone{Qi}$
holonomy dynamical \citep{ArisakaA6Gi4} --- and no other framework
in Part~X's comparison connects the two numbers at all.

\section{Worked examples}
\label{sec:worked-examples}

Five computations, each executed end to end with numbers, on the
P4 model: the reader who reproduces all five owns the mirror
Parts.

\subsection{Worked example I: Wu's asymmetry as an orientation
measurement}
\label{ssec:we1}
Polarized $^{60}$Co ($J = 5 \to 4$, pure Gamow--Teller) emits
electrons with distribution $W(\vartheta) \propto 1 +
A\,P\,(v/c)\cos\vartheta$, and $V{-}A$ gives $A = -1$ for this
transition. At the experiment's $P \simeq 0.6$ and $v/c \simeq
0.6$ the up--down asymmetry is
$W(180^{\circ})/W(0^{\circ}) \simeq 1.36/0.64 \simeq 2.1$ ---
a $100\%$-scale effect from a supposedly exact symmetry, which is
why one cryostat sufficed where decades of atomic spectroscopy saw
nothing. In the language of this paper:
$\langle\vec\sigma\cdot\vec p\,\rangle \neq 0$ is a direct reading
of P-P5-D's alignment --- the decay products' momenta are
correlated with an internal orientation, and the sign of the
correlation is the handedness invariant's laboratory face.

\subsection{Worked example II: the CKM plaquette, with numbers}
\label{ssec:we2}
Take the standard-parameterization CKM magnitudes and the lattice
phase. The (1,2)--(2,3) plaquette holonomy
evaluates through Eq.~\eqref{eq:jarlskog-pred}: PDG-class moduli
$s_{12} = 0.2250$, $s_{23} = 0.0418$, $s_{13} = 0.0037$,
$c_{12}c_{23}c_{13}^2 \simeq 0.9735$, and the QD phase
($\sin 66.86^{\circ} = 0.9195$), so
$\JCKM = 0.9735 \times 0.2250 \times 0.0418 \times 0.0037 \times
0.9195 \simeq 3.1\times10^{-5}$ (measured:
$3.12^{+0.13}_{-0.12}\times10^{-5}$); the first-order phase
($\sin 72^{\circ} = 0.9511$) gives $3.2\times10^{-5}$. All of CP
violation in the quark sector is
this one area, and it is a lattice point's sine times four
measured moduli.

\subsection{Worked example III: \texorpdfstring{$\sin
2\beta$}{sin 2beta} as a holonomy difference}
\label{ssec:we3}
The unitarity-triangle angle $\beta =
\arg(-V_{cd}V_{cb}^{*}/V_{td}V_{tb}^{*})$ is a difference of
plaquette holonomies (Appendix~\ref{app:flavour}). With the lattice phase
$\dCKM^{\mathrm{QD}} = 66.86^{\circ}$ and the PDG-class moduli
($s_{12} = 0.2250$, $s_{23} = 0.0418$, $s_{13} = 0.0037$) the
recomputation returns
$\sin 2\beta \simeq 0.708$ against the measured
$0.709 \pm 0.011$ (mid-2026 average; a $-0.1\sigma$ pull,
watched --- the evaluation layer is stated because the value is
layer-sensitive at the few-percent level, spanning
$0.707$--$0.727$ across the $s_{23}$ and phase-layer choices):
the $B$-factory observable, recomputed as
interferometry of the same single angle as
Worked example~II --- one phase, two mesons, one pentagon.

\subsection{Worked example IV: the neutrino asymmetry at
\texorpdfstring{$196^{\circ}$}{196 deg}}
\label{ssec:we4}
The leptonic CP observable of the long-baseline program is the
$\nu_\mu \to \nu_e$ versus $\bar\nu_\mu \to \bar\nu_e$ rate
asymmetry; at the oscillation maximum it is proportional to
$\JPMNS = c_{13}^2 s_{13} c_{12} s_{12} c_{23} s_{23}
\sin\dPMNS$. With the NuFit angles and the lattice point
($\sin\dPMNS = -8\sqrt3/49 = -0.283$ at first order; $-0.277$
at the QD phase): $\JPMNS \simeq -9.4\times10^{-3}$ --- an
appearance asymmetry $A_{CP} \simeq -0.28\sin\delta \simeq
0.28 \times 0.283 \simeq +8\%$ at the T2K/DUNE baseline
maxima: for $\sin\delta < 0$ in this convention the
\emph{neutrino} rate is enhanced relative to the antineutrino
rate (the interference term of Eq.~\eqref{eq:e3-golden} enters
the neutrino channel as $-\sin\Delta\sin\delta$;
Appendix~\ref{app:nu-appearance}), and at DUNE's baseline the
matter-induced asymmetry is comparable to or larger than this
genuine effect and must be separated by its known $E$ and $L$
dependence (problem H.3.2). DUNE's $\pm15^{\circ}$ resolves
$196^{\circ}$ from the CP-conserving $180^{\circ}$ at
$\sim1\sigma$ and from maximal $270^{\circ}$ at $\sim5\sigma$:
the adjudication of Table~\ref{tab:falsifiers}, in one number.

\subsection{Worked example V: the dial-free door, in matrices}
\label{ssec:we5}
For a weight-$n$ real pair, the symmetric square decomposes as
$\begin{psmallmatrix} a^2 - b^2 & ab \\ ab & b^2 - a^2
\end{psmallmatrix}$-type doublet (rotating with weight $2n$) plus
the invariant $a^2 + b^2$. Under $\alpha = \pi/2n$ the doublet
flips sign: no invariant self-pairing exists for $n \neq 0$. For
$n = 0$ the ``rotation'' is the identity and the reality condition
is the Majorana condition --- three lines of $2\times2$ algebra
carrying T-P5-4, the seesaw's uniqueness, and Part~VIII's door.

\begin{table}[t]
\centering
\caption[Forced values against the world's numbers.]{Forced values against the world's numbers. Pulls are
quoted against current uncertainties; the register column of each
chain applies (\S\ref{sec:claims-ledger}). The $\dCKM$ pulls
carry a one-lattice-step theory width $\pi/35$ in quadrature ---
against experimental error alone the raw $2\pi/5$ sits at
$+4.2\sigma$; the $\JCKM$ band spans the framework's
evaluation-layer choices. No entry was fitted;
every entry can move only by dying.}
\label{tab:predicted-measured}
\small
\renewcommand{\arraystretch}{1.22}
\begin{tabular}{p{3.6cm} p{3.2cm} p{3.2cm} p{3.6cm}}
\toprule
\textbf{Observable} & \textbf{Forced value} & \textbf{Measured} &
\textbf{Status} \\
\midrule
$\sin\theta_{12}^{\mathrm{CKM}}$ & $0.22502$
\eqref{eq:cabibbo} & $0.22501 \pm 0.00068$ & pull $+0.01$;
derived-cond. \\
$\dCKM$ & $2\pi/5 = 1.2566$; QD $13\pi/35 = 1.1669$ &
$1.147 \pm 0.026$ & pulls $+1.17$ / $+0.70$ \\
$\JCKM$ & $3.1\times10^{-5}$ (layer band $2.7$--$3.2$) &
$(3.12^{+0.13}_{-0.12})
\times10^{-5}$ & inside band \\
$\sin 2\beta$ & $0.708$ (WE~III) & $0.709 \pm 0.011$ & pull
$-0.1$; watched \\
$\dPMNS$ & $196.4^{\circ}$; QD $196.06^{\circ}$ & NO:
$177^{\circ}{}^{+19}_{-20}$; IO: $285^{\circ}{}^{+25}_{-28}$ &
$+1.0\sigma$ (NO) \\
$(\alpha_{21}, \alpha_{31})$ & $(0, \pi)$ & unmeasured &
parity data; $0\nu\beta\beta$-testable \\
$\mbb$ & $1.72$~meV & $< 28$--$122$~meV & below reach;
tonne-scale+ target \\
$\JPMNS$ & $-9.4\times10^{-3}$ & sign disfavored $> 0$
\citep{T2K2020} & consistent \\
$\bar\theta$ & $\lesssim 4.7\times10^{-18}$ & $< 10^{-10}$
(nEDM) & eight orders of headroom \\
$\etaB$ & $\approx7\times10^{-10}$ (central $7.34$; $\times3^{\pm1}$) & $(6.12\pm0.04)\times10^{-10}$ & $+20\%$ central; D$^{*}$ \\
\bottomrule
\end{tabular}
\end{table}

\section{What is claimed, and what is inherited}
\label{sec:pentagon-claims}

Concessions first, in full --- and for this Part they must be
made with particular care, because the Part's centerpiece sits
on top of a large inheritance. The rephasing-invariant calculus
--- quartets, the uniqueness of the physical phase at $N = 3$,
the invariant $J$ --- is Jarlskog's and her successors'
\citep{Jarlskog1985}; clauses (i)--(iii) of T-P5-3 geometrize
known mathematics, and the graph packaging is convenience, not
content. The idea that CP violation is \emph{geometric} ---
a property of compact internal structure, broken spontaneously
rather than dialed --- is string theory's, in the line from
Strominger and Witten through Dine--Leigh--MacIntire and
Choi--Kaplan--Nelson (\citealp{StromingerWitten1985};
\citealp{DineLeighMacIntire1992}; \citealp{ChoiKaplanNelson1993}),
where CP is a discrete gauge symmetry broken by moduli values.
Discrete flavor-symmetry model building routinely produces
predicted CP phases, including values of $\dPMNS$. The
relaxation template for the strong angle is Peccei and Quinn's
\citep{PecceiQuinn1977}, with the minimum certified by Vafa and
Witten \citep{VafaWitten1984}. Each of these is cited at its
point of use, and Part~X gives each its own section.

What this Part claims, ranked by confidence, at the grades of
Table~\ref{tab:part4-ledger}. \emph{First --- the load-bearing
claim --- the lattice valuation.} No framework known to this
program \emph{values} the holonomy: the Standard Model leaves
the loop angle continuous and unexplained; string theory's
geometric phases are moduli-valued and landscape-dependent,
predicting that CP violation is possible but never what it is;
discrete flavor models predict phases only after a symmetry
group is chosen from a catalog. The pentagon is not chosen:
$L = 5$ is the ledger's, fixed upstream by twenty-one companion
treatments for independent purposes, and the resulting values
--- $2\pi/5 \to 13\pi/35$, $\pi + 2\arcsin(1/7)$, $(0, \pi)$
--- are forced, falsifiable, and already under test
(Table~\ref{tab:predicted-measured}). \emph{Second, the
two-fate architecture:} the locked weak angle and the relaxed
strong angle as the two fates one alphabet allows --- a
structural connection between the Standard Model's two CP
embarrassments that no comparison framework draws at all.
\emph{Third, the geometrized criterion:} CP violation as
chirality of the texture on the circle --- modest as
mathematics, clarifying as anatomy, and the bridge over which
Part~V's five fates and this Part's lattice meet. \emph{Fourth,
the dial-free door:} T-P5-4's uniqueness theorem, which
converts the seesaw from a model class with a continuum of
dials into a single constitutional channel with none. The first
claim carries the honest residue named in the ledger (the
rigidity of the holonomy moduli, and the flagged QD step); the
second carries lemma L9; the third is derived; the fourth is
derived. Nothing above leans on a grade the ledger does not
print.

\section{Status ledger for the mirror Parts}
\label{sec:part4-ledger}

\begin{table}[h]
\centering
\caption[The mirror Parts at a glance.]{The mirror Parts at a glance. ``Derived'' means proved at the
stated grade on the page or in the cited parent theorem;
``derived-conditional'' means the chain is explicit and one named
step is asserted or outstanding. Eight results, each with its label, its grade, and its named residue, so that the Part's standing can be read down one column rather than reconstructed from the text. Four rows are derived and four derived-conditional, and in every conditional case the residue column says which single step is asserted or outstanding.}
\label{tab:part4-ledger}
\small
\renewcommand{\arraystretch}{1.2}
\begin{tabular}{p{4.3cm} p{2.6cm} p{2.3cm} p{4.6cm}}
\toprule
\textbf{Result} & \textbf{Label} & \textbf{Grade} &
\textbf{Named residue} \\
\midrule
C/P/T table; antiunitary $\Leftrightarrow$ T & T-P5-1 & derived &
--- \\
CPT for real textures & T-P5-2 & derived (quasi-free) & interacting
case on scattering hypotheses; modular route open \\
Holonomy calculus (i)--(iii) & T-P5-3 & derived & (inherited
mathematics, geometrized) \\
Pentagonal lattice (iv) & T-P5-3 & derived-conditional & rigidity
of the holonomy moduli \\
$\dCKM$: $2\pi/5 \to 13\pi/35$ & \S\ref{ssec:ckm} & derived-cond. &
QD step asserted (flagged post-hoc) \\
$\dPMNS$, $(\alpha_{21},\alpha_{31})$, $\mbb$ & \S\ref{ssec:pmns} &
derived-cond. & leptonic assignment lemma \\
Majorana iff weight zero & T-P5-4 & derived & --- \\
$\bar\theta$: no knob $+$ cancellation $+$ Kaxion &
\S\ref{sec:strongcp} & derived-cond. & lemma L9 (determinant
phase) \\
\bottomrule
\end{tabular}
\end{table}

\medskip
\noindent\emph{Part-end summary.} Where CP phases live: in
closed loops, as holonomies no dial can touch. What values them:
one circle, one alphabet, the pentagon --- interior angle in,
central angle out, the golden ratio underwriting the Cabibbo
magnitude on the way. What follows: the quark phase at
$13\pi/35$, the lepton phase at $196^{\circ}$, the Majorana
parity pair, two Jarlskog areas, and a strong-CP angle bounded
at $10^{-18}$ by the alphabet's own integers --- every number a
confrontation, none a fit, the flags flown where the chain is
conditional. The pentagon the ledger forces has priced the
second broken mirror; the neutrino's self-pairing is the unique
open door; and the strong-CP angle is the one the geometry lets
relax. One consequence of the orientation remains --- the
largest: the matter of the universe. It is the next Part.

\GGPart[The origin of matter]{The Origin of Matter}{The number that survived the fire, certified three ways; Sakharov's three locks derived, each already paid; the one channel the conservation laws leave open; the chain, audited step by step; and the resonance question faced in public}
\label{part:matter}

\section[Part~VIII in plain English]{Part~VIII in plain English: why there is something rather than nothing}
About six parts in ten billion of the early universe were matter, and the rest annihilated to light. That number is measured twice, from the light elements and from the microwave background, and the two measurements agree. This Part derives it.

Sakharov's three conditions are met, and for each the Part says who owns it: which are derived here, which are inherited, and at what grade. Leptogenesis is not chosen from a menu --- it is forced, because the door this geometry leaves open is the one the heavy neutrinos walk through. The chain is then run step by step, with the dial count stated in the open: zero fitted parameters.

The honesty sits in the same Part rather than in a later one. The resonance question is faced instead of assumed away; the throughput's exponent is a postulate and is called one; the flavored asymmetry is derived-conditional on a named hypothesis; and the two-route bridge for the baryon density leaves a gap whose owner has not been identified. A chain with no fitted parameters and a named gap is worth more than a chain with one fitted parameter and none, but only if the gap is on the page. The status ledger at the end carries all of it.

\section{The observational facts}
\label{sec:matter-facts}

Everything in this paper has been walking toward this Part. The
twins were forced (Part~IV); the mirrors were defined and their
selective breakage derived (Parts~V--VII); and now the paper
must pay the question it opened with: \emph{why is there
anything?} Let us begin by seeing the event whose outcome we
are. In its first second the universe was a furnace of matter
and antimatter in almost equal numbers --- pairs created and
annihilated in equilibrium, particle for antiparticle, as the
handshake of Part~IV guarantees they come. As it cooled, the
burning stopped being replenished: pair by pair, matter and
antimatter met and left as light. Had the ledger balanced
exactly, that would have been the whole history of the material
universe --- a flash, then radiation thinning forever. It did
not balance. For roughly every billion pairs that burned, one
particle of matter had no partner and survived. Every galaxy,
every star, every atom in every reader is that remainder: we
are not made of the universe's substance so much as of its
\emph{rounding error}. The question of this Part is the deepest
bookkeeping question there is: who wrote the one?

\emph{The number, certified three ways.} Part~I stated the
target; here it is assembled as the boundary condition every
mechanism must hit, with each certificate walked.
\emph{Certificate one --- the CMB.} The baryon-to-photon ratio
is read from the acoustic peaks: baryons load the
photon--baryon fluid, and the relative heights of the
compression and rarefaction peaks weigh that loading directly
--- $\etaB = (6.12\pm0.04)\times10^{-10}$, a $0.7\%$
measurement of a one-in-a-billion number, at $t \simeq
380{,}000$ years \citep{Planck2018}. \emph{Certificate two ---
the concordance.} Primordial deuterium, read in quasar
absorption systems, records the competition between nuclear
burning and expansion at $t \sim 3$ minutes, and returns the
same $\eta$ \citep{Cooke2018}. Two epochs five orders of
magnitude apart in temperature, two entirely different physical
thermometers, one number: the asymmetry was in place before
nucleosynthesis began, and baryon number has been conserved
since. \emph{Certificate three --- the absence.} A universe of
matter and antimatter domains would burn at the seams, and the
diffuse gamma-ray sky shows no such annihilation glow: domain
structure is excluded out to essentially the observable
universe \citep{CohenDeRujulaGlashow1998}; the cosmic rays
contain no confirmed antinuclei \citep{AMS02}. The asymmetry is
global --- not matter here and antimatter hidden elsewhere.

\emph{The counterfactual, computed.} And the excess was
necessary, not decorative. In a baryon-symmetric universe
annihilation continues until the survivors are too dilute to
find each other; the standard freeze-out estimate leaves
$n_B/n_\gamma \sim 10^{-18}$ --- \emph{nine orders of magnitude}
below observation, a universe of pure radiation with a trace of
dust, no stars, no chemistry, no readers. Annihilation is
efficient; only an excess laid down early survives it. One
number, three independent certificates, and a counterfactual:
this is as clean as cosmological boundary conditions come, and
it is the cleanest evidence in all of physics that the Standard
Model --- which predicts essentially $\etaB = 0$ (Part~X) ---
is incomplete.

\section{Sakharov's conditions, and who owns them}
\label{sec:sakharov-custody}

In 1967, in a short paper decades ahead of its audience,
Sakharov identified what \emph{any} dynamical origin of the
excess must contain \citep{Sakharov1967}, and the three
conditions deserve to be derived rather than recited, because
each is a small theorem. \emph{First, baryon-number violation}
--- for the plainest of reasons: if every interaction conserves
$B$, a universe that starts symmetric ($B = 0$) stays symmetric
forever; the one cannot be written by bookkeeping that forbids
writing. \emph{Second, C and CP violation.} Suppose C held:
then for every process making a baryon, the conjugate process
makes an antibaryon at the same rate, and the net vanishes.
Suppose CP held while C broke: then left-handed excesses are
compensated by right-handed ones, and summed over helicities
the net vanishes again. Only if both mirrors are broken can the
rates themselves take sides. \emph{Third, departure from
equilibrium} --- the subtlest, and it is Part~V's unbreakable
product doing the enforcing: in thermal equilibrium the state
is determined by the Hamiltonian's spectrum, CPT pairs every
baryon state with an antibaryon state of equal energy, and the
Boltzmann weights match --- $\langle B\rangle_{\mathrm{eq}} = 0$.
An asymmetry can only be written \emph{while the system is
falling}, never while it rests. Three locks on the vault, and
every proposed origin of matter must pick all three.

Now the striking fact, and the reason this Part can be short on
invention: \emph{the \GG{} architecture had already paid all
three conditions, for reasons that had nothing to do with
cosmology}. In the cosmology companion's formula: \emph{\GRIP{}
owns the angle, \GDOS{} owns its observability, \OGGEP{} owns
the ledger} \citep{ArisakaA10Cosmos}. \emph{Baryon-number
violation:} the bulk baryonic ledger is exactly conserved ---
$\Uone{B}$ is gauged, the proton absolutely stable --- and what
evolves is the projected boundary subsystem's $B{-}L$ budget,
through the unique $\Delta L = 2$ door of T-P5-4: the door was a
uniqueness theorem of Part~VII before it was a cosmological
channel. \emph{C and CP violation:} the CP-odd input is the
lattice-locked loop holonomy of Part~VII --- a real angle of the
inherited flavor geometry, not a tunable phase; it exists
because the pentagon is chiral, not because baryogenesis needed
it. \emph{Departure from equilibrium:} constitutional --- the
boundary law is a forward CPTP semigroup (openness is the
default, AX-3; the same one-way projection that owns the arrow
of time, \S\ref{sec:custody}), and the specific
out-of-equilibrium episode is the decay of the heavy sector at
the lock crossing. Where the Standard Model fails Sakharov on
the arithmetic (Part~X) and its extensions pass by installing
new machinery, here the three locks were opened by keys the
constitution had already cut.

This section now does what the fifty years since 1967 have not
asked of the three conditions: \emph{derive the keys}. Sakharov's
list has been checked, mapped, and satisfied in every
baryogenesis proposal since it was written --- and taken for
granted in all of them: three boxes on a checklist, to be
arranged by model-building, their own origin unquestioned. What
follows traces each condition to the constitutional structure
whose boundary shadow it is --- pure geometry, one condition at a
time --- and then recognizes the three together as something the
paper has already met.

\emph{Origin (i) --- the two books and the still-point door.}
Begin with what makes the bulk ledger \emph{absolute}. A gauged
charge is not conserved by a bookkeeping rule that dynamics might
amend; it is conserved by the geometry of its own field: the
Gauss law reads the total charge as a flux through a distant
surface, so no local process anywhere in the interior can change
it --- gauged conservation is a statement about the arena, not
about the interactions. The bulk's \Uone{B} is of exactly this
kind, and its absoluteness is why the proton is absolutely
stable. Sakharov's first condition would therefore seem
\emph{unpayable} in this geometry --- and here is where the
still point of the circle earns its cosmological office. The
winding bookkeeping of the fiber has exactly one gap: the
weight-zero sector, where the circle moves nothing and the
lepton-number book alone may take a $\Delta L = 2$ entry
(T-P5-4, seeded at the still point of
\S\ref{sec:self-conjugate}). The boundary's evolving budget is
$B{-}L$; the door writes lawful entries into it without
touching the bulk book; and the sphalerons --- boundary
machinery of the projected subsystem, active above the
crossover --- exchange the budget's columns. Condition (i) is
thereby satisfied \emph{without breaking anything}: not a
violated symmetry but the coexistence of two books kept at two
levels of one geometry, with a single lawful door between them
--- the precise content of \emph{\OGGEP{} owns the ledger}.

\emph{Origin (ii) --- the oriented descent and the chiral
lattice.} The second condition asks for broken C and broken CP,
and the paper has already derived both --- what matters here is
\emph{from what}, and how compulsorily. C falls by an absence:
the projection of Part~VI keeps one hand of the spectrum, so
fiber reversal maps kept states onto states that are not there
--- maximal violation, from the descent's one-handedness. CP
survives the absence and falls by an angle: the pentagon of
Part~VII locks the flavor holonomy at $3\pi/5$, which is not a
multiple of $\pi$, so the texture cannot be achiral
(T-P5-3(iv)). Now set this against the standard situation, for
the contrast is the whole point. In the Standard Model,
$\delta = 0$ --- a CP-symmetric world, sterile of receipts ---
is exactly as lawful as the measured value; Sakharov's second
condition holds there by \emph{accident of a dial}. Here it
holds by \emph{theorem}: a \GG{} universe with CP symmetry is
unwritable, because writing one would require a pentagon angle
in $\pi\bZ$, and the lattice contains no such texture with the
forced branch phase. The orientation of the descent, seen twice
--- the hand and the angle --- \emph{is} condition (ii), and it
cannot relax, cannot be tuned away, and was not installed for
cosmology's benefit: \emph{\GRIP{} owns the angle}.

\emph{Origin (iii) --- the always-on projection and the fall.}
The third condition is the subtlest, and the clock's third face
(\S\ref{sec:wick-quartet}) lets us state its geometric content
exactly. What \emph{is} equilibrium, in this geometry? A state
at rest on the clock's thermal shore: periodic in imaginary
time (the KMS condition), invariant under the evolution,
CPT-balanced --- and therefore, by the argument that opened
this section, carrying $\langle B\rangle = 0$ identically. An
asymmetry can be written only by a state that is \emph{off the
shore} --- falling, not resting. And here the constitution does
something no standard framework does: it \emph{denies the
boundary the option of rest}. The boundary law is a forward
CPTP semigroup --- openness is not an episode but the default
(AX-3); the projection is always on and never invertible; and
the cosmology it reads is a Morse descent --- a landscape with
a downhill and a flow strictly along it, a falling system
\emph{by construction}, with the lock crossing as the specific
stumble at which the heavy sector decays out of step with the
bath. In the Standard Model the third condition failed for want
of a first-order transition --- the universe, on that
accounting, never fell hard enough. Here the fall is the
architecture: \emph{\GDOS{} owns the observability}, the
projection owns the arrow, and departure from equilibrium is
what the descent \emph{is}.

Three origins, and now the recognition. Set them beside the
custody map of Part~III:

\begin{center}
\emph{the bulk owns the symmetries} \;---\; the two books and
the door \;---\; Sakharov (i)\\[2pt]
\emph{the descent owns their breaking} \;---\; the hand and the
angle \;---\; Sakharov (ii)\\[2pt]
\emph{the boundary owns their observability} \;---\; the
projection and the fall \;---\; Sakharov (iii)
\end{center}

\noindent The alignment is exact, and it is the deep fact of
this section: \emph{Sakharov's conditions are the custody map,
read as a parts-list.} In 1967 Sakharov proved the necessity
direction --- no receipt without these three. The geometry
supplies the converse in kind --- a universe built of exactly
these three --- and the identification deserves its formal
statement:

\begin{proposition}[\textbf{P-P5-E}: the Sakharov tripod]
\label{prop:sakharov-tripod}
In the \GG{} architecture the three Sakharov conditions are the
boundary shadows of three constitutional structures, each in
place upstream of cosmology and each carried by a named result:
(i) \emph{lawful baryon-budget evolution}, from the two-level
ledger --- exact gauged bulk \Uone{B}, the unique weight-zero
$\Delta L = 2$ door (T-P5-4), and boundary sphaleron
reprocessing; (ii) \emph{compulsory C and CP violation}, from
the descent's orientation --- the one-handed projection
(Part~VI) and the chiral pentagon (T-P5-3(iv)), lattice-locked
and unable to relax; (iii) \emph{constitutional departure from
equilibrium}, from the always-on, non-invertible boundary
projection (AX-3) driving the Morse descent, with equilibrium
itself characterized as the CPT-balanced state on the clock's
thermal shore. Consequently a \GG{} universe cannot fail
Sakharov: the generation of a nonzero baryon asymmetry through
the unique open channel is structural, and only its
\emph{magnitude} (T-P5-5) depends on the named conjecture
C-P5-C.
\end{proposition}

\begin{proof}[Proof, by assembly]
Each clause is the citation of a result already established, and
the proposition's content is the identification, at the grade of
its parts. Clause (i): gauged conservation and proton stability
are ledger results of the parent program; the door is T-P5-4
(derived); sphaleron reprocessing is standard boundary physics
\citep{KuzminRubakovShaposhnikov1985}. Clause (ii): the
projection's one-handedness carries Part~VI's honesty ledger
(mechanism derived, selection graded there); the pentagon's
chirality is T-P5-3(iv) (derived-conditional on the rigidity
residue); jointly they make condition (ii) compulsory at those
grades --- strictly stronger than the Standard Model's
accidental satisfaction, in which the CP-conserving world is
equally lawful. Clause (iii): the forward-semigroup character of
the boundary law is constitutional (AX-3); the equilibrium
characterization is the KMS remark of T-P5-B1 (conditional as
there flagged); the Morse descent is the parent cosmology's
launch architecture. For the consequence: with all three
conditions holding and the lepton channel open
(\S\ref{sec:why-leptogenesis}), the decay of the heavy sector at
the crossing generates a CP-biased lepton asymmetry of
\emph{some} magnitude even arbitrarily far off the resonance
peak --- the self-energy enhancement is suppressed off-peak as
$\Gamma/\Delta M$, not extinguished --- and sphalerons reprocess
a nonzero fraction into baryons. Vanishing exactly would require
a texture angle in $\pi\bZ$ or a closed boundary, both
unwritable here. The observed \emph{size} is the business of the
transport chain and its named conjecture, exactly as
Table~\ref{tab:audit} prints.
\end{proof}

\emph{Physics intuition --- Sakharov, read backward.} For fifty
years the three conditions have been used in one direction: a
checklist that any proposed mechanism must satisfy, three boxes
arranged by adding machinery --- new bosons for the first, new
phases for the second, new phase transitions for the third. Read
in this geometry, the list inverts, and the inversion is the
profound part. The conditions are not requirements the universe
happens to meet; they are the \emph{parts-list of an oriented
descent}: a book that is kept (the ledger, with its one lawful
door), a fall that has a direction (the orientation, worn as a
missing hand and a locked angle), and a readout that cannot be
undone (the projection, whose one-wayness is the arrow of time
itself). Sakharov's theorem said: no universe keeps a receipt of
its history without these three. The custody map replies: this
universe is \emph{made} of these three --- they are its
constitution's three offices, occupied before cosmology posed
its question, for reasons (charge bookkeeping, family structure,
the very existence of records) that had nothing to do with
baryons. That is why the section could say the locks were opened
by keys already cut: the deepest origin of the matter excess is
not a mechanism but a \emph{shape}. The universe is a fall that
keeps books --- and a fall that keeps books leaves a receipt.
The one-in-a-billion of Part~VIII's title is the receipt's size,
and the size must still be computed, conjecture and all; but
that there is a receipt at all was never, in this geometry,
optional. After all, Sakharov wrote his three conditions as the
price of explaining our existence --- and the quiet answer of
the tripod is that the price was paid at the founding, in the
constitution, before the universe had anything to buy.

\section{Why leptogenesis is forced}
\label{sec:why-leptogenesis}

Exact gauged $\Uone{B}$ narrows the field to one mechanism class
before any cosmology is computed, and the narrowing is worth
walking door by door. \emph{The GUT door is welded shut.} The
historical first mechanism --- out-of-equilibrium decay of
superheavy baryon-violating bosons \citep{Yoshimura1978,
Weinberg1979} --- requires operators that violate $B$, and the
bulk's gauged $\Uone{B}$ forbids them exactly: the same ledger
that makes the proton absolutely stable makes GUT baryogenesis
unwritable. The parent program counts this economy a virtue
rather than a loss, for the standard tension runs the other
way: in GUT scenarios the sector that creates baryons is
forever threatening to eat the proton, and the model builder
must arrange the mediators heavy enough to spare it.
Here the baryon-making sector \emph{cannot} touch the proton,
by constitution. \emph{The electroweak door is closed by
inheritance.} Electroweak baryogenesis needs a first-order
phase transition to push bubbles of broken phase through the
plasma; at the physical Higgs mass the transition is a smooth
crossover \citep{Kajantie1996} --- no bubbles, no departure ---
and the \GG{} boundary inherits the Standard Model's Higgs
sector intact; the crossover, and the Standard Model's
$10^{-20}$ CP shortfall besides (\S\ref{sec:road-sm}), close
the door twice. \emph{What remains is the lepton route}
\citep{FukugitaYanagida1986}: create the asymmetry as $L$,
through the unique lawful $\Delta L = 2$ door of T-P5-4, and
let the sphalerons \citep{KuzminRubakovShaposhnikov1985} ---
the electroweak theory's own high-temperature $B{+}L$
reprocessors, active above the crossover --- convert $B{-}L$
into $B$. The framework is, in the cosmology companion's words,
pushed ``almost automatically toward leptogenesis'' --- the only
Sakharov-complete channel its own conservation laws leave open.

\emph{And the scale makes it testable --- the Davidson--Ibarra
point.} Standard thermal leptogenesis with hierarchical heavy
neutrinos carries a famous cost: the CP asymmetry per decay is
bounded by the light and heavy masses, and demanding enough
asymmetry forces $M_N \gtrsim 10^{9}$~GeV
\citep{DavidsonIbarra2002} --- a beautiful mechanism, forever
beyond every collider, its phases decoupled from the measured
PMNS phases. The known escape is \emph{resonant} leptogenesis
\citep{Pilaftsis1997, PilaftsisUnderwood2004}: for
quasi-degenerate heavy states the self-energy enhancement
lifts the asymmetry by orders of magnitude, evading the bound
and admitting TeV-scale heavy neutrinos --- at the price, in
the standard literature, of \emph{engineering} the degeneracy.
The \GG{} chain sits in exactly this regime, but with the
inputs inverted: the TeV scale is not chosen for
accessibility, it is the lock scale $\tfrac12 M_{\mathrm{Pl}}
e^{-35}$ the ledger fixed for its own reasons --- and the
degeneracy question is not engineered away but faced in
public, as the named conjecture of
\S\ref{sec:resonance-faced}. The bound is evaded where the
literature evades it; what is new is who pays, and how
visibly.


\subsection{Why the quark sector cannot pay, in one picture}
\label{sec:quark-cannot-pay}

The preceding section forces leptogenesis by a door argument: sphalerons
preserve $B-L$, so only an asymmetry written in lepton number at high scale
survives reprocessing, and only a Majorana phase can drive it. That
argument is structural and it is complete. It is worth adding the
quantitative statement beside it, because a reader who has just been told
that the laboratory's flashiest $CP$ violation is irrelevant to the origin
of matter will reasonably want to see the size of the thing being set aside.

\begin{figure}[htbp]
\centering
\resizebox{0.66\textwidth}{!}{%
\usebox{\GGplateTwoAreas}}
\caption[The two unitarity triangles at true relative area.]{The quark and
lepton unitarity triangles drawn to true relative area. By the one-area
theorem of
\S\ref{sec:holonomy} a single number prices all of the $CP$
violation in a sector, and the two numbers are both measured: $J_{\rm CKM} =
3.1\times10^{-5}$ against $|J_{\rm PMNS}| \simeq 9.4\times10^{-3}$. The
lepton area is some three hundred times the quark area --- the lepton phase
is large because $\delta_{\rm PMNS}$ sits near $196^\circ$ and, decisively,
because the lepton mixing angles are large where the quark angles are small.
The disparity relevant to baryogenesis is larger still, because the quark
contribution is throttled a second time by the product of quark
mass-squared splittings.}
\label{fig:two-areas-p5}
\end{figure}

The one-area theorem is what makes the comparison legitimate rather than
rhetorical. Because every plaquette of the flavor graph carries the same
area, the whole of a sector's $CP$ violation is one number, and two sectors
can be set side by side without choosing a convention or a channel. The
quark sector's number is real, it is measured to per-cent precision, and it
is a hundredfold too small before the thermal history touches it. What the
quark sector \emph{is} in this framework is the laboratory --- the place
where the pentagon's prediction is confronted at Belle~II and LHCb. What it
is not is the origin of us.

\section{The chain, step by step}
\label{sec:chain}

\begin{proposition}[\textbf{T-P5-5}: the assembled $\etaB$, at
D$^{*}$ grade]
\label{prop:etaB}
On the inherited ledger --- heavy masses at the lock, the dial-free
door, the seesaw-matched Yukawa, the holonomy CP phase --- and
conditional on the on-resonance condition (Conjecture C-P5-C, whose
sharp form is the partial-resonance hypothesis
H-$\phi_{\mathrm{geo}}$-1; \S\ref{sec:resonance-faced}), the
boundary transport yields
$\etaB \simeq 5.9\times10^{-10}$, within the factor-$\sim3$ theory
band of the observed $(6.12\pm0.04)\times10^{-10}$.
\end{proposition}

The chain, each step with its number and grade; the audit is the
point, so no step is skipped.

\emph{1. Heavy scale (derived).} The seesaw partners sit at the
lock: $M_{N_i} = \Mlock(1 + \delta_i)$ with
$\Mlock = \tfrac12 M_{\mathrm{Pl}} e^{-K} \simeq 3.85$~TeV and
geometric splittings $\delta_i = O(1/K)$ --- percent-level spacing
in the few-TeV corridor. No extreme ultraviolet prehistory is
needed, and the derived reheating temperature
$T_{\mathrm{rh}} \simeq 3\times10^{10}$~GeV populates the sector
thermally: the mechanism is selected by the launch, not chosen.

\emph{2. Portal (derived).} T-P5-4: the weight-zero self-pairing is
the unique $\Delta L = 2$ structure, $LHN$ the unique portal. No
choice enters.

\emph{3. Yukawa (derived).} Seesaw matching \eqref{eq:seesaw} to the
inherited light masses and the derived seam coefficient fixes
$Y_\nu^2 \simeq \pi \Mlock/E_{\mathrm{GUT}} \simeq 5.3\times10^{-13}$,
$Y_\nu \simeq 7.3\times10^{-7}$ --- and hence the width
$\Gamma_N = Y_\nu^2 M_N/8\pi \simeq 8\times10^{-11}$~GeV,
$\Gamma_N/M_N \simeq 2\times10^{-14}$. A width this small would doom
hierarchical leptogenesis; the resonant regime is what rescues it.
One bookkeeping convention is declared here: this width is quoted at
the seam-matched one-generation Yukawa, which corresponds to
$\widetilde m \simeq 8.4$~meV, while step~6 runs washout at
$\widetilde m = m_3 \simeq 50.5$~meV --- conventions differing by a
factor $\simeq13$ in $Y_\nu^2$. Run at the washout convention the
width is $\Gamma_N \simeq 1\times10^{-9}$~GeV: an $O(10)$ shift
inside the order-of-magnitude register, absorbed into the effective
width $\Gamma^{\mathrm{eff}}$ that H-$\phi_{\mathrm{geo}}$-1
parameterizes (\S\ref{sec:resonance-faced}); the vacuum off-peak gap
reads eleven-to-twelve orders in either convention.

\emph{4. Resonance (the named input).} At the self-energy resonance
$|M_i - M_j| \simeq \Gamma_j/2$ the enhancement saturates and the
Yukawa magnitude cancels between numerator and denominator
\citep{Pilaftsis1997, PilaftsisUnderwood2004}: the asymmetry is
controlled by the geometric phase alone,
$\epsilon^{\mathrm{res}} \simeq \tfrac12\sin(2\phi_{\mathrm{geo}})$.
\textbf{The on-resonance condition is the chain's one named input}
(C-P5-C); Section~\ref{sec:resonance-faced} quantifies what it
asks.

\emph{5. CP phase (derived-conditional on H-$\phi_{\mathrm{geo}}$-1).}
The unflavored CP invariant of the resonant formula is
$U_{\mathrm{PMNS}}$-independent (the Casas--Ibarra cancellation) and
vanishes exactly for the dial-free real-$R$ door; at the TeV lock
scale the chain is fully flavored, and the inherited phases reach
$\etaB$ only through the flavored asymmetries
$\epsilon_{i\alpha}$, now computed on the ledger inputs (\ATen{}
Appendix~I). The driver is the Majorana parity $\alpha_{31} = \pi$,
which sets the sign of the matter excess (the numerics are
$\JPMNS$-blind at the few-percent level), and the result is nonzero
at $m_1 = 0$: the asymmetry does not ride on the QD layer. The
mechanism brackets the observed $\etaB$ --- on the exact
Breit--Wigner peak it overshoots the required throughput by
$\sim3\times10^{4}$, at vacuum splittings it undershoots by
$\sim4\times10^{6}$ --- and $\epsilon^{\mathrm{res}} \simeq
2.5\times10^{-6}$ is carried as the target value the
partial-resonance hypothesis H-$\phi_{\mathrm{geo}}$-1
($\Delta M/\Gamma^{\mathrm{eff}} \simeq 3\times10^{4}$;
\S\ref{sec:resonance-faced}), now carried by an identified
candidate mechanism (the flavor-measurement regulator; closed
form $1/(35\,c_\tau y_\tau^{2})$; conditions C1--C6, home \ATen{}
Appendix~I.3), must deliver: derived-conditional,
with the finite-temperature physics the place the number must park
(row 9 of the promotion roadmap,
Appendix~\ref{app:promotion-roadmap}).

\emph{6. Washout (derived-conditional).} The equilibrium
benchmark is itself derivable in three lines and is notably
\emph{ledger-free}: equating the inverse-decay rate to the Hubble
rate at $T = M_N$ gives
\begin{equation}
m_\star \;=\; \frac{16\pi^{5/2}\sqrt{\gstar}}{3\sqrt{5}}
\,\frac{v^2}{M_{\mathrm{Pl}}}
\;\simeq\; 1.1\times10^{-3}~\mathrm{eV},
\label{eq:mstar}
\end{equation}
fixed by measured constants alone. (The three lines: the
inverse-decay rate is $\Gamma_{\mathrm{ID}} \sim \Gamma_N \sim
\widetilde m\, M_N^2/(8\pi v^2)$; the Hubble rate at $T = M_N$
is $H \sim \sqrt{\gstar}\, M_N^2/M_{\mathrm{Pl}}$ up to the
displayed numerical factor; equating them turns the comparison
``does the door re-close what it lets out?'' into a pure
neutrino-mass threshold. That $m_\star$ contains only $v$,
$M_{\mathrm{Pl}}$, and $\gstar$ --- no ledger integer, no \GG{}
input --- is why this step is marked ledger-free: nature set
the washout regime before any framework arrived.) With
$\widetilde m \sim m_3 \simeq 5\times10^{-2}$~eV the washout is
strong, $K_{\mathrm{wash}} = \widetilde m/m_\star \simeq 50$, and
the surviving efficiency is $\kappa \simeq 1/K_{\mathrm{wash}}
= 2.0\times10^{-2}$; the chain adopts $\kappa =
1.0\times10^{-2}$ within the uncomputed $O(1)$ --- with the
standard strong-washout blessing that memory of initial
conditions is erased, so the surviving asymmetry depends only on
quantities the ledger fixes. The convention note of step~3 applies
here in reverse: $\widetilde m = m_3$ is the washout convention of
the chain; at the seam-matched $\widetilde m \simeq 8.4$~meV one
would instead have $K_{\mathrm{wash}} \simeq 7.6$ and $\kappa
\simeq 0.13$. The factor-$\simeq12$ spread in $Y_\nu^2$ between the two conventions (the single-convention ratio $2\,m_3/\widetilde m_{\mathrm{seam}} \simeq 12$; the Casas--Ibarra half is bookkeeping, not physics) is declared and priced as an $O(10)$ freedom of
the order-of-magnitude register, carried with the transport
$O(1)$'s.

\emph{7. The throughput identity (the central claim).} The product
\begin{equation}
\epsilon^{\mathrm{res}}\,\kappa
\;\simeq\; (2.5\times10^{-6})(1.0\times10^{-2})
\;\simeq\; 2.51\times10^{-8}
\;=\; e^{-K/2} = e^{-17.5}
\label{eq:throughput}
\end{equation}
reconstructs, to displayed precision, the ledger's throat-depth
exponential --- and the displayed agreement uses the $O(1)$
freedom in $\kappa$: the identity holds at the adopted $\kappa =
1.0\times10^{-2}$, a factor of two inside $1/K_{\mathrm{wash}} =
2.0\times10^{-2}$. The exponent $K/2$ is a \emph{postulate} of the
compression --- geometrically suggestive as $\sqrt{e^{-K}}$, not
derived --- and fixing it from first principles is exactly what
Conjecture C-P5-C (the sphaleron-transport normalization) must do.
What is new is that the independently derived washout efficiency and the candidate regulator's closed form bracket the required throughput --- the identity at displayed precision uses exactly the $\kappa$ freedom declared above and the target status of $\epsilon^{\mathrm{res}}$. And
the reader should notice which integer has appeared, for the
third time in three Parts: the same $K = 35$ that sets the lock
scale ($e^{-K}$ in $\Mlock$) and bounds the strong-CP leakage
($e^{-K}$ in \eqref{eq:thetabar-leakage}) now measures the
matter throughput ($e^{-K/2}$). One throat depth, three
exports: a mass, a bound, and the number of us.

\emph{8. Sphalerons, entropy, and the equilibrium yield (standard).} With the one-doublet coefficient $\tfrac{28}{79}$ \citep{HarveyTurner1990}, $\gstar = 106.75$, and the heavy triplet's equilibrium yield displayed rather than absorbed --- $\sum_i Y_{N_i}^{\mathrm{eq}} \simeq 1.25/\gstar$, i.e.\ $0.42/\gstar$ per Majorana species ($g_N = 2$), each equilibrium species carrying its fair share of the bath's heat (derivation in the parent, \citealp{ArisakaA10Cosmos}, App.~I) ---
\begin{equation}
\etaB \;\simeq\; 7.04\times\frac{28}{79}\times
\frac{1.25\,e^{-K/2}}{\gstar}
\;\simeq\; 7.34\times10^{-10}.
\label{eq:etaB-pred}
\end{equation}

\begin{table}[t]
\centering
\caption[The audit table of the chain.]{The audit table: every number of the chain, its source,
and its grade. One named input (row 4), one flavored projection
carried derived-conditional on H-$\phi_{\mathrm{geo}}$-1 (row 5;
the former $m_1/m_3$-order-factor language is withdrawn as a
reconstruction artifact; the hypothesis is now carried by an
identified candidate mechanism, transport closure of C1--C6
open), one
postulated exponent (row 7), zero fitted parameters. Row 6
adopts $\kappa = 1.0\times10^{-2}$ from $1/K_{\mathrm{wash}} =
2.0\times10^{-2}$ within the uncomputed $O(1)$; the throughput
identity of row 7 uses that freedom.}
\label{tab:audit}
\small
\renewcommand{\arraystretch}{1.22}
\begin{tabular}{p{0.6cm} p{4.2cm} p{3.6cm} p{2.2cm} p{2.6cm}}
\toprule
\# & \textbf{Quantity} & \textbf{Value} & \textbf{Source} &
\textbf{Grade} \\
\midrule
1 & $M_N \simeq \Mlock$ & $3.85$~TeV & $\tfrac12 M_{\mathrm{Pl}}
e^{-35}$ & derived \\
1' & splittings $\delta_i$ & $O(1/K) \sim 1\%$ & ledger &
derived \\
1'' & $T_{\mathrm{rh}}$ & $3\times10^{10}$~GeV & inflation
sector & derived \\
2 & portal & $LHN$, unique & T-P5-4 & derived \\
3 & $Y_\nu$ & $7.3\times10^{-7}$ & seesaw match & derived \\
3' & $\Gamma_N$ & $8\times10^{-11}$~GeV & from $Y_\nu$ &
derived \\
4 & resonance & $|\Delta M| \simeq \Gamma^{\mathrm{eff}}/2$ &
\textbf{named input} & C-P5-C(i) \\
5 & $\epsilon^{\mathrm{res}}$ & $2.5\times10^{-6}$ &
flavored $\epsilon_{i\alpha}$; H-$\phi_{\mathrm{geo}}$-1 target &
derived-cond. (H-$\phi_{\mathrm{geo}}$-1) \\
6 & $\kappa$ & $1.0\times10^{-2}$ & $K_{\mathrm{wash}} \simeq
50$ & derived-cond. \\
7 & $\epsilon\,\kappa$ & $2.51\times10^{-8} = e^{-K/2}$ &
reconstruction & C-P5-C(ii) \\
8 & $c_{\mathrm{sph}}$; $\gstar$ & $28/79$; $106.75$ &
standard & standard \\
8' & $\sum_i Y_{N_i}^{\mathrm{eq}}$ & $1.25/\gstar$ (three species) &
equilibrium statistics & standard \\
9 & $\etaB$ & $\approx7\times10^{-10}$ (central $7.34$) & assembled & D$^{*}$,
$\times3^{\pm1}$ \\
--- & $\etaB^{\mathrm{obs}}$ & $(6.12\pm0.04)\times10^{-10}$ &
\citet{Planck2018} & target \\
\bottomrule
\end{tabular}
\end{table}

\begin{figure}[t]
\centering
\resizebox{0.715\textwidth}{!}{%
\begin{tikzpicture}[font=\sffamily\footnotesize, >=Stealth,
 node distance=3.5mm,
 st/.style={draw=black!55, rounded corners=2pt, align=center, inner sep=4pt,
   minimum width=52mm}]
\node[st, fill=blue!8] (a) {ledger $K = 35$ $\Rightarrow$
 $\Mlock \simeq 3.85$~TeV; splittings $O(1/K)$};
\node[st, fill=blue!8, below=of a] (b) {T-P5-4: unique Majorana
 door; seesaw $\Rightarrow$ $Y_\nu \simeq 7.3\times10^{-7}$};
\node[st, fill=orange!14, below=of b] (c) {\textbf{resonant decay}
 (C-P5-C / H-$\phi_{\mathrm{geo}}$-1): flavored
 $\epsilon_{i\alpha}$, driver $\alpha_{31} = \pi$; target
 $\epsilon^{\mathrm{res}} \simeq 2.5\times10^{-6}$\\
 CP bias from the ledger's inherited Majorana parity};
\node[st, fill=orange!14, below=of c] (d) {strong washout:
 $\kappa \simeq 10^{-2}$;\quad throughput $\epsilon\kappa =
 e^{-K/2}$};
\node[st, fill=green!10, below=of d] (e) {sphalerons
 ($28/79$) $+$ entropy ($\gstar$) $+$ yield ($1.25/\gstar$) $\Rightarrow$
 $\etaB \approx 7\times10^{-10}$};
\node[st, fill=green!10, below=of e] (f) {prompt lock crossing
 ($N_2 = 0$): zero dilution $\to$ CMB reads
 $6.12\times10^{-10}$};
\draw[->, line width=0.80pt] (a) -- (b);
\draw[->, line width=0.80pt] (b) -- (c);
\draw[->, line width=0.80pt] (c) -- (d);
\draw[->, line width=0.80pt] (d) -- (e);
\draw[->, line width=0.80pt] (e) -- (f);
\end{tikzpicture}}
\caption[The leptogenesis chain as a flowchart.]{The leptogenesis chain as a flowchart: blue = inherited
and derived, orange = the transport block carrying the named
conjecture, green = standard reprocessing and conservation. The
one direction of the whole diagram is the descent's. The blocks run top to bottom in the order the chain runs, and every arrow points the same way, because the descent has one direction and the figure is drawn to make that visible rather than to be read from either end. The color coding is what separates the inherited part of the chain from the one block that carries a named conjecture. Read a second way, with the anatomy of \S\ref{sec:anatomy-of-a-theorem} in hand, the same six blocks sort by fiber rather than by provenance: the ledger, the Majorana door and the sphaleron conversion are circle-side --- quantized, protected, the same in every region --- while the lock crossing, the exponential throughput and the entropy bath are interval-side, addresses deposited once along a one-way descent. The figure is not recolored to show this, because its colors are already carrying the conjecture mark, and a picture should not be asked to say two things with one device.}
\label{fig:chain}
\end{figure}

\emph{9. The verdict (D$^{*}$).} Against $\etaB^{\mathrm{obs}} = (6.12\pm0.04)\times10^{-10}$: $+20\%$ central, with an honest factor-$\sim3$ band from the uncomputed $O(1)$'s --- at headline precision, $\etaB \approx 7\times10^{-10}$, band $(2.4\text{--}22)\times10^{-10}$, the observation comfortably inside. The parent paper's own warning is adopted verbatim: closeness of central values at this tier must \emph{not} be read as a precision success; the genuine claim is the bracketing \eqref{eq:throughput} and the dial count --- zero
fitted parameters, one named input, one postulated exponent.

\section{Zero dilution}
\label{sec:zero-dilution}

An asymmetry made at the TeV scale must survive to the CMB, and two
standard killers stand in the way: late inflation and late entropy
release (the moduli problem that afflicts string cosmologies
\citep{CoughlanEtAl1983, BanksKaplanNelson1994}). The \GG{} descent
has neither: the lock crossing is \emph{prompt} ($N_2 = 0$), on
three independent grounds --- the nucleation action sits some
fifty-six orders above the supercooling threshold, the lock
potential is a barrier-free saddle, and the large-$N$ enhancement
that drives supercooling in warped-throat models is structurally
absent \citep{ArisakaA10Cosmos}. No secondary inflation, no stolen
e-folds, no relic dilution: the $B{-}L$ excess is fixed early,
reprocessed by sphalerons before the crossover closes the channel,
and then simply conserved --- \emph{zero dilution} --- into the
number the CMB reads. The intuition is a mint's: an asymmetry
is a coin struck once, and every late episode of inflation or
entropy release is a furnace that re-melts the coinage and
strikes less of it per photon. String cosmologies generically
carry such furnaces (the moduli problem) and so need \emph{more}
baryogenesis than field theory does; the \GG{} descent, by the
prompt-crossing lemma, carries none: the coin struck at the lock
is the coin the CMB counts, an age of the universe later.

\section{The resonance question, faced}
\label{sec:resonance-faced}

\begin{conjecture}[\textbf{C-P5-C} (sphaleron-transport
normalization; the on-resonance condition)]
\label{conj:resonance}
A first-principles transport treatment of the lock-crossing decays
exists in which (i) the effective degeneracy condition
$|M_i - M_j| \simeq \Gamma^{\mathrm{eff}}_j/2$ is realized at the
crossing, and (ii) the coarse-grained throughput of asymmetry to
entropy is $e^{-K/2}$.
\end{conjecture}

\medskip\noindent\textbf{The named hypothesis
H-$\phi_{\mathrm{geo}}$-1 (partial resonance).} \emph{There exists
a finite-temperature transport computation of the lock-crossing
decays in which the effective splitting-to-width ratio at the
crossing satisfies $\Delta M/\Gamma^{\mathrm{eff}} \simeq
3\times10^{4}$ (equivalently, Breit--Wigner factor $\simeq
1.5\times10^{-5}$; $\Gamma^{\mathrm{eff}} \simeq 3\times10^{-3}$~GeV
at $\Delta M \simeq 110$~GeV), so that the flavored, real-$R$
throughput on the inherited phases lands within $O(1)$ of
$e^{-K/2}$.} The flavored recomputation of the projection (the
cosmology companion's Appendix~I) sharpens clause (i): the sharp
form of the condition is partial resonance ---
$\Delta M/\Gamma^{\mathrm{eff}} \simeq 3\times10^{4}$, strictly
weaker than exact degeneracy and strictly stronger than vacuum
numbers; the exact-peak form overshoots the required throughput by
$\sim3\times10^{4}$, and vacuum splittings undershoot it by
$\sim4\times10^{6}$. $\Gamma^{\mathrm{eff}} \sim 3$~MeV is
parametrically plausible for thermal damping at $T \sim M_N$, which
makes H-$\phi_{\mathrm{geo}}$-1 a computation with an acceptance
band ($\pm O(1)$ in $\epsilon_{\mathrm{eff}}$), not a wish; it is
row~9 of the promotion roadmap
(Appendix~\ref{app:promotion-roadmap}), where the row now records
the identified candidate mechanism below at candidate grade.

\begin{figure}[t]
\centering
\resizebox{0.754\textwidth}{!}{%
\begin{tikzpicture}[font=\sffamily\footnotesize, >=Stealth]
 % axes: log10(DeltaM/Gamma) vs log10(enhancement)
 \draw[->] (0,0) -- (10.6,0) node[right]{$\log_{10}(\Delta M/\Gamma)$};
 \draw[->] (0,-3.4) -- (0,1.2) node[above]{$\log_{10}
   \bigl(\epsilon/\tfrac12\sin 2\phi\bigr)$};
 % curve: flat near 0 (peak at DeltaM ~ Gamma/2), falls as Gamma/DeltaM
 \draw[line width=1.10pt, blue]
  (0.0,-0.05) .. controls (0.8,-0.02) .. (1.0,-0.1)
  -- (9.6,-3.2);
 \node[blue, font=\footnotesize] at (2.6,0.55)
   {peak: $\epsilon \to \tfrac12\sin 2\phi$ at
    $\Delta M \simeq \Gamma/2$};
 % markers
 \fill[red] (0.35,-0.05) circle (1.6pt);
 \node[red, align=left, anchor=west, font=\footnotesize] at (5.6,0.55)
   {where the chain \emph{needs} to sit (C-P5-C(i))};
 \draw[->, red] (5.5,0.5) .. controls (2.5,0.35) .. (0.5,0.08);
 \fill[black] (9.1,-3.02) circle (1.6pt);
 \node[align=left, anchor=east] at (10.4,-1.5)
   {where vacuum numbers put it:\\ $\Delta M/\Gamma \sim 10^{12}$
    $\Rightarrow$ $\epsilon$ suppressed $\sim\!10^{-12}$};
 \draw[->] (9.4,-1.85) -- (9.15,-2.85);
 % thermal width arrow
 \draw[->, line width=1.10pt, green!55!black] (9.1,-3.02)
   .. controls (5.5,-3.6) and (2.6,-2.4) .. (0.7,-0.4);
 \node[green!55!black, align=left, anchor=west] at (0.3,-3.15)
   {candidate mechanism: flavor-measurement rate\\
    $\Gamma^{\mathrm{eff}} = c_\tau y_\tau^{2} T$:
    lands $(2.7\text{--}5.3)\times10^{4}$};
\end{tikzpicture}}
\caption[The resonance regime map.]{The resonance regime map (the formula is derived in Appendix~\ref{app:lepto-epsilon}). The
self-energy enhancement saturates at the Breit--Wigner peak and
falls as $\Gamma/\Delta M$ off it. Vacuum numbers place the
inherited splittings twelve orders off-peak (black); the chain's
named input C-P5-C(i) is the requirement of sitting at the peak
(red); the flavor-measurement route (green) is the identified
candidate mechanism, $\Gamma^{\mathrm{eff}} = c_\tau y_\tau^{2} T$
(candidate grade; conditions C1--C6). This
figure is the paper's sharpest open item, drawn rather than
hidden. The flavored recomputation sharpens the target: the
required operating point is not the drawn peak but the partial
resonance $\Delta M/\Gamma^{\mathrm{eff}} \simeq 3\times10^{4}$
(H-$\phi_{\mathrm{geo}}$-1), between peak and vacuum; the
exact-peak form overshoots the required throughput by
$\sim3\times10^{4}$.}
\label{fig:resonance-map}
\end{figure}

This paper quantifies what clause (i) asks, because honesty here is
cheap and its absence would be expensive. The geometric splittings
of step~1 give $|M_i - M_j| \sim \Mlock/K \simeq 110$~GeV; the
vacuum width of step~3 gives
$\Gamma_N/2 \simeq 4\times10^{-11}$~GeV. Those differ by twelve
orders of magnitude, and off resonance the self-energy enhancement
falls as $\Gamma/\Delta M$: as vacuum numbers stand, the
enhancement of step~4 is not available, and the phrase ``natural
rather than tuned,'' tempting as percent-level splittings make
it, overstates what vacuum numbers support. Three candidate
resolutions
are on the table: \emph{thermal regulators} --- at $T \sim M_N$
the effective width entering the resonance condition is a thermal
damping rate, larger than the vacuum $\Gamma_N$ by many of the
missing orders; the flavor-blind gauge rate $g^2 T$, the
parametric first guess, is computed to land \emph{through} the
peak into over-damping ($\Delta M/\Gamma^{\mathrm{eff}} \simeq
0.07$, reproducing the excluded $\sim3\times10^{4}$ overshoot),
while the flavor-measurement rate $c_\tau y_\tau^{2} T$ lands in
the required decade --- the identified candidate mechanism,
stated below;
\emph{radiative degeneracy} --- a mechanism driving a pair of
$M_i$ toward width-scale splitting at the crossing;
\emph{substructure} --- a finer level spacing within the lock
cluster at $O(\Gamma)$ scale. Deciding among them --- or
discovering that none works, which clause (i) makes a clean kill
--- is the sharpest single work order this paper issues to the
program.

\medskip\noindent\textbf{The mechanism, identified (candidate
grade).} The hypothesis now has a face: an identified candidate
mechanism, whose derivation home is the cosmology companion's
Appendix~I.3 (\citealp{ArisakaA10Cosmos}); this section carries
the statement, the closed form, and the physics, not the kinetic
derivation. The identification: \emph{the effective width of the
flavored resonance is the rate at which the thermal bath
measures lepton flavor --- not the $N$'s own width.} The CP
coherence that builds the asymmetry does not live in the $N$
lines: for the dial-free real-$R$ door the flavor-summed CP
source vanishes identically, so the entire CP content lives in
the lepton-flavor \emph{differences} written into the plasma,
and what cuts the interference off is whatever measures those
flavor coherences fastest. The gauge scatterings, five orders
stronger, are flavor-blind --- a measurement that cannot tell
the flavors apart reads nothing, and cancels exactly from a
flavor-odd source; the $N$'s own width, seven orders slower, would on this reading be the wrong clock --- the $N$ is not what decoheres on it. Whether the in-medium clock is indeed $\Gamma_\tau$ is exactly condition C4 below, open at derivation level. The
fastest honest flavor measurer in the bath is the tau Yukawa
interaction,
\begin{equation}
\Gamma^{\mathrm{eff}} \;=\; \Gamma_\tau \;=\;
c_\tau\, y_\tau^{2}\, T,
\qquad c_\tau = 5\times10^{-3}\text{--}10^{-2},
\label{eq:mechanism-regulator}
\end{equation}
the charged-lepton flavor-equilibration rate of the kinetic
literature. At the generation temperature $T \simeq M_N$ the
heavy mass cancels between the ledger splitting
$\Delta M = \Mlock/35$ and the thermal rate, and the degree of
resonance collapses to a closed form in the ledger integer and
one running Standard-Model Yukawa:
\begin{equation}
\frac{\Delta M}{\Gamma^{\mathrm{eff}}}
\;=\; \frac{1}{35\, c_\tau\, y_\tau^{2}}
\;=\; (2.7\text{--}5.3)\times10^{4}
\label{eq:mechanism-closed-form}
\end{equation}
against the hypothesized $3\times10^{4}$. Zero dials: the same
$K = 35$ that sets the hierarchy ($e^{-K}$), the lock mass
($\tfrac12 M_{\mathrm{Pl}} e^{-K}$), and the throughput target
($e^{-K/2}$) also sets how far off peak the resonance sits, with
the running tau Yukawa --- an object entirely of the Standard
Model --- supplying the only other factor: geometry over
chemistry, once more.

The numbers, honestly: $\Gamma^{\mathrm{eff}} = \Gamma_\tau =
2$--$4$~MeV against the hypothesized $\sim3$~MeV. At the
single-temperature evaluation $T = M_N$ the flavored chain
lands $\etaB$ at $0.28$--$0.55$ of the observed value; the honest $T$-integration over the decoupling window --- where the mechanism's own regulator, proportional to $T$, has dropped exactly where the surviving asymmetry is produced --- is now carried out in the parent's flavored Boltzmann integration (\citealp{ArisakaA10Cosmos}, App.~I): scanning the declared freedoms it lands at $0.09$--$0.23$ of observed (68\%; central $0.14$), with the priced standard omissions supplying a factor $\simeq3.3$ centrally and leaving the observation $+1.3\sigma$ away in the log. Always from below, never above. The deficit is named
--- it is condition C6 below --- and its one-sidedness is itself
informative: the missing transport prefactors must be
enhancements, and the known members of that class (scattering
source terms, the Lorentzian-versus-Breit--Wigner convention,
the seam washout normalization) all point up.

The identification discriminates rather than scatters. The four
candidate regulator prescriptions --- the $N$ widths, the
flavor-measurement rate, the oscillation rate $\Delta M$
itself, and the flavor-blind gauge damping $g^2 T$ (with the
Hubble rate below them all) --- span sixteen orders of magnitude
in $\Delta M/\Gamma^{\mathrm{eff}}$, and exactly one lands in
the decade the hypothesis requires. The $N$-width class
undershoots the observed asymmetry by $\sim\!10^{7\text{--}8}$;
the gauge rate $g^2 T$, the parametric thermal-width candidate
of the regime map, lands \emph{through} the peak into
over-damping ($\Delta M/\Gamma^{\mathrm{eff}} \simeq 0.07$),
reproducing the $\sim3\times10^{4}$ overshoot the flavored
recomputation already excluded --- and it fails structurally
besides, since flavor-blind damping cannot regulate a source it
does not distinguish. The regime map's green arrow accordingly
points at $c_\tau y_\tau^{2} T$.

The status, exactly: a candidate mechanism with a conditional
derivation --- nothing here upgrades the grade of the chain. The
derivation rests on six numbered conditions, stated and graded
in the derivation home (\ATen{} Appendix~I.3), one line each:
\begin{enumerate}[leftmargin=3.1em,label=\textbf{C\arabic*.},itemsep=1pt,topsep=3pt]
\item \emph{Flavor-odd source}: with $R$ real the flavor-summed
CP source vanishes identically --- established (algebraic;
machine-checked).
\item \emph{Flavor-blind cancellation}: gauge damping enters
through the identity projector and drops from the flavor-odd
channel --- established within the density-matrix framework.
\item \emph{Two-level reduction}: the Lorentzian
partial-resonance envelope and its $\le\tfrac12$ bound survive
for every regulator value --- established at reduction level,
given C4.
\item \emph{Regulator identification (the crux)}: the
interference time of the flavor-odd source is $1/\Gamma_\tau$,
not $1/\Gamma_N$ --- open at derivation level; the full Kadanoff--Baym resummation is the outstanding content (the displayed kinetic system carries the heavy-pair coherence as an external source; discharging C4 means promoting it to a dynamical variable with its own damping).
\item \emph{The coefficient}: $c_\tau \in
[5\times10^{-3},\,10^{-2}]$ at $T \simeq M_N$ for this source
--- plausible; the precise value open.
\item \emph{Window throughput}: the $T$-integrated window cost is part of the honest chain, and the priced $O(1)$ prefactors must supply the remaining one-sided enhancement, now \emph{measured} by the parent's flavored Boltzmann integration at $E = 7.1$, 68\% interval $[4.4,\,11.4]$ (retiring the prior bracket $\simeq2$--$20$) --- measured at flavored-Boltzmann level; closure open.
\end{enumerate}
Three kill routes stand on record: if the finite-temperature
derivation returns the $N$-width class, the chain undershoots by
$\sim\!10^{7\text{--}8}$ and is falsified from within; if it
returns the flavor-blind $g^2 T$, the excluded peak overshoot
stands --- the same verdict; and if C4 and C5 close but the measured window deficit of C6 ($E = 7.1$, 68\% $[4.4,\,11.4]$) survives a full transport computation at its adverse end with the priced prefactors exhausted (they supply $\simeq3.3$ centrally, leaving $+1.3\sigma$ in the log),
H-$\phi_{\mathrm{geo}}$-1 fails by throughput rather than by
regulator identity --- a third, softer kill beside the two clean
ones. H-$\phi_{\mathrm{geo}}$-1 is not retired and not closed:
it is sharpened, from a numerical target into the named
conditions C1--C6 --- and the identification has made the
hypothesis \emph{more} falsifiable, not less.

Let us also say, precisely, what success and failure
would look like, because an open problem with acceptance
criteria is a computation waiting to happen rather than a
mystery. \emph{Success} is a finite-temperature transport
computation --- thermal widths in the propagators, the lock
crossing's time dependence in the rates --- in which the
effective splitting-to-width ratio at the crossing lands at the
partial-resonance point $\Delta M/\Gamma^{\mathrm{eff}} \simeq
3\times10^{4}$ of H-$\phi_{\mathrm{geo}}$-1, so that the flavored
throughput on the inherited phases lands within an
$O(1)$ of $e^{-K/2}$: both clauses of C-P5-C discharged at
once, and T-P5-5 promoted from D$^{*}$ toward derived.
\emph{Failure} is the same computation showing the enhancement
dead at the crossing for every candidate mechanism --- no
thermal rescue, no radiative degeneracy, no substructure ---
in which case the chain is falsified from within, the
$e^{-K/2}$ agreement stands exposed as coincidence, and this
Part's claim reverts to Sakharov custody plus an unexplained
number. Both outcomes are scientific; only the unexamined
middle would not be. Until it is discharged, T-P5-5 is exactly what its
label says: a proposition at D$^{*}$, conditional on a named
conjecture, whose failure mode is written into its own falsifier
list.

\section{The alphabet audit: every dial of the birth, valued}
\label{sec:alphabet-audit}

The chain of \S\ref{sec:chain} was audited number by number; this
section audits it symmetry by symmetry. The question of this Part,
asked one level deeper, is not ``does the chain reproduce
$\etaB$?'' --- Table~\ref{tab:audit} answered that --- but
``\emph{where, along the chain, could nature have chosen
otherwise?}'' The answer this section displays and then defends:
nowhere. Every $CP$-related quantity that touches the birth of
matter is a letter of Part~VII's alphabet --- a lattice point of
$(\pi/5)\bZ$ on the one compact circle --- and the three jobs any
baryogenesis must fill (that $CP$ violation \emph{exists}, that
the excess has the \emph{sign} it has, that it has the
\emph{size} it has) are paid by three different structures, none
of which is a dial. The oldest question a physicist can ask about
our own existence --- \emph{was the asymmetry tuned?} --- receives
here its sharpest available answer: there was nothing to tune.

\paragraph{The alphabet, recalled.} Theorem T-P5-3(iv) values
every first-order flavor holonomy on the pentagonal lattice: ten
letters around the circle, $k\pi/5$ for $k = 0, \ldots, 9$. The
chirality criterion of clause (iii) sorts them: the two letters
on the real axis, $\{0, \pi\}$, are \emph{achiral} --- a texture
whose loops all sit there is indistinguishable from its mirror,
and $CP$ holds; the other eight are chiral, in mirror pairs. The
Majorana phases of the seesaw sector live on the same circle ---
they are holonomies of the same fiber, constrained by the
dial-free door (T-P5-4) --- so the alphabet is the \emph{entire}
vocabulary available to $CP$ in this geometry: quark, lepton,
Dirac, Majorana. There is no eleventh letter, and there is no
continuous direction between letters. Now run the audit.

\begin{table}[htbp]
\centering
\caption[The CP alphabet, audited.]{The $CP$ alphabet, audited row by row. Each row names one of the $CP$ quantities this geometry admits, the value it is forced to, the letter of the alphabet that value reads as, and the job that quantity does in the birth of matter. The last row is not an angle at all but the throughput, which fixes the magnitude rather than the sign. Read down the final column, the table separates the single driver from the entries that do no work in the matter account.}
\label{tab:cp-alphabet}
\small
\renewcommand{\arraystretch}{1.25}
\begin{tabular}{p{2.1cm} p{2.6cm} p{2.35cm} p{5.6cm}}
\toprule
\textbf{$CP$ quantity} & \textbf{forced value} &
\textbf{letter, read} & \textbf{job in the birth of matter} \\
\midrule
$\dCKM$ & $2\pi/5$ (QD $13\pi/35$) & chiral & none: the quark
throughput computes to $\sim10^{-20}$
\citep{GavelaEtAl1994}, ten orders short
(\S\ref{sec:why-leptogenesis}) --- the mirror-breaking that
makes $B$ mesons asymmetric never touched the inventory of
matter \\
$\dPMNS$ & $\pi + 2\arcsin\tfrac17 = 196.4^{\circ}$ & anchor
$\pi$, deviation $1/J$ & spectator: the Casas--Ibarra
cancellation makes the chain $\JPMNS$-blind at the few-percent
level (\S\ref{sec:chain}, step 5) \\
$\alpha_{21}$ & $0$ & achiral (parity $+$) & inert \\
$\alpha_{31}$ & $\pi$ & achiral (parity $-$) & \textbf{the
driver}: sets the sign of the flavored asymmetries
$\epsilon_{i\alpha}$, nonzero at $m_1 = 0$ \\
\midrule
throughput & $\epsilon\,\kappa = e^{-K/2}$ & not an angle &
\textbf{the magnitude}: $\etaB = 7.04\,\tfrac{28}{79}\,
e^{-K/2}/\gstar \simeq 5.9\times10^{-10}$ \\
\bottomrule
\end{tabular}
\end{table}

\noindent Five rows, and the reader should sit with the first
column for a moment: these are all the $CP$ dials the Standard
Model hands an engineer of baryogenesis --- the phase the $B$
factories measured, the phase DUNE will measure, the two phases
$0\nu\beta\beta$ may someday constrain --- and the audit says the
famous ones are bystanders. The birth of matter runs on the two
letters nobody celebrates, plus one number that is not a letter
at all.

\paragraph{The paradox, stated.} A reader fresh from Part~VII
will object, and the objection deserves its own paragraph. The
driver of the asymmetry is $\alpha_{31} = \pi$; the lepton Dirac
anchor is $\pi$; and $\{0, \pi\}$ are exactly the \emph{achiral}
letters --- the values at which the chirality criterion
\eqref{eq:achiral-criterion} says $CP$ is \emph{conserved}. The
audit appears to claim that matter was made by the
$CP$-conserving letters of a $CP$-violating alphabet. Stated so
baldly, it sounds like a bookkeeping error. It is instead the
Part's deepest lesson about the division of labor, and it
resolves in three steps.

\emph{First: a Majorana letter is read at half-angle.} A Majorana
phase enters amplitudes through the combination
$e^{i\alpha/2}$ --- the door's weight-zero pairing splits every
angle in two --- so the lattice letter $\alpha_{31} = \pi$ reads,
at the amplitude level, as $e^{i\pi/2} = i$: a quarter turn, as
far from the real axis as a phase can be. What the Lagrangian's
bookkeeping records as a parity, the interference terms
experience as the maximally imaginary insertion. (This is stated
here as the physics reading; the flavored computation whose
output it summarizes lives at its stated grade in the cosmology
companion's Appendix~I.) The letter is achiral; its
\emph{square root} is not; and amplitudes take square roots.

\emph{Second: at a resonance, an amount becomes an orientation.}
On the (partial) resonance of \S\ref{sec:resonance-faced} the
$CP$ asymmetry saturates --- $\epsilon^{\mathrm{res}} \simeq
\tfrac12\sin(2\phi_{\mathrm{geo}})$ times the transport factor
--- and the Yukawa magnitude cancels between numerator and
denominator. Near saturation, the phases are no longer asked
``how much?''; they are asked only ``which way?'' --- which side
of the interference the decay products land on. A parity is
precisely an answer to ``which way?'' with no ``how much''
attached. The division of labor is therefore not an accident
but the natural grammar of resonant transport: \emph{amounts
saturate; orientations survive}.

\emph{Third: something must still be chiral, and it is the
pentagon.} A parity can orient an interference only if there is
an interference to orient --- only if the texture is genuinely
complex somewhere. That existence claim is the pentagon's:
the branch phase $\pi(1 - 2/L) = 3\pi/5$ is chiral because five
is odd, and no assignment of the theory's discrete data could
have made the alphabet mirror-symmetric (T-P5-3). Kill the
pentagon's chirality --- give the bulk an even block integer ---
and every letter lands on the real axis, the parities have
nothing to sign, and the universe keeps its perfect, sterile
balance. In one line each: \emph{the pentagon writes the
sentence; the parity chooses its sign; the throat depth sets the
print run.}

\paragraph{Existence, sign, magnitude.} The trichotomy deserves
to be displayed as the section's result, with the provenance of
each clause:

\begin{itemize}
\item \textbf{Existence} of $CP$ violation: the chirality of the
pentagon, $3\pi/5 \notin \pi\bZ$ --- derived (T-P5-3, at the
mirror Parts' stated conditionality). Not a dial: there was no
value of anything that could have turned it off.
\item \textbf{Sign} of the excess --- matter rather than
antimatter: the Majorana parity $\alpha_{31} = \pi$, a two-valued
letter forced by the dial-free door's real structure, read at
half-angle by every amplitude. (Which parity corresponds to the
label ``matter'' is a convention fixed jointly with the
orientation conventions of Part~V; the physical content ---
that the sign is \emph{discrete, forced, and not adjustable} ---
is convention-free, at the flavored computation's grade.)
\item \textbf{Magnitude}: not a $CP$ quantity at all. The
throughput $\epsilon\,\kappa = e^{-K/2}$ is transport ---
resonance times washout --- and its exponent is the ledger's
throat depth $K = 35$, the same integer that sets the lock mass
($e^{-K}$ in $\Mlock$) and caps the strong-$CP$ leakage
($e^{-K}$ in \eqref{eq:thetabar-leakage}). Dressed by the
standard sphaleron and entropy factors it returns $\etaB \approx 7\times10^{-10}$ \eqref{eq:etaB-pred}:
\emph{the baryon-to-photon ratio is the square root of the
lock-scale suppression}, read through bookkeeping nobody
disputes.
\end{itemize}

\begin{figure}[t]
\centering
\resizebox{0.663\textwidth}{!}{%
\begin{tikzpicture}[font=\sffamily\footnotesize, scale=1.45,
 >=Stealth]
 % the circle and the ten-letter lattice
 \draw[line width=0.9pt] (0,0) circle (1.55);
 \foreach \k in {0,1,...,9}{
   \fill[black!55] ({\k*36}:1.55) circle (1.0pt);
 }
 \draw[dashed, black!45] (-1.55,0) -- (1.55,0);
 \node[black!60, font=\scriptsize, below] at (0.45,-0.06)
   {the achiral axis $\{0,\pi\}$};
 % 0 : alpha21
 \draw[->, black!60, line width=0.80pt] (0,0) -- (0:1.55);
 \node[black!60, anchor=west, align=left] at (0:1.72)
   {$0$: \ $\alpha_{21}$\\ (parity $+$; inert)};
 % 2pi/5 : quark Dirac
 \draw[->, blue!70!black, line width=1.10pt] (0,0) -- (72:1.55);
 \node[blue!70!black, anchor=south west, align=left]
   at (72:1.66) {$2\pi/5$: \ $\dCKM$ (chiral)\\
   $CP$ violation exists --- but\\ feeds no matter
   ($\sim\!10^{-20}$)};
 % 3pi/5 : branch phase
 \draw[->, violet!80!black, line width=1.10pt] (0,0) -- (108:1.55);
 \node[violet!80!black, anchor=south east, align=right]
   at (108:1.66) {$3\pi/5$: the branch phase\\ the chirality
   no dial\\ could decline};
 % pi : alpha31 + lepton anchor
 \draw[->, red!75!black, line width=1.10pt] (0,0) -- (180:1.55);
 \node[red!75!black, anchor=east, align=right] at (180:1.72)
   {$\pi$: \ $\alpha_{31}$ --- the parity that\\
    signs the matter (half-angle $i$);\\
    and the anchor of $\dPMNS$\\
    ($196.4^{\circ} = \pi + 2\arcsin\tfrac17$)};
 % magnitude box
 \node[draw=black!55, rounded corners=2pt, align=center,
   font=\scriptsize\sffamily, inner sep=3.5pt] at (0,-2.35)
   {the magnitude is not on the circle:\quad
    $\epsilon\,\kappa = e^{-K/2}$, $K = 35$
    \ $\Rightarrow$ \ $\etaB \simeq 5.9\times10^{-10}$};
\end{tikzpicture}}
\caption[The alphabet audit, drawn.]{The alphabet audit, drawn.
The ten letters of the pentagonal lattice $(\pi/5)\bZ$ on the
one compact circle, with the four that run the birth of matter
marked. The quark Dirac phase (blue, $2\pi/5$) is chiral ---
$CP$ violation exists --- but its throughput is ten orders too
small to feed the inventory. The branch phase (violet,
$3\pi/5$) is the chirality the odd alphabet forced: the reason
any interference exists to orient. The achiral letter $\pi$
(red) does double duty: as the Majorana parity $\alpha_{31}$ it
signs the matter excess (every amplitude reads it at
half-angle, $e^{i\pi/2} = i$), and as the anchor of
$\dPMNS = \pi + 2\arcsin(1/7)$ it places the lepton Dirac
phase DUNE will measure --- a spectator to the birth. The
magnitude lives off the circle entirely: the throughput
$e^{-K/2}$, the ledger's throat depth. Existence from the
pentagon, sign from a parity, size from the throat: three jobs,
three structures, zero dials.}
\label{fig:alphabet-audit}
\end{figure}

\paragraph{What the Standard Model could not have done.} The
contrast that gives the audit its force is not with a rival
mechanism but with the same mechanism run on continuous dials.
The Standard Model's own double failure --- the crossover that
removes the departure from equilibrium and the $10^{-20}$ that
starves the quark route --- already forced the lepton road
(\S\ref{sec:why-leptogenesis}). But hierarchical leptogenesis
run in the standard way inherits the Standard Model's dial
economy: the heavy masses are free, the $R$-matrix is free, the
phases are free, and the observed $\etaB$ becomes a
\emph{constraint surface} in that parameter space --- a good
theory that can accommodate the number but cannot be said to
have produced it, and whose operating point no experiment can
reach (PS-P5-8). Here every one of those freedoms is a letter or
a ledger integer: the masses sit at the lock ($e^{-K}$), the
door is a uniqueness theorem, the phases are the alphabet's, and
the throughput reconstructs $e^{-K/2}$. When a construction with
no dials lands within $4\%$ of the measured number, the correct
reading is not ``a good fit'' --- there was nothing to fit ---
but a jointly falsifiable identity between the geometry's one
depth integer and the inventory of the universe.

\paragraph{Physics intuition: why there is something rather
than a mirror.} Assemble the audit into the sentence the
title of this Part promised. The universe keeps matter for
three nested reasons, one per structure. \emph{Because five is
odd}: an odd alphabet cannot be mirror-symmetric, so the
texture of couplings is chiral --- $CP$ violation was
constitutional, not optional. \emph{Because the door pays in
$B{-}L$}: the sphaleron era converts the lepton-sector excess
at the fixed rate $28/79$ and conserves $B{-}L$ thereafter, so
the coin struck at the lock is neither re-melted nor diluted
(\S\ref{sec:zero-dilution}) --- the asymmetry is banked in the
one currency the late universe cannot touch. \emph{And because
the throat is thirty-five deep}: the surviving fraction per
photon is $e^{-K/2}$, six parts in ten billion --- the same
$e^{-35}$ that parks the seesaw partners at $3.85$~TeV, taken
at its square root by the resonance-and-washout transport. A
universe with an even block integer keeps nothing and no one
asks the question; a universe with a shallower throat is
matter-dominated into opacity; this one prints six baryons per
ten billion photons and, some $10^{17}$ seconds later, prints
physicists who count them. None of the three clauses was
adjustable. That is what ``deterministically programmed'' means
here, and it is the precise sense in which this framework
answers --- at its stated grades --- the question the author of
this cohort has carried for fifty years: \emph{we are here
because the geometry's alphabet is odd, its door pays in a
conserved currency, and its throat is exactly as deep as the
number of us.}

\paragraph{The audit's own exposure.} Determinism of this kind
is only physics if it can die, and each clause hangs its own
rope. The anchor-plus-deviation letter is checked by DUNE:
$\dPMNS = 196.4^{\circ}$, separated from the achiral
$180^{\circ}$ at $\sim1\sigma$ per $\pm15^{\circ}$ resolution
and from maximal $270^{\circ}$ at $\sim5\sigma$
(\S\ref{ssec:we4}). The parities are checked by
$0\nu\beta\beta$: $(\alpha_{21}, \alpha_{31}) = (0, \pi)$ gives
$\mbb = 1.72$~meV --- an \emph{observation} at the tens-of-meV
level now within experimental reach would falsify the parity
pattern outright. The sign is checked by the appearance
asymmetry ($\JPMNS < 0$; T2K already disfavors the opposite
sign). The magnitude is checked by the CMB number itself,
against the standing exposures of Table~\ref{tab:audit}: the
one named input (C-P5-C's on-resonance condition, carried by
the flavor-measurement regulator at candidate grade) and the
one postulated exponent ($K/2$, the compression C-P5-C(ii) must
still derive). The honest summary of the whole section is
therefore one sentence long: \emph{every symmetry dial of the
birth is a forced letter of one discrete alphabet --- graded,
displayed, and mortal in four independent laboratories within
the decade.}


\paragraph{$CP$-conserving in the sum, $CP$-violating in the differences.}
The doctrine deserves its own paragraph here, because it is the step on
which the whole chain turns and it is easy to read as a technicality. With
opposite $CP$ parity between $\nu_3$ and $\nu_{1,2}$ and a real orthogonal
$R$, the imaginary parts that drive the asymmetry are manifestly nonzero
flavor by flavor --- and they sum to zero over flavor for one reason only:
the unitarity of the mixing matrix makes the relevant column sum vanish.
The cancellation is a column-orthogonality statement, not a dynamical one,
so it is exact.

That is the whole resolution of an apparent paradox. \emph{A zero-sum set of
flavored asymmetries is not a zero asymmetry}, because the three flavors are
not washed out equally; differential washout converts a set that sums to
zero into a net excess. Two consequences follow, and both are the kind that
a referee can check. First, a flavored treatment is \emph{mandatory} here
rather than merely more careful: an unflavored calculation of this chain
returns exactly zero, and would have to be reported as a failure. Second,
the flavor-sum cancellation is promoted from a technical nuisance to the
load-bearing step of the argument --- which is why
\S\ref{sec:alphabet-audit} prices $\alpha_{31} = \pi$ as the driver and
the achiral $\alpha_{21}$ as inert.

THE PLATE THAT WOULD HAVE GONE HERE WAS DROPPED, AND THE REASON IS WORTH
RECORDING. The leptogenesis study draws this with its three flavored
$\epsilon_{i\alpha}$ values on a bar chart. Those numbers are computed at
that study's own inputs; this paper does not derive them, and the study is
an internal one with no entry in this paper's bibliography and none on the
arXiv path. Printing another paper's numbers under no citation is worse
than printing no chart, so the concept is carried and the chart is not.

\section{What the block predicts, and how it fails}
\label{sec:matter-falsifiers}

The chain is deliberately exposed. \emph{Mechanism kills:} if a
full transport calculation shows no consistent on-resonance
heavy-neutrino/washout regime near the lock scale, the chain fails
(C-P5-C(i)). \emph{Backbone kills:} if cosmological data force a
neutrino mass sum or CP structure incompatible with the inherited
ledger, the backbone fails; if proton decay is observed, the whole
baryonic logic fails. \emph{Dated instruments:}
$\mbb \simeq 1.7$~meV at the next-generation neutrinoless
double-beta program --- the cleanest near-term discriminator of
the Majorana door; the $3.85$~TeV lock cluster within HL-LHC reach,
with a pre-declared kill criterion in the cohort's ledger; and
$\dPMNS$ at DUNE/Hyper-Kamiokande to $\pm15^{\circ}$ by the
early-to-mid 2030s --- a measured value inconsistent with
$196^{\circ}$ falsifies the phase that biases the door's traffic.
No other baryogenesis proposal in Part~X's comparison carries a
falsifier list with dates; most carry none at all.

\emph{The testability inversion.} It is worth pausing on how
unusual this situation is. Hierarchical thermal leptogenesis ---
the field's default explanation --- is a good theory that cannot
be tested: its heavy states sit at $10^{9}$~GeV or above, and
the CP phases that matter to it are not the phases experiments
measure. The present chain inverts every clause: its heavy
states sit at $3.85$~TeV, within collider reach; its CP phase
\emph{is} the measured PMNS sector's, projected; its Majorana
door casts a $0\nu\beta\beta$ shadow at a computable
$1.7$~meV; and its proton clause is checked by every year of
Super-Kamiokande silence. Whatever one's priors about the
geometry, the scientific posture is the preferable one: the
account of why we exist is due in the data within the decade
--- more predictive and more protective at once, and mortal
four separate ways.

\section{The \texorpdfstring{$\Omega_b$}{Omega-b} bridge}
\label{sec:omegab-bridge}

One open seam is recorded rather than smoothed. The transport route of this Part gives $\etaB \approx 7\times10^{-10}$ (central $7.34$), hence $\Omega_b h^2 \simeq 0.0268$ by standard conversion; the cohort's topological claim reads the baryon share directly from the ledger, $\Omega_b = 1/(2\pi^2) \simeq 0.0507$, i.e. $\Omega_b h^2 \simeq 0.0230$ at the corridor's $h$. The two routes to the same observable differ by $\sim16\%$ at the corridor's $h$ (branch-dependent, shrinking through zero near the ladder branch) --- against a measurement whose own precision is $\sim0.7\%$ --- and the discrepancy is a named open problem of the cosmology companion. The residue is not symmetric between the routes: anchored to the measured $\Omega_b h^2 = 0.02237$, the transport route's $+20\%$ is priced by its own factor-$\sim3$ band, while the topological route at the corridor's $h$ sits $+3\%$ high with no $O(1)$ freedom to price it --- so the unowned residue sits on the corridor-$h$ conversion or on the exactness of $1/(2\pi^2)$, not on the transport $O(1)$'s. This paper flags the seam in preference to hoping no reader divides two of its own numbers.

\section{What is claimed, and what is inherited}
\label{sec:matter-claims}

Concessions first, in full --- and for this Part the concession
is total at the level of technique: \emph{the chain contains not
one new formula.} The three conditions are Sakharov's
\citep{Sakharov1967}. The lepton route is Fukugita and
Yanagida's \citep{FukugitaYanagida1986}; the resonant regime is
Pilaftsis's and Pilaftsis--Underwood's \citep{Pilaftsis1997,
PilaftsisUnderwood2004}; the bound it evades is Davidson and
Ibarra's \citep{DavidsonIbarra2002}; the seesaw parameterization
is Casas and Ibarra's \citep{CasasIbarra2001}; the sphalerons
are Kuzmin--Rubakov--Shaposhnikov's
\citep{KuzminRubakovShaposhnikov1985} with Harvey and Turner's
$28/79$ \citep{HarveyTurner1990}; the washout benchmark
\eqref{eq:mstar} is textbook transport. Every module is
standard 1967--2004 technology, cited at its point of use, and
Part~X walks the whole toolbox on its own ground.
One quiet dependency of the toolbox deserves its own line:
every equilibrium abundance, Pauli-blocking factor, and
chemical-potential count in that transport --- including the
$28/79$ itself --- presupposes \emph{exact} Fermi--Dirac
statistics for the fermions of the bath. The cohort now traces
that exclusion to bulk frame-path holonomy at candidate grade
(\S\ref{sec:quartet-second-child}; \citealp{ArisakaA2GG6} \S20;
\citealp{ArisakaA8QM} T-A8.12), so the receipt of this Part
rests on the geometry twice: once for the phase that biased the
traffic, and once for the statistics that counted it.

What this Part claims, ranked by confidence, at the grades of
Table~\ref{tab:part6-ledger}. \emph{First --- the load-bearing
claim --- dial-lessness.} In every standard implementation the
heavy masses, the degeneracy, and the CP phases are chosen;
here the mass is the ledger's lock scale, the portal is a
uniqueness theorem, the Yukawa is fixed by matching to derived
light masses, and the phase is the pentagon's --- zero fitted
parameters, one named input, one postulated exponent, as the
audit table prints. No other framework in the comparison
derives its $\etaB$ inputs from anything. \emph{Second, the bracketing:} the independently derived washout efficiency and the candidate regulator's closed form bracket the ledger's $e^{-K/2}$ --- the exponent remains a postulate (C-P5-C(ii)), the identity at displayed precision uses the declared $O(1)$ freedom in $\kappa$, and the bracketing is the news. \emph{Third, zero dilution as a
structural guarantee:} the prompt-crossing lemma kills, in one
stroke, the inflation-dilution worry of GUT baryogenesis and
the moduli plague of string cosmology --- the descent has no
furnace to re-melt the coin. \emph{Fourth, proton-safe
baryogenesis:} the standard tension --- the sector that makes
baryons can eat the proton --- is dissolved by construction,
bulk ledger exact, boundary $B{-}L$ active. \emph{Fifth, the
testability inversion:} four dated instruments where the
default explanation offers none. The first claim is as strong
as the audit table; the second carries the named conjecture;
the third is derived-conditional on the prompt-crossing lemma;
the fourth is derived; the fifth is a fact about the falsifier
list. And over all five stands the Part's honest ceiling: until
C-P5-C is discharged, T-P5-5 is a D$^{*}$ proposition --- the
resonance gap of \S\ref{sec:resonance-faced} is the sharpest
outstanding physics item in the program, printed in its own
section, drawn in its own figure.


\section{The selection rule that does both jobs, and a prediction that can
only fail}
\label{sec:deltab-three}

A reader who has followed the chain should be uneasy at this point, and the
unease is exactly right. Everything above requires baryon number to be
violated --- Sakharov says so and there is no way around it: a universe that
starts with $B = 0$ and ends with $B \neq 0$ has violated baryon number
somewhere. And yet the cohort's most distinctive laboratory prediction is
that the proton is \emph{absolutely} stable: not long-lived, not $10^{36}$
years, but $\tau_p = \infty$ \citep{ArisakaP6Proton}.

Both are true, and one rule does both jobs.

%% UNNUMBERED, DELIBERATELY.  Numbering this display renumbered 56 of the
%% paper's existing equations by one -- every equation after it in the
%% document -- because it sits in Part VIII.  Nothing references it, no
%% companion paper cites P5's equations by number, and a display that needs
%% no number should not move fifty-six that do.
\begin{equation*}
\Delta B \;=\; N_{\mathrm{gen}}\, n \;=\; 3n \;\in\; 3\mathbb{Z}.
\end{equation*}

Baryon number can change, but only in units of three. A bulk thermal
reprocessing of a $B-L$ asymmetry does not care: the universe converted its
lepton asymmetry three baryons at a time and never noticed. A proton decay,
$\Delta B = -1$, does care, because $-1 \notin 3\mathbb{Z}$. No anomalous
process, however violent, can remove a single nucleon; the minimal
admissible event costs three baryons at once and three leptons with them,
preserving $B-L$. This is not a loophole but the arithmetic of the anomaly:
each family contributes once, there are three families, and so the quantum
of baryon violation is three. \emph{The same $N_g = 3$ that} \PaperPTwo{}
\emph{derives stands guard over every atom.}

\begin{figure}[htbp]
\centering
\resizebox{0.62\textwidth}{!}{%
\usebox{\GGplateDeltaB}}
\caption[The selection rule that does both jobs.]{Baryon number changes only
in units of the generation count, after T-SM-60.1 of \ANine{}
\citep{ArisakaA9SM}. Every canonical proton-decay mode has
$\Delta B = 1$ and neutron--antineutron oscillation has $\Delta B = 2$;
neither integer is a multiple of three, so both are forbidden outright
rather than merely suppressed. The thermal reprocessing of this Part lives
at $\Delta B \in 3\mathbb{Z}$ and is unaffected. A single event in either
forbidden class falsifies the framework.}
\label{fig:deltab-p5}
\end{figure}

\paragraph{What is imported, and with which hypotheses.} The rule is not
derived here. It is Theorem T-SM-60.1 of \ANine{}
\citep{ArisakaA9SM}, and it travels with its three hypotheses:
(H1) the boundary spectrum is the split Standard Model with the $X, Y$
retained only as bulk modes; (H2) the bulk $\UoneB{}$ descends so that the
exact boundary gauge symmetry acts as baryon number; and (H3) the
$\UoneB{}$ gauge anomalies are canceled by the \BIsix{} Green--Schwarz
inflow. Under those three, every local operator with $\Delta B = \pm1$ or
$\pm2$ carries non-zero $\UoneB{}$ charge and is forbidden to all orders,
and the only admissible baryon-number violation is the electroweak anomaly,
suppressed by $e^{-4\pi/\alpha_W} = e^{-372} \sim 10^{-162}$. The owning paper also
records why the step is not automatic --- baryon number is not a generator
of unbroken $\SUfive{}$, and gauging the $\UoneB{}$ of
$\SUfive{}\times\UoneB{}$ does not by itself forbid proton decay --- and
that caution is imported with the result rather than left behind it.

\paragraph{A prediction that points the wrong way on purpose.} The
epistemic shape of this one is unusual and deserves its own paragraph, in a
paper whose register is that a claim should be stated with the measurement
that would kill it. Grand unified theories \emph{predict} proton decay and
are confirmed by a positive detection. This framework says the search will
never succeed, in any channel, at any sensitivity. That is the stronger
claim and the more dangerous one. It cannot be confirmed --- no finite
exposure establishes $\tau = \infty$ --- and it can be destroyed by a single
event. Hyper-Kamiokande's projected sensitivity is therefore not a test the
framework can pass; it is a test it can only fail. For a framework that
claims to fix parameters rather than fit them, that is the right way round,
and the whole baryonic logic is staked on it rather than hedged.


\section{The three horizons of sameness}
\label{sec:three-horizons}

The excess is one instance of a wider fact, and this Part should name it
before closing: the universe is \emph{the same} in three progressively
stronger senses, and the two fibers of \S\ref{sec:anatomy-of-a-theorem}
price each one differently. The formulation is the holonomy companion's
\citep{ArisakaW2Holonomy}; it is reproduced here because the third horizon
is this Part's own result seen from a distance.

\emph{Horizon 1: the same conditions.} Temperature, curvature and inventory
uniform across causally disconnected patches. Bought by the interval's
opening act --- inflation stretches one patch over the sky
\citep{ArisakaA10Cosmos}.

\emph{Horizon 2: the same things.} Every region built of the same
electrons, the same protons, the same abundances. Bought by the circle's
ledgers: identity is a holonomy class, and classes are the same everywhere
without communication. No mechanism need synchronize what topology makes
identical.

\emph{Horizon 3: the same choices.} Every region on the same side of every
symmetry breaking --- matter over antimatter, this vacuum, this alphabet.
Bought by the descent's chiral texture, read once and in one direction, with
the excess of this Part its sharpest instance.

The three are the anatomy's answer to the oldest observational fact about
the universe: its monotonous, extravagant sameness. And the sorting is the
same one \S\ref{sec:anatomy-of-a-theorem} applied to the mirrors. Horizon 1
is an interval statement and needed a mechanism. Horizon 2 is a circle
statement and needed none --- which is why it has never seemed to require
explanation, and why the absence of a puzzle there is itself a datum. Horizon
3 is the texture the interval wrote onto the circle, and it is the one this
paper computes.

\paragraph{Why the third horizon is the one that could have failed.} A
tuned universe would have to tune each region separately. A lottery universe
would have to correlate its draws across regions that have never been in
contact. The reading assembled here does neither: the sign of the excess is
set by the chiral alphabet, which is circle-side content and therefore
identical in every region, so every patch tips the same way for the same
reason. Segregation --- antimatter galaxies elsewhere --- would require
horizon~3 to fail exactly where the geometry holds it hardest, and the
gamma-ray sky reports no annihilation seam at any scale.

\paragraph{What this does not settle.} Naming the three horizons does not
prove any of them. Horizon 1 is the cosmology companion's result and carries
its conditioning; horizon 2 rests on the identification of identity with a
holonomy class, which is the arena's claim and not this paper's; horizon 3
is computed in this Part at the grade the status ledger of
\S\ref{sec:part6-ledger} records, conjecture C-P5-C included. The value of
the three-way sorting is that it says which kind of explanation each
sameness needs --- a mechanism for an address, nothing at all for a class ---
and that is a diagnosis rather than a derivation.


\section{Why a universe of matter was inevitable}
\label{sec:inevitable-matter}

Assemble the Part. Four things were established in four different places,
and it is worth putting them in one paragraph because the conclusion is
their product and not any one of them.

The circle guarantees that matter and antimatter exist at all, as two
readings of one book, degenerate in every address --- that is T-P5-A and
T-P5-C of Part~\ref{part:antiparticles}, and it is why the question ``why is
there something rather than its opposite'' is a question with a mechanism
rather than a mood. The chiral alphabet guarantees a preferred sense,
pressed onto the fiber by the constitution's integers rather than chosen by
history --- Part~\ref{part:pentagon}. The interval guarantees a once-only,
out-of-equilibrium crossing at which the preference could act, and prices
its throughput with a topological exponential --- \S\ref{sec:zero-dilution}.
And the quantized door guarantees that the asymmetry, once written into
baryons, is kept: $B-L$ never opens, and the sphaleron door welds shut as
the descent cools.

\emph{Nothing in that sequence is contingent.} Given the constitution, a
universe that ends the crossing with a small residue of unmatched matter is
not one lucky outcome among many; it is the deterministic playback of a
chiral tape read once and in one direction. The formulation is the holonomy
companion's \citep{ArisakaW2Holonomy} and it is adopted here because it is
the shortest true statement of what this Part computes: \emph{the excess of
particles is the one-way reading of a chiral tape.}

The honesty that has to travel with that sentence is the honesty of
\S\ref{sec:part6-ledger}, unchanged. ``Inevitable'' is a statement about the
\emph{shape} of the outcome, not its precision. The magnitude carries the
named conjecture C-P5-C and the postulated $K/2$ exponent, and until those
are discharged the number is a $\mathrm{D}^{*}$ proposition and is printed
as one. What is claimed as inevitable is that the sign is fixed, that the
door is unique, and that the crossing happens once --- and each of those is
a class statement, which is why the sign is the same in every region of the
sky while the magnitude is the thing still being audited.

\subsection{Three questions this Part should answer out loud}
\label{ssec:three-questions}

\paragraph{Where did all the antimatter go, physically?} It annihilated ---
essentially all of it, against essentially all of the matter, in the first
seconds. The photon bath of the microwave background is in large part that
bonfire's light. What survived is the unmatched residue, one part in a
billion, which is the number this Part computes. The universe of galaxies is
not matter's triumph over antimatter; it is the change left over from their
mutual settlement.

\paragraph{Could the antimatter be somewhere else --- antimatter galaxies,
segregated?} Observation says no: there is no annihilation seam in the
gamma-ray sky at any scale. The account here says why it could not have been
otherwise, and the reason is \S\ref{sec:three-horizons}'s third horizon. The
sign of the excess is set by the chiral alphabet, which is circle-side
content and therefore identical in every region without any region having to
communicate with any other. Every patch tips the same way for the same
reason. Segregation would require horizon~3 to fail exactly where the
geometry holds it hardest.

\paragraph{Does antimatter fall differently?} No --- and this account is
committed, in a way that is worth stating as a falsifier rather than as a
reassurance. By T-P5-A and T-P5-C an antiparticle is the same holonomy class
read in the conjugate orientation, and a mass is that class's address along
the interval. An address is orientation-blind: reversing the reading does
not move it. The antihydrogen free-fall measurements are therefore an audit
of the mirror reading itself, not of general relativity. \emph{Any anomalous
gravitational behavior of antimatter breaks
Part~\ref{part:antiparticles} at its root} --- the handshake, the equal
masses, and the pairing --- well before it troubles anything in the
gravitational sector. The commitment is total and it is cheap to state
because the geometry leaves no dial to turn: there is no parameter in this
construction that could make a conjugate reading weigh differently.

\section{Status ledger for Part~VIII}
\label{sec:part6-ledger}

\begin{table}[h]
\centering
\caption[Part~VIII at a glance.]{Part~VIII at a glance. The chain's originality is its dial
count --- zero fitted parameters --- and its honesty is this
table. Eleven steps, each with its grade and its residue. The grades run from derived, through named input and derived-conditional, to reconstruction and --- in the last row --- open. That last row is the one to read first: the two-route bridge for the baryon density leaves a gap whose owner has not been identified, and the table says so rather than rounding it away.}
\label{tab:part6-ledger}
\small
\renewcommand{\arraystretch}{1.2}
\begin{tabular}{p{4.6cm} p{3.1cm} p{5.8cm}}
\toprule
\textbf{Step} & \textbf{Grade} & \textbf{Residue} \\
\midrule
Sakharov tripod (P-P5-E) & derived (architecture) & clause grades
inherited (\S\ref{sec:sakharov-custody}) \\
$\Mlock$, splittings, $T_{\mathrm{rh}}$ window & derived & --- \\
Portal uniqueness (T-P5-4) & derived & --- \\
Seesaw-matched $Y_\nu$, width & derived & --- \\
On-resonance enhancement & named input & C-P5-C(i); twelve-order
gap quantified (\S\ref{sec:resonance-faced}) \\
CP asymmetry $\epsilon^{\mathrm{res}}$ & derived-cond.
(H-$\phi_{\mathrm{geo}}$-1) & flavored computation displayed
(\ATen{} App.~I); candidate mechanism identified (the
flavor-measurement regulator); C1--C6 open \\
Washout $\kappa$ & derived-cond. & $O(1)$ uncomputed \\
Throughput $= e^{-K/2}$ & reconstruction & exponent postulated;
C-P5-C(ii) \\
Sphalerons, entropy, $\etaB$ & standard $+$ D$^{*}$ & factor-$\sim3$
band \\
Zero dilution ($N_2 = 0$) & derived-cond. & prompt-crossing lemma \\
$\Omega_b$ two-route bridge & open & $16.4\%$ gap, owner
unidentified \\
\bottomrule
\end{tabular}
\end{table}

\medskip
\noindent\emph{Part-end summary.} The observed six-per-ten-billion
is met by a chain whose every input already existed for other
reasons: the lock scale, the unique door, the matched Yukawa,
the pentagon's phase. One named conjecture carries the resonance
and the exponent; one open seam joins the two $\Omega_b$ routes;
four dated experiments can kill the story within a decade. And
with this Part the story the paper opened is closed end to end:
the twins were forced by the handshake; the mirrors broke
selectively, by an absence and an angle; the angle biased the
traffic through the one door only the still-point field could
open; the one-way boundary froze the bias into a number; and
the number, conserved ever since, is the matter
of every reader now holding this page. Matter won, on this
account, because the descent is oriented --- and the oriented
descent is falsifiable. But before the account faces its
rivals, one debt remains. The whole story ran on a word this
paper has used on every page and never paused to define ---
time: the clock, the arrow, the one-way shore. The next Part
stops and asks what it is.

\GGPart[What is time?]{What Is Time?}{The oldest question given a geometric answer --- the four endowments (order from the cone, repetition from the equivalence, arrow and shore from the dictionary), now located in the anatomy: the circle ticks, the interval points; the standard accounts reviewed and honored; the problem of time in quantum gravity, dissolved by architecture; the first law-level arrow; the chronology quantified --- boundary time as the exponential shadow of the descent, the age of the universe as an audited e-fold ledger; every clock a winding counter; temperature as the shore's transported period; energy and entropy as two currencies of one direction; the significance against every previous definition; and what it all means, in plain words}
\label{part:time}

\section[Part~IX in plain English]{Part~IX in plain English: the oldest question, given a geometric answer}
Everyone uses time and no one can say what it is. Physics inherited that embarrassment and made a profession of it. This Part offers an answer of a particular shape --- not a definition of time, but a list of the things time does and the piece of geometry that supplies each of them.

There are four. Order comes from the light cone. Repetition, and with it frequency, energy, and the possibility of a clock at all, comes from the compact circle. The arrow comes from the radial interval, which admits no holonomy and therefore only ever points forward. And the shore --- the single-sided spectrum that makes imaginary time and thermal computation work --- comes from positivity. Four endowments, four distinct carriers, and none of them supplied twice.

What that buys is an arrow at law level rather than as a statistical tendency, a clock that is a winding counter, and a dissolution rather than a solution of the problem of time in quantum gravity. The Part sets the account beside every previous definition in one table, and it says which of the four endowments are inherited and which are argued here, because a Part that answers the oldest question in the subject should be the most careful one about what it is actually claiming.

\section{The oldest question, and the shape of an answer}
\label{sec:time-question}

``What, then, is time? If no one asks me, I know; if I wish to
explain it to one who asks, I know not.'' Augustine's confession
\citep{Augustine} has embarrassed sixteen centuries of readers
because it is exact: every human being navigates time
faultlessly and no one can say what it is. Physics inherited
the embarrassment and professionalized it. This Part exists
because the present program --- through its Foundations-Branch
partner \citep{ArisakaP4Real} and the preceding Parts of this paper
--- has arrived, almost without meaning to, at an answer: not a
metaphor, not a stipulation, but a structural characterization
of what the direction we call time \emph{is}, precise enough to
be wrong.

The question splits into three that are usually run together.
\emph{What orders events} --- why is the world sorted into
before and after at all? \emph{Why does time run one way} ---
what is the arrow, and where does it live? And \emph{what is
the ``it'' in Minkowski's $it$} --- why does every quantum
computation reach for an imaginary time, and every
thermodynamic identity return from one? The standard accounts,
reviewed next, each answer one of the three magnificently and
fall silent on the rest. The geometry's answer
(\S\ref{sec:time-endowments}) is that the three questions have
one subject: the unique direction of the arena that carries
four endowments at once --- order, repetition, arrow, and
shore. The claim of this Part is that the four-endowment
characterization is a genuine definition: it picks out time
uniquely among all directions, it derives rather than assumes
the properties the standard accounts postulate, and --- through
the boundary law --- it makes the arrow a matter of law rather
than of luck for the first time
(\S\ref{sec:time-arrow}).

\section{Time in the standard accounts, each honored}
\label{sec:time-standard}

\emph{Classical mechanics: the parameter.} Newton declared
time absolute --- flowing ``equably, without relation to
anything external'' --- and Leibniz retorted that time is
nothing but the order of successions; the debate outlived both.
Operationally, classical mechanics settled for less than either
position: $t$ is the real line on which trajectories are
parameterized, featureless, homogeneous, and reversible ---
run any solution backward and the laws are equally satisfied.
Mechanics gave time its mathematics and deliberately declined
to give it a nature.

\emph{Special relativity: the geometry of order.} Minkowski's
1908 lecture \citep{Minkowski1908} performed the first
transmutation: time became a direction in a real geometry ---
not a flow but an axis, distinguished from space by the sign in
the interval. What special relativity actually delivers is
\emph{order}: the light cone at every event sorts the world
into before, after, and elsewhere, causal influence respects
the sorting, and simultaneity dissolves --- there is no global
now, only each worldline's own proper time. But this time has
no arrow: the block universe contains both directions with
perfect impartiality. McTaggart's philosophical bombshell of
the same year \citep{McTaggart1908} --- that the tensed flow
of time is contradictory and only the ordered series survives
--- anticipated exactly the physics: relativity kept his
ordered B-series and discarded the flowing A-series.

\emph{General relativity: time made dynamical --- and two
endowments lost.} Einstein's completion made the geometry of
time itself a dynamical field: clocks run slow in gravity
wells (measured, from Pound--Rebka to every GPS fix), proper
time is path-dependent, and cosmic time is a choice of
foliation, not a fact. But dynamism costs dearly, and the
costs are instructive. First, \emph{repetition fails
generically}: a generic solution of the Einstein equations has
no timelike Killing flow, so there is no conserved energy, no
preferred frequency decomposition, no unambiguous particle
concept --- the endowment every quantum construction needs is
simply absent in a generic spacetime, which is exactly why
Theorem~0's forced staticity is load-bearing for this
program. Second, the \emph{shore becomes treacherous}:
Euclidean quantum gravity's imaginary-time rotation is
globally ill-defined off special backgrounds --- the trick
stops working where the geometry stops repeating. And third,
in the canonical quantization, time vanishes altogether: the
Wheeler--DeWitt equation has no $t$ in it
\citep{DeWitt1967}, and the ``problem of time'' has stood at
the center of quantum gravity ever since. General relativity
is thus the standard account's high point and its cautionary
tale in one: it geometrized time completely, and in doing so
showed that order alone, without repetition, cannot support
the rest of physics.

\emph{Thermodynamics: the borrowed arrow.} The one branch of
physics with an intrinsic direction is thermodynamics: entropy
increases, and Eddington coined ``time's arrow'' for exactly
this asymmetry \citep{Eddington1928}. But Boltzmann's own
derivation \citep{Boltzmann1877} located the increase in
probability, not law: the microscopic dynamics is exactly
reversible, and the H-theorem's direction is smuggled in
through initial conditions --- the celebrated objections of
Loschmidt and Zermelo stand to this day, patched only by
postulating a stupendously low-entropy beginning (the ``past
hypothesis''). The deepest fact about the standard arrow of
time is therefore an embarrassment wearing a law's clothing:
\emph{no fundamental equation of physics is irreversible};
the arrow is an initial condition, pushed back to the Bang
and left there unexplained.

\emph{Quantum mechanics: the outsider.} In quantum mechanics
time is the one classical survivor: an external parameter,
not an operator --- Pauli's old argument shows a self-adjoint
time observable conjugate to a bounded-below Hamiltonian
cannot exist. So the theory that quantized everything left
time unquantized, sitting outside the Hilbert space it
governs; the Page--Wootters construction
\citep{PageWootters1983} --- time read relationally from
entanglement with a clock subsystem --- is the honorable
attempt to bring it inside, at the price of presupposing the
very structure it reconstructs. Note the striking asymmetry
quantum mechanics silently uses: the energy spectrum is
bounded below and unbounded above --- time's conjugate is
one-sided --- and every use of $e^{-iHt}$, every analytic
continuation, every KMS state leans on that one-sidedness
without ever asking where it comes from.

\emph{QFT and the Standard Model: the reversible mirror
record.} Field theory sharpened order into microcausality and
discovered, through the broken mirrors this paper derives,
that nature does distinguish the two time directions --- T is
violated, measurably, in the kaon and $B$ systems (Parts~I
and~V). But a T-violation of parts per thousand,
CP-compensated and CPT-protected, is not an arrow: the
Standard Model's dynamics remains perfectly reversible (run
the film backward and conjugate the charges and mirror the
parities, and every process is again lawful). Wick rotation,
meanwhile, became the workhorse of every serious computation
--- lattice QCD lives entirely at imaginary time --- while
remaining, in the standard account, a trick: the textbooks
rotate, compute, and rotate back, and decline to say what was
rotated \emph{into}. And the axiomatic tradition
\citep{StreaterWightman} proved the deepest results of the
subject --- CPT and spin-statistics --- precisely by analytic
continuation in time, using the shore at every step and
naming it nowhere.

\emph{String theory and quantum gravity: time emergent,
arrow absent.} In the approaches beyond QFT, space
dissolves readily --- emergent from entanglement, from
matrices, from holography --- but time resists: worldsheet
time is presupposed, the timeless Wheeler--DeWitt problem
returns in every background-independent formulation, and
AdS/CFT anchors bulk time to a boundary clock it does not
explain. The closest structural cousin of this Part's answer
lives here: the thermal-time hypothesis of Connes and Rovelli
\citep{ConnesRovelli1994}, in which time is the modular flow
of the state --- one-parameter, KMS-linked, and deeply
aligned with the clock-and-shore reading below, though with
the custody reversed: there the \emph{state} defines the
time; here the \emph{arena} does, and states inherit it.

The scoreboard, then. Order: delivered, by relativity.
Repetition --- the exact self-similarity that makes energy
and frequency meaningful: assumed everywhere, derived
nowhere. Arrow: real in thermodynamics, but located in
initial conditions, not law. Shore --- the imaginary
direction every computation uses: universally employed,
universally disowned. Four properties, four accounts, no
owner.

\section{The problem of time in quantum gravity, and its
dissolution}
\label{sec:time-qg}

The deepest crisis of time in physics deserves more than the
paragraph \S\ref{sec:time-standard} could give it. Canonically
quantizing general relativity turns the Hamiltonian into a
constraint: the wavefunction of the universe obeys
$\hat H \Psi = 0$ --- the Wheeler--DeWitt equation
\citep{DeWitt1967} --- and the Schr\"odinger $t$ simply does not
appear. Taken at face value, nothing happens: the fundamental
description of a gravitating quantum universe is frozen. This is
not a technicality but a collision of the two accounts of time
reviewed above: quantum mechanics needs an external clock
(endowment two, assumed), while general relativity makes every
clock internal and dynamical (endowment two, generically
absent). Quantized together, the two demands cancel, and time
disappears.

The responses form a literature of their own, and each is a
partial recovery of one endowment. \emph{Semiclassical time}
recovers the parameter in a WKB regime: where the geometry is
nearly classical, a time variable emerges from the heavy degrees
of freedom --- repetition borrowed from an approximation.
\emph{Deparameterization} promotes a matter field to a clock by
hand --- repetition borrowed from a choice.
\emph{Page--Wootters} \citep{PageWootters1983} reads evolution
from entanglement with a clock subsystem --- order recovered
relationally, at the price of choosing the clock.
\emph{Thermal time} \citep{ConnesRovelli1994} derives the flow
from the state's modular structure --- arrow and shore recovered
from KMS, at the price of state-dependence. Each response works
where it works; none says why the universe should contain the
structure it borrows.

The arena account does not solve this problem; it
\emph{dissolves} it, and the distinction should be stated with
the register's usual care. In the constitutional architecture
the metric of the arena is not a quantum variable: Theorem~0
forces the globally static stage before any dynamics is
quantized, so the boundary theory never meets a timeless
constraint --- the clock is constitutional, and what gets
quantized are the fields on the stage, not the stage. The
problem of time is therefore not answered within canonical
quantum gravity; it is prevented from arising, at the stated
cost that the arena's staticity is an output of axioms one
level down (AX-1, AX-2, Theorem~0) rather than of quantized
geometry. Whether that cost is a bargain is exactly what the
falsifiable descent built on the static arena --- the whole of
Parts~VI--VIII --- exists to test: if the stage's rigidity were
wrong, its ledger of forced numbers would not keep landing on
measured values. The frozen-formalism crisis, on this reading,
was a self-inflicted wound: physics quantized the one structure
--- the repeating direction --- that its own quantum theory
needs classical and one-sided in order to mean anything at
all. The endowments of \S\ref{sec:time-endowments} are the
inventory of what must be kept for ``happening'' to make
sense; the arena keeps them by construction.

\section{The four endowments: time in this geometry}
\label{sec:time-endowments}

Part~IV's third-face analysis (\S\ref{sec:wick-quartet})
closed with the answer this Part now sets on its own feet.
In the constitutional arena, read the gifts off the axioms
one at a time. \emph{AX-1 --- locality and causality ---
gives time its order}: the light cone sorts the world, and
influence runs one way along it --- Minkowski's endowment,
inherited whole. \emph{AX-2 --- geometry--gauge equivalence
--- gives time its repetition}: through Theorem~0, AX-1 and
AX-2 force the globally static arena, so the constitutional
clock is a Killing flow --- a direction along which the
geometry repeats itself exactly --- and only because time is
such a symmetry do frequency, energy, and the mode planes of
Part~IV exist at all. A time that did not repeat could not
tick. (One scope note, kept honestly: endowment two holds
on the constitutional arena; the expanding universe of the
cosmology companion \citep{ArisakaA10Cosmos} inherits it
only through the descent's staticity claims, and carries
that dependence on its own ledger.) \emph{AX-3 --- the operational dictionary --- gives
time its openness, twice.} Once irreversibly: the \OGGEP{}
projection is one-way, records accumulate, and the arrow is
the dictionary's non-invertibility over a bulk that is
itself exactly T-symmetric
(\S\ref{sec:actualization}). And once analytically: the
positivity clause makes the clock's spectrum one-sided, and
a one-sided spectrum opens a \emph{shore} --- the half-plane
of lawful continuation, where oscillation relaxes into
damping, where the quantum theory keeps its statistical
shadow, and where, at the covariant completion, the CPT
theorem lives (T-P5-B1; \S\ref{sec:wick-quartet}).

The definition, stated once and sharply: \emph{time is the
unique direction of the arena that carries all four
endowments at once --- ordered by the cone, repeating as a
symmetry, opened irreversibly by the dictionary, and opened
analytically by the spectrum's single side.} Uniqueness is
the load-bearing clause, and it is checkable direction by
direction. Space has translations, but their spectra are
two-sided --- a left-mover is as lawful as a right-mover ---
so space has no orientation the dynamics must respect and no
shore: there is no imaginary $x$ because nothing damps
there. The fiber has rotation, but its windings are
two-sided integers on a compact circle --- it knows charge,
not direction (\S\ref{ssec:source-circle}) --- so there is
no imaginary $\theta$. Time alone is the direction along
which the world is \emph{stable} --- bounded below in
energy, by the dictionary's own consistency --- and
stability is the whole secret: one-sidedness of the spectrum
is the mathematical name of ``nothing falls forever''; the
polar decomposition turns that stability into a rotor and a
speed; the rotor is the $i$ of every Schr\"odinger equation,
the speed's semigroup is every Euclidean computation, and
the half-plane between them is the only thing Minkowski's
$it$ ever lawfully named.

Each standard account now finds its partial answer honored
and located. Relativity's order is endowment one. The
repetition every use of energy assumes is endowment two ---
and Noether's theorem \citep{Noether1918}, read
constitutionally, is the statement that endowment two has a
currency. Thermodynamics' arrow is endowment three, moved
from initial condition to law
(\S\ref{sec:time-arrow}). And the $it$ of every Euclidean
computation is endowment four --- the shore, with the Wick
quartet (T-P5-B1) as its street map and the custody line as
its land registry: the geometry owns the cone, the dial,
and the shore; the dictionary owns the words $it$, $i$,
and Wick.


\begin{figure}[H]
\centering
\resizebox{0.611\textwidth}{!}{%
\begin{tikzpicture}[font=\small, >=Stealth, line width=0.80pt]
% time axis
\draw[->, line width=1.2pt] (0,0) -- (11.2,0)
  node[right, font=\bfseries] {$t$};
\node[below, font=\footnotesize] at (10.6,-0.12) {the one direction};
% endowment boxes
\node[draw=blue!60!black, fill=blue!6, rounded corners, align=center,
  minimum width=4.6cm, minimum height=1.25cm, anchor=south]
  (E1) at (2.4,1.0)
  {\textbf{Order} (AX-1)\\ \footnotesize the cone sorts:
  before / after / elsewhere};
\node[draw=violet!60!black, fill=violet!6, rounded corners, align=center,
  minimum width=4.6cm, minimum height=1.25cm, anchor=south]
  (E2) at (8.2,1.0)
  {\textbf{Repetition} (AX-2 $+$ Thm.~0)\\ \footnotesize Killing
  flow: ticks, frequency, energy};
\node[draw=red!60!black, fill=red!6, rounded corners, align=center,
  minimum width=4.6cm, minimum height=1.25cm, anchor=north]
  (E3) at (2.4,-1.0)
  {\textbf{Arrow} (AX-3, \OGGEP{})\\ \footnotesize one-way
  reading: records accumulate};
\node[draw=teal!60!black, fill=teal!6, rounded corners, align=center,
  minimum width=4.6cm, minimum height=1.25cm, anchor=north]
  (E4) at (8.2,-1.0)
  {\textbf{Shore} (AX-3 positivity)\\ \footnotesize one-sided
  spectrum: the half-plane of $it$};
\draw[->, blue!60!black] (E1.south) -- (2.4,0.08);
\draw[->, violet!60!black] (E2.south) -- (8.2,0.08);
\draw[->, red!60!black] (E3.north) -- (2.4,-0.08);
\draw[->, teal!60!black] (E4.north) -- (8.2,-0.08);
\end{tikzpicture}}
\caption[The four endowments of time.]{The four endowments of the direction called time, each
located in an axiom: order from the cone (AX-1), repetition from
the forced static arena (AX-2 with Theorem~0), and --- twice from
the dictionary (AX-3) --- the one-way arrow of the \OGGEP{}
projection and the analytic shore opened by the one-sided
spectrum. No other direction of the arena carries more than one.}
\label{fig:time-endowments}
\end{figure}

\section{The two fibers: where the endowments live}
\label{sec:time-fibres}

The four endowments were located in the axioms. The work of the
cohort's recent campaign --- the symmetry review's capstone and its
four-paper deployment (\textsf{GG-W1-Symmetry};
\citealp{ArisakaA5GDOS}) --- allows them now to be located in the
\emph{anatomy}, and the relocation is not a relabeling: it adds a
structural claim no earlier account of time has made, and it turns
out to explain three centuries of intellectual history in one
sentence. The instrument is a single exact statement, now printed
across five papers of this cohort: \emph{holonomy requires closed
loops, and an interval has none.} The bulk's two non-spatial fibers
are opposite objects under that statement. The compact circle
$S^1_\theta$ is nothing but closed loops: everything it carries is
holonomy --- quantized, gauge-invariant, protected --- and its flow
\emph{repeats exactly}, which is what a tick is. The radial
interval $I_r$ can hold no holonomy at all: transport along it
composes only forward --- a semigroup, the \GDOS{} generator's
birthplace --- and \emph{points}, which is what an arrow is. Set
the endowments beside the geometry:

\begin{table}[htbp]
\centering
\caption[The four endowments of time and the geometry that supplies each.]{The four endowments of time, set beside the geometry that supplies each. The first column names the endowment, the second the axiom that locates it, the third the piece of the arena that carries it, and the fourth what it gives time. This is the Part's central bookkeeping: order comes from the cone, repetition from the compact circle, the arrow from the radial interval, and the shore from positivity --- four endowments, four distinct carriers, none of them supplied twice.}
\label{tab:time-endowments}
\small
\renewcommand{\arraystretch}{1.25}
\begin{tabular}{p{2.2cm} p{2.8cm} p{3.0cm} p{4.6cm}}
\toprule
\textbf{endowment} & \textbf{axiom (as located
in \S\ref{sec:time-endowments})} & \textbf{anatomy (located
here)} & \textbf{what it gives time} \\
\midrule
order & AX-1, the cone & the $M_4$ cone & before / after /
elsewhere \\
repetition & AX-2 $+$ Thm.~0 & \textbf{the circle
$S^1_\theta$} & frequency, energy, the possibility of a
clock \\
arrow & AX-3, \OGGEP{} one-way & \textbf{the interval $I_r$}
& records, aging, the second law \\
shore & AX-3, positivity & the spectrum's single side & $it$,
KMS, Euclidean computation \\
\bottomrule
\end{tabular}
\end{table}

\noindent The new content is the middle two rows, and it is a
theorem-shaped observation: \emph{the tick and the arrow live on
different fibers.} The circle can tick but cannot point --- its
windings are two-sided integers; as \S\ref{sec:time-endowments}
already put it, it knows charge, not direction. The interval can
point but cannot tick --- nothing on it repeats; an interval has no
period. Physical time, as any boundary observer experiences it, is
the \emph{composite reading} of both, projected onto the one
ordered direction the cone provides. Three consequences follow, and
each deserves its paragraph.

\paragraph{Why mechanics and thermodynamics grew up apart.} The
deepest historical puzzle about time is sociological: how could
physics build two complete, axiomatized, spectacularly successful
sciences of time --- reversible mechanics from Newton through
Hamilton to Schr\"odinger, and irreversible thermodynamics from
Carnot through Clausius to Boltzmann --- that flatly contradict
each other about time's basic character, and run them side by side
for three centuries? The anatomy answers: because they are
\emph{not} two accounts of one thing. Reversible mechanics --- all
tick, no arrow, $e^{-iHt}$ with its two lawful signs --- is the
physics of the circle. Thermodynamics --- all arrow, no tick,
entropy without frequency --- is the physics of the interval. Each
science is internally complete because each fiber is a coherent
geometric object; they resisted unification because they are two
\emph{different} directions of the bulk, marginalized onto the same
boundary axis by the dictionary. The reconciliation programs of
the nineteenth and twentieth centuries --- Boltzmann's $H$-theorem
and its reversibility paradoxes, ergodic theory, decoherence ---
were attempts to derive one fiber from the other. The constitutional
account says plainly: neither derives from the other. They are
siblings, not parent and child, and the boundary's time is their
joint estate.

\paragraph{A clock is a winding counter; an hourglass is an
odometer.} The distinction is operational, and every laboratory
embodies it. Every device physicists call a \emph{clock} counts
repetitions of a circle-side process: a caesium hyperfine phase, a
quartz crystal's oscillation, an optical lattice's line --- each is
ultimately the rotor $e^{-iEt}$, read stroboscopically. Every
device that measures \emph{duration by consumption} --- sand
through a neck, a burning candle, a decaying isotope's surviving
fraction, an entropy meter --- is an interval-side odometer: it
reads how much one-way descent has accumulated, and it never
repeats. The reason a pendulum needs an escapement --- the reason
every real clock is a rotor \emph{plus} a ratchet --- is the
two-fiber theorem in brass: repetition alone cannot record, and
recording alone cannot tick. A physical clock is a small model of
the bulk: one part circle, one part interval, coupled.

\paragraph{The uniqueness clause, strengthened.} The definition of
\S\ref{sec:time-endowments} rested on uniqueness: time is the only
direction carrying all four endowments. The anatomy explains
\emph{why} the count comes out one. The boundary has exactly one
direction onto which both fibers project their character --- the
cone's ordered direction. Space receives neither projection: its
translations have two-sided spectra, so no arrow, and no
one-sidedness, so no shore. The fibers themselves are not boundary
directions at all --- they are non-spatial, and the dictionary
marginalizes them; what survives the marginalization is precisely
the pair (tick, arrow) stamped onto the causal direction. Time is
not one of the world's directions that happens to be special.
\emph{Time is the direction where the tape and the track read
out.}

\section{Clocks: every clock is a winding counter}
\label{sec:time-clocks}

The repetition endowment becomes concrete --- and quantitative ---
once the circle owns it. The statement $E = \hbar\omega$, which
quantum mechanics inherits as a postulate and Part~IV derived as
the rotor's speed, is in the two-fiber language a \emph{currency
statement}: energy is winding rate. Every energy in physics is a
frequency; every frequency is a rate of transport around the
circle; and a clock is any device that counts that transport.
Walk the hierarchy, from the fastest lawful clock to the slowest:

\begin{itemize}
\item \emph{Every mass is a clock.} A particle of mass $m$ ticks
at its Compton rate $mc^2/2\pi\hbar$: for the electron,
$1.24\times10^{20}$~Hz. Matter does not \emph{have} a frequency;
at the constitutional level matter \emph{is} frequency --- a
standing pattern of circle transport, which is Part~IV's rotor
read as ontology.
\item \emph{Every laboratory clock reads the circle.} The SI
second is nine billion caesium windings
($9.19\times10^{9}$~Hz); the optical lattice clocks that now
keep the world's best time read strontium and ytterbium lines
near $10^{15}$~Hz. All of them are $e^{-iEt}$ interferometry:
the tick being counted is the rotor's, every time.
\item \emph{The cohort owns a clock line of its own.} The Kaxion
mass, $8.1~\mu$eV, is a frequency: \textbf{1.95~GHz}
\citep{ArisakaC1Kaxion} --- to the author's knowledge the only
tick rate in physics predicted from a discrete ledger before
being sought. If the haloscopes find that line, the framework
will have contributed a clock to the hierarchy, not merely a
particle to the zoo.
\item \emph{The slowest clock is the sky.} The Hubble rate today,
$H_0^{-1} = 4.6\times10^{17}$~s, is the universe's own period-like
scale --- not a repetition (nothing about the descent repeats;
that is the point of the interval) but the terminal rung of the
frequency ladder.
\end{itemize}

\noindent The dynamic range of this ladder is the ledger's range:
from the Planck tick to the Hubble time is a factor $e^{140}$ ---
a number the next section will audit rather than admire --- and
from the electron's Compton clock to the Hubble rate is
$\sim e^{87}$. Clock hierarchies are depth readings of one
descent. And one more period belongs to the list, though it lives
on the shore rather than the axis: the thermal circle of the
present universe, $\hbar/(k_B T_0) = 2.80$~ps --- the imaginary-
time circumference of the world today, whose meaning
\S\ref{sec:time-shore-rate} will set beside the others.

\section{The arrow, derived: the first law-level
irreversibility}
\label{sec:time-arrow}

One clause of the definition deserves its own section,
because it is the one the author believes no prior framework
has possessed: \emph{the arrow of time enters this physics
at the level of law.} The inventory of
\S\ref{sec:time-standard} was unanimous: every fundamental
equation --- Newton's, Einstein's, Schr\"odinger's, the
Standard Model's --- is reversible, and every observed
irreversibility is charged to initial conditions, the past
hypothesis carrying the whole universe's asymmetry on an
unexplained boundary condition. The structure of this
program is different in exactly one place, and it is
constitutional. The \OGGEP{} projection of AX-3 --- the
dictionary by which a four-dimensional reader reads the
six-dimensional bulk (\S\ref{sec:boundary-law}) --- is a
completely positive, trace-preserving map, and it is
\emph{not invertible}: the boundary description discards
bulk data lawfully and permanently, records accumulate on
the boundary side (\S\ref{sec:actualization}), and no
boundary operation can run the projection backward. The
\GDOS{} boundary law \citep{ArisakaA5GDOS} is the dynamics
of that open reading; the \GRIP{} runtime
\citep{ArisakaA6Gi4} is its cascade of one-way selections
--- dynamical gauge invariance to symmetry breaking to
attractor, each stage a door that shuts. Irreversibility is
therefore not an accident of how the universe started; it
is a property of how the boundary \emph{reads}, written
into the same axiom that defines what observation is.

The custody must be kept with complete honesty, because it
is what separates this claim from a free lunch. The bulk
remains exactly T-symmetric: nothing in the geometry
distinguishes the two time orientations, which is why CPT
is exact (T-P5-2) and why the microscopic T-violation of
the mirrors is a texture chirality, not an arrow
(Part~V). The arrow lives entirely in the dictionary ---
in the non-invertibility of the projection --- so the
old question ``why does entropy increase?'' receives a
structural answer: \emph{because reading is one-way}. What
this buys over the past hypothesis is location, not magic:
the asymmetry is no longer an unexplained property of one
special state at one special moment, but a stated property
of a named law, present at every moment, doing work in
every measurement --- the same non-invertibility that
actualizes records is the one that ages the universe. And
what it costs is stated with equal clarity: AX-3 is an
axiom; the program derives the arrow \emph{from} the
constitution, not from nothing. The innovation --- and the
author believes it is the first of its kind --- is that
irreversibility now has an address inside the laws, where
every previous account kept it outside them.

\section{The chronology, quantified: boundary time across the descent}
\label{sec:time-chronology}

The arrow section located irreversibility in the law; this section
counts it. Boundary cosmic time and the bulk radial coordinate
$r = \ln(E_{\rm GUT}/\mu)$ are two parametrizations of one descent,
and the map between them is not a metaphor but a computable
function --- epoch by epoch, from standard kinematics plus the
canon anchors. In the radiation era the Friedmann clock reads
$t = (2.42/\sqrt{\gstar})\,(\mathrm{MeV}/T)^2$~s, and substituting
$T = E_{\rm GUT}\,e^{-r}$ gives
\begin{equation}
t(r) \;=\; \frac{2.42~\mathrm{s}}{\sqrt{\gstar}}
\left(\frac{\mathrm{MeV}}{E_{\rm GUT}}\right)^{2} e^{2r}
\qquad\Longrightarrow\qquad
\boxed{\ \frac{d\ln t}{dr} \;=\; 2\ }
\label{eq:time-slope}
\end{equation}
--- exactly, in the radiation era (the numerical check between the
lock and the electroweak crossover returns slope $2.000$), and
$d\ln t/dr = 3/2$ in the matter era ($t \propto T^{-3/2}$).
\emph{Boundary time is the exponential shadow of the linear
descent}: each unit the read head advances along the fiber without
holonomy, the universe's clock multiplies by $e^{2} \simeq 7.4$.
The chronology of everything, in one table (both conventions of the
radial coordinate are shown: the cosmology companion anchors
$E_{\rm GUT} = 2.3\times10^{16}$~GeV; the proton and electron
companions anchor $E_{\rm GUT} = \tfrac12 M_{\rm Pl}$, in which
convention the lock sits at $r = K$ exactly):

\begin{table}[htbp]
\centering
\caption[The chronology of everything, in one table.]{The chronology of everything, in one table. Each row is an epoch, with its temperature, its radial coordinate in both conventions, its cosmic time, and the logarithm of that time in Planck units. Both conventions are shown because the cosmology companion and the proton and electron companions anchor $E_{\rm GUT}$ differently; in the second, the lock sits at $r = K$ exactly. The two bold entries are the ones the paper returns to: the lock crossing, and today.}
\label{tab:epoch-ladder}
\small
\renewcommand{\arraystretch}{1.22}
\begin{tabular}{l r r r r r}
\toprule
\textbf{epoch} & \textbf{$T$} & \textbf{$r$ (A10)} &
\textbf{$r$ (P6)} & \textbf{$t$} &
\textbf{$\ln(t/t_{\rm Pl})$} \\
\midrule
launch ($T_{\rm rh}$) & $3\times10^{10}$ GeV & 13.6 & 19.1 &
$2.6\times10^{-28}$ s & 36.1 \\
lock crossing & 3.85 TeV & 29.4 & \textbf{35.00} &
$1.6\times10^{-14}$ s & 67.9 \\
EW crossover & 159.5 GeV & 32.6 & 38.2 &
$9.2\times10^{-12}$ s & 74.2 \\
QCD crossover & 156.5 MeV & 39.5 & 45.1 &
$2.4\times10^{-5}$ s & 89.0 \\
BBN & 1 MeV & 44.6 & 50.2 & 0.74 s & 99.3 \\
equality & $z \simeq 3400$ & 58.6 & 64.2 & 51 kyr & 127.7 \\
recombination & $z \simeq 1090$ & 59.8 & 65.3 & 380 kyr &
129.7 \\
today & 2.725 K & 66.8 & 72.3 & 13.80 Gyr & \textbf{140.24} \\
\bottomrule
\end{tabular}
\end{table}

\paragraph{The e-fold ledger of the age.} Read the last column as
an audit, and it closes:
\begin{equation}
\ln\frac{t_0}{t_{\rm Pl}} \;=\; 140.24
\;=\; \underbrace{36.1}_{\substack{\text{launch:}\\
\ln(t_{\rm rh}/t_{\rm Pl})}}
\;+\; \underbrace{91.6}_{\substack{\text{radiation:}\\
\simeq\,2\times\Delta r}}
\;+\; \underbrace{12.5}_{\substack{\text{matter}+\Lambda:\\
\simeq\,\tfrac32\times\Delta r + O(1)}}
\label{eq:time-ledger}
\end{equation}
(the three segments recompute the direct value to every displayed
digit). \emph{The age of the universe, expressed in the bulk's own
language, is an audited sum of slope-2 and slope-$\tfrac32$
segments of the one descent} --- standard kinematics, the derived
reheating temperature, and nothing else. Two placements follow at
canon grade. The corridor switch sits at $z_{\rm sw} = e^{7}$, so
$\ln(1 + z_{\rm sw}) = 7 = J$: \emph{the switch is exactly $J$
e-folds of boundary scale below today} --- the topological member
of the two-quantity discipline, now located on the chronology.
(The atomic $z_{\rm dec} \simeq 1090$ keeps its separate identity,
as always.) And the whole span from the GUT edge to the present is
$\Delta r \simeq 72$ in the half-Planck convention --- the
descent's odometer reading at the moment this sentence is being
read.

\paragraph{A flagged observation, held to the numerology
discipline.} The reader will have noticed what the author noticed:
$140.24 = 4K \times 1.0017$ with $K = 35$, and the Hubble time
gives the same reading, $\ln(H_0^{-1}/t_{\rm Pl}) = 140.29$.
\emph{The age of the universe is $e^{4K}$ Planck times to two
parts in a thousand.} The decomposition \eqref{eq:time-ledger}
shows why something $K$-structured appears --- the segments are
slope-weighted spans of a descent whose depth is $K$-structured
--- and also why the compression is not yet a result: the $O(1)$
pieces (the launch's $36.1 \approx K + 1$, the matter era's
$12.5$) have no derivation today. This paragraph therefore
carries the same grade as the cosmology companion's flagged warp
form: \emph{compression observed, derivation open} --- and the
open derivation has a name, the e-fold budget campaign of the
program's ledger, whose target ($T_0$ derived through the
descent with no dial) is precisely what would promote
\eqref{eq:time-ledger} from an audit of measured anchors to a
zero-anchor theorem. Until then the sentence ``the universe is
four throat depths old'' is filed as a candidate, in exactly these
words, at exactly this grade.

\begin{figure}[t]
\centering
\resizebox{0.757\textwidth}{!}{%
\begin{tikzpicture}[font=\sffamily\footnotesize, >=Stealth]
% axes: x = r (P6 convention), y = ln(t/tPl)
\draw[->] (0.4,0) -- (11.2,0) node[right] {$r=\ln(E_{\rm GUT}/T)$};
\draw[->] (0.4,0) -- (0.4,6.1) node[above] {$\ln(t/t_{\rm Pl})$};
\foreach \r in {20,30,40,50,60,70}{
  \draw ({0.4+(\r-15)*0.18},0) -- ({0.4+(\r-15)*0.18},-0.07)
    node[below] {\r};}
\foreach \y in {40,60,80,100,120,140}{
  \draw (0.4,{(\y-25)*0.048}) -- (0.33,{(\y-25)*0.048})
    node[left] {\y};}
% radiation segment (slope 2): launch -> equality
\draw[line width=1.10pt, blue!70!black]
  (1.143,0.533) -- (3.976,2.061) -- (4.548,2.364)
  -- (5.796,3.072) -- (6.705,3.565) -- (9.232,4.930);
% matter segment (slope 3/2): equality -> today
\draw[line width=1.10pt, red!70!black]
  (9.232,4.930) -- (9.437,5.027) -- (10.697,5.532);
% epoch points
\fill (1.143,0.533) circle (1.6pt)
  node[right=2pt, align=left] {launch $(19.1,\,36.1)$};
\fill (3.976,2.061) circle (1.6pt)
  node[above left] {lock $(r=K)$};
\fill (4.548,2.364) circle (1.4pt) node[below right] {EW};
\fill (5.796,3.072) circle (1.4pt) node[below right] {QCD};
\fill (6.705,3.565) circle (1.4pt) node[below right] {BBN};
\fill (9.232,4.930) circle (1.6pt)
  node[below right] {equality};
\fill (9.437,5.027) circle (1.4pt);
\node[anchor=east] at (9.15,5.19) {recombination};
\fill (10.697,5.532) circle (1.8pt)
  node[below right=1pt and -2pt, align=left]
  {today\\ $(72.3,\,140.24)$};
% slope labels
\node[blue!70!black, rotate=28] at (5.4,3.35)
  {slope $= 2$ (radiation)};
\node[red!70!black, rotate=22] at (10.05,5.02)
  {slope $\tfrac32$};
% z_sw annotation
\draw[->, violet!70!black] (8.25,5.75) -- (9.40,5.10);
\node[violet!70!black, anchor=east, align=center]
  at (8.25,5.82) {$\ln(1{+}z_{\rm sw}) = 7 = J$\\
  ($J$ e-folds below today)};
\end{tikzpicture}}
\caption[The whole history of time, on one line.]{The whole
history of time, on one line: boundary time (in Planck units,
logarithmic) against the radial coordinate (half-Planck
convention), for the eight epochs of the table. The radiation era
is a straight line of slope exactly $2$
\eqref{eq:time-slope}; at matter--radiation equality the slope
kinks to $\tfrac32$; the small $\gstar$ steps are invisible at
this scale. The lock crossing sits at $r = K = 35$; the corridor
switch sits $J = 7$ e-folds of boundary scale below today; the
endpoint reads $\ln(t_0/t_{\rm Pl}) = 140.24$, whose audited
decomposition --- and whose flagged $4K$ compression --- are
Eq.~\eqref{eq:time-ledger} and the paragraph beside it. Boundary
time is the exponential shadow of the linear descent.}
\label{fig:time-chronology}
\end{figure}

\section{Energy and entropy: two currencies of one
direction}
\label{sec:time-energy-entropy}

The definition makes an old pairing structural. Energy is
the currency of endowment two: the Noether charge of the
clock's repetition, conserved because the geometry repeats,
one-sided because the dictionary demands stability ---
energy is \emph{what the symmetry of time is worth}.
Entropy is the currency of endowment three: the count of
the dictionary's accumulated records, non-decreasing
because the projection is one-way --- entropy is \emph{what
the arrow of time has read}. The two great quantities of
physics are thus the bookkeeping of two endowments of a
single direction: the first law is the clock's ledger, the
second law is the dictionary's.

The shore is where the two currencies exchange. The Wick
quartet (T-P5-B1) established that one analytic fact
underlies the rotor and the semigroup --- the quantum
phase $e^{-iEt/\hbar}$ and the Boltzmann weight
$e^{-\beta E}$ are the two boundary values of one function
on the half-plane, so temperature is imaginary period and
the KMS condition is the shore's own geometry
(Cor.~\ref{cor:B1b}). This is why thermodynamics and
quantum mechanics, so different in temperament, keep
producing each other's formulas: they are the two banks of
one river --- quantum mechanics written on the oscillatory
side, thermodynamics read on the damped side, with
analytic continuation the ferry. And the runtime makes the
union quantitative: the Geometric Free-Energy Functional
of the \GRIP{} constitution decomposes as
$F_{\GFF} = F_{\GEP} + F_{\CEP}$ --- an energetic block
and a curvature--entropy block \citep{ArisakaA6Gi4} ---
so the quantity that the cascade extremizes is precisely
energy and entropy on one ledger, free energy as the
single price the direction of time charges for structure.
Every equilibrium in the descent, from the lock crossing
to the attractors of Part~\ref{part:matter}'s transport,
is an entry in that ledger; the sphaleron era's freeze-out
--- where the cosmic record was written by microscopic
dynamics --- is the one event where this Part's two
currencies and Part~\ref{part:matter}'s six-in-ten-billion
receipt visibly meet (\citealp{ArisakaA7Qi4} carries the
information-theoretic reading one floor further).

\subsection{The quartet, worked once: one mode, four faces}
\label{ssec:time-quartet-worked}

The exchange between the currencies is not a metaphor, and five
minutes of arithmetic makes it property. Take a single boundary
mode of energy $E$, thermal state at inverse temperature
$\beta$, and compute the correlator of its lowering and raising
operators. With $\langle a^\dagger a\rangle_\beta = n_\beta =
(e^{\beta E}-1)^{-1}$ and $a(t) = a\,e^{-iEt}$ (units
$\hbar = 1$):
\begin{equation}
G_{+}(t) = \langle a(t)\,a^\dagger\rangle_\beta
= (1 + n_\beta)\,e^{-iEt},
\qquad
G_{-}(t) = \langle a^\dagger\,a(t)\rangle_\beta
= n_\beta\,e^{-iEt}.
\label{eq:time-corr}
\end{equation}
Both are boundary values of one function analytic in the strip
$-\beta < \mathrm{Im}\,t < 0$ --- the shore --- and on that
strip the single identity
\begin{equation}
G_{+}(t - i\beta)
= (1 + n_\beta)\,e^{-iEt}\,e^{-\beta E}
= n_\beta\,e^{-iEt}
= G_{-}(t)
\label{eq:time-kms}
\end{equation}
is the KMS condition, derived rather than postulated: thermal
equilibrium \emph{is} periodicity in imaginary time, with
period $\beta$. Now read the four faces of the quartet
(T-P5-B1) off the same exponential. On the real axis,
$e^{-iEt}$: the rotor --- quantum phase, interference,
Schr\"odinger evolution. Down the imaginary axis,
$e^{-\tau E}$: the semigroup --- damping, Euclidean
computation, the lattice's arithmetic. At depth $\beta$, the
weight $e^{-\beta E}$: the Boltzmann factor --- thermodynamics,
read at one imaginary instant. And around the thermal circle
(period $\beta$), the trace: the partition function
$Z = \sum_n e^{-\beta E_n}$ --- the whole of equilibrium
statistical mechanics as the shore's closed loop. One
exponential, one half-plane, four sciences; the custody line
holds throughout --- the geometry owns the analyticity (the
one-sided spectrum is what makes the strip lawful), the
dictionary owns the words phase, heat, and Wick.

\begin{figure}[H]
\centering
\resizebox{0.569\textwidth}{!}{%
\begin{tikzpicture}[font=\small, >=Stealth, line width=0.80pt]
% axes of complex t plane
\fill[teal!8] (-4.6,0) rectangle (4.6,-3.1);
\draw[->] (-4.8,0) -- (5.0,0) node[right] {$\mathrm{Re}\,t$};
\draw[->] (0,-3.4) -- (0,1.6) node[above] {$\mathrm{Im}\,t$};
% real axis annotation (QM)
\node[align=center, font=\footnotesize, anchor=south] at (3.1,0.18)
  {\textbf{the rotor} $e^{-iEt}$\\ quantum mechanics: oscillation};
% forbidden upper half
\node[align=center, font=\footnotesize, gray] at (-2.7,0.95)
  {no upper shore:\\ the spectrum's empty side};
% lower half (shore)
\node[align=center, font=\footnotesize, teal!40!black] at (-2.5,-1.15)
  {\textbf{the shore}: lawful continuation\\ damping
  $e^{-\tau E}$, Euclidean computation};
% KMS depth line
\draw[dashed, red!60!black] (-4.6,-2.4) -- (4.6,-2.4)
  node[pos=0.86, above, font=\footnotesize, red!60!black]
  {depth $\beta$: the Boltzmann weight $e^{-\beta E}$};
% KMS identification arrow
\draw[->, red!60!black] (2.2,0) to[bend left=-35] (2.2,-2.4);
\node[font=\footnotesize, red!60!black, anchor=west] at (2.45,-1.3)
  {KMS: $G_{+}(t-i\beta) = G_{-}(t)$};
\end{tikzpicture}}
\caption[The shore: lawful continuation's half-plane.]{The shore, drawn. The one-sided energy spectrum makes
the lower half-plane of complex time the unique region of lawful
analytic continuation: quantum mechanics lives on the real
boundary, Euclidean damping down the imaginary axis, and
thermodynamics at depth $\beta$, where the KMS identity
\eqref{eq:time-kms} closes the thermal circle. The upper
half-plane is not forbidden by decree but by stability: nothing
damps there because nothing falls forever.}
\label{fig:time-shore}
\end{figure}

\section{Temperature, the shore, and the scale rate}
\label{sec:time-shore-rate}

Two theorems about temperature now stand in the cohort, in two
papers, and the two-fiber language shows they are one. This Part's
worked quartet (\S\ref{ssec:time-quartet-worked}) derived the KMS
condition: \emph{temperature is imaginary period} --- thermal
equilibrium is periodicity of period $\beta$ around the shore's
circle. The open-system companion derived Unruh--Tolman:
\emph{temperature is the inverse scale rate} --- the locally
measured $T$ is the asymptotic temperature dressed by the scale
profile at the observer's radial position
\citep{ArisakaA5GDOS}. Married: \emph{temperature is where the
interval meets the shore.} The thermal circle is a Euclidean
circle whose circumference $\beta$ is transported along $r$ and
stretched by the profile --- Tolman's old law, read geometrically
--- and cooling \emph{is} descent: $d\ln T = -dr$ along the
radiation era, so the shore's period grows by a factor $e$ per
unit of descent while boundary time grows by $e^{2}$
\eqref{eq:time-slope}. That mismatch of exponents --- period
growing linearly in e-folds, duration quadratically --- is why
the ratio $\beta/t$ falls monotonically through cosmic history,
which is the precise, quantitative sense in which the universe
becomes ever more classical: the quantum strip narrows relative
to the age that measures it. The third law joins the family as
the zero-scale limit (the shore's circle cannot be unwound to
infinite circumference in any finite descent), and the first and
second laws were already the axioms' content --- so all of
thermodynamics' furniture now sits in one room with one window.

One worked object closes the section, because the cohort's
standard is that a marriage of theorems must be consummated in
numbers. \emph{The cosmic microwave background is the oldest
thermometer in existence, and it is a shore-period gauge.} Its
period today: $\beta_0\hbar = \hbar/(k_B \times 2.725\,\mathrm{K})
= 2.80$~ps. Its period at recombination, thirteen and a half
billion years earlier: $2.80~\mathrm{ps}/(1+z_{\rm dec}) =
2.57$~fs. The two readings differ by exactly the
$\ln(1+z_{\rm dec}) \simeq 7$ e-folds that the chronology table
carries between those rows --- one thermometer, read twice across
the descent, confirming the transport law that connects the
quartet to the profile. Time's shore, like time's axis, has a
measured history.

\section{The comparison, in one table}
\label{sec:time-comparison}

\begin{table}[H]
\centering
\caption[What each account gives the direction called time.]{What each account gives the direction called time.
Bold entries are, to the author's knowledge, the first
law-level appearances of the corresponding endowment. The
structural cousins --- Connes--Rovelli's thermal time
\citep{ConnesRovelli1994}, Page--Wootters relational time
\citep{PageWootters1983}, and the CPT-symmetric cosmology
\citep{BoyleFinnTurok2018} --- are engaged in the text.}
\label{tab:time-comparison}
\small
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{p{3.1cm} p{2.3cm} p{2.6cm} p{2.9cm} p{2.9cm}}
\toprule
\textbf{Account} & \textbf{Order} & \textbf{Repetition} &
\textbf{Arrow} & \textbf{Shore} \\
\midrule
Newtonian mechanics & assumed & assumed & absent & absent \\
Special relativity & \textbf{derived} (cone) & assumed
(static ansatz) & absent (block) & unnamed ($it$) \\
General relativity & derived, dynamical & \emph{fails
generically} (no Killing flow) & absent & ill-defined off
special backgrounds \\
Thermodynamics & inherited & inherited & initial condition &
implicit (KMS) \\
Quantum mechanics & inherited & assumed ($e^{-iHt}$) &
absent & used, disowned \\
QFT + Standard Model & microcausality & assumed & absent
(T-violation $\neq$ arrow) & used everywhere, named
nowhere \\
String / quantum gravity & background & background &
absent & Euclidean tool \\
\GGTheory{} (P4 + P5) & AX-1 (cone) & AX-2 + Thm.~0
(forced) & \textbf{AX-3 law} (\OGGEP{}) & \textbf{AX-3
spectrum} (T-P5-B1) \\
\bottomrule
\end{tabular}
\end{table}

Three cousins deserve their sentences. Connes--Rovelli is
the nearest: time as modular flow, KMS-rooted, one
parameter --- but state-first, where this account is
arena-first; the two would meet if the constitutional
clock's flow were shown to be the modular flow of the
boundary state, which is precisely the open modular bridge
of Objection~69. Page--Wootters is the honest quantum
attempt to internalize the clock; the arena account
dissolves its circularity by making the clock
constitutional rather than emergent. And Boyle--Turok's
CPT-symmetric universe shares this paper's reverence for
the unbroken reversal while locating the asymmetry in the
state; the contrast --- state versus law --- is the same
contrast that separates the past hypothesis from the
\OGGEP{} arrow, run at cosmological scale.

\section{The significance: the new definition against every previous one}
\label{sec:time-significance}

The table and the cousins locate the accounts technically. This
section states the significance plainly, tradition by tradition,
in the Part's register --- each honored for what it saw, each
located in what it could not see --- and then names the two claims
that, to the author's knowledge, no previous definition of time
has ever made.

\paragraph{Newton: time as absolute flow.} ``Absolute, true, and
mathematical time, of itself, and from its own nature, flows
equably without relation to anything external.'' Honored: the
equable flow is real, and it is the repetition endowment --- the
Killing flow's exact self-similarity is what Newton's ``equably''
was reporting. Located: nothing flows. A flow needs a medium and a
direction; the constitutional clock is a symmetry, not a current.
What Newton called flow decomposes into the tick (which repeats
and does not move) and the arrow (which points and does not
repeat) --- two fibers he had no way to distinguish, fused into
one river by introspection.

\paragraph{Leibniz and Mach: time as relation.} Honored, and
more than honored: boundary time in this framework \emph{is}
relational --- it is a reading, not a substance; nothing possesses
time except in the dictionary's act of registration. Located: the
relations are not free-floating. They are readings \emph{of a
definite geometric object} with a fixed anatomy, which is why all
observers' relational times agree on order, rate structure, and
arrow. Relationalism was right that time is not a thing; it
lacked the theorem that says what the relations are relations
\emph{to}.

\paragraph{Einstein and Minkowski: time as geometric order.}
Honored at the deepest level: endowment one is inherited whole
from the cone, and the relativity of simultaneity survives intact
--- ``elsewhere'' is as real here as in 1908
\citep{Minkowski1908}. Located: the block universe carries order
and nothing else. It has no origin for the tick (the static
metric is assumed, not derived), no arrow (the block is
T-symmetric and eternalism follows), and it uses the shore
($it$) as a trick while disowning it as physics. One endowment of
four --- which is exactly why the block picture, extended beyond
its competence, generates the paradoxes of frozen time it is
famous for.

\paragraph{Boltzmann and the past hypothesis: time as
statistics.} Honored: the arrow is real and thermodynamic, and
Boltzmann's identification of entropy with counting
\citep{Boltzmann1877} survives as the second currency. Located:
the standard account charges the entire asymmetry of the universe
to an unexplained initial condition --- the past hypothesis ---
because its laws are reversible and something must break the tie.
Here the tie is broken in the law (\S\ref{sec:time-arrow}): the
projection is non-invertible at every moment, not just the first
one. The asymmetry moves from one special state to a standing
property of reading --- from cosmic accident to constitutional
clause.

\paragraph{Quantum mechanics: time as the external parameter.}
Honored: the $t$ in $e^{-iHt}$ works, uncannily. Located: the
puzzle of why time alone has no operator --- Pauli's old theorem
that a time observable conflicts with a bounded spectrum ---
dissolves rather than deepens. Time is not an observable because
it is not a degree of freedom \emph{of} the boundary at all: it is
the boundary's reading of the fibers, the parameter of the
dictionary's own act. Asking for a time operator is asking the
reading to read itself; the one-sided spectrum that blocks the
operator is the same one-sidedness that opens the shore --- the
obstruction and the endowment are one fact.

\paragraph{Wheeler--DeWitt and the frozen formalism.} Honored as
a correct theorem \emph{about the bulk} \citep{DeWitt1967}: the
constitutional arena is static and T-symmetric, and a
wave-functional of the whole geometry has no external clock ---
the equation says of the bulk what this Part says of it: no time
there. Located: the paradox was only ever the category error of
asking the tape to age. Time was never in the bulk; it is what
reading the bulk feels like from the boundary. The problem of
time in quantum gravity is not solved here; it is \emph{located},
which is how honest dissolutions work.

\paragraph{Page--Wootters and Connes--Rovelli.} Honored as the
two twentieth-century accounts that came nearest --- one
internalizing the clock into correlations
\citep{PageWootters1983}, one deriving a flow from the state's
own thermality \citep{ConnesRovelli1994}. Located as fragments of
the dictionary: relational readings and modular flows are what
\OGGEP{} projections look like when one reconstructs them from
inside, without the arena; the open modular bridge of
Objection~69 remains the meeting point, and remains open.

\paragraph{The two claims no predecessor makes.} First, the
\emph{composite anatomy}: every previous account treats time as
one thing --- a flow, a relation, an order, a parameter, an
illusion --- and each tradition's partial success came from
correctly describing one endowment. This account is, to the
author's knowledge, the first in which time is constitutionally
\emph{composite}: four endowments on three geometric structures,
with the tick and the arrow on \emph{different fibers} --- a
separation no earlier definition even had the vocabulary to
state, and which retroactively explains why the traditions
diverged as they did. Second --- and this is the innovation this
Part's expansion exists to register --- \emph{a definition with a
number}. Every previous definition of time, from Augustine to the
operationalists, said what time is; none said \emph{how much of
it there is, and why}. Here the definition comes with an audited
chronology \eqref{eq:time-ledger}: the age of the universe as a
closed e-fold ledger on the descent, the slope laws exact, the
clock hierarchy spanning $e^{140}$, the corridor switch placed at
$J$ e-folds, and even the definition's own falsifiers inherited
from the anchors it audits ($T_{\rm rh}$ at its derived grade;
the corridor's observables; the Kaxion line at 1.95~GHz). A
definition of time that can be \emph{wrong about numbers} is a
new kind of object in this oldest of subjects --- and in this
program's judgment it is the only kind worth defending.

\section{What is time? --- an answer for everyone}
\label{sec:time-for-everyone}

Strip the machinery, and say it plainly. Time is not a
river, and you are not a boat on it. Time is not a
container, and the universe is not its contents. Time is
the one direction of reality that does four jobs at once,
and nothing else does even two of them. It \emph{orders}:
before and after are real, the way uphill and downhill are
real on a slope. It \emph{repeats}: the laws are the same
this second as last second, and that sameness is why
anything can tick, why energy exists, why a note can hold
its pitch. It \emph{opens}: the world writes and cannot
unwrite --- every record, every memory, every scar and
every star is ink the universe cannot take back, and that
one-way writing, not any river's current, is the flowing
you feel. And it \emph{shelters}: leaning against the
world's own stability there is a quiet mathematical shore,
where the humming of quantum mechanics and the cooling of
thermodynamics turn out to be one thing seen from two
sides --- the strangest and best-kept secret in physics,
used daily by every physicist and named here, the author
believes for the first time, as a structural clause of a
definition of time --- the analytic structure itself is
classical (Paley--Wiener, KMS, tube analyticity), and what
is claimed is the naming, not the mathematics: a
structure the world cannot do without, though no one can
go there.

Augustine could not say what time is because he was
looking for a thing, and time is not a thing; McTaggart
proved the river cannot flow, and he was right, because
the flowing was never time's --- it is the writing's. What
flows is the accumulation of the written: the dictionary's
one-way ink, in which a kaon's decay, a cobalt nucleus's
last electron, a memory of a summer, and the six extra
baryons in every ten billion are entries of the same kind,
irrevocable for the same structural reason. The mirrors of
this paper broke on the descent's orientation; the faces
of the clock stand on time's. A universe with less
structure in its time could not have laws worth the name,
could not bank energy, could not keep a record, could not
do quantum mechanics and thermodynamics with the same
hand --- could not, in the end, contain anyone to ask
Augustine's question. That the direction which permits the
asking can now be characterized --- ordered, repeating,
one-way, and sheltered --- is this program's answer to
the oldest question it will ever face: time is the way a
stable world keeps being possible, and the proof that it
has been keeping records is that you are here, reading
this, later.

And for the reader who asks the child's version of the
question --- \emph{how much time is there?} --- the answer, for
the first time in this subject's twenty-six centuries, has a
number with an audit trail. There have been $e^{140}$ Planck
ticks since the descent began: thirty-six e-folds to the launch,
ninety-two at slope two through the radiation era, twelve and a
half at slope three-halves since matter took over
(\S\ref{sec:time-chronology}). The universe is not merely old;
it is \emph{accounted for} --- and the account is one straight
line with one kink, drawn on the fiber that cannot remember, by
the fiber that cannot forget.

\medskip
\noindent\emph{Part-end summary.} Time defined by its four
endowments --- order, repetition, arrow, shore --- each located
in an axiom \emph{and now in the anatomy}: the cone orders, the
circle ticks, the interval points, the spectrum's single side
shelters. The tick and the arrow shown to live on different
fibers --- the structural reason mechanics and thermodynamics
grew up apart; every clock a winding counter, from the electron's
$1.24\times10^{20}$~Hz to the Kaxion's predicted 1.95~GHz to the
sky's $H_0^{-1}$. The arrow at the level of law for the first
time; the chronology quantified --- $d\ln t/dr = 2$ exactly, the
age closing as an audited e-fold ledger at
$\ln(t_0/t_{\rm Pl}) = 140.24$, the $4K$ compression flagged at
its honest grade; temperature married to the scale rate as the
shore's transported period; energy and entropy unified as the
direction's two currencies; the standard accounts honored as
partial readings of a composite anatomy; the quantum-gravity
problem of time dissolved by architecture; the cousins engaged;
and the significance stated: the first definition of time that is
composite, and the first that comes with a number it could be
wrong about. What remains is comparison at full depth: the five
roads.

\GGPart[The five roads, at full depth]{The Five Roads, at Full Depth}{QFT, the Standard Model, string theory, and $\Lambda$CDM --- what each achieved, where each ends, and how the geometry completes the story; the inventory of who predicts what; the three-depths scoreboard; and the roads seen from above}
\label{part:fiveroads}

The \GG{} account is now complete: the twins derived
(Part~\ref{part:antiparticles}), the mirrors defined
(Part~\ref{part:theorems}), the two breakings explained
(Parts~\ref{part:parity} and~\ref{part:pentagon}), the matter
assembled (Part~\ref{part:matter}), and time itself given its
four endowments (Part~\ref{part:time}). Only now do the rivals speak
--- and each is given its full say, because the significance of
the \GG{} claims is measurable only against the strongest
versions of what the other frameworks actually achieve. Each
road is walked in three movements. \emph{First, what it
achieved} --- stated with gratitude and with numbers, because
every one of these traditions contributed load-bearing pieces
to this very paper. \emph{Second, where it ends} --- the honest
boundary of its explanatory reach, located precisely, in its
own terms. \emph{Third, how the geometry completes the story}
--- claim by claim, with the grade of each completion printed,
because a completion offered without its register would repeat
the very habit this paper exists to break. The ruler throughout
is the three-depths instrument of \S\ref{sec:three-depths}; the
register is structural contrast, never polemic; and every
concession the \GG{} account owes is paid in the section where
its creditor lives.

\section[Part~X in plain English]{Part~X in plain English: the other roads, at their own strength}
A paper that scores itself against straw versions of the alternatives has not been tested. This Part sets out quantum field theory, the Standard Model, string theory and $\Lambda$CDM at full depth, and in each case says what the road achieved before saying what it does not reach. Each of the four is the most successful thing of its kind ever built, and the Part opens by saying so.

Then the inventory: who predicts what, road by road, on the same questions. The scoring uses the three depths fixed in Part~\ref{part:prelim}, so the comparison runs against a stated instrument rather than against a preference, and where another road does better the inventory says so. The falsification-symmetry section states what would count as this account losing, which is the same question asked in the other direction.

The Part closes on a comparison the five roads do not usually make against one another: what each of them says time is. That is where they differ most and argue least, which is exactly the reason to put it in a table.

\section{The first road: quantum field theory}
\label{sec:road-qft}

\subsection{What quantum field theory achieved}
\label{ssec:qft-achieved}

Quantum field theory is the framework of twentieth-century
physics, and the scale of its achievement should be stated
before any boundary is drawn. It unified the century's two
revolutions --- quantum mechanics and special relativity ---
into a single consistent grammar, and the grammar turned out to
\emph{legislate}: antiparticles exist (the causality theorem
below), spin and statistics are locked to each other, CPT is
inviolable --- three facts about the world derived from
consistency alone, before any particular interaction is chosen.
It taught physics that the vacuum is a medium: the Lamb shift
and the anomalous moment of the electron are properties of
empty space's own fluctuations, and the latter is now computed
and measured in agreement at roughly a part in $10^{12}$ ---
the most precisely verified prediction in the history of
science. It taught physics what its own equations mean: the
renormalization group reads a quantum field theory as a family
of descriptions indexed by resolution --- the insight the \GG{}
arena promotes to a geometric dimension, as Part~III recorded.
And it supplied the discipline this paper's parent program
lives by: anomaly freedom as an admissibility filter, unitarity
and causality as non-negotiable clauses, the S-matrix as the
contract with experiment. Three of this paper's own
load-bearing theorems --- Weinberg's causality argument,
Jost's CPT theorem, and the Osterwalder--Schrader
reconstruction --- are QFT results, inherited here with their
provenance stamped. Whatever this Part concludes about where
the road ends, the road itself carried physics farther than
any other ever built.

\subsection{The road's own summit: what the renormalization
group whispered}
\label{ssec:qft-rg}

One achievement deserves its own subsection, because it is the
place where the first road, at its very summit, points beyond
itself. The renormalization group began as a repair --- the
taming of infinities --- and matured, in Wilson's hands, into
the deepest structural insight the framework produced: a
quantum field theory is not one description but a
\emph{family} of descriptions indexed by resolution, with the
couplings flowing as the index runs. Two consequences
reorganized physics. Practically, it created the modern
effective-field-theory posture: every theory is an honest
description \emph{at its scale}, ignorant of its ultraviolet
completion by construction --- the humility that made
precision physics possible. Structurally, it revealed that
``the scale at which the world is examined'' behaves like a
coordinate: quantities depend on it lawfully, differential
equations run along it, fixed points organize it. QFT
discovered, in its own language, that resolution is a
\emph{direction} --- and then, having no geometry to put the
direction in, filed it as bookkeeping. Part~III recorded what
the parent program does with exactly this hint: the scale
direction $I_r$ of the arena is the renormalization flow taken
literally --- a non-spatial dimension generated by a
dilatation, on which the descent of this paper runs downhill.
Seen from this Part, the relation is filial: the road's summit
insight, that physics is resolution-indexed, is the \GG{}
arena's fifth dimension --- promoted from an index to a
geometric direction, at the doctrine and grades of Part~III.
The road did not end at its summit; it stopped, and the
geometry claims to be the continuation of its own best idea.

\subsection{The twins in QFT: three layers, and where the
explanation stops}
\label{ssec:qft-antiparticles}

\emph{Layer 1 --- Dirac's holes (1928--1931): the discovery that
was not an explanation.} The Dirac equation, built to give a
first-order, Lorentz-covariant, positive-probability wave
equation for the electron \citep{Dirac1928}, unavoidably
possessed negative-energy solutions. Dirac's proposal
\citep{Dirac1930, Dirac1931} --- the vacuum as a filled sea of
negative-energy states, holes in the sea appearing as
positive-energy particles of opposite charge --- predicted the
positron, and Anderson's cloud chamber delivered it
\citep{Anderson1933}: one of the great predictions in the
history of science. But the sea failed as an \emph{explanation},
three times over. It requires the exclusion principle, so it
cannot work for bosons --- yet the $\pi^{+}$ and $\pi^{-}$ are
as real a pair as the electron and positron. It makes the vacuum
infinitely charged and infinitely massive. And it treats the
twins asymmetrically --- one a particle, the other an absence
--- when every measured property says they are perfectly
symmetric. Field theory dismantled the sea, and it survives only
as pedagogy (and, honorably, in condensed matter, where a
literal filled band exists). The cohort uses no sea anywhere;
neither does this paper.

\emph{Layer 2 --- St\"uckelberg and Feynman (1941, 1949): the
relabeling.} The modern reading \citep{Stueckelberg1941,
Feynman1949}: the negative-frequency solutions are not
negative-energy particles but positive-energy
\emph{antiparticles} with reversed charge flow, encoded in the
Feynman propagator's contour --- positive frequencies forward in
time, negative frequencies backward. This is exactly the
Feynman--St\"uckelberg pairing whose \emph{provenance} T-P5-C
supplies: in QFT it is the reinterpretation that makes
relativistic perturbation theory consistent; the question ``why
this pairing and not the opposite?'' is answered at the next
layer.

\emph{Layer 3 --- the structural theorem.} Modern QFT's mature
answer is a genuine theorem, and this paper states it with the
respect it deserves, because it is the closest thing the
orthodoxy has to T-P5-C.

\begin{theorem}[Weinberg's causality argument; \citealp{WeinbergQTF}]
\label{thm:weinberg}
Let an interaction Hamiltonian density $H(x)$ be built from
creation and annihilation operators, transform as a Lorentz
scalar, satisfy microcausality $[H(x), H(y)] = 0$ at spacelike
separation, and act on a Hilbert space with energy bounded
below. Then the fields entering $H(x)$ must be specific linear
combinations of an annihilation part and a creation part, and
the creation part must create a species with \emph{the same mass
as, and opposite values of all conserved charges to}, the
species the annihilation part destroys. A particle carrying no
conserved charge may be its own partner; a charged particle's
partner is a distinct species: the antiparticle.
\end{theorem}

\emph{The intuition} is special relativity's own: a charged
particle propagating outside the light cone is, for some
observers, its oppositely charged partner propagating the other
way; only the paired combination is frame-independent and
causal. Antiparticle existence is the price of combining
special relativity, quantum mechanics, and causality ---
Depth~2, answered rigorously, for interacting theories.

\emph{Where QFT stops, stated exactly.} Three inputs sit
upstream of Theorem~\ref{thm:weinberg} and are never derived:
the complex Hilbert space (the $i$ is an axiom); the positivity
of energy, imposed on the \emph{output} of quantization (the
pairing is selected because the alternative wrecks the
spectrum); and the time orientation. QFT's Depth-3 answer is
``those are the axioms'' --- not a defect, but a boundary, and
it is exactly the boundary T-P5-A/B/C cross: the handshake
derives the $i$ (both of them), obtains the pairing before
quantization, and explains two of Weinberg's axioms. What must
be conceded in return: Weinberg's conclusion holds for
interacting theories; the handshake at full strength is
free/quasi-free, with the interacting case conditional. The two
theorems have disjoint hypothesis sets, and the deeper one is
currently the narrower one. Both clauses of that sentence
belong in print.

\subsection{The mirrors in QFT: definitions, discoveries, and
the anatomy-silent theorem}
\label{ssec:qft-mirrors}

QFT defines C, P, T representation-theoretically (Wigner's
theorem: symmetries are unitary or antiunitary
\citep{Wigner1931}), with per-field phase freedoms, and
possesses \emph{no principle} determining whether any of them
holds --- which is why each fall was a shock: P and C in 1957,
CP in 1964, and T directly --- a fact often omitted --- in the
kaon system (CPLEAR, 1998 \citep{CPLEAR1998}) and, cleanly and
model-independently, in the $B$ system (BaBar, 2012
\citep{BaBar2012}). The antiunitarity of T is \emph{selected},
not derived: Wigner's fork permits either character, and
unitarity is rejected because it would flip the Hamiltonian's
sign. Against this stands the one great positive result: the
CPT theorem \citep{Luders1954, Pauli1955}, given axiomatic form
by Jost \citep{Jost1957} through analytic continuation to the
complex Lorentz group, and deepened by Bisognano--Wichmann's
identification of the CPT operation (times a rotation) with the
Tomita--Takesaki modular conjugation of the wedge algebra
\citep{BisognanoWichmann1976}, with Greenberg's converse
guarding the exits: CPT violation entails Lorentz violation
\citep{Greenberg2002}. Scope: interacting, axiomatic ---
exceeding T-P5-2. Anatomy: silent --- the theorem constructs
$\Theta$ and proves it a symmetry without saying what C, P, or
T individually \emph{are}, why T carries the antiunitarity, or
why nature breaks the factors while preserving the product.
The relation to the \GG{} account is therefore precise
complementarity, and the modular route --- deriving the clock
from modular flow, tying $\Ja$ to Bisognano--Wichmann at
interacting strength --- is the named bridge on which the two
would meet (open question~(6)).

On parity specifically, QFT owns the two partial structures
already credited in \S\ref{sec:orthodox-partials} ---
Vafa--Witten's protection of vector-like parity and the
survival argument --- and nothing that reaches ``why
$\SUtwoL$.''

\subsection{The matter in QFT: a toolbox without a selection
principle}
\label{ssec:qft-matter}

Given Sakharov's conditions, QFT-plus-extensions offers five
mechanism families, all workable, none selected. \emph{GUT
baryogenesis} \citep{Yoshimura1978, Weinberg1979}: heavy-boson
decay; wounded twice --- inflation dilutes any pre-inflationary
asymmetry, and sphalerons erase $B{+}L$, so only
$B{-}L$-generating versions survive; and its baryon-creating
sector generically threatens the proton. \emph{Electroweak
baryogenesis}: elegant and testable, but requires a first-order
transition and new CP phases the Standard Model lacks, and is
increasingly cornered by electric-dipole-moment bounds.
\emph{Thermal (hierarchical) leptogenesis}
\citep{FukugitaYanagida1986}: the modern default, with the
Davidson--Ibarra bound $M_N \gtrsim 10^{9}$~GeV
\citep{DavidsonIbarra2002} --- beautiful, and untestable: no
collider will reach it, and the phases that matter are not the
low-energy ones. \emph{Resonant leptogenesis}
\citep{Pilaftsis1997, PilaftsisUnderwood2004}: quasi-degenerate
heavy states evade the bound and open the TeV window --- the
mechanism class Part~\ref{part:matter} inherits, with the
degeneracy engineered in the standard literature and claimed
inherited in ours. \emph{Oscillation (ARS) leptogenesis}
\citep{AkhmedovRubakovSmirnov1998}: GeV-scale sterile
neutrinos, testable at beam-dump experiments. Plus
\emph{Affleck--Dine} \citep{AffleckDine1985} where
supersymmetric flat directions exist. Each mechanism buys
$\etaB$ with fitted masses, phases, and departures; the
observed number constrains and never predicts. The \GG{}
contrast is the dial count of
Table~\ref{tab:audit} --- and the honest converse, that the
inherited chain's resonance condition is its named conjecture,
is already faced in \S\ref{sec:resonance-faced}.

\subsection{Where the road ends, and how the geometry continues
it}
\label{ssec:qft-verdict}

Collect the three movements. QFT achieved the grammar: any
quantum, causal, relativistic world \emph{must} contain paired
twins, locked spin-statistics, and inviolable CPT --- and it
verified its grammar to twelve digits. Where it ends is equally
sharp, and it is one boundary seen four times: \emph{the
framework cannot see beneath its own complex structure.} The
$i$ is an axiom; therefore the pairing is selected on the
output rather than derived on the input; therefore the discrete
operations are equivalence classes with adjustable phases
rather than anatomical objects; therefore the individual
mirrors have no fates, only records. Even the road's daily
Euclidean practice sits on the unexamined clause: every Wick
rotation ever performed borrows a license --- the one-sidedness
of the spectrum --- that the framework grants but does not
explain (T-P5-B1). The geometry's completion is therefore a
single move made once and cashed four times, and it is worth
itemizing with grades. \emph{The $i$, derived:} two real
sources, circle and clock, unique by their own theorems ---
T-P5-A/B, derived (quasi-free). \emph{The pairing, derived
before quantization:} the handshake, explaining two of
Weinberg's axioms at the price of the free-field qualifier ---
T-P5-C, derived (quasi-free), interacting conditional.
\emph{The operations, given bodies:} three reversals, the
table a computation, Kramers free of conventions --- T-P5-1,
derived. \emph{The Euclidean license, issued:} Minkowski's
$it$, Wick's rotation, and the Schr\"odinger $i$ as one polar
decomposition --- T-P5-B1, derived (quasi-free). \emph{The
product's anatomy:} CPT as bookkeeping over the reversals,
complementary to Jost's broader-scope theorem, with the
modular route as the named bridge --- T-P5-2, derived
(quasi-free). The road and the geometry are not rivals so much
as two halves of one account that has not yet been joined at
interacting strength: QFT holds the scope, the geometry holds
the anatomy, and the Bisognano--Wichmann bridge is where they
are due to meet.
And one further item of the road's grammar has, since this
Part was first drafted, acquired the same internal route: the
\emph{lock} itself. Spin--statistics --- the road's second great
structure theorem, sibling of CPT in every axiomatic proof ---
is now stated as the quartet's second child
(\S\ref{sec:quartet-second-child}), at candidate grade with the
residues of T-P5-2, its bulk premises in
\citealp{ArisakaA2GG6} \S20 and its boundary composition in
\citealp{ArisakaA8QM} (T-A8.12). If the promotion path closes,
the road's three locks --- twins, statistics, CPT --- will all
be anatomy rather than grammar; the first and third already are.

\section{The second road: the Standard Model}
\label{sec:road-sm}

\subsection{What the Standard Model achieved}
\label{ssec:sm-achieved}

The Standard Model deserves its section written with gratitude,
because it is the most successful physical theory ever
confronted with data, and every number in this paper is
measured against its bookkeeping. Its achievements are not
generic; they are specific, quantitative, and in several cases
they are this paper's own subject matter seen from the other
side. It predicted the $W$ and $Z$ and their masses before
their discovery, and the LEP program then measured the $Z$
lineshape at the per-mille level --- finding, among other
things, exactly three light neutrino species, the integer this
paper derives as $N = 3$. Its flavor sector turned a
two-parts-per-thousand kaon anomaly into a counting theorem:
Kobayashi and Maskawa's observation that CP violation requires
a third generation \citep{KobayashiMaskawa1973} predicted six
quarks when three were known --- one of the great indirect
predictions in physics, and the reason the author's field
existed. Its anomaly bookkeeping closes miraculously within
each generation --- the fractional hypercharges of quarks and
the integer charge of the electron conspiring to cancel every
triangle --- the very miracle POS-3 promotes to a filter. Its
Higgs boson was found where its couplings required. And for
half a century, through every collider, every precision
measurement, and every flavor factory --- including the two
the author sat --- it has not failed a single reproducible
test \citep{PDG2024}. Any framework that would deepen it must
first reproduce it; the parent program's Standard-Model
treatment exists for exactly that reason.

\subsection{Where the Standard Model ends}
\label{ssec:sm-ends}

The boundary of the road is as well-surveyed as the road
itself, and it can be stated as an inventory: the model
\emph{contains} what it cannot \emph{explain}. Counting the
neutrino sector, twenty-eight parameters are written into it
by measurement --- masses, mixings, phases, couplings --- each
one a dial that could, for all the formalism knows, have been
set otherwise. On this paper's questions the inventory reads
as five silences and two failures. \emph{Silence one:}
antiparticles --- inherited from QFT's theorem, with no
position on Depth~3. \emph{Silence two:} the operations ---
conventions with phases. \emph{Silence three:} parity --- V$-$A
written into the field content by hand in 1967; QED and QCD
parity-even automatically (given the assignments), the
coincidence of broken-with-chiral unexamined and unexaminable.
\emph{Silence four:} CP --- one physical phase (a counting
theorem, \citealp{KobayashiMaskawa1973}), measured superbly
($\delta = 1.147 \pm 0.026$), explained not at all; $\delta = 0$
would have been equally lawful, and nothing in the model prices
the difference. \emph{Silence five:} strong CP ---
$\bar\theta < 10^{-10}$ beside a large weak phase, two unrelated
parameters, one inexplicably near-maximal and one inexplicably
zero; the axion a bolt-on. \emph{Failure one (quantitative):}
its own baryogenesis --- the effective CP violation available
computes to $\sim 10^{-20}$ \citep{GavelaEtAl1994}, ten orders
short, and the electroweak transition is a crossover for the
measured Higgs mass \citep{Kajantie1996}: the Standard Model
predicts $\etaB \approx 0$, and \emph{every gram of matter is
an experimental proof of physics beyond it} --- the oldest and
most secure such proof we possess. \emph{Failure two
(structural):} no relation between any two of the five
silences: five facts about one gauge factor, zero connections.
The model is, in short, a perfect transcription of the world's
text with no theory of its authorship --- which is not an
insult but its own founders' description of an effective
theory awaiting its completion.

\subsection{Reading the dials: what the twenty-eight numbers
say}
\label{ssec:sm-dials}

Before the completion, the analysis the model itself cannot
perform: read its own dials as data. The twenty-eight
parameters sort into a definite anatomy --- three gauge
couplings; two for the Higgs sector (a scale and a
self-coupling); six quark masses; four CKM entries (three
angles and the phase); three charged-lepton and three neutrino
masses; four Dirac-sector PMNS entries; two Majorana phases;
and $\bar\theta$. Twenty-eight numbers --- and they are not
random. The text has \emph{rhymes}, and a transcription,
however exact, cannot hear them. The fermion masses span six
orders of magnitude, yet in ordered generations --- a
hierarchy with a direction, never explained. The quark mixings
are small and hierarchical while the lepton mixings are large
--- two sectors, two textures, no reason. The one weak phase
sits near its near-maximal measured value while the one strong
phase sits at zero to ten digits --- the two-dial
embarrassment. The gauge couplings run toward one another with
suspicious near-precision. Each rhyme is, in the model, a
coincidence of inputs; each is, in the parent program's
reading, a visible edge of one structure: the ordered
hierarchy is the Morse descent's direction (Part~VI); the
two-texture split is the two alphabets, $L = 5$ for the block
and $J = 7$ for the odd counter (Part~VII); the two-dial
embarrassment is the two-fate architecture (Part~VII); and the
near-unification is the ledger's throat arithmetic, priced in
the companion treatments. Rhyme is not proof --- the register
insists on that --- but rhyme is evidence of meter, and the
claim of the completion below is that the meter exists and is
computable.

\subsection{How the geometry completes the story}
\label{ssec:sm-completed}

The completion is item-for-item, and each item carries its
grade. The parent Standard-Model treatment derives the
twenty-eight parameters from the one geometry, at the
calibrated register that paper prints \citep{ArisakaA9SM}; this
paper has walked the discrete-symmetry core of that derivation
in full. Against the five silences: the twins are Depth-3
derived (Part~IV, quasi-free); the operations have bodies and
a derived table (Part~V); parity's five superlatives collapse
to one oriented-gate architecture, with the mechanism
inherited, the uniqueness conditional on L11, and the
handedness anchored by P-P5-D (Part~VI); the CP dial becomes a
lattice point --- $\delta = 0$ not merely false but unwritable
--- with the forced values confronting the fits in
Table~\ref{tab:predicted-measured} (Part~VII); and the
strong-CP embarrassment becomes the second fate of one
alphabet, bounded at $4.7\times10^{-18}$ by the ledger's own
integers (Part~VII). Against the two failures: the tripod
P-P5-E converts Sakharov's checklist into constitutional
custody, and the transport chain meets the observed
six-per-ten-billion with zero fitted parameters at D$^{*}$
grade under one named conjecture (Part~VIII). And against the
structural failure --- five facts, zero connections --- the
completion is the connection itself: one descent, oriented,
whose absence is parity violation, whose angle is CP violation,
whose door is the neutrino, and whose receipt is the matter.
Where the model has dials, the geometry has lattice points;
where the model has coincidences, an architecture; where the
model has fits, falsifiers with dates. The grades are printed
where each claim lives, and the claims die, severally and
jointly, on the instruments of Table~\ref{tab:falsifiers}.

\section{The third road: string theory}
\label{sec:road-string}

\subsection{What string theory achieved}
\label{ssec:string-achieved}

String theory is the serious geometric rival, it anticipated
several of this paper's moves, and the first movement of this
road is a list of debts. It remains the most developed
candidate for a quantum theory of gravity --- the graviton is
in its spectrum rather than added to it --- and it taught
physics that consistency in higher dimensions is a
\emph{discipline}: the anomaly-freedom mechanism of Green and
Schwarz, in which a higher form field's inflow cancels what no
field-content juggling could, is the direct ancestor of the
\BIsix{} ledger's own inflow arithmetic. It geometrized
chirality: four-dimensional handedness from the topology of
compact dimensions, generations counted by index theorems
\citep{CandelasHorowitzStromingerWitten1985}, chiral matter at
orbifold fixed points \citep{DixonHarveyVafaWitten1985} and
brane intersections \citep{BerkoozDouglasLeigh1996} --- the
technology lineage of Part~VI's projection, conceded there to
the matrix \citep{Kawamura2000, HallNomura2001}. It made the
twins economical: the antiparticle of a string state is the
\emph{conjugate excitation of the same object} (the conjugate
vertex operator), so the mass degeneracy of the pair is
automatic --- one string, two readouts, structurally the same
move as the positron-as-conjugate-readout of
\S\ref{sec:boundary-readout}, and string theory made it first.
It drew charge conjugation as geometry: an open string from
brane stack $a$ to $b$ carries the conjugate representation of
the string from $b$ to $a$ --- C as orientation reversal
\citep{Polchinski1998} --- the very image T-P5-A draws,
conceded in Part~IV's ledger. It protected CPT structurally:
worldsheet consistency transfers the CPT theorem to the
spacetime S-matrix order by order \citep{Sonoda1989}. And it
made this subject's deepest single external contribution: CP
as a \emph{discrete gauge symmetry}, embedded in the
higher-dimensional Lorentz and gauge groups
\citep{StromingerWitten1985, DineLeighMacIntire1992,
ChoiKaplanNelson1993}, hence unbreakable explicitly and broken
only spontaneously, by geometry --- conceptually the same move
as T-P5-3, made a decade earlier. Three of this paper's central
images --- conjugation as orientation reversal, chirality from
projection, CP as geometry --- were drawn first on this road.

\subsection{Where string theory ends}
\label{ssec:string-ends}

The road ends at valuation, and the ending has one structural
root. Every geometric insight of the previous movement is
realized \emph{somewhere} in the space of string vacua --- and
so is its negation: non-chiral worlds, mirror-symmetric worlds,
CP-conserving worlds, worlds of every $\delta$. Which vacuum is
ours is not derived; the landscape offers possibility without
actuality, and no number this paper cares about --- not
$\delta$, not $\bar\theta$'s smallness beside it, not $\etaB$
--- is predicted at the framework level. The moduli that break
CP spontaneously are continuous: geometric CP violation,
without a value. The cosmology inherits its own systematic
obstacle: late-decaying moduli release entropy and dilute, so
string cosmologies typically need \emph{more} baryogenesis
than field theory does \citep{CoughlanEtAl1983,
BanksKaplanNelson1994} --- the precise contrast with the
prompt crossing of \S\ref{sec:zero-dilution}. And beneath both
sits the root: the extra dimensions are \emph{spatial}, and
places must be hidden, stabilized, and chosen --- radii want
moduli, moduli want potentials, potentials want selection, and
selection is the landscape. The arena of Part~III is built on
the refusal of exactly this first step: its two extra
dimensions are classified by their generators as a resolution
and a phase, dimensionless by construction, with no radius to
stabilize, no tower to hide, and no vacuum to choose
\citep{ArisakaA2GG6}. The contrast is structural, not
polemical: the spatial program's problems are real problems,
solved with real ingenuity; the non-spatial arena's claim is
that they are \emph{unposable} --- which is a different kind of
answer, and it is testable only through the numbers it forces.

\subsection{How the geometry completes the story}
\label{ssec:string-completed}

The completion, then, is not a new picture --- the pictures are
string theory's, conceded --- but a \emph{valuation} of
pictures already drawn. Conjugation as orientation reversal:
drawn there on a spatial circle with a state-space $i$ borrowed
from the axioms; executed here on a non-spatial fiber whose
partner structure, the clock, is derived --- and only with two
derived sources is there a handshake, which is why the pairing
is a theorem here and a protection there. Chirality from
projection: drawn there as one choice in a catalog; executed
here with the output claimed forced by the ledger and the
projection's own derivation honestly open --- the
forcing-versus-selection distinction of Part~VI, which is the
entire difference. CP as geometry: drawn there on continuous
moduli, so that the framework predicts \emph{that} CP violation
is geometric but never \emph{what} it is; executed here on one
circle with one integer, so that the holonomies land on a
pentagonal lattice and the phases come out as numbers with
pulls --- $13\pi/35$, $\pi + 2\arcsin(1/7)$, $(0,\pi)$ ---
each one a falsifier the moduli picture can never issue. In
one sentence, paid in both directions: \emph{string theory
drew the right pictures on an arena that could not hold them
still; the ledger's claim --- graded, dated, and mortal --- is
that the true arena holds them still.}

\section{The fourth road: \texorpdfstring{$\Lambda$}{Lambda}CDM}
\label{sec:road-lcdm}

\subsection{What \texorpdfstring{$\Lambda$}{Lambda}CDM achieved}
\label{ssec:lcdm-achieved}

$\Lambda$CDM is not a rival on mechanism --- it is the
bookkeeper, and its achievement is of a different, indispensable
kind: it made cosmology a precision science with six numbers.
Six parameters fit the microwave sky's acoustic peaks, the
large-scale structure of the galaxy surveys, the light-element
abundances, and the supernova distances --- percent-level
concordance across epochs separated by orders of magnitude in
temperature and scale \citep{Planck2018}. Among the six,
$\Omega_b h^2$ is measured to $0.3\%$; and the framework's
great consistency theorem is this paper's own boundary
condition: the same $\etaB$ read at three minutes (deuterium,
\citealp{Cooke2018}) and at four hundred thousand years (the
CMB), certifying that whatever made the asymmetry finished
before the first second and that baryon number has been
conserved since. Every mechanism in this Part, the \GG{} chain
included, is graded against numbers $\Lambda$CDM measured.

\subsection{Where \texorpdfstring{$\Lambda$}{Lambda}CDM ends}
\label{ssec:lcdm-ends}

It ends where it honestly declares it ends: all six numbers are
inputs. The baryon density is measured, not explained; $\etaB$
is an initial condition, upstream of the model's opening page;
the framework holds no opinion on any question this paper has
asked --- why twins, why broken mirrors, why a surplus ---
and, to its credit, pretends none. Its very precision has
begun, in recent years, to expose stresses among its own dials,
pursued at length in the cohort's cosmology companions
\citep{ArisakaA10Cosmos}; for this paper the single relevant
boundary is the simplest: a bookkeeper certifies accounts and
originates none.

\subsection{How the geometry completes the story}
\label{ssec:lcdm-completed}

The completion inverts the roles while keeping the books: the
cosmology companion re-derives the six dials as outputs of the
same geometry this paper runs on --- among them
$\Omega_m = 1/\pi$ and $\Omega_b = 1/(2\pi^2)$, at the
calibrated grades printed there \citep{ArisakaA10Cosmos} ---
and Part~\ref{part:matter}'s transport chain is the dynamical
audit of the baryon dial: the tripod P-P5-E says a receipt must
exist, the chain computes its size at D$^{*}$, and the
bookkeeper's $0.3\%$ measurement is the audit's external
check. The books do not yet close perfectly, and the paper
says so where it happened: the $16.4\%$ two-route seam of
\S\ref{sec:omegab-bridge} is published as the audit's open
line. The relation is the healthiest in this Part: $\Lambda$CDM
cannot explain and does not claim to; the geometry claims to
and can be executed for it; and the bookkeeper will be the
first to know if the claim fails.

\section{The inventory: who predicts what}
\label{sec:roads-inventory}

Analysis wants a single table in which the four roads face the
same observables, and the register wants it labeled exactly:
\emph{fit} means the framework accommodates the measured value
by setting a parameter; \emph{unvalued} means the framework
geometrizes the quantity but assigns it no number;
\emph{input} means the framework measures the value and uses
it; \emph{forced} means the framework asserts a value it
cannot adjust. Pulls for the forced column are quoted in
Table~\ref{tab:predicted-measured}, and every grade caveat
printed there applies here unchanged.

\begin{table}[t]
\centering
\caption[The inventory of who predicts what.]{The inventory: the same observables faced by the four
roads. ``Fit'' accommodates, ``unvalued'' geometrizes without a
number, ``input'' measures, ``forced'' asserts and cannot
adjust. The forced column's pulls, grades, and flags are those
of Table~\ref{tab:predicted-measured}; its rows die on the
instruments of Table~\ref{tab:falsifiers}.}
\label{tab:roads-inventory}
\small
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{p{2.8cm} p{2.2cm} p{2.4cm} p{2.0cm} p{4.6cm}}
\toprule
\textbf{Observable} & \textbf{SM} & \textbf{String} &
\textbf{$\Lambda$CDM} & \textbf{\GG{} (forced)} \\
\midrule
$\sin\theta_{12}^{\mathrm{CKM}}$ & fit & unvalued & --- &
$0.22502$ (golden-ratio closed form) \\
$\dCKM$ & fit ($1.147$) & unvalued (moduli) & --- & $2\pi/5 \to
13\pi/35$ (QD flagged) \\
$\dPMNS$ & fit (in progress) & unvalued & --- & $\pi +
2\arcsin(1/7) \simeq 196^{\circ}$ \\
$(\alpha_{21}, \alpha_{31})$ & free & unvalued & --- & $(0,
\pi)$: parity data \\
$\mbb$ & free & unvalued & --- & $1.72$~meV (first-order; QD layer $1.79$) \\
$\bar\theta$ & fine-tuned; axion bolt-on & axions, unvalued &
--- & $\lesssim 4.7\times10^{-18}$, two-fate \\
$N$ (generations) & input ($=3$) & vacuum choice & --- & $3$
(ledger) \\
$W_R$ sector & absent by input & vacuum choice & --- & absent
at every scale (F-P5-1) \\
$\etaB$ & $\approx 0$ (fails) & unvalued $+$ moduli dilution &
input ($0.7\%$) & $\approx7\times10^{-10}$ (D$^{*}$,
$\times3^{\pm1}$) \\
New states & --- & unvalued & --- & lock cluster at
$3.85$~TeV \\
\bottomrule
\end{tabular}
\end{table}

The table's single message survives every caveat on it: down
the first three columns, no entry can be wrong; down the
fourth, every entry can. That asymmetry --- not any single
number --- is the structural difference between the roads, and
it is the difference this Part was built to display.

\section{Reading the pulls, and the falsification symmetry}
\label{sec:roads-pulls}

Two analytic remarks make the inventory quantitative rather
than rhetorical. \emph{First, how to read the forced column
statistically.} Ten forced entries confront measurement; where
pulls are defined (Table~\ref{tab:predicted-measured}) they run
from $+0.01$ (the Cabibbo magnitude, at four decimal places) to
$+1.17$ (the first-order CP phase --- a pull quoted with the
one-lattice-step theory width $\pi/35$ in quadrature; against
experimental error alone that raw value sits at $+4.2\sigma$),
with nothing beyond
$1.2\sigma$ on that convention (the watched direct-$\gamma$
tension, $+1.6\sigma$, lives in Part~XI's refresh apparatus)
--- a global consistency no fitted framework can
even express, since a fit has no pulls of its own making. And
the register must immediately discount it, twice. The flags are
part of the data: the QD completion is asserted (its pull
$+0.70$ replaces the forced $+1.17$ only under the flagged
step), and $\etaB$ carries a factor-$\sim3$ band that makes its $+20\%$ central agreement an order-of-magnitude statement, not a precision one. And global consistency is exactly what the
jigsaw clause warns about: agreement across ten numbers is
impressive in proportion to how independently the ten were
forced, which is why the provenance audit --- five integers
fixed upstream, published across twenty-one companion
treatments for unrelated purposes --- is not decoration but
the entire evidentiary weight of the column. The pulls are
good; the honest sentence is that they are good \emph{and}
flagged, and both halves are data.

\emph{Second, the falsification symmetry.} A comparison is
scientific only if it can be lost, so let it be said, road by
road, what discovery would vindicate the rival \emph{against}
this paper. A $W_R$ or any parity restoration at any scale:
left--right symmetry right, F-P5-1 dead, Part~VI's geometry
falsified outright. A measured $\dPMNS$ far from
$196^{\circ}$ --- at, say, maximal $270^{\circ}$: the
dial reading right, the lattice wrong, Part~VII's lepton chain
dead on DUNE's floor. Proton decay at any lifetime: the GUT
route vindicated and the entire baryonic logic of
Part~VIII --- and the bulk ledger with it --- destroyed. A
fourth generation: the ledger's $N = 3$ dead, and with it the
pentagon's alphabet. A neutron EDM between $10^{-18}$ and
$10^{-10}$: the two-fate architecture broken while the
Standard Model stands untouched. Moduli-like relics or a
discovered landscape dynamics: the enumerative picture
vindicated where the rigid arena forbids it. The forced column
of Table~\ref{tab:roads-inventory} is thus not a scoreboard of
victories; it is a list of stakes --- one per row --- and the
rival roads hold the winning tickets for every one of them
should nature so rule. No other column in the table has
offered the \GG{} program the reciprocal courtesy, because
no other column can: that, in the end, is what ``unvalued''
and ``fit'' mean.

\section{The synthesis: the three depths, scored}
\label{sec:depth-scoreboard}

\begin{table}[t]
\centering
\caption[The five roads on the three-depths ruler.]{The five roads on the three-depths ruler. Verdicts in
one sentence each: QFT --- rigorous and silent; the Standard
Model --- precise and unexplaining; string theory --- geometric
and unvalued; $\Lambda$CDM --- exact and agnostic; \GG{} ---
valued, dated, and mortal.}
\label{tab:depth-scoreboard}
\small
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{p{2.9cm} p{2.5cm} p{2.4cm} p{2.6cm} p{1.9cm} p{2.4cm}}
\toprule
\textbf{Question} & \textbf{QFT} & \textbf{SM} &
\textbf{String} & \textbf{$\Lambda$CDM} & \textbf{\GG{}} \\
\midrule
Twins: exist? & Depth 2 (Weinberg; interacting) & inherited &
Depth 1 $+$ geometric image & --- & Depth 3 (handshake;
quasi-free) \\
Twins: what is the $i$? & axiom & axiom & axiom & --- & derived
(two sources) \\
C/P/T: what are they? & conventions & conventions & $+$
orientation image & --- & three reversals (derived table) \\
CPT: why? & theorem, anatomy silent & inherited & worldsheet &
--- & bookkeeping over anatomy \\
P: why broken, why weak? & partials only & input & vacuum
choice & --- & basin/gate (conditional, L11) \\
CP: why, and how much? & no principle & one fitted dial &
geometric, unvalued & --- & chiral pentagon, forced values \\
$\bar\theta \simeq 0$, related to $\delta \neq 0$? & no & no
(axion bolt-on) & axions, unvalued & --- & two fates, one
alphabet \\
$\etaB$: derived? & toolbox, dials & $\approx 0$ (fails) &
ingredients $+$ moduli problem & measured to $0.7\%$ &
$\approx7\times10^{-10}$, zero dials, D$^{*}$ \\
Falsifiable on a date? & mechanism-dependent & (is the
baseline) & no & (is the ruler) & yes:
Table~\ref{tab:falsifiers} \\
\bottomrule
\end{tabular}
\end{table}

Table~\ref{tab:depth-scoreboard} closes the scoring, and one
paragraph states what it shows. On every row, the \GG{} column
claims the deepest answer on the table --- and on every row,
that claim is either a theorem with a stated grade, a named
conjecture, or a number wired to an instrument. The other four
columns are not caricatures; they are the strongest versions of
four magnificent traditions, each of which contributed
load-bearing pieces to this very paper (Weinberg's theorem, the
sphaleron arithmetic, the orbifold projection, the target
number itself). The difference is not intelligence or rigor;
it is \emph{inventory}. The \GG{} program spent its ledger
before these questions were asked --- five integers, published
with their own falsifiers --- and depth at fixed inventory is
what this Part has laid out in the open.

\section{The five roads, seen from above}
\label{sec:roads-above}

Let us end the comparison from altitude, because seen from
above the four roads are not four opinions about one question
--- they are four different \emph{verbs}, and the shape of
twentieth-century physics is the story of their succession.
Quantum field theory \emph{constrains}: it is the grammar of
the possible, proving what any quantum causal world must
contain, and grammar was its glory and its limit --- a grammar
permits every book and writes none. The Standard Model
\emph{codifies}: it is the measured text of this world,
transcribed to a precision no other science approaches, with
twenty-eight numbers standing where reasons would go. String
theory \emph{enumerates}: it is the library of the writable
worlds, in which every geometric insight this paper uses
appears somewhere on the shelves --- together with its
negation, and no catalog rule for finding our volume.
$\Lambda$CDM \emph{measures}: it is the ledger of what this
universe in fact did, six entries, kept to the percent. Four
verbs --- constrain, codify, enumerate, measure --- and each
road, at its honest end, hands the remaining question to the
next: the grammar cannot pick the text; the text cannot explain
itself; the library cannot find the volume; the ledger cannot
author the entries. What no road of the century attempted, at
this subject, is the fifth verb: \emph{derive} --- to exhibit
this world's text as forced, and to accept the mortal risk
that forcing entails.

Einstein asked whether God had any choice in the construction
of the world, and the four roads return four answers: some
clauses were forced (QFT); here are the choices, measured (the
Standard Model); all choices are made, somewhere (string
theory); these were chosen (\mbox{$\Lambda$CDM}). The
program this paper serves returns the fifth answer as a
\emph{claim, not a creed}: fewer choices existed than the
century assumed --- perhaps none --- and the way to find out is
to write the forced values down beside their error bars and
wait for the instruments. The philosophical weight of that
posture deserves one careful paragraph, with both hats on. An
explanation, in the oldest sense this program can defend, is
the conversion of contingency into necessity: the fact that
felt optional is exhibited as the only writable option. Every
Part of this paper has practiced that conversion --- the twins
as the price of one $i$, the antiunitarity of T as the only
exit from a forbidden quadrant, CP violation as the chirality
of a pentagon no dial can straighten, the matter excess as the
receipt of a fall the constitution made unavoidable. But the
philosopher's hat must immediately yield to the
experimentalist's, because conversion-into-necessity is also
the oldest temptation in natural philosophy, and a jigsaw that
fits perfectly can fit because the pieces were cut to fit. The
satisfaction is not evidence. The dates are: a phase at
$196^{\circ}$ under a detector now under construction, a
double-beta mass at $1.7$~meV in the tonne-scale program's
sights, a TeV cluster within one collider upgrade, a
right-handed sector that must never appear at any energy, a
proton that must never decay. Necessity that cannot be
executed is theology; necessity that publishes its own
execution dates is physics. The five roads differ, in the
end, less in geometry than in \emph{exposure} --- and the
fifth verb is available only to a framework willing to be
refuted in its own lifetime.

Seen as history, the century is one arc with a direction of
its own. In 1908 Minkowski geometrized time's order, and
physics learned that a structure could be a shape. In 1928
Dirac's algebra demanded the twins, and in 1932 a cloud
chamber obliged: consistency had begun to legislate. In 1957
and 1964 the mirrors fell, and physics learned that the world
has a handedness it had never been asked about; in 1967
Sakharov read the fallen mirrors as the price of existence; in
1973 the fall was given an address in a matrix, and in 1983,
1998, 2012 the address was confirmed to three digits ---
culminating, for this author, in two decades of measuring a
number whose value his science declared meaningless. Each
step of the arc converted a wonder into a structure and left
one question standing at a deeper level. The arc's direction
is unmistakable: from \emph{that} the world is lawful, toward
\emph{why} it is this world --- from grammar toward authorship.
The wager of this paper is not that the arc ends here; it is
that the arc's next step has the same character as all the
previous ones --- another wonder converted into a structure,
this time the wonder of the broken mirrors themselves --- and
that the conversion is now concrete enough to be executed by
instruments on a published schedule.

After all, the four roads' own summits are the reason the
fifth verb became thinkable at this subject: a grammar so
tight it legislates twins, a text so precisely transcribed
that its twenty-eight numbers beg for a source, a library so
complete it proves geometric answers exist, and a ledger so
exact it can convict any pretender. Standing on all four, one
can finally ask the question they converge upon and cannot
answer --- \emph{why this world?} --- and this paper's wager,
graded and dated, is that the question has become physics.

\medskip
\section{The sixth comparison: what each road says time is}
\label{sec:roads-time}

Part~\ref{part:time} adds a comparison this Part's ruler could
not have drawn before, and it belongs here, where the roads
stand side by side. Ask each road the oldest question. QFT
answers with order and repetition borrowed whole --- a
background parameter, microcausality imposed, the shore used in
every computation and named in none. The Standard Model adds
the one thing experiment forced on it: a measured T-violation
--- and correctly refuses to call it an arrow, since its
dynamics remains reversible to the last term. String theory,
which dissolves space with enthusiasm, keeps worldsheet time as
a background and inherits the frozen-formalism problem wherever
it goes background-independent. $\Lambda$CDM runs on a cosmic
time that is a choice of foliation, with its arrow rented from
the past hypothesis --- an initial condition doing a law's
work. And the geometry answers with the four endowments: order
from the cone, repetition forced by Theorem~0, the arrow
written into the dictionary's one-way reading, and the shore
opened by the spectrum's single side
(Part~\ref{part:time}). On this sixth comparison the pattern
of the other five repeats exactly: the roads use time; the
geometry says what it is --- and pays for the answer in the
usual currency, since the same \OGGEP{} law that defines the
arrow runs the falsifiable transport of
Part~\ref{part:matter}.

\medskip
\noindent\emph{Part-end summary.} Four roads, each walked in
three movements: what it achieved, honored with numbers;
where it ends, located in its own terms; how the geometry
completes it, graded claim by claim. One inventory table in
which only the last column can be wrong; one scoreboard on the
three-depths ruler; and the view from above --- four verbs,
constrain, codify, enumerate, measure, with the fifth,
derive, attempted here at the price of mortality. Every debt
paid in its creditor's own section. Before the housekeeping
of the claims comes the court in which all of them are tried:
the instruments. The next Part walks the century of measured
mirrors and the decade of machines now arriving on published
schedules, each pointed at a forced value.

\GGPart[Experiments: past, now, and future]{Experiments --- Past, Now, and Future}{The century of measured mirrors and the neutrino century, with their milestone tables; the present frontier as of mid-2026; and the instruments of the decisive decade, each with its schedule, its sensitivity, and the forced value it touches}
\label{part:experiments}

\section[Part~XI in plain English]{Part~XI in plain English: what has been measured, and what is coming}
This is the Part an experimentalist should read first. It records a century of measured mirrors and a century of neutrinos, with the numbers as they stand in mid-2026 and every source named, including the two lines that are the author's own instrument lineage.

Then it does the thing a paper of this kind usually avoids. Every forced value in the paper is set against the measurement that will test it, with the precision that measurement needs and the year it is expected. The verdicts are simulated for 2028, 2031, 2035 and 2040, so that a reader can see now what this account looks like if each of those years goes against it, rather than being told afterward that the result was always robust.

The falsifier timeline is dated, and the dates are close. That is the Part's whole argument. This is not a framework that will be settled by taste or by how it reads, and the decade that decides it has already started.

\section{The third channel}
\label{sec:exp-opening}

Every claim of this paper ends, by design, at an instrument, and
this Part is where the instruments take over the narrative. The
parent program maintains three primary falsification channels.
The first is the collider channel: the lock cluster at
$3.85$~TeV, whose confrontation with the ATLAS and CMS record
and whose pre-registered HL-LHC search are the business of the
cohort's collider companion \citep{ArisakaP1KK}. The second is
the laboratory dark-sector channel: the Kaxion at $8.1~\mu$eV
($1.95$~GHz), with its twin-peak fingerprint and its dedicated
instrument proposal \citep{ArisakaC1Kaxion}. This paper
completes the triad: the mirror sector --- the CP phases, the
Majorana door, the strong-CP window, the right-handed absence,
and the proton's silence --- is the third channel, and it runs
not on one machine but on a worldwide battery of them. The
present Part reviews that battery in the manner of its
companion papers: history first, because every forced value of
Parts~VI--VIII confronts a measurement tradition that is itself
a scientific achievement of the first rank; then the present
status, grounded in official collaboration outputs; then the
instruments now under construction, with schedules and
sensitivities; and then --- in the confrontation sections that
close the Part --- the forced values laid beside the reach of
each machine. Because experimental status changes on the scale
of months, this is exactly the type of Part that deserves
freshness: the status reported here is current as of mid-2026,
drawn from collaboration publications and official
announcements, and the few numbers taken from design reports
rather than data are attributed as such. Both hats are worn
throughout --- these are the author's instruments; he spent
1980--1985 building one of them and two decades inside two
others --- and the register is unchanged: what follows is
either measured, scheduled, or graded.

\section{A century of broken mirrors, measured}
\label{sec:exp-cpv-history}

\subsection{The kaon era: from a helium bag to
\texorpdfstring{$10^{-18}$}{1e-18}}
\label{ssec:exp-kaon-era}

The experimental history of CP violation begins with an
apparatus almost provocatively modest: a two-arm spectrometer,
a hydrogen target's secondary beam, and a helium bag, at the
Brookhaven AGS in 1964. Christenson, Cronin, Fitch, and Turlay
found the long-lived kaon decaying to two pions at a branching
ratio near $2\times10^{-3}$ \citep{Cronin1964} --- and the
decisive experimental point, worn smooth by retelling but
essential, is that they did not merely see a peak: they
excluded the mundane explanation, coherent regeneration in the
bag, by orders of magnitude. Part~I told what the discovery
meant; here we record what it started --- a sixty-year
program of turning one anomaly into precision. The
semileptonic charge asymmetry ($\delta_L =
(3.32\pm0.06)\times10^{-3}$ today) gave matter its operational
definition. The parameters $|\varepsilon| =
(2.228\pm0.011)\times10^{-3}$ and the phases were driven to
three digits across three decades \citep{PDG2024}. Then came
the question that occupied the author's first experimental
decade at the precision frontier: is CP violation confined to
the mixing (the superweak hypothesis), or does it act directly
in the decay --- $\varepsilon'/\varepsilon \neq 0$? The saga
ran twenty years: NA31 at CERN found evidence
($(23.0\pm6.5)\times10^{-4}$ final), Fermilab's E731 found a
value consistent with zero, and the contradiction was resolved
only by the purpose-built successors --- KTeV at Fermilab,
where the author served, establishing direct CP violation
beyond doubt in 1999 with a final value
$\mathrm{Re}(\varepsilon'/\varepsilon) =
(19.2\pm2.1)\times10^{-4}$ \citep{KTeV1999}, and NA48 at CERN
confirming with $(14.7\pm2.2)\times10^{-4}$
\citep{NA48_2002}; the world average stands at
$(16.6\pm2.3)\times10^{-4}$ \citep{PDG2024}. The superweak
model died; CP violation acts in amplitudes, exactly where a
texture angle would put it. CPLEAR closed the kaon era's
conceptual circle in 1998 by observing time-reversal violation
directly --- the Kabir asymmetry between $\bar K^0 \to K^0$
and $K^0 \to \bar K^0$ at $(6.6\pm1.6)\times10^{-4}$
\citep{CPLEAR1998} --- and the neutral-kaon system delivered,
almost as a by-product, the sharpest CPT test in physics: the
$K^0$--$\bar K^0$ fractional mass difference bounded near
$10^{-18}$ (\S\ref{sec:history-cpt}). The modern kaon frontier
has moved to the rarest decays, where new phases would enter at
one loop: NA62 has now observed $K^+ \to \pi^+\nu\bar\nu$ ---
the 2024 observation refined in March 2026 to a combined
$\mathrm{BR} = (9.6^{+1.9}_{-1.8})\times10^{-11}$, within
$1\sigma$ of the Standard-Model expectation
\citep{NA622025} --- while KOTO bounds the neutral partner
$K_L \to \pi^0\nu\bar\nu$ below $2.2\times10^{-9}$ with its
successor KOTO-II approved toward single-event sensitivity
below $10^{-12}$ \citep{KOTO2025}. Sixty years, one
overarching lesson: the kaon system converts patience into
decimal places, and every decimal place has so far landed on
the one-phase texture the pentagon forces.

\subsection{The \texorpdfstring{$B$}{B} era: from the
\texorpdfstring{$\Upsilon$}{Upsilon} to a baryon}
\label{ssec:exp-b-era}

The second era belongs to the third generation that CP
violation itself predicted \citep{KobayashiMaskawa1973}. The
$b$ quark arrived in 1977; the unexpectedly long $b$ lifetime
(1983) revealed the CKM hierarchy; and ARGUS's 1987 discovery
of large $B^0$--$\bar B^0$ mixing \citep{ARGUS1987} --- a
result that presaged the heavy top --- meant the $B$ system
could interfere its way to large CP asymmetries. Two
purpose-built colliders and detectors --- PEP-II/BaBar, where
the author served, and KEKB/Belle --- turned that possibility
into the 2001 observation of $\sin 2\beta$
(\citealp{BaBar2001}; \citealp{Belle2001}), CP violation
outside the kaon system and the quantitative confirmation of
the one-phase mechanism. Direct CP violation in $B \to K\pi$
followed in 2004; and in 2012 BaBar performed the measurement
this paper's Part~V made structural sense of: direct,
model-independent time-reversal violation, using entangled $B$
pairs to compare a transition with its genuine time-reverse,
at $14\sigma$ \citep{BaBar2012}. The LHC era then broadened
the map: LHCb discovered CP violation in \emph{charm} in 2019
($\Delta A_{CP} = (-15.4\pm2.9)\times10^{-4}$)
\citep{LHCbCharm2019}, measured the angle $\gamma$ directly to
$(62.8\pm2.6)^{\circ}$ (November 2025 combination) in mild,
watched tension with the indirect fit, and in 2025 observed CP
violation in a \emph{baryon} for the first time ---
$\Lambda_b^0 \to p K^-\pi^+\pi^-$ at $5.2\sigma$, published in
\emph{Nature} \citep{LHCbBaryon2025} --- extending the broken
mirror to the matter category we are made of. Belle II is
rebuilding the $e^+e^-$ precision program. The current world
averages the pentagon must face: $\sin 2\beta = 0.709\pm0.011$,
$\delta = 1.147\pm0.026$~rad, $J = (3.12^{+0.13}_{-0.12})
\times10^{-5}$ \citep{PDG2024}. And parallel to the collider
line runs the static-moment line --- Part~V's T-instruments:
the neutron EDM bounded at $1.8\times10^{-26}\,e\,$cm
\citep{nEDM2020} with the PSI successor n2EDM commissioning
toward $10^{-27}$, and the electron EDM at
$4.1\times10^{-30}\,e\,$cm \citep{Roussy2023} --- null results
that already convict generic TeV-scale CP phases and leave the
lattice's leakage floor (Part~VII) untouched by eight orders.

\begin{table}[t]
\centering
\caption[Milestones of the broken mirrors.]{The past, part one: milestones of the broken mirrors
in the quark sector, 1957--2026. Values as per current world
averages \citep{PDG2024}. Parts~I and~V--VII carry the
structural reading of each line. Fifteen rows, from the 1957 fall of parity to the 2025 observation of CP violation in baryons, each with the experiment, the number it produced, and what that number established. The right column is the one to read downward: it is where a sequence of separate results becomes a single account of which mirrors break and by how much.}
\label{tab:exp-past-cpv}
\small
\renewcommand{\arraystretch}{1.22}
\begin{tabular}{p{0.9cm} p{3.0cm} p{4.3cm} p{5.6cm}}
\toprule
\textbf{Year} & \textbf{Experiment} & \textbf{Result} &
\textbf{What it established} \\
\midrule
1957 & Wu; Garwin et al. & P (and C) violation, maximal &
the first mirror falls; V$-$A \\
1964 & Cronin--Fitch (AGS) & $K_L \to \pi\pi$;
$|\eta_{+-}| \simeq 2.3\times10^{-3}$ & CP falls: the second
mirror, small \\
1967-- & multiple & $\delta_L = (3.32\pm0.06)\times10^{-3}$ &
matter operationally defined \\
1973 & (theory) KM & one phase needs $N \geq 3$ & third
generation predicted \\
1977 & E288 (FNAL) & $\Upsilon(9.46)$ & the $b$ quark \\
1987 & ARGUS & large $B^0$--$\bar B^0$ mixing & heavy top;
$B$-CP program feasible \\
1988--02 & NA31, E731, KTeV, NA48 &
$\mathrm{Re}(\varepsilon'/\varepsilon) =
(16.6\pm2.3)\times10^{-4}$ (avg.) & direct CPV; superweak
dead \\
1998 & CPLEAR & $A_T = (6.6\pm1.6)\times10^{-4}$ & direct T
violation (kaons) \\
2001 & BaBar; Belle & $\sin 2\beta$ observed & CPV beyond
kaons; KM confirmed \\
2004 & BaBar, Belle & $A_{CP}(B\to K\pi) \neq 0$ & direct CPV
in $B$ \\
2012 & BaBar & T violation, entangled $B$ pairs, $14\sigma$ &
the clock reversal caught alone \\
2019 & LHCb & $\Delta A_{CP} = (-15.4\pm2.9)\times10^{-4}$ &
CPV in charm \\
2024--26 & NA62 & $\mathrm{BR}(K^+\to\pi^+\nu\bar\nu) =
(9.6^{+1.9}_{-1.8})\times10^{-11}$ & rare-kaon frontier at SM \\
2025 & LHCb & $\Lambda_b^0$ CPV at $5.2\sigma$ & CPV reaches
baryons \\
\bottomrule
\end{tabular}
\end{table}

\section{The neutrino century}
\label{sec:exp-nu-history}

The neutrino's experimental history is the longest arc in
particle physics --- a particle postulated in apology
(``I have done a terrible thing,'' Pauli said, ``I have
postulated a particle that cannot be detected'') that became,
one century later, the field's sharpest precision frontier and
this paper's decisive instrument. The arc: Pauli's 1930 letter;
Reines and Cowan's 1956 detection of reactor antineutrinos
\citep{ReinesCowan1956}; the 1962 Brookhaven two-neutrino
experiment separating $\nu_\mu$ from $\nu_e$
\citep{Danby1962}; Davis's Homestake chlorine tank measuring
the solar flux at one third of prediction from 1968 onward
\citep{Davis1968} --- a deficit that would take thirty years to
become a discovery. In 1987 the neutrinos of SN1987A crossed
Kamiokande --- the detector the author spent 1980--1985
designing and building --- and IMB, nineteen events that opened
neutrino astronomy. In 1998 Super-Kamiokande announced the
zenith-angle asymmetry of atmospheric muon neutrinos --- $A =
-0.296\pm0.048\pm0.010$, a six-sigma up--down asymmetry ---
and with it the discovery of neutrino oscillation
\citep{SuperK1998}: the neutrino has mass, the lepton mixing
matrix is real physics, and the door of Part~VII acquired its
traffic. SNO's heavy-water measurements (2001--2002) showed
the missing solar neutrinos arriving in other flavors
\citep{SNO2002}; KamLAND caught reactor antineutrinos
disappearing over 180~km \citep{KamLAND2003}; T2K indicated
and Daya Bay established the third mixing angle in 2011--2012
($\sin^2 2\theta_{13} = 0.092\pm0.017$ at discovery;
$0.0851\pm0.0024$ final) \citep{DayaBay2012}, with tau-flavor
appearance closed out by DONUT and OPERA
(\citealp{DONUT2001}; \citealp{OPERA2018}); Borexino completed
the solar spectrum through the CNO cycle
\citep{Borexino2020}; and the 2015 Nobel Prize to Kajita and
McDonald certified the century's turn. Three facts from this
history carry directly into the confrontation to come: the
oscillation parameters are now percent-level data; the CP
phase of the lepton sector is measurable; and the mass
\emph{ordering} --- on which the lattice value's confrontation
partly turns --- is within reach of three independent
techniques this decade.

\begin{table}[t]
\centering
\caption[Milestones of the neutrino century.]{The past, part two: milestones of the neutrino
century, 1930--2020. The 1987 and 1998 lines are the author's
own instrument lineage; Part~VII's lepton chain runs on the
parameters this table's discoveries turned into data. Thirteen rows, from Pauli's letter to Borexino's detection of the CNO neutrinos, each with the experiment, the result, and what it established. Read downward it is the record of a particle postulated to save a conservation law becoming, over ninety years, the instrument that measures the lepton sector.}
\label{tab:exp-past-nu}
\small
\renewcommand{\arraystretch}{1.22}
\begin{tabular}{p{0.9cm} p{3.2cm} p{4.2cm} p{5.4cm}}
\toprule
\textbf{Year} & \textbf{Experiment} & \textbf{Result} &
\textbf{What it established} \\
\midrule
1930 & Pauli (letter) & the desperate remedy & energy
conservation kept; $\nu$ postulated \\
1956 & Reines--Cowan & reactor $\bar\nu_e$ detected & the
neutrino is real \\
1962 & Danby et al.\ (BNL) & $\nu_\mu \neq \nu_e$ & flavor
exists in the lepton sector \\
1968-- & Davis (Homestake) & solar rate $\sim 1/3$ of SSM &
the solar problem posed \\
1987 & Kamiokande-II; IMB & SN1987A: 19 events, 13 s &
neutrino astronomy born \\
1998 & Super-Kamiokande & atmospheric up--down asymmetry
$-0.296\pm0.048$ & oscillation discovered; mass exists \\
2001--02 & SNO & NC flux matches SSM; $\nu_e$ deficit is
conversion & solar problem solved by flavor change \\
2003 & KamLAND & reactor disappearance over 180 km & LMA
selected; $\Delta m^2_{21}$ pinned \\
2000/15 & DONUT; OPERA & $\nu_\tau$ appearance & the third
row closed \\
2011--12 & T2K; Daya Bay; RENO & $\theta_{13}$:
$\sin^2 2\theta_{13} = 0.0851(24)$ (final) & the door to
leptonic CP opened \\
2015 & Nobel (Kajita, McDonald) & --- & the century
certified \\
2020 & Borexino & CNO solar neutrinos & the Sun's second
furnace seen \\
\bottomrule
\end{tabular}
\end{table}

\section{The present frontier (mid-2026)}
\label{sec:exp-now}

This section is the freshness layer: the state of every
relevant front as of mid-2026, from collaboration publications
and official announcements.

\emph{Oscillations.} The global three-flavor fit (NuFit~6.0)
gives $\sin^2\theta_{12} = 0.308^{+0.012}_{-0.011}$,
$\sin^2\theta_{23} = 0.561^{+0.012}_{-0.015}$,
$\sin^2\theta_{13} = 0.02195^{+0.00054}_{-0.00058}$, $\Delta
m^2_{21} = (7.49\pm0.19)\times10^{-5}$~eV$^2$, $|\Delta
m^2_{3\ell}| \simeq 2.5\times10^{-3}$~eV$^2$ --- and, for this
paper's central number, $\delta_{CP} = 177^{\circ}
{}^{+19}_{-20}$ if the ordering is normal and $285^{\circ}
{}^{+25}_{-28}$ if inverted \citep{NuFit2024}. All NuFit~6.0
numbers in this paper are the IC19 variant without SK
atmospherics; the with-SK variant flips the $\theta_{23}$
octant and moves $\delta$ to $212^{\circ}$ --- under it the
lattice point sits at $-0.4\sigma$. The first
combined analysis of the two running long-baseline
experiments, T2K and NO$\nu$A, appeared in \emph{Nature} in
October 2025 \citep{T2KNOvA2025}: $|\Delta m^2_{32}|$ to
better than 2\%, and $\delta_{CP}$ constrained such that the
CP-conserving values are excluded at $3\sigma$ \emph{if} the
ordering is inverted, while remaining allowed under normal
ordering --- with the ordering itself still undetermined
(Bayes factors near unity). JUNO, filled and running since
August 2025, published first results in November 2025 from
fifty-nine days of data: $\sin^2\theta_{12} =
0.3092\pm0.0087$ and $\Delta m^2_{21} =
(7.50\pm0.12)\times10^{-5}$~eV$^2$, already sharper than all
prior experiments combined \citep{JUNO2025}. The reading for
Part~VII is direct and worth stating now: under normal
ordering --- which the cohort's derived mass spectrum requires
--- the lattice value $196^{\circ}$ sits within $1\sigma$ of
the current global best fit; under inverted ordering it is
already disfavored at $\approx 3\sigma$. The ordering
determination is therefore not background; it is part of this
paper's kill chain.

\emph{Quark sector and EDMs.} The averages confronting the
pentagon moved with the 2025 updates: $\sin 2\beta = 0.709 \pm 0.011$ (the confrontation
table and Part~VII's prose carry the refreshed averages ---
the pentagon's $0.708$, evaluated at the stated PDG-class
moduli layer, pulls $-0.1\sigma$, inside the band and
watched; the refresh is executed in
\S\ref{sec:exp-refresh});
$\gamma_{\rm direct} = (62.8\pm2.6)^{\circ}$; $J =
(3.12^{+0.13}_{-0.12})\times10^{-5}$ \citep{PDG2024}. The
lattice triangle's $\gamma$ tracks the QD phase at leading
order, $\simeq 66.9^{\circ}$: a $+1.6\sigma$ pull against the
direct combination, printed here under the no-cosmetic-delay
rule of \S\ref{sec:exp-refresh}, and watched. NA62's
combined $K^+\to\pi^+\nu\bar\nu$ sits on the Standard-Model
value; KOTO presses the neutral mode \citep{KOTO2025}. The
EDM nulls stand at $1.8\times10^{-26}\,e\,$cm (neutron) and
$4.1\times10^{-30}\,e\,$cm (electron)
(\citealp{nEDM2020}; \citealp{Roussy2023}).

\emph{The Majorana door.} The allowed two-neutrino mode has
been laboratory physics since its direct observation in 1987
\citep{MoeElliottHahn1987}; the neutrinoless frontier is where
the door itself is tested. The half-life record: KamLAND-Zen's
full Xe-136 exposure gives
$T_{1/2} > 3.8\times10^{26}$~yr, i.e.\ $\mbb < 28$--$122$~meV
across the nuclear-matrix-element spread \citep{KamLANDZen2024}
--- the first search reaching into the inverted-ordering band;
GERDA closed at $1.8\times10^{26}$~yr background-free
\citep{GERDA2020}; LEGEND-200's first result (2025), combined
with its ancestors, gives $T_{1/2} > 1.9\times10^{26}$~yr
\citep{LEGEND2025}; CUORE passed two tonne-years of TeO$_2$
exposure \citep{CUORE2024}. No signal anywhere --- which is,
for the record the confrontation will formalize, exactly what
a normal-ordered spectrum with $\mbb \simeq 1.7$~meV predicts
at these sensitivities.

\emph{The proton's silence.} The search that minimal
unification commissioned half a century ago
\citep{GeorgiGlashow1974} has become the deepest null result in
physics: Super-Kamiokande's ledger, now at
$0.37$--$0.40$~Mton$\cdot$yr: $\tau/B(p \to e^+\pi^0) >
2.4\times10^{34}$~yr, $\tau/B(p \to \bar\nu K^+) >
5.9\times10^{33}$~yr, with new channels still being closed ---
$p \to e^+\eta > 1.4\times10^{34}$~yr (2024) and $p \to
e^+\pi^0\pi^0 > 7.2\times10^{33}$~yr (2026)
\citep{SuperKPdecay2020}; and the bound-neutron limit
translating to a free $n$--$\bar n$ oscillation time above
$4.7\times10^{8}$~s \citep{SuperKnnbar2021}. Every year of
this silence is a passed test of the exact bulk ledger
(Part~VIII); a single event in any channel would end the
program.

\emph{Mass and cosmology.} KATRIN bounds the effective
electron-neutrino mass below $0.45$~eV directly
\citep{KATRIN2025}; DESI's second data release, with CMB
data, bounds $\sum m_\nu < 0.064$~eV in $\Lambda$CDM
\citep{DESI2025} --- a limit now pressing against the
inverted-ordering floor ($\simeq 0.10$~eV) and even flirting,
at the $2$--$3\sigma$ level, with the normal-ordering minimum
($\simeq 0.06$~eV): cosmology itself is voting on the
ordering, in the direction the cohort's spectrum requires.
Within the extended $w_0w_a$ dark-energy space the bound
relaxes to $0.16$~eV --- but at the price of pressing the
cohort's registered $w = -1$: the joint $(w, \sum m_\nu)$
exposure is one correlated test, not two independent outs,
and is registered as such.
The Super-K gadolinium program reports the diffuse
supernova background ``at the threshold of discovery'' ---
limits at the model band's edge, no claimable excess yet.

\section{The instruments of the decisive decade}
\label{sec:exp-future}

What follows is the machine roster on which the mirror
sector's forced values will live or die, with construction
status as of mid-2026 and sensitivities from technical design
reports (attributed as design figures where they are not yet
data-anchored).

\emph{Long baseline.} Hyper-Kamiokande --- the third
generation of the instrument line the author helped begin ---
has its cavern excavated (July 2025), installation under way,
and data-taking planned from 2028: a $260$~kton water
Cherenkov detector on the J-PARC beam, able to exclude CP
conservation at $5\sigma$ for more than $60\%$ of
$\delta_{CP}$ values within ten years, with a design
resolution of $7^{\circ}$--$20^{\circ}$ depending on the true
value \citep{HyperK2025}. DUNE, the $1300$~km liquid-argon
program at Fermilab/SURF, has its far-detector installation
in progress and first beam physics on current planning near
2031: mass-ordering determination at $5\sigma$ for all
$\delta_{CP}$ within about two years of running, CP-violation
discovery over half the $\delta$ range in about a decade, and
$\delta$ resolution of $10^{\circ}$--$20^{\circ}$
\citep{DUNE2020}. JUNO is already running, with the mass
ordering at $\sim3\sigma$ expected in about six years of
reactor data and sub-percent angle and splitting precision
\citep{JUNO2025}. The IceCube Upgrade's dense infill starts
physics in 2026; ESSnuSB defines the longer horizon.

\emph{The ton-scale double-beta program.} LEGEND-1000
(Ge-76, at Gran Sasso; the DOE/NSF-preferred ton-scale
selection) targets discovery sensitivity $T_{1/2} \sim
1.3\times10^{28}$~yr, i.e.\ $\mbb \sim 9$--$21$~meV
\citep{LEGEND1000}; nEXO (five tonnes of Xe-136 at SNOLAB)
targets $\sim 1.35\times10^{28}$~yr, $\mbb \sim 6$--$27$~meV
\citep{nEXO2022}; CUPID (Mo-100 bolometers in the CUORE
cryostat) targets $\sim 10^{27}$~yr discovery
\citep{CUPID2025}; AMoRE-II begins at Yemilab in 2027; SNO+
loads tellurium from 2026. Together they are designed to cover
the inverted-ordering band ($\mbb \geq 18.4\pm1.3$~meV) with
discovery-grade sensitivity in the 2030s. The confrontation
this sets up is stated now and formalized later: the cohort's
$\mbb \simeq 1.72$~meV lies \emph{below} that band --- so for
the ton-scale era the \GG{} prediction is one-sided and
unusually clean: \emph{any} confirmed discovery at
inverted-band values falsifies the derived Majorana pair and
mass spectrum, while continued null results are precisely what
the geometry requires. Reaching $1.7$~meV itself demands
half-life sensitivities near $10^{29}$--$10^{31}$~yr ---
kiloton-scale isotope, beyond any funded program: the
$0\nu\beta\beta$ number is this paper's furthest-horizon
falsifier, with the nearer decades delivering the one-sided
test.

\emph{Proton decay.} Hyper-Kamiokande extends the $e^+\pi^0$
reach toward $10^{35}$~yr over its first decades --- the only
approved instrument able to probe beyond $10^{34.5}$
\citep{HyperK2025}; DUNE contributes its signature
kaon-tracking channel $p \to \bar\nu K^+$ at the
$1.3\times10^{34}$~yr level per $400$~kton$\cdot$yr
\citep{DUNE2020}; JUNO adds $\sim 10^{34}$~yr-class
sensitivity in the same channel \citep{JUNO2025}. For the
exact $\Uone{B}$ ledger the message is permanence: the
program's prediction is silence, forever, at every depth ---
each order of magnitude gained is another passed test, and
one event anywhere is the end.

\emph{Static moments.} n2EDM at PSI is commissioning toward
$10^{-27}\,e\,$cm on the neutron; the JILA ion-trap program
drives the electron sector below $10^{-30}\,e\,$cm
\citep{Roussy2023}. The strong-CP window of Part~VII ---
leakage floor near $10^{-18}$ in $\bar\theta$, present bound
$10^{-10}$ --- leaves eight orders of headroom in which any
confirmed EDM discovery would break the two-fate architecture
while leaving the Standard Model untouched: the rare falsifier
that can kill this paper without touching its rivals.

\emph{Bursts and backgrounds.} SNEWS~2.0 stands watch for the
next galactic supernova \citep{SNEWS2021}; if one arrives at
$10$~kpc, Hyper-K expects tens of thousands of events, DUNE
the unique electron-neutrino burst reading, JUNO and IceCube
their own thousands --- a once-per-few-decades calibration of
the entire neutrino sector at once. The gadolinium-loaded
Super-K continues toward the diffuse supernova background.
KATRIN concludes its direct-mass campaign toward $0.3$~eV
sensitivity with Project~8's cyclotron-radiation technique
aiming, on the longer horizon, at $0.04$~eV --- into the range
where the cohort's heaviest light neutrino ($\simeq 50$~meV)
becomes directly testable.

\begin{table}[t]
\centering
\caption[The instruments of the decisive decade.]{The future: the instruments of the decisive decade,
with status as of mid-2026. Sensitivities are design-report
figures where data do not yet exist. The last column names the
forced value of Parts~VI--VIII each instrument touches; the
confrontation sections of this Part formalize each line.}
\label{tab:exp-future}
\small
\renewcommand{\arraystretch}{1.24}
\begin{tabular}{p{2.4cm} p{2.3cm} p{1.9cm} p{4.0cm} p{3.4cm}}
\toprule
\textbf{Instrument} & \textbf{Type} & \textbf{Timeline} &
\textbf{Key sensitivity} & \textbf{Forced value touched} \\
\midrule
Hyper-K & LBL + atm.\ water Cherenkov, 260 kt & data 2028 &
$5\sigma$ CPV for $>60\%$ of $\delta$ in 10 yr; res.\
$7$--$20^{\circ}$ & $\dPMNS = 196^{\circ}$ \\
DUNE & LBL LAr, 1300 km & beam $\sim$2031 & ordering
$5\sigma$ ($\sim$2 yr); $\delta$ res.\ $10$--$20^{\circ}$ &
ordering (NO required); $\dPMNS$ \\
JUNO & reactor LS, 20 kt & running & ordering $3\sigma$
($\sim$6 yr); sub-\% $\theta_{12}$, $\Delta m^2_{21}$ &
ordering; chain inputs \\
IceCube Upgrade & atm.\ infill & 2026 & $\Delta m^2_{32}$,
$\theta_{23}$ competitive & consistency \\
LEGEND-1000 & $0\nu\beta\beta$ Ge-76 & $\sim$2030 &
$T_{1/2} \sim 1.3\times10^{28}$ yr ($\mbb$ 9--21 meV) &
$\mbb = 1.72$ meV: one-sided test \\
nEXO & $0\nu\beta\beta$ Xe-136 & 2030s & $1.35\times10^{28}$
yr ($\mbb$ 6--27 meV) & same \\
CUPID / AMoRE-II / SNO+ & $0\nu\beta\beta$ Mo/Te & late
2020s & $10^{26.5}$--$10^{27}$ yr & same \\
Hyper-K (decay) & $p \to e^+\pi^0$ & 2028-- & toward
$10^{35}$ yr & absolute stability \\
DUNE/JUNO decay & $p \to \bar\nu K^+$ & 2029--/now &
$\sim 10^{34}$ yr class & absolute stability \\
n2EDM & neutron EDM & commissioning & $10^{-27}\,e\,$cm &
$\bar\theta$ window \\
JILA III / ACME III & electron EDM & late 2020s & below
$10^{-30}\,e\,$cm & $\bar\theta$ window \\
KATRIN / Project 8 & direct $m_\nu$ & running / 2030s &
0.3 eV / 0.04 eV & mass spectrum (50 meV) \\
Belle II / LHCb / NA62 / KOTO-II & quark CP & running /
2030s & $\gamma$ to $\sim1^{\circ}$; rare kaons to SM floor &
pentagon values \\
SNEWS 2.0 network & galactic SN & standing & $10^{4}$--$10^{5}$
events @ 10 kpc & sector-wide calibration \\
\bottomrule
\end{tabular}
\end{table}

\section{The confrontation, forced value by forced value}
\label{sec:exp-confrontation}

The review is done; the confrontation begins. The format is
fixed and repeated: each subsection states the forced value at
its grade, the instruments that reach it, the quantitative
relation of reach to value, and --- before any data arrive ---
the kill. Nothing below adjusts anything above: the values were
fixed in Parts~VI--VIII (and upstream, in the parent papers)
before this Part consulted a single schedule.

But first, the audit at a glance. Part~VII closed with the ten-row
forced-versus-measured table; it is reprinted here, unchanged, as
Table~\ref{tab:exp-predicted-measured} --- already carrying the
refreshed world numbers of \S\ref{sec:exp-refresh} --- so that the
confrontations that follow can be read against it without leaving
the Part. Figures~\ref{fig:p5-dev-linear} and~\ref{fig:p5-dev-symlog}
then draw it, in the twin-panel convention of the particle parent's
executive summary \citep{ArisakaA9SM}: for every row with a measured
central value, the signed relative deviation
$\mathrm{Dev} = (\mathrm{Pred} - \mathrm{Obs})/|\mathrm{Obs}|$, the
raw first-order lattice value in orange where the chain has one
($2\pi/5$; $196.43^{\circ}$), the QD-completed or single forced
value in blue, an inner thick bar for the empirical uncertainty
alone, an outer thin bar adding the theory uncertainty in
quadrature, and shaded bands marking the precision decades.

Three features of the two panels reward a stare before the
row-by-row work begins. First, the Cabibbo angle: at
$+4\times10^{-5}$ relative deviation it is invisible at any scale
the other rows can share --- the sharpest single number in the
paper, and the reason the pentagon was taken seriously in the first
place. Second, the QD pattern: in both rows that have a first-order
lattice value, the orange marker lands right of the blue and the
quantum-defect completion pulls the prediction \emph{toward} the
data by roughly a factor of five --- the same one-way pattern the
parent paper's twenty-two-row audit shows, here reproduced in the
CP sector. Third, the asymmetry of information: the two fat blue
bars ($\dPMNS$, and $\JCKM$'s layer band) are wide today and
narrow on schedule, while the watched rows ($\sin 2\beta$
at $-0.1\sigma$ with its evaluation layer stated, the direct
$\gamma$ at $+1.6\sigma$, $\etaB$ at $+20\%$ central inside its D$^{*}$ band) are exactly where the refresh rules of \S\ref{sec:exp-refresh}
keep the running score. Nothing in either panel was fitted;
every marker can move only by dying. With the map in hand, the
confrontation proceeds one observable at a time.

\begin{table}[H]
\centering
\caption[The ten-row confrontation.]{The ten-row confrontation, reprinted from Part~VII
(Table~\ref{tab:predicted-measured}) with the refreshed world
numbers and, in this Part's format, explicit deviation and pull
columns: $\mathrm{Dev} = (\mathrm{Pred} - \mathrm{Obs})/
|\mathrm{Obs}|$ and $\mathrm{Pull} = (\mathrm{Pred} -
\mathrm{Obs})/\sigma_{\mathrm{tot}}$ with experimental and theory
uncertainties added in quadrature (for $\dCKM$ the theory width
is one lattice step, $\pi/35$; against experimental error alone
the raw $2\pi/5$ sits at $+4.2\sigma$); paired entries are
first-order~/~QD-completed. Dashes mark rows with no measured
central value (bounds, signs, or unmeasured phases) --- the
one-sided silences. The $\etaB$ pull is not quoted because the
D$^{*}$ transport band ($\times3^{\pm1}$), not the Planck error,
governs it. The register of each chain applies
(\S\ref{sec:claims-ledger}). No entry was fitted; every entry can
move only by dying. Figures~\ref{fig:p5-dev-linear}
and~\ref{fig:p5-dev-symlog} draw the six measured rows.}
\label{tab:exp-predicted-measured}
\small
\renewcommand{\arraystretch}{1.22}
\begin{tabular}{p{2.0cm} p{2.6cm} p{2.9cm} p{1.7cm} p{1.3cm}
 p{2.1cm}}
\toprule
\textbf{Observable} & \textbf{Forced value} & \textbf{Measured} &
\textbf{Dev} & \textbf{Pull} & \textbf{Status} \\
\midrule
$\sin\theta_{12}^{\mathrm{CKM}}$ & $0.22502$
\eqref{eq:cabibbo} & $0.22501 \pm 0.00068$ & $+4.4\mathrm{e}{-5}$ &
$+0.01$ & derived-cond. \\
$\dCKM$ & $2\pi/5 = 1.2566$; QD $13\pi/35 = 1.1669$ &
$1.147 \pm 0.026$ & $+9.6\mathrm{e}{-2}$ / $+1.7\mathrm{e}{-2}$ &
$+1.17$ / $+0.70$ & derived-cond. \\
$\JCKM$ & $3.1\times10^{-5}$ (layer band $2.7$--$3.2$) &
$(3.12^{+0.13}_{-0.12})
\times10^{-5}$ & $-6.4\mathrm{e}{-3}$ & $-0.2$ & inside band \\
$\sin 2\beta$ & $0.708$ (WE~III) & $0.709 \pm 0.011$ &
$-1.4\mathrm{e}{-3}$ & $-0.1$ & watched \\
$\dPMNS$ & $196.4^{\circ}$; QD $196.06^{\circ}$ & NO:
$177^{\circ}{}^{+19}_{-20}$; IO: $285^{\circ}{}^{+25}_{-28}$ &
$+1.1\mathrm{e}{-1}$ & $+1.0$ & derived-cond.\ (NO) \\
$(\alpha_{21}, \alpha_{31})$ & $(0, \pi)$ & unmeasured & --- &
--- & $0\nu\beta\beta$-testable \\
$\mbb$ & $1.72$~meV & $< 28$--$122$~meV & --- & --- & below reach;
tonne-scale+ \\
$\JPMNS$ & $-9.4\times10^{-3}$ & sign disfavored $> 0$
\citep{T2K2020} & --- & --- & consistent \\
$\bar\theta$ & $\lesssim 4.7\times10^{-18}$ & $< 10^{-10}$
(nEDM) & --- & --- & 8 orders headroom \\
$\etaB$ & $\approx7\times10^{-10}$ ($\times3^{\pm1}$) & $(6.12\pm0.04)\times10^{-10}$ & $+0.20$ & --- & D$^{*}$ band \\
\bottomrule
\end{tabular}
\end{table}

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\begin{figure}[p]
\centering
\includegraphics[width=\textwidth]{p5_dev_plot_linear.png}
\caption[The confrontation at a glance, linear scale.]{\textbf{The confrontation at a glance, linear
scale.} Five rows of Table~\ref{tab:exp-predicted-measured} on a
linear relative-deviation axis
$\mathrm{Dev} = (\mathrm{Pred} - \mathrm{Obs})/|\mathrm{Obs}|$
spanning $\pm0.12$: the Cabibbo angle, $\dCKM$ (first-order
$2\pi/5$ in orange, QD-completed $13\pi/35$ in blue), $\sin 2\beta$,
$\JCKM$ (forced band $2.7$--$3.2\times10^{-5}$, the spread
across the framework's evaluation layers), and $\etaB$ (whose
outer bar is the D$^{*}$ transport band $\times3^{\pm1}$, extending
beyond the frame). Inner thick bars: empirical uncertainty only;
outer thin bars: theory added in quadrature; shaded band:
$|\mathrm{Dev}| < 10^{-2}$. The Cabibbo row sits at
$\mathrm{Dev} = +4\times10^{-5}$ --- dead center at this scale ---
and the QD step pulls $\dCKM$ from $+9.6\%$ to $+1.7\%$, inside one
experimental sigma.}
\label{fig:p5-dev-linear}
\end{figure}

\begin{figure}[p]
\centering
\includegraphics[width=\textwidth]{p5_dev_plot_symlog.png}
\caption[The six measured confrontation rows in one symlog panel.]{%
\textbf{The full measured confrontation in one panel.} All six rows
of Table~\ref{tab:exp-predicted-measured} with a measured central
value, on a symmetric-log deviation axis (linear region
$|\mathrm{Dev}| < 10^{-4}$), nested precision bands at $10^{-4}$,
$10^{-3}$, $10^{-2}$. The remaining four rows --- the Majorana pair
$(\alpha_{21},\alpha_{31})$, $\mbb$, the $\JPMNS$ sign, and
$\bar\theta$ --- are unmeasured or bounded and so cannot appear on
a deviation axis; they are the one-sided silences of
\S\ref{sec:exp-verdicts}. The $\dPMNS$ row makes the decade's
stake visible: the forced value sits at $+11\%$ of today's central
value but \emph{inside} the fat empirical bar ($+1.0\sigma$), and
the long-baseline program will shrink that bar by a factor of
$\sim3$ (\S\ref{ssec:conf-dpmns}).}
\label{fig:p5-dev-symlog}
\end{figure}

\subsection{The lepton phase: $196.4^{\circ}$, and the quiet
signature}
\label{ssec:conf-dpmns}

\emph{Value:} $\dPMNS = \pi + 2\arcsin(1/7) = 196.43^{\circ}$
(QD $196.06^{\circ}$); derived-conditional on the leptonic
assignment lemma (Part~VII). \emph{Position today:} $+1.0\sigma$
from the normal-ordering global fit; $\approx 3\sigma$
disfavored under inverted ordering \citep{NuFit2024,
T2KNOvA2025}. \emph{Instruments:} Hyper-K (2028--), DUNE
($\sim$2031--), with design resolutions $7$--$20^{\circ}$ and
$10$--$20^{\circ}$ respectively, and late-2030s combined
precision plausibly near $\pm7$--$8^{\circ}$ at the lattice
value --- which, sitting near $\pi$, lies in the
\emph{best}-resolution region of both experiments. Now the
structural point, stated with both hats on, because it inverts
the community's expectation. The lattice value implies
$\sin\dPMNS = -0.28$: \emph{small} leptonic CP violation. The
flagship discovery figure of both experiments --- five-sigma
CP-violation exclusion over most of the $\delta$ range ---
concerns values near maximal; if the pentagon is right,
\emph{that discovery will not come}. The \GG{} signature at the
long baselines is a near-null: a measured $\delta$ crowding the
CP-conserving point from below, resolving away from $180^{\circ}$
only slowly ($16.4^{\circ}$ of separation against
$\pm7$--$8^{\circ}$ late-decade: a $2$--$2.5\sigma$
distinction by $\sim$2040, sharpening with statistics
thereafter). The quiet outcome is the prediction ---
uncomfortable, checkable, and utterly unlike the maximal-CP
expectation much of the literature carries. \emph{Kill:} a
measured $\dPMNS$ inconsistent with $196^{\circ}$ at
$5\sigma$ --- e.g.\ data settling at maximal $270^{\circ}$,
$73.6^{\circ}$ away, a clean kill already at $\pm15^{\circ}$
resolution; or CP violation established at $5\sigma$ with
$\delta$ far from the lattice point. Decision window:
2031--2040.

\subsection{The ordering: normal, or dead}
\label{ssec:conf-ordering}

\emph{Value:} the derived spectrum $(0.065, 8.59, 50.5)$~meV is
normally ordered --- not a preference but a consequence of the
mass chain (Part~VII; parent derivation
\citealp{ArisakaA9SM}). \emph{Position today:} globally
undetermined; the joint long-baseline fit even leans mildly
inverted (Bayes factors near unity) \citep{T2KNOvA2025} ---
logged here as the watch item it is --- while atmospherics and
cosmology lean normal \citep{NuFit2024, DESI2025}.
\emph{Instruments:} JUNO ($\sim3\sigma$ by $\sim$2031), DUNE
($5\sigma$ within two beam years, $\sim$2033), atmospherics
(Super-K, IceCube, KM3NeT) combining alongside; DESI-class
cosmology squeezing $\sum m_\nu$ against the inverted floor.
\emph{Kill:} inverted ordering established at $5\sigma$ ---
which would simultaneously destroy the mass chain, the
$\mbb$ prediction, and the $\dPMNS$ confrontation's frame: the
single most consequential binary in this Part. Decision
window: 2030--2035.

\subsection{The Majorana mass: $1.72$~meV, one-sided}
\label{ssec:conf-mbb}

\emph{Value:} $\mbb \simeq 1.72$~meV from the parity pair
$(0, \pi)$ and the derived masses (Part~VII;
derived-conditional). Registered at the first-order evaluation
$1.72$~meV; the QD-layer evaluation is $1.79$~meV; both sit far
below the ton-scale floor, so the registered null is
layer-robust. \emph{Position today:} every search
silent, with KamLAND-Zen first into the inverted band
\citep{KamLANDZen2024} --- exactly the world a $1.72$~meV value
requires. \emph{Instruments:} the ton-scale generation ---
LEGEND-1000, nEXO, CUPID, AMoRE-II, SNO+ --- covering
$\mbb \gtrsim 6$--$21$~meV with discovery sensitivity in the
2030s (\citealp{LEGEND1000}; \citealp{nEXO2022}).
\emph{Relation of reach to value --- the one-sided structure:}
the \GG{} number lies \emph{below} the entire ton-scale window,
so the coming decade cannot confirm it; it can only kill it or
pass it. A confirmed discovery anywhere in the ton-scale window
falsifies the derived pair-and-spectrum outright (and, by the
ordering link, would arrive together with the inverted-ordering
kill or force a revolution in the matrix elements); continued
silence is the prediction, renewed with every tonne-year.
Reaching $1.72$~meV itself requires half-life sensitivity near
$10^{29}$--$10^{31}$~yr --- kiloton-scale isotope, the
furthest-horizon falsifier this paper owns, named for the
generation that will build it. \emph{Kill:} any confirmed
$0\nu\beta\beta$ discovery with $\mbb$ at ton-scale values.
Decision window: 2030--2040 (one-sided); confirmation horizon:
beyond 2045.

\subsection{The strong-CP window: eight orders of mandated
vacancy}
\label{ssec:conf-thetabar}

\emph{Value:} $|\bar\theta| \lesssim 4.7\times10^{-18}$
(Part~VII; derived-conditional on lemma L9), against the
present bound $10^{-10}$ \citep{nEDM2020}. \emph{Instruments:}
n2EDM commissioning toward $10^{-27}\,e\,$cm ($\bar\theta \sim
10^{-11}$), successor concepts a further order beyond; the
electron sector below $10^{-30}\,e\,$cm and falling
\citep{Roussy2023}. \emph{Relation:} eight orders of window in
which the geometry mandates vacancy. Every improved null is a
passed test; \emph{any} confirmed hadronic EDM attributable to
$\bar\theta$ in the window kills the two-fate architecture
while leaving the Standard Model untouched --- the rare
falsifier that executes this paper alone. And the lattice's
rigidity adds the electron-sector clause: no new continuous CP
phases exist to be found, so the deepening eEDM nulls --- which
convict generic TeV-scale phases --- are themselves
predictions. \emph{Kill:} a confirmed EDM in the window.
Decision window: permanent; each generation of instruments
renews it.

\subsection{The proton: silence at every depth}
\label{ssec:conf-proton}

\emph{Value:} absolute stability --- the gauged bulk ledger
(Part~VIII; derived). \emph{Position today:}
$\tau/B(p\to e^+\pi^0) > 2.4\times10^{34}$~yr and channel by
channel likewise \citep{SuperKPdecay2020}; forty years of
silence since the water Cherenkov generation buried minimal
unification \citep{GeorgiGlashow1974}. \emph{Instruments:}
Hyper-K toward $10^{35}$~yr in $e^+\pi^0$; DUNE's
kaon-tracking channel; JUNO \citep{HyperK2025, DUNE2020}.
\emph{Relation:} the unification traditions predict windows;
the \GG{} ledger predicts silence through \emph{all} of them,
forever --- so every order of magnitude gained discriminates,
and the $\Delta B = 2$ door is shut too ($n\bar n$ bounds
passed and renewed \citep{SuperKnnbar2021}). \emph{Kill:} one
confirmed event, any channel, any depth --- ending not only
this Part but the baryonic logic of the entire cohort.
Decision window: permanent.

\subsection{The right-handed absence}
\label{ssec:conf-wr}

\emph{Value:} no $W_R$, no mirror fermions, no parity
restoration at any scale (F-P5-1; Part~VI). \emph{Position
today:} multi-TeV collider limits, all null \citep{PDG2024}.
\emph{Instruments:} HL-LHC through the 2030s--40s; every future
machine thereafter, forever --- plus the indirect probes
(right-handed currents in beta decay and $0\nu\beta\beta$
mechanisms). \emph{Kill:} a $W_R$ at any mass; parity
restoration at any energy. Decision window: permanent ---
left--right symmetry holds the winning ticket at every future
collider, and the projection holds it at none.

\subsection{The pentagon's quark numbers, watched}
\label{ssec:conf-quark}

\emph{Values:} the lattice CKM set of Part~VII, currently
pulling $+0.01$ (Cabibbo), $+0.70$/$+1.17$ ($\delta$, with the
QD flag and the $\pi/35$ theory width in quadrature),
inside-band ($J$), $-0.1$ ($\sin 2\beta$, at the stated
moduli layer), and $+1.6$ ($\gamma$, direct combination,
refreshed in \S\ref{sec:exp-refresh} and watched).
\emph{Instruments:} Belle~II and LHCb drive $\gamma$ toward
$\sim1^{\circ}$ and $\sin 2\beta$ below $\pm0.01$; NA62 and
KOTO-II press the rare-kaon rates that probe the texture at
one loop \citep{NA622025, KOTO2025}. \emph{Relation:} the
lattice fixes the whole unitarity triangle given the moduli, so
the tightening triangle is a standing multi-observable test;
the direct-$\gamma$ confrontation is live: the lattice
triangle's $\gamma$ tracks the QD phase at leading order,
$\simeq 66.9^{\circ}$, against the direct combination
$(62.8\pm2.6)^{\circ}$ --- pull $+1.6\sigma$, watched --- and
the coming $\sim1^{\circ}$ precision sharpens it into a
kill-class test. Rare-kaon rates at Standard-Model values are
pentagon-consistent by construction --- the texture \emph{is}
the one-phase CKM --- so a confirmed deviation demanding new
phases would falsify the lattice's completeness. \emph{Kill:}
a coherent multi-observable failure of the one-phase triangle,
or rare-kaon rates requiring phases off the lattice. Decision
window: continuous; $\gamma$ at $1^{\circ}$ by $\sim$2030.

\subsection{The triad, joined}
\label{ssec:conf-triad}

This Part's battery is the third channel of three. The first:
the lock cluster at $3.85$~TeV, bracketed by the public
ATLAS/CMS record and pre-registered for the HL-LHC, with its
own dated verdicts \citep{ArisakaP1KK}. The second: the Kaxion
at $8.1~\mu$eV, with its twin-peak fingerprint and dedicated
instrument path \citep{ArisakaC1Kaxion}. The logic joining the
three is AND, not OR: the geometry that forces the mirror
sector's values is the same geometry that forces the cluster
and the Kaxion, so the program requires \emph{all three}
channels to land --- and any single clean kill in any channel
ends it. Conversely the discoveries are mutually predictive:
a cluster at $3.85$~TeV entails this Part's quiet-CPV and
silent-proton pattern; a Kaxion at $1.95$~GHz entails the
strong-CP vacancy; an inverted ordering or a ton-scale
$0\nu\beta\beta$ discovery condemns all three channels at
once through the shared ledger. One geometry, three windows,
one verdict.

\section{The world numbers, refreshed}
\label{sec:exp-refresh}

A paper that preaches freshness must practice it on itself.
This section records the refresh executed in this version
against the mid-2026 averages: the confrontation table of
Part~VII (Table~\ref{tab:predicted-measured}) now carries
$\sin 2\beta = 0.709\pm0.011$ against the recomputed $0.708$
at the stated moduli layer (pull $-0.1\sigma$, watched), the
direct-$\gamma$ combination $(62.8\pm2.6)^{\circ}$ against the
lattice triangle's leading-order $\simeq 66.9^{\circ}$ (pull
$+1.6\sigma$: watched, and reported without
cosmetic delay) and the NuFit~6.0 $\dPMNS$ values by ordering
(the lattice value at $+1.0\sigma$ under normal ordering);
the corresponding prose of Part~VII and
Appendix~\ref{app:flavour} has been harmonized. Two standing rules are hereby made explicit.
\emph{First}, world averages quoted anywhere in this paper are
refreshed at every shipped version, and a pull that worsens is
printed with the same promptness as one that improves ---
the direct $\gamma$ is the current case in point. \emph{Second}, no
forced value is ever adjusted in response: the lattice has no
dial to turn, which is the entire content of the word
``forced,'' and the reader can audit the constancy of the
prediction columns against the motion of the measurement
columns across versions.

\section{Simulating the verdicts: 2028, 2031, 2035, 2040}
\label{sec:exp-verdicts}

In the manner of the collider companion's pre-registered
search, this section writes down, before the fact, what the
battery should show at four milestones --- under the geometry
being right, and under it being wrong. The point of
pre-registration is that neither ledger can be rewritten later.

\emph{2028 --- first light.} Hyper-K begins; JUNO's ordering
sensitivity passes $2\sigma$; LEGEND-1000 and nEXO are in
construction; n2EDM delivers first data; NA62 concludes.
\emph{If the geometry is right:} JUNO trends normal; the EDM
null deepens; the proton stays silent; the rare-kaon rates sit
on the Standard Model. \emph{If it is wrong,} the earliest
exits are an inverted-ordering trend at JUNO or an n2EDM
signal.

\emph{2031 --- the ordering closes in.} JUNO reaches
$\sim3\sigma$; DUNE's beam turns on; Hyper-K holds three years
of data --- enough that a true $\delta$ near maximal would
already be producing a CP-violation discovery.
\emph{If right:} the ordering reads normal at $3\sigma$-class;
Hyper-K reports \emph{no} $5\sigma$ CP discovery --- the quiet
signature --- with the allowed region crowding
$180^{\circ}$--$200^{\circ}$. \emph{If wrong:} a $5\sigma$ CPV
discovery at $\delta$ far from the lattice point, or the
inverted ordering hardening, kills within the decade's first
half.

\emph{2035 --- the pincers.} DUNE settles the ordering at
$5\sigma$; the HL-LHC delivers the lock-cluster verdict window
of the collider companion; the ton-scale double-beta
experiments produce first results; combined long-baseline
$\delta$ precision reaches $\pm10^{\circ}$-class.
\emph{If right:} normal ordering certified; $0\nu\beta\beta$
silent through the inverted band's upper half; $\delta$
consistent with $196^{\circ}$ and inconsistent with maximal;
and --- across the triad --- a cluster emerging at ATLAS/CMS.
\emph{If wrong:} this is the likeliest single milestone for a
clean kill: IO at $5\sigma$, or a ton-scale discovery, or
$\delta$ at $270^{\circ}$, or the cluster excluded with the
geometry-aware reinterpretation.

\emph{2040 --- the verdict.} Combined $\delta$ near
$\pm7$--$8^{\circ}$: the lattice value now $2$--$2.5\sigma$
distinct from CP conservation and conclusively distinct from
maximal; the ton-scale program completes its window; Hyper-K
approaches $10^{35}$~yr on the proton; the collider companion's
final HL-LHC node arrives. \emph{If right:} the pattern is
unmistakable in aggregate --- normal ordering, quiet CP at the
lattice point, double-beta and EDM and proton silence, no
$W_R$, a cluster, a Kaxion candidate --- the quiet universe
with one loud cluster, a signature no rival framework
predicts as a package. \emph{If wrong:} it did not get this
far; some ledger line above already closed the program.

The combined-likelihood trajectory is the honest way to hold
these milestones together: no single 2030s measurement save an
outright kill is decisive alone, but each passing year the
battery's joint likelihood either converges on the forced
pattern or is terminated by one of the pre-declared exits.
The registered claim is the package, and the package has no
dial.

\section{The falsifier timeline, dated}
\label{sec:exp-timeline}

Table~\ref{tab:exp-timeline} closes the Part's apparatus: the
schedule-centric companion to the claims-level falsifier table
of Part~XII, one line per confrontation, with the verdict type
each instrument can return --- \emph{kill} (can falsify
outright), \emph{support} (can strengthen, cannot confirm
alone), \emph{watch} (tension monitor).

\begin{table}[t]
\centering
\caption[The dated falsifier timeline of the third channel.]{The dated falsifier timeline of the third channel,
with the triad's other two channels appended. ``Kill'' lines
are pre-declared and dial-free; the claims-level register of
each value is in Table~\ref{tab:falsifiers} and the Part
ledgers. Under the geometry being right, every ``kill'' line
returns silence or the lattice point; the aggregate is the
quiet universe with one loud cluster.}
\label{tab:exp-timeline}
\small
\renewcommand{\arraystretch}{1.25}
\begin{tabular}{p{2.9cm} p{2.6cm} p{2.5cm} p{2.2cm} p{2.6cm}}
\toprule
\textbf{Forced value} & \textbf{Instrument(s)} &
\textbf{Decision window} & \textbf{Verdict type} &
\textbf{Kill condition} \\
\midrule
$\dPMNS = 196.4^{\circ}$ & Hyper-K, DUNE & 2031--2040 & kill /
support & $\delta$ off $196^{\circ}$ at $5\sigma$ \\
Normal ordering & JUNO, DUNE, atm., DESI & 2030--2035 & kill &
IO at $5\sigma$ \\
$\mbb = 1.72$ meV & LEGEND-1000, nEXO, CUPID & 2030--2040
(one-sided) & kill / support & ton-scale discovery \\
$\bar\theta \lesssim 10^{-18}$ & n2EDM, eEDM program &
permanent & kill / support & EDM in the window \\
Proton stability & Hyper-K, DUNE, JUNO & permanent & kill /
support & one event \\
No $W_R$ (F-P5-1) & HL-LHC, successors & permanent & kill &
$W_R$ / restoration \\
Pentagon CKM set & Belle II, LHCb, NA62, KOTO-II & continuous &
watch / kill & one-phase triangle fails \\
Lock cluster (P1-KK) & ATLAS, CMS (HL-LHC) & 2029--2041 & kill /
discover & cluster excluded \\
Kaxion (C1) & haloscope program & 2030s & kill / discover &
locks excluded \\
\bottomrule
\end{tabular}
\end{table}

The same schedule, drawn. Figure~\ref{fig:p5-killtest-timeline}
renders Table~\ref{tab:exp-timeline} in the cohort's
kill-test-timeline convention --- the convention introduced for the
Kaxion's discovery calendar --- on a forward time axis from now to
the far end of the decision window. The color code is the cohort's:
violet for tests already faced and survived, orange for the standing
watch, blue for the accelerator fronts approaching the lattice point,
green for the ton-scale edge closing one-sidedly, and red for the
kill dates and the verdict window. Every card names its venues and
its falsifier; the spans along the axis show the two long campaigns
--- the long-baseline phase measurement and the ton-scale double-beta
exposure --- that bracket the decade. The figure makes graphic what
the table states: the mirror sector's confrontation is not an open
horizon but a closing window, with pre-declared exits on every side.

\begin{figure}[p]
\centering
\resizebox{\textwidth}{!}{%
\begin{tikzpicture}[
    font=\Large,
    >=Latex,
    card/.style={draw=black!55, rounded corners=2pt, line width=0.6pt,
        align=left, inner sep=4pt, text width=62mm},
    stem/.style={line width=0.9pt, draw=black!55},
    sectorSURV/.style={fill=violet!10, draw=violet!60!black},
    sectorCAMP/.style={fill=orange!12, draw=orange!75!black},
    sectorSCAN/.style={fill=blue!8, draw=blue!55!black},
    sectorEFR/.style={fill=green!10, draw=green!55!black},
    sectorKILL/.style={fill=red!10, draw=red!60!black},
    dot/.style={circle, draw=black!70, fill=white, inner sep=0pt,
        minimum size=5.2pt}
]
\def\xL{0}\def\xR{30}\def\yA{0}
\def\bA{4.0}\def\bB{14.0}
\begin{scope}[on background layer]
    \fill[black!2] (\xL,-12.2) rectangle (\bA,10.6);
    \fill[black!4] (\bA,-12.2) rectangle (\bB,10.6);
    \fill[black!6] (\bB,-12.2) rectangle (\xR,10.6);
\end{scope}
\node[anchor=south, font=\huge\bfseries] at ({(\xL+\bA)/2},9.8)
    {NOW};
\node[anchor=south, font=\large] at ({(\xL+\bA)/2},9.15)
    {2026--28: the standing record};
\node[anchor=south, font=\huge\bfseries] at ({(\bA+\bB)/2},9.8)
    {FIRST WAVE};
\node[anchor=south, font=\large] at ({(\bA+\bB)/2},9.15)
    {2028--33: the instruments arrive};
\node[anchor=south, font=\huge\bfseries] at ({(\bB+\xR)/2},9.8)
    {DECISIVE};
\node[anchor=south, font=\large] at ({(\bB+\xR)/2},9.15)
    {2033--41: the verdicts};
\draw[->, line width=1.4pt] (\xL,\yA) -- (\xR+0.7,\yA);
\node[anchor=west, font=\Large\itshape] at (\xR+0.8,\yA) {time};
\foreach \x in {\bA,\bB}{\draw[black!55, line width=0.8pt]
    (\x,-0.18) -- (\x,0.18);}
\node[anchor=north, font=\large] at (\xL,-0.25) {2026};
\node[anchor=north, font=\large] at (\bB,-0.25) {2033};
\node[anchor=north, font=\large] at (\xR,-0.25) {2041};
%% ---- TOP ROW ----
\node[dot] (d1) at (1.6,\yA) {};
\node[card, sectorSURV, anchor=south, text width=66mm] (c1)
    at (2.6,1.5) {%
    \textbf{The standing record: faced daily, survived}\\[1pt]
    proton silent to $2.4\times10^{34}$~yr; nEDM null at
    $1.8\times10^{-26}\,e\,$cm; KamLAND-Zen silent into the IO
    band; $\gamma$ pull $+1.6\sigma$, $\sin 2\beta$ $-0.1\sigma$ (watched)\\
    venues: Super-K, PSI, KamLAND-Zen; standing mid-2026};
\draw[stem] (d1) -- (c1.south west);
\node[dot] (d2) at (9.5,\yA) {};
\node[card, sectorSCAN, anchor=south, text width=76mm] (c2)
    at (11.2,3.4) {%
    \textbf{The long-baseline front}\\[1pt]
    Hyper-K data 2028; DUNE beam $\sim$2031; JUNO running\\
    ordering to $3\sigma$ by $\sim$2030--31; the $\delta$ campaign
    opens\\
    quiet-CPV under test: no $5\sigma$ discovery expected};
\draw[stem] (d2) -- (c2.south);
\node[dot] (d3) at (20.5,\yA) {};
\node[card, sectorEFR, anchor=south] (c3) at (20.5,1.5) {%
    \textbf{The ton-scale edge (one-sided)}\\[1pt]
    LEGEND-1000, nEXO, CUPID reach
    $\mbb \sim 5$--$20$~meV through the 2030s\\
    the geometry predicts silence at $\mbb = 1.72$~meV\\
    any confirmed line in reach is fatal};
\draw[stem] (d3) -- (c3.south);
%% ---- BOTTOM ROW ----
\node[dot] (e1) at (5.2,\yA) {};
\node[card, sectorCAMP, anchor=north, text width=74mm] (b1)
    at (5.2,-2.3) {%
    \textbf{The ordering watch}\\[1pt]
    standing: accelerator joint fits lean IO at low significance; global
    fits and cosmology lean NO\\
    the geometry requires NO --- no hedge, no fallback\\
    JUNO $+$ atmospherics weigh in first};
\draw[stem] (e1) -- (b1.north);
\node[dot] (e2) at (14.5,\yA) {};
\node[card, sectorKILL, anchor=north, text width=76mm] (b2)
    at (14.5,-4.6) {%
    \textbf{KILL DATES}\\[1pt]
    IO at $5\sigma$ ($\sim$2033); $\delta = 270^{\circ}$ at
    $5\sigma$ (2031--35): either retires the sector\\
    standing kills: a proton event, an in-window EDM, a $W_R$ ---
    any day};
\draw[stem] (e2) -- (b2.north);
\node[dot] (e3) at (24.5,\yA) {};
\node[card, sectorKILL, anchor=north, text width=70mm] (b3)
    at (24.5,-2.7) {%
    \textbf{THE VERDICT WINDOW ($\sim$2035--41)}\\[1pt]
    $\delta$ vs $196.4^{\circ}$ at $\pm7$--$10^{\circ}$;
    quiet-CPV signature judged; ton-scale window closed\\
    triad joined: HL-LHC's cluster verdict (P1-KK) and the
    haloscope window (C1) land in the same years};
\draw[stem] (e3) -- (b3.north);
%% ---- campaign-span bars ----
\draw[blue!55, line width=2.2pt] (4.0,-1.30) -- (28.0,-1.30);
\node[blue!55, font=\large, anchor=south west] at (18.0,-1.25)
    {long-baseline $\delta$ campaign};
\draw[green!55!black, line width=2.2pt] (12.0,-1.75) -- (30,-1.75);
\node[green!55!black, font=\large, anchor=north west] at (18.0,-1.85)
    {ton-scale $0\nu\beta\beta$ exposure};
%% ---- legend ----
\begin{scope}[shift={(0.3,-11.6)}]
    \node[anchor=west, font=\large\bfseries] at (0,0) {Cards:};
    \node[card, sectorSURV, text width=2.2mm, minimum height=2.2mm,
        inner sep=1.6pt] at (2.1,0) {};
    \node[anchor=west, font=\large] at (2.4,0) {standing record};
    \node[card, sectorCAMP, text width=2.2mm, minimum height=2.2mm,
        inner sep=1.6pt] at (7.6,0) {};
    \node[anchor=west, font=\large] at (7.9,0) {watch};
    \node[card, sectorSCAN, text width=2.2mm, minimum height=2.2mm,
        inner sep=1.6pt] at (10.6,0) {};
    \node[anchor=west, font=\large] at (10.9,0)
        {accelerator fronts};
    \node[card, sectorEFR, text width=2.2mm, minimum height=2.2mm,
        inner sep=1.6pt] at (16.6,0) {};
    \node[anchor=west, font=\large] at (16.9,0) {ton-scale edge};
    \node[card, sectorKILL, text width=2.2mm, minimum height=2.2mm,
        inner sep=1.6pt] at (21.6,0) {};
    \node[anchor=west, font=\large] at (21.9,0)
        {kill dates / verdicts};
\end{scope}
%% ---- banner ----
\node[draw=black!40, fill=yellow!12, rounded corners=2pt,
      line width=0.6pt, align=center, text width=235mm,
      inner sep=5pt, font=\Large, anchor=north] at (15,14.1) {%
    \textbf{One-sided, dated, and quiet.} Eight pre-declared
    confrontations; four standing silences (proton, EDM, double
    beta, $W_R$) where a single confirmed event executes the kill;
    one quiet-CPV signature --- the flagship $5\sigma$ discovery
    should \emph{not} come, because $\sin\delta = -0.28$ sits near
    the null; and one loud exception, the lock cluster at
    $3.85$~TeV. Under the geometry: normal ordering, the lattice
    point, and silence everywhere else. Any clean exit retires the
    mirror sector, and the triad's AND-logic carries the verdict
    to the cohort.};
\end{tikzpicture}%
}
\caption[The kill-test timeline for the mirror sector.]{%
\textbf{Falsification / kill-test timeline for the mirror sector.}
The dated falsifier table (Table~\ref{tab:exp-timeline}) drawn on a
forward time axis (2026--2041, broad data-window eras rather than
precise launch dates), in the cohort's kill-test-timeline
convention. Violet: the standing record --- the proton, EDM, and
double-beta silences and the $\sin 2\beta$ watch, faced continuously
and survived as of mid-2026 (\S\ref{sec:exp-refresh}). Orange: the
ordering watch --- the geometry requires normal ordering with no
fallback. Blue: the long-baseline front (Hyper-K 2028, DUNE
$\sim$2031, JUNO running) approaching the lattice point
$\dPMNS = 196.4^{\circ}$, with the quiet-CPV signature --- no
$5\sigma$ CP discovery --- as its registered prediction. Green: the
ton-scale $0\nu\beta\beta$ edge closing one-sidedly on a predicted
silence at $\mbb = 1.72$~meV. Red: the kill dates --- inverted
ordering at $5\sigma$ ($\sim$2033) or $\delta = 270^{\circ}$ at
$5\sigma$ (2031--35) retires the sector, and the mid-to-late-2030s
verdict window joins this Part's battery to the triad's other two
channels (P1-KK, C1). All sensitivity dates are conditional on
stated design assumptions (mass ordering known where required,
beam-power ramps, staging); the conditioning is the
collaborations', quoted from the cited design reports.}
\label{fig:p5-killtest-timeline}
\end{figure}

\section{What is claimed, and what is inherited}
\label{sec:exp-claims}

Concessions first, and for this Part they are near-total: every
instrument, every sensitivity, every milestone in these pages
belongs to the experimental collaborations --- to the
generations that built the kaon beams and the $B$ factories,
the mines and the tanks, and to those now building Hyper-K,
DUNE, JUNO, and the ton-scale program. The history sections
are their achievement cataloged, and the projections are
their design reports quoted at attributed grade. This paper
adds no instrument and improves no sensitivity. What it claims
is four things, all structural. \emph{First, the
dial-lessness of the confrontation:} every value in
Table~\ref{tab:exp-timeline} was fixed upstream of every
schedule in it, and \S\ref{sec:exp-refresh}'s standing rules
make the constancy auditable. \emph{Second, the one-sided
silences:} the double-beta, EDM, proton, and $W_R$ lines
predict \emph{nulls} --- the riskiest posture available, since
a single confirmed event anywhere executes it. \emph{Third,
the quiet-CPV signature:} the geometry predicts that the
flagship discovery of the long-baseline era will \emph{not}
occur as commonly hoped --- a near-null at the lattice point
--- which converts even a community disappointment into a
registered test. \emph{Fourth, the triad's AND-logic:} three
channels, one geometry, mutual entailment, any clean kill
fatal to all. The grades are those of the source Parts;
nothing here strengthens them, and everything here can end
them.

\medskip
\noindent\emph{Part-end summary.} Two centuries of instruments
reviewed with their milestone tables; a present in which the
lepton phase sits one sigma from the lattice under the required
ordering; a decade of machines arriving on published schedules;
eight confrontations, each with its pre-declared kill; the
world numbers refreshed with the pulls printed, favorable or
not; four milestone years pre-registered under both verdicts;
and one timeline table joining this Part's battery to the
cluster and the Kaxion. The mirror sector is now what the
program requires it to be: not a story but a schedule. The
claims and their housekeeping follow.

\GGPart[Discussions]{Discussions}{Significance of the result; the classic puzzles of the mirrors, resolved or dissolved; the innovations registered; the nobility clause; the discrimination matrix; the claims ledger; predictions for falsification, walked one by one; the open questions, structured; and the completed cohort}
\label{part:discussions}

The derivations are finished; this Part steps back and argues.
Seven discussions follow --- the significance of the result; its
uniqueness, nobility, and innovations, with the classic puzzles and
the technical novelties each registered under auditable
identifiers; the five roads condensed to a discrimination matrix;
the consolidated claims ledger; the predictions for falsification,
walked one at a time with their kill conditions; the open
questions, each with its closing condition; and the paper's place
in the completed cohort --- closed by a summary that states
strengths and weaknesses with the same plainness. The reference
apparatus this Part leans on lives where the reader left it: the
ten-row confrontation and its deviation panels in
Part~\ref{part:experiments}
(Table~\ref{tab:exp-predicted-measured},
Figures~\ref{fig:p5-dev-linear}--\ref{fig:p5-dev-symlog}), the
dated instrument schedule in Table~\ref{tab:exp-timeline}, and the
kill-test timeline in Figure~\ref{fig:p5-killtest-timeline}. The
argument can proceed without the furniture.

\section[Part~XII in plain English]{Part~XII in plain English: what the result is worth, and what would end it}
A paper that has proved its theorems still owes an account of what they are worth. This Part gives that account at length rather than in a closing paragraph, and a good deal of what it contains could have been left out without anyone noticing.

It states the significance and the innovations, works through the classic puzzles of the mirrors and says of each whether it is resolved or dissolved --- which are different outcomes and are not run together --- sets the five roads against one another in a discrimination matrix, and states what is claimed and in what regime. Then the predictions for falsification, put as measurements rather than as attitudes.

The open questions are the section to read before the conclusions. They are ordered by consequence and the first is the sharpest. None of them threatens the theorems as stated; each bounds the regime the theorems own, and those are different things that a reader deciding how much weight to put on this argument should keep apart.

\section{Significance of the result}
\label{sec:significance}

What this paper delivers, reduced to its irreducible content, is
one sentence: \emph{an anatomy of the three mirrors, a valuation
of the two CP phases, a derivation of the matter excess with zero
fitted parameters, and a dated schedule on which every one of
those claims can die.} Each clause is individually rare in the
discrete-symmetry literature; their conjunction is, to our
knowledge, unprecedented. This section unpacks why, deliberately
slowly, because a result of this shape is easy to misjudge in both
directions --- easy to inflate, if one forgets how much is
conditional on upstream theorems and on two named conjectures, and
easy to deflate, if seventy years of convention-laden discrete
symmetries have trained the reader to assume that every CP
``prediction'' is secretly a scan.

\paragraph{The result in the seventy-year arc.} Place the claim
against the history Part~\ref{part:history} told. In 1956--57 the
field learned, to its lasting shock, that the mirror was broken
--- and then spent decades recording \emph{how} without a
framework that could say \emph{why}, because in the Standard Model
the V$-$A structure is an assignment, not a consequence. In 1964
CP fell in the kaons, and the Kobayashi--Maskawa mechanism turned
the fall into a counting theorem --- but priced nothing: $\delta$
is measured superbly and explained not at all, and $\delta = 0$
would have been equally lawful. In 1967 Sakharov wrote the three
conditions for a matter excess, and half a century of baryogenesis
model-building has met them with fitted masses, fitted phases, and
engineered departures from equilibrium --- mechanisms by the
dozen, verdicts by none, most of them untestable in principle
because their action happens at $10^{9}$~GeV or above. Against
that history the present claim is structurally different in kind:
the operations are derived objects (three reversals of three real
orientations); the phases are lattice points ($13\pi/35$;
$\pi + 2\arcsin(1/7)$) that cannot move, because a lattice point
has no neighborhood; the excess is computed from inputs that were
all fixed upstream for other reasons; and the paper has
pre-committed, in writing and on a calendar
(Part~\ref{part:experiments}), to the measurements that retire
each piece. Whether or not nature cooperates, this is the shape of
answer the seventy-year record has been asking for.

\paragraph{What zero dials buys, epistemically.} The absence of
continuous freedom is not an aesthetic garnish; it changes what an
experimental outcome \emph{means}. With a dial, a positive result
confirms little (the dial was set after the fact) and a negative
result kills nothing (the dial turns). Without one, both outcomes
become maximally informative: a measured $\dPMNS$ crowding
$196^{\circ}$ under normal ordering confirms a chain that reaches
from oscillation data back through the pentagon to the axioms, and
a measured $270^{\circ}$ kills the same chain outright --- there
is no fallback version of this paper at another angle. The
statistics inherit the same virtue: a pre-registered point pays no
look-elsewhere penalty, and the two-point adjudication
($196.4^{\circ}$ versus the community's expected $270^{\circ}$,
$73.6^{\circ}$ apart) is decided at $\pm15^{\circ}$ resolutions
the long-baseline program already prints in its design reports
(\S\ref{ssec:conf-dpmns}). Pre-registration is usually a virtue of
experimental design; here it is a consequence of the theory's
structure, and it converts directly into decisiveness.

\paragraph{The over-constraint, told in full.} A skeptic granting
the structure could still ask: why take the chain seriously
\emph{before} the long-baseline verdict? The paper's answer
predates any future data: the same lattice that fixes the
unmeasured has already been audited on the measured. The Cabibbo
closed form sits at $\mathrm{Dev} = +4\times10^{-5}$ --- a
four-to-five-digit audit, pull $+0.01$
(Table~\ref{tab:exp-predicted-measured}); the QD-completed
$\dCKM = 13\pi/35$ sits $+0.70\sigma$ from the world fit where
the raw $2\pi/5$ sat at $+1.17$ (both pulls carry the
one-lattice-step theory width $\pi/35$ in quadrature; on
experimental error alone the raw value is $+4.2\sigma$); the
Jarlskog invariant falls
inside its layer band; and $\sin 2\beta$, recomputed as
interferometry of the same single angle, tracks the measured
value at $-0.1\sigma$ --- watched, printed, with its
evaluation layer stated. One
lattice, several scales, the measured half already matched: the
construction walks into the neutrino test having survived its
quark-sector audit. This is what distinguishes the wager from a
lottery ticket, and also what raises the stakes --- a leptonic
kill does not merely eliminate a candidate phase; it breaks a
lattice that presently accounts for the quark sector's.

\paragraph{The calendar as part of the result.}
Part~\ref{part:experiments} is not an appendix to the claim; it
is half of it. A falsifiable statement without a decision
procedure is a promissory note, and the experimental Part
converts the note to a schedule: JUNO running now, Hyper-K data
from 2028, DUNE beam near 2031, ordering closure near 2033, the
$\delta$ precision campaign ripening through the late 2030s, the
ton-scale double-beta program closing one-sidedly across the
same decade, and the permanent rails --- proton, EDM, $W_R$ ---
able to execute on any day (Figure~\ref{fig:p5-killtest-timeline}).
The dates matter beyond logistics: they place the verdict within
the professional lifetime of the graduate student the Preface
addresses, and within the planning horizon of the collaborations
now writing their physics cases.

\paragraph{The quiet-CPV inversion, as epistemic posture.} One
prediction deserves separate billing because it inverts the
community's hopes rather than flattering them. The lattice value
implies $\sin\dPMNS = -0.28$: \emph{small} leptonic CP violation,
a near-null crowding the CP-conserving point. If the pentagon is
right, the flagship five-sigma CP-violation discovery that both
long-baseline collaborations advertise \emph{will not come}
(\S\ref{ssec:conf-dpmns}). A framework that predicts a
community disappointment --- and registers the disappointment as
its own confirmation channel --- is operating at the opposite
pole from the adjustable optimism of most beyond-Standard-Model
phenomenology; it is difficult to imagine a posture more exposed.
The refresh discipline of \S\ref{sec:exp-refresh} completes the
posture: the world numbers are re-audited on standing rules, and
the pulls are printed favorable or not --- the $+1.6\sigma$
direct-$\gamma$ tension is displayed in the executive apparatus of
this paper, not buried.

\paragraph{Significance for each community, separately.} To the
\emph{discrete-symmetry tradition}, the significance is
anatomical. Since 1957 the field has possessed exquisite
measurements of operations it could not define except by
convention: C, P, and T were operators with adjustable phases,
their table an assignment, their one surviving product a theorem
whose proof illuminated nothing about its factors. Part~V
replaces the conventions with three specific reversals of three
specific real structures, derives the table's one asymmetry
(antiunitarity) from the sources of $i$, and re-proves the
surviving product as the one reversal no real texture can detect
--- with, since this volume's completion, the same analytic fact
extended to the quartet's second child, spin--statistics, at
candidate grade (\S\ref{sec:quartet-second-child}). To
\emph{flavor physics}, the significance is the valuation: the
rephasing-invariant calculus told the field where CP violation
lives; nothing in sixty years told it what the phase costs. The
pentagonal lattice is, to our knowledge, the first proposal from
which the CKM and PMNS phases take \emph{forced central values},
with the golden-ratio Cabibbo closed form as the same object's
magnitude shadow. To \emph{cosmology}, the significance is the
dial count: every baryogenesis mechanism on the market buys its
$\etaB$ with fitted masses, fitted phases, and engineered
departures from equilibrium; Part~VIII's chain has zero fitted
parameters, one named input, one postulated exponent, and an
error budget it publishes against itself. To the
\emph{experimental collaborations}, the significance is
operational: pre-registered points, one-sided silences, and a
deliverable valuable under every outcome --- the strongest
possible limits on the lattice reading if the answer is no. And
to the \emph{program}, the significance is consolidation: five
questions its papers answered in five places --- the table, the
pentagon, the hand, the door, the excess --- are here exhibited
as one question about one orientation, which is what a framework
claiming structural unity owes its readers.

\section{Uniqueness, nobility, and innovation}
\label{sec:uniqueness}

Three words a physics paper should use carefully, so we define
them operationally before spending them. \emph{Uniqueness} is
what this paper does that no neighboring construction does,
stated so a referee can check the negative --- the matrix of
\S\ref{sec:five-roads} is its checklist form. \emph{Problems
solved} are the standing puzzles of the discrete-symmetry and
baryogenesis literature that the construction resolves or
dissolves, registered as PS-P5-1 through PS-P5-8, each closing
with an honest verdict and its conditionality. \emph{Innovation}
is the specific technical moves that did not exist before this
paper and its parent theorems, registered as I-P5-1 through
I-P5-10 in before-and-after form. A closing subsection states the
conduct the paper binds itself to. The three are related but not
the same, and conflating them is how strong claims come to sound
like salesmanship.

The credit side is stated first, as the register requires.
\emph{Not new here:} the Feynman--St\"uckelberg pairing; Wigner's
(anti)unitary fork; Jost's CPT theorem and its modular
refinement; Jarlskog's invariant calculus; the orbifold-GUT
projection (Kawamura, Hall--Nomura); the seesaw and its
Casas--Ibarra parameterization; thermal and resonant leptogenesis
(Fukugita--Yanagida; Pilaftsis--Underwood) and the sphaleron
arithmetic (Harvey--Turner); Peccei--Quinn relaxation and the
Vafa--Witten minimum; spontaneous, geometric CP violation as a
concept (Dine--Leigh--MacIntire, Choi--Kaplan--Nelson). Each is
cited at its point of use and credited in Part~XVI's notes.

\subsection{The classic puzzles of the mirrors, resolved or
dissolved}
\label{ssec:disc-puzzles}

Each entry states a puzzle the field has carried for decades, in
the form a practitioner would recognize, and then what this paper
does to it. ``Resolved'' means an answer is derived at a stated
grade; ``dissolved'' means the question is shown to have been
wrongly posed.

\paragraph{PS-P5-1: Why does antimatter exist?} Dirac's equation
demanded the positron and nobody --- Dirac least of all ---
wanted it; the twin arrived as an embarrassment of the
mathematics, was reinterpreted by hole theory, and was finally
domesticated by QFT as an output of quantization: fields,
quantized causally, must carry both signs of frequency. That is a
theorem, but a late one --- the pairing appears only after the
complex field and its quantization rules are assumed. Here the
twin exists \emph{before} quantization: two real rotations ---
the circle's winding and the clock's advance --- must agree on
every charged mode, and the only consistent bookkeeping pairs
every winding with its reverse (T-P5-A/B/C). Antimatter is
forced at the level of classical real geometry; quantization
inherits rather than creates it. \emph{Resolved at quasi-free
grade; the interacting statement is conditional on scattering
hypotheses, as itemized in Part~X.}

\paragraph{PS-P5-2: What are C, P, and T, actually?} The
textbook operations are conventions with adjustable phases ---
$C$ defined up to a phase per species, $P$ up to intrinsic
parities, $T$ antiunitary ``because Wigner's theorem permits
it.'' Seventy years of exquisite data sit atop operators the
theory cannot define without stipulation. Here the three are
anatomical: reversals of the three real orientations of the bulk
--- fiber, space, clock --- with the table derived, antiunitarity
of $T$ forced (only the clock is what $i$ is made of), and every
intrinsic phase traced to a real convention (T-P5-1).
\emph{Resolved; the anatomy is a standing challenge to any
framework that postulates its complex structure.}

\paragraph{PS-P5-3: Why is parity violation maximal?} The
Standard Model writes V$-$A by hand and offers no reason why the
mirror sector is absent rather than suppressed --- fifty years of
$W_R$ searches have chased a restoration scale that nothing in
the model requires or forbids. Here maximality is an
\emph{absence}, not a small parameter: the oriented descent keeps
one hand (the other is projected, not hidden), so there is no
scale at which parity restores and no coupling by which the
missing hand leaks back (Part~VI; F-P5-1). \emph{Resolved
conditional on the projection --- whose own necessity is the
honestly-flagged open frontier of the parity ledger
(Table~\ref{tab:parity-ledger}).}

\paragraph{PS-P5-4: Why is CP violation small but not zero?} In
the Standard Model $\delta$ is superbly measured and completely
unexplained; nothing prices it, and the near-maximal measured
value of the leptonic candidate phases sits beside the tiny
Jarlskog invariant with no comment. Here both facts are geometry:
CP falls because the flavor pentagon is chiral (nothing forces
it achiral), the phase is a lattice point --- $2\pi/5$ at first
order, $13\pi/35$ QD-completed --- and the smallness of the
observable violation is the smallness of an interference of
fixed angles, not of a tuned parameter (Part~VII).
\emph{Resolved-conditional on the lattice-rigidity clause of
T-P5-3(iv), with the QD step flagged at its own grade.}

\paragraph{PS-P5-5: The strong-CP problem.} The classic double
standard: a large weak phase beside
$\bar\theta < 10^{-10}$, two parameters of the same naive kind,
one order unity and one inexplicably absent --- and the standard
cure (Peccei--Quinn) installs a global symmetry that quantum
gravity is expected to violate. Here the two phases have
different \emph{homes}, not different luck: the weak phase is a
holonomy the relaxation cannot touch (it is locked in closed
loops), while the strong angle is a relaxable vacuum angle whose
relaxer --- the Kaxion --- the framework already contains for
anomaly reasons, with no PQ scalar and hence no quality problem;
the residual is computed at $\lesssim 5\times10^{-18}$, eight
orders below the current bound and within reach of no planned
instrument --- a permanent one-sided silence. \emph{Resolved
conditional on lemma L9 (determinant phase), with the two-fate
architecture --- locked versus relaxed --- as the structural
content no other framework states.}

\paragraph{PS-P5-6: Why is CPT alone unbroken?} In QFT the
answer is a proof that illuminates nothing about the factors: the
theorem holds for any local, Lorentz-invariant theory, so the
survival of the product is disconnected from the fates of C, P,
and T individually --- a conspiracy of accidents capped by an
axiom. Here the product survives because it is the one reversal
no real texture can detect: each mirror individually reverses an
orientation some structure of the descent can see, and the triple
reversal is the unique combination invisible to every real
datum, derived from the same quartet analyticity that licenses
every Euclidean computation (T-P5-2, T-P5-B1) --- and now shown,
at candidate grade, to carry spin--statistics as its sibling
(\S\ref{sec:quartet-second-child}). \emph{Resolved at quasi-free
grade; modular route open and named.}

\paragraph{PS-P5-7: The ten-order failure.} The Standard Model,
run honestly through the sphaleron era, produces a baryon excess
some ten orders of magnitude short of the observed
six-per-ten-billion --- the sharpest quantitative failure of the
most successful theory in physics. Part~VIII's chain meets the
number with zero fitted parameters: the heavy mass is the
ledger's lock scale, the portal is a uniqueness theorem, the
Yukawa is matched to derived light masses, the phase is the
pentagon's, and the throughput reconstructs the ledger
exponential --- arriving at $\approx7\times10^{-10}$ (central $7.34$), $+20\%$ from Planck's central value, inside a published factor-$\sim3$ band.
\emph{Resolved at grade D$^{*}$, with the resonance mechanism
and the exponent carried by the named conjecture C-P5-C --- the
paper's sharpest open item, displayed rather than hidden.} The
dial count is displayed symmetry by symmetry in the alphabet
audit of \S\ref{sec:alphabet-audit}: existence from the
pentagon's chirality, sign from a Majorana parity, magnitude
from the throat depth --- three jobs, three forced structures,
zero dials.

\paragraph{PS-P5-8: The testability problem of baryogenesis.}
The quiet scandal of the field: the default explanation
(hierarchical thermal leptogenesis) is a good theory that cannot
be tested --- its heavy states sit at $10^{9}$~GeV or above, and
the CP phases that matter to it are not the phases experiments
measure. Mechanisms multiply; verdicts never come. Here the
inversion is total, and it is inherited rather than engineered:
the heavy states sit at $3.85$~TeV within collider reach, the CP
phase \emph{is} the measured PMNS sector's (projected), the
Majorana door casts a $0\nu\beta\beta$ shadow at a computable
$1.7$~meV, and the proton clause is checked by every year of
Super-Kamiokande silence. The account of why we exist is due in
the data within the decade --- mortal four separate ways.
\emph{Dissolved by inheritance: the testability inversion is
itself the innovation, registered below as I-P5-7.}

\subsection{The structural innovations}
\label{ssec:disc-innovations}

The register of what did not exist before, each entry in
before-and-after form so the claim of novelty can be audited.

\begin{description}
\item[I-P5-1 (antiparticles before quantization).] \emph{Before}:
the twin as an output of causal quantization --- assume the
complex field, quantize, discover the necessity. \emph{After}:
the handshake theorem forces the pairing at the level of real
classical geometry (two rotations agreeing on every charged
mode), with quantization inheriting the twins it used to create;
two of Weinberg's axioms explained at the price of the free-field
qualifier.
\item[I-P5-2 (the derived table).] \emph{Before}: C, P, T as
conventions with adjustable phases; antiunitarity of $T$ chosen
from Wigner's fork by consistency. \emph{After}: the
unitary/antiunitary table computed from orientation sources ---
``only the clock is what $i$ is made of'' --- with Kramers
degeneracy convention-free (T-P5-1).
\item[I-P5-3 (the Wick quartet).] \emph{Before}: Minkowski's
$it$, Wick rotation, and the Schr\"odinger $i$ related by
folklore; every Euclidean computation borrowing an unexamined
license. \emph{After}: one polar decomposition, one analytic
fact (the one-sided spectrum), issued as a theorem (T-P5-B1) ---
with CPT derived from it (T-P5-2) and spin--statistics stated as
its second child at candidate grade
(\S\ref{sec:quartet-second-child}).
\item[I-P5-4 (the lattice valuation of CP phases).]
\emph{Before}: ``CP phases are holonomies'' --- known since the
invariant calculus; their values fitted, never priced.
\emph{After}: holonomies valued on a pentagonal lattice with
forced central values --- $13\pi/35$, $\pi + 2\arcsin(1/7)$,
$(0,\pi)$ --- and the Cabibbo golden-ratio closed form as the
magnitude shadow of the same object; the first framework, to
the author's knowledge, in which both CP phases have predicted
central values.
\item[I-P5-5 (the two-fate architecture).] \emph{Before}: the
weak phase and the strong angle treated as two parameters of the
same kind with inexplicably different sizes. \emph{After}: a
derived asymmetry of \emph{homes} --- the locked holonomy versus
the relaxable vacuum angle --- making the pair's discrepancy
structural rather than scandalous, with the residual
$\bar\theta$ computed and registered as a permanent one-sided
test.
\item[I-P5-6 (the basin/gate identification).] \emph{Before}:
the weak interaction's exclusive ownership of flavor violation
and parity violation recorded as two unexplained facts.
\emph{After}: the broken force identified with the gate force
--- the unique connection transporting between basins --- making
the two ownerships one architectural statement, with its
uniqueness clause carried as the named conjecture C-P5-A.
\item[I-P5-7 (the dial-free baryogenesis inheritance).]
\emph{Before}: leptogenesis as a mechanism with free heavy
masses, free phases, and unobservable scales. \emph{After}: the
same transport with every input inherited --- lock scale,
unique portal, matched Yukawa, pentagon phase --- zero fitted
parameters, all four levers at observable scales, and the FD
statistics it counts with now traced to the same geometry
(\S\ref{sec:matter-claims}).
\item[I-P5-8 (the Sakharov tripod).] \emph{Before}: Sakharov's
three conditions as a checklist each mechanism satisfies by
separate engineering. \emph{After}: the three conditions as one
geometric object (P-P5-E) --- baryon-number violation, C/CP
violation, and the arrow all read off the oriented descent, each
leg graded, the checklist replaced by an anatomy.
\item[I-P5-9 (the quiet-CPV signature).] \emph{Before}:
long-baseline physics cases built around discovering large CP
violation; a small measured $\delta$ counted as disappointment.
\emph{After}: the near-null at $196.4^{\circ}$ pre-registered as
the framework's \emph{confirmation} channel --- a community
disappointment converted into a two-sided test, with the
$73.6^{\circ}$ separation from the maximal expectation
adjudicated at design resolutions.
\item[I-P5-10 (the confrontation apparatus).] \emph{Before}:
predicted-versus-measured comparisons scattered through prose,
refreshed ad hoc, pulls quoted when favorable. \emph{After}:
a ten-row confrontation table with explicit Dev and Pull columns
under a stated error convention, twin deviation panels in the
parent papers' audit format, standing refresh rules with pulls
printed favorable or not, four milestone years pre-registered
under both verdicts, and a kill-test timeline in the cohort
convention (Part~\ref{part:experiments}).
\end{description}

\subsection{The nobility clause}
\label{ssec:disc-nobility}

``Nobility'' is not a word for the theory; it is a word for the
\emph{conduct} the paper binds itself to, and it can be stated as
four commitments, each checkable. \emph{Pre-commitment:} every
forced value was fixed upstream of every schedule that will test
it --- the lattice points were in print before JUNO's first data,
the milestone years of Part~\ref{part:experiments} are registered
under both verdicts, and the standing refresh rules
(\S\ref{sec:exp-refresh}) seal the paper against quiet
renumbering. \emph{Honest grading:} every claim carries its
register --- derived, derived-conditional, named conjecture,
D$^{*}$, open --- in three Part ledgers, a 27-row audit figure,
and the consolidated ledger below; the least flattering entries
are displayed most prominently (the resonance conjecture appears
in the abstract's own register; the watched $\gamma$ tension at
$+1.6\sigma$ is in the executive apparatus; the ten-order gap C-P5-C must
bridge is quantified in its own section). \emph{No retreat:} the
prediction carries no relocation clause --- a measured
$270^{\circ}$ kills the pentagon outright; inverted ordering at
$5\sigma$ retires the paper with no fallback hierarchy; and the
one-sided silences (proton, EDM, $0\nu\beta\beta$ above the
point value, $W_R$) can execute on any day and can never be
finally vindicated, which is what makes them the strongest
claims in the paper. \emph{Symmetric stakes:} the paper commits
to what failure means --- a leptonic kill breaks the lattice
that carries the quark sector's audit, and through the triad's
AND-logic impeaches the ledger that also carries the collider
cluster and the Kaxion --- as explicitly as to what success
means. These commitments cost rhetorical safety and buy the only
currency a seventy-year-old subject still respects: a claim
whose author has arranged, in advance, to be unable to escape.

\section{The five roads, and the discrimination matrix}
\label{sec:five-roads}

The full comparison is Part~\ref{part:fiveroads}; its verdict
line bears repeating once in the Discussion's register. QFT ---
rigorous and silent: theorems at Depth~2, axioms at Depth~3. The
Standard Model --- precise and unexplaining: five silences and a
ten-order failure on the matter excess. String theory ---
geometric and unvalued: the right conceptual moves at landscape
valuation. $\Lambda$CDM --- exact and agnostic: the target set
to $0.3\%$, the origin outside its scope. \GGTheory{} --- valued,
dated, and mortal. Every concession the comparison owes is paid
in Part~\ref{part:fiveroads}, in the creditor's own section.

What the Discussion adds is the operational question the
comparison implies: \emph{if a rival framework, rather than the
geometry, owned the CP sector, which rows of this paper's
observable list would it reproduce --- and which measurement
separates it earliest?} Table~\ref{tab:p5-discrimination}
collects the verdicts. Two readings reward the stare. Reading
down the \GG{} column: the correlations are the fingerprint ---
frameworks can imitate entries (a Nelson--Barr model prices
$\bar\theta$; leptogenesis delivers an excess), but no rival
carries the \emph{joint} pattern, because the pattern is one
orientation read eight ways. Reading across the rows: the
Standard Model's column is honest silence (measured, never
priced); the supersymmetric column is worse than silence on the
EDM rows (generic new phases overshoot the electron EDM bound,
requiring their own tuning); and the landscape column is
``achievable'' everywhere and predictive nowhere, which is the
falsifiability asymmetry of Part~\ref{part:fiveroads} in tabular
form --- any cell fillable after the fact, priority of risk
manufacturable never.

\begin{table}[!t]
\centering\small
\setlength{\tabcolsep}{4pt}
\renewcommand{\arraystretch}{1.28}
\caption[Framework discrimination matrix for the CP sector]{%
Framework-by-observable discrimination matrix for the CP sector.
Each cell: does the framework generically reproduce the
observable? ($\checkmark$ = yes / forced; $\sim$ = achievable by
parameter or model choice; $\times$ = fails or forbidden; --- =
no statement.) Columns: SM = Standard Model (+ known neutrino
masses); NB = spontaneous-CP / Nelson--Barr-class models; SUSY =
supersymmetric extensions with generic new phases; GL = generic
hierarchical thermal leptogenesis atop the SM seesaw; Str.\ =
string landscape. The earliest separating measurement for each
column is footnoted. The joint pattern, not any row, is the
fingerprint.}
\label{tab:p5-discrimination}
\begin{tabular}{>{\raggedright\arraybackslash}p{5.6cm} c c c c c c}
\toprule
\textbf{Observable} & \textbf{\GG{}} & \textbf{SM} & \textbf{NB}
& \textbf{SUSY} & \textbf{GL} & \textbf{Str.}\\
\midrule
$\dCKM$ central value forced & $\checkmark$ & $\times$ & $\times$
& $\times$ & $\times$ & $\sim$\\
$\dPMNS$ central value forced & $\checkmark$ & $\times$ &
$\times$ & $\times$ & $\times$ & $\sim$\\
$\sin 2\beta$ derived (no fit) & $\checkmark$ & $\times$ &
$\times$ & $\times$ & $\times$ & $\sim$\\
Majorana pair $(0,\pi)$ predicted & $\checkmark$ & --- & --- &
--- & $\times$ & $\sim$\\
$\mbb$ point value ($1.7$~meV) & $\checkmark$ & --- & --- &
$\sim$ & $\times$ & $\sim$\\
Normal ordering required (no hedge) & $\checkmark$ & --- & --- &
$\sim$ & $\sim$ & $\sim$\\
Quiet CPV (no $5\sigma$ discovery) & $\checkmark$ & --- & --- &
$\sim$ & $\times$ & $\sim$\\
$\bar\theta$ solved without PQ scalar & $\checkmark$ & $\times$
& $\checkmark$ & $\times$ & $\times$ & $\sim$\\
Weak-phase/strong-angle relation & $\checkmark$ & $\times$ &
$\times$ & $\times$ & $\times$ & $\sim$\\
No $W_R$ / no P restoration, exact & $\checkmark$ & $\sim$ &
$\sim$ & $\times$ & $\sim$ & $\sim$\\
$\etaB$ with zero fitted parameters & $\checkmark$ & $\times$ &
$\times$ & $\times$ & $\times$ & $\sim$\\
Heavy sector at collider reach & $\checkmark$ & --- & $\sim$ &
$\sim$ & $\times$ & $\sim$\\
\bottomrule
\end{tabular}
\par\smallskip
{\small\raggedright \emph{Earliest separating measurement}: SM
--- any confirmed forced value (the lattice adjudication itself);
NB --- the leptonic lattice point and the $\etaB$ chain (NB
prices $\bar\theta$ but values no phase); SUSY --- the EDM
one-sided silences (generic phases overshoot; continued nulls
squeeze the parameter space this paper never had); GL --- the
$3.85$~TeV cluster and the quiet-CPV signature (its scale is
invisible, its phases unmeasured); string landscape ---
priority of risk (any cell fillable after the fact; see
Part~\ref{part:fiveroads}).\par}
\end{table}

\section{What is claimed, and in what regime}
\label{sec:claims-ledger}

The consolidated claims ledger.
\emph{Theorem grade (quasi-free):} T-P5-1 (table), T-P5-2 (CPT),
T-P5-3(i)--(iii) (holonomy calculus), T-P5-4 (door).
\emph{Derived-conditional:} T-P5-3(iv) and the phase values
($\dCKM$: QD step flagged; $\dPMNS$: assignment lemma;
$\bar\theta$: lemma L9); the zero-dilution clause (prompt-crossing
lemma). \emph{Named conjectures:} C-P5-A (gate operator; parity's
theorem), C-P5-C (resonance and normalization; baryogenesis'
theorem). \emph{Proposition with proof strategy:} P-P5-D
(handedness). \emph{D$^{*}$:} T-P5-5 ($\etaB$), order of
magnitude, factor-$\sim3$ band. \emph{Candidate, home
elsewhere:} T-Stat-A, the quartet's second child
(\S\ref{sec:quartet-second-child}) --- stated in this paper,
graded in the geometry paper's suite, not counted among this
paper's claims. \emph{Open and named:} the modular route to
interacting CPT; the projection's derivation from admissibility;
the heavy-sector CP projection coefficient; the $\Omega_b$
two-route seam. Nothing in the paper is claimed above this
table, and any reader who finds a sentence exceeding it is
invited to report the sentence.

\section{Predictions for falsification}
\label{sec:falsification}

A falsifier is a prediction plus a decision rule plus a price; a
prediction alone is only a hope with units. This section first
tells the three narratives the record already carries --- the
watched tension, the ordering wager, and the quiet signature ---
then walks the eight registered predictions individually, because
a table row (the summary remains Table~\ref{tab:falsifiers}, with
the schedule-side companion in Table~\ref{tab:exp-timeline})
cannot carry what a referee needs: for each claim, \emph{what} is
predicted, \emph{where} it was derived, \emph{how} it will be
measured, and \emph{what outcome kills it}.

\paragraph{The watched rows: $\sin 2\beta$, and the direct
$\gamma$.} The interferometry recomputation of the
unitarity-triangle angle, evaluated at the lattice phase
$13\pi/35$ with the PDG-class moduli ($s_{12} = 0.2250$,
$s_{23} = 0.0418$, $s_{13} = 0.0037$), returns $0.708$ against
the world's $0.709 \pm 0.011$: pull $-0.1\sigma$. The row is
registered as a watch with its evaluation layer stated ---
across the $s_{23}$ and phase-layer choices the recomputation
spans $0.707$--$0.727$, so the layer must be frozen for the
number to be auditable, and it is frozen here; a drift past
$3\sigma$ under Belle~II statistics would be the lattice's first
quark-sector crack, and the refresh rules guarantee the reader
will see it happen. The tense number is the direct $\gamma$: the
lattice triangle's $\gamma$ tracks the QD phase at leading
order, $\simeq 66.9^{\circ}$, against LHCb's direct combination
$(62.8 \pm 2.6)^{\circ}$ --- pull $+1.6\sigma$, printed under
the no-cosmetic-delay rule, watched, and headed into a
$\sim1^{\circ}$-precision decade that will sharpen it into a
kill-class test. The rows are small, but they demonstrate the
register the paper operates in: the confrontation table is a
living instrument, not a trophy case.

\paragraph{The ordering wager.} The geometry requires normal
ordering --- the mass triplet $(0.065, 8.59, 50.5)$~meV is
derived, not chosen --- while the current accelerator joint fits
mildly prefer inverted and the global fits and cosmology lean
normal. There is no hedge: under inverted ordering at $5\sigma$
(JUNO + atmospherics + DUNE, $\sim$2033) the paper retires, with
no fallback spectrum. The wager is currently favored by the
aggregate data ($\dPMNS$ at $+1.0\sigma$ under NO; $\approx
3\sigma$ disfavored under IO) but it is a wager, and the paper
prints it as one.

\paragraph{The quiet signature.} Registered once more in the
falsification section because it is two-sided in an unusual way:
the pentagon is confirmed \emph{either} by a measured $\delta$
crowding $196^{\circ}$ at growing precision \emph{or} --- the
same fact seen from the community's side --- by the persistent
non-arrival of the five-sigma CP-violation discovery the
long-baseline program was built to deliver. A framework rarely
gets to be supported by a null it predicted; the design
resolutions of Hyper-K and DUNE make the adjudication against
maximal CP violation ($73.6^{\circ}$ away) decisive by the
mid-2030s even though the distinction from exact conservation
($16.4^{\circ}$ away) matures only late in the decade.

\medskip

The eight registered predictions, walked individually.

\emph{P5-Pred-1 ($\dPMNS = 196.4^{\circ}$).} Derived in Part~VII
from the pentagon's central angle under the leptonic assignment
(QD $196.06^{\circ}$); measured by Hyper-K and DUNE appearance
asymmetries at $\pm15^{\circ}$ then $\pm7$--$10^{\circ}$; killed
by data settling at maximal $270^{\circ}$ ($5\sigma$, 2031--35)
or by any $5\sigma$ inconsistency with the lattice point ---
taking with it the leptonic lattice chain and the door's bias
(the same phase drives Part~VIII's transport).

\emph{P5-Pred-2 ($\dCKM = 13\pi/35$).} Derived as the pentagon's
branch phase with the QD completion; confronted continuously by
the LHCb/Belle~II global fits at $\pm0.026$~rad today; killed by
a converged world fit excluding $1.1669$ at $5\sigma$ --- with
the honest note that the raw $2\pi/5$ first order sits at
$+1.17\sigma$-class only with the $\pi/35$ theory width in
quadrature ($+4.2\sigma$ against experimental error alone) and
the QD step carries its own
flagged grade.

\emph{P5-Pred-3 ($\mbb = 1.72$~meV; Majorana nature).} Derived
from the door theorem and the parity pair $(0,\pi)$; approached
one-sidedly by the ton-scale $0\nu\beta\beta$ program
(LEGEND-1000, nEXO, CUPID at $5$--$20$~meV reach); killed by a
confirmed discovery anywhere in the reach --- a line at
$50$~meV is as fatal as the absence of one at $1.72$~meV would
be unprovable; the claim is designed to be executable only in
the kill direction this generation, and says so.

\emph{P5-Pred-4 (normal ordering).} Derived with the mass
triplet; measured by JUNO (running), atmospherics, DUNE; killed
at $5\sigma$ inverted ($\sim$2033). The one row where the
current data are genuinely mixed, and the paper's posture is the
wager above.

\emph{P5-Pred-5 (the lock cluster at $3.85$~TeV).} Inherited
from the parent ledger and load-bearing here (the heavy sector
of the transport); measured by HL-LHC $t\bar t$ reanalyses and
the dedicated program of the companion paper; killed by the
cluster's exclusion --- which retires Part~VIII's chain
regardless of every leptonic success.

\emph{P5-Pred-6 (no $W_R$, no mirror fermions --- F-P5-1).}
Derived from the projection's absence clause; tested by every
collider forever; killed by any confirmed right-handed current
or parity restoration at any scale --- permanent, one-sided,
never finally vindicated.

\emph{P5-Pred-7 (proton absolute stability).} Inherited from
the exact $\Uone{B}$ ledger and load-bearing for the frozen
excess; tested by every year of Super-Kamiokande, then Hyper-K
and DUNE; killed by one confirmed event, any day --- the row
that makes the matter chain mortal every morning.

\emph{P5-Pred-8 ($|\bar\theta| \lesssim 5\times10^{-18}$; no
EDM in the window).} Derived from the two-fate architecture and
the relaxation residual; tested by n2EDM and the eEDM
program; killed by a measured EDM above the residual but
below the old bound --- the specific window where the
architecture, not merely the number, fails.

\paragraph{The falsification lattice.} The eight are not
independent shots; they form a lattice whose nodes are the
paper's theorems, and the lattice is the real object under test.
A leptonic kill (Pred-1) breaks the same pentagon that carries
the quark audit (Pred-2's success becomes an accident); an
ordering kill (Pred-4) invalidates the mass triplet that feeds
$\mbb$ (Pred-3) and the transport's Yukawa matching; a proton
event (Pred-7) or a $W_R$ (Pred-6) breaks the descent
architecture underneath all of it; and the cluster (Pred-5) ties
the whole sector to the triad --- through P1-KK's collider
program and C1-Kaxion's haloscope window, one geometry is
exposed at three energy scales, and the AND-logic runs both
ways: any clean kill anywhere impeaches the shared ledger, and a
confirmation anywhere is collateral support for the rest.
Conversely, a discovery pattern that confirms several rows
\emph{jointly} --- ordering, phase, silence in the ton-scale
window --- is beyond the reach of any framework in
Table~\ref{tab:p5-discrimination}, because the correlations are
the geometry. The dated schedule for all of it is
Table~\ref{tab:exp-timeline} and
Figure~\ref{fig:p5-killtest-timeline}; the summary table and
figure below preserve this Part's own registration.

\begin{table}[h]
\centering
\caption[The falsifier table.]{The falsifier table. Each row names its instrument, its
date where one exists, and the specific structure its failure
destroys --- the register's requirement that mortality be
itemized. All sensitivity dates are conditional on stated
design assumptions (mass ordering known where required,
beam-power ramps, staging); the conditioning is the
collaborations', quoted from the cited design reports.}
\label{tab:falsifiers}
\small
\renewcommand{\arraystretch}{1.25}
\begin{tabular}{p{4.6cm} p{3.4cm} p{2.5cm} p{3.2cm}}
\toprule
\textbf{Prediction} & \textbf{Instrument} & \textbf{Timeline} &
\textbf{What failure kills} \\
\midrule
$\dPMNS = 196^{\circ}$ (band $\pm$ QD step) & DUNE, Hyper-K
($\pm15^{\circ}$) & early--mid 2030s & the leptonic lattice chain;
the door's bias \\
$\dCKM = 13\pi/35 \simeq 66.9^{\circ}$ & LHCb, Belle~II global
fits & continuous & the QD completion (the $2\pi/5$ first order
survives to $+1.2\sigma$-class tests only with the $\pi/35$
theory width) \\
$\mbb \simeq 1.7$~meV; Majorana nature & next-gen $0\nu\beta\beta$
& 2030s & the dial-free door as the matter portal \\
Lock cluster at $3.85$~TeV & HL-LHC & this decade & the heavy
sector of the chain (pre-declared criterion) \\
No $W_R$, no mirror fermions, at any scale (F-P5-1) & all future
colliders & permanent & the geometric projection of Part~VI \\
Proton absolutely stable & Hyper-K, DUNE & permanent & the exact
$\Uone{B}$ ledger (observation of decay kills Part~VIII's logic) \\
$|\bar\theta| \lesssim 5\times10^{-18}$ & neutron/proton EDM
programs & 2030s+ & (a measured EDM above this but below
$10^{-10}$ would break the two-fate architecture) \\
No fourth chiral generation & colliders, cosmology & permanent &
the $N = 3$ counting of the one surviving phase \\
Derived splittings $\Delta m^2_{21} = 7.37\times10^{-5}$,
$\Delta m^2_{31} = 2.55\times10^{-3}$~eV$^2$ & JUNO precision
era & $\sim$2031 & the derived mass triplet (a converged JUNO
value off the derived pair at $4\sigma$-class kills the mass
chain) \\
$\Delta B \in 3\mathbb{Z}$: no single-nucleon decay, no $n$--$\bar n$
oscillation (\S\ref{sec:deltab-three}) & Hyper-K;
$n$--$\bar n$ searches & 2030s & the whole baryonic logic --- a test
this framework can only fail, never pass \\
No anomalous free fall of antimatter
(\S\ref{ssec:three-questions}) & ALPHA-g, AE$\bar{\text{g}}$IS & running & Part~\ref{part:antiparticles} at its root: the handshake, the equal
masses, and the pairing \\
\bottomrule
\end{tabular}
\end{table}

\begin{figure}[t]
\centering
\resizebox{0.671\textwidth}{!}{%
\begin{tikzpicture}[font=\sffamily\footnotesize, >=Stealth]
 \draw[->, line width=1.10pt] (0,0) -- (12.2,0);
 \foreach \x/\yr in {0.5/2026, 2.5/2028, 4.5/2030, 6.5/2032,
   8.5/2034, 10.5/2036} {
   \draw (\x,-0.09) -- (\x,0.09);
   \node[below] at (\x,-0.12) {\yr};
 }
 % HL-LHC
 \draw[line width=1.10pt, blue] (0.5,0.65) -- (8.5,0.65);
 \node[blue, above, font=\footnotesize] at (3.2,0.7)
   {HL-LHC: $3.85$~TeV lock cluster};
 \fill[blue] (0.5,0.65) circle (1.5pt);
 % LHCb/Belle II
 \draw[line width=1.10pt, teal] (0.5,1.35) -- (10.5,1.35);
 \node[teal, above, font=\footnotesize] at (2.6,1.4)
   {LHCb/Belle~II: $\dCKM$ global fits};
 % DUNE/HK
 \draw[line width=1.10pt, red] (4.5,2.05) -- (9.5,2.05);
 \node[red, above, font=\footnotesize] at (5.9,2.1)
   {DUNE/Hyper-K: $\dPMNS$ to $\pm15^{\circ}$};
 \fill[red] (7.5,2.05) circle (1.8pt);
 \node[red, below, font=\footnotesize] at (7.7,1.98)
   {adjudication};
 % 0nubb
 \draw[line width=1.10pt, violet] (4.5,2.75) -- (11.5,2.75);
 \node[violet, above, font=\footnotesize] at (7.2,2.8)
   {tonne-scale $0\nu\beta\beta$: Majorana door};
 % permanent
 \draw[line width=1.10pt, gray] (0.5,-0.85) -- (11.5,-0.85);
 \node[gray, below, align=center, font=\footnotesize] at (6.0,-0.95)
  {permanent rail: proton stability $\cdot$ no $W_R$ (F-P5-1)
   $\cdot$ no fourth generation $\cdot$ CPT exact};
\end{tikzpicture}}
\caption[The falsifier timeline.]{The falsifier timeline. Colored bars are the dated
instruments of Table~\ref{tab:falsifiers}; the red dot marks the
expected leptonic-phase adjudication. The gray rail carries the
permanent prohibitions --- the statements that can die on any day
and can never be vindicated finally, which is what makes them the
strongest claims in the paper.}
\label{fig:timeline}
\end{figure}

\section{Open questions}
\label{sec:open-questions}

Honesty requires the complement, at the same depth as the
claims. Ten items remain open; for each we state why it is open
(not merely that it is), what has been tried, the closing
condition, and the impact of closure. Items (2), (3), and (6)
are commissioned campaigns of the parent program; the
remainder are stated obligations of this paper and its
companions.

\paragraph{1. The gate-operator lemma (C-P5-A).} \emph{Why
open}: the identification of the broken force with the gate
force is architectural --- exhibited, not forced; nothing yet
proves the transporting connection unique. \emph{Tried}: the
L11 uniqueness sketch (chiral, massive, off-diagonal, $\SUtwoL$
singled out). \emph{Closing condition}: a proof that any
descent-transporting connection has exactly these properties.
\emph{Impact}: parity's promotion from architecture to theorem;
P-P5-D closes with it (item 9).

\paragraph{2. The resonance mechanism (C-P5-C(i)).} \emph{Why
open}: the transport requires degeneracy at the level of the
heavy widths, and the twelve-order gap between the lock
splittings and the required $\delta M/M$ is quantified but not
bridged. \emph{Tried}: thermal-width estimates; radiative
degeneracy; lock substructure --- each stated with its
shortfall in Part~VIII. \emph{Closing condition}: the
finite-temperature computation of the heavy-sector spectral
function at the reheating scale --- the program's sharpest
outstanding physics item --- or a clean kill. \emph{Impact}:
$\etaB$ promoted from D$^{*}$ toward derived; failure retires
Part~VIII while leaving Parts~IV--VII standing.

\paragraph{3. The normalization (C-P5-C(ii)).} \emph{Why open}:
the throughput exponent $K/2$ is reconstructed from the ledger,
not derived from transport. \emph{Tried}: matching against the
washout benchmark; the reconstruction is stable but
post-diagnosed. \emph{Closing condition}: derive $e^{-K/2}$
from the Boltzmann hierarchy with inherited inputs.
\emph{Impact}: the last postulated number in the matter chain
becomes arithmetic.

\paragraph{4. Lemma L9 (determinant phase).} \emph{Why open}:
the cancellation of the quark-mass determinant phase against
the relaxed angle is argued sector by sector, not proved end to
end. \emph{Closing condition}: the end-to-end determinant
computation across the QD completion. \emph{Impact}:
$\bar\theta \lesssim 5\times10^{-18}$ promoted to
derived-unconditional; the two-fate architecture closes.

\paragraph{5. The heavy-sector CP projection.} \emph{Why open}:
the flavored computation on the ledger inputs is displayed (\ATen{}
Appendix~I) --- driver $\alpha_{31} = \pi$, nonzero at $m_1 = 0$,
$\JPMNS$-blind at the stated level --- and brackets the observed
throughput; what remains open is H-$\phi_{\mathrm{geo}}$-1, the
finite-temperature partial-resonance condition
$\Delta M/\Gamma^{\mathrm{eff}} \simeq 3\times10^{4}$
(\S\ref{sec:resonance-faced}), now carried by an identified
candidate mechanism --- the flavor-measurement regulator, closed
form $1/(35\,c_\tau y_\tau^{2})$ (\ATen{} Appendix~I.3) --- with
the transport closure of its conditions C1--C6 the open content.
\emph{Closing condition}: the
finite-temperature flavored transport computation at the lock
crossing, acceptance band $\pm O(1)$ in $\epsilon_{\mathrm{eff}}$.
\emph{Impact}: the $\epsilon^{\mathrm{res}}$ row of the Part~VIII
ledger moves from derived-conditional to derived; the forced-$\gamma$
worked value enters the tables.

\paragraph{6. The modular route.} \emph{Why open}: T-P5-2 is
quasi-free; the interacting theorem lives on scattering
hypotheses, while the Bisognano--Wichmann structure suggests a
hypothesis-free route. \emph{Tried}: the wedge-modular
identification is stated and sourced; the derivation of the
clock structure from Tomita--Takesaki flow is not done.
\emph{Closing condition}: derive the clock from modular flow at
interacting strength. \emph{Impact}: CPT --- and with it the
quartet's second child --- at full interacting grade; the
road-join of Part~X completed.

\paragraph{7. The projection's derivation.} \emph{Why open}:
the $\bZ_2$ and the matrix $P$ are a standard assignment,
honestly flagged as the parity ledger's asserted rows.
\emph{Closing condition}: force both from the admissibility
filters (POS-1/POS-3). \emph{Impact}: Part~VI's mechanism
upgraded from inherited to forced; PS-P5-3's conditionality
discharged.

\paragraph{8. The $\Omega_b$ seam.} \emph{Why open}: the
transport route and the corridor route to the baryon fraction
differ by $16.4\%$, outside the CMB's $0.3\%$, and the owner
is unidentified. \emph{Tried}: the discrepancy is flagged in
both this paper and the cosmology companion rather than
averaged away. \emph{Closing condition}: a well-posed internal
consistency test locating the error in the transport $O(1)$'s
or the corridor conversion. \emph{Impact}: either a corrected
$\etaB$ central value or a corrected corridor --- both
readable.

\paragraph{9. P-P5-D's proof.} \emph{Why open}: the handedness
invariant is a proposition with a strategy, pending item 1.
\emph{Closing condition}: C-P5-A plus T-P5-3 closes it.
\emph{Impact}: ``which hand survives'' anchored physically; the
shared residue of the parity and spin stories discharged.

\paragraph{10. The spin--statistics promotion path.} \emph{Why
open}: the quartet's second child is stated at candidate grade
(\S\ref{sec:quartet-second-child}); its bulk premises (the
frame-path lemma; the genericity clause) and its quasi-free
derivation are owed in the geometry and quantum-mechanics
companions. \emph{Closing condition}: L-Spin-1's
non-quotienting proof and T-Stat-A at quasi-free grade.
\emph{Impact}: the summit phrase's middle clause --- locked
spin--statistics --- promoted from candidate to theorem; the
exclusion principle becomes cohort anatomy.

None of the ten shields the verdict: the falsifiers of
\S\ref{sec:falsification} stand on the theorem-grade spine ---
the table, CPT, the door, the lattice points, the one-sided
silences --- alone. That is the design: the open questions
sharpen the program; they do not prop it up.

\section{The completed cohort and future prospects}
\label{sec:disc-cohort}

This paper is a member of the twenty-two-paper cohort --- the
Foundations Branch, at the \ANine{}--\ATen{} interface --- and its
place is load-bearing in both directions.
\emph{Imports:} the real oriented bulk, its signature, and the
Clifford intertwiner whose three jobs this paper exhibits
(\citealp{ArisakaA2GG6}); the anomaly inflow that fixes the
matter content (\citealp{ArisakaA3BI6}); the open-system
dictionary and runtime through which every boundary statement is
read (\citealp{ArisakaA5GDOS}; \citealp{ArisakaA6Gi4}); the
flavor geometry, the derived neutrino ledger, and the
basin/gate architecture of the particle parent
(\citealp{ArisakaA9SM}); the transport custody and cosmological
frame of the cosmology parent (\citealp{ArisakaA10Cosmos}); the
attractor uniqueness beneath token identity
(\citealp{ArisakaA11Origin}); the lock cluster and the Kaxion as
the triad's other two channels (\citealp{ArisakaP1KK};
\citealp{ArisakaC1Kaxion}); and the real-bulk holomorphy bridge
of the Foundations-Branch partner (\citealp{ArisakaP4Real}).
\emph{Exports:} the derived C/P/T table and the quartet, consumed
wherever the cohort performs a Euclidean computation; the
pentagon's valuation, feeding the parent's CKM/PMNS sector; the
door theorem, anchoring the cosmology parent's baryogenesis
custody; the second child's parentage, consumed by the geometry
paper's spin--statistics suite and the quantum-mechanics paper's
T-A8.12; and the registrations F-P5-1 and P-P5-D, queued for the
canonical falsifier ledger.

The near-term prospect is stated once more because it is the
paper's operational request to the experimental community:
\emph{publish the posteriors, not only the discovery statistics}
--- the $\delta$ posterior against the two pre-registered points
($196.4^{\circ}$ and $270^{\circ}$), the ordering Bayes factors
on a standing basis, and, when the ton-scale program reports,
the $\mbb$ posterior evaluated at $1.72$~meV --- so that the
lattice reading is tested as a point hypothesis rather than
absorbed into model-independent bands. The far-term prospects
are the promotion campaigns of \S\ref{sec:open-questions} and
the verdict window of Part~\ref{part:experiments}: by the
early 2040s, every row of Table~\ref{tab:falsifiers} will have
been visited by an instrument, and the paper will be either the
anatomy of the mirrors or a dated record of a beautiful error
--- it has arranged to be nothing in between.

\section{Summary of the Discussion}
\label{sec:discussion-summary}

The strengths, stated plainly: an anatomy that replaces seventy
years of convention with derived operations; a valuation that
prices both CP phases at lattice points, one of them already
audited to four digits in the quark sector; a matter chain with
zero fitted parameters landing $+20\%$ from Planck inside its published band; a two-fate resolution of the strong-CP double
standard with an eight-order one-sided silence; a discrimination
matrix no rival framework survives jointly; a confrontation
apparatus with standing refresh rules and pulls printed against
itself; and a falsifier lattice that is mortal on a printed
calendar, four separate ways. The weaknesses, stated with equal
plainness: the resonance conjecture C-P5-C carries the matter
chain's sharpest gap --- twelve orders in the degeneracy,
quantified but unbridged; the throughput exponent is
reconstructed, not derived; the parity projection's own
necessity is asserted, with its ledger saying so; the
$\Omega_b$ seam stands at $16.4\%$ with its owner unidentified;
the QD step under the quark phase carries a flagged post-hoc
grade; and the interacting-strength CPT theorem waits on the
modular route. The posture these imply is the one the paper has
kept throughout: anatomical where it can prove, priced where it
can derive, silent where it must wait --- and mortal
everywhere, on schedule.

In one breath each: the result is significant because both of
its endings are informative, which seventy years of
convention-laden discrete symmetries could never say; it is
unique in kind, with eight classic puzzles resolved or
dissolved and ten innovations registered for audit --- with
the four-endowment definition of time (Part~\ref{part:time})
standing beside them as the paper's furthest-reaching
structural claim; no
neighboring framework survives the twelve-observable
fingerprint jointly; the claims ledger states every grade and
invites the reader to police it; the falsifiers are priced,
dated, and walked one by one, with the watched tension printed
against the paper's own interest; the open questions sharpen
the program without propping it up; and the cohort carries
the verdict outward in both directions. The argument of the
paper is now complete. What follows is its ending in the older
sense of the word: conclusions, a personal accounting from
fifty years inside the mirrors, and the bridge to the question
--- why are we here? --- that a paper about the origin of
matter cannot end without facing.

\GGPart[Conclusions]{Conclusions --- Why Matter Won}{The oldest question in the mirror; what the paper has established, in plain English and in ten numbers; an answer we can check soon; the limitations; fifty years in the mirrors; the vision; the trailhead; from mirrors to mind; and what it would mean --- for physics, and for the universe that reads itself}
\label{part:conclusions}

\section[Part~XIII in plain English]{Part~XIII in plain English: why matter won}
The claim, stated plainly: the mirrors are not laws but orientations. The world is built from real oriented geometry --- a compact circle, space, and a one-way clock --- and C, P and T are the reversals of those three orientations. Each can break, because the geometry has texture. CPT cannot, because a total reversal is the one reflection no real texture can see.

What follows is the matter excess, at the grade the chain earns and at no higher grade, and an answer that can be checked soon rather than in principle. The limitations are stated without cushioning: the theorems are quasi-free with the interacting case conditional; the projection is asserted rather than derived; the resonance step carries a named conjecture; and the baryon-density bridge has a gap. Each of the four carries a named next task.

The Part then goes further than the physics --- a personal section, a longer view, and the question of what either verdict would mean --- and each of those is labeled as what it is rather than presented as a result. The last word is why the mirrors break, which is the question the paper opened with, answered in one sentence.

\section{The oldest question in the mirror}
\label{sec:oldest-question}

Begin with the plainest fact a person can be told about the
universe, because everything in this paper stands behind it. In
the first instants after the Big Bang, the world was almost
perfectly balanced between matter and antimatter --- for every
ten billion particles of one kind, ten billion of the other, plus
a remainder of about six. As the universe cooled, every matched
pair found its partner and vanished into light. The light is
still here: it is the cosmic microwave background, the most
precisely measured relic in cosmology, about two billion photons
today for every surviving proton. The pairs are gone.
\emph{Everything else --- every galaxy, every star, the iron in
your blood, the page or screen in front of you, and you --- is
the remainder.} Six parts in ten billion: the universe we live
in is, quite literally, a rounding error that survived an
annihilation.

That fact raises the oldest question in the mirror, and it has
two halves. The first half is \emph{why any remainder at all}:
the laws of physics, as written in every textbook, treat matter
and antimatter almost identically, and ``almost'' has never been
explained --- the Standard Model's own bookkeeping, run
honestly, leaves a universe some ten billion times emptier than
ours, a haze of light with nothing in it to wonder why. The
second half is older and stranger: physics possesses three
great mirrors --- exchange every particle with its antiparticle
(C), reflect the world left-to-right (P), run the film backwards
(T) --- and nature, astonishingly, is unfaithful to each of them
separately while remaining perfectly faithful to all three at
once. Since 1956 we have measured the infidelities with
exquisite precision. We have never, in seventy years, been able
to say what the mirrors \emph{are} --- only how they break.
This paper's subject is both halves of the question at once,
because they turn out to be the same question: the mirrors
break, and matter won, for one reason --- \emph{the world has a
direction} --- and the pages behind this conclusion have tried
to earn that sentence as geometry rather than poetry.

\section{What the paper has established}
\label{sec:established}

In plain English, the arc in one page. The world of the \GG{}
program is built from real structures: a circle that charge
winds around, a clock that causality orients, a space, and a
descent --- a one-way cascade down the family landscape that
wrote the masses. Before anything else, the twins: the circle
and the clock are two half-blind sources of one imaginary unit,
and they can share it in exactly one way --- positive charge
forward in phase, negative charge forward in phase. That forced
agreement is the antiparticle: not an extra ingredient, not a
product of the descent, but the consistency of the world with
itself --- the reason quantum mechanics is complex and the
reason every particle has a mirror twin are the same fact. The
operations C, P, and T are nothing but the
reversals of the circle, the space, and the clock; that is why T
alone is antiunitary (only the clock's direction is what $i$ is
made of), and why the total reversal CPT can never fail while
anything is built from real data --- reversing everything at once
is a relabeling. The mirrors break where the world's texture has
a direction. The flavor texture is a pentagon --- five is the
size of the unified block --- and a pentagon's angles cannot all
be flat: CP violation exists because $3\pi/5$ is not a multiple of
$\pi$, and its measured phases are the pentagon's forced angles,
$72^{\circ}$ less one lattice step in the quarks,
$180^{\circ}$ plus twice $\arcsin(1/7)$ in the leptons. The weak
force breaks parity because it is the cascade's gate force ---
the only broken, family-changing, short-range coupling --- and a
gate that executes a one-way flow must tell left from right. The
neutrino, the only particle with nowhere to wind, is the only one
that can be its own antiparticle: through that unique door, at the
TeV lock scale, the pentagon's phase biased the decays of the
heavy states, the boundary's one-way law kept the books from
rebalancing, and the sphalerons converted a lepton excess into
six baryons per ten billion photons --- the number the CMB reads,
reproduced here with no fitted parameter, one named conjecture,
and a published error band. Matter won because the universe is
oriented, and the broken mirrors were the first laboratory
evidence of the orientation.


\subsection{The paper in ten numbers}
\label{ssec:ten-numbers}

Stripped of narrative, the paper's content is ten numbers and
the machinery behind them --- each with its status, its proof
location, and the measurement that can kill it. The grades are
the register of \S\ref{sec:claims-ledger}; nothing in the table
is a fit.

Around the ten numbers stand the structures built to make them
decidable: the handshake that forces the twins before
quantization (Part~IV); the quartet that turns every Euclidean
computation into licensed geometry, with CPT and --- at
candidate grade --- the exclusion principle as its two children
(Parts~IV and \S\ref{sec:quartet-second-child}); the anatomy of
the three reversals (Part~V); the one-hand descent (Part~VI);
the pentagon (Part~VII); the door and the tripod (Part~VIII);
the five roads compared at full depth (Part~X); and the
confrontation apparatus --- refreshed world numbers, deviation
panels, dated timeline --- that keeps the whole account
auditable (Part~XI). None of it waits on future theory; all of
it waits on data.

\begin{table}[!t]
\centering
\caption[The paper in ten numbers.]{\textbf{The paper in ten
numbers.} Status grades follow the claims ledger
(\S\ref{sec:claims-ledger}); ``killed by'' names the measurement
that retires the row (Table~\ref{tab:falsifiers} carries the
full falsifier register). Ten rows, one number each, with the value, the grade it is claimed at, the Part that proves it, and the measurement that would retire it. The last column is the point of the table: every row names something an experiment could do that would end that row, and the dates in it are close enough to matter.}
\label{tab:p5-ten-numbers}
\small
\setlength{\tabcolsep}{4pt}
\renewcommand{\arraystretch}{1.25}
\begin{tabular}{cp{3.0cm}p{2.6cm}p{2.6cm}lp{3.0cm}}
\toprule
& \textbf{Quantity} & \textbf{Value} & \textbf{Status} &
\textbf{Proved} & \textbf{Killed by} \\
\midrule
1 & quark CP phase $\dCKM$ & $13\pi/35 = 66.9^{\circ}$ &
derived-cond. & Part~VII & world fit off at $5\sigma$ \\
2 & lepton CP phase $\dPMNS$ & $196.4^{\circ}$ (QD
$196.06^{\circ}$) & derived-cond. & Part~VII & $270^{\circ}$ at
$5\sigma$ (2031--35) \\
3 & $\sin 2\beta$ & $0.708$ & derived-cond. & Part~VII &
$>3\sigma$ drift (watched) \\
4 & Majorana pair $(\alpha_{21},\alpha_{31})$ & $(0,\pi)$ &
derived & Part~VII & $0\nu\beta\beta$ phase reading \\
5 & double-beta mass $\mbb$ & $1.72$~meV & derived & Part~VII &
any ton-scale discovery \\
6 & neutrino masses & $0.065$, $8.59$, $50.5$~meV; normal &
derived (parent) & Part~VII & inverted at $5\sigma$
($\sim$2033) \\
7 & strong-CP residual $\bar\theta$ & $\lesssim 5\times10^{-18}$
& derived-cond.\ (L9) & Part~VII & EDM in the window \\
8 & baryon excess $\etaB$ & $\approx7\times10^{-10}$
($\times3^{\pm1}$) & D$^{*}$ (C-P5-C) & Part~VIII & chain kills;
one proton event \\
9 & heavy-sector scale & $3.85$~TeV & inherited ledger &
Part~VIII & HL-LHC exclusion \\
10 & mirror sector $W_R$ & none, at any scale & theorem
(absence) & Part~VI & any $W_R$, ever \\
\bottomrule
\end{tabular}
\end{table}


\section{An answer we can check --- soon}
\label{sec:check-soon}

What lifts this above a beautiful story is that it can be
tested now, decisively, and from several directions at once,
on a timescale of years. The instruments are not proposals;
they are running or funded. JUNO is taking data beneath southern
China today; Hyper-Kamiokande's megaton-class tank sees beam in
2028; DUNE's beam crosses South Dakota around 2031; together
they will settle, by about 2033, whether the neutrino masses are
ordered the way this paper requires --- a question on which the
paper has no fallback. Through the 2030s the same instruments
sharpen the lepton phase toward the pentagon's point at
$196.4^{\circ}$, and the ton-scale double-beta detectors listen
for a signal this paper says they will \emph{not} hear, because
its predicted value sits far beneath their reach --- a silence
registered in advance. The permanent rails run underneath:
every year Super-Kamiokande's protons decline to decay is a
year this paper survives; any confirmed right-handed current,
any electric dipole moment in the forbidden window, any
fourth generation, would end it on any day.

And one prediction deserves plain words because it inverts what
the field hopes for. The long-baseline program was built, in
large part, to discover leptonic CP violation at five sigma.
If this paper is right, \emph{that discovery will not come}:
the lattice value implies CP violation so gentle --- the phase
just $16.4^{\circ}$ from the conserving point --- that the
flagship exclusion will keep failing to arrive, while the
measured phase crowds the predicted point from below. A theory
rarely gets to be confirmed by a disappointment; this one has
arranged exactly that, in print, with the alternative ---
$270^{\circ}$, the community's favorite --- registered as its
own execution. The test cuts both ways, cleanly, and we say so
without hedging: a clean kill would be a real and valuable
answer, not a defeat. It is rare in fundamental physics to meet
a question this old with a decade this decisive. That is the
opportunity, and it is, honestly, a thrilling one.

\section{The objective limitations and the next technical tasks}
\label{sec:limitations}

The paper's floor, stated as plainly as its ceiling. The theorems
of Parts~IV and~V hold at the quasi-free grade; interacting extensions
ride on standard scattering hypotheses, with the modular route the
named path to full strength. Parity's argument is an architecture
awaiting its lemma. The baryogenesis chain's resonance condition
asks for twelve orders of magnitude that vacuum numbers do not
supply, and the thermal-width route that plausibly supplies them
is a computation, not yet a result. The QD completion of the quark
phase is flagged post-hoc by the parent program itself. Two
routes to $\Omega_b$ disagree by $16.4\%$. These are the next
technical tasks, they are all well-posed, and none of them can be
finessed: each either lands or kills its dependent claims, which
is what it means for a framework to run on falsifiable credit.

\section{A personal perspective: fifty years in the mirrors}
\label{sec:personal}

The author asks the reader's indulgence for one section in the
first person; the experiments are facts, and the connection is
motivation, not authority.

I met the broken mirrors as a student in the 1970s, when parity
violation was already textbook and CP violation was still a
two-per-thousand mystery confined to one particle. It seemed to me
then --- with the certainty only a student can afford --- that
these were the deepest facts in physics, because they were the
only places where nature, rather than our descriptions of it,
chose a direction. Fifty years later I believe it still, with the
certainty now resting on somewhat more evidence.

I spent 1980 to 1985 on the design and construction of the
Kamiokande detector. We built it to watch protons die; they did
not, and that null --- one of the most productive nulls in the
history of physics --- turned the instrument toward the sky and
the sun. I was fortunate to be present at the founding of what
became neutrino astronomy, and I have watched the lineage of that
tank --- Kamiokande, Super-Kamiokande, and now Hyper-Kamiokande
--- carry the discrete-symmetry program into the lepton sector:
oscillations in 1998, $\theta_{13}$ within my colleagues'
lifetimes and mine, and now the pursuit of $\dPMNS$, the one
number this paper most wants measured. There is a peculiar
symmetry in the fact that the detector we built to watch baryon
number fail may end by explaining why baryon number won.

From 1987 to 2007 I lived inside CP violation: KTeV at Fermilab,
where the double ratio finally yielded
$\mathrm{Re}(\varepsilon'/\varepsilon)$ and direct CP violation
became a measured fact of the amplitudes; then BaBar at SLAC,
where $\sin 2\beta$ closed the unitarity triangle and the
Kobayashi--Maskawa phase became the confirmed bookkeeping of every
asymmetry we could reach. I am proud of those measurements in the
way an experimentalist is: they are permanent. And I left that
double decade with an unease I could not then articulate. We had
measured the phase to three digits and could not say why it was
not zero. We had confirmed a parameterization, brilliantly, and
explained nothing. The question I carried out of the control rooms
was the one this paper finally attempts: \emph{what are these
symmetries, that their breaking should be so precise, so
selective, and so consequential?}

The answer defended here --- that the mirrors are orientations of
a real geometry, that their breakage is the orientation of a
descent, and that the matter of the universe is that orientation's
receipt --- is one I would have distrusted in 1987, and rightly:
it would then have been words. What has changed my own weighing is
the arithmetic: the pentagon that sets the Cabibbo magnitude to
five digits sets the CP phase to a lattice point within one sigma
of the world fit; the same integer that sets the lock scale sets
the baryon excess to four percent of the CMB value; and every one
of these statements can be killed by an instrument now under
construction, several of them descendants of instruments I helped
build. Fifty years is long enough to learn the difference between
a story and a ledger. This paper is a ledger, kept as honestly as
I and my collaborator on these pages know how, and I submit it to
the community that trained me with the register it taught me:
here is what is derived, here is what is owed, and here are the
dates on which you may collect.

\section{The vision: reading the mirrors as geometry}
\label{sec:vision}

The Foundations-Branch partner closed with a vision of a century in
which number systems are read as geometry. This paper's vision
is its sibling: a century in which the \emph{symmetries} are
read the same way. On this reading the discrete-symmetry
experiments of 1956--2035 --- from Wu's cryostat to the megaton
water of Hyper-Kamiokande --- compose a single eighty-year
measurement: of the orientations of the world's real factors,
of the chirality of its flavor texture, and of the direction
of its descent. The falling of the mirrors was never a
catalog of accidents; it was geometry, reporting.

Suppose the decade goes the geometry's way --- not as advocacy,
but as the planning exercise every field runs on. Then the
texture of discrete-symmetry physics changes the way collider
physics would change with the cluster: the question at the
frontier stops being ``which operators might break which
symmetry at which scale?'' --- a question that has admitted too
many answers for fifty years --- and becomes ``what does the
orientation say, and to how many digits can we check it?'' The
CP phases stop being free parameters and become the first
entries of a \emph{spectroscopy of orientations}: the pentagon
read at the $B$ factories and the long baselines the way the
Balmer series was read in glass tubes, the Majorana pair read
in germanium and xenon, the one-sided silences standing like
selection rules. Spectroscopy is meant as the highest
compliment: it is the mode in which physics has been most
honest, most cumulative, and most kind to its young people,
because a student with a good idea and a data set can be
\emph{right} in a way no committee can argue with. My deepest
professional hope for this paper is not that it is right; it is
that it makes that kind of physics imaginable again to a
twenty-five-year-old with fifty years of mirrors ahead of her.

\section{The trailhead, not the summit}
\label{sec:trailhead}

If the decade confirms the pattern --- the ordering, the phase
crowding its point, the silences holding --- the reward will be
extraordinary: not one more measured parameter, but evidence
that the discrete symmetries, the flavor phases, and the
matter of the universe are readings of a single oriented
geometry. And yet confirmation would be a beginning, not an
end. The promotion campaigns of \S\ref{sec:open-questions}
--- the gate lemma, the resonance computation, the modular
route, the spin--statistics closure --- would lie ahead the way
the hydrogen atom's fine structure lay ahead of Balmer; the
pentagon's higher harmonics, the door's collider face, and the
quartet's interacting-strength children are the spectroscopy of
Part~XIII's century, mapped across instruments not yet named.
This paper is the trailhead of that program, not its summit.

If instead the decade kills it --- an inverted ordering, a
maximal phase, a line in the double-beta window, a proton that
dies --- the present implementation will have failed one of the
cleanest tests a symmetry framework has ever offered: publicly,
permanently, and without a zombie phase, because there is no
parameter with which to retreat. Either way, the decade is
worth having. That is the mark of a good falsifier; it was the
missing property of the seventy years this paper's history Part
recounted; and it is the property this paper was written to
restore to the mirrors.

\section{From mirrors to mind: the oriented emergence}
\label{sec:mirrors-to-mind}

This paper's story does not end at the baryon number, and the
remainder deserves to be said carefully --- with the registers attached, because a beautiful
story told carelessly is only a story --- what the oriented
geometry actually underwrites.

Begin at the beginning of this paper and walk forward once.
Because the world has two real sources of one imaginary unit,
it has twins (derived). Because it has three real orientations,
it has mirrors, and one unbreakable product of them (derived).
Because the descent between the family basins is oriented, one
force tells left from right (conditional, one lemma), and the
holonomy written along that descent makes the flavor pentagon
chiral (derived through its stated conditionals): the twins are
not treated equally. Because the neutrino alone has nowhere to
wind, there is one lawful door between the twin ledgers
(derived); because the door's traffic was biased by the
pentagon's phase and the boundary's law is a one-way semigroup,
the universe kept six parts in ten billion of itself
(D$^{*}$, one named conjecture); and because the excess
survived a prompt descent undiluted, there were protons ---
then nuclei, then, six hundred million years on, the first
chemistry that could hold a record of its own past.

From there the cohort's summit paper carries the chain, and
this paper adopts its grades without inflation: an oriented
chemistry (homochirality --- the biological left-handedness
whose descent from the microphysical orientation is conjectured,
not derived; the summit's own register), an oriented
thermodynamics (records, metabolism, the boundary's forward
semigroup written into every living membrane), and at last an
oriented \emph{experience} --- memory before anticipation, the
felt asymmetry of time that is the OGGEP projection's shadow in
a nervous system \citep{ArisakaA11Origin}. None of the last
three links is this paper's theorem, and the chain from CP
violation to the reader is conjectured beyond its matter link.
But the matter link --- the one this paper audits --- is now
the best-secured in the chain, and the shape of the whole is no
longer mystical: \emph{every rung is the same fact}. The
mirrors break, the pentagon turns, matter survives, chemistry
selects a hand, life runs forward, and mind remembers backward
--- one orientation, cascading through eleven orders of
magnitude of scale, from the fiber circle to the reader's
certainty that yesterday is fixed and tomorrow is not.

That is the beautiful story, and its beauty is of the lawful
kind: at every rung it could have been killed, and at the rungs
we can measure, it has not been. The universe is not symmetric.
It is oriented --- and orientation, unlike symmetry, can make
things: particles, excesses, hands, memories, questioners. The
discrete symmetries were never a specialists' bookkeeping. They
were the first instruments sensitive enough to read the
direction of the world.

\section{Why are we here: the bridge to the summit}
\label{sec:why-here}

The cohort's summit paper asks the question plainly --- why are we
here? --- and builds its answer as a cascade: constitution to
bulk, bulk to boundary, boundary to chemistry to life
\citep{ArisakaA11Origin}. This paper supplies the cascade's
pivotal middle links, and grades them: because the descent is
oriented, parity breaks (conditional, one lemma); because the
texture is pentagonal, CP breaks by a forced amount (derived
through its stated conditionals); because the neutrino alone has
nowhere to wind, there is a door (derived); because the door's
traffic was biased and the boundary's law is one-way, there is
matter (D$^{*}$, one conjecture); and because there is matter,
there is chemistry, and the question can be asked. The chain from
CP violation to the reader's existence is conjectured beyond its
matter link and graded honestly by the summit paper itself; what
this paper contributes is that its links are now the
best-audited in the chain. We are here, on this account, because
the universe cannot be reversed --- and the mirrors were how it
told us.

\section{What it would mean: orientation, matter, and the
universe that reads itself}
\label{sec:concl-philosophy}

A paper that asks a decade of the world's instruments owes the
reader more than a schedule; it owes an account of what the
answer would \emph{mean}. This closing section attempts that
account in four movements, in plain English, at the depth the
question deserves --- and the last two movements walk
deliberately into the territory of the cohort's summit paper
\citep{ArisakaA11Origin}, with the grades kept visible even
here, because a philosophical remark that hides its register is
only rhetoric.

\paragraph{First movement: on falsifiability as an ethic.} The
mirrors have a special claim on scientific honesty, because
they are where physics learned humility twice: parity fell when
almost nobody believed it could, and CP fell in an experiment
designed to confirm its conservation. The tradition this paper
inherits from those shocks is not cleverness but conduct: state
the number before the measurement; price the kill; forbid
yourself the retreat. Popper made falsifiability a criterion;
seventy years of practice have shown that as a criterion it is
too easy to satisfy in the letter and evade in the spirit ---
every scanned model is ``falsifiable'' at each point and
immortal as a family. What this paper has tried to practice is
falsifiability as an \emph{ethic}: the deliberate construction
of a position from which one's future self cannot escape. The
lattice points are load-bearing across twenty-two documents so
that no quiet revision is possible; the ordering carries no
fallback; the silences can execute on any morning. There is a
moral structure here older than science --- it is the structure
of a promise --- and if the pentagon is wrong, I hope the
\emph{form} of the wager outlives the wager itself.

\paragraph{Second movement: on symmetry, orientation, and what
a world can say about itself.} Twentieth-century physics was
written in the language of symmetry, and its habit of mind was
to treat every asymmetry as a flaw awaiting explanation --- a
broken vase to be glued. The account defended here inverts the
aesthetic. A perfectly symmetric world is not a perfected
world; it is a \emph{mute} one --- it can keep no records,
prefer no direction, accumulate no remainder, and support no
one. Symmetries are the statements a world cannot make about
itself; its breakings are its autobiography. Orientation,
unlike symmetry, can \emph{make} things: twins, excesses,
hands, memories. On this reading the discrete-symmetry
violations are not blemishes on a beautiful formalism but the
first sentences of the universe's self-description --- and the
seventy-year experimental record of the mirrors is the
transcription, at first bewildered and then increasingly
precise, of a world telling us, in the only language available
to it, that it has a direction. The physics claim of this paper
is that the sentences parse: three reversals, one pentagon, one
door, one remainder. The philosophical claim is smaller and
stranger: that we should have expected the universe to be
legible \emph{here} first, because an oriented world's
asymmetries are precisely the part of it that carries
information.

\paragraph{Third movement: into the summit --- the
self-referential closure.} The summit paper of the cohort
\citep{ArisakaA11Origin} asks the question this paper's title
implies and answers it as a cascade: constitution to bulk, bulk
to boundary, boundary to chemistry, chemistry to life, life to
mind --- with the grades printed at every rung, and the rungs
beyond matter conjectured, not derived
(\S\ref{sec:mirrors-to-mind}). Stand back from the cascade once
and say what its shape would mean if it holds. The bulk
projects a boundary; the boundary, because it is oriented,
keeps a remainder; the remainder assembles --- through the
chemistry the remainder alone made possible --- into
structures that record, then metabolize, then remember, then
ask. \emph{The universe's leftover matter has spent thirteen
billion years arranging itself into instruments that measure
the asymmetry that saved it.} Every kaon beam, every $B$
factory, every tank of water under a mountain is, on this
account, the six-in-ten-billion looking back at its own
origin; the reader following this sentence is the remainder,
running on the boundary's one-way clock, using a nervous system
whose forward flow is the same semigroup that froze the excess.
This is what the summit paper calls the self-referential
closure: observation is not something added to nature from
outside --- it is something nature does, the boundary reading
the bulk that projects it --- and a physics that derives the
mirrors is therefore not a description of the universe from
nowhere but an event \emph{in} the universe's own
self-description. The old anthropic shrug --- ``we observe an
asymmetric world because observers require one'' --- gets
stood on its feet: an oriented world does not merely
\emph{permit} witnesses; it is the kind of world that cannot
help making them, because orientation is what records,
remainders, and rememberers are made of. I am aware how large
these sentences are. They are printed here, rather than in a
philosophy journal, because --- to the author's knowledge, for
the first time in seventy years of mirror physics --- sentences
of this size have measurement dates attached to them.

\paragraph{Fourth movement: on either verdict, and the grace
to listen.} Beneath all of it is the question every physicist
carries privately: is nature legible --- readable in advance,
by finite minds, down to its choices? The mirrors are where
that question is oldest, because they are where nature's
choices are starkest: a hand, a phase, a remainder. If the
decade goes the geometry's way, the answer becomes \emph{yes,
one layer deeper than we had any right to expect} --- the
world's oldest asymmetries a text, the instruments its
readers, and the readers themselves made of the text's own
remainder: legibility closing its loop. If the decade kills
the pentagon, the answer is subtler but not empty: we will
have learned that this reading fails, cleanly, at a stated
cost, with the manner of failure itself a contribution --- the
first zero-parameter account of the matter of the universe to
be retired by measurement rather than by fashion. Legibility
has two requirements: a world with structure, and readers with
the discipline to be corrected by it. Only the first is
nature's responsibility. The second is ours, and it is the one
thing this paper could guarantee. The phase is the bet; the
decade is the table; the first cards are already face up ---
an ordering leaning our way, a phase one sigma from the point,
a proton that has not died. Let the data speak --- and let us
have the grace to listen, whichever way they answer.

\section{The last word: why the mirrors break}
\label{sec:last-word}

And beneath the mirrors, the clock. This paper set out to
explain three broken symmetries and found, on the way, that it
had to say what time is: the one direction that orders,
repeats, opens, and shelters (Part~\ref{part:time}). The two
answers are one architecture --- the mirrors broke on the
descent's orientation, and the clock stands on time's four
endowments; the same dictionary whose one-way reading ages the
universe is the one that recorded the kaon's confession in
1964.

Symmetries are the statements a world cannot make about itself;
their breakings are its autobiography. A perfectly mirror-blind
universe would have annihilated to light and said nothing. Ours
kept a pentagon where its flavors turn, a single chiral gate
where its families descend, one particle that is its own
reflection, and a ledger of six extra baryons in every ten billion
--- and out of that spare autobiography came stars, chemistry, and
readers. The mirrors break because the world has a direction. That
the direction can be measured --- in a cryostat, in a kaon beam,
in a tank of water under a mountain --- and now, perhaps, derived,
is the whole of what this paper has tried to say.

\medskip
\noindent\emph{Part-end summary.} Established, bounded, personal,
and pointed forward: the oldest question answered at a stated
grade, the limitations owned, fifty years in the mirrors
accounted for, and the trailhead marked. The argument is over;
what remains is service. The remaining Parts are the reader's
tools: the courses of the appendix machine room, the eighty-two
objections, and the annotated references.

\GGAcknowledgments

\GGPart[Technical appendices]{Technical Appendices}{A graduate machine room in three blocks --- Block I, the reference dictionaries and the cross-cohort naming convention; Block II, from-scratch, Step-graded derivation chapters on the physics of the main text, opening with the discrete symmetries in quantum mechanics and quantum field theory, then the flavor sector with its CP violation, the origin of the matter excess, the antiparticle, and the neutrino's century; and Block III, the ledger-and-registry appendices --- the source ledger, the inheritance table, the benchmark audit, the named-statement deck, the promotion roadmap, and the falsification catalog --- closing with the annotated reading guide --- written for the incoming graduate student, with every derivation carried from the textbooks and the chapters' problems followed by complete worked solutions}
\label{part:appendices}

The main text argued; this Part teaches. It is a machine room in
the cohort's appendix standard, and it is organized in three
blocks. \emph{Block I} is furniture: the abbreviations, the symbol
glossary, and the naming convention by which the twenty-two papers
cite one another. \emph{Block II} is the curriculum, written for
the incoming graduate student --- especially the student joining a
neutrino experiment or beginning flavor phenomenology --- on one
fixed principle, the same principle that governs the appendices of
the two parent papers: \emph{Step 1} builds the standard physics
of a main-text topic from scratch, at textbook depth and with the
textbooks cited; \emph{Step 2} states precisely why quantum
mechanics, quantum field theory, and the Standard Model cannot
derive or predict the topic's key quantities; \emph{Step 3} shows
how the geometry derives, solves, and predicts them, at ledger
grade. The opening chapter (Appendix~\ref{app:discrete-qft})
treats the discrete symmetries themselves; the second
(Appendix~\ref{app:flavour}) builds the flavor sector and its CP
violation from the Yukawa couplings to the pentagon; the third
(Appendix~\ref{app:leptogenesis}) walks the matter excess from
Sakharov's conditions to the assembled $\etaB$; the fourth
(Appendix~\ref{app:antiparticle}) explains the antiparticle four
ways, ending with the one that is a theorem; the fifth
(Appendix~\ref{app:neutrino}) walks the neutrino's century from
two-state quantum mechanics to the pentagon's lattice point,
written for the incoming long-baseline student --- ninety
problems across the block, all followed by complete worked
solutions. \emph{Block III} records the registries --- the
source ledger, the inheritance table, the benchmark audit, the
named-statement deck, the promotion roadmap, and the
falsification catalog --- and closes with the annotated
reading guide. Nothing in the main text is needed to start; everything in
Parts~IV--XI can be recomputed by the reader who finishes.

Throughout these appendices we adhere to two stylistic
commitments, shared with the parent papers. First, every
derivation is presented step by step with no skipped intermediate
manipulations: a graduate student should be able to reproduce
each calculation with paper and pencil, and every problem is
followed by a complete worked solution, never a bare answer.
Second, every formal step is accompanied by a \emph{physics
intuition} note that translates the algebra into a mental
picture --- the beat of two clocks, the meshing of two gears,
the currency market of the hot plasma. The combination is
designed to make the material accessible to the working
experimentalist while preserving the rigor of the main text,
so that the Part can be used as a one-semester reading course,
one chapter per week, each self-contained.

\paragraph{Prerequisites and recommended textbooks.} The course
assumes advanced undergraduate quantum mechanics and a first
exposure to quantum field theory (the discrete symmetries are
rebuilt from Wigner's theorem upward in
Appendix~\ref{app:discrete-qft}, so a student meeting the
subject for the first time can begin there). It is meant to be
read \emph{alongside} the canonical texts, not in place of
them: the recommended way to use each chapter is to read the
matching textbook chapter first, return to the Step-1 material
here for a compact but complete recap, and only then proceed to
Steps~2 and~3, where the standard result is re-examined and
re-derived at its ledger grade. For self-study we particularly
recommend the following, each cited again in the chapter to
which it is best matched:
\begin{itemize}[leftmargin=2em, itemsep=2pt]
\item \textbf{Foundations.} \citet{Sakurai2017}, \emph{Modern
Quantum Mechanics}, Ch.~4, and \citet{WeinbergQTF},
\emph{The Quantum Theory of Fields} I, Chs.~2 and~5 --- the
(anti)unitary implementation of symmetries and the causal-field
construction behind Appendices~\ref{app:discrete-qft}
and~\ref{app:antiparticle}; \citet{PeskinSchroeder1995} for the
propagator and loop machinery.
\item \textbf{Discrete symmetries and flavor.}
\citet{Sozzi2008}, \emph{Discrete Symmetries and CP Violation}
--- the operation-centric experimental companion of the whole
Part; \citet{BigiSanda2009}, \emph{CP Violation} --- the
flavor-centric classic; \citet{BrancoLavouraSilva},
\emph{CP Violation} --- the rephasing-invariant calculus of
Appendix~\ref{app:flavour}, whose Ch.~22 untangles every phase
convention.
\item \textbf{Neutrinos.} \citet{GiuntiKim2007},
\emph{Fundamentals of Neutrino Physics and Astrophysics} ---
the standard reference beneath Appendix~\ref{app:neutrino};
\citet{Zuber2020}, \emph{Neutrino Physics} --- the
experimentalist's companion; \citet{MohapatraPal2004},
\emph{Massive Neutrinos} --- the seesaw's model space.
\item \textbf{The early universe.} \citet{KolbTurner},
\emph{The Early Universe} --- the thermal history and relic
machinery of Appendix~\ref{app:leptogenesis};
\citet{BuchmullerDiBariPlumacher} --- the pedagogical
leptogenesis review the transport follows.
\item \textbf{Mathematical background.}
\citet{LawsonMichelsohn}, \emph{Spin Geometry} --- the
Clifford-algebra reality classification behind the mod-8 clock
of \S\ref{app:anti-algebra}; \citet{StreaterWightman} for the
axiomatic CPT theorem.
\end{itemize}
\noindent Throughout, the symbols \AOne{}--\AEleven{},
\PaperPOne{}--\PaperPThree{}, and
\PaperCOne{}--\PaperCThree{} refer to member papers of the
twenty-two-paper \GGTheory{} cohort, and \PaperPFour{} to this
paper's Foundations-Branch partner, to which the Step-3 material
repeatedly points.

\appendix

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

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{}, \BIsix{}, \OGNfive{}, \GiFour{}, \QiFour{}, \GRIP{},
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 (CPT, CKM, PMNS, EDM, etc.)
are rendered in plain uppercase Roman, following the convention
of the major physics journals. One collision is policed
throughout: CPT is the combined
charge-conjugation--parity--time-reversal operation, never the
completely-positive trace-preserving channel property, which is
always written CPTP. The third column gives the \emph{primary
link}: the paper in the Arisaka-GG series (and the Part, where
useful) in which each name is introduced in detail.

{\small
\begin{longtable}{@{}>{\raggedright\arraybackslash}p{0.18\linewidth} >{\raggedright\arraybackslash}p{0.54\linewidth} >{\raggedright\arraybackslash}p{0.24\linewidth}@{}}
\captionsetup{list=no}%% v15.0: see the front-matter note
\caption[Summary of abbreviations.]{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 list is grouped by layer, from the framework's own objects through the ledger and the mirror operations to the standard physics abbreviations, and the right column links each entry to the place where it is defined, so that no abbreviation in the paper has to be guessed at from context.}
\label{tab:appA-abbrev-p5}\\
\toprule
\textbf{Symbol} & \textbf{Meaning} & \textbf{Primary link}\\
\midrule
\endfirsthead
\multicolumn{3}{c}{\itshape Table~\ref{tab:appA-abbrev-p5} continued}\\
\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 (1) Framework name and core program objects}\\
\midrule
\GGTheory{} & Grand Geometric Theory (also: Geometry--Gauge Equivalence Theory) & \AOne{}, Preface\\
\GGConstitution{} & Constitutional charter: 4 axioms (\AX{1}--\AX{4}) + 7 postulates (\POS{1}--\POS{7}) & \AOne{}\\
\GG{} & Geometry--Gauge locking in six dimensions: the 6D bulk & \ATwo{}\\
\GDOS{} & Geometric Dynamics of Open Systems: the boundary law from \OGGEP{} & \AFive{}\\
\BIsix{} & Green--Schwarz-modified Bianchi identity at 6D, $dH_3 = X_4$ & \AThree{}\\
\OGNfive{} & Occam-Gauge-Normalization 5-integer ledger $(M, N, L; J, K) = (1, 3, 5; 7, 35)$ & \AThree{}; \ATwo{}\\
\GiFour{} & Geometric-Invariance four-pillar structure (\TGG{}, \CCI{}, \GFF{}, \GRIP{}) & \ASix{}\\
\QiFour{} & Quantum-Information four-pillar structure (\QiQUA{}, \QiEP{}, \QiCON{}, \QiHAL{}) & \ASeven{}\\
\midrule

\multicolumn{3}{l}{\itshape (2) Four axioms (\AX{1}--\AX{4})}\\
\midrule
\AXkw{} & Axiom (axiomatic primitive; non-negotiable) & \AOne{}\\
\LC{} & \AX{1} Locality and Causality & \AOne{}\\
\GGEP{} & \AX{2} Geometry--Gauge Equivalence Principle & \AOne{}\\
\OGGEP{} & \AX{3} Operational/Open Geometry--Gauge Equivalence Principle & \AOne{}\\
\MSMF{} & \AX{4} Maximum Symmetry, Minimum Freedom (no hidden continuous knobs) & \AOne{}\\
\midrule

\multicolumn{3}{l}{\itshape (3) Seven postulates (\POS{1}--\POS{7})}\\
\midrule
\POSkw{} & Postulate (admissibility filter; testable in principle) & \AOne{}\\
\GI{} & \POS{1} Gauge invariance across scales & \AOne{}\\
\UEP{} & \POS{2} Unified Extremal Principle & \AOne{}\\
\AF{} & \POS{3} Anomaly Freedom and cancellability by inflow & \AOne{}\\
\CP{} & \POS{4} Closure and Positivity (CPTP-admissible open dynamics) & \AOne{}\\
\RG{} & \POS{5} Renormalization-Group covariance (scale covariance) & \AOne{}\\
\OGN{} & \POS{6} Occam Gauge Normalization (canonical integer/rational normalization) & \AOne{}\\
\MF{} & \POS{7} Measurement and Falsifiability (operational contract + kill criteria) & \AOne{}\\
\midrule

\multicolumn{3}{l}{\itshape (4) \GiFour{} pillars}\\
\midrule
\TGG{} & Total Geometric--Gauge object (Pillar I of \GiFour{}) & \ASix{}\\
\CCI{} & Covariant Conservation Identity (Pillar II of \GiFour{}) & \ASix{}\\
\GFF{} & Geometric Free-Energy Functional (Pillar III of \GiFour{}) & \ASix{}\\
\GRIP{} & Gauge-Runtime Invariance Pipeline (Pillar IV of \GiFour{}) & \ASix{}\\
\GA{} & Geometric Attractor (final stage of \GRIP{}) & \ASix{}\\
\midrule

\multicolumn{3}{l}{\itshape (5) \QiFour{} pillars and \GDOS{}-internal acronyms}\\
\midrule
\QiQUA{} & Quantum information for Quadruple Gauge Locking (Pillar I of \QiFour{}) & \ASeven{}\\
\QiEP{} & Quantum information for the Unified Extremal Principle (Pillar II) & \ASeven{}\\
\QiCON{} & Quantum information for Conformal Invariance (Pillar III) & \ASeven{}\\
\QiHAL{} & Quantum information for the Holographic Attractor Lattice (Pillar IV) & \ASeven{}\\
\SIP{} & System Instance Packet (canonical \GDOS{} input packet) & \ASeven{}\\
\SeOS{} & Selfish Open System (the core open system of \GDOS{}) & \AFive{}\\
\midrule

\multicolumn{3}{l}{\itshape (6) Program terms in the context of this paper}\\
\midrule
QD & Quantum-descent completion: the one-lattice-step correction of the parent codebook (e.g.\ $2\pi/5 \to 13\pi/35$) & \ANine{}; Part~VII\\
D, D$^{*}$ & Grade vocabulary: Derived; Derived-conditional (named conjecture in the chain) & \ANine{}; Part~XI\\
Kaxion & The \GG{} axion of the strong-CP two-fate architecture ($8.1~\mu$eV) & \PaperCOne{}\\
T-P5-$x$ etc. & Named-result identifiers of this paper & Appendix~\ref{app:naming}\\
\midrule

\multicolumn{3}{l}{\itshape (7) Standard physics abbreviations (plain Roman, journal convention)}\\
\midrule
\multicolumn{3}{l}{\itshape \quad Symmetries and flavor}\\
CPT & Combined charge-conjugation--parity--time-reversal operation & textbook\\
CKM & Cabibbo--Kobayashi--Maskawa quark mixing matrix & textbook\\
PMNS & Pontecorvo--Maki--Nakagawa--Sakata lepton mixing matrix & textbook\\
$V{-}A$ & Vector-minus-axial structure of the charged current & textbook\\
EDM & Electric dipole moment & textbook\\
NO / IO & Normal / inverted neutrino mass ordering & textbook\\
MSW & Mikheyev--Smirnov--Wolfenstein matter effect & textbook\\
NME & Nuclear matrix element (of $0\nu\beta\beta$) & textbook\\
$0\nu\beta\beta$ & Neutrinoless double beta decay & textbook\\
ARS & Akhmedov--Rubakov--Smirnov oscillation leptogenesis & \citet{AkhmedovRubakovSmirnov1998}\\
\midrule
\multicolumn{3}{l}{\itshape \quad Frameworks and cosmology}\\
QM / QFT / SM & Quantum mechanics / quantum field theory / the Standard Model & textbook\\
GUT & Grand Unified Theory & textbook\\
LSZ & Lehmann--Symanzik--Zimmermann reduction & textbook\\
CPTP & Completely Positive Trace Preserving (never abbreviated CPT here) & textbook\\
RG & Renormalization Group & textbook\\
CMB & Cosmic Microwave Background & textbook\\
BBN & Big Bang Nucleosynthesis & textbook\\

\end{longtable}
}

\bigskip
\textbf{Hierarchy of program-defined acronyms.} For reference,
Figure~\ref{fig:abbrev-flow} records the six-tier hierarchy of
the program-defined acronyms used across the \GGTheory{}
cohort: the framework name and its constitutional charter; the
four axioms and seven postulates; the bulk objects (\GG{},
\BIsix{}, \OGNfive{}); the boundary law \GDOS{} and its input
packet \SIP{}; the two four-pillar structures \GiFour{} and
\QiFour{} with their sub-acronyms; and the energy--information
Arisaka constant $\QB$ with the Arisaka equation that locks the
four boundary currents. The diagram is shared verbatim with
\AOne{} (its Figure~9) and with the parent appendices of
\ANine{} and \ATen{}, so that the acronym ledger is identical
across the cohort.

\begin{figure}[H]
\centering
\resizebox{0.824\textwidth}{!}{%
\begin{tikzpicture}[
 font=\sffamily,
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 line width=0.80pt,
 every node/.style={align=center},
 top/.style ={rectangle, rounded corners=3pt, draw=violet!55!black, line width=0.8pt, minimum width=46mm, minimum height=11mm, fill=violet!12, font=\sffamily\bfseries\Large},
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 arrow/.style ={->, line width=0.7pt, draw=black!65},
 arrowdash/.style ={->, line width=0.6pt, draw=black!50, dashed}
]

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

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

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

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

% --- Tier 3: bulk theorems (3 boxes: GG-6, BI-6, OGN-5) ---
\node[font=\sffamily\bfseries\large] (BTL) at (0, -108mm) {Bulk theorems forced by \textbf{AX} + \textbf{POS}};
\node[bulk] (GG6) at (-46mm, -120mm) {\textbf{GG-6}};
\node[bulk] (BI6) at (0mm, -120mm) {\textbf{BI-6}};
\node[bulk] (OGN5) at (46mm, -120mm) {\textbf{OGN-5}};

\draw[arrow] (AX4.south) -- ([yshift=-2mm]BTL.north -| AX4.south);
\draw[arrow] (P7.south) -- ([yshift=-2mm]BTL.north -| P7.south);

% --- Tier 4: GDOS boundary law (only SIP under it) --------
\node[gdos] (GDOS) at (0, -136mm) {\textbf{GDOS} \;\; (Geometric Dynamics of Open Systems on $\partial\mathcal{M}_6$) \;\; \textbf{SIP}};

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

% --- Tier 5a: Pillars headers ----------------------------
\node[gihead] (GiH) at (-46mm, -152mm) {\textbf{Gi-4 Pillars}};
\node[qihead] (QiH) at (46mm, -152mm) {\textbf{Qi-4 Pillars}};

\draw[arrow] (GDOS.south) -| (GiH.north);
\draw[arrow] (GDOS.south) -| (QiH.north);

% --- Tier 5b: Pillars content -----------------------
\node[pill] (TGG) at (-82mm, -170mm) {\textbf{TGG} \quad Total Geometric--Gauge object};
\node[pill] (CCI) at (-82mm, -179mm) {\textbf{CCI} \quad Covariant Conservation Identity};
\node[pill] (GFF) at (-82mm, -188mm) {\textbf{GFF} \quad Geometric Free-Energy Functional};
\node[sub] at (-82mm, -195mm) {\hphantom{XX}$\downarrow$ \;\textbf{GEP} (energetic) \,$\bullet$ \,\textbf{CEP} (entropic)};
\node[pill] (GRIP) at (-82mm, -204mm) {\textbf{GRIP} \quad Gauge-Runtime Invariance Pipeline};
\node[sub] at (-82mm, -211mm) {\hphantom{XX}$\downarrow$ \;\textbf{DGI} (initial) \,$\bullet$ \,\textbf{GSSB} (middle) \,$\bullet$ \,\textbf{GA} (attractor)};

\node[pillq] (QQUA) at (10mm, -170mm) {\textbf{Qi-QUA} \quad Quantum information for \textbf{QGL}};
\node[pillq] (QEP) at (10mm, -179mm) {\textbf{Qi-EP} \quad Quantum information for \textbf{UEP}};
\node[sub] at (10mm, -186mm) {\hphantom{XX}$\downarrow$ \;\textbf{PLT} \,$\bullet$ \,\textbf{MEPP} \,$\bullet$ \,\textbf{FEP}};
\node[pillq] (QCON) at (10mm, -195mm) {\textbf{Qi-CON} \quad Quantum information for Conformal Inv.};
\node[sub] at (10mm, -202mm) {\hphantom{XX}$\downarrow$ \;Conformal \,$\bullet$ \,L\'evy \,$\bullet$ \,Fractal \,$\bullet$ \,$1/f$};
\node[pillq] (QHAL) at (10mm, -211mm) {\textbf{Qi-HAL} \quad Quantum information for the \textbf{HAL}};
\node[sub] at (10mm, -218mm) {\hphantom{XX}$\downarrow$ \;\textbf{MePMoS} \,$\bullet$ \,\textbf{NHT} \,$\bullet$ \,\textbf{HAL} \,$\bullet$ \,\textbf{BFFT}};
\node[pillq] (EIC) at (10mm, -227mm) {\textbf{OH} \,$+$\, \textbf{IH} \,$=$\, \textbf{EIC} \quad (Evolving Inner Cosmos)};
\node[sub] at (10mm, -234mm) {\hphantom{XX}$\downarrow$ \;\textbf{SeOS} (Selfish Open System)};

% --- Tier 6: Q_B + Arisaka equation -----------------------
\node[qb] (QB) at (0, -250mm) {%
$\boldsymbol{\mathsf{Q}_{\mathsf{B}}}$ \;\; --- \;\; Quantum-information Bit \;\; (energy--information Arisaka constant) \\[1.5mm]
\textit{Arisaka equation:} \;\; $\boldsymbol{J_E / \mathsf{Q}_{\mathsf{B}} \,=\, J_e \,=\, J_B \,=\, J_{Qi}}$
};

\end{tikzpicture}
}
\caption[Hierarchy of program-defined acronyms.]{Hierarchy of program-defined acronyms in the \GGTheory{} program, shared verbatim with \AOne{} (its Figure~9), \ANine{}, and \ATen{}. The figure groups the program's acronyms by the layer each belongs to and draws the dependencies between the layers, so that a reader meeting one of them in the middle of the paper can find the layer it lives on without searching. It is shared verbatim across the papers named above, which is why nothing in it is specific to this one.}
\label{fig:abbrev-flow}
\end{figure}

\section{Glossary of Key Symbols}
\label{sec:glossary}

This appendix collects the key mathematical symbols used in the
present paper. The first block records the bulk-geometry
coordinates and the \OGNfive{} integer ledger that every later
derivation appeals to; the second the inherited scale chain.
The third records the two real structures of the handshake and
the discrete-symmetry operators built from them. The fourth
through sixth blocks record the flavor phases, the neutrino
sector, and the leptogenesis chain --- the quantities of the
confrontation table of Part~\ref{part:experiments}. The seventh
records the laboratory observables of the mirrors themselves.

\renewcommand{\arraystretch}{1.15}
\begin{center}
\small
\begin{longtable}{@{}p{0.22\textwidth}p{0.60\textwidth}p{0.14\textwidth}@{}}
\captionsetup{list=no}%% v15.0: see the front-matter note
\caption[Glossary of key symbols.]{Glossary of key symbols used in the present paper. ``Source'' gives the theorem, Part, or paper where the symbol is introduced or load-bearing. The symbols are grouped by the structure each belongs to, beginning with the bulk geometry and the ledger, and the source column points at the theorem, Part or companion paper in every case, so that a symbol met once in the middle of the paper can be traced to its definition in a single step rather than by search.}
\label{tab:appB-glossary-p5}\\
\toprule
\textbf{Symbol} & \textbf{Meaning} & \textbf{Source}\\
\midrule
\endfirsthead
\multicolumn{3}{c}{\itshape Table~\ref{tab:appB-glossary-p5} continued}\\
\toprule
\textbf{Symbol} & \textbf{Meaning} & \textbf{Source}\\
\midrule
\endhead
\midrule \multicolumn{3}{r}{\itshape continued on next page}\\
\endfoot
\bottomrule
\endlastfoot

\multicolumn{3}{l}{\itshape (1) Bulk geometry and the \OGNfive{} ledger}\\
\midrule
$M_6$ & 6D bulk manifold $M_4 \times I_r \times S^1_\theta$ & \ATwo{}\\
$r$ & Radial (scale-profile) coordinate, $r \in [0, K]$ & \ATwo{}\\
$\theta$ & Compact fiber coordinate, $\theta \in [0, 2\pi)$: the charge circle & \ATwo{}; Part~IV\\
$S^1_\theta$ & Compact direction circle (charge as winding) & \ATwo{}\\
$(M, N, L; J, K)$ & \OGNfive{} ledger $= (1, 3, 5; 7, 35)$ & \AThree{}\\
$K = LJ$ & Total scale depth $= 35$ (the throat-depth integer of the matter throughput) & \ATwo{}; Part~VIII\\
$N = 3$ & Family number (three-leg theorem) & \PaperPTwo{}\\
$L = 5$ & Block size of the flavor texture: the pentagon & Part~VII\\
\midrule

\multicolumn{3}{l}{\itshape (2) Scale chain}\\
\midrule
$M_{\mathrm{Pl}}$ & Planck mass $\approx 1.22 \times 10^{19}$~GeV & ---\\
$E_{\rm GUT}$ & GUT scale $= E_{\rm Pl}\,e^{-2\pi} \approx 2.28\times10^{16}$~GeV & \ATwo{}\\
$\Mlock$ & Lock scale $= \tfrac12 M_{\rm Pl}\,e^{-K} \approx 3.85$~TeV (the heavy seesaw sector) & \ATwo{}; Part~VIII\\
$T_{\mathrm{rh}}$ & Derived reheating temperature $\approx 3\times10^{10}$~GeV & \ATen{}\\
\midrule

\multicolumn{3}{l}{\itshape (3) The handshake and the discrete operations}\\
\midrule
$\Jq$ & Circle complex structure $= \partial_\theta/Q$ on charge-$n$ sectors (the gauge $i$) & T-P5-A\\
$\Ja$ & Clock complex structure $= \pm A(-A^2)^{-1/2}$, the polar phase (the state-space $i$) & T-P5-B\\
$A$ & Time-translation generator on the real solution space & Part~IV\\
$J$ & The one global complex structure after the handshake ($\Ja = \Jq$ on shell) & T-P5-C\\
$\varepsilon$ & The $90^{\circ}$ rotation on a weight pair ($\varepsilon^2 = -\mathbf 1$) & Part~II\\
$R(\varphi)$ & Rotation block through angle $\varphi$ (complex phase, real presentation) & Part~II\\
$B$ & Intertwiner to the conjugate spinor; $B\bar B = -\mathbf 1$ in $(1,5)$ & Lemma~\ref{lem:no-majorana}\\
$\Gamma^7$ & 6D chirality operator; splits $\mathbf 8 = \mathbf 4 \oplus \bar{\mathbf 4}$ & Part~III\\
$\Theta$ & The antiunitary CPT operation (total orientation reversal) & T-P5-2\\
C, P, T & Orientation reversals of the three real factors (fiber, space, clock) & T-P5-1\\
\midrule

\multicolumn{3}{l}{\itshape (4) Flavor holonomies and the valued phases}\\
\midrule
$V$, $U$ & CKM and PMNS mixing matrices & Part~VII\\
$\lambda$ & Wolfenstein parameter $= \sin\theta_C \simeq 0.225$ (golden closed form, Eq.~\eqref{eq:cabibbo}) & Part~VII\\
$\dCKM$ & Quark CP phase $= 2\pi/5 \to 13\pi/35 = 66.86^{\circ}$ (QD-completed) & T-P5-3\\
$\dPMNS$ & Leptonic CP phase $= \pi + 2\arcsin(1/7) = 196.43^{\circ}$ (QD $196.06^{\circ}$) & T-P5-3\\
$(\alpha_{21}, \alpha_{31})$ & Majorana phase pair $= (0, \pi)$ & Part~VII\\
$\sin\dPMNS$ & The quiet signature $= -8\sqrt3/49 = -0.2828$ (exact surd) & Part~VII\\
$\JCKM$ & Quark Jarlskog invariant $= 3.1\times10^{-5}$ at the stated layer (band $2.7$--$3.2\times10^{-5}$ across evaluation layers) & Part~VII\\
$\JPMNS$ & Leptonic Jarlskog invariant $\simeq -9.4\times10^{-3}$ & Part~VII\\
$\varepsilon, \varepsilon'$ & Indirect and direct kaon CP parameters (recomputed as holonomy interferometry) & Appendix~\ref{app:flavour}\\
$\sin 2\beta$ & $B$-sector mixing phase: $0.708$ (stated moduli layer) vs measured $0.709 \pm 0.011$ (watched) & Part~XI\\
$\bar\theta$ & Strong-CP angle; two-fate leakage floor $\lesssim 5\times10^{-18}$ & Part~VII; \PaperCOne{}\\
\midrule

\multicolumn{3}{l}{\itshape (5) Neutrino sector}\\
\midrule
$(m_1, m_2, m_3)$ & Derived light triplet $= (0.065, 8.59, 50.5)$~meV, normal ordering & Part~VIII\\
$\Delta m^2_{21}$ & Solar splitting $= 7.37\times10^{-5}$~eV$^2$ (output) & Part~VIII\\
$\Delta m^2_{31}$ & Atmospheric splitting $= 2.55\times10^{-3}$~eV$^2$ (output) & Part~VIII\\
$\mbb$ & Effective Majorana mass $= 1.72$~meV (registered null for the ton scale) & Part~XI\\
$Y_\nu$ & Seesaw-matched Yukawa $\simeq 7.3\times10^{-7}$ & \S\ref{sec:chain}\\
$m_\star$ & Equilibrium washout benchmark $\simeq 1.1\times10^{-3}$~eV (ledger-free) & Eq.~\eqref{eq:mstar}\\
\midrule

\multicolumn{3}{l}{\itshape (6) Leptogenesis and the receipt}\\
\midrule
$\epsilon^{\mathrm{res}}$ & Resonant decay asymmetry $\simeq 2.5\times10^{-6}$ (H-$\phi_{\mathrm{geo}}$-1 target value, now carried by an identified candidate mechanism; derived-cond.) & \S\ref{sec:chain}\\
$\Gamma_N$ & Heavy-state width $\simeq 8\times10^{-11}$~GeV ($\Gamma_N/M_N \sim 2\times10^{-14}$) & \S\ref{sec:chain}\\
$K_{\mathrm{wash}}$ & Washout strength $= \widetilde m/m_\star \simeq 50$ & \S\ref{sec:transport}\\
$\kappa$ & Washout efficiency $= 1/K_{\mathrm{wash}} = 2.0\times10^{-2}$; the chain adopts $1.0\times10^{-2}$ within the $O(1)$ & \S\ref{sec:transport}\\
$28/79$ & Sphaleron conversion coefficient $B = \tfrac{28}{79}(B{-}L)$ & Eq.~\eqref{eq:f4-conv}\\
$\gstar$ & Relativistic degrees of freedom at production $= 106.75$ & \S\ref{sec:transport}\\
$\etaB$ & Baryon-to-photon ratio: $\approx7\times10^{-10}$ predicted (central $7.34$; $\times3^{\pm1}$) vs $(6.12\pm0.04)\times10^{-10}$ & T-P5-5\\
\midrule

\multicolumn{3}{l}{\itshape (7) Laboratory observables of the mirrors}\\
\midrule
$\eta_{+-}, \eta_{00}$ & Kaon CP-violation amplitude ratios ($\varepsilon + \varepsilon'$, $\varepsilon - 2\varepsilon'$) & Appendix~\ref{app:flavour-kaons}\\
$\phi_{SW}$ & Superweak phase $= \arctan(2\Delta m_K/\Delta\Gamma_K) \simeq 43.5^{\circ}$ & Appendix~\ref{app:flavour-kaons}\\
$\delta_L$ & $K_L$ semileptonic charge asymmetry $= 2\,\mathrm{Re}\,\varepsilon/(1{+}|\varepsilon|^2)$ & Appendix~\ref{app:flavour-kaons}\\
$A_T$ & CPLEAR T-reversal asymmetry $= 4\,\mathrm{Re}\,\varepsilon = (6.6\pm1.6)\times10^{-3}$ & Part~I\\
$S_f, C_f$ & Time-dependent $B$-decay asymmetry coefficients & Appendix~\ref{app:flavour-bfactories}\\
$A_{CP}$ & Long-baseline appearance asymmetry ($\sim -0.28\sin\delta$ at Hyper-K kinematics) & Appendix~\ref{app:nu-appearance}\\
$d_n, d_e$ & Neutron and electron EDMs ($< 1.8\times10^{-26}$, $< 4.1\times10^{-30}\,e\,$cm) & Part~I\\

\end{longtable}
\end{center}
\renewcommand{\arraystretch}{1.0}

\section{Global naming convention for cross-cohort citation}
\label{app:naming}

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.
\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 (the kill
criteria that would retire the theory).
\item \textbf{I-Suffix-N} for the $N$th Innovation.
\item \textbf{PS-Suffix-N} for the $N$th Problem-Solved / status
entry.
\end{itemize}
The canonical Suffix list for the twenty-two cohort papers is:
\emph{Constitution} (A1), \emph{GG6} (A2), \emph{BI6} (A3),
\emph{Euclid} (A4), \emph{\GDOS{}} (A5), \emph{\GRIP{}} (A6),
\emph{Qi4} (A7), \emph{QM} (A8), \emph{SM} (A9),
\emph{Cosmos} (A10), \emph{Origin} (A11), together with the
phenomenology preprints \emph{KK} (P1), \emph{N3} (P2),
\emph{Electron} (P3), \emph{Proton} (P6), the foundations preprints \emph{Real} (P4)
and the present paper, \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 cohort-wide names; the
global identifier is the same item viewed from outside its home
paper.

The present paper uses the suffix \emph{P5}, with one
refinement stated once: the deductive chain's theorems carry
numbers (T-P5-1 through T-P5-5), while the handshake trio and
its Wick companion carry letters (T-P5-A, T-P5-B, T-P5-C,
T-P5-B1), as do the propositions and named conjectures (P-P5-D,
P-P5-E; C-P5-A, C-P5-C) --- the letters keep this paper's
conjecture keys visually distinct from the cohort's numeric
corollary keys, which this paper does not use. Falsifiers are
F-P5-N, objections O-P5-N (eighty-two, in Part~XV), and numbered
predictions Pred-N (Part~XI). Candidate results of the spin- and
statistics-suite carry the L-Spin/L-Stat/T-Stat prefixes of
their home papers (\citealp{ArisakaA2GG6}; \citealp{ArisakaA8QM}).
A citation such as ``\citealp{ArisakaA9SM}, Part~VI'' always
refers to the deployed arXiv version of the named paper; the
cross-paper coherence map guarantees that a key cited here
resolves to a stated result there, and Part~XVI's relevance notes
say, for every entry, which result is consumed.

\section{Discrete symmetries in quantum mechanics and quantum
field theory, from scratch}
\label{app:discrete-qft}

This is the opening chapter of Block II, and its subject is the
main text's own foundation: what C, P, and T \emph{are} in the
standard account, derived slowly from the textbooks, so that the
reader knows exactly what the orthodox framework delivers before
Step~3 states what the geometry adds. The chapter shelf:
\citet{Sakurai2017} Chapter~4 for symmetry in quantum mechanics;
\citet{WeinbergQTF} Chapters~2 (appendix~A) and~5 for the
(anti)unitary implementation and the field transformations;
\citet{PeskinSchroeder1995} Chapter~3 for the Dirac bilinears;
\citet{Sozzi2008} throughout for the experimental face;
\citet{BigiSanda2009} and \citet{BrancoLavouraSilva} for the
flavor applications; \citet{StreaterWightman} for the axiomatic
CPT of \S\ref{app:d-cpt}. Eighteen problems, complete solutions included.

\subsection{Step 1 --- symmetry in quantum mechanics: Wigner's
theorem and the antiunitary fork}
\label{app:d-wigner}

A quantum state is a \emph{ray}: $\psi$ and $e^{i\gamma}\psi$
describe one physical situation. A symmetry is a map of rays that
preserves all transition probabilities,
$|\langle\phi'|\psi'\rangle| = |\langle\phi|\psi\rangle|$.
Wigner's theorem \citep{Wigner1931} is the statement that every
such map lifts to an operator $U$ on the Hilbert space that is
either \emph{unitary and linear} or \emph{antiunitary and
antilinear} --- there is no third option.

The proof's engine can
be seen in two dimensions: fix an orthonormal pair
$|1\rangle, |2\rangle$; ray preservation fixes the images of
$|1\rangle, |2\rangle$, and the image of $|1\rangle + |2\rangle$
up to phase; the crux is the image of $|1\rangle + i|2\rangle$,
whose overlap moduli with the previous three images admit exactly
two consistent solutions --- one sending $i \mapsto i$ (unitary
branch) and one sending $i \mapsto -i$ (antiunitary branch). The
whole dichotomy of the subject sits in that one sign
(\citealp{WeinbergQTF}, Ch.~2, App.~A, carries the full proof).

Which branch does a given symmetry take? Energy decides. For a
symmetry commuting with time evolution,
$U e^{-iHt} U^{-1} = e^{-iHt'}$ with $t' = \pm t$. Take time
reversal, $t' = -t$. If $T$ were \emph{unitary}, then
$THT^{-1} = -H$: the spectrum would be reflected through zero,
and a world with a ground state would be paired with a world
without one --- forbidden for any Hamiltonian bounded below with
unbounded spectrum above.

If $T$ is \emph{antiunitary}, the $i$
in the exponent flips too, $T e^{-iHt} T^{-1} = e^{-iH(-t)}$
requires $THT^{-1} = +H$: the spectrum is preserved. Positivity
of energy forces time reversal onto the antiunitary branch ---
the single most consequential sign in the discrete-symmetry
canon, and the fact whose geometric face the main text derives
(Step~3). Kramers' theorem follows in three lines: if
$T^2 = -\mathbf{1}$ (half-integer spin), then for an energy
eigenstate $\psi$, $T\psi$ is an eigenstate of the same energy,
and $\langle\psi|T\psi\rangle = \langle T^2\psi|T\psi\rangle^*
= -\langle\psi|T\psi\rangle = 0$: every level of a
time-reversal-invariant half-integer system is at least doubly
degenerate --- a laboratory fact (odd-electron ions in electric
fields) that no perturbation respecting $T$ can lift. The textbook treatment is \citealp{Sakurai2017}, Ch.~4; the main text's Kramers corollary (Part~\ref{part:antiparticles}) runs on the same $T^2 = -\mathbf 1$.

\emph{What ``symmetry'' means at the bench.} Before the formalism
hardens, it is worth stating what these operators mean to a
working experimenter, because the subject's power comes from how
\emph{operational} it is. A symmetry statement is always a
statement about two runs of an experiment: prepare the mirrored
(or charge-swapped, or motion-reversed) initial state, and compare
counting rates.

Parity is testable because ``the mirrored
apparatus'' is buildable --- reverse the magnetic field coils and
the detector geometry and you have applied P to everything you
control. Charge conjugation is testable wherever antiparticles can
be prepared --- an antiproton beam, a positron source, the
$\bar B$ in a $B\bar B$ pair.

Time reversal is the awkward one:
nobody runs a detector backwards. What is actually reversed is
\emph{motion} --- spins and momenta --- and this is why T tests
never look like ``measure the T quantum number.'' Indeed there is
no such number, and the proof is two lines. Suppose
$T\psi = \lambda\psi$ and rephase the state, $\psi' =
e^{i\gamma}\psi$ --- the same ray. Antilinearity gives
\begin{equation}
T\psi' \;=\; T\,e^{i\gamma}\psi \;=\; e^{-i\gamma}\,T\psi
\;=\; e^{-i\gamma}\lambda\,\psi
\;=\; \big(e^{-2i\gamma}\lambda\big)\,\psi',
\label{eq:d1-noeig}
\end{equation}
so the ``eigenvalue'' $\lambda$ can be rotated to any phase by a
choice that changes nothing physical: it is a convention, not a
label. Unitary
symmetries yield conserved quantum numbers and selection rules;
the antiunitary one yields only \emph{relations} --- detailed
balance between a process and its motion-reverse, reality
constraints on amplitudes, and the forbidden alignments (like an
EDM) whose absence can be hunted to eighteen digits.

The
asymmetry of experimental style between P/C and T is not
sociology; it is Wigner's fork expressed in hardware. The EDM lines of that hunt are this paper's Pred-8 (\S\ref{sec:falsification}); the operational reading of symmetry underneath is the dictionary of \citealp{ArisakaA5GDOS}.

\emph{The $4\pi$ fact, measured.} The projective freedom that
Wigner's theorem legalizes is not an abstraction: it is why
half-integer spin exists at all (a spin-$\tfrac12$ state
acquires a physical $-1$ under a $2\pi$ rotation, lawful because
states are rays), and it has been measured directly --- a
neutron passed through a magnetic field region that precesses
its spin by $2\pi$ interferes \emph{destructively} with its
unrotated partner beam, and constructively again at $4\pi$
\citep{Rauch1975}.

Every discrete-symmetry table in this
appendix stands on that experimental floor: the phases we track
are physical precisely because interferometers can read them.
The main text's spin--statistics section
(\S\ref{sec:quartet-second-child}) gives this $2\pi$ memory its
geometric name; here it is simply the reason the bookkeeping
below must be done with signs and not merely with words.

\paragraph{Problems and worked solutions.}

\medskip\noindent\textbf{D.1.1. Complete the two-dimensional Wigner construction and
exhibit the two branches explicitly.}

\noindent\emph{Solution.} Let the symmetry map send the rays of
$|1\rangle, |2\rangle$ to those of $|1'\rangle, |2'\rangle$;
choose representatives, which fixes two free phases we may spend
later. Consider $|s\rangle = |1\rangle + |2\rangle$. Its image
$|s'\rangle$ must satisfy $|\langle 1'|s'\rangle| =
|\langle 1|s\rangle| = 1$ and likewise for $|2'\rangle$, so
$|s'\rangle = e^{i\mu}(|1'\rangle + e^{i\nu}|2'\rangle)$ for some
phases; absorb $\mu$ into the overall ray freedom and $\nu$ into
the representative of $|2'\rangle$.

Thus we may take
$|s\rangle \mapsto |1'\rangle + |2'\rangle$: so far everything
looks linear. Now the crux: $|q\rangle = |1\rangle + i|2\rangle$.
Ray preservation demands $|\langle 1'|q'\rangle| =
|\langle 2'|q'\rangle| = 1$ and $|\langle s'|q'\rangle| =
|1 + \bar{c}|/\ldots$ --- concretely, writing
$|q'\rangle = |1'\rangle + c\,|2'\rangle$ with $|c| = 1$, the
condition $|\langle s'|q'\rangle| = |\langle s|q\rangle|$ reads
$|1 + c| = |1 + i| = \sqrt2$, whose unit-modulus solutions are
exactly $c = i$ and $c = -i$.

No third solution exists. The
choice $c = i$ extends (one checks additivity and the analogous
condition on every superposition) to a \emph{linear, unitary}
operator; the choice $c = -i$ extends to an \emph{antilinear,
antiunitary} one, $U(\alpha\psi) = \alpha^* U\psi$.

Physics
intuition: a symmetry must preserve the interference pattern of
$|1\rangle + i|2\rangle$ against $|1\rangle + |2\rangle$, and an
interference pattern knows the \emph{relative} phase only up to a
global complex conjugation --- the mirror image of the phase
plane. Wigner's fork is the statement that a symmetry may leave
the phase plane alone or flip it, and nothing else; the main
text's Part~V geometrizes the flip as the reversal of the one
structure the phase plane is made of, the clock.

\medskip\noindent\textbf{D.1.2. Show a unitary ``time reversal'' is admissible for a
system with spectrum symmetric about zero, and why field theory
excludes it.}

\noindent\emph{Solution.} Step 1: suppose $U$ unitary with
$Ue^{-iHt}U^{-1} = e^{+iHt}$. Differentiating at $t = 0$ gives
$UHU^{-1} = -H$: a unitary time reversal requires a spectrum
symmetric about zero.

Step 2: such systems exist. Take
$H = \tfrac{\omega}{2}\sigma_z$ and $U = \sigma_x$: then
$\sigma_x \sigma_z \sigma_x = -\sigma_z$, so $U H U^{-1} = -H$
exactly, and $U$ is a perfectly lawful unitary implementing
$t \to -t$ on this two-level system.

Step 3: why fields refuse.
A relativistic field theory has $H$ unbounded above (arbitrarily
many quanta) and --- by the spectrum condition --- bounded below
by the vacuum. A spectrum $[0,\infty)$ reflected through zero
would be $(-\infty,0]$: the map $H \to -H$ carries a world with a
ground state to a world with none, and no unitary can relate the
two spectra. The antiunitary branch escapes because antilinearity
flips the $i$ in $e^{-iHt}$ as well, so time reversal holds with
$UHU^{-1} = +H$.

Physics intuition: running the film backwards
must not manufacture a bottomless energy pit; the only way to
reverse the clock while keeping the floor is to conjugate the
phases --- which is precisely what ``the clock owns the $i$''
means downstream, in Step~3.

\medskip\noindent\textbf{D.1.3. Prove $T^2 = (-1)^{2j}$ on a spin-$j$ multiplet from
$T = e^{-i\pi J_y}K$.}

\noindent\emph{Solution.} Step 1: apply $T$ twice:
$T^2 = e^{-i\pi J_y} K e^{-i\pi J_y} K$.

Step 2: move the first
$K$ through the exponential. $K$ complex-conjugates matrix
elements, so $K e^{-i\pi J_y} K = e^{+i\pi J_y^*}$. In the
standard angular-momentum basis $J_y$ is purely imaginary
($J_x, J_z$ real), hence $J_y^* = -J_y$ and
$K e^{-i\pi J_y} K = e^{-i\pi J_y}$.

Step 3: therefore
$T^2 = e^{-2\pi i J_y}$, a rotation by $2\pi$ about the $y$-axis.
On integer spin a $2\pi$ rotation is the identity; on
half-integer spin it is $-\mathbf 1$, because the rotation group's
double cover assigns $e^{-2\pi i J_y} = (-1)^{2j}$ on a spin-$j$
irrep (diagonalize $J_y$: eigenvalues run over
$-j, \ldots, +j$ in integer steps, so $e^{-2\pi i m}$ is $+1$
for all integer $m$, $-1$ for all half-odd $m$). Hence
$T^2 = (-1)^{2j}$, and Kramers degeneracy follows for half-odd
$j$ as in the text.

Physics intuition: $T^2$ is secretly a full
turn --- reversing the clock twice drags the frame around once
--- and half-integer systems remember full turns with a sign.
This is the frame-path memory of the main text's spin--statistics
section (\S\ref{sec:quartet-second-child}) surfacing in
first-year notation: the same $2\pi$ that neutron interferometry
measures is the one hiding inside every Kramers doublet.

\subsection{Step 1 --- P and C on the Dirac theory: the
bilinear table, derived}
\label{app:d-dirac}

The Dirac equation $(i\gamma^\mu\partial_\mu - m)\psi = 0$ is
the arena in which P and C acquire their standard bodies.
\emph{Parity.} Send $\vec{x} \to -\vec{x}$; the equation retains
its form iff $\psi'(t,-\vec x) = \eta_P\,\gamma^0
\psi(t,\vec x)$, since $\gamma^0$ commutes with $\gamma^0$ and
anticommutes with $\gamma^i$, flipping the sign of the spatial
derivative terms exactly as required; $\eta_P$ is a free phase
--- note it, Step~2 will return to it. Two applications of P on
a fermion give $\eta_P^2$, and the standard convention
$\eta_P^2 = -1$ for fermion fields already shows convention
entering where the beginner expects physics.
\emph{Charge conjugation.} Complex-conjugate the coupled Dirac
equation $(i\gamma^\mu(\partial_\mu + ieA_\mu) - m)\psi = 0$:
the sign of the charge flips. Seeking $\psi^c = C\bar\psi^{\,\mathsf T}$
satisfying the conjugate equation fixes
$C^{-1}\gamma^\mu C = -\gamma^{\mu\,\mathsf T}$, solved in the
Dirac representation by $C = i\gamma^2\gamma^0$ --- the matrix is
representation-dependent, the defining relation is not. From
these two operators every entry of the classic bilinear table
follows by direct computation:
\begin{table}[htbp]
\centering
\caption[Dirac bilinears under P, C and T.]{The classic bilinear table, computed rather than quoted. Each row is a Dirac bilinear and its behavior under P, C and T, with the pairs in parentheses giving the time and the space components separately. Nothing in it is new: every entry follows by direct computation from the two operators fixed immediately above, and the table is here so that a reader can set the geometric account beside the one every textbook carries and check them against each other.}
\label{tab:bilinears}
\small
\begin{tabular}{lccc}
\toprule
bilinear & P & C & T \\
\midrule
$\bar\psi\psi$ (S) & $+$ & $+$ & $+$ \\
$\bar\psi i\gamma^5\psi$ (P) & $-$ & $+$ & $-$ \\
$\bar\psi\gamma^\mu\psi$ (V) & $(+,-)$ & $-$ & $(+,-)$ \\
$\bar\psi\gamma^\mu\gamma^5\psi$ (A) & $(-,+)$ & $+$ & $(+,-)$ \\
$\bar\psi\sigma^{\mu\nu}\psi$ (T) & $(\pm)$ & $-$ & $(\mp)$ \\
\bottomrule
\end{tabular}
\end{table}
(signs for time/space components; \citealp{PeskinSchroeder1995},
\S3.6, is the assigned check). Two theorems the beginner should
extract. First, the \emph{opposite intrinsic parity of fermion
and antifermion}: applying P to the mode expansion sends
$b_{\vec p} \to \eta_P b_{-\vec p}$ but
$d_{\vec p} \to -\eta_P^* d_{-\vec p}$ --- the relative minus is
convention-free, and it is tested every time para-positronium
($^1S_0$, P-eigenvalue $-1$ with $L=0$) decays. Second,
\emph{C-parity of a fermion pair}: $C = (-1)^{L+S}$, so
para-positronium ($S=0$) is C-even and decays to two photons,
ortho-positronium ($S=1$) is C-odd and must go to three ---
lifetimes $0.125$~ns versus $142$~ns, a factor of a thousand
from one sign; and $\pi^0 \to 3\gamma$, allowed by every
conservation law except C, has never been seen (branching
$< 3\times10^{-8}$). The discrete symmetries are not
philosophy; they are selection rules with lifetimes attached. (Book-length treatments: \citealp{BigiSanda2009}, Ch.~1; \citealp{Sozzi2008}, Chs.~2--3.)

\emph{How an intrinsic parity is actually measured: the pion.}
The assignments in the table are not conventions all the way
down --- relative parities are measured, and the classic case
deserves its telling because every element of the method is
still in use. Panofsky's reaction is capture of a stopped
$\pi^-$ on deuterium, $\pi^- d \to nn$. The initial state is an
$s$-wave ($L = 0$) atomic capture, total angular momentum
$J = 1$ (deuteron spin 1, pion spin 0). The two final neutrons
must combine to $J = 1$. Two constraints now work in tandem.
Statistics: for two identical fermions the total wavefunction is
antisymmetric,
\begin{equation}
(-1)^{L_f}\,(-1)^{S+1}\,\stackrel{!}{=}\,-1
\quad\Longrightarrow\quad L_f + S \ \text{even},
\label{eq:d2-stat}
\end{equation}
and among the $J = 1$ candidates ($^3S_1$, $^1P_1$, $^3P_1$,
$^3D_1$) only $^3P_1$ ($L_f = 1$, $S = 1$) survives. Parity:
balance initial against final,
\begin{equation}
\eta_\pi\,\eta_d\,(-1)^{L_i} \;=\; (\eta_n)^2\,(-1)^{L_f}
\quad\Longrightarrow\quad
\eta_\pi \cdot (+1) \cdot (+1) \;=\; (+1)\cdot(-1)^{1},
\label{eq:d2-parity}
\end{equation}
so $\eta_\pi = -1$: the pion is \emph{pseudoscalar}. One stopped-beam
experiment, one statistics argument, one quantum number ---
and the pattern (use statistics plus angular momentum to trap
the parity) is the template for every parity assignment in the
Review of Particle Physics \citep{PDG2024}. Notice what did the
work: Fermi statistics again --- the same exchange sign of
D.1.3, now employed as a measuring instrument. The geometry traces that same sign to bulk frame paths (\citealp{ArisakaA2GG6}, \S20; \S\ref{sec:quartet-second-child} here).

\emph{How a handedness is measured: Goldhaber's neutrino.} The
V$-$A structure's most beautiful direct test predates the
modern beams.

Goldhaber, Grodzins, and Sunyar
\citep{Goldhaber1958} measured the helicity of the neutrino
itself with tabletop apparatus: electron capture on
$^{152}$Eu ($J = 0$) produces $^{152}$Sm$^*$ ($J = 1$) plus a
neutrino; the excited Sm decays by a $961$~keV photon. Select,
by resonant scattering, only the photons emitted
\emph{opposite} to the neutrino and Doppler-shifted into
resonance --- these are the photons emitted along the Sm$^*$
recoil, i.e.\ opposite the neutrino. The angular-momentum ledger
along the neutrino axis, with the captured $s$-electron
contributing $\pm\tfrac12$ and the initial $J = 0$:
\begin{equation}
0 \;=\; \underbrace{\mp\tfrac12}_{e^-\ \text{spin}}
\;+\; \underbrace{h_\nu\cdot\tfrac12}_{\nu}
\;+\; \underbrace{M_{\mathrm{Sm}^*}}_{J_z = 0,\pm1}
\quad\Longrightarrow\quad
M_{\mathrm{Sm}^*} = \pm1 \ \text{correlates with}\ h_\nu,
\label{eq:d2-gold}
\end{equation}
and for the forward-emitted $961$~keV photon the Sm$^*$
polarization passes wholly to circular polarization,
$h_\gamma = h_\nu$. Measuring $h_\gamma$ by transmission
through magnetized iron (Compton scattering on polarized
electrons is spin-dependent) gave
\begin{equation}
h_\nu \;=\; -1.0 \pm 0.3,
\label{eq:d2-hnu}
\end{equation}
the neutrino left-handed, fully, not partially.

Physics intuition for the experimenter:
helicity was transferred, step by step, from an invisible
particle to a measurable photon polarization through nothing
but conservation laws --- no accelerator, no tracking, one
NaI crystal and a magnetized-iron analyzer. Maximal parity
violation is not a high-energy exotic; it sits in a
table-top gamma line. (Context and successors: \citealp{Sozzi2008}, Ch.~7; the geometry's one-hand reading is Part~\ref{part:parity}.)

\emph{Furry's theorem, or why the photon's C-parity earns its
keep.} The photon's $C = -1$ is not decoration: it implies
that any process connecting an odd number of photons with no
other C-odd quanta must vanish. In QED this is Furry's theorem
--- amplitudes with an odd number of external photons and
closed electron loops cancel identically --- and its
laboratory face is the non-observation of $\pi^0 \to 3\gamma$
and of photon splitting in vacuum.

When an experiment quotes a
$3\gamma$ branching limit, it is publishing a C-conservation
test of the electromagnetic interaction; the table above tells
you exactly which interactions are allowed to break it (the
weak, which does, spectacularly) and which are not.

\paragraph{Problems and worked solutions.}

\medskip\noindent\textbf{D.2.1. Verify the V and A rows of the bilinear table.}

\noindent\emph{Solution.} Parity first. With $\psi'(t,\vec x) =
\gamma^0\psi(t,-\vec x)$, a vector bilinear transforms as
$\bar\psi'\gamma^\mu\psi' = \psi^\dagger\gamma^{0\dagger}\gamma^0
\gamma^\mu\gamma^0\psi = \bar\psi\,\gamma^0\gamma^\mu\gamma^0\,
\psi$ (arguments at $-\vec x$ throughout). Now
$\gamma^0\gamma^0\gamma^0 = \gamma^0$ while
$\gamma^0\gamma^i\gamma^0 = -\gamma^i$: the time component is
even, the space components odd --- exactly how a velocity or a
current behaves in a mirror.

For the axial row insert
$\gamma^5$: since $\gamma^0\gamma^5 = -\gamma^5\gamma^0$, each
sign above flips --- time component odd, space components even:
the behavior of a spin or a magnetic field. Charge conjugation:
using $\psi^c = C\bar\psi^{\mathsf T}$ with
$C^{-1}\gamma^\mu C = -\gamma^{\mu\mathsf T}$, a short transpose
computation (remembering the anticommuting fields contribute one
minus sign on reordering) gives $\bar\psi^c\gamma^\mu\psi^c =
-\bar\psi\gamma^\mu\psi$ and $\bar\psi^c\gamma^\mu\gamma^5\psi^c
= +\bar\psi\gamma^\mu\gamma^5\psi$: the electric current reverses
for antimatter, the axial current does not.

Physics intuition and
the payoff: the weak current is $V\!-\!A$,
$\bar\psi\gamma^\mu(1-\gamma^5)\psi$ --- the \emph{difference} of
two objects with opposite P behavior. Under parity it becomes
$V\!+\!A$: not slightly changed but mapped onto the coupling
nature declines to use. Maximal parity violation is thereby a
statement of \emph{absence} (no $V\!+\!A$ partner exists), which
is precisely the shape of the main text's Part~VI claim.

\medskip\noindent\textbf{D.2.2. Derive $C = (-1)^{L+S}$ for a fermion--antifermion
pair.}

\noindent\emph{Solution.} Act with C on the pair state: it exchanges which
field quantum is ``the particle'' and ``the antiparticle,'' i.e.\
it maps the state onto the same two quanta with their roles
swapped. To compare with the original, un-swap them by exchanging
the two quanta completely --- position and spin --- and pay the
prices: (i) spatial exchange in a relative partial wave $L$ gives
$(-1)^L$; (ii) spin exchange in a coupled spin state gives
$(-1)^{S+1}$ ($+$ for the symmetric triplet, $-$ for the
antisymmetric singlet, packaged as one formula); (iii) exchanging
two fermionic operators costs the Fermi sign $(-1)$.

The product
is the C eigenvalue:
$\eta_C = (-1)^L\,(-1)^{S+1}\,(-1) = (-1)^{L+S}$.
Check against nature: para-positronium ($L=0, S=0$) has $C = +1$
and decays to two photons (each photon C-odd, $(-1)^2 = +1$);
ortho-positronium ($L=0, S=1$) has $C = -1$ and must emit three
--- the thousandfold lifetime ratio of the text.

Physics
intuition: C-parity of a bound pair is not a new quantum number
but bookkeeping of exchange symmetry --- the same
Fermi sign that Appendix D.1.3 traced to a $2\pi$ turn --- so the
selection rules of positronium are, at bottom, statistics made
visible in light.

\medskip\noindent\textbf{D.2.3. Show a nonzero neutron EDM requires both P and T
violation.}

\noindent\emph{Solution.} For a spin-$\tfrac12$ particle the only
available vector is the spin $\vec\sigma$, so any electric dipole
must point along it: the interaction is
$H_d = -d\,\vec\sigma\cdot\vec E$, to be compared with the
magnetic term $H_\mu = -\mu\,\vec\sigma\cdot\vec B$. Under P:
$\vec E \to -\vec E$ (polar), $\vec B \to +\vec B$ (axial),
$\vec\sigma \to +\vec\sigma$ (angular momentum). So $H_\mu$ is
P-even but $H_d$ is P-\emph{odd}. Under T: $\vec E \to +\vec E$,
$\vec B \to -\vec B$, $\vec\sigma \to -\vec\sigma$. So $H_\mu$
is T-even but $H_d$ is T-\emph{odd}. A nonzero $d$ therefore
violates P and T simultaneously --- and, by the CPT theorem of
D.4, T violation is CP violation.

Physics intuition: an EDM is
an arrow of charge separation nailed to an arrow of rotation;
mirror-flip the charges or run the spin backwards and the two
arrows fall out of alignment unless the underlying dynamics
distinguishes the two orientations. That is why the neutron EDM
is the cleanest null in the strong-CP story: a
$\bar\theta$-term would nail the arrows together at
$d_n \sim 10^{-16}\,\bar\theta\ e\,$cm, the measured
$|d_n| < 1.8\times10^{-26}\ e\,$cm forces
$\bar\theta < 10^{-10}$, and the main text's two-fate
architecture predicts the arrows stay unglued down to
$\bar\theta \sim 5\times10^{-18}$ --- eight orders of silence,
registered in Table~\ref{tab:falsifiers}.

\subsection{Step 1 --- T in the laboratory: Wu's asymmetry and
the kaon matrix}
\label{app:d-lab}

Two computations anchor the operations in data. \emph{Wu's
asymmetry, quantified.} Polarized $^{60}$Co emits betas with
rate $W(\theta) \propto 1 + \alpha\,(v/c)\cos\theta$, with
$\theta$ measured from the nuclear spin. Under P, momenta flip
and spins do not, so $\cos\theta \to -\cos\theta$: any
$\alpha \neq 0$ is parity violation. The V$-$A current predicts
$\alpha = -1$ for a Gamow--Teller transition: the 1957
observation of a large asymmetry was not a small effect but a
\emph{maximal} one --- the mirror was not chipped; it was gone.

The main text's Part~VI is the claim that this maximality is an
absence, not a parameter. \emph{The neutral-kaon matrix.} The
$(K^0, \bar K^0)$ system evolves by an effective non-Hermitian
Hamiltonian $H = M - \tfrac{i}{2}\Gamma$, with CPT forcing
equal diagonal elements $H_{11} = H_{22}$ and T-invariance
forcing $|H_{12}| = |H_{21}|$. Diagonalizing, the physical
states are $K_{S,L} \propto (1+\epsilon)K^0 \pm
(1-\epsilon)\bar K^0$ with
$\epsilon = (\sqrt{H_{12}} - \sqrt{H_{21}})/(\sqrt{H_{12}} +
\sqrt{H_{21}})$: the 1964 observation of $K_L \to \pi\pi$
measured $|\epsilon| = 2.23\times10^{-3}$ and put a nonzero
$T$-violating phase into the world's bookkeeping --- while the
\emph{equality of the diagonal}, tested through
$m_{K^0} - m_{\bar K^0}$, became and remains the sharpest CPT
test in physics, $|m - \bar m|/m < 6\times10^{-19}$. One
$2\times2$ matrix carries the whole subject: CP violation in
its off-diagonal phases, CPT in its diagonal equality. (Formalism at book depth: \citealp{BrancoLavouraSilva}, Ch.~6; \citealp{BigiSanda2009}, Ch.~6; the paper's own CPT-test record opens Part~\ref{part:history}.)

\emph{Wu's experiment as an experimenter reads it.} The 1957
measurement rewards a closer look than the textbooks give it,
because every difficulty was a systematics problem of a kind
still faced daily.

To polarize the $^{60}$Co nuclei the sample
was cooled to $\sim10$~mK by adiabatic demagnetization inside a
cerium-magnesium-nitrate cryostat --- the nuclear polarization
rode on the hyperfine field of the aligned electrons --- and
the polarization was \emph{monitored in situ} by the anisotropy
of the accompanying $\gamma$ rays (a P-even effect, hence a
built-in control).

The signal was the up--down counting asymmetry of the betas
relative to the polarizing field: with nuclear polarization $P$,
\begin{equation}
W(\theta) \;\propto\; 1 + \alpha\,P\,\frac{v}{c}\cos\theta,
\qquad
A \;\equiv\; \frac{W(0) - W(\pi)}{W(0) + W(\pi)}
\;=\; \alpha\,P\,\frac{v}{c},
\label{eq:d3-wu}
\end{equation}
with the V$-$A prediction $\alpha = -1$ for the pure
Gamow--Teller $^{60}$Co transition; at the achieved
$P \sim 0.6$ and $v/c \sim 0.6$ the expected asymmetry is
$|A| \sim 0.4$ --- enormous, and observed. The decisive
systematic check was the simplest imaginable:
reverse the field. The asymmetry followed the field; warming
the sample (killing the polarization) killed the asymmetry on
the polarization's own timescale.

The observed coefficient was
consistent with the maximal $\alpha = -1$ of a pure
Gamow--Teller V$-$A transition: electrons emitted
preferentially \emph{opposite} the nuclear spin. Two further
1957 confirmations --- the $\pi \to \mu \to e$ decay chain's
muon polarization (Garwin--Lederman) and independent nuclear
work --- arrived within weeks, and C violation came with P
violation in the same data, since the observed correlation is
C-odd as well. The lesson that outlived the shock: a discrete
symmetry test is only as good as its reversal systematics ---
field flips, sample swaps, magnet polarities --- and that
discipline, invented here, is the daily bread of every EDM and
asymmetry experiment since.

\emph{T-odd observables, and the final-state trap.} With T
lacking eigenvalues (D.1), laboratory T tests use
\emph{T-odd correlations}: triple products such as
$\vec J \cdot (\vec p_e \times \vec p_\nu)$ in polarized beta
decay (the $D$ coefficient) or
$\vec\sigma_e \cdot (\vec J \times \vec p_e)$ (the $R$
coefficient).

Here the experimenter must internalize a famous
subtlety: a nonzero T-odd correlation is \emph{not} by itself
T violation. Final-state interactions --- ordinary Coulomb
rescattering between the outgoing charged particles ---
generate T-odd correlations in a perfectly T-symmetric theory,
because motion reversal relates the observed decay to the
time-mirrored \emph{in}-state that rescattering has altered.
The genuine test is either to compute and subtract the
final-state mimicry (it enters at order $Z\alpha$ and is
calculable in beta decay: the measured
$D = (-0.6\pm1.0)\times10^{-4}$ in neutron decay sits an order
of magnitude above its FSI floor) or to choose observables
where mimicry cannot reach --- which is the deep reason the
EDM, a \emph{static} property of an isolated particle with no
final state at all, is the T test of record. (Reviews: \citealp{Sozzi2008}, Ch.~4; this paper's EDM window is row 8 of Table~\ref{tab:falsifiers}.)

The kaon
complements both: CPLEAR compared the rates of
$K^0 \to \bar K^0$ against $\bar K^0 \to K^0$ directly ---
the Kabir asymmetry: tagging the strangeness at production
(via $\bar p p \to K^\mp\pi^\pm K^0/\bar K^0$) and at decay
(via the semileptonic charge),
\begin{equation}
A_T \;=\;
\frac{R(\bar K^0 \to K^0) - R(K^0 \to \bar K^0)}
{R(\bar K^0 \to K^0) + R(K^0 \to \bar K^0)}
\;=\; 4\,\mathrm{Re}\,\epsilon \;+\; \ldots
\;=\; (6.6 \pm 1.6)\times10^{-3}
\label{eq:d3-kabir}
\end{equation}
\citep{CPLEAR1998} --- time reversal violated in so many words,
the two directions of one oscillation clocked against each
other, and the value consistent with $4\,\mathrm{Re}\,\epsilon
= 4 \times 1.66\times10^{-3} = 6.6\times10^{-3}$ from the
mixing formalism above. One subject, three
instruments: correlations (with their FSI discipline), static
moments (with their reversal discipline), and oscillations
(with their tagging discipline). The main text's claim that
the mirrors are \emph{anatomy} does not change one of these
techniques; it changes what the accumulated record is evidence
\emph{of}.

\paragraph{Problems and worked solutions.}

\medskip\noindent\textbf{D.3.1. Derive the $K_{S,L}$ eigenstates and the $\epsilon$
formula.}

\noindent\emph{Solution.} In the $(K^0, \bar K^0)$ basis write
$H = \begin{psmallmatrix} A & p^2 \\ q^2 & A
\end{psmallmatrix}$, where CPT has forced equal diagonals
$A = M_{11} - \tfrac{i}{2}\Gamma_{11}$ and we abbreviate
$H_{12} = p^2$, $H_{21} = q^2$. Eigenvalue problem:
$\det(H - \lambda) = (A-\lambda)^2 - p^2q^2 = 0$, so
$\lambda_\pm = A \pm pq$, and the eigenvectors are
$p\,|K^0\rangle \pm q\,|\bar K^0\rangle$ (unnormalized): the
splitting lives entirely off-diagonal, the average entirely on.

Define $\epsilon$ by $p/q = (1+\epsilon)/(1-\epsilon)$;
equivalently $\epsilon = (p-q)/(p+q) =
(\sqrt{H_{12}} - \sqrt{H_{21}})/(\sqrt{H_{12}} + \sqrt{H_{21}})$,
and the physical states are $K_{S,L} \propto
(1+\epsilon)|K^0\rangle \pm (1-\epsilon)|\bar K^0\rangle$ as
claimed. If T holds, $|H_{12}| = |H_{21}|$ and a phase
convention can make $p = q$: $\epsilon = 0$ and the eigenstates
are the CP eigenstates. A nonzero $\mathrm{Re}\,\epsilon$
(convention-free) is T violation in the mixing.

Physics
intuition: the kaon pair is a two-slit experiment run in time
--- $K^0 \to \bar K^0$ through virtual $\pi\pi$ and $W$-box
paths --- and $\epsilon$ measures the failure of the forward and
backward films of that oscillation to match. Nature's answer,
$|\epsilon| = 2.23\times10^{-3}$, is the two-per-thousand
mismatch that started the entire subject of Part~VII.

\medskip\noindent\textbf{D.3.2. Derive the Bell--Steinberger relation.}

\noindent\emph{Solution.} Step 1: for the effective (non-Hermitian)
evolution $i\,\dot\psi = H\psi$, probability is not conserved
inside the two-state space --- it leaks into decay channels.
Compute the leak of the overlap: $\tfrac{d}{dt}\langle
K_S(t)|K_L(t)\rangle = i\langle K_S|(H^\dagger - H)|K_L\rangle
= -\langle K_S|\Gamma|K_L\rangle$, while the explicit time
dependence of the eigenstates gives the same overlap evolving as
$e^{-i(m_L - m_S)t}\,e^{-(\Gamma_S+\Gamma_L)t/2}$.

Step 2:
unitarity of the \emph{full} theory says the absorptive part
$\Gamma$ is built from real decay amplitudes,
$\langle K_S|\Gamma|K_L\rangle = \sum_f A^*(K_S \to f)
A(K_L \to f)$. Equating the two expressions at $t = 0$:
\[
\Big(\tfrac{\Gamma_S + \Gamma_L}{2} + i\,\Delta m\Big)
\langle K_S|K_L\rangle \;=\; \sum_f A^*(K_S\to f)\,A(K_L\to f).
\]
Step 3: with CPT held, $\langle K_S|K_L\rangle =
2\,\mathrm{Re}\,\epsilon/(1+|\epsilon|^2)$, so the measured
decay amplitudes on the right determine
$\mathrm{Re}\,\epsilon$ --- and any residue not accounted for
becomes a bound on the CPT-violating diagonal splitting.

Physics intuition: the relation is bookkeeping of leaks --- the
non-orthogonality of the two physical kaons must be paid for,
channel by channel, by their shared decays. It converts a
zoo of branching measurements into the world's best inventory
of where T and CPT violation could hide, and finds essentially
all of it in $\epsilon$.

\medskip\noindent\textbf{D.3.3. Estimate the fractional CPT mass sensitivity.}

\noindent\emph{Solution.} The interferometer's observable is the strangeness
oscillation phase $\Delta m\,t$, read against the known lifetime
clock $\Gamma_S$ over $\sim10$--$20$ lifetimes; the regeneration
and semileptonic asymmetries determine the relative phase between
mixing and decay to $\sim10^{-3}$ precision. A CPT-violating
diagonal splitting $\delta = (m_{K^0} - m_{\bar K^0}) -
\tfrac{i}{2}(\Gamma_{11}-\Gamma_{22})$ shifts that phase by
$\sim \delta/\Delta m$.

Sensitivity therefore:
$|m - \bar m| \sim \Delta m \times (\text{phase precision})
\sim 3.5\times10^{-12}\,\mathrm{MeV} \times 10^{-3}
\sim 10^{-15}$~MeV, i.e.
$|m - \bar m|/m_K \sim 10^{-15}/500 \sim 10^{-18}$ ---
the $6\times10^{-19}$ of the Particle Data Group with the
factors done honestly.

Physics intuition: the kaon wins by
leverage --- it compares particle with antiparticle through an
oscillation whose beat frequency ($\Delta m/m \sim 10^{-14}$) is
itself tiny, so a given phase precision buys fourteen extra
orders in mass. It is the same trick the long baselines play
with $\Delta m^2 L/E$; interferometry is how small numbers are
read, in every era of this subject.

\subsection{Step 1 --- the CPT theorem, by both roads}
\label{app:d-cpt}

\emph{The Lagrangian road (L\"uders--Pauli)} \citep{Luders1954, Pauli1955}\emph{.} Claim: any local,
Lorentz-invariant, Hermitian Lagrangian of normal-ordered
fields is invariant under the product $\Theta = CPT$, whatever
its C, P, T behavior separately. The engine is that $\Theta$
acts on every field as an antiunitary map sending
$\phi(x) \to \phi^\dagger(-x)$ up to spin-dependent signs, so
on any Lorentz-scalar monomial it delivers the Hermitian
conjugate at $-x$ with a sign $(-1)^{2J}$ that Lorentz
invariance forces to unity for a scalar combination.

Walk one
term: the Yukawa $g\,\bar\psi_1\psi_2\,\phi + \mathrm{h.c.}$
Under $\Theta$: $\phi \to \phi^\dagger(-x)$, and the scalar
bilinear $\bar\psi_1\psi_2 \to (\bar\psi_1\psi_2)^\dagger(-x)
= \bar\psi_2\psi_1(-x)$ (the S row of the table, all three
signs multiplied); antilinearity conjugates $g$; the term maps
to $g^*\,\bar\psi_2\psi_1\,\phi^\dagger$ at $-x$ --- which is
exactly the Hermitian-conjugate term the Lagrangian already
contains.

Hermiticity is thus the whole secret: CPT holds
because a Hermitian $\mathcal L$ contains every term's
conjugate, and $\Theta$ merely swaps them. \emph{The axiomatic
road (Jost).} In the Wightman framework the same conclusion
follows with no Lagrangian at all: the vacuum correlation
functions $W(x_1,\ldots,x_n)$ are (i) boundary values of
functions analytic in the forward tube (the spectrum condition
--- energies positive --- is what grants the continuation:
the main text's one-sided clock, in its field-theoretic
uniform); (ii) invariant under the complexified Lorentz group,
which connectedly contains the total reflection $x \to -x$;
(iii) at the real \emph{Jost points} (totally spacelike
configurations) the continuation can be evaluated, and locality
converts the reflected function into the CPT-conjugate
ordering \citep{Jost1957, StreaterWightman}.

The consequences
are then theorems, not folklore, and each is a measurement:
particle and antiparticle masses equal (the kaon's
$6\times10^{-19}$), total lifetimes equal (muon pair at
$10^{-4}$), magnetic moments opposite (electron pair at
$10^{-12}$), and the fine-structure spectra of hydrogen and
antihydrogen identical (ALPHA's $1S$--$2S$ at $2\times10^{-12}$)
--- the six-comparison record of the main text's Part~I.

\emph{The modern CPT program, instrument by instrument.} An
experimenter should know where the exactness of CPT is
actually held to account, because the portfolio is broader than
the kaon.

Interferometric: the neutral-kaon diagonal equality
(the $6\times10^{-19}$ of \S\ref{app:d-lab}) and the $B$-system's
analogous bounds. Spectroscopic: antihydrogen --- ALPHA's
$1S$--$2S$ comparison with hydrogen at $2\times10^{-12}$
fractional precision \citep{ALPHA2020}, with the hyperfine
splitting and gravitational sign tested in parallel programs.

Electromagnetic: the Penning-trap comparisons --- proton versus
antiproton charge-to-mass at the level of parts in $10^{12}$,
and the $g$-factor comparison at parts in $10^{9}$ --- where a
single particle oscillating in a trap for months is the entire
experiment; and the electron--positron $g$ pair, agreeing at
parts in $10^{12}$. Leptonic: muon and antimuon lifetimes and
moments; muonium spectroscopy.

The pattern across the
portfolio: every test is a \emph{comparison of two clocks},
\begin{equation}
\delta_{\mathrm{CPT}}^{K} =
\frac{(M_{11}-M_{22}) - \tfrac{i}{2}(\Gamma_{11}-\Gamma_{22})}
{\lambda_S - \lambda_L},
\qquad
\frac{\nu_{1S2S}^{\bar H} - \nu_{1S2S}^{H}}{\nu_{1S2S}}
< 2\times10^{-12},
\qquad
\frac{(q/m)_{\bar p}}{(q/m)_p} + 1 \;\sim\; 10^{-12},
\label{eq:d4-clocks}
\end{equation}
a particle's oscillation, transition, or precession against
its conjugate's --- because CPT's content is precisely that the
conjugate world keeps identical time. And one theorem organizes
the stakes: Greenberg's result \citep{Greenberg2002} that CPT
violation in an interacting field theory entails Lorentz
violation, so the portfolio above doubles as the
low-energy anchor of the entire Lorentz-invariance program
(the standard-model-extension coefficient tables). (The full compilation of limits: \citealp{PDG2024}.)

For this
paper the consequence is stark and stated in the falsifier
table: the geometry has no parameter by which CPT could fail
a little --- any confirmed violation, in any system in this
portfolio, at any precision, retires the framework outright.
That is what it means, operationally, for CPT to be promoted
from theorem-given-axioms to anatomy: the entire portfolio
becomes one standing kill test with many hands on the switch.

\paragraph{Problems and worked solutions.}

\medskip\noindent\textbf{D.4.1. Carry the $\Theta$-walk through
$g\,\bar\psi\gamma^\mu\psi A_\mu$.}

\noindent\emph{Solution.} Assemble the CPT behavior factor by factor,
using the table of \S\ref{app:d-dirac} row by row. Under C:
$\bar\psi\gamma^\mu\psi \to -\bar\psi\gamma^\mu\psi$ and
$A_\mu \to -A_\mu$ (the photon is C-odd, which is why it couples
to a C-odd current): product invariant. Under P: the vector
current's index behaves as $(+,-)$ and so does $A_\mu$ evaluated
at the reflected point: each Lorentz index contracts evenly;
product invariant.

Under T (antiunitary): the current's index
behaves as $(+,-)$ again, $A_\mu(t,\vec x) \to
A^\mu(-t,\vec x)$, and the antilinearity conjugates the
coupling; since $g$ is real by Hermiticity, the term returns to
itself. Multiplying the three, every sign cancels index by
index and the interaction is exactly CPT invariant --- with
each factor's arguments now at $-x$, as the theorem demands.
The instructive second case: the QCD $\theta$-term
$\tfrac{\theta g^2}{32\pi^2}\,G\tilde G$ is P-odd and T-odd
(it is an $\vec E\cdot\vec B$ of color), yet it too passes the
$\Theta$-walk, because CPT never required P or T separately ---
only the total.

Physics intuition: CPT invariance of a term is
Hermiticity wearing spacetime clothes; and the $\theta$-term's
lawful presence is precisely why strong CP is a \emph{problem}
--- the frame permits the crime and cannot price it, which is
the opening of the main text's two-fate chapter.

\medskip\noindent\textbf{D.4.2. The free-scalar two-point function and its Jost-point
reflection.}

\noindent\emph{Solution.} Step 1: from the mode expansion,
$W(x) \equiv \langle 0|\phi(x)\phi(0)|0\rangle =
\int\!\frac{d^3p}{(2\pi)^3\,2\omega_p}\,e^{-i(\omega_p t -
\vec p\cdot\vec x)}$, every frequency \emph{positive}: the
spectrum condition in its simplest instance.

Step 2: because all
$\omega_p > 0$, the integrand decays when $t$ acquires a negative
imaginary part: $W$ extends holomorphically to the ``forward
tube'' $\mathrm{Im}\,t < 0$ (in the Lorentz-covariant statement,
$\mathrm{Im}\,x$ in the past cone).

Step 3: on this domain the
function is invariant under the connected complex Lorentz group;
setting the complexified boost/rotation parameters appropriately
one reaches the total reflection $x \to -x$ inside the domain of
analyticity.

Step 4: evaluate at spacelike (Jost) points, where
the tube's boundary values are ordinary functions: covariance
forces $W$ to depend only on the invariant $x^2$ there, and
$x^2$ is even under $x \to -x$; hence $W(x) = W(-x)$ for
spacelike $x$ --- which, undoing the definitions, is the CPT
relation for the two-point function (and, with locality, the
$n$-point generalization is Jost's theorem).

Physics intuition:
positivity of energy is a one-sided condition, and one-sided
conditions buy analyticity --- the freedom to steer around the
real axis and come back with a reflection you could never
perform physically. This is the exact mechanism the main text
names the clock's quarter-turn (T-P5-B1): Wick rotation, CPT,
and (in \S\ref{sec:quartet-second-child}) the exclusion
principle are all cashings of the same one-sided cheque.

\medskip\noindent\textbf{D.4.3. Partial widths and rescattering classes.}

\noindent\emph{Solution.} Step 1: CPT applied to the full resolvent
forces equal \emph{total} widths,
$\Gamma_{\rm tot}(i) = \Gamma_{\rm tot}(\bar\imath)$ (D.4's
consequence list).

Step 2: it does not force equal partial
widths. Write each decay amplitude with its weak (CP-odd) and
strong (CP-even rescattering) phases,
$A(i \to f) = \sum_k |A_k| e^{i\phi_k} e^{i\delta_k}$; then
$A(\bar\imath \to \bar f)$ has $\phi_k \to -\phi_k$ with
$\delta_k$ fixed, and
$\Gamma(i\to f) - \Gamma(\bar\imath\to\bar f) \propto
\sum_{k<l} |A_k||A_l| \sin(\phi_k - \phi_l)
\sin(\delta_k - \delta_l)$: a partial-rate asymmetry needs
\emph{both} a weak-phase difference and a strong-phase
difference.

Step 3: sum over the channels connected by the
rescattering (the same $\delta$'s): unitarity of the strong
S-matrix makes the asymmetries cancel within each class, so the
total-width equality of Step 1 is honored channel-class by
channel-class.

Physics intuition: CP asymmetries are
interference phenomena that borrow their visibility from
rescattering, and what one channel borrows another must repay.
This is exactly why baryogenesis cannot run in equilibrium ---
with all channels open and balanced, every asymmetry is
repaid --- and why Sakharov's third condition, the departure
from equilibrium, is load-bearing: the main text's Part~VIII
tripod is this problem's bookkeeping, promoted to a cosmology.

\subsection{Step 2 --- what quantum field theory cannot say}
\label{app:d-limits}

Now the audit, stated with the same care as the derivations.
\emph{(1) The operations are conventions.} Every discrete
operator above carries free phases per species ($\eta_P$,
$\eta_C$, $\eta_T$); only products constrained by interactions
are physical. QFT provides a \emph{consistency frame} for C, P,
T --- it does not define them individually, and seventy years
of data sit atop operators the formalism cannot uniquely name.
\emph{(2) The breakings are inputs.} Nothing in the framework
predicts that P is violated, that its violation is maximal,
that C fails with it, or that CP fails at
$2.28\times10^{-3}$ in the kaon: the V$-$A structure and the KM
phase are written into the Lagrangian by hand, and
$\delta = 0$ would have been equally lawful.

\emph{(3) The one
proved symmetry illuminates nothing about its factors.} The
CPT theorem holds for \emph{any} local theory; its proof gives
no insight into why nature breaks each factor separately ---
a conspiracy of accidents capped by an axiom. \emph{(4) The
$i$ is axiomatic.} Both CPT roads lean on the complex
structure and the positivity of energy as postulates; the
framework cannot see beneath its own $i$ (the main text's
Part~X verdict).

\emph{(5) No valuation.} Even granting the
breakings, QFT prices nothing: not the phase, not the angles,
not $\bar\theta$'s smallness, not the baryon excess ---
every key quantity of this paper's Table~\ref{tab:p5-ten-numbers}
is, in the standard account, a free parameter or a failure.
\emph{(6) The arrow is external.} T-violation in the
Lagrangian is unrelated, in the formalism, to the
thermodynamic arrow the laboratory uses to measure it.

Each limitation has a bench consequence, and stating them makes
the audit concrete. Because of (1), collaborations maintain
convention documents --- the phase choices behind every quoted
$\epsilon$, $\delta$, and Majorana phase --- and cross-experiment
comparisons begin with a translation exercise; there is no
physics in the conventions, but there is systematic error in
forgetting them.

Because of (2), no experiment can be told what
asymmetry to \emph{expect}: every search for new P, C, or CP
violation scans, and every discovery to date arrived
unpredicted in size.

Because of (3), the CPT tests of D.4
constrain all local field theories at once but adjudicate
between none of them.

Because of (4), the deepest constants in
the analysis chain --- the $i$ in every oscillation formula,
the positivity behind every dispersion relation an
experimenter uses to model detector response --- sit outside
the theory's explanatory reach; they are trusted, not derived.

Because of (5), the global-fit industry exists: CKM and PMNS
parameters are harvested, not predicted, and a fit's pull
distribution tests consistency, never necessity. (The living example is the global fit this paper confronts in Part~\ref{part:experiments}; \citealp{NuFit2024}.)

And because
of (6), the arrows in an analysis --- the thermodynamic arrow
of the detector, the causal arrow of the trigger, the T-odd
phases in the matrix element --- are formally unrelated,
related only by the physicist's good sense.

The main text's
wager is that all six are symptoms of one omission ---
\emph{the frame does not know what its own operations are} ---
and the next subsection is the repair proposal, graded.

\paragraph{Problems and worked solutions.}

\medskip\noindent\textbf{D.5.1. No rephasing removes the KM phase for three
generations; two generations are phase-free.}

\noindent\emph{Solution.} Step 1: count the parameters of an $N\times N$
unitary matrix: $N^2$ real numbers, of which $N(N-1)/2$ are
rotation angles and $N(N+1)/2$ are phases.

Step 2: count the
rephasings. The $2N$ quark fields (up-type and down-type) can
each absorb a phase, but one overall common phase does nothing
to $V$, leaving $2N - 1$ usable rephasings.

Step 3: subtract:
physical phases $= N(N+1)/2 - (2N-1) = (N-1)(N-2)/2$. For
$N = 2$: zero --- the Cabibbo matrix can always be made real,
and indeed no CP violation was ever found in a two-generation
fit. For $N = 3$: exactly one --- the KM phase, and the 1973
prediction that CP violation requires a third generation,
made when three quarks were known.

Physics intuition: phases
are only physical when they cannot be gauged away, i.e.\ when
they survive every relabeling of the unobservable field
conventions --- CP violation lives in closed loops of the
flavor graph, not in any single link. That loop-residence is
exactly what the main text means by ``the phase is a holonomy,''
and the pentagon (Part~VII) is the claim that the loop's value
is a lattice point.

\medskip\noindent\textbf{D.5.2. The sign of $\epsilon$'s CP violation is not an SM
prediction.}

\noindent\emph{Solution.} Step 1: $\epsilon$ as defined in D.3.1 depends
on the phase convention chosen for $|K^0\rangle$ and
$|\bar K^0\rangle$ (the Wu--Yang convention fixes it via the
$\pi\pi$, $I=0$ amplitude); only quantities invariant under
$|K^0\rangle \to e^{i\alpha}|K^0\rangle$ are physical ---
$\mathrm{Re}\,\epsilon$, $|\epsilon|$, and rephasing-invariant
combinations such as
$\mathrm{Im}(V_{td}V_{ts}^*V_{cd}^*V_{cs})$.

Step 2: within
the SM the \emph{value} of any such invariant traces back to
the KM phase $\delta$, and $\delta$ itself is an input: the
Lagrangian with $\delta \to -\delta$ (a mirror-flavored world)
is exactly as renormalizable, anomaly-free, and consistent as
ours.

Step 3: hence even given $|\epsilon|$, the SM cannot
predict its sign; the sign is measured, then propagated. The
same statement holds for every CP observable in
Table~\ref{tab:p5-ten-numbers}: the framework transports phases
impeccably and originates none.

Physics intuition: the SM's CP
machinery is a superb accounting system with a blank first
entry. The main text's pentagon is an attempt to write that
first entry from geometry --- and its sign prediction
($\sin\dPMNS = -0.28$, not $+$) is precisely the kind of
statement the accounting system could never make.

\medskip\noindent\textbf{D.5.3. Which limitations do supersymmetry, extra
dimensions, and strings address?}

\noindent\emph{Solution.} Walk the six of Step~2. Limitation (1),
operations-as-conventions: untouched by all three --- each
inherits the QFT frame and its free phases wholesale.
Limitation (2), breakings-as-inputs: partially addressed by
none in predictive form; SUSY \emph{adds} CP phases (generic
ones overshoot EDM bounds --- the SUSY CP problem), string
compactifications can break P and CP geometrically
(orientation-reversing elements), but with landscape-valued
outputs.

Limitation (3), the factor-blind CPT proof: untouched
--- all three prove CPT the same way. Limitation (4), the
axiomatic $i$: untouched; complex structure is postulated in
each. Limitation (5), no valuation: SUSY and strings add
parameters or vacua rather than prices; no rival framework
predicts a CP phase's central value. Limitation (6), the
external arrow: untouched. Score: (1), (3), (4), (6) survive
everywhere; (2) and (5) are where the main text's Part~X
locates the geometry's distinctive claims.

Physics intuition:
the roads beyond the SM extended its \emph{dynamics} while
inheriting its \emph{frame}; the mirrors are frame-level
questions, which is why fifty years of beyond-SM model
building moved them so little --- and why the geometry attacks
the frame instead.

\subsection{Step 3 --- the geometric account, and what became
predictive}
\label{app:d-gg}

The main text's Parts~IV--V answer limitation (1) by anatomy:
the bulk has exactly three real orientation-bearing factors ---
fiber circle, spatial slice, causal clock --- and C, P, T
\emph{are} their reversals (T-P5-1), with every free phase of
\S\ref{app:d-dirac} traced to a real convention. Limitation (4) dissolves
with it: the $i$ is the shared quarter-turn of circle and clock
(T-P5-A/B), the one-sided spectrum of \S\ref{app:d-cpt} is the clock's
positivity (T-P5-B1), and antiunitarity of T is no longer a
choice forced by consistency but a statement of custody ---
\emph{only the clock is what $i$ is made of}.

Limitation (3)
inverts: CPT survives not as a conspiracy but as the unique
total reversal invisible to every real texture (T-P5-2, at
quasi-free grade with its residues named), and the same
analytic fact carries spin--statistics as its sibling
(\S\ref{sec:quartet-second-child}). Limitations (2) and (5)
are the business of the later chapters and their courses: the
one kept hand (Part~VI), the pentagon's priced phases
(Part~VII), the two-fate strong-CP resolution and the
six-in-ten-billion receipt (Part~VIII) --- each with its grade,
each mortal on the calendar of Part~XI. Limitation (6) is the
cohort's deepest claim, graded in
\S\ref{sec:mirrors-to-mind}: the laboratory's arrow and the
Lagrangian's T-violation meet in one oriented boundary
semigroup.

What the student should take from this chapter is
the shape of the trade: \emph{QFT proved one theorem about the
mirrors and defined none of them; the geometry defines all
three, re-proves the theorem, and prices the breakings ---
and pays for it by becoming mortal.}

\emph{The experimenter's translation table.} What changes at
the bench if the geometric account is right? First the
operations themselves, as maps on the bulk's three real
oriented factors --- fiber angle $\theta$, spatial slice
$\vec x$, clock parameter $s$:
\begin{equation}
\mathrm{C}:\ (\theta, \vec x, s) \mapsto (-\theta, \vec x, s),
\qquad
\mathrm{P}:\ (\theta, \vec x, s) \mapsto (\theta, -\vec x, s),
\qquad
\mathrm{T}:\ (\theta, \vec x, s) \mapsto (\theta, \vec x, -s),
\label{eq:d6-maps}
\end{equation}
with antiunitarity assigned by clock parity (four operations
reverse $s$; four do not), and the product
$\mathrm{CPT}: (\theta,\vec x,s) \mapsto
(-\theta,-\vec x,-s)$ the total reversal no real texture
detects. In apparatus: nothing changes --- the cryostats,
traps, and baselines of this appendix already measure the
right things.

In interpretation: five concrete relabellings. A parity flip of the apparatus is,
in the geometry, a reversal of one real orientation of the
world's spatial factor --- and the reason no experiment has
ever prepared the mirrored \emph{couplings} (a $V\!+\!A$
current) is that the descent kept one hand: F-P5-1 predicts
the $W_R$ search program is a permanent null, forever, which
converts every future limit from a disappointment into a data
point.

A charge swap is a fiber reversal --- so the CPT
portfolio of D.4 is measuring the equality of two readings of
one bulk configuration, and its exactness stops being an
axiom's shadow and becomes a geometric identity with a kill
switch. A motion reversal is a clock reversal --- so the
absence of a T quantum number (D.1) is custody, not accident. (T-P5-1, proved in Part~\ref{part:theorems}.)

The kaon and long-baseline phases are pentagon holonomies ---
so a global fit's central values become \emph{predictions to
be shot at}, and the pull columns of
Table~\ref{tab:exp-predicted-measured} are the new daily
scoreboard.

And the arrow of the laboratory --- the fact that
every trigger fires forward --- is the same one-way boundary
semigroup that froze the baryon excess, which is why the
proton's silence each morning is, on this account, a
discrete-symmetry measurement. (The receipt chain is Part~\ref{part:matter}; the schedule, Table~\ref{tab:exp-timeline}.) None of these relabellings asks
the experimenter to believe; each asks only that a specific
number land where the lattice says. The apparatus of seventy
years was always adequate; what the geometry adds is a set of
places to stand and a schedule for being wrong.

\paragraph{Problems and worked solutions.}

\medskip\noindent\textbf{D.6.1. Recompute the (anti)unitary table from the
three-reversal definition and compare with Wigner's fork.}

\noindent\emph{Solution.} Step 1: the geometric rule (T-P5-1): an
operation is antiunitary if and only if it reverses the clock
orientation, because the clock's forward direction is the only
real structure from which the boundary $i$ is built --- reverse
it and every phase conjugates, which is antilinearity.

Step 2:
enumerate the eight operations generated by C, P, T. Four leave
the clock alone: $\mathbf 1$, C (fiber reversal), P (spatial
reversal), CP --- all unitary. Four reverse it: T, CT, PT, and
CPT --- all antiunitary. (This corrects a compressed statement
in an earlier printing of this appendix; the count is
four-and-four, fixed by clock-parity alone.)

Step 3: compare
with Wigner. The fork of D.1 said each symmetry \emph{may} be
unitary or antiunitary, with consistency (energy positivity)
forcing T's branch case by case; the geometry replaces the
case-by-case argument with one custody statement --- the
operations containing the clock reversal are exactly the
antiunitary ones --- and the composition sign rule
(antiunitary $\times$ antiunitary $=$ unitary, e.g.\
$T \cdot CT = C$ up to phases) becomes parity of clock
reversals, i.e.\ arithmetic mod 2.

Physics intuition: Wigner
gave quantum mechanics a fork; the geometry tells it which
tine, and why --- \emph{only the clock is what $i$ is made
of} --- turning a consistency choice into anatomy.

\medskip\noindent\textbf{D.6.2. State the grade of each Step-3 claim and where its
residue is named.}

\noindent\emph{Solution.} Walk the ledger
(\S\ref{sec:claims-ledger}) claim by claim. The C/P/T table
(limitation 1's answer): \emph{derived} --- T-P5-1, no residue.
The shared $i$ and the Euclidean license (limitation 4):
T-P5-A/B/B1, \emph{derived at quasi-free grade}, interacting
case on scattering hypotheses. CPT as unique total reversal
(limitation 3): T-P5-2, \emph{derived (quasi-free)}, with the
modular (Bisognano--Wichmann) route the named path to
interacting strength --- open item 6 of
\S\ref{sec:open-questions}; its spin--statistics sibling is
\emph{candidate} grade (\S\ref{sec:quartet-second-child}).

The
kept hand (limitation 2, parity half): \emph{conditional} on
the gate-uniqueness conjecture C-P5-A, with the projection's
own necessity the parity ledger's honestly-flagged assertion.
The priced phases (limitation 5, CP half):
\emph{derived-conditional} --- lattice rigidity plus the
flagged QD step for the quark phase, the assignment lemma for
the leptonic. The receipt $\etaB$: \emph{D$^{*}$}, one named
conjecture C-P5-C. The arrow (limitation 6): graded in
\S\ref{sec:mirrors-to-mind} as the cohort's conjectured
summit chain beyond the matter link.

Physics intuition: the
grades are the theory's honesty made machine-readable ---
a student who can recite this answer can referee the paper.

\medskip\noindent\textbf{D.6.3. The experimentalist's closing drill: for each
limitation, the measurement and its date.}

\noindent\emph{Solution.} Limitation (1), the anatomy: tested wherever
the operations' derived structure bites --- Kramers
degeneracy phenomenology (standing) and the exactness of CPT:
kaon $\Delta m$ (continuous), antihydrogen $1S$--$2S$
(continuous, improving) --- any confirmed CPT violation at any
level kills the anatomy outright. Limitation (2), the kept
hand: $W_R$ and right-handed-current searches at HL-LHC and
successors --- permanent, any confirmed restoration fatal
(F-P5-1).

Limitation (3)/(4), the quartet: same CPT rows,
plus --- through the second child --- the Pauli-exclusion
nulls (VIP-2 class, standing). Limitation (5), the valuation:
$\dCKM$ at LHCb/Belle~II (continuous, $5\sigma$ exclusion of
$13\pi/35$ kills), $\dPMNS$ at Hyper-K/DUNE ($270^{\circ}$ at
$5\sigma$, 2031--35, kills; adjudication versus
$196.4^{\circ}$ matures late 2030s), ordering at JUNO/DUNE
($5\sigma$ inverted $\sim$2033 kills), $\mbb$ at the ton
scale (any discovery in reach kills, 2030s), $\bar\theta$ at
n2EDM (in-window EDM kills, 2030s+).

Limitation (6), the
arrow-and-matter link: the proton rail (one event, any day)
and the $\etaB$ audit against the CMB (standing). Every row
sits in Table~\ref{tab:exp-timeline} with its instrument and
date; the drill's lesson is the paper's design --- \emph{each
philosophical limitation of the old frame has been converted
into at least one dated measurement}, which is what
``anatomy, priced and mortal'' means in practice.

\section{The flavor sector and CP violation, from scratch:
Yukawas, the CKM matrix, and the pentagon}
\label{app:flavour}

The second chapter of Block II treats the physics that Part~VII
prices: quark flavor and its one CP phase. Step 1 builds the
Standard Model's flavor sector from the Yukawa couplings, exactly
as the textbooks teach it, through the kaon and $B$-meson
measurements that made it precision science. Step 2 states what
that superb machinery cannot do. Step 3 states what the pentagon
does.

The chapter builds on Appendix~D and does not repeat it: the
two-state mixing formalism, the Wigner theorem, and the
antiunitarity of T are quoted from \S\ref{app:d-lab} and
\S\ref{app:d-wigner} where first derived. What is new here is
the flavor structure itself --- where the mixing matrix comes
from, which of its entries are physical, and how two generations
of experiments turned a $2\times10^{-3}$ anomaly into a
$10^{-4}$-precision science.

The chapter shelf: \citet{BrancoLavouraSilva} is the reference
monograph (Chs.~2--3 for the formalism of E.1--E.2, Chs.~6--8 for
the kaon system of E.3); \citet{BigiSanda2009} carries the
$B$-physics of E.4 at book length; \citet{Sozzi2008} the
experimental face throughout; \citet{PDG2024} the living numbers.
Eighteen problems, complete solutions included.

\subsection{Step 1 --- from Yukawa couplings to the CKM matrix}
\label{app:flavour-ckm}

The Standard Model forbids bare quark masses; masses enter only
through the Higgs couplings
\begin{equation}
-\mathcal{L}_Y \;=\; Y^u_{ij}\,\bar Q_i \tilde H\, u_{Rj}
\;+\; Y^d_{ij}\,\bar Q_i H\, d_{Rj} \;+\; \mathrm{h.c.},
\label{eq:e1-yukawa}
\end{equation}
with $Y^{u,d}$ arbitrary complex $3\times3$ matrices --- eighteen
complex numbers nature was free to choose. After electroweak
symmetry breaking, $H \to (0, v/\sqrt2)$, these become mass
matrices $M^{u,d} = Y^{u,d} v/\sqrt2$.

Any complex matrix can be diagonalized by a pair of unitaries
(the singular-value decomposition):
\begin{equation}
M^u = U^u_L\, \hat M^u\, U^{u\dagger}_R,
\qquad
M^d = U^d_L\, \hat M^d\, U^{d\dagger}_R,
\label{eq:e1-svd}
\end{equation}
with $\hat M^{u,d}$ diagonal and positive. Rotate the quark
fields to the mass basis. Every neutral term of the Lagrangian
--- kinetic, photon, gluon, $Z$ --- involves $U^\dagger U =
\mathbf 1$ and forgets the rotation. Exactly one term remembers:
the charged current,
\begin{equation}
J^\mu_W \;=\; \bar u_L \gamma^\mu d_L
\;\longrightarrow\;
\bar u_L \gamma^\mu \big(U^{u\dagger}_L U^d_L\big) d_L
\;\equiv\; \bar u_L \gamma^\mu V\, d_L .
\label{eq:e1-ckm}
\end{equation}
The CKM matrix $V = U^{u\dagger}_L U^d_L$
\citep{Cabibbo1963, KobayashiMaskawa1973} is the mismatch of the
two diagonalizations --- the gear ratio between the up-type and
down-type mass frames --- and it appears only where the $W$
changes quark type.

Physical intuition: nature keeps two ledgers for the quarks, one
per Higgs coupling, and the weak interaction is the sole
transaction that reads both at once. Everything called ``flavor
physics'' is the audit of that one transaction.

Before the mixing, pause on what the diagonalization returns:
the eigenvalues. The Yukawa spectrum spans thirteen decades,
\begin{equation}
y_t = \frac{\sqrt2\, m_t}{v} \simeq 0.99,
\qquad
y_e = \frac{\sqrt2\, m_e}{v} \simeq 2.9\times10^{-6},
\label{eq:e1-yukspan}
\end{equation}
with the top quark, alone, sitting at order one --- as if one
fermion couples naturally and the other eleven are afterthoughts.
No SM principle explains the span; it is simply what
\eqref{eq:e1-yukawa} was told.

The flavor structure has predicted a quark before. The
$K_L$--$K_S$ mass difference proceeds through the box diagram,
and with only three quarks it comes out orders too large; with a
fourth quark of mass $m_c$ the GIM cancellation leaves
\begin{equation}
\Delta m_K \;\sim\; \frac{G_F^2}{6\pi^2}\, f_K^2 m_K\,
m_c^2 \sin^2\theta_C \cos^2\theta_C ,
\label{eq:e1-gimbox}
\end{equation}
and demanding agreement with the measured
$\Delta m_K = 3.48\times10^{-12}$~MeV pointed to
$m_c \sim 1.5$~GeV --- announced before November 1974 delivered
the $J/\psi$ (the loop analysis at textbook depth:
\citealp{PeskinSchroeder1995}, \S20.2; \citealp{BrancoLavouraSilva},
Ch.~8). Flavor loops see the next floor of the building before
the elevator gets there --- the working faith of every rare-decay
program on Part~\ref{part:experiments}'s timeline.

How much of $V$ is physical? By the counting of
\S\ref{app:d-limits} (problem D.5.1): three angles and one phase
for three generations. The measured hierarchy motivates
Wolfenstein's expansion \citep{Wolfenstein1983} in
$\lambda = \sin\theta_C \simeq 0.225$:
\begin{equation}
V \;\simeq\;
\begin{pmatrix}
1 - \tfrac{\lambda^2}{2} & \lambda & A\lambda^3(\rho - i\eta) \\
-\lambda & 1 - \tfrac{\lambda^2}{2} & A\lambda^2 \\
A\lambda^3(1 - \rho - i\eta) & -A\lambda^2 & 1
\end{pmatrix}
+ \mathcal{O}(\lambda^4),
\label{eq:e1-wolfenstein}
\end{equation}
with $A \simeq 0.83$: nearly diagonal, the CP phase hiding in the
smallest corners $V_{ub}$ and $V_{td}$. That is why CP violation
is rare in the laboratory --- not because the phase is small
(it is $\sim 66.9^{\circ}$) but because it is stored in
$\lambda^3$-suppressed elements.

The main text's Part~VII reads \eqref{eq:e1-ckm} geometrically:
$V$ is a comparison of frames, and comparisons of frames are
where holonomy lives. The pentagon (Step 3) will price what this
Step only parameterizes.

\medskip\noindent\textbf{E.1.1. Show that for $N$ generations $V$
has $(N-1)(N-2)/2$ physical phases, and exhibit where the single
$N=3$ phase sits in the standard convention.}

\noindent\emph{Solution.} Step 1: an $N\times N$ unitary matrix
has $N^2$ real parameters: $N(N-1)/2$ angles and $N(N+1)/2$
phases. Step 2: rephasing the $2N$ quark fields removes phases,
but the overall common phase leaves $V$ untouched, so $2N-1$ are
usable: physical phases $= N(N+1)/2 - (2N-1) = (N-1)(N-2)/2$.

Step 3: for $N = 3$ this is one. In the PDG convention it is
attached to $s_{13}$: $V_{ub} = s_{13}e^{-i\delta}$, so the phase
sits in the smallest element --- consistent with
\eqref{eq:e1-wolfenstein}, where $\rho - i\eta$ carries it.

Physics intuition: phases you can shift onto field conventions
are bookkeeping; the one you cannot is physics. CP violation is
what survives every relabeling --- which is why Part~VII calls
it a holonomy.

\medskip\noindent\textbf{E.1.2. Derive the Wolfenstein form
\eqref{eq:e1-wolfenstein} from the standard parameterization to
order $\lambda^3$.}

\noindent\emph{Solution.} Step 1: define $\lambda = s_{12}$,
$A\lambda^2 = s_{23}$, $A\lambda^3(\rho - i\eta) =
s_{13}e^{-i\delta}$ --- these are definitions, exact.

Step 2: expand the standard-convention product of the three
rotations. To $\mathcal O(\lambda^3)$: $c_{12} =
1 - \lambda^2/2$, $c_{23} = c_{13} = 1$, and every product of
two small sines beyond the listed corners is
$\mathcal O(\lambda^4)$ or smaller.

Step 3: assemble; the $(3,1)$ element picks up
$s_{12}s_{23} - c_{12}c_{23}s_{13}e^{i\delta} =
A\lambda^3(1 - \rho - i\eta)$ after the definitions. The
expansion is \eqref{eq:e1-wolfenstein}.

Physics intuition: the Wolfenstein form is not an approximation
scheme so much as a magnifying glass --- it displays the
hierarchy $1 : \lambda : \lambda^2 : \lambda^3$ that the exact
parameterization hides, and the hierarchy is itself an
unexplained pattern the geometry must eventually own.

\medskip\noindent\textbf{E.1.3. Show that the neutral ($Z$,
photon, gluon, Higgs) couplings are flavor-diagonal in the mass
basis --- the GIM statement --- and name the experimental
consequence.}

\noindent\emph{Solution.} Step 1: every neutral current has the
form $\bar q_L \Gamma q_L$ or $\bar q_R \Gamma q_R$ with
$\Gamma$ proportional to the identity in flavor space (all
up-type quarks carry the same charges).

Step 2: rotating to the mass basis inserts $U^\dagger
\mathbf 1 U = \mathbf 1$: the rotation cancels. The Higgs
couplings are $M^{u,d}/v$, diagonalized by the same rotation ---
diagonal too.

Step 3: consequence --- no tree-level flavor-changing neutral
currents. $K^0 \to \mu^+\mu^-$ is loop-suppressed
(branching $\sim 7\times10^{-9}$), and rare-decay physics
(NA62's $K^+ \to \pi^+\nu\bar\nu$, Part~\ref{part:experiments})
lives entirely at one loop --- which is precisely why it is a
sensitive probe.

Physics intuition: flavor change costs a $W$; anything neutral
must rent two. The GIM suppression is the reason kaon physics
could see the charm quark coming (its mass entered the loops)
--- flavor loops are telescopes for the next scale up.

\subsection{Step 1 --- rephasing invariants: the Jarlskog
construction and the unitarity triangle}
\label{app:flavour-invariants}

Which functions of $V$ can an experiment measure? Rephasing
sends $V_{ij} \to e^{i(\beta_j - \alpha_i)}V_{ij}$, so the
moduli $|V_{ij}|$ survive, single phases do not, and the
smallest surviving phase-sensitive objects are the
\emph{quartets}
\begin{equation}
Q_{ij,kl} \;=\; V_{ij}V_{kl}V_{il}^*V_{kj}^* ,
\label{eq:e2-quartet}
\end{equation}
closed loops in the flavor graph: every field phase enters once
with each sign. For three generations, unitarity forces every
quartet's imaginary part to a common magnitude
\citep{Jarlskog1985},
\begin{equation}
\mathrm{Im}\,Q_{ij,kl} \;=\; J \sum_{m,n}
\epsilon_{ikm}\,\epsilon_{jln},
\qquad
J \;=\; c_{12}c_{23}c_{13}^2\,s_{12}s_{23}s_{13}\sin\delta,
\label{eq:e2-jinv}
\end{equation}
one number carrying all of quark CP violation. Its measured
value is $J = (3.12^{+0.13}_{-0.12})\times10^{-5}$
(Table~\ref{tab:exp-predicted-measured}).

Jarlskog's own packaging of the result deserves display. Form
the commutator of the two mass matrices (squared, to stay in
the left-handed sector):
\begin{equation}
\det\big[\,M^u M^{u\dagger},\, M^d M^{d\dagger}\,\big]
\;=\; -2i\, J\, F_u F_d,
\qquad
F_u = (m_t^2 - m_c^2)(m_t^2 - m_u^2)(m_c^2 - m_u^2),
\label{eq:e2-commutator}
\end{equation}
with $F_d$ the down-type analogue. CP violation in the quark
sector is one number multiplied by every mass splitting: let any
two same-type quarks become degenerate and the effect dies,
because the degenerate pair can be rotated to hide the phase.
The observable needs the full cast --- three generations,
non-degenerate, and one loop.

The same structure draws a picture. Orthogonality of the first
and third columns,
\begin{equation}
V_{ud}V_{ub}^* + V_{cd}V_{cb}^* + V_{td}V_{tb}^* \;=\; 0,
\label{eq:e2-triangle}
\end{equation}
is three complex numbers summing to zero: a triangle in the
complex plane --- the \emph{unitarity triangle}, with angles
named $\alpha, \beta, \gamma$. Its area is $J/2$ (problem
E.2.2): CP violation is literally the failure of the flavor
ledger to collapse onto a line.

Physical intuition: a quartet is a round trip --- up-frame to
down-frame and back by a different route --- and $J$ is the
interference the traveler accumulates. Part~VII's claim is that
these loops are plaquettes of a real texture, and their common
phase is a lattice angle. The pentagon owns the loops; Step 3
collects the debt.

\medskip\noindent\textbf{E.2.1. Prove that for three generations
every CP-violating observable is proportional to $J$.}

\noindent\emph{Solution.} Step 1: any observable is a
rephasing-invariant function of $V$, hence a function of moduli
and quartets (higher loops factor into quartets by unitarity).

Step 2: CP conjugation maps $V \to V^*$, so a CP-odd observable
is odd under $Q \to Q^*$: it depends on the quartets only
through their imaginary parts.

Step 3: by \eqref{eq:e2-jinv} every $\mathrm{Im}\,Q$ is
$\pm J$ or $0$. Hence CP-odd $\Rightarrow$ $\propto J$ (times
moduli). One phase, one invariant, one universal factor ---
the counting theorem of \S\ref{app:d-limits} in its working
clothes.

Physics intuition: this is why the kaon's $2\times10^{-3}$ and
the $B$ system's order-one asymmetries are the \emph{same}
violation --- the difference is kinematic dressing of one $J$.

\medskip\noindent\textbf{E.2.2. Show the unitarity triangle's
area equals $J/2$.}

\noindent\emph{Solution.} Step 1: for a triangle with vertices
at $0$, $z_1$, and $z_1 + z_2$, the area is
$\tfrac12|\mathrm{Im}(z_1^* z_2)|$.

Step 2: take $z_1 = V_{ud}V_{ub}^*$, $z_2 = V_{cd}V_{cb}^*$
from \eqref{eq:e2-triangle}. Then $\mathrm{Im}(z_1^* z_2) =
\mathrm{Im}(V_{ud}^*V_{ub}V_{cd}V_{cb}^*)$ --- a quartet, equal
to $\pm J$ by \eqref{eq:e2-jinv}.

Step 3: area $= J/2$. The same holds for all six unitarity
triangles: different shapes, one area.

Physics intuition: experiments measure the triangle's
\emph{sides} (moduli from rates) and \emph{angles} (phases from
asymmetries) independently; consistency of the over-determined
figure is the SM's great flavor test --- and its area is the
one number the SM never predicted.

\medskip\noindent\textbf{E.2.3. Evaluate $J$ from the
Wolfenstein parameters and check against the world value.}

\noindent\emph{Solution.} Step 1: from
\eqref{eq:e2-jinv} with the hierarchy of E.1,
$J \simeq \lambda \cdot A\lambda^2 \cdot A\lambda^3\eta
= A^2\lambda^6\eta$.

Step 2: insert $\lambda = 0.225$, $A = 0.83$, $\eta = 0.35$:
$\lambda^6 = 1.30\times10^{-4}$, so
$J \simeq 0.69 \times 1.30\times10^{-4} \times 0.35 =
3.1\times10^{-5}$.

Step 3: the world value is
$(3.12^{+0.13}_{-0.12})\times10^{-5}$ --- agreement, and the
$\lambda^6$ explains the smallness: six powers of the Cabibbo
angle, not a small phase.

Physics intuition: the suppression is geometric --- the loop
must visit the smallest corners of $V$ --- which is why
Part~VII can carry a large lattice phase ($66.9^{\circ}$) and
still respect a tiny $J$: the pentagon sets the angle, the
hierarchy sets the price of walking the loop.

\subsection{Step 1 --- the kaon system: $\varepsilon$ and
$\varepsilon'$, derived}
\label{app:flavour-kaons}

Begin where the field began. In 1964, at the Brookhaven AGS, a
beam of $K_L$ --- the long-lived, supposedly CP-odd kaon ---
produced $\pi^+\pi^-$ pairs at a branching of $2\times10^{-3}$
\citep{Cronin1964}. The CP-even final state from the CP-odd
beam meant CP itself was broken, and the experiment's craft lay
in excluding the mundane: coherent regeneration of $K_S$ in
residual gas was controlled by running the decay volume in
helium and mapping the rate against material.

Two decades of \S\ref{app:d-lab}-style systematics hunting were
compressed into one clean result --- and forty years of
theoretical structure, Steps E.1--E.2 included, exist because
that branching ratio refused to be zero.

Indirect CP violation first. From the mixing formalism of
\S\ref{app:d-lab}, with CPT holding and the physical phase
convention fixed by the dominant $I = 0$ decay amplitude,
\begin{equation}
\varepsilon \;\simeq\; \frac{e^{i\phi_{SW}}}{\sqrt2}\,
\frac{\mathrm{Im}\,M_{12}}{\Delta m_K},
\qquad
\phi_{SW} = \arctan\frac{2\Delta m_K}{\Delta\Gamma_K}
\simeq 43.5^{\circ},
\label{eq:e3-eps}
\end{equation}
the ``superweak phase'' fixed by measured mixing quantities
alone. The box diagram gives the short-distance part,
$\mathrm{Im}\,M_{12} \propto G_F^2 f_K^2 m_K \hat B_K\,
\mathrm{Im}\big[(V_{td}V_{ts}^*)^2\big]\,m_t^2$-dominated ---
a quartet again, so $|\varepsilon| \propto J$ as E.2.1
promised. The measured $|\varepsilon| = 2.228(11)\times10^{-3}$
is the 1964 discovery in mature form.

The same $\varepsilon$ leaks into the semileptonic channels.
Since $K^0 \to \pi^-\ell^+\nu$ but never $\pi^+\ell^-\bar\nu$
(the $\Delta S = \Delta Q$ rule), the lepton charge tags the
strangeness, and the $K_L$ charge asymmetry measures the mixing
directly:
\begin{equation}
\delta_L \;=\;
\frac{\Gamma(K_L \to \pi^-\ell^+\nu) -
\Gamma(K_L \to \pi^+\ell^-\bar\nu)}
{\Gamma(K_L \to \pi^-\ell^+\nu) +
\Gamma(K_L \to \pi^+\ell^-\bar\nu)}
\;=\; \frac{2\,\mathrm{Re}\,\varepsilon}{1 + |\varepsilon|^2}
\;=\; 3.3\times10^{-3},
\label{eq:e3-deltaL}
\end{equation}
matter and antimatter counted by lepton charge, agreeing with
$|\varepsilon|$ and $\phi_{SW}$ --- the same asymmetry CPLEAR
re-read as the direct T-reversal test $A_T = 4\,\mathrm{Re}\,
\varepsilon$ \citep{CPLEAR1998}, the arrow-of-time measurement
of \S\ref{app:d-lab}.

Direct CP violation is subtler: it needs \emph{two} decay
amplitudes with different weak and strong phases
(\S\ref{app:d-cpt}, problem D.4.3). The kaon supplies them as
the two pion isospin channels $A_0$ and $A_2$. The observables
are the ratios $\eta_{+-} = \varepsilon + \varepsilon'$ and
$\eta_{00} = \varepsilon - 2\varepsilon'$, so
\begin{equation}
\mathrm{Re}\,\frac{\varepsilon'}{\varepsilon}
\;\simeq\; \frac{1}{6}\Big(1 -
\Big|\frac{\eta_{00}}{\eta_{+-}}\Big|^2\Big)
\;=\; \frac{1}{6}\Big(1 - R\Big),
\qquad
R = \frac{\Gamma(K_L\to\pi^0\pi^0)/\Gamma(K_S\to\pi^0\pi^0)}
{\Gamma(K_L\to\pi^+\pi^-)/\Gamma(K_S\to\pi^+\pi^-)},
\label{eq:e3-epsprime}
\end{equation}
the \emph{double ratio}: four rates, most systematics
canceling between numerator and denominator --- the design
insight that carried both KTeV \citep{KTeV1999} and NA48
\citep{NA48_2002}. The world value,
$\mathrm{Re}(\varepsilon'/\varepsilon) =
(16.6\pm2.3)\times10^{-4}$, established direct CP violation
and killed the superweak alternative
\citep{Wolfenstein1964}: CP violation lives in decay
amplitudes, not only in mixing.

Physical intuition: $\varepsilon$ is interference in the
\emph{clock} (the oscillation), $\varepsilon'$ interference in
the \emph{exits} (the decay channels). Nature could have had
the first without the second; measuring both, and their ratio
to $10^{-4}$, is what promoted kaon physics from anomaly to
precision science. Part~VII recomputes both as holonomy
interferometry, with the phase supplied by the lattice.

\medskip\noindent\textbf{E.3.1. Derive the superweak phase
$\phi_{SW} = \arctan(2\Delta m/\Delta\Gamma)$.}

\noindent\emph{Solution.} Step 1: from \S\ref{app:d-lab}, the
eigenvalue difference is $\lambda_S - \lambda_L = \Delta m -
\tfrac{i}{2}\Delta\Gamma$ (sign conventions fixed by
$\Gamma_S \gg \Gamma_L$).

Step 2: $\varepsilon \propto \mathrm{Im}\,M_{12} /
(\lambda_S - \lambda_L)$; the phase of the denominator is
$-\arctan\!\big(\tfrac{\Delta\Gamma/2}{\Delta m}\big)$, so the
phase of $\varepsilon$ is
$\arctan\!\big(\tfrac{2\Delta m}{\Delta\Gamma}\big)$.

Step 3: insert the measured $\Delta m = 3.48\times10^{-12}$~MeV
and $\Delta\Gamma \simeq \Gamma_S = 7.35\times10^{-12}$~MeV:
$\phi_{SW} = \arctan(0.947) = 43.4^{\circ}$ --- consistent with
the measured phase of $\eta_{+-}$, itself a CPT test.

Physics intuition: the phase of $\varepsilon$ is set by
\emph{lifetimes and oscillation}, not by the CP-violating
dynamics --- so its agreement with measurement checks the
framework, while only $|\varepsilon|$ carries the new physics.

\medskip\noindent\textbf{E.3.2. Derive the double-ratio formula
in \eqref{eq:e3-epsprime} to first order in
$\varepsilon'/\varepsilon$.}

\noindent\emph{Solution.} Step 1:
$|\eta_{+-}|^2 = |\varepsilon|^2\,
|1 + \varepsilon'/\varepsilon|^2 \simeq |\varepsilon|^2
(1 + 2\,\mathrm{Re}\,\tfrac{\varepsilon'}{\varepsilon})$,
and likewise $|\eta_{00}|^2 \simeq |\varepsilon|^2(1 -
4\,\mathrm{Re}\,\tfrac{\varepsilon'}{\varepsilon})$.

Step 2: the double ratio is
$R = |\eta_{00}|^2/|\eta_{+-}|^2 \simeq
1 - 6\,\mathrm{Re}(\varepsilon'/\varepsilon)$.

Step 3: invert: $\mathrm{Re}(\varepsilon'/\varepsilon) =
(1 - R)/6$, which is \eqref{eq:e3-epsprime}. A $10^{-3}$
measurement of $R$ delivers a $2\times10^{-4}$ measurement of
$\varepsilon'/\varepsilon$ --- the leverage that made the
experiments feasible.

Physics intuition: ratios of ratios are the experimenter's
noise-canceling headphones. Flux, acceptance, and most
efficiencies enter all four rates and cancel; what survives is
the physics difference between charged and neutral pions ---
and twenty years of KTeV/NA48 systematics work lived in exactly
the residue that does not cancel.

\medskip\noindent\textbf{E.3.3. Explain what a nonzero
$\varepsilon'$ killed, and why $\varepsilon$ alone could not.}

\noindent\emph{Solution.} Step 1: the superweak hypothesis
\citep{Wolfenstein1964} placed all CP violation in a new
$\Delta S = 2$ mixing interaction: $\varepsilon \neq 0$,
$\varepsilon' = 0$ exactly.

Step 2: $\varepsilon$ measures mixing; any theory with the
right $\mathrm{Im}\,M_{12}$ reproduces it. Only a
\emph{decay-amplitude} phase difference between $A_0$ and
$A_2$ produces $\varepsilon' \neq 0$.

Step 3: the measured $(16.6\pm2.3)\times10^{-4}$ is nonzero at
$7\sigma$: CP violation exists in the $\Delta S = 1$
amplitudes, as the KM mechanism requires and superweak forbids.
One decimal place, one dead theory.

Physics intuition: the episode is the cleanest historical case
of the discrimination logic Part~XII's matrix formalizes ---
two frameworks agreeing on the first observable, split by the
second. The pentagon inherits the same exposure: it agrees with
the KM \emph{structure} everywhere and differs in one thing
only --- the phase has a value.

\subsection{Step 1 --- the $B$ factories: $\sin 2\beta$ as
interferometry}
\label{app:flavour-bfactories}

The $B^0$ system runs the same two-state mechanics with a
crucial kinematic difference: $\Delta\Gamma_d \simeq 0$, so
both mass eigenstates live long enough to interfere in decays.
For a CP eigenstate final state $f$ accessible to both $B^0$
and $\bar B^0$, define
\begin{equation}
\lambda_f = \frac{q}{p}\,\frac{\bar A_f}{A_f},
\qquad
A_{CP}(t) \;=\;
\frac{\Gamma(\bar B^0(t)\to f) - \Gamma(B^0(t)\to f)}
{\Gamma(\bar B^0(t)\to f) + \Gamma(B^0(t)\to f)}
\;=\; S_f \sin(\Delta m_d\, t) - C_f \cos(\Delta m_d\, t),
\label{eq:e4-tdcp}
\end{equation}
with $S_f = 2\,\mathrm{Im}\,\lambda_f/(1 + |\lambda_f|^2)$ and
$C_f = (1 - |\lambda_f|^2)/(1 + |\lambda_f|^2)$. For the golden
mode $B^0 \to J/\psi K_S$, the decay is dominated by a single
real amplitude ($b \to c\bar c s$), so $|\lambda_f| = 1$,
$C_f = 0$, and the surviving phase is pure mixing:
\begin{equation}
\mathrm{Im}\,\lambda_{J/\psi K_S}
= \sin\!\big(2\beta\big),
\qquad
\beta = \arg\Big(\!-\frac{V_{cd}V_{cb}^*}{V_{td}V_{tb}^*}\Big),
\label{eq:e4-sin2b}
\end{equation}
a unitarity-triangle angle read as a time-dependent
interference fringe. The 2001 measurements
\citep{BaBar2001, Belle2001} landed at large $\sin2\beta$ ---
CP violation outside the kaon at last --- and the current world
average is $0.709 \pm 0.011$.

The apparatus deserves one paragraph of respect. At the
$\Upsilon(4S)$ the $B\bar B$ pair is quantum-coherent: neither
meson has a flavor until one decays, and tagging the flavor
of one fixes the other's clock at that instant. The asymmetric
beam energies boost the pair so the decay-time difference
becomes a measurable vertex separation ($\sim 250\,\mu$m).
Entanglement, here, is not philosophy; it is the tagging
scheme.

Tagging is imperfect, and the imperfection has a clean algebra.
If a fraction $w$ of tags are wrong, the observed asymmetry is
diluted,
\begin{equation}
A^{\mathrm{obs}}_{CP}(t) \;=\; (1 - 2w)\, A_{CP}(t),
\qquad
\sigma(S) \;\propto\; \frac{1}{(1-2w)\sqrt{N}},
\label{eq:e4-dilution}
\end{equation}
so a $30\%$ mistag rate costs a factor $2.5$ in effective
statistics --- kaon-charge and lepton tags are combined into an
effective tagging power near $30\%$, and every point of it was
fought for. The mixing frequency itself, $\Delta m_d$, is
measured from the same time-dependent machinery run on
flavor-specific final states, where the oscillation appears in
the mixed-versus-unmixed rate rather than in a CP asymmetry.

With $\Delta m_d = 0.51\,\mathrm{ps}^{-1}$ and
$\tau_B = 1.52$~ps, the oscillation completes
$\Delta m_d \tau_B / 2\pi \simeq 0.12$ of a cycle per lifetime
--- the fringe is caught rising, which is why time resolution,
not statistics, was the design driver. The program has since
widened: LHCb carried the interferometry into the $B_s$ system
and, in 2019, into charm --- $\Delta A_{CP} = (-15.4\pm2.9)
\times10^{-4}$, CP violation in up-type quarks at last
\citep{LHCbCharm2019} --- while Belle~II now owns the
statistical future of the golden mode itself. The main text's
Part~VII recomputes \eqref{eq:e4-sin2b} with the lattice phase
and the stated PDG-class moduli, returning $0.708$ --- the
watched row of
Table~\ref{tab:exp-predicted-measured}.

\medskip\noindent\textbf{E.4.1. Derive the time-dependent
asymmetry \eqref{eq:e4-tdcp} from the mixing formalism.}

\noindent\emph{Solution.} Step 1: with $\Delta\Gamma = 0$ the
flavor states evolve as $B^0(t) = g_+(t)B^0 + (q/p)g_-(t)
\bar B^0$, $g_\pm(t) = e^{-imt}e^{-\Gamma t/2}
\{\cos, i\sin\}(\Delta m\, t/2)$.

Step 2: the decay amplitude to $f$ is $A_f\big[g_+(t) +
\lambda_f g_-(t)\big]$; square it, and likewise for
$\bar B^0(t)$ with $\lambda_f \to 1/\lambda_f$ and
$q/p \leftrightarrow p/q$.

Step 3: form the asymmetry; the $e^{-\Gamma t}$ prefactors
cancel, the cross terms deliver $\mathrm{Im}\,\lambda_f
\sin(\Delta m t)$, the moduli difference delivers the cosine
term, and normalization gives $S_f$ and $C_f$ as stated.

Physics intuition: it is a two-slit experiment in time ---
path one, decay directly; path two, oscillate first, then
decay --- and $S_f$ is the fringe visibility. The mixing phase
is read the way all phases are read: by interference against a
reference path.

\medskip\noindent\textbf{E.4.2. Show that for $b \to c\bar cs$
the weak phase of $\lambda_f$ is $2\beta$ in the standard
convention.}

\noindent\emph{Solution.} Step 1: the mixing ratio from the
box diagram is $q/p \simeq
(V_{tb}^*V_{td})/(V_{tb}V_{td}^*)$ --- phase $-2\arg(V_{td})
= +2\beta$ modulo the convention.

Step 2: the decay ratio $\bar A/A$ for $b\to c\bar cs$ carries
$(V_{cb}V_{cs}^*)/(V_{cb}^*V_{cs})$ --- real to
$\mathcal O(\lambda^4)$ in \eqref{eq:e1-wolfenstein}. The
$K_S$ in the final state adds the kaon-mixing factor, real in
the same convention.

Step 3: hence $\arg\lambda_f = 2\beta$ with $\beta$ as in
\eqref{eq:e4-sin2b}: the golden mode measures a pure mixing
phase, uncontaminated at the percent level.

Physics intuition: the mode is golden because the amplitude
is a single CKM combination --- no competing path, no
strong-phase dependence; the theory error lives two orders
below the measurement, a luxury almost nowhere else in flavor
physics.

\medskip\noindent\textbf{E.4.3. Estimate the penguin pollution
of $\sin 2\beta$ and state why it matters at current
precision.}

\noindent\emph{Solution.} Step 1: the $b\to s$ penguin
amplitude carries $V_{tb}V_{ts}^*$, whose phase differs from
the tree's at $\mathcal O(\lambda^2)$; its magnitude relative
to the tree is loop-suppressed, a few percent.

Step 2: the induced shift is $\Delta S \sim 2\,r\,
\cos2\beta\,\sin\gamma\,\cos\delta_s$ with $r$ the
penguin-to-tree ratio: estimates give
$|\Delta S| \lesssim 0.01$.

Step 3: the world average carries $\pm 0.011$: pollution and
precision have met. Any future drift of the watched
row (Part~XII's first narrative) must be argued
\emph{after} this correction --- which is exactly how the
refresh rules of \S\ref{sec:exp-refresh} instruct the
confrontation to proceed.

Physics intuition: every precision observable eventually
meets its penguin. The discipline --- quote the pollution
with the measurement --- is the flavor community's version
of the reversal systematics of \S\ref{app:d-lab}.

\subsection{Step 2 --- what the Standard Model cannot price}
\label{app:flavour-limits}

Now the audit, in the register of \S\ref{app:d-limits}.
\emph{(1) The phase is free.} Nothing in the SM prefers
$\delta = 66.9^{\circ}$ over $0^{\circ}$ or $180^{\circ}$; a
CP-conserving world was equally renormalizable, and the
measured value is harvested by the global fits, never
predicted (problem D.5.2).

\emph{(2) The hierarchy is free.} The pattern
$1 : \lambda : \lambda^2 : \lambda^3$ of
\eqref{eq:e1-wolfenstein} --- arguably the most striking
regularity in the fermion sector --- is an input: the Yukawa
matrices of \eqref{eq:e1-yukawa} are arbitrary.

\emph{(3) No quark--lepton bridge.} The SM contains no
relation between $\delta_{\rm CKM}$ and $\dPMNS$; they live in
disconnected Yukawa sectors. A framework in which both descend
from one texture makes a statement the SM cannot even pose.

\emph{(4) The triangle test checks consistency, not origin.}
The over-determined unitarity triangle is a magnificent
audit of the \emph{parameterization} --- its closure would
survive any value of the phase. Consistency is not valuation.

\emph{(5) The conventions bite.} $\beta$, $\varepsilon$, and
the Wolfenstein phase are convention-laden; cross-experiment
combinations begin with translation exercises
(\citealp{BrancoLavouraSilva}, Ch.~22) --- the bench
consequence of \S\ref{app:d-limits}(1), now with real
person-years attached.

\emph{(6) The other phase is a scandal.} QCD admits its own
CP-violating angle, $\bar\theta$, and the neutron EDM bounds it
at $\bar\theta \lesssim 5\times10^{-18}$ in this cohort's
accounting --- while the flavor phase sits at
$66.9^{\circ}$. Two CP phases, one enormous, one absurdly
absent, and the SM prices neither; the geometric account of
the second is the parent program's axion sector
(\citealp{ArisakaC1Kaxion}), confronted at
$8.1\,\mu$eV in its own paper.

\medskip\noindent\textbf{E.5.1. Exhibit the CP-mirror world:
show the SM with $\delta \to -\delta$ is exactly as consistent
as ours.}

\noindent\emph{Solution.} Step 1: complex-conjugating the
Yukawa matrices in \eqref{eq:e1-yukawa} maps $V \to V^*$,
i.e.\ $\delta \to -\delta$, leaving all moduli, masses, and
gauge couplings fixed.

Step 2: renormalizability, anomaly cancellation, and unitarity
are blind to the conjugation; every SM consistency proof goes
through verbatim.

Step 3: hence $J \to -J$: a world with all CP asymmetries
reversed is an equally lawful SM. The sign of the observed
asymmetries --- and through Part~VIII, the sign of the matter
excess --- is not an SM prediction.

Physics intuition: this is the sharpest form of the valuation
gap. The SM cannot even say why \emph{this} universe rather
than its CP mirror; the pentagon, whose lattice point carries
a sign, can --- and is falsifiable for it.

\medskip\noindent\textbf{E.5.2. Show that grand unification
alone does not fix the CKM phase either.}

\noindent\emph{Solution.} Step 1: in $\mathrm{SU}(5)$, the
Yukawa couplings remain free matrices in the $\mathbf{5}^*$
and $\mathbf{10}$ representations; unification relates some
down-type and lepton Yukawas but constrains no phase to a
value.

Step 2: the counting of E.1.1 is representation-independent:
one physical phase survives for three generations regardless
of the unified group.

Step 3: hence GUTs inherit limitation (1) wholesale ---
consistent with Part~X's verdict that the roads extend the
dynamics while inheriting the frame.

Physics intuition: making the gauge group bigger organizes
the \emph{charges}, not the \emph{textures}. Valuing a phase
needs a geometry of the flavor space itself --- which is
Step 3's opening line.

\medskip\noindent\textbf{E.5.3. Quantify what ``the fits
harvest'' means: how would a fit respond if
$\delta$ drifted?}

\noindent\emph{Solution.} Step 1: the global fit maximizes a
likelihood over $(\lambda, A, \rho, \eta)$ given rates and
asymmetries; $\delta$ is a derived angle of the fitted
$(\rho, \eta)$ vertex.

Step 2: if future data moved the vertex, the fit would follow
smoothly --- no fit output can falsify the parameterization,
only its internal consistency (limitation 4).

Step 3: contrast Part~VII: the pentagon pins
$\delta = 13\pi/35$ with no vertex freedom; a drifted world
fit \emph{kills} it (Table~\ref{tab:falsifiers}, row 2). The
same data stream that the SM absorbs, the geometry stakes its
life on.

Physics intuition: the operational difference between a
parameterization and a theory is what a surprise does to it
--- absorb versus die. Part~XII's nobility clause is this
problem, stated as conduct.

\subsection{Step 3 --- the pentagon: the phases priced}
\label{app:flavour-pentagon}

The geometry's answer to Step 2 is developed in
Part~\ref{part:pentagon}; this Step states it in the
chapter's vocabulary. The quartets \eqref{eq:e2-quartet} are
plaquettes --- closed loops on the flavor graph --- and in
the \GG{} descent the graph is not an abstraction but a real
texture whose block size is $L = 5$: a pentagon. A pentagon's
angles cannot all be flat, so the loops carry an irreducible
holonomy: CP violation exists because $3\pi/5 \neq \pi$
(\citealp{ArisakaA9SM} for the parent derivation; the concept
of geometric CP violation descends from
\citealp{DineLeighMacIntire1992}).

The valuation follows from the lattice. The branch phase is
the pentagon's first-order angle, quantum-defect-completed by
one lattice step:
\begin{equation}
\dCKM \;=\; \frac{2\pi}{5} \;\xrightarrow{\ \mathrm{QD}\ }\;
\frac{2\pi}{5} - \frac{\pi}{35} \;=\; \frac{13\pi}{35}
\;=\; 66.86^{\circ},
\qquad
\text{world fit: } 65.7^{\circ} \pm 1.5^{\circ},
\label{eq:e6-dckm}
\end{equation}
a pull of $+0.70$ (Table~\ref{tab:exp-predicted-measured});
the raw first-order value sits at $+1.17$ with the
one-lattice-step theory width $\pi/35$ in quadrature, at
$+4.2\sigma$ against the experimental error alone. The Cabibbo
magnitude is the same object's modulus shadow, the golden
closed form of \eqref{eq:cabibbo}, at
$\mathrm{Dev} = +4\times10^{-5}$. The leptonic phase is the
central angle's completion,
$\dPMNS = \pi + 2\arcsin(1/7) = 196.43^{\circ}$
(QD-completed $196.06^{\circ}$), with the Majorana pair
$(0, \pi)$ --- sitting inside the current global-fit preference
\citep{NuFit2024} and awaiting the long-baseline verdict ---
Part~VII end to end, confronted in
Part~\ref{part:experiments}.

What became falsifiable: every free number of Steps 1--2 now
has a forced central value and a kill condition --- the phase
by the world fit (row 2 of Table~\ref{tab:falsifiers}), the
watched $\sin2\beta$ row by Belle~II statistics, the leptonic
point by the long-baseline decade. The chapter's arc, in one
sentence: \emph{Step 1 built the machine that measures the
phase; Step 2 showed the machine cannot say what the phase
should be; Step 3 is a geometry that says one number --- and
stands or falls with it.}

\medskip\noindent\textbf{E.6.1. Verify the lattice-step
arithmetic of \eqref{eq:e6-dckm} and compute the pull against
the world fit.}

\noindent\emph{Solution.} Step 1: $2\pi/5 = 14\pi/35$, so one
lattice step down is $14\pi/35 - \pi/35 = 13\pi/35$ --- the QD
completion is exactly one unit of the $(\pi/35)$-lattice that
clause (iv) of T-P5-3 forces.

Step 2: in degrees, $13\pi/35 = 13 \times 180/35 =
66.857^{\circ}$; in radians $1.1670$.

Step 3: against $1.147 \pm 0.026$ rad: pull $=
(1.1670 - 1.147)/0.0283 = +0.70$ with the theory width in
quadrature --- the value printed in
Table~\ref{tab:exp-predicted-measured}.

Step 4: the raw first-order value, with the missing step
displayed: against the experimental error alone,
$(1.2566 - 1.147)/0.026 = +4.2$; adding the one-lattice-step
theory width $\pi/35 = 0.0898$ in quadrature,
$\sigma_{\mathrm{tot}} = \sqrt{0.026^2 + 0.0898^2} = 0.0935$,
and the pull is $(1.2566 - 1.147)/0.0935 = +1.17$ --- the
quoted number. The $+1.17$ presumes the theory width; without
it the raw lattice point is excluded at $4\sigma$-class, which
is why the QD completion carries a flag and not a footnote.

Physics intuition: the QD step is not a fit --- it is the same
one-lattice-unit correction the parent program applies
throughout its codebook, and its direction (down, toward the
data) was not adjustable.

\medskip\noindent\textbf{E.6.2. From the pentagon's geometry,
derive why the \emph{branch} phase is $2\pi/5$ and the
\emph{central} completion appears in the leptonic sector.}

\noindent\emph{Solution.} Step 1: a regular pentagon has
central angle $2\pi/5 = 72^{\circ}$ and interior angle
$3\pi/5 = 108^{\circ}$; a loop of the flavor graph that turns
at one vertex picks up the exterior/branch angle $2\pi/5$.

Step 2: the quark loops of E.2 close within the texture's
branches --- their holonomy is the branch angle, QD-completed
to \eqref{eq:e6-dckm}.

Step 3: the leptonic assignment (Part~VII's conditional lemma)
routes the lepton loop through the center: its phase is built
on $\pi$ (the straight angle) displaced by the $1/7$-arcsine
completion, $\pi + 2\arcsin(1/7)$. Two sectors, one figure,
two different circuits of it.

Physics intuition: quarks and leptons do not carry two
unrelated phases; they walk two different paths around one
shape. That is the quark--lepton bridge of Step 2's limitation
(3) --- unposeable in the SM, a theorem-with-a-residue here.

\medskip\noindent\textbf{E.6.3. Evaluate $J$ at the lattice
point and confirm it lands in the measured band.}

\noindent\emph{Solution.} Step 1: from E.2.3, $J =
A^2\lambda^6\eta$ with $(\rho, \eta)$ now \emph{derived} from
the fixed $\delta$: $\eta = |V_{ub}/(A\lambda^3)|
\sin\delta$.

Step 2: with $\sin(66.86^{\circ}) = 0.9196$ and the measured
moduli, $\eta = 0.354$, so $J = 0.69 \times 1.30\times10^{-4}
\times 0.354 \times 0.9196/0.9186 \simeq 3.2\times10^{-5}$.

Step 3: the band quoted in
Table~\ref{tab:exp-predicted-measured} is
$2.7$--$3.2\times10^{-5}$ --- the spread across the
framework's evaluation layers; the world value
$3.12\times10^{-5}$ sits inside. One lattice angle, one
invariant, one confrontation row.

Physics intuition: this is the chapter closing its own loop
--- the invariant that Step 1 constructed, Step 2 declared
unpriceable, and Step 3 priced, now checked against the data
stream the student's experiment feeds. The remaining rows of
that confrontation are the neutrino course's business, and
the decade's.

\section{The origin of the matter excess, from scratch:
Sakharov, the seesaw, and the six-in-ten-billion receipt}
\label{app:leptogenesis}

The third chapter of Block II treats the physics of
Part~\ref{part:matter}: why there is anything rather than an
equal mixture of thing and antithing. Step 1 builds the standard
machinery from the textbooks --- Sakharov's conditions, the
seesaw, the decay asymmetry, the transport --- deriving each
formula the main text's chain later calls. Step 2 states what
that machinery, for all its beauty, cannot compute. Step 3
states what the ledger computes, and at what honesty grade.

The chapter builds on Appendices~D and~E and does not repeat
them: the CPT theorem and the antiunitarity of T are quoted from
\S\ref{app:d-cpt}, and the two-amplitude requirement for a rate
asymmetry --- one weak phase, one rephasing-even phase --- is
the same lesson \S\ref{app:flavour-kaons} taught for
$\varepsilon'$, now wearing thermal clothes. The number to be
explained is fixed by observation:
\begin{equation}
\etaB \;=\; \frac{n_B - n_{\bar B}}{n_\gamma}
\;=\; (6.12 \pm 0.04)\times10^{-10},
\label{eq:f1-etaBobs}
\end{equation}
six baryons per ten billion photons, measured two independent
ways --- the CMB acoustic peaks and primordial deuterium ---
which agree \citep{Planck2018}.

The chapter shelf: \citet{KolbTurner} is the standard cosmology
text (Ch.~6 for baryogenesis); \citet{BuchmullerDiBariPlumacher}
the pedagogical review of thermal leptogenesis this chapter's
transport follows; \citet{MohapatraPal2004} the seesaw at book
length; \citet{PDG2024} the living numbers. Eighteen problems,
complete solutions included.

\subsection{Step 1 --- Sakharov's three conditions, derived}
\label{app:lepto-sakharov}

In 1967, Sakharov stated what any dynamical explanation of
\eqref{eq:f1-etaBobs} must contain \citep{Sakharov1967}: baryon
number violation, C and CP violation, and a departure from
thermal equilibrium. Each is a small theorem, and each is worth
deriving, because each names a place where a proposal can
quietly fail.

\emph{Condition three first, since it is the sharpest.} In
equilibrium the density operator is $\rho = e^{-\beta H}/Z$.
CPT (Theorem \S\ref{app:d-cpt}) supplies an antiunitary
$\Theta$ with $\Theta H \Theta^{-1} = H$ and $\Theta B
\Theta^{-1} = -B$: baryon number is CPT-odd. Then
\begin{equation}
\langle B\rangle_{\mathrm{eq}}
= \mathrm{Tr}\!\left[\tfrac{e^{-\beta H}}{Z}\,B\right]
= \mathrm{Tr}\!\left[\tfrac{e^{-\beta H}}{Z}\,
\Theta^{-1}(-B)\Theta\right]
= -\langle B\rangle_{\mathrm{eq}}
\;=\; 0,
\label{eq:f1-eqzero}
\end{equation}
an exact statement: a universe that stays in equilibrium keeps
no matter, whatever its couplings. The excess must be minted
during a departure --- an expansion outrunning a reaction ---
and frozen before equilibrium returns.

C and CP must \emph{both} break, and the reason is the mirror
pair of Part~\ref{part:parity}'s lesson: C alone relates a
left-handed quark to a right-handed antiquark, so a C-breaking,
CP-respecting theory can still produce baryons and antibaryons
at equal total rates, chirality by chirality (problem F.1.2
runs the two-channel proof). And $B$ violation is needed for
the trivial reason that $B = 0$ initially --- inflation having
diluted any primordial charge --- and $B$ commuting with $H$
keeps it zero forever.

The Standard Model, remarkably, has all three ingredients and
still fails, twice over, quantitatively. Its CP violation is
the single Jarlskog invariant of \S\ref{app:flavour-invariants},
and the dimensionless asymmetry it can feed the plasma is
suppressed by the quark-mass differences over the electroweak
temperature to $\sim10^{-20}$ --- ten orders short. And its
departure from equilibrium fails too: at the measured Higgs
mass the electroweak transition is a smooth crossover, not the
first-order bubble bath the mechanism needs \citep{Kajantie1996}.
The excess is real physics beyond the Standard Model --- one of
the few claims in this subject on which every framework agrees.

One more inherited theorem redirects the search. Above the
electroweak crossover the Standard Model itself violates
$B{+}L$: 't~Hooft's anomaly ties baryon number to the gauge
vacuum's winding \citep{tHooft1976}, the sphaleron is the pass
between windings \citep{KlinkhamerManton1984}, and the hot
plasma walks over it freely
\citep{KuzminRubakovShaposhnikov1985}. Consequence: anything
stored as pure $B{+}L$ --- the old GUT-era asymmetries --- is
erased; only $B{-}L$ survives the sphaleron era. The 1985
lesson moved the field's attention from baryons to leptons, and
it is why this chapter is about leptogenesis
\citep{FukugitaYanagida1986} at all.

\medskip\noindent\textbf{F.1.1. Prove the equilibrium theorem
\eqref{eq:f1-eqzero} carefully, stating where antiunitarity
matters.}

\noindent\emph{Solution.} Step 1: for antiunitary $\Theta$ and
Hermitian $A$, $\mathrm{Tr}[\Theta^{-1} A \Theta] =
\overline{\mathrm{Tr}[A]} = \mathrm{Tr}[A]$: conjugation by an
antiunitary preserves traces of Hermitian operators (it
transposes and conjugates the matrix, and the trace of a
Hermitian matrix is real).

Step 2: insert $\Theta^{-1}\Theta$ pairs:
$\mathrm{Tr}[e^{-\beta H} B] =
\mathrm{Tr}[\Theta^{-1} e^{-\beta H}\Theta\,
\Theta^{-1} B \Theta] = \mathrm{Tr}[e^{-\beta H}(-B)]$, using
$[\Theta, H] = 0$ (CPT of the dynamics) and CPT-oddness of $B$.

Step 3: hence $\langle B\rangle = -\langle B\rangle = 0$. The
theorem needs exact CPT and exact equilibrium; break either and
the conclusion goes. (Breaking CPT instead of equilibrium is a
logically open route the data does not favor --- the CPT-test
record of Part~\ref{part:history} bounds it.)

Physics intuition: equilibrium is the state that has forgotten
its history, and an asymmetry is a memory. Sakharov's third
condition says: to keep a memory, write it while the system is
not looking --- during a departure --- and freeze it before
the plasma re-reads.

\medskip\noindent\textbf{F.1.2. Show C violation alone is
insufficient: exhibit the chirality bookkeeping.}

\noindent\emph{Solution.} Step 1: let a heavy boson $X$ decay
to quark pairs of definite chirality. C maps
$X \to \bar X$ and $q_L \to \bar q_L$ conjugate --- but the
combination CP maps $q_L \to \bar q_R$: the two operations
distribute the asymmetry differently over chirality channels.

Step 2: suppose C is broken but CP exact. CP equality gives
$\Gamma(X \to q_L q_L) = \Gamma(\bar X \to \bar q_R \bar q_R)$
and $\Gamma(X \to q_R q_R) = \Gamma(\bar X \to \bar q_L
\bar q_L)$.

Step 3: sum over chirality: $\Gamma(X \to qq)_{\mathrm{tot}} =
\Gamma(\bar X \to \bar q\bar q)_{\mathrm{tot}}$. Equal numbers
of baryons and antibaryons are minted even though C is broken
in each channel separately. Only breaking CP as well lets the
\emph{total} rates differ.

Physics intuition: C violation biases who is left-handed; CP
violation biases who exists. The excess needs the second. This
is Part~\ref{part:matter}'s requirement that the pentagon's
phase --- not merely the descent's handedness --- enter the
chain.

\medskip\noindent\textbf{F.1.3. Estimate the Standard Model's
CP shortfall: why $10^{-20}$?}

\noindent\emph{Solution.} Step 1: the only
reparameterization-invariant CP quantity is $J$ times the
non-degeneracy factors
of \eqref{eq:e2-commutator}; the dimensionless combination
available to the hot plasma is
$d_{CP} \sim J\,\prod_{u} (m^2_{u_i} - m^2_{u_j})
\prod_{d} (m^2_{d_i} - m^2_{d_j})/T_{EW}^{12}$.

Step 2: insert $T_{EW} \sim 100$~GeV and the quark masses: the
twelve powers of small masses over the electroweak temperature
crush $J \sim 3\times10^{-5}$ to $d_{CP} \sim 10^{-20}$.

Step 3: compare the target $\sim 10^{-10}$
\eqref{eq:f1-etaBobs}: ten orders short --- with the crossover
failure of condition three on top. The verdict is structural,
not numerical accident: light quarks cannot bias a hot plasma.

Physics intuition: the same suppression that made CP violation
hard to \emph{discover} (E.2's $\lambda^6$) makes it useless
for \emph{cosmology} inside the Standard Model. Whatever wrote
the matter excess used a bigger pen --- which is the seesaw
sector's cue to walk on stage.

\subsection{Step 1 --- the seesaw: the heavy sector derived}
\label{app:lepto-seesaw}

If new physics generates neutrino masses, its lowest-energy
shadow is fixed by symmetry alone: the unique dimension-five
operator built from Standard-Model fields,
\begin{equation}
\mathcal{O}_5 \;=\; \frac{c_{ij}}{\Lambda}\,
(\ell_i H)(\ell_j H),
\qquad
m_\nu \;=\; c\,\frac{v^2}{\Lambda},
\label{eq:f2-weinberg}
\end{equation}
the Weinberg operator: Majorana masses, $\Delta L = 2$,
scale-suppressed. Measured $m_\nu \sim 50$~meV with
$c \sim 1$ points
at $\Lambda \sim 6\times10^{14}$~GeV; smaller couplings buy
lower scales --- the trade the TeV chain of Step 3 lives on.

The simplest ultraviolet completion is the type-I seesaw
\citep{Minkowski1977, YanagidaSeesaw, GellMannRamondSlansky,
MohapatraSenjanovic1980}: add singlet fermions $N$ with Majorana
mass $M_N$ and Yukawa $Y_\nu\,\bar\ell \tilde H N$. The full
neutral mass matrix in the $(\nu, N)$ basis is block
off-diagonal plus heavy diagonal, and block-diagonalizing it
(problem F.2.1) gives
\begin{equation}
\mathcal{M} =
\begin{pmatrix} 0 & m_D \\ m_D^{T} & M_N \end{pmatrix}
\;\longrightarrow\;
m_\nu \simeq -\,m_D\, M_N^{-1} m_D^{T},
\qquad m_D = \frac{Y_\nu v}{\sqrt2},
\label{eq:f2-block}
\end{equation}
the main text's \eqref{eq:seesaw}: light masses inversely
proportional to the heavy scale --- one end of the seesaw goes
up because the other went up.

Which Yukawas reproduce the measured light sector? Inverting
\eqref{eq:f2-block} is the Casas--Ibarra parameterization
\citep{CasasIbarra2001},
\begin{equation}
Y_\nu \;=\; \frac{\sqrt2}{v}\, M_N^{1/2}\, R\,
\hat m_\nu^{1/2}\, U_{\mathrm{PMNS}}^{\dagger},
\label{eq:f2-ci}
\end{equation}
with $R$ any complex orthogonal matrix ($R^T R = \mathbf 1$).
Every choice of $R$ fits the same low-energy data --- the
matrix is precisely the seesaw's hidden-sector freedom, and it
returns in Step 2 as the reason measured phases do not pin the
cosmology.

Physical intuition: lepton number dies quietly, of coexistence.
Neither $M_N$ (which violates $L$ by two but touches only
singlets) nor $m_D$ (which conserves $L$) is dangerous alone;
together they let a neutrino become its own antiparticle two
insertions later. That coexistence is the ``door'' of
Part~\ref{part:matter} --- and T-P5-4's content is that the
descent's ledger leaves exactly one lawful way to build it.

\medskip\noindent\textbf{F.2.1. Derive the seesaw formula
\eqref{eq:f2-block} by block diagonalization, including the
sign and the size of the corrections.}

\noindent\emph{Solution.} Step 1: seek a rotation
$\mathcal{U} = \exp\begin{pmatrix} 0 & \rho \\
-\rho^\dagger & 0\end{pmatrix}$ with $\rho$ small, such that
$\mathcal{U}^T \mathcal{M}\, \mathcal{U}$ is block diagonal.
Expanding to first order, the off-diagonal block vanishes for
$\rho \simeq m_D M_N^{-1}$ --- small of order $m_D/M_N$.

Step 2: the light block becomes $0 - \rho M_N \rho^T -
(\rho m_D^T + m_D \rho^T) + \ldots = -\,m_D M_N^{-1} m_D^T$ at
leading order; the heavy block is $M_N$ plus
$O(m_D^2/M_N)$ shifts.

Step 3: corrections enter at relative order
$(m_D/M_N)^2 \sim m_\nu/M_N$ --- for the TeV chain,
$5\times10^{-11}/3.85\times10^{3}$~GeV ratios, i.e.
$10^{-14}$-class: the formula is exact for every purpose in
this paper. The minus sign is physical but unobservable --- it
is absorbed by a Majorana phase redefinition.

Physics intuition: the light neutrino is a heavy state seen
through a keyhole --- twice through the Yukawa, once through
the heavy propagator, $Y (1/M) Y$. Small masses are not small
couplings; they are big denominators.

\medskip\noindent\textbf{F.2.2. Show lepton-number violation
requires \emph{both} $M_N$ and $m_D$ (the coexistence
argument).}

\noindent\emph{Solution.} Step 1: with $M_N = 0$, assign
$L(N) = 1$: every term in the Lagrangian conserves $L$; the
would-be Majorana sector is Dirac in disguise, and light
neutrinos pair with singlets into $L$-conserving Dirac states.

Step 2: with $m_D = 0$, assign $L(N) = 0$: $M_N$ violates
nothing that talks to the visible sector; lepton number of the
doublets is exactly conserved.

Step 3: only with both present does no assignment work ---
$m_D$ forces $L(N) = 1$, $M_N$ then breaks $L$ by two units.
The physical signature is the Weinberg operator
\eqref{eq:f2-weinberg} with coefficient $m_\nu/v^2$: at cold
temperatures, $L$-violation is neutrino-mass-suppressed --- the
cold universe conserves lepton number to $10^{-11}$-class
precision while the hot one, with real $N$'s in the bath, did
not. That is exactly the freeze-out structure the chain uses.

Physics intuition: the algebra is the same ``two insertions''
that gives $0\nu\beta\beta$ its amplitude --- the door that
made the matter also makes the double-beta rate, $\mbb =
1.72$~meV in this paper's chain, one hinge with two readouts
(Part~\ref{part:experiments}).

\medskip\noindent\textbf{F.2.3. Run the numbers: what Yukawa
does the lock scale need, and how does it compare with the
chain's seam value?}

\noindent\emph{Solution.} Step 1: from \eqref{eq:f2-block}
one-generation style, $Y_\nu^2 = 2 m_3 M_N/v^2$ with
$m_3 = 50.5$~meV $= 5.05\times10^{-11}$~GeV, $M_N =
3.85\times10^{3}$~GeV, $v = 246$~GeV.

Step 2: $Y_\nu^2 = 2 \times 5.05\times10^{-11}\times
3.85\times10^{3}/6.05\times10^{4} \simeq 6.4\times10^{-12}$,
so $Y_\nu \simeq 2.5\times10^{-6}$.

Step 3: the chain's seam-matched value is $Y_\nu \simeq
7.3\times10^{-7}$ (\S\ref{sec:chain}, step 3) --- same decade
within the $R$-matrix and projection freedom the audit band
carries. Note the scaling $Y_\nu \propto \sqrt{M_N}$: the TeV
seesaw needs \emph{feebler} couplings than the
$10^{9}$~GeV one --- the origin of the tiny width
$\Gamma_N/M_N \sim 2\times10^{-14}$ that will make resonance
obligatory in the next Step. (This is the chain's convention
spread made explicit: \S\ref{sec:chain} quotes the width at the
seam value $7.3\times10^{-7}$ and runs washout at $\widetilde m =
m_3$, whose one-generation Yukawa is this Step's
$2.5\times10^{-6}$ --- a factor $\simeq13$ in $Y_\nu^2$, declared
and priced at both chain sites, steps~3 and~6.)

Physics intuition: lowering the seesaw scale does not buy
visibility for free --- the couplings shrink as the square
root, and the heavy states become narrow, feeble, and
displaced-vertex-like: exactly the collider signature class
the falsifier table bets on
(Table~\ref{tab:falsifiers}).

\subsection{Step 1 --- the decay asymmetry: interference,
the bound, and the resonance}
\label{app:lepto-epsilon}

The working part is the CP asymmetry per decay,
$\epsilon = [\Gamma(N \to \ell H) - \Gamma(N \to \bar\ell\bar
H)]/[\Gamma(N \to \ell H) + \Gamma(N \to \bar\ell\bar H)]$.
At tree level it vanishes identically: the two rates are
$|Y|^2$ times identical kinematics, and CPT plus unitarity
forbids a lowest-order difference (problem F.3.1). The
asymmetry is an interference effect --- tree times loop ---
and it needs the same two ingredients as every rate asymmetry
in this paper (\S\ref{app:d-cpt}, \S\ref{app:flavour-kaons}):
a relative \emph{weak} phase, supplied by the Yukawa matrix,
and a relative \emph{rephasing-even} phase, supplied here by
the absorptive part of the loop --- the on-shell intermediate
states the optical theorem counts.

Carrying out the one-loop computation (vertex plus
self-energy) gives the standard result
\citep{CoviRoulet1996},
\begin{equation}
\epsilon_i \;=\; \frac{1}{8\pi}\sum_{j\neq i}
\frac{\mathrm{Im}\big[(Y^\dagger Y)_{ij}^2\big]}
{(Y^\dagger Y)_{ii}}\;
f\!\left(\frac{M_j^2}{M_i^2}\right),
\label{eq:f3-cr}
\end{equation}
with $f$ the sum of the vertex and self-energy loop functions.
The structure is worth reading before the algebra:
$\mathrm{Im}[(Y^\dagger Y)_{ij}^2]$ is the flavor-space
quartet --- the same closed-loop invariance logic as
\eqref{eq:e2-quartet}, now in the heavy sector --- and $f$
carries the absorptive kinematics.

For hierarchical heavy masses the loop function is
$O(M_i/M_j)$ and the asymmetry is bounded by the light and
heavy spectra --- the Davidson--Ibarra bound
\citep{DavidsonIbarra2002},
\begin{equation}
|\epsilon_1| \;\lesssim\; \frac{3}{8\pi}\,
\frac{M_1\, m_3}{\bar v^2},
\qquad \bar v = 174~\mathrm{GeV},
\label{eq:f3-di}
\end{equation}
so demanding the observed asymmetry forces
$M_1 \gtrsim 10^{9}$~GeV: beautiful, thermal, and forever
beyond every collider. The known escape is quasi-degeneracy.
When $|M_i - M_j|$ approaches the width, the self-energy
contribution is resonantly enhanced
\citep{Pilaftsis1997, PilaftsisUnderwood2004}:
\begin{equation}
\epsilon \;\propto\;
\frac{\Delta M\,\Gamma}
{\Delta M^2 + \Gamma^2/4}\,\sin(2\phi)
\;\;\xrightarrow{\ \Delta M = \Gamma/2\ }\;\;
\tfrac12\sin(2\phi),
\label{eq:f3-bw}
\end{equation}
a Breit--Wigner in the mass splitting: the Yukawa magnitude
cancels between numerator and denominator, and the asymmetry
is controlled by the CP phase alone. TeV-scale leptogenesis
lives here --- and so does its honesty problem: off the peak
the same formula \emph{suppresses} as $\Gamma/\Delta M$, and
the vacuum numbers of the chain sit twelve orders off-peak
(Figure~\ref{fig:resonance-map}). A finite-temperature width
large enough to close the gap must also be flavor-sensitive to
count: the computed candidate is the flavor-measurement rate
$\Gamma^{\mathrm{eff}} = c_\tau y_\tau^{2} T$ (the flavor-blind
$g^2 T$ lands through the peak and overshoots), and computing it
honestly is the named conjecture C-P5-C
(\S\ref{sec:resonance-faced}). The CP factor also has a stated
provenance: at the TeV scale the operative object is the flavored
asymmetry $\epsilon_{i\alpha}$ (the unflavored invariant is
$U_{\mathrm{PMNS}}$-independent and vanishes for real $R$), and the
chain's $\epsilon^{\mathrm{res}} \simeq 2.5\times10^{-6}$ is the
H-$\phi_{\mathrm{geo}}$-1 target value the partial-resonance
condition must deliver, not a derived output (\S\ref{sec:chain},
step~5), now carried by an identified candidate mechanism
(transport closure of C1--C6 open). (A distinct TeV route ---
asymmetry from singlet \emph{oscillations} rather than decays
--- is the ARS mechanism
\citep{AkhmedovRubakovSmirnov1998}; the chain does not use it,
but the student will meet it in every review.)

\medskip\noindent\textbf{F.3.1. Prove $\epsilon = 0$ at tree
level from CPT and unitarity.}

\noindent\emph{Solution.} Step 1: CPT relates amplitudes, not
rates: $\mathcal{M}(N \to \ell H) =
\mathcal{M}(\bar H \bar\ell \to N)^*$ up to spin bookkeeping.
At tree level $\mathcal{M} = Y\,(\text{real kinematics})$, so
$|\mathcal{M}(N\to\ell H)|^2 = |Y|^2 K$ and
$|\mathcal{M}(N\to\bar\ell\bar H)|^2 = |Y^*|^2 K = |Y|^2 K$:
equal.

Step 2: more generally, $S = 1 + iT$ unitarity gives
$T - T^\dagger = i T^\dagger T$: the difference between a rate
and its reverse is second order in $T$ --- any first-order
(tree) asymmetry integrates to zero over final states.

Step 3: hence a nonzero $\epsilon$ requires the $O(T^2)$ term:
an absorptive loop --- physical on-shell intermediate states
--- interfering with the tree. Coupling phases alone are
rephasing; the loop's imaginary part is what makes them
measurable, exactly as the strong phases did for
$\varepsilon'$ in \S\ref{app:flavour-kaons}.

Physics intuition: CP asymmetries are always two-path
interference, and one path must be ``delayed'' by something
CP-even --- a strong phase, an oscillation, a thermal cut.
Nature reuses one trick from the kaon tank to the primordial
plasma.

\medskip\noindent\textbf{F.3.2. Maximize the resonance factor
in \eqref{eq:f3-bw} and evaluate the off-peak suppression at
$\Delta M = 10\,\Gamma$ and $10^{12}\,\Gamma$.}

\noindent\emph{Solution.} Step 1: let $g(\Delta M) =
\Delta M\,\Gamma/(\Delta M^2 + \Gamma^2/4)$. Then
$g'= \Gamma(\Gamma^2/4 - \Delta M^2)/(\Delta M^2 +
\Gamma^2/4)^2 = 0$ at $\Delta M = \Gamma/2$.

Step 2: at the peak, $g(\Gamma/2) = (\Gamma/2)\,\Gamma/
(\Gamma^2/4 + \Gamma^2/4) = 1$: the ceiling
$\epsilon = \tfrac12\sin 2\phi$ of \eqref{eq:f3-bw}.

Step 3: off peak, $g \simeq \Gamma/\Delta M$: at
$10\,\Gamma$, a tenth; at $10^{12}\,\Gamma$, $10^{-12}$ ---
the twelve-orders cliff the vacuum numbers of the chain sit
on, drawn as Figure~\ref{fig:resonance-map}. Sitting at the
peak is worth twelve orders of magnitude, which is why
C-P5-C is a \emph{named} input and not a footnote.

Physics intuition: resonance is nature's amplifier of last
resort, and its price is tuning --- unless something sets the
splitting. The chain's claim is that the ledger's
$O(1/K)$ splittings plus the thermal width do the setting;
the honesty is that the second half is a conjecture with a
stated discharge route.

\medskip\noindent\textbf{F.3.3. Evaluate the Davidson--Ibarra
ceiling at the lock scale and show hierarchical TeV
leptogenesis is dead without resonance.}

\noindent\emph{Solution.} Step 1: insert $M_1 = 3.85\times
10^{3}$~GeV, $m_3 = 5.05\times10^{-11}$~GeV, $\bar v =
174$~GeV into \eqref{eq:f3-di}.

Step 2: $|\epsilon_1| \lesssim (3/8\pi)\times 3.85\times10^3
\times 5.05\times10^{-11}/3.03\times10^{4} \simeq
8\times10^{-13}$.

Step 3: the chain needs $\epsilon \sim 2.5\times10^{-6}$ --- the
H-$\phi_{\mathrm{geo}}$-1 target value
(\S\ref{sec:chain}, step 5): the hierarchical ceiling is more
than six orders too low. Either the heavy scale climbs to
$10^{9}$~GeV --- and out of every collider's reach --- or the
degeneracy escape of \eqref{eq:f3-bw} is engaged. The ledger,
which fixed $M_N$ at the lock for its own reasons, has no
choice in the matter; that the required escape is the one its
$O(1/K)$ splittings naturally furnish is either a coincidence
or a design, and Part~\ref{part:matter} prices which.

Physics intuition: the bound is why ``leptogenesis is
untestable'' became folklore --- and why a framework whose
heavy sector is \emph{forced} to the TeV corridor turns the
folklore into a falsifiable schedule
(Table~\ref{tab:exp-timeline}).

\subsection{Step 1 --- transport: washout, sphalerons, and the
conversion arithmetic}
\label{app:lepto-transport}

Between the asymmetry per decay and the asymmetry per photon
stand three pieces of plumbing, each standard, each derivable.

\emph{Washout.} The same Yukawas that produce the asymmetry
can erase it: inverse decays $\ell H \to N$ re-absorb leptons
while the bath is hot. The competition is governed by one
comparison --- the decay rate against the expansion rate at
$T = M_N$ --- and it turns into a pure neutrino-mass
threshold: equating $\Gamma_N \sim \widetilde m M_N^2/(8\pi
v^2)$ with $H \simeq 1.66\sqrt{\gstar}\,T^2/M_{\mathrm{Pl}}$
at $T = M_N$ (the derivation is problem F.4.1) gives the main
text's \eqref{eq:mstar}: $m_\star \simeq 1.1\times10^{-3}$~eV,
containing only $v$, $M_{\mathrm{Pl}}$, and $\gstar$ ---
notably ledger-free.

With the atmospheric mass in the seat,
$K_{\mathrm{wash}} = \widetilde m/m_\star \simeq 50$: strong
washout, efficiency $\kappa \simeq 1/K_{\mathrm{wash}} =
2.0\times10^{-2}$, adopted as $10^{-2}$ up to $O(1)$
\citep{BuchmullerDiBariPlumacher}. (Convention note, as declared at
\S\ref{sec:chain} steps~3/6: $\widetilde m = m_3$ is the chain's
washout convention; the seam-matched $\widetilde m \simeq 8.4$~meV
would give $K_{\mathrm{wash}} \simeq 7.6$, $\kappa \simeq 0.13$
--- a factor-$\simeq13$ spread in $Y_\nu^2$, priced as $O(10)$ in
the order-of-magnitude register.)

The Boltzmann bookkeeping behind that shorthand is a pair of
equations in $z = M_N/T$,
\begin{equation}
\frac{dN_N}{dz} = -D\,(N_N - N_N^{\mathrm{eq}}),
\qquad
\frac{dN_{B-L}}{dz} = -\,\epsilon\, D\,(N_N -
N_N^{\mathrm{eq}}) \;-\; W\, N_{B-L},
\label{eq:f4-boltz}
\end{equation}
source minus washout: every decay out of equilibrium mints
$\epsilon$ units of $B{-}L$, and the washout term $W$ eats a
fraction. In the strong regime the surviving asymmetry
forgets its initial conditions --- thermal or vanishing $N$
abundance converge --- which is the standard strong-washout
blessing: the answer depends only on late-time quantities.

\emph{Sphaleron conversion.} The plasma's chemical equilibrium
redistributes any $B{-}L$ endowment over every species that
shares a reaction with it. Imposing Yukawa equilibrium,
sphaleron equilibrium, and hypercharge neutrality on the
Standard-Model plasma and solving the linear system (problem
F.4.2) gives the Harvey--Turner coefficient
\citep{HarveyTurner1990},
\begin{equation}
B \;=\; \frac{8 N_f + 4 N_H}{22 N_f + 13 N_H}\,(B - L)
\;\;\xrightarrow{\ N_f = 3,\ N_H = 1\ }\;\;
\frac{28}{79}\,(B - L),
\label{eq:f4-conv}
\end{equation}
the number every baryogenesis mechanism in existence must
multiply by.

\emph{Entropy.} Finally the photon bath: the asymmetry is
diluted by the entropy carried in $\gstar = 106.75$
relativistic degrees of freedom at production, $\etaB \simeq
7.04\,Y_B$ in the standard normalization
\citep{KolbTurner}. Nothing in the plumbing is exotic: how
fast does it decay, does the crowd undo it, who converts the
currency, how much did the room grow --- the four household
questions of \S\ref{sec:transport}, now each with its
derivation attached.

\medskip\noindent\textbf{F.4.1. Derive $m_\star$ in closed
form.}

\noindent\emph{Solution.} Step 1: the decay width with
effective mass $\widetilde m$ is $\Gamma_N = \widetilde m\,
M_N^2/(8\pi v^2)$ --- this \emph{defines} $\widetilde m$ as
the neutrino-mass-like combination
$(m_D^\dagger m_D)_{11}/M_1$.

Step 2: Hubble in the radiation era is $H = 1.66\sqrt{\gstar}
\,T^2/M_{\mathrm{Pl}}$; evaluate at $T = M_N$ and set
$\Gamma_N = H$: the $M_N^2$ cancels on both sides ---
the comparison is scale-free.

Step 3: solving for $\widetilde m$ gives $m_\star = 8\pi\times
1.66\sqrt{\gstar}\, v^2/M_{\mathrm{Pl}}$, which assembled with
the standard numerical bookkeeping is the closed form of
\eqref{eq:mstar}, $\simeq 1.1\times10^{-3}$~eV. Only $v$,
$M_{\mathrm{Pl}}$, and $\gstar$ enter: nature fixed the
washout regime before any theory of the heavy sector arrived,
and the measured $m_3 \simeq 50$~meV puts \emph{every} seesaw
in the strong-washout class.

Physics intuition: $m_\star$ is the neutrino mass at which the
door re-closes exactly what it lets out. Fifty times over
threshold means the plasma erases its own first draft --- and
paradoxically that is good: what survives is insensitive to
how the story started.

\medskip\noindent\textbf{F.4.2. Solve the chemical-potential
system: derive $28/79$, then move to two Higgs doublets.}

\noindent\emph{Solution.} Step 1: above the crossover, assign
chemical potentials to the quark and lepton doublets and
singlets and the Higgs; fast Yukawa reactions equate
$\mu_{q} - \mu_{u} + \mu_H = 0$ etc.; the sphaleron imposes
$\sum_f (3\mu_{q_f} + \mu_{\ell_f}) = 0$; hypercharge
neutrality closes the system.

Step 2: with $N_f$ families and $N_H$ doublets the linear
system solves to $B = \tfrac{8N_f + 4N_H}{22N_f + 13N_H}
(B{-}L)$ --- Harvey--Turner's \eqref{eq:f4-conv}. For
$N_f = 3, N_H = 1$: $28/79 \simeq 0.354$.

Step 3: for $N_H = 2$: $(24+8)/(66+26) = 32/92 = 8/23 \simeq
0.348$ --- a $2\%$-class shift on the coefficient,
$8\%$-class on some intermediate conventions, comfortably
inside the chain's factor-$\sim3$ band. The measured $\etaB$
does, in principle, care about the scalar sector --- a
pleasingly concrete reminder that cosmology audits particle
content.

Physics intuition: the plasma is a currency market with fast
arbitrage; the sphaleron is the one desk that prints and
burns $B{+}L$. Equilibrium then fixes the exchange rate
between the conserved $B{-}L$ and the baryons we are made of
--- a rational number, computed from bookkeeping, no dynamics
required.

\medskip\noindent\textbf{F.4.3. Why does \emph{strong} washout
make the prediction more robust, not less?}

\noindent\emph{Solution.} Step 1: in weak washout
($K_{\mathrm{wash}} < 1$) the surviving asymmetry depends on
the initial $N$ abundance --- thermal, zero, or dominant give
different answers: the prediction inherits cosmological
initial-condition freedom.

Step 2: in strong washout the $W N_{B-L}$ term of
\eqref{eq:f4-boltz} drives all initial conditions to a common
attractor before freeze-out; the surviving $\kappa \simeq
1/K_{\mathrm{wash}}$ depends only on $\widetilde m$ --- a
measured-class quantity.

Step 3: the cost is the erasure itself, $\kappa \sim 10^{-2}$
--- two orders of asymmetry paid for initial-condition
independence. The chain's throughput identity
\eqref{eq:throughput} spends exactly this factor and still
lands, which is the content of its $e^{-K/2}$ claim.

Physics intuition: an attractor is the theorist's best
friend: it converts ``it depends on the beginning'' into ``it
depends on the measured masses.'' The strong-washout blessing
is inherited intact by any seesaw whose light masses are the
measured ones --- the ledger's included.

\subsection{Step 2 --- what the Standard Model plus seesaw
cannot price}
\label{app:lepto-limits}

Now the audit, in the register of \S\ref{app:d-limits} and
\S\ref{app:flavour-limits}. \emph{(1) The scale is free.}
$M_N$ ranges over thirteen decades of viable model space, from
ARS-style GeV singlets to GUT-adjacent $10^{14}$~GeV; the
seesaw relation prices only the product $Y^2/M$, never the
factors.

\emph{(2) The measured phases do not reach the cosmology.} By
Casas--Ibarra \eqref{eq:f2-ci}, the asymmetry depends on the
$R$-matrix, which low-energy data never fixes: a measured
$\dPMNS$ is compatible with either sign and any magnitude of
$\epsilon$ (problem F.5.3 exhibits it). DUNE could measure the
leptonic phase perfectly and standard leptogenesis would
predict nothing new.

\emph{(3) The resonance is engineered.} In the TeV literature
the quasi-degeneracy $\Delta M \sim \Gamma$ is imposed by
hand or by a bespoke symmetry --- a dial, tuned to twelve
digits from the natural splittings, with nothing upstream to
set it.

\emph{(4) The sign is not predicted.} Matter rather than
antimatter is, in every standard realization, a choice of
phase convention dressed as a result (the CP-mirror argument
of problem F.5.1).

\emph{(5) The number is a target, not an output.} Across the
model space, $\etaB$ spans decades; the observed
$6.12\times10^{-10}$ selects a surface in parameter space
rather than testing a prediction. The mechanism is
falsifiable in almost none of its realizations ---

\emph{(6) --- and largely untestable.} Hierarchical thermal
leptogenesis lives at $10^{9+}$~GeV, beyond any collider
horizon; its CP phases are the hidden ones of item (2). The
one Standard-Model-owned alternative, electroweak
baryogenesis, is closed by the crossover
\citep{Kajantie1996}. The field's most beautiful mechanism
has spent four decades officially plausible and officially
out of reach.

\medskip\noindent\textbf{F.5.1. Exhibit the CP-mirror
universe: show every standard leptogenesis realization has a
twin with $\etaB \to -\etaB$.}

\noindent\emph{Solution.} Step 1: conjugate every Yukawa and
Majorana phase, $Y_\nu \to Y_\nu^*$, $M_N \to M_N^*$: all
masses, mixings, and $|$couplings$|$ are unchanged; every
$\mathrm{Im}[\cdot]$ in \eqref{eq:f3-cr} flips sign, so
$\epsilon \to -\epsilon$.

Step 2: the transport \eqref{eq:f4-boltz} is linear in
$\epsilon$ and in $N_{B-L}$: the solution maps to minus
itself; sphalerons and entropy are sign-blind.

Step 3: hence $\etaB \to -\etaB$ with every laboratory
observable held fixed except the CP-odd ones, which flip
consistently. An antimatter universe is an equally lawful
seesaw. Only a framework in which the phase is \emph{valued}
--- carried with its sign from something structural --- can
say why this orientation (Part~\ref{part:matter}'s claim,
graded there).

Physics intuition: this is E.5.1's argument promoted from an
aesthetic gap to a cosmological one: the SM cannot say why
\emph{this} universe, and the seesaw alone cannot either.

\medskip\noindent\textbf{F.5.2. Count the seesaw's parameters
against its low-energy observables.}

\noindent\emph{Solution.} Step 1: three-generation type-I: the
complex $3\times3$ matrix $Y_\nu$ carries 18 real parameters
and the heavy eigenvalues 3 more; removing the 3 unphysical
rephasings of the lepton doublets leaves 18 physical
high-scale parameters.

Step 2: the light sector delivers 3 masses + 3 angles + 1
Dirac + 2 Majorana phases = 9 observables.

Step 3: 18 - 9 = 9 parameters invisible at low energy ---
exactly the 3 complex angles of the orthogonal $R$-matrix
\eqref{eq:f2-ci} plus the 3 heavy masses. The cosmology
depends on them; experiments below the heavy scale do not
constrain them. The gap is structural, not experimental
impatience.

Physics intuition: the seesaw is a funnel --- eighteen
numbers in, nine out. Leptogenesis lives in the part that
drained away, which is why it is simultaneously the most
natural explanation of \eqref{eq:f1-etaBobs} and the least
testable: a lesson in how a mechanism can be convincing and
unfalsifiable at once.

\medskip\noindent\textbf{F.5.3. Exhibit two seesaws with
identical low-energy data and opposite-sign asymmetries.}

\noindent\emph{Solution.} Step 1: fix $\hat m_\nu$,
$U_{\mathrm{PMNS}}$, and the heavy spectrum; choose
$R(\omega)$ a complex rotation with angle $\omega$ in one
plane.

Step 2: by \eqref{eq:f2-ci}, $(Y^\dagger Y)$ depends on
$\omega$; the flavor-summed asymmetry carries
$\mathrm{Im}[(Y^\dagger Y)_{ij}^2] \propto$ odd functions of
$\mathrm{Im}\,\omega$. The replacement $\omega \to
-\omega^*$ (well inside the allowed orthogonal family) flips
the sign while $R^T R = \mathbf 1$ still holds.

Step 3: both realizations reproduce every measured mass,
angle, and phase; their $\epsilon$'s differ by a sign, their
$\etaB$'s land on opposite sides of zero. Low-energy
completeness, cosmological opposites --- the sharpest form of
limitation (2), and the precise thing a valued, structural
phase assignment (Step 3) must and does forbid.

Physics intuition: what the funnel drained away was not
noise; it was the cosmology. Any program claiming to
predict $\etaB$ must first close the $R$-matrix freedom ---
which is exactly what an inherited, dial-free door does.

\subsection{Step 3 --- the ledger's receipt}
\label{app:lepto-descent}

The geometry's answer to Step 2 is
Part~\ref{part:matter}; this Step states it in the chapter's
vocabulary, limitation by limitation. \emph{The scale} (1) is
not chosen: the heavy sector sits at the lock,
$\Mlock = \tfrac12 M_{\mathrm{Pl}} e^{-K} \simeq 3.85$~TeV,
$K = 35$ the ledger's throat depth, with the derived
reheating temperature $T_{\mathrm{rh}} \simeq
3\times10^{10}$~GeV populating it thermally
(\citealp{ArisakaA10Cosmos}).

\emph{The door} is dial-free:
T-P5-4 proves the weight-zero self-pairing is the unique
lawful $\Delta L = 2$ structure --- the $R$-matrix freedom of
(2) is not fitted but closed, which is why the measured
$\dPMNS$ reaches the cosmology here and nowhere else.
\emph{The phase and its sign} (4) descend from the ledger with
orientation attached: in the flavored channel the TeV scale
mandates, the driver is the Majorana parity $\alpha_{31} = \pi$
(the numerics are $\JPMNS$-blind at the few-percent level), and
$\epsilon^{\mathrm{res}} \simeq 2.5\times10^{-6}$ is the
H-$\phi_{\mathrm{geo}}$-1 target value the partial-resonance
condition must deliver (\S\ref{sec:resonance-faced}) --- a target
value now carried by an identified candidate mechanism, the
flavor-measurement regulator, with transport closure of C1--C6
open.

\emph{The resonance} (3) is the chapter's honesty test, and
the chain does not hide it: the on-resonance condition is the
one named input, Conjecture C-P5-C, with the vacuum numbers
sitting twelve orders off-peak and the flavor-measurement rate
$\Gamma^{\mathrm{eff}} = c_\tau y_\tau^{2} T$ named, at candidate
grade, as the discharge route (the flavor-blind $g^2 T$
overshoots) --- faced in public in \S\ref{sec:resonance-faced},
priced into the grade.

\emph{The number} (5): washout
$\kappa \simeq 10^{-2}$ (ledger-free $m_\star$; the $O(1)$
adoption from $1/K_{\mathrm{wash}} = 2.0\times10^{-2}$ stated
there), throughput
$\epsilon\,\kappa \simeq 2.5\times10^{-8} = e^{-K/2}$ --- the
identity of \eqref{eq:throughput}, its exponent a postulate
of the compression, flagged as such --- then sphalerons and
entropy \eqref{eq:f4-conv} deliver the main text's
\eqref{eq:etaB-pred}:
$\etaB \approx 7\times10^{-10}$ (central $7.34$), against the observed $(6.12\pm0.04)\times10^{-10}$ --- $+20\%$ central, inside a factor-$\sim3$ audit band, at grade D$^{*}$: derived on the
inherited ledger, conditional on one named conjecture.

What became falsifiable (6): the same door predicts
$\mbb = 1.72$~meV for the tonne-scale $0\nu\beta\beta$ era;
the TeV cluster is HL-LHC's to find or exclude; the leptonic
phase is DUNE and Hyper-K's decade
(Table~\ref{tab:exp-timeline}). The chapter's arc, in one
sentence: \emph{Step 1 built the machinery that could mint a
universe; Step 2 showed the machinery will mint any universe
you ask of it; Step 3 is a geometry that asks for exactly one
--- six in ten billion, matter-signed --- and posts the
schedule on which that request can be refused.}

\medskip\noindent\textbf{F.6.1. Verify the throughput identity
numerically, and state precisely what is derived and what is
postulated.}

\noindent\emph{Solution.} Step 1: $\epsilon^{\mathrm{res}}
\kappa = (2.5\times10^{-6})(1.0\times10^{-2}) =
2.5\times10^{-8}$ --- with $\kappa = 1.0\times10^{-2}$ adopted
from $1/K_{\mathrm{wash}} = 2.0\times10^{-2}$ within the $O(1)$
transport freedom: the displayed agreement uses that freedom,
and would read $5.0\times10^{-8}$ at the unadjusted
$1/K_{\mathrm{wash}}$.

Step 2: $e^{-K/2} = e^{-17.5} = 2.51\times10^{-8}$: agreement
to displayed precision --- the ledger's throat-depth
exponential, reconstructed by an explicit
seesaw-plus-washout chain.

Step 3: audit: $\epsilon$ is derived-conditional on
H-$\phi_{\mathrm{geo}}$-1: the flavored computation is displayed
(\ATen{} Appendix~I; driver $\alpha_{31} = \pi$, nonzero at
$m_1 = 0$, $\JPMNS$-blind at the stated level; the former
$m_1/m_3$-order language is withdrawn), and the $2.5\times10^{-6}$
of Step~1 is the target value the partial-resonance condition
$\Delta M/\Gamma^{\mathrm{eff}} \simeq 3\times10^{4}$ must
deliver, not a derived output, now carried by an identified
candidate mechanism with C1--C6 open (resonance C-P5-C); $\kappa$ is
derived-conditional (ledger-free $m_\star$, $O(1)$ transport
prefactor uncomputed and used above); the \emph{identification} of the
product with $e^{-K/2}$ is a postulate of the compression ---
geometrically suggestive as $\sqrt{e^{-K}}$, not derived. The
same $K = 35$ now prices a mass ($\Mlock$), a bound
($\bar\theta$ leakage), and the matter throughput: one
integer, three exports --- either the ledger's deepest
success or its most seductive numerology, and
\S\ref{sec:chain} says which experiments decide.

Physics intuition: when three unrelated observables return
the same integer, you have found either the mechanism or the
mirror in your own instrument. The falsifier table is the
procedure for telling them apart.

\medskip\noindent\textbf{F.6.2. Propagate the three uncomputed
$O(1)$'s through to $\etaB$ and reconstruct the audit band.}

\noindent\emph{Solution.} Step 1: the chain carries three
honest $O(1)$'s: the projection coefficient
$c \in [0.25, 1]$, the washout prefactor $[0.5, 2]$, and the
transport normalization $[0.5, 2]$.

Step 2: the product's extreme range is $[0.25\times0.5\times
0.5,\ 1\times2\times2] = [0.0625, 4]$ relative to the central
chain; applied to $7.34\times10^{-10}$ this spans roughly
$[0.5, 29]\times10^{-10}$.

Step 3: the published factor-$\sim3$ band is the one-sigma
core of this propagation (the extremes require all three
factors conspiring); the $+20\%$ central landing therefore
buys no extra credit --- and is claimed as none. The
confrontation row (Table~\ref{tab:exp-predicted-measured})
carries the band, not the landing.

Physics intuition: an audit band you constructed yourself is
worth more than a bull's-eye you cannot justify. The
deflation is deliberate: the chapter's credibility spends
better on the falsifiable entries --- the phase, the
double-beta target, the cluster --- than on a central value
inside its own honest slop.

\medskip\noindent\textbf{F.6.3. Suppose DUNE returns
$\dPMNS = 270^{\circ}$. Walk the chain and state what dies
and what survives.}

\noindent\emph{Solution.} Step 1: the leptonic lattice point
dies: $270^{\circ}$ is outside the $196^{\circ}$ band by far
more than the QD width, killing the lepton chain of T-P5-3
and with it the \emph{forced} value of the heavy-sector
phase.

Step 2: the door (T-P5-4), the lock scale, the washout, and
the sphaleron arithmetic survive untouched --- they never
used the phase's value. $\etaB$ reverts from D$^{*}$ toward
ordinary fitted resonant leptogenesis: mechanism intact,
valuation lost.

Step 3: the quark pentagon, the CPT custody, and the
confrontation table's quark rows are unaffected --- the
falsifier table's row structure
(Table~\ref{tab:falsifiers}) is exactly this walk,
pre-computed, which is what ``modular kill conditions''
means in practice.

Physics intuition: a good theory fails in parts, and says in
advance which parts. The student on a long-baseline shift is
not testing ``the framework''; she is testing row by row ---
and this chapter has now handed her the derivation behind
every row she can reach.

\section{The antiparticle, from scratch: the sea, the
contour, the causality theorem, and the handshake}
\label{app:antiparticle}

The fourth chapter of Block II treats the physics of
Part~\ref{part:antiparticles}: where the twin comes from. The
question has received three great answers in a century ---
Dirac's sea, Feynman and St\"uckelberg's relabeling, and
Weinberg's causality theorem --- and this chapter derives each
from the textbooks (Step 1), states exactly what each cannot say
(Step 2), and then works the fourth answer, the handshake, in
which the pairing is a consistency condition between two
independently forced complex structures (Step 3).

The chapter builds on its predecessors and does not repeat
them: Wigner's theorem and antiunitarity are quoted from
\S\ref{app:d-wigner}, the bilinear machinery of the Dirac
theory from \S\ref{app:d-dirac}, and the CPT identities the
twin must satisfy from \S\ref{app:d-cpt} and
Table~\ref{tab:cpt-tests}. The three-depths ruler of
\S\ref{sec:three-depths} --- spectral, structural, ontological
--- is the chapter's grading instrument throughout.

The chapter shelf: \citet{WeinbergQTF}, Vol.~I Ch.~5, is the
definitive modern treatment G.3 follows; the original papers
--- \citet{Dirac1928, Dirac1930, Dirac1931},
\citet{Anderson1933}, \citet{Stueckelberg1941},
\citet{Feynman1949} --- are short, readable, and assigned;
\citet{LawsonMichelsohn} carries the spinor algebra of G.4;
\citet{PeskinSchroeder1995}, Ch.~2--3, the propagator
machinery of G.2. Eighteen problems, complete solutions
included.

\subsection{Step 1 --- the Dirac sea, derived and dismantled}
\label{app:anti-sea}

Dirac's 1928 equation \citep{Dirac1928} bought the electron's
spin and magnetic moment at a price displayed in its spectrum:
\begin{equation}
(i\gamma^\mu \partial_\mu - m)\,\psi = 0
\quad\Longrightarrow\quad
E \;=\; \pm\sqrt{\mathbf p^2 + m^2},
\label{eq:g1-spectrum}
\end{equation}
a negative branch, unbounded below, into which every atom
should radiate its electrons in a flash. Dirac's 1930 repair
\citep{Dirac1930} was audacious bookkeeping: declare the
negative-energy states already \emph{filled} --- the sea ---
so exclusion blocks the catastrophe. The observable excitations
are then electrons above the sea and \emph{holes} in it, and a
hole behaves as a particle of positive energy and opposite
charge (problem G.1.1): the antiparticle, predicted
\citep{Dirac1931} before Anderson's cloud chamber found it
\citep{Anderson1933} --- perhaps the greatest successful
prediction ever made by bookkeeping.

And yet the sea is wrong, and its wrongness is worth deriving
with the same care as its triumph, because each failure is a
theorem. \emph{Bosons.} The $\pi^{+}$ and $\pi^{-}$ are
antiparticles of each other and obey Bose statistics: no
exclusion principle, no blocking, no sea --- yet the twin
exists (problem G.1.2). Whatever explains antiparticles cannot
be exclusion. \emph{The vacuum.} The sea's charge density
diverges cubically with the cutoff (problem G.1.3);
subtracting it by decree concedes that the sea was never
physical.

\emph{The asymmetry.} The sea makes the electron
fundamental and the positron a defect --- yet every CPT
identity of Table~\ref{tab:cpt-tests} finds the two exactly
interchangeable, to $10^{-18}$ in the kaon system's mass
identity. An asymmetric ontology carrying a perfectly
symmetric phenomenology is an ontology on notice.

Physical intuition: the sea is condensed-matter thinking ---
holes in semiconductors are real and useful --- imported to a
place where the ``band'' is infinite and the ``lattice'' is
the vacuum. It gave the right answer once, for fermions,
because exclusion happened to mimic the real mechanism's
bookkeeping. The rest of the chapter is the search for what
it mimicked.

\medskip\noindent\textbf{G.1.1. Show that a hole in the filled
negative-energy band has charge $+e$ and positive effective
mass.}

\noindent\emph{Solution.} Step 1: the filled sea is declared
the vacuum: charge $Q_0$, energy $E_0$, momentum zero, all by
fiat of subtraction.

Step 2: remove one electron of charge $-e$, energy
$-E < 0$, momentum $-\mathbf p$. The remaining state has
charge $Q_0 - (-e) = Q_0 + e$, energy $E_0 - (-E) = E_0 + E$,
momentum $+\mathbf p$: relative to the vacuum, a quantum of
charge $+e$, energy $+E$, momentum $\mathbf p$.

Step 3: its dynamics follow the same subtraction: the hole
accelerates in a field as would a positive charge with
positive mass $m$ (the band's curvature enters with the
removed state's sign twice, restoring the particle sign). The
hole is the positron, kinematically complete.

Physics intuition: a bubble in water rises --- absence,
bookkept against a full background, behaves like presence
with inverted quantum numbers. The derivation is impeccable;
the question Step 2 will press is whether the ``water'' ---
an infinite, unobservable, charged background --- was ever
there.

\medskip\noindent\textbf{G.1.2. Prove no bosonic sea can
exist, and state what the $\pi^{\pm}$ pair therefore
demonstrates.}

\noindent\emph{Solution.} Step 1: the sea mechanism requires
the negative-energy states to be \emph{fully occupied} so
that decay into them is Pauli-blocked.

Step 2: bosons have no exclusion principle: occupation is
unbounded, ``full'' does not exist, and a bosonic
negative-energy branch triggers unbounded cascade --- each
quantum lowers the energy, and stimulated emission
\emph{accelerates} the collapse. No bosonic sea is even
formally available.

Step 3: yet the charged pion pair, spin zero, exists, with
$\pi^{-}$ the antiparticle of $\pi^{+}$, produced and
annihilated in exact charge balance. Antiparticles exist
where the sea cannot: the sea is refuted as the
\emph{explanation} even where it works as a mnemonic.

Physics intuition: one clean counterexample outranks a
thousand confirmations. The pion did to the sea what the
crossover did to electroweak baryogenesis
(\S\ref{app:lepto-sakharov}): closed the mechanism while
leaving the phenomenon standing --- the surest sign the
explanation lives elsewhere.

\medskip\noindent\textbf{G.1.3. Compute the sea's charge
density with cutoff $\Lambda$ and watch it diverge.}

\noindent\emph{Solution.} Step 1: the sea fills both spin
states of every negative-energy momentum mode up to
$|\mathbf k| = \Lambda$:
\begin{equation}
\rho_{\mathrm{sea}}
\;=\; (-e)\times 2 \int^{\Lambda}
\frac{d^3k}{(2\pi)^3}
\;=\; -\,\frac{e\,\Lambda^3}{3\pi^2}.
\label{eq:g1-seacharge}
\end{equation}

Step 2: at $\Lambda = m_e$ this is already $\sim 10^{30}\,e$
per cm$^3$; at any Planck-adjacent cutoff it is beyond
astronomical --- and gravity, which couples to the energy
density, would see the sea's (equally divergent) energy.

Step 3: the standard repair --- normal ordering --- subtracts
the divergence by definition. That is a consistent
\emph{procedure} and an abandoned \emph{ontology}: one cannot
both believe in the sea and define it away. Modern QFT keeps
the subtraction and drops the sea; the chapter's Step 3 will
supply what the subtraction was silently standing in for.

Physics intuition: infinities in physics are usually the
theory's way of saying ``I was asked about something that is
not there.'' The sea's divergences are not technical blemishes
--- they are the ontology billing for itself.

\subsection{Step 1 --- Feynman--St\"uckelberg: the contour and
the relabeling}
\label{app:anti-contour}

Modern quantum field theory replaced the sea with a reading of
the propagator. The free propagator is an integral with two
poles, and its meaning is fixed by how the energy contour
threads them:
\begin{equation}
\Delta_F(x - y) \;=\; \int\!\frac{d^4p}{(2\pi)^4}\,
\frac{i\,e^{-ip\cdot(x-y)}}{p^2 - m^2 + i\epsilon},
\label{eq:g2-feynman}
\end{equation}
the $i\epsilon$ encoding one choice among four (retarded,
advanced, Feynman, anti-Feynman --- problem G.2.1). Closing
the energy contour splits the Feynman function into its two
time-ordered pieces,
\begin{equation}
\Delta_F(x - y) \;=\;
\theta(t - t')\,\Delta_{+}(x - y)
\;+\; \theta(t' - t)\,\Delta_{+}(y - x),
\label{eq:g2-timeorder}
\end{equation}
positive frequencies forward in time and negative frequencies
backward, and St\"uckelberg
\citep{Stueckelberg1941} and Feynman \citep{Feynman1949} read
the backward piece physically: \emph{a negative-energy
solution propagating backward in time is a positive-energy
antiparticle propagating forward} (problem G.2.2). One
function then computes both particle and antiparticle
processes, and \emph{crossing symmetry} --- the analytic
continuation that turns an incoming particle into an outgoing
antiparticle (problem G.2.3) --- becomes the pairing's
analytic face: annihilation, pair creation, and scattering
are one amplitude read in different channels.

This is enormously better than the sea: no infinite
background, bosons and fermions treated identically, and the
zigzag picture of pair creation --- one worldline, turning in
time --- correctly encodes every rate. What it is \emph{not}
is an explanation, and the chapter will press the point in
Step 2: the contour is chosen, not derived; ``backward in
time'' is a label for an analytic continuation, not a
mechanism; and the question \emph{why} the consistent choice
pairs opposite charge with positive energy is exactly the
question the formalism answers by prescription. The main
text's verdict (\S\ref{ssec:qft-antiparticles}): a
bookkeeping identity of genius, standing in for a mechanism
it does not contain.

\medskip\noindent\textbf{G.2.1. Evaluate the four contours for
the free scalar and identify which yields causal,
positive-energy propagation.}

\noindent\emph{Solution.} Step 1: the $p^0$ integrand of
\eqref{eq:g2-feynman} has poles at $p^0 = \pm E_{\mathbf p}$.
Threading the contour above both gives the retarded
propagator (vanishes for $t < t'$); below both, advanced
(vanishes for $t > t'$); above the negative pole and below
the positive one, Feynman; the reverse, anti-Feynman
(Dyson).

Step 2: close the contour downward for $t > t'$ and upward
for $t < t'$: the Feynman choice picks up $e^{-iE(t-t')}$
(positive frequency) forward and $e^{+iE(t-t')}$ backward ---
the unique prescription in which only positive energies flow
forward while the amplitude remains nonzero at spacelike
separation in the charge-symmetric combination causality
requires.

Step 3: retarded and advanced are causal Green functions but
mix frequencies both ways (classical signaling, not
amplitudes); anti-Feynman propagates negative energies
forward, wrecking stability. One contour survives the joint
demand of causal amplitudes and bounded energy.

Physics intuition: the contour menu is the theory showing
its four possible worlds; three are inconsistent with
quantum-plus-relativity, and the surviving one \emph{contains
antiparticles automatically}. The formalism selects the twin
--- it just cannot say what selected the formalism, which is
Step 3's opening.

\medskip\noindent\textbf{G.2.2. Derive the antiparticle
interpretation of the $t' < t$ piece.}

\noindent\emph{Solution.} Step 1: take the backward piece of
the Feynman function: it carries $e^{+iE(t-t')}$, a
negative-frequency oscillation transported against the time
order.

Step 2: couple the field to the electromagnetic current. On
the backward piece the current's charge enters with reversed
sign relative to the forward piece: transporting charge $-e$
backward from $y$ to $x$ is, line by line in the amplitude,
identical to transporting charge $+e$ forward from $x$ to
$y$.

Step 3: relabel: define the forward-moving, positive-energy,
charge-$+e$ quantum as the antiparticle. Every amplitude is
unchanged --- the relabeling is exact --- and the
negative-energy problem dissolves: nothing physical ever
moves backward or carries $E < 0$. The cost, recorded for
Step 2 of this chapter: ``define'' did the work.

Physics intuition: the zigzag electron --- scattering into
the past and back --- is the most famous cartoon in QFT, and
its content is precisely this relabeling. Cartoons are
memory aids for identities; the identity is real physics,
the cartoon's arrow of time is not.

\medskip\noindent\textbf{G.2.3. Show crossing maps the
$s$-channel particle amplitude to the $u$-channel
antiparticle amplitude.}

\noindent\emph{Solution.} Step 1: write the amplitude
$\mathcal{M}(s, t, u)$ for $a b \to c d$ with the Mandelstam
invariants; LSZ expresses it through one analytic
correlation function.

Step 2: continue an incoming momentum $p_a \to -p_a$: the
invariants permute ($s \leftrightarrow u$), and the continued
amplitude is, by the same LSZ formula, the physical amplitude
for the process with $a$ replaced by its outgoing
antiparticle $\bar a$.

Step 3: analyticity (a causality consequence) guarantees the
continuation exists and is unique: particle and antiparticle
processes are boundary values of \emph{one} analytic
function. The twin is woven into the amplitude's analytic
structure --- which is why no consistent relativistic
S-matrix without antiparticles has ever been built.

Physics intuition: crossing is the experimentalist's daily
bread --- the same vertex measured in scattering and in
annihilation --- and the deepest orthodox hint that particle
and antiparticle are one object described twice, the hint
Step 3 promotes to a theorem.

\subsection{Step 1 --- Weinberg's causality theorem, with its
axioms displayed}
\label{app:anti-weinberg}

The mature orthodox answer is a theorem, and the main text
states it with respect as Theorem~\ref{thm:weinberg}
\citep{WeinbergQTF}: if the interaction density is a Lorentz
scalar built from creation and annihilation operators,
commuting with itself at spacelike separation, on a Hilbert
space with energy bounded below, then every field in it must
combine an annihilation part with a creation part of a
species of \emph{equal mass and opposite charges}. The
mechanism of the proof is worth owning (problem G.3.1): write
the candidate charged field and impose causality,
\begin{equation}
\phi(x) \;=\; \kappa\,\phi^{(+)}_a(x)
\;+\; \lambda\,\phi^{(-)}_b(x),
\qquad
[\phi(x), \phi^\dagger(y)]\big|_{\mathrm{spacelike}}
\;\stackrel{!}{=}\; 0 :
\label{eq:g3-causalfield}
\end{equation}
the spacelike commutator of the annihilation part alone does
not vanish --- relativity forbids localizing a
positive-frequency function --- and the only lawful
cancellation is against a creation part whose mass matches
exactly and whose charges oppose. A neutral particle may satisfy the cancellation
self-conjugately (problem G.3.2): the photon is its own twin,
and whether the neutrino is remains the 4D experimental
question this cohort's $\mbb = 1.72$~meV addresses.

This is Depth~2 on the chapter's ruler, answered rigorously:
\emph{any} interacting relativistic quantum theory with
bounded energy contains antiparticles. And its boundary is
equally sharp (problem G.3.3): three inputs sit upstream,
underived --- the complex Hilbert space (the $i$ is an
axiom), the positivity of energy (imposed on the
\emph{output} of quantization: the pairing is selected
because the alternative wrecks the spectrum), and the time
orientation. Weinberg's theorem is the strongest sentence the
orthodoxy can utter, and its three axioms are the exact
coordinates of where it stops --- the two doors, plus the
arrow, that T-P5-C walks through in Step 3.

\medskip\noindent\textbf{G.3.1. Carry the causal-field proof
for a charged scalar and extract the mass equality.}

\noindent\emph{Solution.} Step 1: write the candidate field
$\phi(x) = \int\! d^3p\,[\,\kappa\, a_{\mathbf p}
e^{ip\cdot x} + \lambda\, b^\dagger_{\mathbf p}
e^{-ip\cdot x}\,]$ with $a$ annihilating a species of mass
$m_a$ and $b^\dagger$ creating a species of mass $m_b$;
charge conservation forbids mixing $a$ with $a^\dagger$ in a
charged field.

Step 2: compute the commutator of \eqref{eq:g3-causalfield}:
\begin{equation}
[\phi(x), \phi^\dagger(y)] \;=\;
|\kappa|^2\, \Delta_{+}^{(m_a)}(x - y)
\;-\; |\lambda|^2\, \Delta_{+}^{(m_b)}(y - x).
\label{eq:g3-commutator}
\end{equation}
At spacelike separation
$\Delta_{+}(x-y) = \Delta_{+}(y-x) \neq 0$: each term alone
survives, and the commutator vanishes iff $m_a = m_b$ and
$|\kappa| = |\lambda|$.

Step 3: the $b$ species must also carry opposite values of
every conserved charge (the field must have a definite
charge for the current to be well defined). Conclusion: the
causal charged field \emph{must} annihilate a particle and
create its equal-mass, opposite-charge twin --- existence
and mass equality in one stroke, for interacting theories.

Physics intuition: relativity will not let charge be
localized without a partner process --- for a moving
observer, absorption at $x$ and emission at $y$ swap order,
and only the twin keeps the bookkeeping frame-independent.
The antiparticle is causality's change for a broken
symmetry of temporal order.

\medskip\noindent\textbf{G.3.2. Show a particle with no
conserved charge may satisfy causality self-conjugately.}

\noindent\emph{Solution.} Step 1: with no charge to conserve,
the field may mix $a$ and $a^\dagger$: set $b = a$ in the
G.3.1 expansion, i.e.\ $\phi = \phi^\dagger$, a real field.

Step 2: the commutator computation repeats with a single
species: $[\phi(x), \phi(y)] = \Delta_{+}(x-y) -
\Delta_{+}(y-x)$, which vanishes at spacelike separation
identically --- the cancellation is between the particle and
\emph{itself}.

Step 3: hence neutral quanta may be their own antiparticles
(photon, $\pi^0$, $Z$), and for neutral \emph{fermions} in
four dimensions the same option is the Majorana question ---
open experimentally, and the entire subject of the
$0\nu\beta\beta$ program (\S\ref{app:lepto-seesaw};
Part~\ref{part:experiments}). Note for Step 3 of this
chapter: in the $(1,5)$ bulk this option is closed by
algebra, not experiment.

Physics intuition: the twin is charge's shadow. No charge,
no obligatory shadow --- and nature's use or refusal of the
self-conjugate option for the neutrino is worth a
tonne-scale experiment precisely because in 4D no theorem
decides it.

\medskip\noindent\textbf{G.3.3. Identify precisely where the
complex Hilbert space, output-positivity, and time
orientation enter Weinberg's argument as assumptions.}

\noindent\emph{Solution.} Step 1: \emph{the $i$.} The
argument lives on a complex Hilbert space from the first
line --- creation operators, phases $e^{ip\cdot x}$,
interference. Nothing in it derives complexity; a
real-Hilbert quantum mechanics would have to be refuted
separately (and is instead \emph{selected against} by the
network Bell tests \citealp{RenouEtAl2021} --- experiment,
not theorem).

Step 2: \emph{positivity as output.} The pairing ``creation
part = opposite charges'' is enforced because the alternative
assignment, quantized, yields an energy spectrum unbounded
below: the criterion inspects the \emph{result} of
quantization and rejects the sick one. Nothing upstream
explains why the healthy assignment exists at all.

Step 3: \emph{time orientation.} ``Energy bounded below''
and ``positive frequency'' presuppose a chosen arrow;
reverse it and the roles of particle and antiparticle swap
consistently. Three assumptions, none derived, all used ---
the exact hypothesis set T-P5-A/B/C is built to discharge
(the circle supplies the charged $i$, the clock the neutral
$i$ and the arrow, the handshake the pairing before
quantization).

Physics intuition: a theorem is a machine that converts
axioms into necessity. Weinberg's converts three axioms into
the antiparticle; the handshake's claim, priced in
\S\ref{sec:anti-ledger}, is that two of those axioms were
themselves conclusions one level down.

\subsection{Step 1 --- the spinor algebra: the mod-8 clock,
read from the textbooks}
\label{app:anti-algebra}

The fourth Step-1 body of knowledge is pure mathematics, and
the student should meet it as such before Step 3 uses it. The
reality type of a spinor representation depends on the
signature $(t, s)$ only through $(t{-}s) \bmod 8$ --- the
Clifford-algebra periodicity \citep{LawsonMichelsohn} --- and
the physical content is carried by the intertwiner $B$
relating a spinor to its complex conjugate:
\begin{equation}
\psi^{c} \;=\; B\,\psi^{*},
\qquad
B\bar B \;=\; \pm\,\mathbf 1
\quad\text{according to the hour of }(t{-}s) \bmod 8 .
\label{eq:g4-intertwiner}
\end{equation}
When $B\bar B = +\mathbf 1$ the condition $\psi = B\psi^*$ is
consistent: Majorana spinors exist, and a fermion \emph{may}
be its own conjugate. When $B\bar B = -\mathbf 1$ the
would-be projection squares to $-1$ and is empty: no Majorana
condition exists, and fermion and conjugate are necessarily
distinct. Four-dimensional spacetime, $(1,3)$, sits at
$t - s \equiv 6$: Majorana is \emph{available} (hence the
neutrino question). The \GG{} bulk, $(1,5)$, sits at
$t - s \equiv 4$: $B\bar B = -\mathbf 1$, and the option is
\emph{closed} --- the main text's
Lemma~\ref{lem:no-majorana}, here recognized as a line read
off a standard table (problem G.4.1).

Two more textbook facts complete the kit. In $(1,5)$ the
chirality operator squares to $+\mathbf 1$ and splits the
eight-component Dirac spinor into conjugate Weyl quartets,
$\mathbf 8 = \mathbf 4 \oplus \bar{\mathbf 4}$
\eqref{eq:weyl-split}, exchanged by conjugation. And in
$4k{+}2$ dimensions chirality is a \emph{signed currency}:
the anomaly eight-form of a chirality-flipped species is the
negative of its partner's, \eqref{eq:anomaly-flip}, the
classic Alvarez-Gaum\'e--Witten structure
\citep{AlvarezGaumeWitten1984} (problem G.4.2). Consistency
--- the vanishing of the total --- therefore forces
conjugate-signed content into any lawful spectrum: the books
cannot balance in one sign (\citealp{ArisakaA3BI6} executes
the full \BIsix{} arithmetic).

Physical intuition --- the eight-hour clock, retold for the
drill. A spinor is a square root of a vector, and square
roots negotiate with reality the way they always have:
sometimes the root of a real thing is real, sometimes it
comes in conjugate pairs. The negotiation cycles with
dimension, period eight (Bott). Four dimensions happens to
sit at an hour where self-conjugacy is on the menu;
six sits four hours on, where it is not. No physics was
chosen in either statement --- which is exactly why Step 3
can build physics on them.

\medskip\noindent\textbf{G.4.1. Verify $B\bar B = -\mathbf 1$
for $(1,5)$ from the mod-8 classification, and locate $(1,3)$
for contrast.}

\noindent\emph{Solution.} Step 1: the standard table
\citep{LawsonMichelsohn} assigns to each $(t{-}s) \bmod 8$
the algebra type: hours $0, 6$ carry $B\bar B = +\mathbf 1$
on the relevant chiral pieces (real hours); hours $2, 4$
carry $B\bar B = -\mathbf 1$ (quaternionic hours); odd hours
carry no chiral split.

Step 2: $(1,5)$: $t - s = 1 - 5 = -4 \equiv 4 \pmod 8$ ---
a quaternionic hour, $B\bar B = -\mathbf 1$, and the
would-be Majorana projection closes on itself:
\begin{equation}
\psi = B\psi^{*}
\;\Longrightarrow\;
\psi = B\,(B\psi^{*})^{*} = B\bar B\,\psi = -\,\psi
\;\Longrightarrow\; \psi = 0,
\label{eq:g4-empty}
\end{equation}
the condition is empty. Lemma~\ref{lem:no-majorana},
verified.

Step 3: $(1,3)$: $t - s = -2 \equiv 6$ --- a real hour:
$B\bar B = +\mathbf 1$, Majorana spinors exist, and the
neutrino's nature is physics, not algebra. One clock, two
hours, two epistemic situations --- option versus verdict.

Physics intuition: this is the rare place where ``we live in
six dimensions'' has an immediate, checkable consequence of
pure algebra: the bulk \emph{cannot} hide the twin inside
the particle. The antiparticle content is forced to be
genuinely present --- Step 3 collects.

\medskip\noindent\textbf{G.4.2. Derive the anomaly flip: show
a chirality-flipped Weyl species contributes the opposite
eight-form.}

\noindent\emph{Solution.} Step 1: the anomaly of a Weyl
fermion in $4k{+}2$ dimensions is computed by the index
density of the chiral Dirac operator: the eight-form (for
$d = 6$) $I_8[\rho, \sigma]$ depends on the representation
$\rho$ and enters multiplied by the chirality
$\sigma = \pm 1$.

Step 2: flipping chirality at fixed $\rho$ flips the
projector $P_\pm = \tfrac12(1 \pm \Gamma^7)$, and the index
density changes sign --- the descent equations carry the sign
through: $I_8[D^-] = -I_8[D^+]$, which is
\eqref{eq:anomaly-flip}.

Step 3: consequence: anomaly freedom $\sum_s \sigma_s
I_8[\rho_s] = 0$ is a \emph{signed} sum. A spectrum written
all in one chirality with nontrivial content cannot cancel;
conjugate-signed species are obligatory bookkeeping ---
which the \BIsix{} ledger turns into the specific reservoir
the main text counts (\S\ref{sec:bulk-encoding}).

Physics intuition: in six dimensions the bookkeeper cannot
be bribed --- every chiral species signs its contribution.
Anomaly cancellation is the second, independent forcing of
antiparticle content, and it is arithmetic, not preference.

\medskip\noindent\textbf{G.4.3. The 4D hour: state exactly
what the mod-8 clock does and does not decide about the
neutrino, and connect to the double-beta program.}

\noindent\emph{Solution.} Step 1: at hour 6, Majorana is
\emph{available}: a neutral 4D fermion may satisfy
$\psi = B\psi^*$. Algebra offers the option and stops.

Step 2: whether nature exercises it is dynamical: the
boundary neutrino inherits a Majorana mass through the
unique weight-zero door (T-P5-4,
\S\ref{app:lepto-seesaw}), so \emph{this} framework answers
yes --- with the readout $\mbb = 1.72$~meV, below the
tonne-scale floor: nature says yes, quietly.

Step 3: the epistemic contrast is the drill's point: in the
bulk the question is \emph{closed} by algebra
($B\bar B = -\mathbf 1$); on the 4D boundary it is
\emph{open} by algebra and closed only by mechanism. When
the student's collaboration debates ``is the neutrino its
own antiparticle,'' the honest answer is: in six dimensions
nothing is; in four, this one is --- by inheritance, at a
stated grade, with a falsifiable size.

Physics intuition: the same question changes character with
the dimension in which it is asked --- a working example of
why ``what the twin \emph{is}'' (Depth 3) cannot be settled
by 4D formalism alone.

\subsection{Step 2 --- what the three answers cannot say}
\label{app:anti-limits}

Now the audit, on the three-depths ruler of
\S\ref{sec:three-depths}. \emph{(1) The sea} answers Depth 1
--- the spectrum contains the twin --- and answers it
\emph{wrongly} at Depth 3: its ontology fails for bosons,
diverges in the vacuum, and asymmetrizes what experiment
finds exactly symmetric (\S\ref{app:anti-sea}).

\emph{(2) The contour} answers Depth 1 exactly and gestures
at Depth 2: it \emph{prescribes} the pairing rather than
deriving it. The Feynman $i\epsilon$ is selected because the
alternatives fail; the selection criterion (causal
amplitudes, bounded energy) is imported, not explained; and
``backward in time'' is an evocative label for an analytic
identity. \emph{(3) Weinberg's theorem} is the real Depth-2
result: the twin is forced in any interacting, causal,
stable, relativistic quantum theory. Its boundary is its
three axioms --- the $i$, positivity-on-output, the arrow
(G.3.3) --- each used, none derived.

\emph{(4) CPT protects, never explains.} The exact identities
of Table~\ref{tab:cpt-tests} --- kaon mass to $10^{-18}$,
moments to $2\times10^{-12}$ --- are \emph{bounds enforced
between two objects}. Nothing in the orthodox account says
why the twin is so exactly the particle's equal that
``identity between two things'' looks suspiciously like
``one thing counted twice.''

\emph{(5) The Majorana undecidability.} In 4D the formalism
offers self-conjugacy as an option (G.4.3) and cannot close
it; forty years of $0\nu\beta\beta$ experiments are the
price of an algebra that shrugs. \emph{(6) The two disjoint
literatures.} ``Why is quantum mechanics complex?'' and
``why do antiparticles exist?'' are pursued by communities
that do not cite each other. An account in which they are
one question would explain, at a stroke, why the twin is
exact: because conjugation is not a relation between two
objects but an operation on one. That account is Step 3.

\medskip\noindent\textbf{G.5.1. Grade the sea, the contour,
and the causality theorem on the three-depths ruler, with
justifications.}

\noindent\emph{Solution.} Step 1: \emph{sea}: Depth 1
(spectral) --- it produces the twin in the spectrum --- and
a \emph{negative} Depth 3: it asserts an ontology, and the
assertion is refuted (bosons, vacuum charge, asymmetry).
Being wrong at Depth 3 is worse than silence; it is the only
one of the three that overclaims.

Step 2: \emph{contour}: Depth 1 with Depth-2 gestures. It is
exact bookkeeping with no failed ontology --- and no
ontology at all: the prescription is selected by criteria it
does not derive.

Step 3: \emph{Weinberg}: Depth 2, fully earned --- necessity
from stated axioms, for interacting theories. Depth 3 is
explicitly renounced: ``those are the axioms.'' The grading
motivates the chapter's shape: what remains open is exactly
Depth 3, and \S\ref{sec:anti-ledger} prices the handshake's
claim to it --- including the concession that its full
strength is free/quasi-free where Weinberg's is
interacting.

Physics intuition: keeping the depths straight prevents both
cheap criticism (the contour is not \emph{wrong}; it is
silent) and cheap victory (a Depth-3 claim must pay in
theorems, not vocabulary). The ruler is the honesty
instrument of Part~\ref{part:fiveroads}, drilled once by
hand.

\medskip\noindent\textbf{G.5.2. Show that CPT bounds the
electron--positron identities but cannot explain them, and
say precisely what would.}

\noindent\emph{Solution.} Step 1: the CPT theorem
(\S\ref{app:d-cpt}) proves $\Theta$ exists and commutes with
the dynamics; consequences: $m_{e^+} = m_{e^-}$,
$g_{+} = g_{-}$, equal lifetimes --- \emph{as identities
between the spectra of two species}. Experiment confirms to
$2\times10^{-12}$ (moments) and $10^{-18}$ (kaon masses),
Table~\ref{tab:cpt-tests}.

Step 2: but the theorem is anatomically silent: it does not
say what the pair \emph{is}, why nature contains it, or why
the identity is exact rather than approximate --- it
constrains whatever exists, given the axioms.

Step 3: an \emph{explanation} would be a construction in
which the two spectra are one spectrum read twice --- where
mass equality is trivial because there are not two masses.
That is precisely the conjugate-readout structure of
\S\ref{ssec:positron-readout}: Depth 3 answering what
Depth 2 can only enforce, with the honest debt (D-P3-4)
that the readout picture's full promotion is registered,
owed, and unclaimed.

Physics intuition: ``why are these two things exactly
alike?'' has only ever had one fully satisfying answer in
physics --- \emph{they are not two things}. The chapter's
arc bends toward earning that answer rather than asserting
it.

\medskip\noindent\textbf{G.5.3. Show Weinberg's three inputs
are mutually independent: none implies the others.}

\noindent\emph{Solution.} Step 1: \emph{the $i$ without
positivity}: complex quantum mechanics with the wrong
particle--antiparticle assignment is kinematically
well-defined --- quantization proceeds --- and delivers a
Hamiltonian unbounded below. Complexity does not imply the
healthy pairing.

Step 2: \emph{positivity without the $i$}: real-Hilbert
quantum theory admits bounded-below dynamics; it fails
elsewhere (tensor-product correlations --- the network Bell
refutation \citealp{RenouEtAl2021}), showing positivity does
not manufacture the complex structure.

Step 3: \emph{the arrow independent of both}: reversing the
time orientation preserves complexity and positivity while
swapping the particle and antiparticle labels globally ---
the assignment's \emph{content} survives, its \emph{naming}
flips. Three independent axioms; a derivation of the twin
from fewer assumptions must therefore supply new structure
--- which is exactly what the bulk's circle and clock are
(Step 3).

Physics intuition: axiom-counting is how one measures the
depth of a theorem. Weinberg spends three; the handshake
claims two of them as outputs by spending geometry instead
--- and \S\ref{sec:anti-ledger} audits whether the purchase
is real.

\subsection{Step 3 --- the handshake, the forced content, and
the readouts}
\label{app:anti-handshake}

The geometry's answer is Part~\ref{part:antiparticles}; this
Step works it in the chapter's vocabulary, and the student
should do the computation once with a pencil ---
\S\ref{ssec:handshake-blocks} is the full rehearsal.

\emph{Two half-blind sources of $i$.} T-P5-A: on each
charge-$n$ sector the normalized circle generator
$\Jq = \partial_\theta/Q$ squares to $-\mathbf 1$: charge
supplies a complex structure, with conjugation as
orientation reversal of the circle --- but it is blind on
neutral fields and blind to its own sign. T-P5-B: the polar
phase of the time-translation generator,
$\Ja = \pm A(-A^2)^{-1/2}$, is the unique positive-energy
complex structure commuting with the dynamics
\citep{AshtekarMagnon1975, Kay1978}: the clock supplies the
$i$ for everything and knows the arrow --- but knows nothing
of charge. Where charge lives, the two must agree.

\emph{The handshake.} T-P5-C, in one displayed line each
way:
\begin{equation}
\begin{aligned}
\cos(n\theta - Et)&:\quad
\Jq \to -\sin,\quad \Ja \to -\sin
\quad\text{(agree)};\\
\cos(n\theta + Et)&:\quad
\Jq \to -\sin,\quad \Ja \to +\sin
\quad\text{(anti-agree)},
\end{aligned}
\label{eq:g6-handshake}
\end{equation}
so on the physical pairing the two structures are one $J$,
and on the wrong pairing no global $J$ exists
(problems G.6.1--G.6.2). The unique repair is to read
negative-frequency charge-$n$ modes as positive-frequency
charge-$(-n)$ modes --- \emph{the Feynman--St\"uckelberg
assignment, derived}, one level below quantization, fixing
the circle's sign as it goes. The contour prescription of
G.2 was this agreement, encoded; Weinberg's
positivity-on-output was this selection, inspected after
the fact. \emph{The existence of antiparticles is the
consistency condition between the sixth dimension and
causal time.}

\emph{The content, forced twice.} Independently of the
pairing, the bulk cannot help containing the conjugate
sector: the mod-8 hour closes the Majorana escape
(G.4.1), and the anomaly ledger cannot balance in one
chirality (G.4.2) --- algebra and arithmetic, agreeing.
What the handshake pairs is therefore genuinely there.

\emph{The readouts.} The positron is the conjugate readout
of the electron's one basin --- same level, opposite winding
--- so mass equality is trivial rather than enforced
(\S\ref{ssec:positron-readout}; the promotion to ``the
positron as a theorem'' is the registered debt D-P3-4).
Wheeler's one-electron universe --- right that the twin is
one object doubly read, wrong custodian: the identification
lives in each mode's geometry, not in a global worldline
(problem G.6.3; \S\ref{ssec:wheeler}).

The Hawking partner
is the same conjugate readout with the \OGGEP{} trace as
the reader (\S\ref{ssec:hawking-partner}). And custody
throughout is the cohort's: \emph{the dictionary owns the
sectors; the geometry owns the $i$.} The chapter's arc, in
one sentence: \emph{Step 1 gave three answers that work;
Step 2 showed none of them says what the twin is; Step 3 is
a geometry in which the twin is the other way around one
object --- and the concessions, free/quasi-free at full
strength, are printed where they belong
(\S\ref{sec:anti-ledger}).}

\medskip\noindent\textbf{G.6.1. Fill the $4\times2$ table of
$(\Jq, \Ja)$ actions on all four modes and locate the unique
agreeing pairing.}

\noindent\emph{Solution.} Step 1: write the four labeled
modes as $u = \cos(n\theta - \omega t)$ with $n = \pm1$ and
$\omega = \pm E$, $E > 0$. The circle acts by
$\Jq u = \tfrac{1}{n}\partial_\theta u =
-\sin(n\theta - \omega t)$ --- the $n$ from the chain rule
cancels the $1/n$ of the normalization, so $\Jq$ returns
$-\sin(\mathrm{arg})$ on every mode.

Step 2: the clock advances the phase a quarter period in
the direction its arrow fixes: on positive-frequency modes
($\omega = +E$) it returns $-\sin(n\theta - Et)$; on
negative-frequency modes ($\omega = -E$) it returns
$+\sin(n\theta + Et)$. Compare: $(n, \omega{=}{+}E)$ agree
for both $n = \pm1$; $(n, \omega{=}{-}E)$ anti-agree for
both.

Step 3: the agreeing set is exactly $\{$charge $+1$ with
positive frequency, charge $-1$ with positive
frequency$\}$: both species run forward, nothing runs
backward --- the Feynman--St\"uckelberg pairing, read off a
$4\times2$ table of signs. The drill is the theorem
(T-P5-C); no step used quantization.

Physics intuition: two gears can mesh in exactly one
relative orientation. Once the student has watched the
$\pm\sin$'s align, ``why do antiparticles exist'' stops
being a mystery and becomes a memory of a table.

\medskip\noindent\textbf{G.6.2. Show the relabeling restores
agreement globally, and that under the wrong pairing no
global $J$ exists.}

\noindent\emph{Solution.} Step 1: the relabeling reads
$(n, \omega{<}0)$ as $(-n, \omega{>}0)$. Under it
$Q \to -Q$, so $\Jq = \partial_\theta/Q$ flips sign on
exactly the modes where it anti-agreed with $\Ja$:
agreement becomes universal, and one global
$J = \Ja = \Jq$ acts on every physical mode.

Step 2: suppose instead one keeps the wrong pairing and
tries to patch: flip $\Jq \to -\Jq$ on the offending modes
by hand. The flip is a choice of sign per mode --- a
1-cochain --- and consistency of $J$ on products and sums
of modes (composite states) requires the signs to be
constant on connected sectors: the mismatch between the
agreeing and anti-agreeing modes is a cocycle no global
choice trivializes.

Step 3: hence no single complex structure serves the wrong
pairing --- the obstruction is structural, not aesthetic.
The theory does not merely prefer the
Feynman--St\"uckelberg assignment; it is inconsistent
without it. This is positivity-on-output (G.3.3), caught
one level earlier, as geometry.

Physics intuition: ``you cannot choose the signs
consistently'' is the same species of argument as a
M\"obius band having no global normal --- the wrong pairing
builds a twisted bundle, and the twist is the answer to
``why not the other way?''

\medskip\noindent\textbf{G.6.3. Second basin or second
orientation: state what experiment separates the two
positron pictures, and what Wheeler's picture gets right and
wrong.}

\noindent\emph{Solution.} Step 1: a \emph{second-basin}
positron --- an independent object coincidentally paired ---
could differ from the electron in any property CPT does not
protect: a slightly different binding in exotic matter, a
drift in the identities of Table~\ref{tab:cpt-tests}. A
\emph{second-orientation} positron cannot differ in
anything: every identity is exact by construction. The
experimental separator is the whole CPT-test column ---
every tightening null is evidence of one object, and any
confirmed violation would resurrect two.

Step 2: Wheeler's one-electron universe carries the same
insight in the wrong custody: one worldline threading time
forward and backward predicts equal electron and positron
counts and zero net lepton number --- refuted by every
baryon in the room (and by Part~\ref{part:matter}'s six in
ten billion). What it got right --- the twin as one object
doubly read --- survives with the custody corrected: the
identification is made mode by mode, by the circle's two
orientations, not by a global worldline's two directions.

Step 3: the Hawking partner completes the family: the
in-falling ``partner'' is not a second object behind the
horizon but the conjugate readout under the \OGGEP{} trace
--- the two-mode-squeezed state read from one side. Three
puzzles, one resolution, custody stated each time: the
dictionary owns the sectors; the geometry owns the $i$.

Physics intuition: the century's antiparticle puzzles ---
Dirac's twin, Wheeler's telephone call, Hawking's partner
--- are one lesson learned three times: where experiment
finds perfect twins, ontology should count one. The student
who has filled G.6.1's table owns that lesson as
arithmetic.

\section{The neutrino's century, from scratch: oscillations,
the matrix, the mass, and the lattice point}
\label{app:neutrino}

The fifth chapter of Block II is written for a specific reader:
the incoming graduate student who has just joined a long-baseline
or double-beta collaboration, or who is beginning flavor
phenomenology, and who wants to own --- not merely quote ---
every neutrino number this paper confronts. Step 1 builds the
standard physics from two-state quantum mechanics to the
double-beta master formula, exactly as the textbooks teach it;
Step 2 marks what that physics assumes but cannot explain;
Step 3 walks the lattice point end to end, at its ledger grade.
Nothing in the main text is needed to start; everything in
Part~\ref{part:experiments} can be recomputed by the reader who
finishes.

The chapter shelf, annotated. \citet{GiuntiKim2007} is the
field's standard reference --- oscillations in vacuum and
matter, field-theory foundations, and the kinematic bounds;
read Chapters 6--10 alongside
\S\ref{app:nu-osc}--\ref{app:nu-appearance}.
\citet{Zuber2020} is the experimentalist's companion ---
detectors, backgrounds, and the double-beta chapter that
\S\ref{app:nu-mass} drills. \citet{MohapatraPal2004} carries
the seesaw and its model space at book length.

\citet{BrancoLavouraSilva} is the CP-violation monograph ---
rephasing invariants, Majorana phases, and the conventions
(their Ch.~22 disambiguates the phase conventions that trap
every beginner). And the Particle Data Group's neutrino review
\citep{PDG2024} is the living document: the reader should
recompute this chapter's numbers against each new edition, in
the spirit of \S\ref{sec:exp-refresh}. Eighteen problems,
complete solutions included.

\subsection{Step 1 --- oscillations from two-state quantum
mechanics}
\label{app:nu-osc}

A neutrino is produced by the weak interaction as a
\emph{flavor} state --- the partner of a charged lepton ---
but propagates as a superposition of \emph{mass} states. Two
flavors suffice for the physics. Let
\begin{equation}
\begin{pmatrix} \nu_\alpha \\ \nu_\beta \end{pmatrix}
=
\begin{pmatrix} \cos\theta & \sin\theta \\
-\sin\theta & \cos\theta \end{pmatrix}
\begin{pmatrix} \nu_1 \\ \nu_2 \end{pmatrix},
\end{equation}
and let each mass state evolve with its own phase. For an
ultrarelativistic neutrino of momentum $p$,
$E_i = \sqrt{p^2 + m_i^2} \simeq p + m_i^2/2E$, so after a
distance $L$ the two components have slipped in phase by
$\Delta\phi = \Delta m^2 L/2E$. Projecting back onto flavor,
the appearance probability is
\begin{equation}
P(\nu_\alpha \to \nu_\beta)
= \sin^2 2\theta \,\sin^2\!\Big(\frac{\Delta m^2 L}{4E}\Big)
= \sin^2 2\theta \,\sin^2\!\Big(
1.267\,\frac{\Delta m^2[\mathrm{eV}^2]\,L[\mathrm{km}]}
{E[\mathrm{GeV}]}\Big),
\label{eq:e1-twoflavour}
\end{equation}
the master formula of the field; the $1.267$ is
$1/(4\hbar c)$ in the stated units, and the reader should
derive it once by hand (problem H.1.1). Physical intuition:
this is the beat note of two clocks. The two mass states are
two pendulums swinging at slightly different rates; the
flavor content is where the beat happens to stand when the
neutrino is caught. Nothing oscillates in the neutrino itself
--- what oscillates is the \emph{alignment} between the basis
nature uses to make neutrinos and the basis it uses to move
them.

\emph{The worked number.} T2K's baseline is $L = 295$~km and
its beam peaks at $E \simeq 0.6$~GeV. With
$|\Delta m^2_{31}| = 2.5\times10^{-3}\,\mathrm{eV}^2$:
$1.267 \times 2.5\times10^{-3} \times 295 / 0.6 = 1.56 \simeq
\pi/2$ --- the experiment sits at its first oscillation
maximum, by design. The same arithmetic at DUNE ($1300$~km,
$2.5$~GeV) gives $1.65$: also near maximum. Two very
different machines, one dimensionless tuning --- the reader
who can do this one line owns the design logic of every
long-baseline experiment on Earth.

\emph{Three flavors and the discovery history.} With three
mass states the amplitude is a coherent sum,
$A_{\alpha\beta} = \sum_i U^*_{\alpha i} e^{-i m_i^2 L/2E}
U_{\beta i}$, and two independent splittings drive two beat
notes: the ``solar'' $\Delta m^2_{21} \simeq
7.4\times10^{-5}\,\mathrm{eV}^2$ and the ``atmospheric''
$|\Delta m^2_{31}| \simeq 2.5\times10^{-3}\,\mathrm{eV}^2$,
thirty-four times apart. The history is a course in itself
(Part~\ref{part:experiments} tells it): Super-Kamiokande's
1998 zenith-angle deficit, SNO's flavor-blind
neutral-current measurement, KamLAND's reactor beat --- three
Nobel-line measurements, one matrix.

Matter modifies the story: coherent forward scattering on
electrons adds an effective potential
$V = \sqrt{2}\, G_F N_e$ for $\nu_e$ only (Wolfenstein; the
MSW resonance), which regenerates or suppresses conversion
depending on the \emph{sign} of $\Delta m^2$ --- the handle
by which DUNE's long, dense baseline reads the mass ordering
(\S\ref{app:nu-appearance}). Textbook custody:
\citet{GiuntiKim2007}, Chs.~7--9; \citet{Zuber2020}, Ch.~8.

\medskip\noindent\textbf{H.1.1. Restore $\hbar$ and $c$ in
\eqref{eq:e1-twoflavour} and derive the $1.267$.}

\noindent\emph{Solution.} Step 1: in natural units the
argument is $\Delta m^2 L/4E$; restoring dimensions, phase
$= \Delta m^2 c^4\, L/(4\hbar c\, E)$ --- one $\hbar c$
because the phase is (energy difference)$\times$(time)
$= (\Delta m^2 c^4/2E)(L/c)/\hbar$, halved again into the
$\sin^2$ convention.

Step 2: insert units with $\hbar c = 197.327$~eV\,nm
$= 197.327\times10^{-9}$~eV\,m:
$\dfrac{(1~\mathrm{eV})^2 (1~\mathrm{km})}
{4\,\hbar c\,(1~\mathrm{GeV})}
= \dfrac{10^{3}}
{4 \times 197.327\times10^{-9} \times 10^{9}}
= \dfrac{10^{3}}{789.3} = 1.267$.

Step 3: hence the argument is $1.267\,\Delta m^2[\mathrm{eV}^2]
L[\mathrm{km}]/E[\mathrm{GeV}]$, dimensionless as a phase must
be. The number every proposal, thesis, and design report in
the field carries is a unit conversion of $\hbar c$ --- worth
deriving once so it is never mysterious again.

Physics intuition: oscillation physics is interferometry, and
$1.267$ is the instrument's fringe-spacing constant. Baselines
and beam energies are chosen by inverting this one line ---
which is exactly what the worked number above did.

\medskip\noindent\textbf{H.1.2. Compute the first-maximum
energies at JUNO's $52.5$~km reactor baseline for both
splittings, and describe the spectrum JUNO reads.}

\noindent\emph{Solution.} Step 1: set the argument of
\eqref{eq:e1-twoflavour} to $\pi/2$ and solve for $E$:
$E = 1.267\,\Delta m^2 L/(\pi/2)$.

Step 2: solar splitting:
$E = 1.267 \times 7.4\times10^{-5} \times 52.5/1.571
= 3.1\times10^{-3}$~GeV $= 3.1$~MeV --- squarely inside the
reactor antineutrino spectrum (2--8~MeV): JUNO sits at the
\emph{solar} first maximum, and the reader who has heard
looser values quoted should trust the arithmetic.

Step 3: atmospheric splitting: the same formula gives
$E = 106$~MeV --- far above reactor energies, so at 2--8~MeV
the atmospheric phase is large and rapidly varying: a fast
wiggle riding on the deep solar envelope. The \emph{phase} of
that fast wiggle differs between the orderings, which is
precisely how JUNO reads the ordering with no accelerator, no
matter effect, and no $\delta$ dependence: pure vacuum
interferometry at the percent level.

Physics intuition: JUNO is a radio tuned between two
stations: the slow solar note sets the volume, the fast
atmospheric note carries the message, and resolving it costs
the $3\%/\sqrt{E}$ energy resolution that drove the entire
detector design.

\medskip\noindent\textbf{H.1.3. Show that for
$\Delta\phi \gg 2\pi$ with energy smearing the probability
averages to $\tfrac12\sin^2 2\theta$, and identify where
Part~XI uses this.}

\noindent\emph{Solution.} Step 1: average
\eqref{eq:e1-twoflavour} over a spread of $E$ (or $L$): the
argument winds many cycles, and
$\langle \sin^2 x \rangle = \tfrac12$ for any smooth
distribution wide compared to a period.

Step 2: hence $\langle P \rangle = \tfrac12 \sin^2 2\theta$:
the interference washes out, the flavor conversion does not
--- an incoherent mixture at the averaged value.

Step 3: Super-Kamiokande's high-$L/E$ atmospheric bins live
in exactly this regime, and the averaged formula is what the
atmospheric-sector numbers of Part~\ref{part:experiments}
fit in those bins; the coherent wiggle survives only in the
resolved-$L/E$ analysis that made the 1998 discovery plot.

Physics intuition: interferometers lose fringes when the
source is broadband; the residual $\tfrac12$ is the
incoherent shadow of the mixing. Knowing which regime an
analysis lives in is the first question a new collaborator
should ask of any oscillation plot.

\subsection{Step 1 --- the PMNS matrix, its conventions, and
the invariant}
\label{app:nu-pmns}

The leptonic mixing matrix in the standard parameterization
is
\begin{equation}
U = \underbrace{\begin{pmatrix} 1 & 0 & 0 \\
0 & c_{23} & s_{23} \\ 0 & -s_{23} & c_{23}
\end{pmatrix}}_{\text{atmospheric}}
\underbrace{\begin{pmatrix} c_{13} & 0 & s_{13}e^{-i\delta} \\
0 & 1 & 0 \\ -s_{13}e^{i\delta} & 0 & c_{13}
\end{pmatrix}}_{\text{reactor + phase}}
\underbrace{\begin{pmatrix} c_{12} & s_{12} & 0 \\
-s_{12} & c_{12} & 0 \\ 0 & 0 & 1
\end{pmatrix}}_{\text{solar}}
\,\mathrm{diag}\big(1, e^{i\alpha_{21}/2},
e^{i\alpha_{31}/2}\big),
\label{eq:e2-pmns}
\end{equation}
with $s_{ij} = \sin\theta_{ij}$. The three angles are
measured by three different classes of experiment (the labels
above), the Dirac phase $\delta$ by appearance asymmetries
(\S\ref{app:nu-appearance}), and the two Majorana phases ---
physical only if the neutrino is its own antiparticle --- by
nothing yet except double beta (\S\ref{app:nu-mass}).
Convention warning, worth a lecture of its own
(\citealp{BrancoLavouraSilva}, Ch.~22): where $\delta$
multiplies $s_{13}$ is a choice, and comparing papers without
fixing it has wasted real person-years.

\emph{The numbers, mid-2026.} $\sin^2\theta_{12} =
0.307(12)$, $\sin^2\theta_{13} = 0.0219(6)$,
$\sin^2\theta_{23} = 0.56(2)$ (upper octant preferred), and
NuFit~6.0's $\delta = 177^{\circ\,+19}_{\;\;-20}$ under
normal ordering (\S\ref{sec:exp-refresh} carries the refresh
rules and pulls). The reader should learn to read a global
fit the way Part~XI does: as a likelihood over the oscillation
data with correlated systematics, whose preferred octant,
ordering, and phase move with each dataset --- which is why
the cohort's confrontation prints pulls with dates.

\emph{The invariant.} All CP-violating appearance effects
are proportional to one rephasing-invariant, the leptonic
Jarlskog (\S\ref{app:flavour-invariants} derives its
uniqueness):
\begin{equation}
\JPMNS = c_{13}^2 s_{13} c_{12} s_{12} c_{23} s_{23}
\sin\delta .
\label{eq:e2-jarlskog}
\end{equation}
With the angles above the prefactor is $0.0332$--$0.0335$,
depending on the fit variant and rounding; the
\emph{maximal} leptonic Jarlskog is thus
$3.3\times10^{-2}$, a thousand times the quark sector's
$3.1\times10^{-5}$ --- large mixing buys the lepton sector
its loud interferometry.

\medskip\noindent\textbf{H.2.1. Count the physical phases of
the lepton mixing matrix for Dirac versus Majorana
neutrinos.}

\noindent\emph{Solution.} Step 1: a general $3\times3$
unitary matrix carries three angles and six phases. The
charged-lepton fields can always be rephased --- three
phases removed, in either case.

Step 2: if the neutrinos are Dirac, their fields can be
rephased too; two further phases go (the overall common
phase is unphysical and already spent), leaving
$6 - 3 - 2 = 1$: the single Dirac phase $\delta$ of the
quark-sector counting (problem E.1.1 of
Appendix~\ref{app:flavour}).

Step 3: if the neutrinos are Majorana, the mass term
$\tfrac12 m\,\overline{\nu^c}\nu$ is quadratic in the same
field: a rephasing $\nu \to e^{i\beta}\nu$ shifts its phase
and is \emph{not} a symmetry. The neutrino rephasings are
lost, and $6 - 3 = 3$ phases survive: $\delta$ plus the two
Majorana phases of \eqref{eq:e2-pmns} --- observable only in
lepton-number-violating interference, which is why
\S\ref{app:nu-mass}'s coherent sum is their sole known
laboratory.

Physics intuition: phases live where rephasings die.
Majorana-ness removes freedom, and removed freedom becomes
physics --- two extra dials that only the
$0\nu\beta\beta$ interferometer can read.

\medskip\noindent\textbf{H.2.2. Show $\JPMNS =
\mathrm{Im}(U_{e1}U_{\mu2}U^*_{e2}U^*_{\mu1})$ reproduces
\eqref{eq:e2-jarlskog} in the standard parameterization.}

\noindent\emph{Solution.} Step 1: read off the elements from
\eqref{eq:e2-pmns}: $U_{e1} = c_{12}c_{13}$,
$U_{e2} = s_{12}c_{13}$,
$U_{\mu1} = -s_{12}c_{23} - c_{12}s_{23}s_{13}e^{i\delta}$,
$U_{\mu2} = c_{12}c_{23} - s_{12}s_{23}s_{13}e^{i\delta}$
(the Majorana factors cancel in any quartet --- each
neutrino index appears once conjugated, once not).

Step 2: multiply. The product
$U_{e1}U_{\mu2}U^*_{e2}U^*_{\mu1}$ contains four terms; the
real-times-real and $|e^{i\delta}|^2$ pieces are real, and
the imaginary part survives only in the cross terms,
collecting to
$c_{12}s_{12}c_{23}s_{23}c_{13}^2 s_{13}
\,\mathrm{Im}(e^{i\delta}) \times (c_{12}^2 + s_{12}^2)$.

Step 3: hence $\mathrm{Im}\,Q = c_{13}^2 s_{13} c_{12}s_{12}
c_{23}s_{23}\sin\delta$, which is \eqref{eq:e2-jarlskog} ---
and the exercise's real lesson: quartets are
convention-proof, parameterizations are not; every claimed
CP effect should be expressible through $\JPMNS$ or it is an
artifact of a chart.

Physics intuition: this is the leptonic copy of
\S\ref{app:flavour-invariants}'s machinery --- one
invariant, sector by sector; nature keeps exactly one
CP dial per Dirac sector and hides the Majorana pair from
oscillations altogether.

\medskip\noindent\textbf{H.2.3. Propagate the NuFit
uncertainties into $\JPMNS^{\max}$, identifying the dominant
contribution.}

\noindent\emph{Solution.} Step 1: logarithmic derivatives of
$\JPMNS^{\max} = c_{13}^2 s_{13}\, c_{12}s_{12}\,
c_{23}s_{23}$. For $\theta_{13}$:
$J \propto s_{13}$ to leading order, so
$\delta J/J \simeq \tfrac12\,\delta(s^2_{13})/s^2_{13}
= \tfrac12(0.0006/0.0219) = 1.4\%$.

Step 2: for $\theta_{12}$: $J \propto \sin 2\theta_{12}$,
$\delta J/J = 2\cot(2\theta_{12})\,\delta\theta_{12}$; with
$s^2_{12} = 0.307(12)$, $\theta_{12} = 33.6^{\circ}$,
$\delta\theta_{12} = 0.013$, this gives $\simeq 1.1\%$. For
$\theta_{23}$: $J \propto \sin 2\theta_{23}$ with
$2\theta_{23} \simeq 97^{\circ}$ --- nearly maximal, so
$\cot(2\theta_{23}) = -0.12$ and the contribution is only
$\simeq 0.5\%$ despite the fat error on the octant.

Step 3: in quadrature: $\delta J/J \simeq 1.9\% \approx 2\%$,
dominated by $\theta_{13}$ and $\theta_{12}$ --- \emph{not}
by $\theta_{23}$, however often the octant is blamed:
near-maximality \emph{protects} $J$ from the octant
uncertainty, because $\sin 2\theta$ is flat at its peak. The
corrected budget is the one to quote.

Physics intuition: a product is most sensitive to the
factors far from their extrema. The octant war that
dominates conference talks barely touches $\JPMNS$ ---
a two-line calculation that saves the reader from a common
and confidently repeated error.

\subsection{Step 1 --- the appearance asymmetry: how a phase
becomes a rate}
\label{app:nu-appearance}

The long-baseline observable is $\nu_\mu \to \nu_e$
appearance against its antineutrino mirror. Expanding the
three-flavor probability to first order in the small
quantities $\alpha \equiv \Delta m^2_{21}/\Delta m^2_{31}
\simeq 0.03$ and $s_{13}$, with
$\Delta \equiv \Delta m^2_{31}L/4E$ (the Cervera--Freund
expansion; \citealp{GiuntiKim2007}, \S13):
\begin{equation}
\begin{aligned}
P(\nu_\mu \to \nu_e) \simeq{}& s_{23}^2 \sin^2 2\theta_{13}
\sin^2\Delta \\
&+ 8\,\JPMNS^{\mathrm{red}}\,\alpha\,\Delta\,\sin\Delta\,
\big(\cos\Delta\cos\delta - \sin\Delta\sin\delta\big)
+ \mathcal{O}(\alpha^2),
\end{aligned}
\label{eq:e3-golden}
\end{equation}
where $\JPMNS^{\mathrm{red}}$ is the Jarlskog prefactor
without $\sin\delta$. The reader should derive the exact
expression once (problem H.3.1) and then trust the
structure: \emph{the atmospheric amplitude is the carrier
wave, the solar amplitude is the modulation, and $\delta$
is the phase between them.} At an oscillation maximum
($\Delta = \pi/2$) the $\cos\delta$ term dies and the
asymmetry is cleanest --- $A_{CP} \sim -0.28\sin\delta$ at
T2K/Hyper-K kinematics: the asymmetry scale of
\S\ref{ssec:we4} ($\sim28\%$ at maximal $\sin\delta$;
$\simeq +8\%$ at the lattice point $\sin\delta = -0.283$),
derived rather than quoted.

Matter complicates and enriches: the potential enters as
$a = 2\sqrt2\, G_F N_e E/\Delta m^2_{31}$, which
\emph{fakes} CP violation (matter is not CP symmetric ---
the Earth is made of electrons, not positrons) but does so
with a known $E$ and $L$ dependence and with opposite sign
for the two orderings. Hence the division of labor the
reader will live inside: Hyper-K at $295$~km
($a \simeq 0.05$, matter-thin) reads $\delta$ cleanly; DUNE
at $1300$~km ($a \simeq 0.2$, matter-thick) reads the
ordering first and $\delta$ second; JUNO reads the ordering
with no $\delta$ dependence at all. Three machines, three
systematics, one phase.

\medskip\noindent\textbf{H.3.1. Derive \eqref{eq:e3-golden}
from the three-flavor amplitude.}

\noindent\emph{Solution.} Step 1: write the appearance
amplitude $A_{\mu e} = \sum_i U^*_{\mu i} U_{e i}
e^{-i m_i^2 L/2E}$, factor out the overall phase of the
first mass state, and expand in the two small quantities:
the atmospheric piece
$A_{\mathrm{atm}} \simeq s_{23}\sin 2\theta_{13}
e^{-i(\Delta + \delta)}\sin\Delta$ and the solar piece
$A_{\mathrm{sol}} \simeq c_{23}\sin 2\theta_{12}\,
\alpha\,\Delta$ (to first order each).

Step 2: square the sum:
$P = |A_{\mathrm{atm}}|^2 + |A_{\mathrm{sol}}|^2 +
2\,\mathrm{Re}(A_{\mathrm{atm}}A^*_{\mathrm{sol}})$. The
first term is the carrier line of \eqref{eq:e3-golden}; the
cross term carries $\cos(\Delta + \delta) =
\cos\Delta\cos\delta - \sin\Delta\sin\delta$ --- the
displayed interference, with the full prefactor assembling
to $8\JPMNS^{\mathrm{red}}\alpha\Delta\sin\Delta$.

Step 3: the antineutrino probability follows by
$\delta \to -\delta$ (and $a \to -a$ in matter): the
$\sin\Delta\sin\delta$ interference flips sign --- the
entire CP asymmetry lives in one term, and checking the
result against the literature's Eq.~(1) is part of the
training.

Physics intuition: a phase is invisible until something
interferes with it. The solar amplitude is the reference
beam of the leptonic interferometer; without the ``small''
solar splitting there would be no CP measurement at any
baseline, at any cost.

\medskip\noindent\textbf{H.3.2. Show matter's fake asymmetry
grows with $L$ and reverses with the ordering, and estimate
it at DUNE.}

\noindent\emph{Solution.} Step 1: in constant-density
matter the atmospheric piece is modified by
$\Delta \to \Delta(1 - a)$ and $\sin 2\theta_{13} \to
\sin 2\theta_{13}/(1 - a)$ (leading order): the appearance
probability is enhanced for neutrinos and suppressed for
antineutrinos in the normal ordering --- and the reverse in
the inverted, because $a \propto 1/\Delta m^2_{31}$ flips
sign with the ordering.

Step 2: at DUNE, $E = 2.5$~GeV and crustal
$N_e$ give $a \simeq 0.2$: the induced neutrino--%
antineutrino rate difference is of order $2a/(1-a^2) \sim
40\%$-class on the atmospheric term --- comparable to or
larger than the genuine $-0.28\sin\delta$ effect
($\simeq +8\%$ at the lattice point).

Step 3: the resolution is the known $E$ and $L$ dependence:
the fake grows linearly with $L$ and with $E$, the genuine
effect does not --- so DUNE fits ordering and phase jointly,
and its spectral shape separates them. This is why DUNE
\emph{must} be matter-thick (the ordering is its first
deliverable) while Hyper-K \emph{must} be matter-thin (the
phase is its first deliverable).

Physics intuition: the Earth is a polarizer inserted in the
interferometer --- it biases the fringes, but predictably,
and a two-detector program with different thicknesses
turns the nuisance into a cross-check.

\medskip\noindent\textbf{H.3.3. At the lattice point,
estimate the bias if an analysis assumes
$\delta = 270^{\circ}$-centered priors.}

\noindent\emph{Solution.} Step 1: the likelihood in
$\delta$ near a non-maximal true value is asymmetric: the
appearance rates depend on $\sin\delta$ and $\cos\delta$,
and at $\delta_{\mathrm{true}} = 196^{\circ}$ the
$\sin\delta$ observable sits at $-0.28$, far from the
boundary values $\pm1$.

Step 2: a prior (or a test statistic) centered on maximal
violation weights the $\sin\delta = \pm 1$ region; with
finite statistics the posterior mode pulls toward maximal
--- the standard boundary bias of constrained fits, here
aggravated because the discovery statistic
(``CP conservation excluded'') is optimized for
$270^{\circ}$.

Step 3: the operational request of Part~XII follows:
publish posteriors (or full likelihood profiles) in
$\delta$, not only discovery significances --- under which
the lattice point and the maximal point are two stated
hypotheses compared on equal footing
(\S\ref{ssec:conf-dpmns} carries the two-point
adjudication).

Physics intuition: instruments inherit the questions they
were funded to ask. A discovery statistic tuned to maximal
CP violation is the right tool for one hypothesis and a
biased tool for another --- the fix is not new hardware but
published likelihoods.

\subsection{Step 1 --- Dirac, Majorana, the seesaw, and
double beta}
\label{app:nu-mass}

A Dirac mass couples left to right,
$m_D\,\bar\nu_L \nu_R$, and conserves lepton number; a
Majorana mass couples a field to its own conjugate,
$\tfrac12 M\,\overline{N^c} N$, and violates lepton number
by two --- lawful only for a fermion with zero conserved
charge. With both present for one generation, the mass
matrix in the $(\nu_L, N^c)$ basis is
\begin{equation}
\mathcal{M} = \begin{pmatrix} 0 & m_D \\ m_D & M
\end{pmatrix},
\qquad
m_{1,2} = \frac{M}{2}\Big(1 \mp \sqrt{1 + 4m_D^2/M^2}\Big)
\;\xrightarrow{M \gg m_D}\;
-\frac{m_D^2}{M},\;\; M .
\label{eq:e4-seesaw}
\end{equation}
The light eigenvalue is the seesaw of
\S\ref{app:lepto-seesaw}, here diagonalized by hand
(problem H.4.1); the light state
$\nu \simeq \nu_L - (m_D/M)N^c$ is almost entirely the
active field. \emph{The worked number:} set $M = 3.85$~TeV
and ask what Dirac coupling gives $m_3 = 50.5$~meV:
$m_D = \sqrt{m_3 M} = 0.44$~MeV, a Yukawa
$y = \sqrt2\,m_D/v \simeq 2.5\times10^{-6}$ ---
electron-class smallness with a TeV partner, no desert
required. This one line is why the cohort's leptogenesis
lives at collider scales rather than at $10^{9}$~GeV.

If the neutrino is Majorana, certain even-even nuclei
($^{76}$Ge, $^{136}$Xe, $^{100}$Mo) can decay
$(A,Z) \to (A,Z{+}2) + 2e^-$ with no neutrinos: one emitted
virtually, reabsorbed as its own antiparticle. The rate is
\begin{equation}
\big[T^{0\nu}_{1/2}\big]^{-1}
= G^{0\nu}(Q,Z)\,\big|M^{0\nu}\big|^2
\Big(\frac{\mbb}{m_e}\Big)^{2},
\qquad
\mbb = \Big|\sum_i U_{ei}^2\, m_i\Big|,
\label{eq:e5-rate}
\end{equation}
with $G^{0\nu}$ the calculable phase space, $M^{0\nu}$ the
nuclear matrix element (NME), and $\mbb$ the effective
Majorana mass --- the coherent sum in which the Majorana
phases finally become physical. Written out in the
parameterization of \eqref{eq:e2-pmns}:
\begin{equation}
\mbb = \big| c_{12}^2 c_{13}^2\, m_1
+ s_{12}^2 c_{13}^2\, m_2\, e^{i\alpha_{21}}
+ s_{13}^2\, m_3\, e^{i(\alpha_{31} - 2\delta)} \big| .
\label{eq:e5-mbb}
\end{equation}
Physical intuition: three paths through the nucleus, one per
mass state, added as \emph{amplitudes}; the Majorana phases
set the interference. The state of the art is
KamLAND-Zen's $T_{1/2} > 3.8\times10^{26}$~yr, translating
--- through the NME spread, the honest systematic of the
field, a factor $\sim2$--$3$ between many-body methods
(\citealp{Zuber2020}, Ch.~7) --- to
$\mbb < 28$--$122$~meV; the ton-scale program
(LEGEND-1000, nEXO, CUPID) reaches
$\mbb \sim 5$--$20$~meV in the 2030s
(\S\ref{sec:exp-future}).

\medskip\noindent\textbf{H.4.1. Diagonalize
\eqref{eq:e4-seesaw} exactly and verify
$\tan 2\theta = 2m_D/M$; evaluate the active--sterile
mixing at the worked numbers.}

\noindent\emph{Solution.} Step 1: for the symmetric matrix
$\begin{pmatrix} 0 & m_D \\ m_D & M\end{pmatrix}$ the
rotation angle diagonalizing it obeys
$\tan 2\theta = 2m_D/(M - 0) = 2m_D/M$; the eigenvalues are
$\tfrac{M}{2}\big(1 \mp \sqrt{1 + 4m_D^2/M^2}\big)$, as
displayed --- the light one negative (a Majorana phase
absorbs the sign, as in problem F.2.1).

Step 2: expand for $M \gg m_D$: $m_{\mathrm{light}} \simeq
-m_D^2/M$, $m_{\mathrm{heavy}} \simeq M$, and
$\theta \simeq m_D/M$.

Step 3: at the worked numbers,
$\theta \simeq 0.44~\mathrm{MeV}/3.85~\mathrm{TeV} =
1.1\times10^{-7}$: active--sterile mixing at the
$10^{-14}$ level in probability --- invisible to every
oscillation experiment, which is why the heavy states hide
from DUNE while running the early-universe door of
Appendix~\ref{app:leptogenesis}.

Physics intuition: the seesaw hides its own evidence ---
the lighter it makes the neutrino, the more invisible it
makes the partner. The only laboratories left are lepton
number itself ($0\nu\beta\beta$) and the cosmology; the
chapter's remaining problems live in both.

\medskip\noindent\textbf{H.4.2. Evaluate \eqref{eq:e5-mbb}
with the ledger inputs and reproduce the point value.}

\noindent\emph{Solution.} Step 1: inputs: masses
$(m_1, m_2, m_3) = (0.065, 8.59, 50.5)$~meV, phases
$(\alpha_{21}, \alpha_{31}) = (0, \pi)$,
$\delta = 196.06^{\circ}$, and the NuFit angles of
\S\ref{app:nu-pmns}.

Step 2: term by term:
$c_{12}^2 c_{13}^2 m_1 = 0.677 \times 0.065 = 0.044$~meV;
$s_{12}^2 c_{13}^2 m_2\,e^{i\cdot 0} = 0.301 \times 8.59 =
2.59$~meV (the first two add);
$s_{13}^2 m_3 = 0.0219 \times 50.5 = 1.11$~meV with phase
$\alpha_{31} - 2\delta = 180^{\circ} - 392.1^{\circ} \equiv
147.9^{\circ}$, contributing $(-0.94 + 0.59\,i)$~meV.

Step 3: sum: $(2.63 - 0.94) + 0.59\,i = 1.69 + 0.59\,i$,
modulus $\sqrt{1.69^2 + 0.59^2} = 1.79$~meV --- the QD-layer
evaluation ($\delta = 196.06^{\circ}$ with measured angles),
four percent above Part~VII's registered first-order
evaluation $1.72$~meV; the residue is the evaluation layer ---
the $\delta$-convention and angle-central choices (the
answer's last digit moves with the layer, which is itself a
lesson of \S\ref{app:nu-pmns}). Both evaluations sit a factor
of a few below the ton-scale floor: the registered null is
layer-robust.

Physics intuition: the third path, phase-rotated by the
lattice's $(\pi, 196^{\circ})$ pair, \emph{destructively}
clips the dominant solar pair --- the geometry does not
merely predict a small $\mbb$, it predicts a specific
interference, and the number is the fingerprint of the
phase set.

\medskip\noindent\textbf{H.4.3. Show that in the inverted
ordering \eqref{eq:e5-mbb} is bounded below by
$\simeq 19$~meV.}

\noindent\emph{Solution.} Step 1: in the inverted ordering
the two heavy states are quasi-degenerate at the
atmospheric scale: $m_1 \simeq m_2 \simeq
\sqrt{|\Delta m^2_{31}|} = 50$~meV, and $m_3$ is light.

Step 2: the worst case is maximal cancellation between the
two dominant terms ($\alpha_{21} = \pi$):
$\mbb \geq c_{13}^2\,(c_{12}^2 - s_{12}^2)\,
\sqrt{|\Delta m^2_{31}|}$ minus the small $m_3$ term.

Step 3: numerically: $0.978 \times (0.693 - 0.307) \times
50 = 18.9 \simeq 19$~meV --- the famous IO floor, inside
the ton-scale reach. Hence the double exposure this paper
volunteers: it requires normal ordering \emph{and} silence
at the ton scale; an IO determination or a confirmed line
each kills it independently
(Table~\ref{tab:falsifiers}).

Physics intuition: the IO floor exists because degenerate
amplitudes cannot fully cancel through the measured solar
angle --- interferometry with a guaranteed residue. The
ton-scale program is decisive for IO and, for this
paper's NO-plus-silence prediction, decisive in the other
direction too.

\subsection{Step 2 --- what the campaign cannot price}
\label{app:nu-limits}

Now the audit, in the register of the four chapters before
this one. \emph{(1) Six dials, zero explanations.} Two
splittings, three angles, one phase: all measured, none
explained. The parameterization \eqref{eq:e2-pmns} is a
coordinate chart, not a theory --- the oscillation
formalism is a beautiful interferometer with no opinion
about what it interferes.

\emph{(2) Why three? Why tiny? Why large mixing?} The
formalism takes the flavor count, the mass scale, and the
striking contrast with the quark sector's small angles as
inputs; no Standard-Model principle addresses any of them.

\emph{(3) The phase is free, and the campaign is priced on
its freedom.} A billion-dollar program measures $\delta$
precisely because any value is equally lawful; sensitivity
is advertised against maximal violation
($270^{\circ}$) because nothing prefers anything ---
the exposure of problem H.3.3.

\emph{(4) The Majorana question cannot even be posed.}
Whether the neutrino is its own antiparticle is, in the 4D
formalism, an option with no principle behind it
(\S\ref{app:anti-algebra}: the mod-8 hour makes it
available and stops). \emph{(5) The funnel is structural.}
Without masses and phases, \eqref{eq:e5-mbb} is a band
spanning two orders of magnitude; a null refutes nothing
--- the field's deepest frustration is that it measures a
coherent sum whose every term is free. \emph{(6) No
quark--lepton bridge.} $\delta_{\mathrm{CKM}}$ and
$\dPMNS$ live in disconnected Yukawa sectors (limitation
(3) of \S\ref{app:flavour-limits}); the SM cannot even ask
whether $66.9^{\circ}$ and $196.4^{\circ}$ are related.

\medskip\noindent\textbf{H.5.1. Make limitation (1)
precise: count what the oscillation program measures
against what any theory must eventually supply.}

\noindent\emph{Solution.} Step 1: the oscillation sector
delivers six numbers (two splittings, three angles, one
phase) plus two discrete verdicts (ordering, octant); the
kinematic and cosmological sectors add the absolute scale;
double beta adds the nature and, if Majorana, two more
phases.

Step 2: the Standard Model plus seesaw \emph{parameterizes}
all of it (eighteen high-scale parameters, problem F.5.2)
and \emph{predicts} none of it: every measured number maps
to an inequivalent choice of Yukawa texture.

Step 3: a framework claiming explanatory depth must
therefore output, not input, at least: the mass triplet,
the ordering, the phase set, and the nature --- which is
exactly the list Step 3 confronts, entry by entry, with
grades. The counting is the specification of what
``explanation'' would mean here.

Physics intuition: the neutrino sector is physics' cleanest
case of a complete measurement program with an empty
theory column --- which is precisely why it is the right
place to stake a falsifiable geometry.

\medskip\noindent\textbf{H.5.2. Propagate a factor-2 NME
uncertainty into the half-life needed to reach
$\mbb = 1.72$~meV in $^{136}$Xe.}

\noindent\emph{Solution.} Step 1: from \eqref{eq:e5-rate},
$T_{1/2} \propto 1/(|M^{0\nu}|^2\,\mbb^2)$: scaling from
KamLAND-Zen's $3.8\times10^{26}$~yr at
$\mbb = 28$--$122$~meV (the spread \emph{is} the NME
band).

Step 2: $T_{1/2}(1.72~\mathrm{meV}) = 3.8\times10^{26}
\times (28/1.72)^2 = 1.0\times10^{29}$~yr at the
favorable NME edge, and $3.8\times10^{26}\times
(122/1.72)^2 = 1.9\times10^{30}$~yr at the unfavorable ---
call it $10^{29}$--$10^{31}$~yr with the factor-2 NME
freedom folded in.

Step 3: the ton-scale program's design reach is
$\sim10^{28}$~yr: the lattice point sits one to two
orders beyond it (three at the unfavorable NME edge with
its factor-2 freedom) --- ``below reach'' is arithmetic, not
pessimism, and the registered-null prediction of Step 3 is
therefore safe from the NME systematic in \emph{both}
directions: no NME choice brings $1.72$~meV into this
generation's view.

Physics intuition: half-life scales as the inverse square
of the amplitude --- precision bought in $\mbb$ costs
quadratically in exposure. The NME band moves the goalposts
by a factor of a few; the lattice point sits a stadium
away.

\medskip\noindent\textbf{H.5.3. Show that in the SM plus
seesaw, $\delta_{\mathrm{CKM}}$ and $\dPMNS$ are
independent inputs: no measurement of one constrains the
other.}

\noindent\emph{Solution.} Step 1: the quark phase lives in
$V = U_L^{u\dagger}U_L^d$, built from the up and down
Yukawas; the lepton phase lives in
$U = U_L^{e\dagger}U_\nu$, built from the charged-lepton
Yukawa and the seesaw output \eqref{eq:e4-seesaw}.

Step 2: the four Yukawa matrices are independent inputs of
the Lagrangian; any $(\delta_{\mathrm{CKM}}, \dPMNS)$ pair
is realized by some choice, and the renormalization-group
mixing between sectors is negligible at measured couplings.
Hence the joint parameter space is a product: measuring one
phase updates nothing about the other.

Step 3: contrast Step 3's claim: both phases are circuits
of \emph{one} pentagon --- branch for quarks, center for
leptons (problem E.6.2 of Appendix~\ref{app:flavour}) ---
so their values are correlated outputs of one texture. The
SM cannot pose the question; the geometry answers it with
two numbers already in the confrontation table.

Physics intuition: unrelated in the orthodoxy, cousins in
the geometry --- the pair
$(66.9^{\circ}, 196.4^{\circ})$ is either a coincidence
twice over or one shape read twice. The falsifier table
prices the difference.

\subsection{Step 3 --- the lattice point, end to end}
\label{app:nu-lattice}

The geometry's answers, in the order the campaign will test
them. \emph{The masses} (limitations 1--2): the three light
masses are derived --- $(0.065, 8.59, 50.5)$~meV, normal
ordering, from the seesaw with the matched Yukawa and the
lock scale (Part~\ref{part:matter}; parent derivation in
\citealp{ArisakaA9SM}) --- so both splittings are outputs:
$\Delta m^2_{21} = 7.37\times10^{-5}\,\mathrm{eV}^2$ and
$\Delta m^2_{31} = 2.55\times10^{-3}\,\mathrm{eV}^2$, with
the pulls printed: the solar value sits at $-0.6\sigma$
against NuFit~6.0's $(7.49\pm0.19)\times10^{-5}$ and at
$-1.0\sigma$ against JUNO's $(7.50\pm0.12)\times10^{-5}$;
the atmospheric value at $+0.7$ to $+1.9\sigma$ against the
$2.51$--$2.53\times10^{-3}$ class, depending on the fit
variant. The interferometer of
\S\ref{app:nu-osc}, on this account, has been measuring
ledger entries all along.

\emph{The phase set} (limitation 3): $\dPMNS = \pi +
2\arcsin(1/7) = 196.43^{\circ}$ (QD $196.06^{\circ}$),
$(\alpha_{21}, \alpha_{31}) = (0, \pi)$ --- so
$\sin\dPMNS = -0.283$ exactly (problem H.6.1), $\JPMNS =
-9.4\times10^{-3}$ (problem H.6.2), and the campaign
becomes a two-point discrimination: the lattice point
against the maximal expectation, $73.6^{\circ}$ apart,
decided at the late-2030s $\pm7$--$10^{\circ}$ resolutions
(\S\ref{ssec:conf-dpmns}). The flagship $5\sigma$
CP-violation discovery becomes the quiet signature: it
should \emph{not} arrive.

\emph{The nature} (limitations 4--5): the door theorem
T-P5-4 makes the neutrino Majorana and unique --- the 4D
option converted to a verdict by inheritance
(\S\ref{app:anti-algebra}) --- and the funnel collapses to
the point $\mbb = 1.72$~meV (problem H.4.2): a registered
null for the ton-scale era, doubly exposed through the IO
floor (problem H.4.3). \emph{The bridge} (limitation 6):
one pentagon, two circuits ---
$66.9^{\circ}$ and $196.4^{\circ}$ as correlated outputs
(problem H.5.3). And the same door, run at the lock scale
in the early universe, hands the sphalerons their $28/79$
--- Appendix~\ref{app:leptogenesis}, grade D$^{*}$ --- and
the six extra baryons per ten billion photons are the
receipt.

\emph{What each instrument adjudicates.} JUNO: the
ordering, $\delta$-blind --- a clean kill channel.
Hyper-K: the phase, matter-thin --- the quiet signature's
home. DUNE: ordering then phase, matter-thick --- the
consistency check that matter effects cannot fake both.
Ton-scale $0\nu\beta\beta$: the Majorana architecture,
one-sided. Each column of Table~\ref{tab:exp-timeline} is
a row of this chapter.

\emph{The reader's checklist.} By the end of this chapter
the reader should be able to: derive
\eqref{eq:e1-twoflavour} and tune any baseline; write
\eqref{eq:e2-pmns} in a stated convention and translate
between conventions; expand \eqref{eq:e3-golden} and
separate genuine from matter-faked CP violation;
diagonalize \eqref{eq:e4-seesaw} and run the
matched-Yukawa arithmetic; evaluate \eqref{eq:e5-mbb} at
the ledger point; and recompute every pull in
Table~\ref{tab:exp-predicted-measured} from the current
NuFit release. That last exercise --- rerun the
confrontation at each data release, in the discipline of
\S\ref{sec:exp-refresh} --- is not homework; it is the
profession this paper hopes its reader joins.

\medskip\noindent\textbf{H.6.1. From $\dPMNS = \pi +
2\arcsin(1/7)$, derive $\sin\dPMNS = -8\sqrt3/49$
exactly.}

\noindent\emph{Solution.} Step 1:
$\sin(\pi + x) = -\sin x$ with $x = 2\arcsin(1/7)$.

Step 2: $\sin(2\arcsin u) = 2u\sqrt{1 - u^2}$ with
$u = 1/7$:
$2 \cdot \tfrac17 \cdot \sqrt{1 - \tfrac1{49}}
= \tfrac{2}{7}\cdot\tfrac{\sqrt{48}}{7}
= \tfrac{2 \cdot 4\sqrt3}{49} = \tfrac{8\sqrt3}{49}$.

Step 3: hence $\sin\dPMNS = -8\sqrt3/49 = -0.28278$ --- an
exact surd. The ``quiet signature'' --- the moderate,
negative-sign asymmetry every long-baseline fit will
converge toward if the geometry is right --- is not a fitted
decimal but a closed form, as befits a lattice point.

Physics intuition: when a theory's prediction is a surd,
the experiment is measuring arithmetic. The reader should
keep the closed form in mind when the collaboration's
posterior narrows: the target has zero width.

\medskip\noindent\textbf{H.6.2. Assemble $\JPMNS$ from the
prefactor and the surd, and compare with the worked value.}

\noindent\emph{Solution.} Step 1: the prefactor of
\eqref{eq:e2-jarlskog} at the mid-2026 angles is
$0.0332$--$0.0335$ across the current fit variants
(\S\ref{app:nu-pmns}), with the $\sim2\%$ budget of
problem H.2.3.

Step 2: multiply by the surd:
$\JPMNS = 0.0332 \times (-0.28278) = -9.4\times10^{-3}$; the
prefactor spread gives $-9.4$ to $-9.5\times10^{-3}$ --- the
paper quotes $-9.4\times10^{-3}$, in agreement with the worked
example \S\ref{ssec:we4} inside
the angle uncertainties.

Step 3: the sign is physical and load-bearing: it sets the
direction of the T2K/Hyper-K asymmetry and, upstream, the
sign of the lepton ledger the sphalerons process
(Appendix~\ref{app:leptogenesis}) --- one surd, laboratory
and cosmology both.

Physics intuition: $\JPMNS$ is the lepton sector's entire
CP budget in one number; that this paper writes it as
(measured angles) $\times$ (exact surd) is the cleanest
statement of what ``valuing the phase'' means.

\medskip\noindent\textbf{H.6.3. Show the pair (ordering,
$\mbb$) is internally redundant for this paper: either
measurement alone can kill it; both together cannot save
it.}

\noindent\emph{Solution.} Step 1: the chain requires normal
ordering (the derived triplet is NO); an experimental IO
determination --- JUNO's vacuum wiggle, DUNE's matter
lever, or their combination --- kills the mass chain
directly, independent of any double-beta result.

Step 2: the chain requires ton-scale silence
($\mbb = 1.72$~meV, below reach --- problem H.5.2); a
confirmed $0\nu\beta\beta$ line in the ton-scale window
kills the door architecture directly, independent of the
ordering.

Step 3: conversely, NO confirmed \emph{and} silence heard
are each merely consistent; neither, nor both, proves the
point value --- proof-grade discrimination waits on the
phase measurement and the $10^{29+}$-yr era. The AND-logic
of \S\ref{sec:falsification} is deliberate: kill channels
are independent, confirmation channels are not ---
redundancy is the design, not an accident.

Physics intuition: a falsifiable framework wires its
failure modes in parallel and its success modes in series.
The reader joining any one of JUNO, DUNE, Hyper-K, or
LEGEND holds a kill switch; no single collaboration holds
a confirmation --- which is how it should be.

\section{Internal source ledger for this paper}
\label{app:source-ledger}

The derivation, notation, and theorem flow of this paper are
taken from its cohort companions listed below; historical
literature is cited for orientation and confrontation, never as
a derivational premise. The two \emph{parents} --- the particle
paper and the cosmology paper --- carry the heaviest load, and
the Foundations-Branch partner \PaperPFour{} supplies the
complex-structure machinery the handshake runs on.

\begin{table}[H]
\centering
\captionsetup{list=no}%% v15.0: see the front-matter note
\caption[Internal source ledger.]{Internal source ledger used in the present paper. One row per upstream preprint, with the role it plays in this paper's chain and a sentence saying what is actually taken from it rather than what the source paper is about. The ledger exists so that every inherited premise in the argument can be traced to the paper that owns it, and so that nothing is inherited silently.}
\label{tab:source-ledger-p5}
\renewcommand{\arraystretch}{1.2}
\small
\begin{tabular}{p{0.15\textwidth}p{0.24\textwidth}p{0.51\textwidth}}
\toprule
Preprint & Role in the chain & Relevance \\
\midrule
\AOne{} & Constitutional foundation & Fixes the four axioms, seven postulates, and \OGGEP{} semantics; AX-1 supplies the arrow and AX-3 the positivity clause that select the clock's complex structure. \\
\ATwo{} & Six-dimensional stage & Supplies the \GG{} bulk, the spinor content $\mathbf 4 \oplus \bar{\mathbf 4}$, the weight arithmetic, and the frame-path spin registers (its \S20) behind the candidate spin suite. \\
\AThree{} & Anomaly ledger & Proves the \BIsix{} bookkeeping: the anomaly flip, the reservoir arithmetic $(8,4,48,12,2)$, and the $n_F = 32\Omega$ counting that forces the conjugate content. \\
\AFive{} & Boundary law & Derives the \GDOS{} boundary reduction: the \OGGEP{} projection whose one-way trace is this paper's arrow, partner, and sphaleron custody. \\
\AEight{} & Quantum-mechanics companion & Carries the real-bulk series (R1--R15), the Born-rule status, and the statistics theorem T-A8.12 this paper's quartet argument feeds. \\
\ANine{} & Particle parent & Supplies the flavor geometry: the pentagon and QD codebook, the Majorana-door parent theorem, the derived neutrino ledger (masses, $\dPMNS$, Majorana parity), and the honesty-grade vocabulary. \\
\ATen{} & Cosmology parent & Supplies the transport: the descent and $T_{\mathrm{rh}}$, the Sakharov custody, the resonant-leptogenesis chain, and the zero-dilution clause audited in Part~VIII. \\
\PaperPFour{} & Foundations partner & Owns ``why complex numbers?'': the circle and clock as the two real sources of $i$ (its T-P4-1 through T-P4-4), restated in Part~III as this paper's inputs. \\
\PaperCOne{} & Strong-CP companion & Carries the Kaxion two-fate architecture behind the $\bar\theta$ leakage floor and its haloscope window. \\
\PaperWTwo{} & Holonomy white paper & Supplies the two-fiber anatomy used in \S\ref{sec:anatomy-of-a-theorem} and the three-horizon sorting of \S\ref{sec:three-horizons}; developed there out of this paper's own reversals and returned here as a way of reading them. \\
\bottomrule
\end{tabular}
\end{table}

\section{Cross-paper inheritance table (compact)}
\label{app:inheritance-table}

This appendix compresses the inheritance narrative of
\S\ref{sec:inheritance} into a one-page table for quick
reference: what the present paper draws from each cohort paper,
and which of its Parts consume the contribution.

\begin{table}[H]
\centering
\captionsetup{list=no}%% v15.0: see the front-matter note
\caption[Compact cross-paper inheritance.]{Compact cross-paper inheritance: what the present paper draws from each upstream paper. ``Used by'' lists the Parts of this paper that depend on each contribution. Fourteen rows, one per upstream paper. Where the source ledger says what each paper contributes, this table says where the contribution is consumed, so that a reader following a single Part can see which companions that Part depends on without having to read the rest of the paper first.}
\label{tab:inheritance-compact-p5}
\renewcommand{\arraystretch}{1.18}
\small
\begin{tabular}{p{0.14\textwidth}p{0.53\textwidth}p{0.25\textwidth}}
\toprule
\textbf{Paper} & \textbf{Key contribution to this paper} & \textbf{Used by (Parts)} \\
\midrule
\AOne{} & Axioms and postulates; the arrow (AX-1) and the positivity clause (AX-3) & III, IV \\
\ATwo{} & Bulk, spinor split, weight arithmetic, \S20 frame-path registers & III, IV, X \\
\AThree{} & Anomaly flip; reservoir arithmetic; conjugate content forced & III, IV \\
\AFive{} & \OGGEP{} projection: arrow, partner trace, boundary custody & V, VIII, X \\
\ASix{} & \GRIP{} runtime vocabulary for the descent's selections & VI, VIII \\
\AEight{} & Real-bulk series; Born status; statistics theorem T-A8.12 & III, IV, X \\
\ANine{} & Pentagon + QD codebook; Majorana-door parent; neutrino ledger; grade register & VII, VIII, XI \\
\ATen{} & Descent, $T_{\mathrm{rh}}$; Sakharov custody; leptogenesis chain; honesty objections & VIII, XI, XII \\
\PaperPOne{} & KK-tower context of the compact directions & III \\
\PaperPTwo{} & $N = 3$: the family count behind the one surviving phase & VII \\
\PaperPThree{} & Electron basin; the conjugate-readout debt D-P3-4 & IV \\
\PaperPFour{} & Two real sources of $i$; the representation machinery of the handshake & III, IV \\
\PaperCOne{} & Kaxion; the $\bar\theta$ two-fate architecture and leakage floor & VII \\
\bottomrule
\end{tabular}
\end{table}

\section{Consolidated benchmark audit}
\label{app:benchmark-audit}

The main text develops the benchmark problems Part by Part;
this appendix compresses the same logic into a single
operational audit sheet: for each problem the paper claims to
address, the inherited mechanism that carries it and the
principal kill switch that would destroy it.

\begin{longtable}{p{0.22\textwidth}p{0.34\textwidth}p{0.28\textwidth}}
\captionsetup{list=no}%% v15.0: see the front-matter note
\caption[Consolidated benchmark audit.]{Consolidated benchmark audit for the present paper. Fourteen problems, each with the mechanism this program brings to it and the single measurement that would retire that mechanism. The third column is the reason the table exists: a benchmark that cannot be failed is not a benchmark, and every row here names the observation that would fail it.}
\label{tab:benchmark-audit-p5}\\
\toprule
Problem & Inherited mechanism & Principal kill switch \\
\midrule
\endfirsthead
\toprule
Problem & Inherited mechanism & Principal kill switch \\
\midrule
\endhead
\midrule \multicolumn{3}{r}{\itshape continued on next page}\\
\endfoot
\bottomrule
\endlastfoot
The C/P/T table & Orientation reversals of the three real factors; the antiunitary fork as clock-content (T-P5-1) & A discrete operation measured outside the forced unitary/antiunitary table \\
CPT survival & Total orientation reversal, invisible to any real texture (T-P5-2) & Any confirmed CPT violation (Table~\ref{tab:cpt-tests}); via Greenberg, the arena itself \\
Antiparticle existence & The handshake: two half-blind sources of $i$ forced to agree (T-P5-A/B/C), content forced twice (Lemma~\ref{lem:no-majorana}; anomaly ledger) & Drift in any CPT-protected identity; a consistent relativistic S-matrix without the pairing \\
Parity violation & Orientation of the family descent through one chiral gate (Part~VI; conditional on C-P5-A) & $W_R$ or mirror fermions at any scale (F-P5-1) \\
Quark CP phase & Pentagonal lattice holonomy, QD-completed to $13\pi/35$ (T-P5-3) & World fit excluding the lattice point (the raw first order survives to $+1.2\sigma$-class only, with the $\pi/35$ theory width in quadrature) \\
Leptonic CP phase & Central-circuit completion $\pi + 2\arcsin(1/7)$ (T-P5-3, conditional on the assignment lemma) & DUNE/Hyper-K posterior excluding the $196^{\circ}$ band \\
Strong CP & Two-fate architecture with the Kaxion fate and the $e^{-K}$ leakage floor & A measured EDM between the leakage floor and $10^{-10}$ \\
Majorana nature & The weight-zero door: Majorana iff nowhere to wind (T-P5-4) & A Dirac neutrino established; an inverted ordering; a ton-scale $0\nu\beta\beta$ line \\
The matter excess & Resonant leptogenesis on the inherited ledger, throughput $e^{-K/2}$ (T-P5-5, conditional on C-P5-C) & Proton decay observed; ordering, phase, or cluster kills; a confirmed line \\
Spin and statistics & The Wick quartet's second child: exchange as half-turn composition (T-Stat-A, candidate) & A consistent quasi-free counterexample to the quartet route \\
\end{longtable}

\section{Named-theorem-and-conjecture deck}
\label{app:statement-deck}

This appendix lists every formal named statement introduced or
restated in this paper, with its label and a one-line summary.
Full statements appear at the referenced labels; the inventory
line at the end is counted from the source, not by hand.

\begingroup
\renewcommand{\arraystretch}{1.20}\small
\begin{longtable}{p{0.13\textwidth}p{0.14\textwidth}p{0.61\textwidth}}
\captionsetup{list=no}%% v15.0: see the front-matter note
\caption[Named-statement deck.]{Named-statement deck of the present paper. Every named statement in the paper, in one list: theorems, lemmas, propositions, conjectures and the imported results, each with its label and a one-line summary. The type column is the honest part. It separates what is proved here from what is conjectured and from what is carried in from elsewhere, and the deck is meant to be read whole before any single entry is leaned on.}
\label{tab:statement-deck-p5}\\
\toprule
\textbf{Type} & \textbf{Name} & \textbf{One-line summary} \\
\midrule
\endfirsthead
\toprule
\textbf{Type} & \textbf{Name} & \textbf{One-line summary} \\
\midrule
\endhead
\midrule \multicolumn{3}{r}{\itshape continued on next page}\\
\endfoot
\bottomrule
\endlastfoot
Theorem & T-P5-A & The circle: the charge-normalized generator is the unique gauge $i$ on charged sectors, up to sign (Thm.~\ref{thm:p5-A}). \\
Theorem & T-P5-B & The clock: the polar phase of time translation is the unique positive-energy state-space $i$ (Thm.~\ref{thm:p5-B}). \\
Theorem & T-P5-C & The handshake: the two structures agree only on the Feynman--St\"uckelberg pairing; antiparticles as the consistency condition (Thm.~\ref{thm:p5-C}). \\
Theorem & T-P5-B1 & The Wick quartet: one analytic fact behind rotation, statistics, and the thermal clock (Thm.~\ref{thm:p5-B1}). \\
Theorem & T-P5-C1 & One shared $J$ extends to composites and Fock space with no new complex input (Thm.~\ref{thm:p5-C1}). \\
Theorem & T-P5-C2 & Fermionic conjugation factorizes: circle-orientation reversal composed with a fixed real spinor map (Thm.~\ref{thm:p5-C2}). \\
Theorem & T-P5-1 & The C/P/T table from orientation: which reversals are unitary, which antiunitary, computed not assigned (Thm.~\ref{thm:p5-1}). \\
Theorem & T-P5-2 & CPT as total reversal: every real texture preserves the product; the identities of Table~\ref{tab:cpt-tests} follow (Thm.~\ref{thm:p5-2}). \\
Theorem & T-P5-3 & Flavor holonomy and the chirality criterion: CP violation iff the texture is chiral; the lattice values (Thm.~\ref{thm:p5-3}). \\
Theorem & T-P5-4 & Majorana iff weight zero: the unique lawful $\Delta L = 2$ door (Thm.~\ref{thm:p5-4}). \\
Proposition & T-P5-5 & The assembled $\etaB \simeq 5.9\times10^{-10}$ at D$^{*}$ grade, conditional on C-P5-C (Prop.~\ref{prop:etaB}). \\
Proposition & P-P5-D & The handedness invariant of the descent (Prop.~\ref{prop:handedness}). \\
Proposition & P-P5-E & The Sakharov tripod: the three conditions as boundary shadows of the constitutional structure (Prop.~\ref{prop:sakharov-tripod}). \\
Conjecture & C-P5-A & The Gate-Operator Lemma (L11): the missing uniqueness clause of the parity chain (Conj.~\ref{conj:gate}). \\
Conjecture & C-P5-C & Sphaleron-transport normalization and the on-resonance condition: the chain's one named input (Conj.~\ref{conj:resonance}). \\
Lemma & --- & No Majorana condition in $(1,5)$: $B\bar B = -\mathbf 1$ at the quaternionic hour (Lemma~\ref{lem:no-majorana}). \\
Corollary & B1-a & The boundary's three shorthands of the quartet (Cor.~\ref{cor:B1a}). \\
Corollary & B1-b & Custody: imaginary time belongs to the clock (Cor.~\ref{cor:B1b}). \\
Corollary & --- & Kramers' degeneracy with no phase choice (Cor.~\ref{cor:kramers}). \\
Restated & T-Stat-A & Spin--statistics as the quartet's second child (candidate grade; home paper \AEight{}) (Thm.~\ref{thm:quartet-second-child}). \\
Imported & --- & Weinberg's causality argument, stated with respect as the orthodoxy's Depth-2 summit (Thm.~\ref{thm:weinberg}). \\
\end{longtable}
\endgroup

\textbf{Inventory.} 10 owned theorems + 3 propositions + 2 named
conjectures + 1 lemma + 3 corollaries, plus one restated
candidate (T-Stat-A) and one imported theorem (Weinberg); 8
numbered predictions (Pred-1--Pred-8, Part~XI); 9 falsifier rows
(Table~\ref{tab:falsifiers}); 82 answered objections
(O-P5-1--O-P5-82, Part~XV).

\section{Named-conjecture promotion roadmap}
\label{app:promotion-roadmap}

This appendix states the open targets that, if discharged, would
promote graded entries --- from conditional to derived, or from
candidate to theorem. Each row names the item, what its
discharge closes, the venue where the work belongs, and its
status. It is the place where the grade register stops being a
table of labels and becomes a research program.

%% v15.0: table[H] -> longtable.  This grid ran 79.35 pt past the
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%% that had none to give.  A `table' cannot break, so the honest
%% answer is a float that can: longtable, which is what the four other
%% reference tables of this appendix already are, with the same
%% \endfirsthead/\endhead/\endfoot pattern.  NOT ONE ROW, VALUE OR
%% COLUMN WIDTH IS CHANGED, and the caption is not shortened back.
\begin{longtable}{p{0.04\textwidth}p{0.30\textwidth}p{0.30\textwidth}p{0.12\textwidth}p{0.12\textwidth}}
\captionsetup{list=no}%% v15.0: see the front-matter note
\caption[Named-conjecture promotion roadmap.]{Named-conjecture promotion roadmap for the present paper. Venue ``this paper'' = work that belongs here and remains open; a paper code = the companion where the work naturally lives; ``open'' = open research target. Ten named items, with the work each discharge would close, the venue where that work belongs, and its status. The first row is the paper's sharpest open item and is placed first for that reason. The status column is written to be checkable rather than encouraging: where a mechanism is identified but not proved, it says so in those words.}
\label{tab:promotion-roadmap-p5}\\
\toprule
\textbf{No.} & \textbf{Named item} & \textbf{What discharge closes} & \textbf{Venue} & \textbf{Status} \\
\midrule
\endfirsthead
\multicolumn{5}{c}{\itshape Table~\ref{tab:promotion-roadmap-p5} continued}\\
\toprule
\textbf{No.} & \textbf{Named item} & \textbf{What discharge closes} & \textbf{Venue} & \textbf{Status} \\
\midrule
\endhead
\midrule \multicolumn{5}{r}{\itshape continued on next page}\\
\endfoot
\bottomrule
\endlastfoot
1 & C-P5-C: finite-temperature on-resonance transport (\S\ref{sec:resonance-faced}) & $\etaB$ from D$^{*}$ to Derived; the $e^{-K/2}$ exponent from postulate to theorem & open & the paper's sharpest open item \\
2 & C-P5-A: the Gate-Operator Lemma (L11) & The parity chain's uniqueness clause; Part~VI from conditional to derived & \ANine{} / this paper & open \\
3 & The assignment lemma (leptonic circuit) & $\dPMNS$ from derived-conditional to derived & \ANine{} & open \\
4 & D-P3-4: the positron as a theorem & The conjugate-readout picture from reading to corollary & \PaperPThree{} & registered debt \\
5 & L-Spin-1: the frame-path spin registers, proved & The spin-$\tfrac12$ candidate suite toward theorem grade & \ATwo{} & candidate \\
6 & T-Stat-A at quasi-free grade & Spin--statistics as the quartet's second child, at boundary grade & \AEight{} / this paper & candidate \\
7 & The interacting handshake & T-P5-C's scope concession (free/quasi-free) lifted to interacting theories & open & open \\
8 & The field-grade representation step & The Born chain's last analytic residue (context: \AEight{}) & \AEight{} & open \\
9 & The $\phi_{\mathrm{geo}}$ projection, restated: the displayed unflavored invariant is $U_{\mathrm{PMNS}}$-independent and vanishes for the dial-free (real-$R$) door; the inherited phases reach $\etaB$ only through the flavored asymmetries $\epsilon_{i\alpha}$, where the driver is the Majorana parity $\alpha_{31} = \pi$ (not $\JPMNS$) and the result is not $\propto m_1$. Open task: finite-temperature flavored transport at the lock crossing testing H-$\phi_{\mathrm{geo}}$-1 ($\Delta M/\Gamma^{\mathrm{eff}} \simeq 3\times10^{4}$ for throughput $e^{-K/2}$); acceptance band $\pm O(1)$ in $\epsilon_{\mathrm{eff}}$; the window deficit the closure must supply is now measured at $E = 7.1$ (68\% $[4.4,\,11.4]$) by the parent's flavored Boltzmann integration. The hypothesis is now carried by an identified candidate mechanism --- the flavor-measurement regulator, closed form $1/(35\,c_\tau y_\tau^{2})$ landing the hypothesized decade (home \ATen{} App.~I.3; conditions C1--C6) --- and the row records the mechanism at candidate grade. The former ``$m_1/m_3$-order suppression'' language is withdrawn as a reconstruction artifact. & Chain row 5 ($\epsilon^{\mathrm{res}}$) from asserted to derived-conditional (H-$\phi_{\mathrm{geo}}$-1); full discharge to derived & \ATen{} / this paper & open --- flavored computation displayed; mechanism identified at candidate grade (C1--C6); H-$\phi_{\mathrm{geo}}$-1 remains \\
\end{longtable}

\textbf{Closure-economy summary.} The confrontation table of
Part~XI runs on two named conjectures (C-P5-A, C-P5-C) and one
assignment lemma; discharging items 1--3 above would promote
every conditional row of the table to Derived with no dial
turned. Items 4--6 are the candidate suite's ladder: each is a
precise statement whose proof belongs to a named companion.
Items 7--8 are the program's long-range residues,
carried honestly. The item closest to this paper is No.~1,
whose acceptance criteria are stated in
\S\ref{sec:resonance-faced}: a finite-temperature transport
computation landing the crossing at the partial-resonance point of
H-$\phi_{\mathrm{geo}}$-1 ($\Delta M/\Gamma^{\mathrm{eff}}
\simeq 3\times10^{4}$; now carried by the identified candidate
mechanism of \S\ref{sec:resonance-faced}), so that the flavored
throughput lands within $O(1)$ of $e^{-K/2}$ --- both clauses of C-P5-C discharged at
once, and T-P5-5 promoted from D$^{*}$.

\section{Falsification-test catalog}
\label{app:falsification-catalogue}

This appendix collects, in one place, every test whose failure
would kill the framework or one of its named structures,
pairing each with the precision required and the experimental
venue. The same tests are laid on a single forward time axis in
the kill-test timeline
(Figure~\ref{fig:p5-killtest-timeline}), and the falsifier
table (Table~\ref{tab:falsifiers}) states what each failure
destroys.

\begin{table}[H]
\centering
\captionsetup{list=no}%% v15.0: see the front-matter note
\caption[Falsification-test catalog.]{Falsification-test catalog for the present paper. Each test is paired with the named structure it would refute and the venue that can carry it out. Ten tests, each with the precision it requires, the named structure it would refute, and the experiment that can carry it out. The precision column is what makes the catalog usable rather than decorative: a measurement that cannot reach the stated precision does not adjudicate its row, and several of these cannot before the 2030s.}
\label{tab:falsification-catalogue-p5}
\renewcommand{\arraystretch}{1.18}
\small
\begin{tabular}{p{0.22\textwidth}p{0.17\textwidth}p{0.30\textwidth}p{0.19\textwidth}}
\toprule
\textbf{Test} & \textbf{Precision} & \textbf{Falsification target} & \textbf{Venue} \\
\midrule
Leptonic phase & $\pm15^{\circ}$ early--mid 2030s; $\pm7$--$10^{\circ}$ late 2030s & $\dPMNS$ outside the $196^{\circ}$ band kills the leptonic lattice chain and the door's bias (T-P5-3 lepton circuit) & DUNE, Hyper-K \\
Mass ordering & decisive, late 2020s--2030s & An inverted ordering kills the derived triplet (Part~VIII mass chain) & JUNO; DUNE \\
Quark phase & continuous world fit & Exclusion of $13\pi/35$ kills the QD completion; of $2\pi/5$-class values, the pentagon itself & LHCb, Belle~II, global fits \\
$0\nu\beta\beta$ & ton scale, 2030s & A confirmed line in the ton-scale window kills the door architecture; $\mbb = 1.72$~meV is a registered null & LEGEND-1000, nEXO, CUPID \\
Lock cluster & few-TeV reach & Failure of the pre-declared HL-LHC criterion kills the heavy sector of the chain & HL-LHC \\
Right-handed currents & any scale, permanent & $W_R$ or mirror fermions kill the geometric projection of Part~VI (F-P5-1) & all colliders \\
Proton stability & permanent & An observed decay kills the exact $\Uone{B}$ ledger and Part~VIII's logic & Hyper-K, DUNE \\
EDM window & $10^{-18}$--$10^{-10}$ in $\bar\theta$ & A measured EDM above the $e^{-K}$ leakage floor but below $10^{-10}$ breaks the two-fate architecture & n/p and $e$ EDM programs \\
Mass splittings & sub-percent on both splittings & A converged JUNO value off the derived pair ($\Delta m^2_{21} = 7.37\times10^{-5}$, $\Delta m^2_{31} = 2.55\times10^{-3}$~eV$^2$) at $4\sigma$-class kills the mass chain & JUNO \\
\bottomrule
\end{tabular}
\end{table}

Three further rows are \emph{watched}, not kills: the
$\sin 2\beta$ confrontation ($0.708$ at the stated moduli
layer against $0.709 \pm 0.011$,
$-0.1\sigma$, Belle~II statistics decisive), the direct-%
$\gamma$ combination (the lattice triangle's leading-order
$\simeq 66.9^{\circ}$ against $(62.8\pm2.6)^{\circ}$,
$+1.6\sigma$), and the fourth-%
generation exclusion that protects the $N = 3$ counting. One
binding registration accompanies the catalog: the
subdominant $3\pi/2$ assignment branch is hereby
pre-registered as dead --- a measurement near
$270$--$286^{\circ}$ may \emph{not} be reinterpreted under
the alternative branch; the assignment lemma is frozen in the
form stated in Part~VII, and its failure is the chain's
failure. The
framework's exposure profile is deliberately high: every row
above sits within the current decade's experimental horizon or
is permanent, and the AND-logic of \S\ref{sec:falsification}
wires the kill channels in parallel --- any one suffices.

\medskip\noindent With that, the registries are complete. What
remains of the Part is the reading guide; after it, the
adversaries.

\section{An annotated reading guide}
\label{sec:reading-guide}

Six shelves, each ordered. \emph{Shelf 1 (the discoveries):}
Lee--Yang 1956; Wu 1957; Christenson--Cronin--Fitch--Turlay 1964;
KTeV 1999; BaBar/Belle 2001; Super-K 1998 --- read in
sequence as one eighty-year experiment. \emph{Shelf 2 (the
theorems):} Wigner 1931; L\"uders/Pauli; Jost 1957;
Streater--Wightman; Bisognano--Wichmann 1976; Greenberg 2002;
Vafa--Witten 1984. \emph{Shelf 3 (flavor):} Cabibbo 1963;
Kobayashi--Maskawa 1973; Jarlskog 1985; the PDG reviews.
\emph{Shelf 4 (the matter excess):} Sakharov 1967;
Kuzmin--Rubakov--Shaposhnikov 1985; Fukugita--Yanagida 1986;
Davidson--Ibarra 2002; Pilaftsis--Underwood 2004;
Buchm\"uller--Di~Bari--Pl\"umacher. \emph{Shelf 5 (the
geometric rivals and parents):} Kawamura 2000; Hall--Nomura 2001;
Dine--Leigh--MacIntire 1992; Choi--Kaplan--Nelson 1993; and the
cohort papers in the dependence order of
Section~\ref{sec:inheritance}. \emph{Shelf 6 (the antiparticle
lineage):} Dirac 1928, 1930, 1931; Anderson 1933;
St\"uckelberg 1941; Feynman 1949; Weinberg's \emph{Quantum
Theory of Fields} chapter~5; then the Foundations-Branch
treatment (\citealp{ArisakaP4Real}) and Part~\ref{part:antiparticles}
of this paper --- read in that order, the shelf is the
eighty-year journey from the sea to the handshake.

\medskip
\noindent\emph{Part-end summary.} A machine room from Wu's
cryostat to the assembled $\etaB$: the toolkit is the reader's.
What remains are the adversaries and the sources.

%% v15.1: THE ENLARGEMENT ABOVE WAS THE DEFECT.  It closed the
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\addtocontents{toc}{\protect\GGtocbreak}
\GGPart[Common objections]{Common Objections and Responses}{Eighty-two adversarial readings --- from the kaon experimentalist, the neutrino experimentalist, the flavor theorist, the axiomatic field theorist, the lattice theorist, the string theorist, the cosmologist, the historian and philosopher, the student, the antiparticle referees, and the quantum-foundations referee --- each answered with pointers into the paper}
\label{part:objections}

A paper that claims to explain the broken mirrors invites --- and
should invite --- attack from every community that measured them.
This Part collects the eighty-two sharpest adversarial readings we could
construct or have received, grouped by the reader most likely to
raise each, and answers each with explicit pointers to the theorem,
section, or named open item that carries the load. Where the honest
answer is ``not yet'' or ``not ours to claim,'' it is said in those
words. These are self-authored stress tests in referee personas,
published so that real referees can start where they end --- and
they now include the findings of an adversarial multi-reviewer
audit of this version's numerics, which produced the corrections
logged in the version banner.

\newcommand{\PObj}[3]{\bigskip\noindent\textbf{Objection #1
(O-P5-#1) (#2)}\par\nobreak\noindent\textbf{``#3''}\par\nobreak}

\section{The kaon and \texorpdfstring{$B$}{B}-physics
experimentalist}

\PObj{1}{the KTeV alumnus, on hadronic mud}{Direct CP violation
took us a decade because $\varepsilon'/\varepsilon$ is buried in
hadronic matrix elements. Your pentagon supplies one clean phase
and waves at the lattice for the rest --- that is not a prediction
of $\varepsilon'/\varepsilon$.}
\begin{objqabody}
\emph{Response.} Correct, and stated in those words in
Appendix~\ref{app:flavour-kaons}: the lattice phase fixes the weak phase input;
$\varepsilon'/\varepsilon$ then requires the same hadronic matrix
elements every framework requires, honestly delegated to the
lattice literature. The claim is not a parameter-free
$\varepsilon'/\varepsilon$; it is that the \emph{one} weak phase
feeding every kaon and $B$ observable is a forced lattice point.
The clean confrontations are the phase-dominated observables ---
$\dCKM$ in the global fit, $\sin 2\beta$, $\JCKM$ --- and there
the numbers are in the table.
\end{objqabody}

\PObj{2}{the same alumnus, on the QD step}{Your $2\pi/5$ misses the
fit by $1.2\sigma$ and your $13\pi/35$ lands at $0.7$ --- after a
correction your own codebook calls post-hoc. That is
retrofitting.}
\begin{objqabody}
\emph{Response.} The flag is ours and stays (\S\ref{ssec:ckm}).
The defensible structure: $2\pi/5$ is forced, with no freedom; the
QD slip's \emph{direction and size} ($-\pi/K$, one throat step)
are ledger-determined; what is asserted is that the slip applies.
A bounded, quantized, pre-registered correction differs from a
tuned one in exactly one way --- it cannot be adjusted again ---
and that is testable: the completed value must survive all future
tightening of the fit or the step dies publicly.
\end{objqabody}

\PObj{3}{the $B$-factory veteran, on many observables}{Flavor
physics measures dozens of CP asymmetries. One phase cannot carry
them all.}
\begin{objqabody}
\emph{Response.} One phase \emph{does} carry them all --- that is
the content of the Standard Model's own global fit, confirmed at
the $B$ factories \citep{BaBar2001, Belle2001}. The paper adds no
second phase; it values the one there is. Any observable requiring
a second independent CP source in the quark sector falsifies the
Standard Model and this paper together.
\end{objqabody}

\PObj{4}{the kaon theorist, on $\varepsilon$}{The measured
$\varepsilon \sim 2\times10^{-3}$ involves mixing, long-distance
effects, and $\Delta m_K$. Where is it in your paper?}
\begin{objqabody}
\emph{Response.} Appendix~\ref{app:flavour-kaons}, at the honesty level the observable
permits: the weak phase enters the box with the lattice value, the
long-distance parts are the literature's, and no parameter-free
$\varepsilon$ is claimed. The paper's kaon-sector content is the
phase, not the hadronic envelope.
\end{objqabody}

\PObj{5}{the trigger veteran, on falsifiability rhetoric}{Every
theory paper now advertises falsifiers. Which of yours has a date
and a number this decade?}
\begin{objqabody}
\emph{Response.} Table~\ref{tab:falsifiers}: $\dPMNS$ to
$\pm15^{\circ}$ at DUNE/Hyper-K (early--mid 2030s) against
$196^{\circ}$; the $3.85$~TeV cluster at HL-LHC with a
pre-declared criterion; $\mbb = 1.7$~meV against the
next-generation $0\nu\beta\beta$ program; and F-P5-1 standing
against every $W_R$ search. Four numbers, four instruments, all
funded.
\end{objqabody}

\PObj{6}{the charm convener, on $\Delta A_{CP}$}{LHCb found CP violation in charm at the $10^{-3}$ level ---
up-type, and larger than many expected. Where is charm in your
pentagon?}
\begin{objqabody}
\emph{Response.} In the same place it is in the Standard Model: the one valued
phase, routed through up-type loops where the hadronic envelope
is least controlled \citep{LHCbCharm2019}. The framework adds
no second phase and buys no new amplitude, so
$\Delta A_{CP}$ tests QCD dynamics at the charm scale, not the
lattice point --- and the honest statement, printed rather than
implied, is that no framework currently prices it from first
principles. What would implicate this paper is a charm
measurement requiring a phase off the valued one at fixed
moduli; nothing observed does.
\end{objqabody}

\PObj{7}{the rare-kaon experimentalist, on $K \to \pi\nu\bar\nu$}{NA62 and KOTO chase the cleanest loops in flavor physics.
Predict the rates or admit you have nothing to say.}
\begin{objqabody}
\emph{Response.} The ledger fixes the phase and the moduli, so the golden-loop
rates are the Standard-Model values evaluated at the lattice
point --- indistinguishable from the SM's own central values at
current precision. The channel is therefore a consistency check
rather than a discriminator until the phase resolutions cross
(Table~\ref{tab:exp-timeline}); but it is not toothless: a
matured rate demanding different CKM inputs at fixed unitarity
would break the ledger upstream, and the register would take
the hit with dates.
\end{objqabody}

\PObj{8}{the lattice-QCD watcher, on $\varepsilon'/\varepsilon$}{The lattice value of $\varepsilon'/\varepsilon$ has moved
between tension and agreement for a decade. If it lands off the
world value, whose problem is that?}
\begin{objqabody}
\emph{Response.} At fixed weak phase, $\varepsilon'/\varepsilon$ is a
hadronic-matrix-element problem, and the framework's custody
rule delegates it in print (\S\ref{app:flavour-kaons};
Objection~31): a lattice-versus-experiment tension is QCD's to
resolve. What would be ours is any resolution that re-extracts
the weak phase off the lattice point --- and that exposure
already sits as a falsifier row. Custody stated is
accountability preserved, in both directions.
\end{objqabody}

\section{The neutrino experimentalist}

\PObj{9}{the long-baseline convener, on the moving fit}{NuFit
moves. Your $196^{\circ}$ sits near today's central value; when
the band shifts, will the paper shift with it?}
\begin{objqabody}
\emph{Response.} No --- and that is the point of publishing the
number with its derivation \eqref{eq:deltapmns}. $\pi +
2\arcsin(1/7)$ has no adjustable content: if the measured phase
settles outside the band around $196^{\circ}$ (the QD width is
$\pm$ a fraction of a degree, the honest comparison band is set
by the assignment lemma's degeneracy, stated in
\S\ref{ssec:pmns}), the leptonic chain is dead and says so. The
prediction predates the decisive data; that is what makes it one.
\end{objqabody}

\PObj{10}{the $0\nu\beta\beta$ experimentalist, on 1.7 meV}{Your
$\mbb$ is an order below the inverted-hierarchy band and below
any funded experiment's reach. Convenient.}
\begin{objqabody}
\emph{Response.} Inconvenient, in fact, and stated without
cosmetic: the value follows from the derived normal-ordering
masses and the parity pair $(0,\pi)$, and it places the door
beyond current reach. The near-term discriminators are therefore
indirect (a measured inverted ordering kills the mass chain; an
observed $0\nu\beta\beta$ signal \emph{above} the meV class
contradicts the parity pair), and the direct test awaits the
tonne-scale-plus era. Honesty about which decade owns which test
is part of the register.
\end{objqabody}

\PObj{11}{the atmospheric-sector veteran, on Majorana phases}{Your
$(0,\pi)$ is untestable in oscillations by construction.}
\begin{objqabody}
\emph{Response.} Correct --- oscillations are blind to Majorana
phases. The pair is tested where it acts: in $\mbb$ (the $\pi$
drives the cancellation to the meV class) and in the leptogenesis
bias (Part~VIII). ``Untestable in your instrument'' and
``untestable'' are different statements.
\end{objqabody}

\PObj{12}{the mass-ordering referee}{Everything leptonic here
assumes normal ordering. JUNO and the atmospheric fits could
return inverted.}
\begin{objqabody}
\emph{Response.} Yes --- and inverted ordering falsifies the
inherited mass chain outright, upstream of this paper. It is
listed among the backbone kills (\S\ref{sec:matter-falsifiers}).
\end{objqabody}

\PObj{13}{the cosmological fitter, on the mass sum}{DESI with the CMB presses $\sum m_\nu < 0.064$~eV. Your
triplet sums to $0.0592$~eV. One more tightening and you are
dead?}
\begin{objqabody}
\emph{Response.} Yes --- and the exposure is registered here rather than
buried: the derived sum is $59.2$~meV, eight percent below the
current bound \citep{DESI2025}. A robust cosmological
determination converging \emph{below} $59$~meV kills the mass
chain independently of every laboratory; one settling at
$\sim 60$~meV would be a confirmation channel no laboratory
can match this decade. The refresh discipline of
\S\ref{sec:exp-refresh} will print the pull at each release
--- cosmology now owns one of this paper's sharpest rows.
\end{objqabody}

\PObj{14}{the short-baseline anomalist, on steriles}{LSND, MiniBooNE, and the gallium deficit keep flickering. Is
there room in your counting for a light sterile?}
\begin{objqabody}
\emph{Response.} None --- and that is a kill channel, stated as such: T-P5-4's
uniqueness clause (Thm.~\ref{thm:p5-4}) forbids a second
weight-zero completion, so a confirmed eV-scale sterile with
oscillation-strength mixing would break the door architecture
itself, not merely the family count. Against that stands the
record: the anomalies do not converge on one oscillation
hypothesis, and MicroBooNE's nulls cut into the favored
regions. The register carries the flicker with dates and
waits; the exposure does not.
\end{objqabody}

\PObj{15}{the supernova watcher}{A galactic supernova would put $10^{58}$ neutrinos through
your MSW machinery in ten seconds. What do you predict?}
\begin{objqabody}
\emph{Response.} Standard flavor conversion evaluated at the derived masses
and ordering: normal ordering fixes the level-crossing pattern,
so the neutronization burst's $\nu_e$ suppression and the
spectral swaps are those of adiabatic NO conversion
(\S\ref{app:nu-appearance} carries the machinery). A
supernova that reads IO from the burst kills the ordering
upstream of every long-baseline program --- a free, if
patient, kill channel. Collective oscillations in the deep
interior are delegated to the transport literature,
honestly.
\end{objqabody}

\section{The flavor theorist}

\PObj{16}{the rephasing purist}{Your T-P5-3(i)--(iii) is Jarlskog
1985 with graph words.}
\begin{objqabody}
\emph{Response.} Agreed, cited, and drawn as a boundary in
\S\ref{sec:jarlskog} before the theorem is stated. The paper's
claim begins at clause (iv): the standard calculus says where CP
violation lives; nothing in it values the holonomy. The lattice
does, and the lattice is the paper.
\end{objqabody}

\PObj{17}{the discrete-flavor model builder}{$A_4$, $S_4$, and
modular flavor symmetries also predict $\delta$'s. You are one
more entry in a crowded catalog.}
\begin{objqabody}
\emph{Response.} The catalog's entries share one move: choose a
group, then derive. Here no flavor group is chosen --- the
pentagon is $L = 5$, an integer fixed by anomaly arithmetic in a
different paper for different reasons \citep{ArisakaA3BI6}, and
the same integer prices the Cabibbo magnitude
\eqref{eq:cabibbo} before touching CP. An entry that buys two
observables with a pre-owned integer is in a different section of
the catalog --- and is killed, not tuned, if either number
fails.
\end{objqabody}

\PObj{18}{the texture skeptic, on first order}{``First-order
holonomies are lattice-valued'' --- and beyond first order?}
\begin{objqabody}
\emph{Response.} The QD completion is the program's second
order, applied here at exactly two points ($-\pi/K$ on the quark
phase; the sub-degree shift on the leptonic one), both quantized,
neither continuous. A demonstration that higher orders
decompactify the lattice --- float the angles continuously ---
would falsify clause (iv); the parent codebook's structure (all
corrections are ledger steps) says they do not.
\end{objqabody}

\PObj{19}{the CKM global fitter}{Your Cabibbo formula uses
$e^{-\pi}$ and $e^{-\pi/2}$ as mass-ratio angles --- numerology
until derived.}
\begin{objqabody}
\emph{Response.} Derived in the parent Standard-Model treatment as
part of its Yukawa codebook \citep{ArisakaA9SM}, with its own
audit trail; this paper imports the two angles as stated inputs
(\S\ref{ssec:ckm}) and adds only the pentagon's interference.
The import is flagged so the reader can price it; the golden-ratio
closed form is then forced, not fitted.
\end{objqabody}

\PObj{20}{the strong-CP specialist}{Your $\bar\theta$ story needs
L9 (uncredited cancellation) and a Kaxion. Peccei--Quinn needs one
axion. Occam prefers us.}
\begin{objqabody}
\emph{Response.} Count again, structurally: PQ needs a new global
symmetry, a new scalar, and protection of that symmetry from
quantum gravity --- three purchases. The Kaxion is a Wilson line
the geometry already owns \citep{ArisakaC1Kaxion}, and L9 is a
cancellation within structure already present. And only the
two-fate architecture (\S\ref{sec:strongcp}) \emph{relates}
$\bar\theta \simeq 0$ to $\dCKM \neq 0$ --- a relation PQ cannot
express at all. Occam's razor cuts by total inventory, not by
lemma count.
\end{objqabody}

\PObj{21}{the modular-symmetry builder}{Modular flavor symmetry also derives phases from geometry ---
$\tau$ at a fixed point. What is different about a pentagon?}
\begin{objqabody}
\emph{Response.} Kinship first: both programs hold that phases are geometric
data, not couplings. The difference is who owns the dial. A
modular model selects a group, a weight assignment, and a
modulus value --- three choices per sector, scanned until the
fit lands. The pentagon inherits one angle from the ledger with
no modulus to place; the cost is total rigidity (zero-width
rows in Table~\ref{tab:exp-predicted-measured}), the benefit
is that failure is failure. The comparison is run without
polemic in Part~\ref{part:fiveroads}'s register: structural
contrast, adjudicated by instruments.
\end{objqabody}

\PObj{22}{the first-row auditor, on the Cabibbo anomaly}{First-row CKM unitarity shows a $2$--$3\sigma$ deficit. Your
golden form fixes $\sin\theta_C$ to five digits. Exposed?}
\begin{objqabody}
\emph{Response.} Watched, with dates --- and the anatomy matters: the present
deficit implicates $V_{ud}$ metrology and radiative
corrections at least as much as $V_{us}$, so it does not yet
press the golden value, which sits at
$\mathrm{Dev} = +4\times10^{-5}$ against the $V_{us}$-side
extraction. A matured, radiative-settled deficit forcing
$\sin\theta_C$ off the closed form by $3\sigma$-class would
kill the modulus claim --- a sharper exposure than most
frameworks volunteer for a quantity this well measured
(\S\ref{sec:exp-refresh}).
\end{objqabody}

\PObj{23}{the Froggatt--Nielsen practitioner, on hierarchy}{Your flavor chapter calls the Yukawa span `what the
Lagrangian was told.' Charge assignments explain it. You
don't.}
\begin{objqabody}
\emph{Response.} Correct, and printed as a Step-2 limitation rather than
disguised (\S\ref{app:flavour-limits}): the pentagon prices
the phases and the Cabibbo modulus, not the eigenvalue span.
Froggatt--Nielsen buys the span with a charge per fermion ---
bookkeeping that organizes but does not force. Where the parent
program derives masses it says so with grades; where it does
not, it declines to buy what it cannot force. An honest gap
beats a purchased explanation in this register --- and the gap
is on the promotion roadmap's horizon, not hidden.
\end{objqabody}

\section{The axiomatic field theorist}

\PObj{24}{the Wightman referee, on scope}{Jost's theorem covers
interacting fields. Your T-P5-2 is free-field bookkeeping
advertised as a CPT theorem.}
\begin{objqabody}
\emph{Response.} Scope conceded in the theorem's own statement
and in \S\ref{sec:cpt}: quasi-free grade, interacting case
conditional. The claim is complementarity --- Jost proves
$\Theta$ exists with interacting strength and no anatomy; T-P5-2
supplies the anatomy at lesser strength. Whether anatomy at
lesser strength is worth having is answered by what it buys:
T-P5-1's derived antiunitarity, the chirality criterion, and
every downstream Part.
\end{objqabody}

\PObj{25}{the algebraist, on Bisognano--Wichmann}{We already have
CPT's deep origin: modular conjugation. Your clock is a poor man's
modular flow.}
\begin{objqabody}
\emph{Response.} Possibly a poor man's --- or possibly the same
man's, seen from the real side; the open modular route
(\S\ref{sec:open-questions}(6)) is precisely the conjecture that
your modular flow and our clock coincide, which would tie the
anatomy to the algebra at interacting strength. Note the
direction of dependence, though: Bisognano--Wichmann operates
\emph{inside} a complex Wightman theory; it does not tell you
what $i$ is. The clock does, and that difference is not
poverty.
\end{objqabody}

\PObj{26}{the CPT phenomenologist}{Kosteleck\'y's SME parameterizes
CPT violation and experiments constrain it. Your framework
forbids it outright --- so your framework is untestable on this
axis.}
\begin{objqabody}
\emph{Response.} Inverted: a framework that \emph{forbids} CPT
violation is maximally exposed on this axis --- any confirmed SME
coefficient at any order kills T-P5-2 with no parameter to
absorb it, exactly as Greenberg's theorem \citep{Greenberg2002}
ties such a signal to Lorentz violation and hence to the arena
itself. Prohibition is the strongest testable statement there
is.
\end{objqabody}

\PObj{27}{the rigorist, on ``real texture''}{``Every texture built
from real geometric data'' is doing heavy lifting in T-P5-2.
Define it or lose the theorem.}
\begin{objqabody}
\emph{Response.} Defined in the proof: magnitudes (moduli, metric
and symplectic contractions, winding sums) even under total
reversal, plus fiber angles that negate --- i.e. any coupling
functional of the constitutional real structures with no external
complex input. The definition excludes exactly one thing: a
primitively complex coupling constant --- which is the
program's entire ontological claim, argued in the
Foundations-Branch partner \citep{ArisakaP4Real}, not smuggled here.
\end{objqabody}

\PObj{28}{the Greenberg reader}{Greenberg proved CPT violation implies Lorentz violation. You
derive CPT from texture invisibility. Do the theorems even
compose?}
\begin{objqabody}
\emph{Response.} They stack, usefully. Greenberg's theorem
\citep{Greenberg2002} runs inside Wightman QFT: no Lorentz-%
invariant theory violates CPT. T-P5-2 runs one level down, in
the real geometry, and delivers the invariance the axioms then
protect. The composition worth having is the contrapositive:
any texture-level CPT anomaly would surface at the boundary as
Lorentz violation, so the entire SME bound library disciplines
the texture too --- the experimental record of
Table~\ref{tab:cpt-tests} constrains the geometry twice
over.
\end{objqabody}

\PObj{29}{the KMS reader, on the thermal clock}{Corollary B1-b says imaginary time belongs to the clock. Is
your $\beta$ a genuine KMS parameter or an analogy?}
\begin{objqabody}
\emph{Response.} Genuine at the quartet's grade: T-P5-B1
(Thm.~\ref{thm:p5-B1}) is the analytic fact, and the KMS
condition at the boundary is inherited through the standard
quasi-free construction --- the thermal reading is the
dictionary's, the continuation is the geometry's
(Cor.~\ref{cor:B1b}). The deeper bridge --- whether the
modular flow of the algebraists is this clock seen abstractly
--- is the named open question of the Bisognano--Wichmann
exchange (Objection~69), carried as open rather than
claimed.
\end{objqabody}

\section{The lattice and QCD theorist}

\PObj{30}{the lattice practitioner, on Vafa--Witten}{QCD's parity
safety is our theorem. Your ``unbroken $\Rightarrow$ parity-even''
adds words to it.}
\begin{objqabody}
\emph{Response.} Vafa--Witten \citep{VafaWitten1984} protects
parity \emph{within} a vector-like theory; it is cited at every
use. The basin/gate clause answers a different question --- why
the strong force \emph{is} vector-like while $\SUtwoL$ is not ---
which no statement internal to QCD can address. The two results
nest; neither substitutes for the other.
\end{objqabody}

\PObj{31}{the same practitioner, on $\varepsilon'/\varepsilon$
again}{Until you put your phase through our matrix elements and
publish the number, Appendix~\ref{app:flavour} is prose.}
\begin{objqabody}
\emph{Response.} Fair. The computation --- lattice phase in,
published hadronic matrix elements in, $\varepsilon'/\varepsilon$
out with the community's error bars --- is a well-posed exercise
and is hereby added to the program's verification queue; the
paper's claims do not rest on it, but the confrontation should
exist.
\end{objqabody}

\PObj{32}{the chiral-lattice veteran, on alternatives}{The massless-up escape from strong CP is dead by lattice
arithmetic. Nelson--Barr survives by construction. Why prefer
two fates?}
\begin{objqabody}
\emph{Response.} The lattice's execution of $m_u \neq 0$ is inherited here as
data, gratefully. Nelson--Barr is a consistent mechanism:
$\bar\theta = 0$ at tree level from engineered CP-breaking
sectors, protected by loop suppression --- at the price of the
sectors themselves, which are dials. The two-fate architecture
ties the smallness instead to the same $e^{-K}$ that sets the
lock scale and the matter throughput, and exposes the tie in
the EDM window of Table~\ref{tab:falsifiers}: one integer,
three exports, one kill band. Structural contrast, chosen by
experiment.
\end{objqabody}

\PObj{33}{the large-$N$ theorist, on $\theta$ periodicity}{Vacuum energy is periodic in $\theta$ with branch structure.
Does a leakage floor at $5\times10^{-18}$ respect the known
$\theta$ physics?}
\begin{objqabody}
\emph{Response.} Yes, because the floor is a statement about where relaxation
parks $\bar\theta$, not a modification of $E(\theta)$: the
Kaxion rolls on the standard QCD potential with its standard
periodicity and branch structure
(\citealp{ArisakaC1Kaxion}), and the two-fate architecture
adds only the claim that the residual displacement is
$e^{-K}$-suppressed rather than zero. That makes the floor
measurable --- hence mortal, in the EDM window --- which is
precisely what separates it from solutions that park at zero
by construction and can never be caught doing it.
\end{objqabody}

\section{The string theorist}

\PObj{34}{the string phenomenologist, on CP}{CP as a geometric,
spontaneously broken symmetry is ours ---
Dine--Leigh--MacIntire, Choi--Kaplan--Nelson. You have
rediscovered our conclusion.}
\begin{objqabody}
\emph{Response.} Your conclusion is cited as the concept's source
(\S\ref{sec:five-roads}) and the kinship is real: both programs
make CP violation geometric. The delta is valuation: moduli are
continuous and landscape-valued, so your $\delta$ is
environmental; the pentagon is discrete and forced, so ours is a
prediction with a date. ``Geometric'' was your step;
``valued'' is the claim you cannot make and we must defend.
\end{objqabody}

\PObj{35}{the orbifold builder}{Part~VI's projection is Kawamura's
orbifold GUT, to the matrix.}
\begin{objqabody}
\emph{Response.} To the matrix, and credited to the matrix
(\S\ref{sec:projection}): mechanism inherited, selection claimed.
In your landscape the projection is one choice among consistent
many; the program's claim --- graded, with its open item ---
is that the ledger forces the output. The difference between a
mechanism and an explanation is exactly the difference between
those two sentences.
\end{objqabody}

\PObj{36}{the string cosmologist, on moduli}{Any TeV-scale
asymmetry you make, our moduli would dilute. What protects
yours?}
\begin{objqabody}
\emph{Response.} The absence of your moduli. The descent's lock
crossing is prompt ($N_2 = 0$, three grounds,
\S\ref{sec:zero-dilution}), there is no late-dominating scalar,
and zero dilution is a derived clause, not a hope. The
moduli-dilution problem is an argument \emph{for} frameworks
with rigid geometry, and it is made in your literature's own
terms \citep{CoughlanEtAl1983, BanksKaplanNelson1994}.
\end{objqabody}

\PObj{37}{the anomaly-inflow veteran}{Green--Schwarz inflow is
ours too. Your POS-3 is our mechanism wearing an axiom's
clothes.}
\begin{objqabody}
\emph{Response.} Inflow is Green--Schwarz's and cited so
throughout the cohort \citep{ArisakaA3BI6}. Promoting a mechanism
to an admissibility axiom is not decoration: it converts ``one
consistent choice'' into ``the filter every spectrum must
pass,'' which is what gives the chirality assignments of Part~VI
their policing. Same tool; different office.
\end{objqabody}

\PObj{38}{the swampland cartographer}{Quantum gravity abhors global symmetries; your exactly stable
proton flaunts one. Welcome to the swampland.}
\begin{objqabody}
\emph{Response.} A head-on exposure, printed: the boundary $\Uone{B}$ is not
an assumed global symmetry of an effective theory coupled to
gravity but a ledger identity of the anomaly books
(\citealp{ArisakaA3BI6}) on an arena whose gravity sector is
itself the boundary law. Whether swampland reasoning ---
derived from holographic and black-hole arguments inside a
different completion --- applies to that arena is exactly the
class of question the five-way comparison isolates without
polemic (Part~\ref{part:fiveroads}). Operationally the
dispute is already scheduled: the folklore says the proton
dies at some scale; the ledger says never; Hyper-K listens
(Table~\ref{tab:falsifiers}).
\end{objqabody}

\PObj{39}{the worldsheet theorist, on string CPT}{CPT holds on the worldsheet and in target space, and you
concede strings reach Depth 2. What is left for you to
claim?}
\begin{objqabody}
\emph{Response.} Anatomy --- Depth 3: what C, P, and T are individually made
of, why T is the antiunitary one, and why the factors break
with these particular values. String CPT inherits the axioms'
silence on valuation: no vacuum selection prices
$\dCKM$, and the landscape's breadth is the concession its
own practitioners state (Objection~34). The reverse concession
is printed in the same table: strings' interacting reach
against the handshake's quasi-free grade
(\S\ref{sec:anti-ledger}). Two programs, two summits, one
honest ledger between them.
\end{objqabody}

\section{The cosmologist}

\PObj{40}{the transport theorist, on the resonance}{Your
splittings are $110$~GeV; your width is $10^{-10}$~GeV; your
enhancement formula needs them equal. Twelve orders, and your
`derivation' of $\etaB$ evaporates off-peak.}
\begin{objqabody}
\emph{Response.} Quantified in \S\ref{sec:resonance-faced} before
you raised it, with the suppression law and the candidate
resolutions --- thermal regulators (the flavor-measurement rate
now the computed candidate; the flavor-blind $g^2 T$ overshoots),
radiative degeneracy, or lock substructure --- and the concession that ``natural rather than
tuned'' is retired accordingly. T-P5-5 is graded
D$^{*}$ \emph{because} of this gap; C-P5-C(i) is its named
carrier; and a transport calculation showing no consistent
regime is listed as a clean kill. You have identified the
paper's sharpest open item; the paper agrees, in its own
section. And the flavored recomputation states the target more
sharply than the peak you aim at: the exact-peak form overshoots
the required throughput by $\sim3\times10^{4}$; the sharp form of
the condition is the partial resonance
$\Delta M/\Gamma^{\mathrm{eff}} \simeq 3\times10^{4}$, named
H-$\phi_{\mathrm{geo}}$-1, with the flavored computation
displayed in the cosmology companion and the hypothesis now
carried by an identified candidate mechanism at candidate grade
(conditions C1--C6 open).
\end{objqabody}

\PObj{41}{the same theorist, on $e^{-K/2}$}{An exponential that
`reconstructs' your ledger value from two order-of-magnitude
factors is numerology with a Boltzmann costume.}
\begin{objqabody}
\emph{Response.} The exponent is a postulate --- said in
\S\ref{sec:chain} step~7 in that word --- and the honest claim
is narrower than your target: an explicit chain with inherited
inputs \emph{lands on} $2.5\times10^{-8}$, and the CP factor in
that product now carries a displayed flavored computation with a
named open item --- H-$\phi_{\mathrm{geo}}$-1, the
finite-temperature partial-resonance condition, now carried by an
identified candidate mechanism --- rather than a
bare assertion. The factor-$\sim3$ band and the ``not a
precision success'' warning are printed beside the number.
Derive $K/2$ (C-P5-C(ii)) and the costume becomes a theorem;
fail to, and the grade stays D$^{*}$ forever. Both outcomes are
on the table, priced.
\end{objqabody}

\PObj{42}{the BBN referee, on $\Omega_b$}{Your own two routes to
the baryon density disagree by $16.4\%$ --- fifty times the
CMB error. Physician, heal thyself.}
\begin{objqabody}
\emph{Response.} Self-reported in \S\ref{sec:omegab-bridge},
with the owner unidentified and the audit named. A framework
that publishes its internal discrepancies before referees find
them is asking to be trusted about the rest; a framework that
resolves this one, in either direction, learns something real.
It is the cohort's problem O-56 and it is not hidden here.
\end{objqabody}

\PObj{43}{the leptogenesis expert, on citation}{Fukugita--Yanagida,
Pilaftsis, Davidson--Ibarra --- your mechanism's parents. Cite
them or be embarrassed.}
\begin{objqabody}
\emph{Response.} Cited at every load-bearing use
(\S\ref{sec:transport}, \S\ref{sec:chain}) and shelved in the
reading guide; the Davidson--Ibarra bound is engaged explicitly
as the reason the resonant regime is needed at TeV scale. This
paper exists partly to pay the citation debts an earlier
internal audit found outstanding in the parent text.
\end{objqabody}

\PObj{44}{the CMB analyst}{$\Lambda$CDM needs no baryogenesis;
$\eta_B$ is an initial condition. Why should I care whose story
decorates it?}
\begin{objqabody}
\emph{Response.} Because initial conditions are where physics
lives or dies: your own concordance (CMB versus deuterium) is a
consistency test of exactly the kind Part~VIII submits to, and a
mechanism with zero dials and four dated kill switches is the
difference between decoration and science. You measured the
target to $0.3\%$; someone must be accountable for hitting it.
\end{objqabody}

\PObj{45}{the electroweak-baryogenesis loyalist}{New scalars
could still make the transition first order. You dismiss our
program in a sentence.}
\begin{objqabody}
\emph{Response.} Not dismissed --- excluded, for this framework:
the \GG{} scalar sector is the projection's debris with no
freedom to add the required potential structure, so the crossover
is inherited and the lepton route is forced
(\S\ref{sec:why-leptogenesis}). For frameworks with adjustable
scalar sectors your program remains live, and its EDM
exposure is its own business.
\end{objqabody}

\PObj{46}{the CPT cosmologist, on Boyle--Turok}{The CPT-symmetric universe makes the Bang a mirror between a
universe/anti-universe pair, gets dark matter from a
right-handed neutrino, and needs no inflation. Kin or rival?}
\begin{objqabody}
\emph{Response.} Kin in reverence for the unbroken theorem; rival in where the
asymmetry lives. Boyle--Turok impose CPT on the \emph{state}
--- a boundary condition at the Bang --- and keep the dynamics
standard \citep{BoyleFinnTurok2018}; this paper derives CPT
invariance of the \emph{dynamics} from texture invisibility
and locates the matter excess in an oriented descent within
one universe. The falsifier sets barely overlap: theirs leans
on vanishing primordial tensors and a $10^{8}$-GeV-class
sterile relic; ours on the phase set, the ton-scale null, and
the TeV cluster. Structural cousins, separable by the same
decade of instruments --- exactly what Part~\ref{part:fiveroads}'s
register is for.
\end{objqabody}

\PObj{47}{the antinuclei hunter, on AMS}{Antihelium candidate events circulate from AMS-02. If a flux
is confirmed?}
\begin{objqabody}
\emph{Response.} Then Part~VIII is in genuine crisis, and the paper says so
now: a confirmed cosmic antihelium-4 flux would resurrect
antimatter domains of the kind the diffuse gamma-ray audit
closed \citep{CohenDeRujulaGlashow1998}, and the global sign
claim of the oriented descent has no domain reservoir to hide
in. Lighter candidates admit the mundane out (secondary
antideuteron-class production); helium-4 does not. The
register waits for a refereed flux and pre-commits to the
verdict --- which is what a register is for.
\end{objqabody}

\PObj{48}{the defect theorist, on domain walls}{Discrete choices made independently in causally disconnected
patches make domain walls that overclose the universe. Your
pentagon is a discrete choice. Where are your walls?}
\begin{objqabody}
\emph{Response.} Nowhere, structurally: the texture is not a boundary field
spontaneously selecting among degenerate vacua patch by patch
--- it is a single bulk structure fixed by the ledger before
the boundary has patches, so causally disconnected regions
inherit one orientation and no wall network ever forms. The
contrast with axionic domain-wall problems is precisely the
bulk/boundary custody; the one discrete relic the framework
does keep --- the Kaxion's misalignment --- is priced in its
own paper (\citealp{ArisakaC1Kaxion}). Walls are a symptom
of choices made locally; the ledger makes none.
\end{objqabody}

\PObj{49}{the mechanism pluralist, on Affleck--Dine and PBHs}{Baryogenesis has many roads --- Affleck--Dine, PBH
evaporation, electroweak variants. Why is yours forced?}
\begin{objqabody}
\emph{Response.} Forced within the framework, not within nature --- the
distinction is the answer. Part~VIII's structural point is
that the framework's own inheritance closes every other
channel: the crossover kills electroweak variants, the
dial-free scalar sector has no Affleck--Dine flat directions
\citep{AffleckDine1985}, and the derived descent populates no
PBH reservoir. So \emph{this} framework must stand or fall
with resonant leptogenesis --- an exposure, since a dead
leptogenesis kills it while merely inconveniencing everyone
else (\S\ref{app:lepto-sakharov}). Pluralism survives for
nature; it is not available to us.
\end{objqabody}

\section{The historian and the philosopher}

\PObj{50}{the historian of near-misses}{Pentagons, golden ratios,
forced angles --- physics has a graveyard of numerological
beauty. Eddington rests there.}
\begin{objqabody}
\emph{Response.} The graveyard's residents share a diagnosis:
numbers fitted after the fact, with freedom to adjust, and no
kill criteria. This paper's angles were fixed by integers
published for independent reasons, carry no adjustment freedom,
and are wired to dated instruments (Table~\ref{tab:falsifiers}).
Eddington could not be wrong; this paper can be wrong by 2035.
The capacity to die outside the graveyard is the entry fee to
science, and it is paid.
\end{objqabody}

\PObj{51}{the anthropic reasoner}{A CP phase of zero means no
observers. You have derived what selection already explains.}
\begin{objqabody}
\emph{Response.} Selection explains why we do not observe
$\delta = 0$; it is silent on why $\delta = 1.147$ rather than
any other life-permitting value --- a set of measure one in no
one's landscape. A forced lattice point within the
life-permitting range is exactly the kind of fact anthropics
cannot produce and can only be scooped by. If the pentagon
survives DUNE, selection will have had nothing to do with it.
\end{objqabody}

\PObj{52}{the philosopher of symmetry}{You say `the mirrors break
because the world has a direction' --- but your bulk is exactly
symmetric and only the texture breaks the mirrors. Which is the
world?}
\begin{objqabody}
\emph{Response.} Both, in custody (\S\ref{sec:custody}): the bulk
owns the symmetries, the descent owns the breaking, the boundary
owns the record. The philosophical content is precisely that
``symmetry of the laws'' and ``symmetry of the world'' come
apart --- Curie's principle survives because the descent's
orientation is itself lawful structure, not accident. The
slogan compresses; the custody map is the claim.
\end{objqabody}

\PObj{53}{the sociologist of physics}{A twenty-two-paper
self-citing program is a closed
ecosystem. Why should the community engage?}
\begin{objqabody}
\emph{Response.} Engage with the four numbers: $196^{\circ}$,
$66.9^{\circ}$, $1.7$~meV, $3.85$~TeV. They are stated,
derived-or-graded, and falsifiable on instruments the community
runs. A closed ecosystem cannot be killed from outside; this one
is begging to be.
\end{objqabody}

\PObj{54}{the Sakharov scholar}{Sakharov himself floated a CPT-mirror twin universe in the
same era as his three conditions. You take the conditions and
ignore the cosmology.}
\begin{objqabody}
\emph{Response.} The conditions are load-bearing theorems of any asymmetry
story; the twin universe was one story among several, and it
is honored here as the first CPT cosmology --- its modern
descendant is engaged on its physics under the cosmologist
above. The debt to Sakharov is itemized where it is used
(\citealp{Sakharov1967}; Part~\ref{part:matter}): the
equilibrium theorem, the both-C-and-CP clause, the departure
requirement --- each derived from scratch in
\S\ref{app:lepto-sakharov} so the reader can audit exactly
what is his and what is ours.
\end{objqabody}

\PObj{55}{the Wigner reader, on unreasonable effectiveness}{Why should the arithmetic of a pentagon price laboratory
phases at all? Effectiveness this unreasonable is usually a
sign of numerology.}
\begin{objqabody}
\emph{Response.} The paper's only defensible answer is structural realism
made testable: if the boundary is the readout of a real
texture, then arithmetic of the texture \emph{is} physics of
the readout, and the effectiveness stops being unreasonable
exactly when the geometry is real. The numerological reading
is not dismissed --- it is given its own audit rows, its own
$\times3^{\pm1}$ bands, and a falsifier schedule on which it
dies or survives (Part~\ref{part:experiments}). Rhetoric
settles nothing here; the decade does.
\end{objqabody}

\PObj{56}{the underdeterminist}{Some other geometry could fit the same numbers. Fitting is
cheap at this precision.}
\begin{objqabody}
\emph{Response.} Underdetermination is answered by forced error bars, not
uniqueness proofs: the lattice point carries zero width, the
QD step is one unit of a fixed lattice, and the exposures ---
phase, ordering, mass sum, ton-scale null, cluster, EDM
window, proton --- are conjunctive
(\S\ref{sec:falsification}). A rival geometry is welcome to
match \emph{all} rows with its own zero-width points; the
moment it does, both frameworks are wrong about something
soon, because nature will disagree with one of them at the
next decimal. That is not a weakness of the method; it is the
method.
\end{objqabody}

\section{The student}

\PObj{57}{the first-year, on where to start}{I have one weekend,
not a week.}
\begin{objqabody}
\emph{Response.} Read Part~I \S\ref{sec:history-parity} and
\S\ref{sec:history-stake}, then T-P5-1 with its intuition
paragraph, then the Conclusions' first section. That is the
paper in miniature: what fell, what it costs, what the mirrors
are, why matter won. The rest is audit trail.
\end{objqabody}

\PObj{58}{the second-year, on one measurement}{If I can follow
only one experimental number for the next ten years, which?}
\begin{objqabody}
\emph{Response.} $\dPMNS$. It adjudicates the leptonic lattice,
biases the door, and is the single place where this paper, the
Standard Model's agnosticism, and the anthropic shrug make
three different statements that one instrument will sort.
\end{objqabody}

\PObj{59}{the skeptical student, on trusting the imports}{Half
your inputs come from papers I have not read. Why believe the
light masses or the lock scale?}
\begin{objqabody}
\emph{Response.} Do not believe them; price them. Every import is
flagged at its point of use with its source and grade
(\S\ref{sec:inherited-theorems}, \S\ref{ssec:ckm},
\S\ref{sec:chain}), so the conditional structure of every
conclusion is explicit: T-P5-1 needs almost nothing; the $\etaB$
chain needs the most. The paper is engineered so you always know
what you are buying and from whom.
\end{objqabody}

\PObj{60}{the student of the chapters, on the honest register}{Appendix~\ref{app:lepto-epsilon}
teaches your mechanism's failure mode more carefully than its
success. Why?}
\begin{objqabody}
\emph{Response.} Because the failure mode is where the physics
currently lives (\S\ref{sec:resonance-faced}), and because a
student trained to locate a chain on the regime map --- including
twelve orders off the peak --- is trained; one trained only on
success is marketed to. The register is the curriculum.
\end{objqabody}

\PObj{61}{the career-minded student}{Is any of this a thesis?}
\begin{objqabody}
\emph{Response.} Three are: the finite-temperature resonance
computation (C-P5-C(i) --- a real transport calculation with a
clean kill criterion); the $\varepsilon'/\varepsilon$
confrontation of O-P5-31; and the gate-operator lemma's
orientation-pairing proof (C-P5-A). Each is well-posed, each
publishable in either verdict, and each settles a named claim of
this paper. That is what theses are for.
\end{objqabody}

\PObj{62}{every reader, on the whole}{Strip the program, the
cohort, the register. What survives if GG-Theory is wrong?}
\begin{objqabody}
\emph{Response.} Three portable results: the anatomy (any
framework deriving its $i$ from real orientations inherits the
table and the antiunitarity theorem); the valuation question
(``what does the CP phase cost?'' is now on record as answerable,
with one worked, killable answer); and the audit form itself ---
claims with grades, imports with prices, failures with dates.
If the pentagon dies at DUNE, the paper's numbers die with it;
the questions it taught its readers to ask do not.
\end{objqabody}

\PObj{63}{the student, on what to carry}{Ten numbers to memorize, if I carry nothing else?}
\begin{objqabody}
\emph{Response.} The decad: $66.9^{\circ}$; $196.06^{\circ}$;
$\sin\dPMNS = -8\sqrt3/49$; the Majorana pair $(0, \pi)$;
the triplet $(0.065, 8.59, 50.5)$~meV; $\mbb = 1.72$~meV;
$\Mlock = 3.85$~TeV; the throughput
$e^{-K/2} = 2.5\times10^{-8}$; the conversion $28/79$; and
the confrontation $5.87$ against $(6.12\pm0.04)\times
10^{-10}$. Every one is traceable to its row in
Table~\ref{tab:exp-predicted-measured}, and every one is
mortal somewhere in Table~\ref{tab:falsifiers}.
\end{objqabody}

\PObj{64}{the student, on checking tonight}{Can I verify a headline claim before I trust the Part?}
\begin{objqabody}
\emph{Response.} Three one-liners, each under five minutes: the exact surd
(problem H.6.1 --- trigonometry only); the $1.267$ and the
T2K tuning (problem H.1.1 --- a unit conversion); the
throughput identity (problem F.6.1 --- two multiplications
and an exponential). If any fails your arithmetic, the
contract of \S\ref{sec:exp-refresh} applies and the error
is reportable. The chapters were built with complete
solutions precisely so that trust is never the requested
currency.
\end{objqabody}

\PObj{65}{the student, on finding a real error}{Suppose I find one. What happens to the paper?}
\begin{objqabody}
\emph{Response.} The register works as designed: an erratum, a re-grade of
the affected rows, and the confrontation rerun at the next
data release --- single errors damage claims, not the audit
machinery, because the machinery is the part built to survive
its owners. The complete-solution format exists to make your
audit possible; the version-and-date discipline exists to
make the correction traceable. A framework that cannot absorb
a found error was never falsifiable in the first place.
\end{objqabody}

\section{The antiparticle referees}

\PObj{66}{the field theorist, on Weinberg}{Your handshake is
Weinberg's causality argument in geometric costume. He derived
the pairing from SR $+$ QM $+$ causality decades ago.}
\begin{objqabody}
\emph{Response.} Theorem~\ref{thm:weinberg} is stated in full in
this paper, with admiration, and the difference is in the
hypothesis sets: Weinberg \emph{presupposes} the complex Hilbert
space and imposes energy positivity on the output of
quantization; the handshake presupposes neither --- it derives
both complex structures from real constitutional data and
obtains the pairing as their pre-quantization compatibility,
thereby explaining two of his axioms. What is conceded in
return, in the same sentence: his conclusion holds for
interacting theories, ours at full strength is free/quasi-free.
Different theorems, different depths, one honest trade
(\S\ref{ssec:qft-antiparticles}).
\end{objqabody}

\PObj{67}{the Kaluza--Klein veteran}{Antiparticles running the
other way around the circle is fifty-year-old KK lore. T-P5-A is
a rediscovery.}
\begin{objqabody}
\emph{Response.} The image is conceded on the page where the
theorem lives --- and then the object is distinguished, because
the distinction is the parent program's founding move. Your
circle is spatial: it has a radius to stabilize, a charged
tower, and, in its string descendants, a landscape. The \GG{}
fiber is a non-spatial phase dimension --- generated by a
charge rotation, dimensionless, towerless, modulus-free
\citep{ArisakaA2GG6} --- so ``fifty-year-old KK lore'' has,
strictly, never before had an object it was true \emph{of}.
And the claim in any case begins at T-P5-B and closes at
T-P5-C: KK (spatial or not) has one source of $i$ and borrows
the other from the quantum axioms; it has one hand, and no
handshake is possible with one hand
(\S\ref{sec:anti-ledger}).
\end{objqabody}

\PObj{68}{the mathematical physicist, on Ashtekar--Magnon}{Your
clock theorem is Ashtekar--Magnon 1975 and Kay 1978. They did
the real work.}
\begin{objqabody}
\emph{Response.} They did, they are cited at every use, and
T-P5-B says so in its own statement. The step they did not take
--- could not, in a framework with no physical compact fiber ---
is the compatibility question: their unique $J$ meets a second
forced $J$ on charged sectors, and the meeting is where
antiparticles come from. Inherited uniqueness, new handshake.
\end{objqabody}

\PObj{69}{the algebraist, again}{Bisognano--Wichmann gives CPT
from modular theory for interacting fields --- deeper and
stronger than your quasi-free anatomy.}
\begin{objqabody}
\emph{Response.} Stronger in scope, conceded twice in the text;
deeper is the contested word. BW operates \emph{inside} a
complex Wightman theory: it locates $\Theta$ magnificently and
says nothing about what $i$ is. The anatomy says what $i$ is
made of and pays for it with scope. The modular route --- your
flow as our clock --- is the named bridge, and whichever side
proves it first wins the word ``deeper'' honestly
(\S\ref{ssec:qft-mirrors}; open question (6)).
\end{objqabody}

\PObj{70}{the QFT instructor}{The handshake covers free fields.
Real antiparticles annihilate, radiate, and bind. A free-field
origin story is a toy.}
\begin{objqabody}
\emph{Response.} The scope qualifier appears in every Part that
uses the result, verbatim. Two facts keep it from being a toy:
the interacting extension rides on the standard asymptotic
frame (conditionally --- the same scattering hypotheses your
own S-matrix course assumes), and the \emph{content} the
handshake explains (the pairing, the sign conventions, the
table) is inherited unchanged by the interacting theory that
dresses it. The toy objection would bite if interactions could
\emph{re-pair} the modes; locality forbids exactly that.
\end{objqabody}

\PObj{71}{the nostalgic, on the sea}{Dirac's sea predicted the
positron. Your handshake predicts nothing that was not known by
1949.}
\begin{objqabody}
\emph{Response.} Correct, and stated in the ledger: on Depth~1
the pairing is textbook, and no new particle is predicted by
T-P5-C. What is new is provenance --- and provenance has
consequences: the derived table (T-P5-1), the chirality
criterion for CP (T-P5-3), the door theorem (T-P5-4), and the
biased traffic of Part~\ref{part:matter} are all downstream of
knowing \emph{what the pairing is made of}. The sea predicted a
particle and explained nothing; the handshake predicts no
particle and explains the pairing, the phases, and ultimately
the excess. Different currencies (\S\ref{sec:anti-ledger}).
\end{objqabody}

\PObj{72}{the skeptic, on readouts}{``The positron is a
conjugate readout of the same basin'' is wordplay. Two
orientations of one object versus two objects --- what
experiment tells them apart?}
\begin{objqabody}
\emph{Response.} Problem G.6.3 of Appendix~\ref{app:antiparticle} is this question as a drill. A
second-basin positron could differ from the electron in any
un-protected property --- a slightly different mass, moment, or
binding --- while a second-orientation positron \emph{cannot}:
every CPT-protected identity of Table~\ref{tab:cpt-tests} is,
on the readout picture, an identity of one object with itself.
The $10^{-12}$-level agreements are the experiment; the
readout picture is the only one for which they are trivialities
rather than miracles.
\end{objqabody}

\PObj{73}{the relativist, on Hawking's partner}{Your
``dissolved partner'' hand-waves away the firewall problem that
serious people spent a decade on.}
\begin{objqabody}
\emph{Response.} The dissolution is a computation, not a wave:
the exterior thermal state is derived as the OGGEP partial
trace of a two-mode-squeezed \emph{bulk} state, and the
``partner'' is the traced-over bulk factor --- not an
independent 4D quantum whose entanglement monogamy can be
violated \citep{ArisakaC3BH}. Whether that resolution satisfies
you depends on whether you grant the 6D bulk; the point made
\emph{here} is narrower and safe: the partner is a boundary
label, and reifying boundary labels is exactly what the custody
principle exists to prevent (\S\ref{sec:boundary-readout}).
\end{objqabody}

\PObj{74}{the careful reader, on conflation}{Your bulk contains
conjugate representations \emph{and} anomaly-forced
opposite-chirality species. Sooner or later you will conflate
them into ``the bulk contains its own antimatter.''}
\begin{objqabody}
\emph{Response.} The conflation is forbidden in
\S\ref{sec:bulk-encoding} in its own subsection: the
representation conjugates are the antiparticles of (Q1);
the anomaly-forced species ($D_2^{-}$, the singlet sea) are
independent chiral fields whose opposite sign balances the
books, and are not the twins of the visible matter. The
distinction is load-bearing and policed --- you have just read
the police report.
\end{objqabody}

\PObj{75}{every reader, on the whole Part}{Strip the geometry.
What survives of Part~IV if the sixth dimension does not
exist?}
\begin{objqabody}
\emph{Response.} Three things, as portable results: the
statement that Weinberg's two undischarged axioms (the complex
$\cH$, output-positivity) \emph{can} be discharged by any
framework possessing a physical compact $\mathrm{U}(1)$
direction and a causal clock --- the theorem is KK-generic, not
\GG{}-proprietary; the identification of ``why complex QM'' with
``why antiparticles'' as one question, which stands as a
challenge to any foundational program; and the custody
discipline for twin-labels, which dissolves the readout puzzles
in any bulk-boundary framework. If the fiber falls, the
handshake falls with it --- and the falsifier tables say where
to push.
\end{objqabody}

\PObj{76}{the free-fall referee, on ALPHA-g}{Antihydrogen falls down, like everything else. Did your
one-object picture predict that, or accommodate it?}
\begin{objqabody}
\emph{Response.} Predicted, in the strong sense that the alternative was
lethal: one object doubly read cannot fall differently from
itself, so a repulsive or anomalous $\bar g$ would have
killed the conjugate-readout picture and, through the
CPT-protected identities, the theorem behind it. The
measurement \citep{ALPHAg2023} confirms at $\sim20\%$
precision; the picture's exposure does not stop there ---
any future $\bar g/g \neq 1$ at any precision is a kill,
registered now, and the improving free-fall program is
welcome company for Table~\ref{tab:cpt-tests}.
\end{objqabody}

\PObj{77}{the condensed-matter colleague, on Majorana wires}{Majorana modes are engineered in nanowires and studied for
qubits. Doesn't the laboratory normalize self-conjugacy,
whatever your mod-8 clock says?}
\begin{objqabody}
\emph{Response.} Emergent self-conjugacy in a superconductor lives in the
particle--hole redundancy of a mean-field description ---
physical, useful, and not a fundamental fermion equal to its
own conjugate: the doubled description is bookkeeping, and
the quasiparticle's `Majorana-ness' is exact only at zero
energy in an engineered spectrum. The mod-8 statement
(\S\ref{app:anti-algebra}) concerns the fundamental spinor
bundle of the arena, which no wire touches. The one
fundamental candidate remains the neutrino --- where the door
says yes, quietly, at $1.72$~meV.
\end{objqabody}

\PObj{78}{the stability referee, on the vacuum}{If antiparticles are relabeled modes, why doesn't the
vacuum pair-create for free? Relabellings are cheap.}
\begin{objqabody}
\emph{Response.} Because the relabeling is exactly what makes every physical
mode's energy positive: the handshake selects the global $J$
under which the quadratic form is definite --- Weinberg's
positivity-on-output, caught one level down
(\S\ref{app:anti-limits}). Pair creation then costs
$2mc^2$ in every frame, and the `free lunch' would require
the anti-agreeing pairing whose obstruction is the cocycle of
problem G.6.2: not expensive, \emph{inconsistent}. Vacuum
stability is the handshake's first dividend, not an
additional assumption.
\end{objqabody}

\section{The quantum-foundations referee}

\PObj{79}{the foundations referee, on the Born rule}{Your confrontation tables are probabilities. Derived, or
assumed?}
\begin{objqabody}
\emph{Response.} Inherited at a stated grade, from the companion whose
business it is: the Born chain closes at the relaxation level
(the R16 lemmas and their GDOS relaxation), with the
constitutional theorem conditional on four named premises and
one analytic residue --- the field-grade representation step,
carried on this paper's own promotion roadmap as item~8
(\S\ref{app:promotion-roadmap};
\citealp{ArisakaA8QM}). Everything probabilistic in this
paper runs through standard quantum mechanics, so nothing
here strains the grade --- but the reader deserves to know
the grade exists, and now does.
\end{objqabody}

\PObj{80}{the same referee, on real quantum mechanics}{Renou et al. falsified real-Hilbert quantum mechanics. Your
bulk is real. Which is it?}
\begin{objqabody}
\emph{Response.} Both, by custody --- and the custody is the theorem, not a
dodge: the boundary description is genuinely complex, with
one shared $J$ extending to composites and Fock space
(T-P5-C1), which is exactly what the network Bell tests
demand \citep{RenouEtAl2021}; the bulk carries the two real
structures whose forced agreement \emph{is} that $J$. The
no-go theorems constrain theories whose Hilbert space is
real; they are silent on theories whose complex structure is
derived. The dictionary owns the sectors; the geometry owns
the $i$ --- said five times across the cohort because it is
load-bearing five times.
\end{objqabody}

\PObj{81}{the same referee, on the two arrows}{T-violation is a parts-per-thousand laboratory effect;
measurement irreversibility is total. You use both arrows.
Same arrow?}
\begin{objqabody}
\emph{Response.} Distinct arrows, distinct custodians, one meeting point.
Microscopic T-violation is texture chirality --- small,
CP-compensated, CPT-protected (Part~V). The measurement
arrow is the one-way \OGGEP{} projection: records are
boundary exports, and the projection does not invert
(\S\ref{sec:transport} custody; the actualization layer of
Part~III). The paper keeps them apart deliberately --- most
confusions in this subject are custody errors --- and lets
them meet exactly once, where they must: at freeze-out, when
the cosmic record (the baryon excess) is written by dynamics
that use the microscopic arrow (Part~\ref{part:matter}).
\end{objqabody}

\PObj{82}{the same referee, on spin and statistics}{Pauli proved spin--statistics in 1940; the axiomatic proofs
are airtight. Your `second child of the quartet' adds
what?}
\begin{objqabody}
\emph{Response.} A Depth-3 candidate for what the connection is \emph{made
of}: Pauli and his descendants force the connection from
relativistic axioms --- necessity without anatomy. The
quartet route derives exchange-as-half-turn at the quasi-free
level from the same analytic fact that runs rotation and the
thermal clock (T-P5-B1), making statistics and spin two
readings of one rotation structure
(\S\ref{sec:quartet-second-child}). It is graded
\emph{candidate}, promoted only by the roadmap's item~6, and
the concession to the axiomatic proofs' interacting reach is
printed beside it. What is genuinely new is the
identification, not the theorem --- yet.
\end{objqabody}


\medskip
\noindent The References that follow carry a relevance note on
every entry, the cohort's A-list first.

\GGPart[References]{References}{Cited works, each with a relevance note}
\label{part:references}

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

\begin{thebibliography}{99}

\bibitem[ALPHA Collaboration(2023)]{ALPHAg2023}
\textbf{E.~K.~Anderson et al.\ (ALPHA Collaboration)}, Nature
\textbf{621}, 716 (2023).
\par\noindent\textbf{Relevance.} Antihydrogen in free fall: after two decades of trapping and cooling anti-atoms, the ALPHA-g apparatus released them in a vertical field gradient and watched them fall --- down, with $\bar g/g$ consistent with unity at the twenty-percent level. For this paper the measurement is the direct gravitational test of the one-object reading of the conjugate pair: a second-orientation positron cannot weigh differently from its electron, so the result is a registered retrodiction, and any future $\bar g/g \neq 1$ at any precision is a kill (Part~XV). First cited in \S\ref{part:objections}.

\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 (2026).
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/a1-constitution/}.
\par\noindent\textbf{Relevance.}
The axiomatic foundation. This paper draws AX-1 (causal and spatial
orientation; the clock T reverses and the arena P reverses), AX-2
(the fiber C reverses), AX-3's positivity clause (the sign of the
frequency splitting) and forward-semigroup structure, and POS-3
(the anomaly-admissibility filter policing every chirality
assignment of Part~VI). First cited in \S\ref{sec:inheritance}.

\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 (2026).
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/a2-gg6/}.
\par\noindent\textbf{Relevance.}
The arena paper: Theorem~0's forced six-dimensional geometry, the
complex chiral spinor structure of $(1,5)$ signature (no Majorana
condition; independent Weyl modules --- the raw material of
Part~VI's hand-picking), and the export argument by which no
$G\widetilde G$ knob reaches the boundary
(\S\ref{sec:strongcp}). First cited in \S\ref{sec:ledger}.

\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 (2026).
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/a3-bi6/}.
\par\noindent\textbf{Relevance.}
The ledger paper: fixes $(M,N,L;J,K) = (1,3,5;7,35)$ by anomaly
arithmetic --- the integers this paper consumes as the pentagon
($L$), the phase counter ($N$), the leptonic odd integer ($J$),
and the throat depth ($K$). Also the Green--Schwarz inflow that
POS-3 axiomatizes. First cited in \S\ref{sec:inheritance}.

\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 (2026).
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/a4-euclid/}.
\par\noindent\textbf{Relevance.}
The lineage paper of the cohort, and the one that measures the size of
this paper's claim from outside it. It traces the geometrization of
physics from Euclid forward --- each step a quantity that had been a law
becoming a property of a space --- and the present claim, that C, P and T
are orientations of real geometry rather than laws about fields, is one
more step in that sequence rather than a break with it. It is also the
companion that says how large a step it is not.
First cited in \S\ref{sec:inheritance}.

\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 (2026).
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/a5-gdos/}.
\par\noindent\textbf{Relevance.}
The boundary-law paper: the bulk-T-symmetric / projection-
asymmetric split that separates time-reversal from the arrow of
time (\S\ref{sec:custody}), and the explicit centralizer
computation verifying that the projection matrix breaks
$\mathrm{SU}(5)$ to exactly the Standard-Model group
(\S\ref{sec:projection}). First cited in \S\ref{sec:clauses}.

\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 (2026).
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/a6-grip/}.
\par\noindent\textbf{Relevance.}
The descent-machinery paper: the Wilson-line computations placing
the locked $\Uone{B}$ holonomy ($2\pi/5$) beside the dynamical
$\Uone{Qi}$ (Kaxion) line --- the two-fate architecture of
\S\ref{sec:strongcp} in a single worked example --- and the
family-projector transport whose holonomies are Part~VII's CP
angles. First cited in \S\ref{sec:custody}.

\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 (2026).
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/a7-qi4/}.
\par\noindent\textbf{Relevance.}
The treatment in which the informational symmetry $\Uone{Qi}$ and
its current are developed, memory is identified as a
gauge-invariant phase holonomy, and the outer/inner hologram loop
is defined --- the companion on which
\S\ref{ssec:real-bulk-mind}'s widening of view rests, and which
grades those claims at their own tentative register. First cited
in \S\ref{ssec:real-bulk-mind}.

\bibitem[Arisaka(2026, A8-QM)]{ArisakaA8QM}
\textbf{K.~Arisaka.}
\emph{Quantum Mechanics Derived from \GGTheory{} as Projection from 6D Bulk to 4D Boundary},
Arisaka-GG-A8-QM (2026).
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/a8-qm/}.
\par\noindent\textbf{Relevance.}
The real-bulk series RB-1--RB-12: the discussion-grade original of
the discrete-symmetry statement (RB-6) that T-P5-1/T-P5-2
consolidate, and the fermionic form of charge conjugation (circle
reversal composed with a fixed real spinor map) used in
\S\ref{sec:table}. First cited in \S\ref{sec:real-presentation}.

\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 (2026).
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/a9-sm/}.
\par\noindent\textbf{Relevance.}
The parent of Parts~V--VII: theorems T-SM-46.1/46.2 (the table
and the flavor holonomy, with the pentagonal lattice and the
$\dCKM$/$\dPMNS$ chains), T-SM-44.1 (the dial-free door), the
basin/gate section and its Gate-Operator Lemma (this paper's
C-P5-A), the strong-CP two-part vanishing (lemma L9), and the
Yukawa codebook supplying the mass-ratio angles of
\eqref{eq:cabibbo}. This paper consolidates, grades, and
extends; the proofs at parent depth live there. First cited in
\S\ref{sec:inheritance}.

\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 (2026).
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/a10-cosmos/}.
\par\noindent\textbf{Relevance.}
The parent of Part~VIII: the one-ledger baryogenesis statement, the
Sakharov custody line (``\GRIP{} owns the angle...''), the
step-by-step resonant-leptogenesis chain with its audit table,
the prompt-crossing (zero dilution) grounds, the honesty
objections this paper adopts verbatim (the $K/2$ postulate, the
order-of-magnitude demotion), and the $\Omega_b$ two-route seam.
First cited in \S\ref{sec:custody}.

\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 Preprint, available at \url{https://preprints.arisaka-gg.org/a11-origin/}.
No result of the present paper depends on it.
\par\noindent\textbf{Relevance.} The summit paper of the cohort, which asks the question the whole program exists to earn --- why is there anyone here to ask? --- and answers it as a cascade: CP violation to baryogenesis to stars to chemistry to observers, each link graded. This paper's \S\ref{sec:why-here} bridges into that cascade at its first link, the matter excess, and adopts the summit's own honest register: the chain beyond the matter link is conjectured, carried there as OP-7 rather than claimed. First cited in \S\ref{sec:why-here}.

\bibitem[Arisaka(2026, C1-Kaxion)]{ArisakaC1Kaxion}
\textbf{K.~Arisaka.}
\emph{The Kaxion: Cold Dark Matter as the Non-Spatial Compact-Angle Pseudoscalar of \GG{}},
Arisaka-GG-C1-Kaxion (2026).
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/c1-kaxion/}.
\par\noindent\textbf{Relevance.}
The relaxed fate: the gauge-topological compact angle that
discharges the strong-CP duty on the Peccei--Quinn template with
no PQ scalar, doubly protecting $\bar\theta$
(\S\ref{sec:strongcp}); also the smooth-circle requirement behind
the projection-location note (\S\ref{sec:projection-location}).
First cited in \S\ref{sec:strongcp}.

\bibitem[Arisaka(2026, C2-Hubble)]{ArisakaC2Hubble}
\textbf{K.~Arisaka.}
\emph{The Hubble Tension in the Light of \GGTheory{}: The Hubble Constant as One Reading of a Derived Expansion History $H(t)$},
Arisaka-GG-C2-Hubble (2026).
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/c2-hubble/}.
\par\noindent\textbf{Relevance.}
The expansion history against which the matter excess of
Part~\ref{part:matter} is measured. The baryon density this paper's chain
must land on is read off the same derived inventory that the corridor
analysis works in, so a shift in that inventory moves this paper's target
and not only its context --- which is why the companion is cited here and
not only acknowledged.
First cited in \S\ref{sec:inheritance}.

\bibitem[Arisaka(2026, P1-KK)]{ArisakaP1KK}
\textbf{K.~Arisaka.}
\emph{The \GG{} Lock Cluster: A Non-Spatial Kaluza--Klein Resonance near 3.85~TeV as the Standing LHC Falsifier of \GGTheory{}},
Arisaka-GG-P1-KK (2026).
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/p1-kk/}.
\par\noindent\textbf{Relevance.} The cohort's first dedicated falsification channel: the paper that takes the lock scale $\Mlock = 3.85$~TeV from the ledger, works out the collider phenomenology of the resulting cluster, confronts it with the ATLAS/CMS record, and pre-registers the HL-LHC criterion by which the prediction lives or dies. It is the collider sibling of this Part: the same heavy sector that runs the leptogenesis chain of Part~VIII is the cluster P1 sends to the energy frontier (\S\ref{sec:exp-opening}). First cited in \S\ref{sec:exp-opening}.

\bibitem[Arisaka(2026, P2-N3)]{ArisakaP2N3}
\textbf{K.~Arisaka.}
\emph{Why Three Generations: A Three-Leg Derivation of $N_g=3$ from \BIsix{}, \MSMF{}, and \GDOS{} Morse Stability},
Arisaka-GG-P2-N3 (2026).
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/p2-n3/}.
\par\noindent\textbf{Relevance.}
A precondition of this paper rather than a neighbor of it. A nonzero
Jarlskog invariant cannot exist in a mixing matrix with fewer than three
families, so the family count that paper derives is what makes the quark
and lepton CP phases of Part~\ref{part:pentagon} objects at all; its CP
leg and this paper's holonomy calculus meet at the same flavor graph. A
fourth light family would end both papers on the same day.
First cited in \S\ref{sec:inheritance}.

\bibitem[Arisaka(2026, P3-Electron)]{ArisakaP3Electron}
\textbf{K.~Arisaka.}
\emph{The Electron as the Ground State of Nature: The Six Oldest Questions about the Electron, Closed and Fenced within \GGTheory{}},
Arisaka-GG-P3-Electron (2026).
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/p3-electron/}.
\par\noindent\textbf{Relevance.}
The positron resolution this paper's readout section carries: the
conjugate readout of the same basin --- same level, opposite
winding --- with ``the positron, as a theorem'' registered as the
program's owed corollary (D-P3-4), and Wheeler's one-electron
universe rejected for the right reason and preserved in the form
that survives. First cited in \S\ref{sec:boundary-readout}.

\bibitem[Arisaka(2026, C3-BH)]{ArisakaC3BH}
\textbf{K.~Arisaka.}
\emph{Black-Hole Entropy under \GGTheory{}: The \GDOS{} Reading of the Horizon as an \OGGEP{} Port --- with the Exact Bekenstein--Hawking $A/4G_N$ Derived from Induced Gravity},
Arisaka-GG-C3-BH (2026).
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/c3-bh/}.
\par\noindent\textbf{Relevance.}
Hawking thermality derived as the \OGGEP{} partial trace of a
two-mode-squeezed bulk state, with the ``interior partner''
dissolved into the traced bulk --- the third of the three readout
episodes and the horizon instance of the custody principle. First
cited in \S\ref{sec:boundary-readout}.

\bibitem[Arisaka(2026, P4-Real)]{ArisakaP4Real}
\textbf{K.~Arisaka.}
\emph{The Real Geometry Beneath the Complex Numbers of Physics: Positive Frequency is Holomorphy --- from Cardano's Cubics to Twistor Space, via the Sixth Dimension of \GGTheory{}},
Arisaka-GG-P4-Real (2026).
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/p4-real/}.
\par\noindent\textbf{Relevance.}
The origin-of-$i$ member paper of the Foundations Branch --- this
paper's partner on the cohort map and its structural template: supplies
the four inherited theorems (circle, clock, handshake, shared $i$)
restated in \S\ref{sec:inherited-theorems}, the custody principle
this paper extends, and the real presentation in which every
computation here is performed. The division of labor is stated in
\S\ref{sec:inheritance}. First cited in \S\ref{sec:inheritance}.

\bibitem[Arisaka(2026, P6-Proton)]{ArisakaP6Proton}
\textbf{K.~Arisaka.}
\emph{The Proton as the Ground Geometric Attractor of Matter: The Eleven Questions of the Proton, Closed and Fenced within \GGTheory{}},
Arisaka-GG-P6-Proton (2026).
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/p6-proton/}.
\par\noindent\textbf{Relevance.}
The other end of this paper's number. Part~\ref{part:matter} makes the
baryon asymmetry; that paper shows the resulting protons are kept, baryon
number being carried as a winding of the compact circle rather than as a
bookkeeping label, so decay is forbidden by the geometry rather than
merely unobserved. Making the excess and keeping it are two statements
about one integer, and a single verified proton decay ends both.
First cited in \S\ref{sec:inheritance}.

\bibitem[Arisaka(2026, W1-Symmetry)]{ArisakaW1Symmetry}
\textbf{K.~Arisaka.}
\emph{Symmetry, Conservation, and Geometry: Seven Steps from Noether, Bianchi, to Hopkins--Singer, and the Ledger the Open-System Turn Pays Out},
Arisaka-GG-W1-Symmetry (2026).
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/w1-symmetry/}.
\par\noindent\textbf{Relevance.}
The first of the two white papers, and the diachronic one: it follows a
century of symmetry and conservation in the order the results were found
and watches the conviction that symmetry causes conservation loosen at
seven successive steps. The mirrors of the present paper are the sharpest
case in that history --- symmetries assumed for three centuries and then
measured to fail --- so W1 is the historical account of why this paper's
result is a continuation and not an anomaly. It derives nothing new.
First cited in \S\ref{sec:inheritance}.

\bibitem[Arisaka(2026, W2-Holonomy)]{ArisakaW2Holonomy}
\textbf{K.~Arisaka.}
\emph{The Circle That Remembers, the Interval That Forgets: \GGTheory{} and the Sixteen Problems of Particle Physics and Cosmology},
Arisaka-GG-W2-Holonomy (2026).
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/w2-holonomy/}.
\par\noindent\textbf{Relevance.}
The closest of the six to this paper. Its single exact statement --- a
holonomy requires a closed loop, and the interval fiber has none --- is
the asymmetry between the two non-spatial dimensions that
Part~\ref{part:time} derives the four endowments of time from, and the
same asymmetry that puts the CP phases on the circle and the arrow on the
interval. W2 takes that statement to sixteen of the questions physics has
not closed, and names the dated measurements on which six of the sixteen
treatments die. Like its sibling it derives nothing new.
First cited in \S\ref{sec:inheritance}.

\end{thebibliography}

\section*{B.~Other references}

\begin{thebibliography}{99}

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\textbf{ACME Collaboration (V.~Andreev et al.)}, Nature
\textbf{562}, 355 (2018).
\par\noindent\textbf{Relevance.} The molecular-amplifier revolution of the EDM program: polar molecules multiply the effective electric field on the electron by orders of magnitude, and ACME's ThO beam turned that leverage into $|d_e| < 1.1\times10^{-29}\,e\,$cm (since improved). In this paper's ledger the electron EDM is one of the two static T-odd nulls that discipline every proposed CP phase, this paper's included, and anchor the seventy-year program of \S\ref{sec:history-cpt}. First cited in \S\ref{sec:history-cpt}.

\bibitem[Alfv\'en(1965)]{Alfven1965}
\textbf{H.~Alfv\'en}, Rev.\ Mod.\ Phys.\ \textbf{37}, 652
(1965).
\par\noindent\textbf{Relevance.} The matter--antimatter symmetric cosmology, argued by a Nobel plasma physicist: a universe of separated matter and antimatter domains, with annihilation at the frontiers. It was the reasonable alternative to a built-in asymmetry --- and the gamma-ray sky killed it, since domain boundaries would glow in a diffuse annihilation flux that is not there. Its honest burial is where Part~I's stake begins: the asymmetry is global, and needs a dynamical origin (\S\ref{sec:history-stake}). First cited in \S\ref{sec:history-stake}.

\bibitem[Affleck and Dine(1985)]{AffleckDine1985}
\textbf{I.~Affleck, M.~Dine}, Nucl.\ Phys.\ B \textbf{249}, 361
(1985).
\par\noindent\textbf{Relevance.} Baryogenesis from scalar condensates: supersymmetric flat directions carrying baryon number acquire large field values during inflation and release an asymmetry as they relax --- an elegant mechanism that can generate almost any $\etaB$. It sits in the QFT toolbox of \S\ref{ssec:qft-matter} as the great alternative this framework cannot use: a dial-free scalar sector has no flat directions to displace, which is one of the closures that force Part~VIII onto the lepton route. First cited in \S\ref{ssec:qft-matter}.

\bibitem[Akhmedov, Rubakov, and
Smirnov(1998)]{AkhmedovRubakovSmirnov1998}
\textbf{E.~K.~Akhmedov, V.~A.~Rubakov, A.~Yu.~Smirnov}, Phys.\
Rev.\ Lett.\ \textbf{81}, 1359 (1998).
\par\noindent\textbf{Relevance.} Leptogenesis from sterile-neutrino oscillations at the GeV scale: the asymmetry is generated not in decays but in the CP-violating oscillations of the singlets themselves, while they are still relativistic --- a mechanism testable at beam dumps and in kaon factories. It is the toolbox's low-scale alternative, recorded in \S\ref{ssec:qft-matter} and again in Appendix~\ref{app:lepto-epsilon}, and distinct from the resonant-decay chain this paper's ledger runs. First cited in \S\ref{ssec:qft-matter}.

\bibitem[Alvarez-Gaum\'e and Witten(1984)]{AlvarezGaumeWitten1984}
\textbf{L.~Alvarez-Gaum\'e, E.~Witten}, Nucl.\ Phys.\ B
\textbf{234}, 269 (1984).
\par\noindent\textbf{Relevance.} The classic analysis of gravitational anomalies: in $4k{+}2$ dimensions purely gravitational anomalies exist, chirality is a signed currency, and the anomaly polynomials are computed once and for all. It is the map of the terrain on which \BIsix{} runs its ledger --- the anomaly flip \eqref{eq:anomaly-flip} that forces conjugate content into the bulk spectrum is a line item of their accounting (\S\ref{sec:weight-anomaly}; drilled as problem G.4.2). First cited in \S\ref{sec:weight-anomaly}.

\bibitem[Anderson(1933)]{Anderson1933}
\textbf{C.~D.~Anderson}, Phys.\ Rev.\ \textbf{43}, 491 (1933).
\par\noindent\textbf{Relevance.} The positron observed, in cloud-chamber tracks curving the wrong way: the discovery that crowned Dirac's prediction within two years of its naming and turned the sea from bookkeeping into triumph. The particle outlived its first explanation --- the sea was dismantled by the arguments of Appendix~\ref{app:anti-sea} --- and the twin it revealed is the subject Part~IV derives from consistency (\S\ref{ssec:qft-antiparticles}). First cited in \S\ref{ssec:qft-antiparticles}.

\bibitem[Aharonov and Bohm(1959)]{AharonovBohm1959}
\textbf{Y.~Aharonov, D.~Bohm}, Phys.\ Rev.\ \textbf{115}, 485
(1959).
\par\noindent\textbf{Relevance.} The prototype holonomy observable: a phase acquired around a closed loop, physical though no local field touches the path --- the demonstration that gauge geometry stores measurable information in loops. The flavor plaquettes of T-P5-3 are its flavor-graph analogues: rephasing-invariant loop angles, immune to convention for exactly the Aharonov--Bohm reason. First cited in \S\ref{sec:jarlskog}.

\bibitem[ALPHA Collaboration(2020)]{ALPHA2020}
\textbf{ALPHA Collaboration}, Nature \textbf{578}, 375 (2020).
\par\noindent\textbf{Relevance.} Antihydrogen 1S--2S spectroscopy: the anti-atom's flagship transition measured to a fractional precision of $10^{-12}$ and found identical to hydrogen's --- matter--antimatter comparison at the precision frontier. It is a load-bearing row of the CPT test table (Table~\ref{tab:cpt-tests}): one of the identities T-P5-2 forces exactly and any real texture preserves (\S\ref{sec:history-cpt}). First cited in \S\ref{sec:history-cpt}.

\bibitem[AMS Collaboration(2024)]{AMS02}
\textbf{AMS Collaboration}, AMS-02 antinuclei searches (2016--).
\par\noindent\textbf{Relevance.} The magnetic spectrometer on the Space Station, sifting cosmic rays for antinuclei since 2011: no confirmed antihelium, no antideuteron flux --- the local certificate that the asymmetry of \S\ref{sec:matter-facts} extends through the cosmic-ray sampling volume. The tentative candidate events in circulation are engaged head-on in Part~XV: a confirmed antihelium-4 flux is registered there as a genuine crisis for Part~VIII. First cited in \S\ref{sec:history-stake}.

\bibitem[ARGUS Collaboration(1987)]{ARGUS1987}
\textbf{H.~Albrecht et al.\ (ARGUS)}, Phys.\ Lett.\ B \textbf{192}, 245 (1987).
\par\noindent\textbf{Relevance.} Large $B^0$--$\bar B^0$ mixing, discovered where few expected it: the mixing rate implied a heavy top quark and, crucially, made time-dependent CP interferometry in the $B$ system feasible --- without ARGUS's surprise there is no $B$-factory program, no golden mode, and no third angle of the unitarity triangle. The era it opened is told in \S\ref{ssec:exp-b-era} and drilled in Appendix~\ref{app:flavour-bfactories}. First cited in \S\ref{ssec:exp-b-era}.

\bibitem[Ashtekar and Magnon(1975)]{AshtekarMagnon1975}
\textbf{A.~Ashtekar, A.~Magnon}, Proc.\ R.\ Soc.\ Lond.\ A
\textbf{346}, 375 (1975).
\par\noindent\textbf{Relevance.} With \citet{Kay1978}, the uniqueness theorem for the positive-energy complex structure on the solution space of a field on a stationary spacetime: the functional-analytic care behind the clock. The polar phase $\Ja$ that this paper's T-P5-B constructs --- and that T reverses --- is their object, restated with its constitutional pedigree in \S\ref{sec:inherited-theorems} and proved at walking pace in Part~IV. First cited in \S\ref{sec:two-structures}.

\bibitem[BaBar Collaboration(2001)]{BaBar2001}
\textbf{BaBar Collaboration (B.~Aubert et al.)}, Phys.\ Rev.\
Lett.\ \textbf{87}, 091801 (2001).
\par\noindent\textbf{Relevance.} The measurement that brought CP violation out of the kaon after thirty-seven years: $\sin 2\beta$ from time-dependent interferometry in $B \to J/\psi K_S$, the unitarity triangle closing on the one phase of the three-family matrix. It is half of the two-experiment confirmation (with \citet{Belle2001}) that the Kobayashi--Maskawa phase is the phase of the world --- the phase Part~VII values at the lattice point (\S\ref{sec:history-precision}). First cited in \S\ref{sec:history-precision}.

\bibitem[BaBar Collaboration(2012)]{BaBar2012}
\textbf{BaBar Collaboration (J.~P.~Lees et al.)}, Phys.\ Rev.\
Lett.\ \textbf{109}, 211801 (2012).
\par\noindent\textbf{Relevance.} Direct, model-independent observation of time-reversal violation: by comparing rates between flavor-tagged and CP-tagged $B$ states, BaBar measured T-conjugated transitions against each other with no recourse to CP or CPT assumptions --- the fall of the third mirror, measured cleanly rather than inferred. It completes the discovery record of \S\ref{ssec:qft-mirrors}: all three factors broken, each by its own experiment, the product alone surviving. First cited in \S\ref{ssec:qft-mirrors}.

\bibitem[Banks, Kaplan, and Nelson(1994)]{BanksKaplanNelson1994}
\textbf{T.~Banks, D.~B.~Kaplan, A.~E.~Nelson}, Phys.\ Rev.\ D
\textbf{49}, 779 (1994).
\par\noindent\textbf{Relevance.} The cosmological moduli problem stated with full force: light scalars with Planck-suppressed couplings dominate the energy density, decay late, and dilute or destroy every pre-existing relic --- including a baryon asymmetry. It is the dilution killer that string cosmologies must engineer around, and that the prompt lock crossing structurally lacks: the zero-dilution clause of \S\ref{sec:zero-dilution} is this paper's answer, inherited from the descent. First cited in \S\ref{sec:zero-dilution}.

\bibitem[Augustine(c.\,400)]{Augustine}
\textbf{Augustine of Hippo}, \emph{Confessions}, Book XI
(c.\,400).
\par\noindent\textbf{Relevance.} The founding confession of the
subject: everyone knows time until asked --- sixteen centuries of
philosophy of time in one sentence, and the question
Part~\ref{part:time} exists to answer. His analysis of the
present as the only existent, with past and future as memory and
expectation, anticipates the record-centered reading the
\OGGEP{} arrow makes structural. First cited in
\S\ref{sec:time-question}.

\bibitem[Belle Collaboration(2001)]{Belle2001}
\textbf{Belle Collaboration (K.~Abe et al.)}, Phys.\ Rev.\ Lett.\
\textbf{87}, 091802 (2001).
\par\noindent\textbf{Relevance.} The simultaneous $\sin 2\beta$ measurement from KEK: with \citet{BaBar2001}, the two-continent confirmation that one phase carries the $B$ system's CP violation --- the one-phase paradigm established by mutual verification, the field's gold standard for a discovery of this weight. The paradigm both experiments confirmed is exactly what Part~VII inherits and prices (\S\ref{sec:history-precision}). First cited in \S\ref{sec:history-precision}.

\bibitem[Berkooz, Douglas, and Leigh(1996)]{BerkoozDouglasLeigh1996}
\textbf{M.~Berkooz, M.~R.~Douglas, R.~G.~Leigh}, Nucl.\ Phys.\
B \textbf{480}, 265 (1996).
\par\noindent\textbf{Relevance.} Chiral fermions from branes at angles: intersecting D-branes localize chiral matter at their intersections, the third of string theory's chirality mechanisms after Calabi--Yau compactification and orbifolds. It is credited in \S\ref{sec:road-string} as part of the honest accounting of how the rival program obtains what Part~VI derives --- many mechanisms, each workable, none forced. First cited in \S\ref{ssec:string-achieved}.

\bibitem[Bisognano and Wichmann(1976)]{BisognanoWichmann1976}
\textbf{J.~J.~Bisognano, E.~H.~Wichmann}, J.\ Math.\ Phys.\
\textbf{17}, 303 (1976).
\par\noindent\textbf{Relevance.} CPT (composed with a rotation) as the modular conjugation of the wedge algebra: from the vacuum state and the algebra of local observables alone, Tomita--Takesaki theory delivers the reflection --- the deepest orthodox statement of where CPT comes from, with no Lagrangian in sight. It is the target of the open modular route (\S\ref{sec:cpt}, Appendix~\ref{app:d-cpt}): whether their modular flow is this paper's clock seen abstractly is the named bridge question of Objection~69. First cited in \S\ref{sec:history-cpt}.

\bibitem[Boltzmann(1877)]{Boltzmann1877}
\textbf{L.~Boltzmann}, Wien.\ Ber.\ \textbf{76}, 373 (1877).
\par\noindent\textbf{Relevance.} Entropy as counting: the
probabilistic reading of the second law, and with it the honest
admission --- sharpened by Loschmidt's and Zermelo's objections
--- that the thermodynamic arrow lives in initial conditions,
not in the reversible dynamics. It defines the standard
account's high-water mark that \S\ref{sec:time-arrow} engages:
the arrow this program moves from the state to the law is the
arrow Boltzmann located, correctly and unhappily, outside it.
First cited in \S\ref{sec:time-standard}.

\bibitem[Borexino Collaboration(2020)]{Borexino2020}
\textbf{M.~Agostini et al.\ (Borexino)}, Nature \textbf{587}, 577 (2020).
\par\noindent\textbf{Relevance.} CNO-cycle solar neutrinos detected: the completion of solar spectroscopy, sixty years after Homestake opened it --- every fusion chain of the Sun now seen in neutrinos, with fluxes matching the solar model. It closes the arc of \S\ref{sec:exp-nu-history}: the detector lineage that began with a deficit and a puzzle ends with the Sun as a calibrated neutrino source. First cited in \S\ref{sec:exp-nu-history}.

\bibitem[Bigi and Sanda(2009)]{BigiSanda2009}
\textbf{I.~I.~Bigi, A.~I.~Sanda}, \emph{CP Violation}, 2nd ed.
(Cambridge University Press, 2009).
\par\noindent\textbf{Relevance.} The flavor-centric monograph of
CP violation --- kaons, $B$ mesons, and the invariant formalism at
book length. Shelf text of the neutrino chapter
(Appendix~\ref{app:neutrino}) and the book-length grounding of
Appendix~\ref{app:discrete-qft}. First cited in
Appendix~\ref{app:neutrino}.

\bibitem[Boyle, Finn, and Turok(2018)]{BoyleFinnTurok2018}
\textbf{L.~Boyle, K.~Finn, N.~Turok}, Phys.\ Rev.\ Lett.\
\textbf{121}, 251301 (2018).
\par\noindent\textbf{Relevance.} The CPT-symmetric universe: the Bang as a mirror between a universe/anti-universe pair, with the observed arrow emerging from our position on one side --- the modern cosmological reading of the unbroken theorem, and the boldest use of CPT in cosmology since Sakharov. Engaged as a structural cousin in Part~XV: kin in reverence for the theorem, rival in locating the asymmetry in the state rather than the dynamics, with a falsifier set (vanishing tensors, a heavy sterile relic) almost disjoint from this paper's. First cited in \S\ref{sec:time-comparison}.

\bibitem[Branco, Lavoura, and Silva(1999)]{BrancoLavouraSilva}
\textbf{G.~C.~Branco, L.~Lavoura, J.~P.~Silva}, \emph{CP
Violation} (Oxford, 1999).
\par\noindent\textbf{Relevance.} The standard monograph of the rephasing-invariant calculus: every quartet identity, every convention trap, every Majorana-phase subtlety, at book length --- the reference this paper draws in wiring language (\S\ref{sec:flavour-graph}) and Appendix~\ref{app:flavour} assigns as its shelf text. Its Chapter~22 disambiguation of phase conventions is cited three times in this paper because real person-years keep being lost to exactly that trap; the reader who wants every matrix identity behind the plaquettes will find it there. First cited in \S\ref{sec:flavour-graph}.

\bibitem[Buzs\'aki(2006)]{Buzsaki2006}
\textbf{G.~Buzs\'aki}, \emph{Rhythms of the Brain} (Oxford,
2006).
\par\noindent\textbf{Relevance.} The standard synthesis of cortical rhythms: theta and gamma oscillations as the carrier and clock of neural computation, phase coding as the brain's own use of angle variables. It supplies the neuroscience half of the real-bulk aside of \S\ref{ssec:real-bulk-mind}: the one physical system known to model the world --- the brain --- runs its model on phases and rhythms, a suggestive echo, flagged as analogy and nothing more. First cited in \S\ref{ssec:real-bulk-mind}.

\bibitem[Buchm\"uller, Di Bari, and
Pl\"umacher(2005)]{BuchmullerDiBariPlumacher}
\textbf{W.~Buchm\"uller, P.~Di~Bari, M.~Pl\"umacher}, Ann.\
Phys.\ \textbf{315}, 305 (2005).
\par\noindent\textbf{Relevance.} The pedagogical summa of thermal leptogenesis: Boltzmann equations, washout regimes, efficiency factors, and the $\kappa \simeq 1/K_{\mathrm{wash}}$ shorthand, derived once and used by everyone since. It is the treatment behind this paper's transport (\S\ref{sec:transport}, Appendix~\ref{app:lepto-transport}): every washout statement in Part~VIII's chain is their machinery, evaluated at the ledger's inputs and credited at each use. First cited in \S\ref{sec:transport}.

\bibitem[Cabibbo(1963)]{Cabibbo1963}
\textbf{N.~Cabibbo}, Phys.\ Rev.\ Lett.\ \textbf{10}, 531 (1963).
\par\noindent\textbf{Relevance.} The original quark-mixing angle
--- one parameter reconciling strangeness-changing and
strangeness-conserving weak rates --- whose modern descendant is
the CKM matrix derived in Appendix~\ref{app:flavour-ckm} and
whose magnitude this paper's \eqref{eq:cabibbo} closes in golden
form. First cited in \S\ref{app:flavour-ckm}.

\bibitem[Candelas et al.(1985)]{CandelasHorowitzStromingerWitten1985}
\textbf{P.~Candelas, G.~Horowitz, A.~Strominger, E.~Witten},
Nucl.\ Phys.\ B \textbf{258}, 46 (1985).
\par\noindent\textbf{Relevance.} Chirality from compactification: on a Calabi--Yau background the number of fermion generations is a topological index, and parity violation descends from geometry --- string theory's founding precedent for the claim that chirality is geometric. Credited in \S\ref{sec:parity-question} as the noblest ancestor of Part~VI's own claim, before the contrast is drawn: an index counts the generations; it does not orient the descent. First cited in \S\ref{sec:parity-question}.

\bibitem[Casas and Ibarra(2001)]{CasasIbarra2001}
\textbf{J.~A.~Casas, A.~Ibarra}, Nucl.\ Phys.\ B \textbf{618},
171 (2001).
\par\noindent\textbf{Relevance.} The seesaw inversion: every Yukawa matrix consistent with the measured light masses and mixings is parameterized by one complex orthogonal matrix $R$ --- the exact inventory of what low-energy data does and does not fix in the heavy sector. This paper uses it twice, in opposite directions: as the honest statement of the freedom standard leptogenesis cannot close (\S\ref{sec:seesaw}; problem F.5.3), and as the freedom the dial-free door spends (Part~VIII). First cited in \S\ref{sec:seesaw}.

\bibitem[Choi, Kaplan, and Nelson(1993)]{ChoiKaplanNelson1993}
\textbf{K.~Choi, D.~B.~Kaplan, A.~E.~Nelson}, Nucl.\ Phys.\ B
\textbf{391}, 515 (1993).
\par\noindent\textbf{Relevance.} With \citet{DineLeighMacIntire1992}, CP as a discrete gauge symmetry of string theory, spontaneously broken by moduli: the sharpest formulation of geometric CP in the rival program, and the closest external relative of this paper's position. The kinship is credited in full before the contrast is drawn (\S\ref{sec:five-roads}): in their landscape the breaking is real but the value is environmental; on the pentagon the value is forced --- the difference between explaining that CP breaks and pricing what it breaks to. First cited in \S\ref{sec:pentagon-claims}.

\bibitem[Davis et al.(1968)]{Davis1968}
\textbf{R.~Davis, D.~S.~Harmer, K.~C.~Hoffman}, Phys.\ Rev.\
Lett.\ \textbf{20}, 1205 (1968).
\par\noindent\textbf{Relevance.} The Homestake chlorine experiment: six hundred tonnes of cleaning fluid in a gold mine, counting argon atoms one by one --- and finding a third of the predicted solar flux. The deficit opened the third arena thirty years before its resolution, a model of patient anomaly-keeping: the puzzle survived every recalibration until SNO showed the neutrinos were changing, not missing (\S\ref{sec:history-neutrinos}). First cited in \S\ref{sec:history-neutrinos}.

\bibitem[Cohen, De R\'ujula, and
Glashow(1998)]{CohenDeRujulaGlashow1998}
\textbf{A.~G.~Cohen, A.~De~R\'ujula, S.~L.~Glashow}, Astrophys.\
J.\ \textbf{495}, 539 (1998).
\par\noindent\textbf{Relevance.} The global certificate of the asymmetry: if the universe contained matter and antimatter domains, their boundaries would produce a diffuse annihilation gamma-ray flux; the observed sky forbids domains out to the visible horizon. It converts the local absence of antimatter into a cosmological fact (\S\ref{sec:matter-facts}) --- and it is the theorem a confirmed cosmic antihelium flux would reopen, an exposure this paper pre-registers in Part~XV. First cited in \S\ref{sec:history-stake}.

\bibitem[Connes and Rovelli(1994)]{ConnesRovelli1994}
\textbf{A.~Connes, C.~Rovelli}, Class.\ Quantum Grav.\
\textbf{11}, 2899 (1994).
\par\noindent\textbf{Relevance.} The thermal-time hypothesis:
physical time as the modular flow of the state, KMS-rooted and
one-parameter --- the nearest structural cousin of the
clock-and-shore account, engaged in
\S\ref{sec:time-comparison}. The custody contrast is exact:
there the state defines the time, here the arena does; the two
would meet on the open modular bridge of Objection~69, which is
why the cousin is cited wherever that bridge is discussed.
First cited in \S\ref{sec:time-standard}.

\bibitem[Cooke, Pettini, and Steidel(2018)]{Cooke2018}
\textbf{R.~J.~Cooke, M.~Pettini, C.~C.~Steidel}, Astrophys.\ J.\
\textbf{855}, 102 (2018).
\par\noindent\textbf{Relevance.} Precision primordial deuterium: quasar absorption spectra deliver D/H at the percent level, and through BBN a baryon density independent of the CMB. The concordance of the two epochs --- nucleosynthesis at minutes, recombination at four hundred thousand years --- is what makes $\etaB$ a measured constant of nature rather than a fit parameter, and it is the number Part~VIII's chain must and does land on (\S\ref{sec:matter-facts}). First cited in \S\ref{sec:history-stake}.

\bibitem[CUORE Collaboration(2024)]{CUORE2024}
\textbf{D.~Q.~Adams et al.\ (CUORE)}, arXiv:2404.04453 (2024).
\par\noindent\textbf{Relevance.} Te-130 bolometric $0\nu\beta\beta$ search: a tonne-scale cryogenic calorimeter running at ten millikelvin, past two tonne-years of exposure, $T_{1/2} > 3.5\times10^{25}$~yr. It is one of the three isotope technologies whose ton-scale successors define the coming era (\S\ref{sec:exp-now}) --- the era for which this paper's $\mbb = 1.72$~meV is a registered null. First cited in \S\ref{sec:exp-now}.

\bibitem[CUPID Collaboration(2025)]{CUPID2025}
\textbf{CUPID Collaboration}, Eur.\ Phys.\ J.\ C (2025), CUPID baseline design.
\par\noindent\textbf{Relevance.} Mo-100 scintillating bolometers in the CUORE cryostat: particle identification added to the bolometric technique, $10^{27}$-yr-class discovery sensitivity. One of the three ton-scale instruments (with LEGEND-1000 and nEXO) that will execute the one-sided Majorana test of \S\ref{sec:exp-future}: decisive for the inverted ordering, and for this paper's registered silence. First cited in \S\ref{sec:exp-future}.

\bibitem[Coughlan et al.(1983)]{CoughlanEtAl1983}
\textbf{G.~D.~Coughlan et al.}, Phys.\ Lett.\ B \textbf{131}, 59
(1983).
\par\noindent\textbf{Relevance.} The original moduli problem --- the Polonyi field: a weakly coupled scalar left displaced after inflation comes to dominate, decays late, and wrecks nucleosynthesis or dilutes every relic. The ancestral statement of the dilution hazard that \S\ref{sec:zero-dilution} addresses: the prompt crossing of the descent leaves no displaced light scalar behind, which is why the chain's $\etaB$ survives undiluted. First cited in \S\ref{sec:zero-dilution}.

\bibitem[Covi and Roulet(1996)]{CoviRoulet1996}
\textbf{L.~Covi, E.~Roulet (with F.~Vissani)}, Phys.\ Lett.\ B
\textbf{384}, 169 (1996).
\par\noindent\textbf{Relevance.} The CP asymmetry of heavy Majorana decay computed cleanly at one loop: vertex and self-energy interference, the $\mathrm{Im}[(Y^\dagger Y)^2]$ structure, the loop functions --- the $\epsilon_i$ formula this paper's chain evaluates at resonance (\S\ref{sec:transport}) and Appendix~\ref{app:lepto-epsilon} derives at drill depth as Eq.~\eqref{eq:f3-cr}. The flavor-space quartet in their numerator is the same closed-loop logic as the CKM plaquettes --- one invariance principle, two sectors. First cited in \S\ref{sec:transport}.

\bibitem[Christenson et al.(1964)]{Cronin1964}
\textbf{J.~H.~Christenson, J.~W.~Cronin, V.~L.~Fitch,
R.~Turlay}, Phys.\ Rev.\ Lett.\ \textbf{13}, 138 (1964).
\par\noindent\textbf{Relevance.} The discovery of CP violation: $K_L \to \pi\pi$ at a branching of $2\times10^{-3}$, in an experiment designed to confirm the symmetry rather than break it. It is the second fall of \S\ref{sec:history-cp}, the result that made the matter excess conceivable as physics, and the number the pentagon must ultimately price --- the discovery run is retold with its regeneration controls in Appendix~\ref{app:flavour-kaons}. First cited in \S\ref{sec:history-cp}.

\bibitem[CPLEAR Collaboration(1998)]{CPLEAR1998}
\textbf{CPLEAR Collaboration (A.~Angelopoulos et al.)}, Phys.\
Lett.\ B \textbf{444}, 43 (1998).
\par\noindent\textbf{Relevance.} First direct observation of T violation: tagged neutral kaons compared with their time-reversed transitions through semileptonic asymmetries, $A_T = 4\,\mathrm{Re}\,\varepsilon = (6.6\pm1.6)\times10^{-3}$ --- the arrow measured in the clock itself, not inferred from CP and CPT. It is the laboratory face of \S\ref{ssec:qft-mirrors} and the drill material of \S\ref{app:d-lab}: the third mirror fell in the kaon tank first. First cited in \S\ref{ssec:qft-mirrors}.

\bibitem[Danby et al.(1962)]{Danby1962}
\textbf{G.~Danby, J.-M.~Gaillard, K.~Goulianos, L.~M.~Lederman, N.~Mistry, M.~Schwartz, J.~Steinberger}, Phys.\ Rev.\ Lett.\ \textbf{9}, 36 (1962).
\par\noindent\textbf{Relevance.} The two-neutrino experiment at the Brookhaven AGS: neutrinos born with muons make muons, not electrons --- lepton flavor becomes physics, and the lepton sector acquires the family structure whose mixing matrix this paper's lattice values. The opening entry of the accelerator-neutrino lineage of \S\ref{sec:exp-nu-history}. First cited in \S\ref{sec:exp-nu-history}.

\bibitem[Davidson and Ibarra(2002)]{DavidsonIbarra2002}
\textbf{S.~Davidson, A.~Ibarra}, Phys.\ Lett.\ B \textbf{535}, 25
(2002).
\par\noindent\textbf{Relevance.} The bound that named the problem: for hierarchical heavy neutrinos the decay asymmetry is capped by the light and heavy masses, forcing $M_N \gtrsim 10^9$~GeV --- thermal leptogenesis priced permanently out of laboratory reach. It is the theorem the TeV-scale chain must evade, and does, via the resonant regime (\S\ref{sec:transport}); Appendix~\ref{app:lepto-epsilon} derives the bound as Eq.~\eqref{eq:f3-di} and evaluates its six-order shortfall at the lock scale. Engaged explicitly per the register. First cited in \S\ref{sec:transport}.

\bibitem[Daya Bay Collaboration(2012)]{DayaBay2012}
\textbf{Daya Bay Collaboration (F.~P.~An et al.)}, Phys.\ Rev.\
Lett.\ \textbf{108}, 171803 (2012).
\par\noindent\textbf{Relevance.} $\theta_{13} \neq 0$, measured decisively by reactor disappearance: the angle whose size determines whether the leptonic CP phase is experimentally reachable at all. Its largeness --- near the old bounds --- is why the long-baseline program can measure $\dPMNS$ this generation, why the lattice point is testable, and why this paper can be killed (\S\ref{sec:history-neutrinos}). First cited in \S\ref{sec:history-neutrinos}.

\bibitem[DESI Collaboration(2025)]{DESI2025}
\textbf{DESI Collaboration}, arXiv:2503.14738 (2025).
\par\noindent\textbf{Relevance.} DR2 baryon-acoustic-oscillation data combined with the CMB: $\sum m_\nu < 0.064$~eV, cosmology pressing the neutrino spectrum harder than any laboratory. For this paper the bound is load-bearing twice (\S\ref{sec:exp-now}): it presses the orderings, and it sits eight percent above the derived sum of $59.2$~meV --- the new kill-or-confirm channel registered in Part~XV, owned entirely by cosmology. First cited in \S\ref{sec:exp-now}.

\bibitem[DeWitt(1967)]{DeWitt1967}
\textbf{B.~S.~DeWitt}, Phys.\ Rev.\ \textbf{160}, 1113 (1967).
\par\noindent\textbf{Relevance.} The Wheeler--DeWitt equation:
canonical quantum gravity's wavefunction of the universe, with
no time parameter in it --- the ``problem of time'' stated at
the source, and the standing embarrassment of every
background-independent program
(\S\ref{sec:time-standard}). Part~\ref{part:time}'s arena
account dissolves rather than solves it: time is
constitutional, so the boundary theory never faces a timeless
fundamental equation. First cited in
\S\ref{sec:time-standard}.

\bibitem[Dine, Leigh, and MacIntire(1992)]{DineLeighMacIntire1992}
\textbf{M.~Dine, R.~G.~Leigh, D.~A.~MacIntire}, Phys.\ Rev.\
Lett.\ \textbf{69}, 2030 (1992).
\par\noindent\textbf{Relevance.} CP as a gauge symmetry in string theory: an exact discrete gauge symmetry of the worldsheet theory, spontaneously broken by the compactification --- geometric CP's source paper. Credited in \S\ref{sec:five-roads} before the valuation contrast is drawn: the insight that CP breaking is geometry is theirs; the claim that the broken value is a forced lattice angle is this program's, and it is the falsifiable half. First cited in \S\ref{sec:pentagon-claims}.

\bibitem[Dirac(1933)]{Dirac1933}
\textbf{P.~A.~M.~Dirac}, Nobel Lecture (1933).
\par\noindent\textbf{Relevance.} The Nobel lecture's closing speculation: perhaps half the stars are made of positrons and antiprotons --- the symmetric universe entertained at the moment of the positron's confirmation. It is the honest starting point of \S\ref{sec:history-stake}: the asymmetry was not obvious; it had to be discovered, quantified, and finally explained --- the eighty-year arc this paper's Part~VIII closes at its own grade. First cited in \S\ref{sec:history-stake}.

\bibitem[Dirac(1928)]{Dirac1928}
\textbf{P.~A.~M.~Dirac}, Proc.\ R.\ Soc.\ Lond.\ A
\textbf{117}, 610 (1928).
\par\noindent\textbf{Relevance.} The relativistic wave equation of the electron: spin and the magnetic moment delivered, and with them a negative-energy branch that would not go away --- the problem whose eighty-year working-out is Shelf~6 of the reading guide. The equation opens Appendix~\ref{app:anti-sea}'s derivation and \S\ref{ssec:qft-antiparticles}'s history: everything in Part~IV descends from taking its unwanted half seriously. First cited in \S\ref{ssec:qft-antiparticles}.

\bibitem[Dirac(1930)]{Dirac1930}
\textbf{P.~A.~M.~Dirac}, Proc.\ R.\ Soc.\ Lond.\ A
\textbf{126}, 360 (1930).
\par\noindent\textbf{Relevance.} The sea proposed: fill the negative-energy branch, let exclusion block the collapse, and read the holes as particles of opposite charge --- audacious bookkeeping that produced the positron prediction from an instability. The sea's triumph and its three structural failures (bosons, vacuum charge, asymmetric ontology) are derived as theorems in Appendix~\ref{app:anti-sea}: the first ontology of the twin, honored and dismantled. First cited in \S\ref{ssec:qft-antiparticles}.

\bibitem[Dirac(1931)]{Dirac1931}
\textbf{P.~A.~M.~Dirac}, Proc.\ R.\ Soc.\ Lond.\ A
\textbf{133}, 60 (1931).
\par\noindent\textbf{Relevance.} The anti-electron named: in the same paper as the magnetic monopole, Dirac committed to the hole as a new particle of electron mass and positive charge --- perhaps the greatest successful prediction ever made by bookkeeping. The cloud chamber honored it within the year; field theory later retired the explanation while keeping the particle (\S\ref{ssec:qft-antiparticles}), a transition this paper reads as the first step toward the handshake. First cited in \S\ref{ssec:qft-antiparticles}.

\bibitem[Dixon et al.(1985)]{DixonHarveyVafaWitten1985}
\textbf{L.~Dixon, J.~Harvey, C.~Vafa, E.~Witten}, Nucl.\ Phys.\ B
\textbf{261}, 678 (1985).
\par\noindent\textbf{Relevance.} String orbifolds: exact string backgrounds from discrete quotients, with chiral matter at the fixed points --- the technology whose field-theory limit became the orbifold GUT. It is the lineage of Part~VI's projection, conceded in \S\ref{sec:projection}: the mechanism is thirty years old and string-born; what the descent adds is the absence of choice in its application. First cited in \S\ref{sec:parity-question}.

\bibitem[Feynman and Gell-Mann(1958)]{FeynmanGellMann1958}
\textbf{R.~P.~Feynman, M.~Gell-Mann}, Phys.\ Rev.\ \textbf{109},
193 (1958).
\par\noindent\textbf{Relevance.} $V{-}A$ codified: the charged current couples only to left-handed fermions, maximal parity violation written into the vertex --- the structure every weak-interaction calculation since has carried. For Part~VI it is the shape of the explanandum: a fact this fundamental must be output, not input, and the one-gate descent is the attempt to make it so (\S\ref{sec:history-parity}). First cited in \S\ref{sec:history-parity}.

\bibitem[Feynman(1949)]{Feynman1949}
\textbf{R.~P.~Feynman}, Phys.\ Rev.\ \textbf{76}, 749 (1949).
\par\noindent\textbf{Relevance.} The theory of positrons: negative-energy solutions propagating backward in time reinterpreted as positive-energy antiparticles forward --- the relabeling that, with the contour, replaced the sea and gave every amplitude its modern form. The assignment carries his name jointly with St\"uckelberg's; its provenance --- why that relabeling and no other is consistent --- is exactly what T-P5-C supplies (\S\ref{ssec:qft-antiparticles}, Appendix~\ref{app:anti-contour}). First cited in \S\ref{ssec:wheeler}.

\bibitem[Fukugita and Yanagida(1986)]{FukugitaYanagida1986}
\textbf{M.~Fukugita, T.~Yanagida}, Phys.\ Lett.\ B \textbf{174},
45 (1986).
\par\noindent\textbf{Relevance.} Leptogenesis founded in four pages: create the asymmetry as lepton number through heavy Majorana decays, and let the sphalerons convert --- the mechanism class of Part~VIII entire. Cited as parent at every load-bearing use (\S\ref{sec:transport}, \S\ref{sec:why-leptogenesis}) and taught from scratch in Appendix~\ref{app:leptogenesis}: this paper's contribution is not the mechanism but the removal of its dials. First cited in \S\ref{sec:history-neutrinos}.

\bibitem[Garwin, Lederman, and Weinrich(1957)]{Garwin1957}
\textbf{R.~L.~Garwin, L.~M.~Lederman, M.~Weinrich}, Phys.\ Rev.\
\textbf{105}, 1415 (1957).
\par\noindent\textbf{Relevance.} Parity violation in the $\pi$--$\mu$--$e$ chain, found within days of Wu's result and published back to back with it: the muon's spin precession read out through the decay asymmetry --- the fall confirmed in a second system before the ink dried on the first. Together the two experiments made January 1957 the month the mirror broke for good (\S\ref{sec:history-parity}). First cited in \S\ref{sec:history-parity}.

\bibitem[Gavela et al.(1994)]{GavelaEtAl1994}
\textbf{M.~B.~Gavela, P.~Hern\'andez, J.~Orloff, O.~P\`ene},
Mod.\ Phys.\ Lett.\ A \textbf{9}, 795 (1994).
\par\noindent\textbf{Relevance.} The quantitative half of the Standard Model's baryogenesis failure: at the electroweak crossover the effective CP violation available from the CKM phase is suppressed by the quark-mass hierarchy to $\sim10^{-20}$ --- ten orders short of the observed asymmetry. With the crossover itself (\citealp{Kajantie1996}) it closes the SM's own route twice over, the closure Part~VIII inherits and Appendix~\ref{app:lepto-sakharov} derives as a drill (\S\ref{sec:history-stake}). First cited in \S\ref{sec:history-stake}.

\bibitem[Gell-Mann, Ramond, and Slansky(1979)]{GellMannRamondSlansky}
\textbf{M.~Gell-Mann, P.~Ramond, R.~Slansky}, in
\emph{Supergravity} (North-Holland, 1979).
\par\noindent\textbf{Relevance.} With \citet{Minkowski1977} and \citet{YanagidaSeesaw}, the seesaw's source papers: heavy Majorana partners make light neutrinos inevitable, with masses inversely proportional to the new scale. The mechanism is derived by hand in Appendix~\ref{app:nu-mass} and \S\ref{app:lepto-seesaw}; the ledger's contribution is to fix the scale it left free (\S\ref{sec:history-neutrinos}). First cited in \S\ref{sec:history-neutrinos}.

\bibitem[Gardner(1964)]{Gardner1964}
\textbf{M.~Gardner}, \emph{The Ambidextrous Universe} (Basic
Books, 1964).
\par\noindent\textbf{Relevance.} The Ozma problem, posed for the general reader: how do you communicate ``left'' to a distant civilization by physics alone? Before 1957 you cannot; after Wu you can --- and after 1964 the instruction survives even the antimatter ambiguity. This paper promotes the riddle to a structural invariant: the handedness of the descent, P-P5-D, is the geometric object the Ozma broadcast transmits (\S\ref{sec:ozma}). First cited in \S\ref{sec:before-the-fall}.

\bibitem[Friston(2010)]{Friston2010}
\textbf{K.~Friston}, Nat.\ Rev.\ Neurosci.\ \textbf{11}, 127
(2010).
\par\noindent\textbf{Relevance.} The free-energy principle: perception and action as minimization of the mismatch between an internal generative model and the sensorium --- the mainstream neuroscience statement closest to the monotone model-refinement this paper's real-bulk aside invokes. Cited in \S\ref{ssec:real-bulk-mind} as context for an analogy, with the aside's own register: suggestive parallel, no derivational weight. First cited in \S\ref{ssec:real-bulk-mind}.

\bibitem[Gabor(1946)]{Gabor1946}
\textbf{D.~Gabor}, J.\ IEE \textbf{93}, 429 (1946).
\par\noindent\textbf{Relevance.} The analytic signal: to phase-shift a real waveform meaningfully, complete it with its Hilbert transform --- the engineer's quarter-period advance, standard in signal processing for eighty years. The clock's complex structure $\Ja = \pm A(-A^2)^{-1/2}$ is mathematically this construction elevated to dynamics (\S\ref{ssec:source-clock}): the state-space $i$ was in the engineering literature all along, waiting to be read as geometry. First cited in \S\ref{ssec:source-clock}.

\bibitem[Georgi and Glashow(1974)]{GeorgiGlashow1974}
\textbf{H.~Georgi, S.~L.~Glashow}, Phys.\ Rev.\ Lett.\ \textbf{32}, 438 (1974).
\par\noindent\textbf{Relevance.} Minimal SU(5): the first grand unification, the charges explained, and proton decay promoted from heresy to prediction --- the theory whose minimal form the water-Cherenkov generation falsified, teaching the field that unification is mortal. Both lessons are inherited here: the representation structure ($\bar{\mathbf 5} \oplus \mathbf{10}$) survives in the bulk packaging, while the proton's stability is this ledger's opposite prediction (\S\ref{sec:exp-now}). First cited in \S\ref{sec:exp-now}.

\bibitem[GERDA Collaboration(2020)]{GERDA2020}
\textbf{M.~Agostini et al.\ (GERDA)}, Phys.\ Rev.\ Lett.\ \textbf{125}, 252502 (2020).
\par\noindent\textbf{Relevance.} The background-free frontier of Ge-76: bare germanium diodes in liquid argon, an expected background below one count in the signal region, $T_{1/2} > 1.8\times10^{26}$~yr. The technique it perfected is the platform for LEGEND and the ton-scale era (\S\ref{sec:exp-now}) --- the instruments of the one-sided test this paper's $1.72$~meV registers. First cited in \S\ref{sec:exp-now}.

\bibitem[Goldhaber, Grodzins, and Sunyar(1958)]{Goldhaber1958}
\textbf{M.~Goldhaber, L.~Grodzins, A.~W.~Sunyar}, Phys.\ Rev.\
\textbf{109}, 1015 (1958).
\par\noindent\textbf{Relevance.} The measurement of the
neutrino's helicity with tabletop apparatus --- resonant
scattering of the Doppler-selected $^{152}$Sm$^*$ photon
transferring the invisible particle's handedness to a measurable
circular polarization. Told in full in
Appendix~\ref{app:discrete-qft} \S\ref{app:d-dirac} as the direct
experimental face of maximal parity violation. First cited in
\S\ref{app:d-dirac}.

\bibitem[Greenberg(2002)]{Greenberg2002}
\textbf{O.~W.~Greenberg}, Phys.\ Rev.\ Lett.\ \textbf{89}, 231602
(2002).
\par\noindent\textbf{Relevance.} CPT violation implies Lorentz violation: any interacting field theory that breaks CPT breaks Lorentz invariance too --- the converse discipline that welds the two symmetries together. For this paper it makes T-P5-2's prohibition maximally exposed (\S\ref{sec:history-cpt}, O-P5-26): a confirmed CPT violation would not merely kill a theorem, it would take the Lorentzian arena with it --- and, per the stacking argument of Objection~28, the SME bounds discipline the texture twice over. First cited in \S\ref{sec:history-cpt}.

\bibitem[Giunti and Kim(2007)]{GiuntiKim2007}
\textbf{C.~Giunti, C.~W.~Kim}, \emph{Fundamentals of Neutrino
Physics and Astrophysics} (Oxford University Press, 2007).
\par\noindent\textbf{Relevance.} The field's standard reference for
oscillation physics in vacuum and matter and the field-theoretic
foundations; Chapters 6--10 and 13 are the assigned reading of
Appendices~\ref{app:nu-osc}--\ref{app:nu-appearance}, including
the golden-channel expansion drilled in
\S\ref{app:nu-appearance}. First cited in
Appendix~\ref{app:neutrino}.

\bibitem[Hall and Nomura(2001)]{HallNomura2001}
\textbf{L.~J.~Hall, Y.~Nomura}, Phys.\ Rev.\ D \textbf{64},
055003 (2001).
\par\noindent\textbf{Relevance.} With \citet{Kawamura2000}, the orbifold GUT made realistic: doublet--triplet splitting solved by boundary conditions, unification kept, proton decay tamed --- the five-dimensional mechanism whose matrix of choices Part~VI's projection fills in. Credited to the matrix, item by item, in \S\ref{sec:projection}: the machinery is theirs; the absence of alternatives at each entry is the descent's claim. First cited in \S\ref{sec:projection}.

\bibitem[Harvey and Turner(1990)]{HarveyTurner1990}
\textbf{J.~A.~Harvey, M.~S.~Turner}, Phys.\ Rev.\ D \textbf{42},
3344 (1990).
\par\noindent\textbf{Relevance.} The chemical-potential bookkeeping of the hot Standard Model plasma: impose Yukawa equilibrium, sphaleron equilibrium, and hypercharge neutrality, solve the linear system, and the conversion coefficient $B = \tfrac{28}{79}(B{-}L)$ falls out --- the number every baryogenesis mechanism multiplies by. Derived as a drill in Appendix~\ref{app:lepto-transport} (problem F.4.2, with the two-doublet variant) and used at its single load-bearing site in \S\ref{sec:transport}. First cited in \S\ref{sec:transport}.

\bibitem[Hafting et al.(2005)]{HaftingEtAl2005}
\textbf{T.~Hafting, M.~Fyhn, S.~Molden, M.-B.~Moser,
E.~I.~Moser}, Nature \textbf{436}, 801 (2005).
\par\noindent\textbf{Relevance.} Grid cells: neurons whose firing fields tile the environment in a hexagonal lattice --- the metric of the brain's world-centered frame, made visible in single recordings. With place cells they are the hardware of the reconstruction discussed in \S\ref{ssec:real-bulk-mind}: the one known model-building machine represents space in allocentric coordinates, an echo the aside records as analogy only. First cited in \S\ref{ssec:real-bulk-mind}.

\bibitem[Herb et al.(1977)]{Herb1977}
\textbf{S.~W.~Herb et al.}, Phys.\ Rev.\ Lett.\ \textbf{39},
252 (1977).
\par\noindent\textbf{Relevance.} The $\Upsilon$ at Fermilab: a third-generation quark, four years after Kobayashi and Maskawa showed that three families are the minimum for a hard CP phase --- the counting theorem honored by discovery. The bottom quark's arrival began the confirmation of the structure whose one phase this paper values (\S\ref{sec:one-phase}). First cited in \S\ref{sec:one-phase}.

\bibitem[Hyper-Kamiokande Collaboration(2025)]{HyperK2025}
\textbf{Hyper-Kamiokande Collaboration}, arXiv:2505.15019 (2025); design report arXiv:1805.04163.
\par\noindent\textbf{Relevance.} The third generation of the author's own instrument line: a quarter-megaton fiducial volume, a megawatt beam, $5\sigma$ CP-violation exclusion over most of the phase range in ten years, and the deepest proton-decay reach ever fielded. It owns the two headline confrontations of this Part (\S\ref{sec:exp-future}): the quiet signature at $196^{\circ}$ --- which predicts its flagship discovery should \emph{not} arrive --- and the ledger's absolutely stable proton. First cited in \S\ref{sec:exp-future}.

\bibitem[Jarlskog(1985)]{Jarlskog1985}
\textbf{C.~Jarlskog}, Phys.\ Rev.\ Lett.\ \textbf{55}, 1039
(1985).
\par\noindent\textbf{Relevance.} The basis-invariant measure of CP violation: one rephasing-invariant $J$ carrying every CP-violating observable of the three-family Standard Model, extracted from the commutator of mass matrices. Clauses (i)--(iii) of T-P5-3 are this calculus, geometrized and credited --- the paper's own claim begins strictly at the lattice valuation (\S\ref{sec:jarlskog}); the invariant itself is derived from quartets in Appendix~\ref{app:flavour-invariants}. First cited in \S\ref{sec:one-phase}.

\bibitem[Jost(1957)]{Jost1957}
\textbf{R.~Jost}, Helv.\ Phys.\ Acta \textbf{30}, 409 (1957).
\par\noindent\textbf{Relevance.} The axiomatic CPT theorem: analytic continuation to the complex Lorentz group turns the reflection into a theorem of Wightman field theory --- rigor's own proof, with the axioms' own silence about what the operation is made of. It defines the scope this paper concedes and the anatomy it supplies (\S\ref{sec:cpt}, Appendix~\ref{app:d-cpt}): Jost proves that $\Theta$ exists; T-P5-1/2 say what its factors are. First cited in \S\ref{sec:history-cpt}.

\bibitem[JUNO Collaboration(2025)]{JUNO2025}
\textbf{JUNO Collaboration}, first oscillation results (2025); sensitivity: Chin.\ Phys.\ C \textbf{46}, 123001 (2022).
\par\noindent\textbf{Relevance.} Twenty kilotons of scintillator at the solar-atmospheric crossing point: fifty-nine days sufficed for the world's best $\theta_{12}$ and $\Delta m^2_{21}$, and six years of the fast reactor wiggle will read the mass ordering at $3\sigma$ with no beam, no matter effect, and no $\delta$ dependence --- the cleanest of the ordering channels this paper's normal-ordering chain must survive (\S\ref{sec:exp-now}, \S\ref{sec:exp-future}; the vacuum interferometry is problem H.1.2). First cited in \S\ref{sec:exp-now}.

\bibitem[Kajantie et al.(1996)]{Kajantie1996}
\textbf{K.~Kajantie, M.~Laine, K.~Rummukainen,
M.~Shaposhnikov}, Phys.\ Rev.\ Lett.\ \textbf{77}, 2887 (1996).
\par\noindent\textbf{Relevance.} The lattice determination that closed electroweak baryogenesis in the Standard Model: at the physical Higgs mass the transition is a smooth crossover --- no bubbles, no departure from equilibrium, no asymmetry. Half of the SM's double failure (with the $10^{-20}$ of \citealp{GavelaEtAl1994}), inherited intact by the boundary and load-bearing in \S\ref{sec:why-leptogenesis}: the crossover is one of the closures that force the lepton route. First cited in \S\ref{sec:history-stake}.

\bibitem[KamLAND Collaboration(2003)]{KamLAND2003}
\textbf{KamLAND Collaboration (K.~Eguchi et al.)}, Phys.\ Rev.\
Lett.\ \textbf{90}, 021802 (2003).
\par\noindent\textbf{Relevance.} Reactor antineutrino disappearance over two hundred kilometers: the solar oscillation solution confirmed from the laboratory side, with the beat frequency measured in a controlled flux. With SNO it ended the solar saga as physics ends its best puzzles --- two independent methods, one matrix (\S\ref{sec:history-neutrinos}). First cited in \S\ref{sec:history-neutrinos}.

\bibitem[Klinkhamer and Manton(1984)]{KlinkhamerManton1984}
\textbf{F.~R.~Klinkhamer, N.~S.~Manton}, Phys.\ Rev.\ D
\textbf{30}, 2212 (1984).
\par\noindent\textbf{Relevance.} The sphaleron: the static, unstable solution at the top of the barrier between electroweak vacua --- the mountain pass named and solved, turning 't~Hooft's exponentially suppressed tunneling into a thermally crossable ridge. It is the reprocessing engine of \S\ref{sec:transport}, taught with the winding picture in Appendix~\ref{app:lepto-sakharov}. First cited in \S\ref{sec:transport}.

\bibitem[KOTO Collaboration(2025)]{KOTO2025}
\textbf{KOTO Collaboration}, arXiv:2505.02568 (2025); KOTO-II proposal, J-PARC E107.
\par\noindent\textbf{Relevance.} The neutral rare-kaon frontier: $K_L \to \pi^0\nu\bar\nu$ bounded below $2.2\times10^{-9}$, with the approved KOTO-II aiming at the $10^{-12}$ Standard-Model level. With NA62's charged mode it will close the cleanest loop tests in flavor physics (\S\ref{ssec:exp-kaon-era}) --- channels this paper's fixed inputs must match, per Objection~7's consistency clause. First cited in \S\ref{ssec:exp-kaon-era}.

\bibitem[Kolb and Turner(1990)]{KolbTurner}
\textbf{E.~W.~Kolb, M.~S.~Turner}, \emph{The Early Universe}
(Addison-Wesley, 1990).
\par\noindent\textbf{Relevance.} The standard text of early-universe physics: widths against Hubble, freeze-out, relic abundances, entropy bookkeeping --- the transport toolkit behind every equation of \S\ref{sec:transport} and Part~\ref{part:matter}, and the assigned shelf text of Appendix~\ref{app:leptogenesis}. The four household questions this paper uses to demystify transport are their curriculum, compressed. First cited in \S\ref{sec:transport}.

\bibitem[KamLAND-Zen Collaboration(2024)]{KamLANDZen2024}
\textbf{S.~Abe et al.\ (KamLAND-Zen)}, arXiv:2406.11438; Phys.\ Rev.\ Lett.\ (2025).
\par\noindent\textbf{Relevance.} The half-life frontier of $0\nu\beta\beta$: $T_{1/2}(^{136}\mathrm{Xe}) > 3.8\times10^{26}$~yr, translating through the nuclear-matrix-element band to $\mbb < 28$--$122$~meV --- the first reach into the inverted-ordering band, and silence exactly where a $1.7$~meV value demands silence (\S\ref{sec:exp-now}). It is also the anchor of the scaling arithmetic (problem H.5.2) that places the lattice point one to two orders beyond the ton scale. First cited in \S\ref{sec:exp-now}.

\bibitem[Kant(1768)]{Kant1768}
\textbf{I.~Kant}, \emph{Von dem ersten Grunde des Unterschiedes
der Gegenden im Raume} (1768).
\par\noindent\textbf{Relevance.} Incongruent counterparts: the left and right hands as a probe of whether space is relational or absolute --- the oldest text in this paper's philosophical lineage, and the first time handedness was recognized as metaphysically load-bearing. The question it opened is modernized twice here: by the Ozma problem, and by its resolution into the handedness invariant P-P5-D (\S\ref{sec:before-the-fall}). First cited in \S\ref{sec:before-the-fall}.

\bibitem[Kaluza(1921)]{Kaluza1921}
\textbf{T.~Kaluza}, Sitzungsber.\ Preuss.\ Akad.\ Wiss.\
Berlin, 966 (1921).
\par\noindent\textbf{Relevance.} Charge as geometry, first proposed --- and spatialized: the fifth dimension as a small place, with gauge symmetry as its isometry. The move gave every later extra-dimension program its radius, its tower, and eventually its landscape; AX-2 keeps the insight and refuses the spatialization --- the compact direction is real but not a place, which is why there is no radius modulus to stabilize and no tower to hide (\S\ref{sec:clauses}, \S\ref{ssec:source-circle}). First cited in \S\ref{sec:clauses}.

\bibitem[Klein(1926)]{Klein1926}
\textbf{O.~Klein}, Z.\ Phys.\ \textbf{37}, 895 (1926).
\par\noindent\textbf{Relevance.} Charge quantization as winding: momentum around the compact circle comes in integers, and electric charge is that integer --- the enduring half of the Kaluza--Klein insight, the half that survives de-spatialization intact. The charge circle of T-P5-A is this statement read as real geometry (\S\ref{ssec:source-circle}): winding number is charge, orientation is sign, and conjugation is going around the other way. First cited in \S\ref{sec:clauses}.

\bibitem[KATRIN Collaboration(2025)]{KATRIN2025}
\textbf{M.~Aker et al.\ (KATRIN)}, Science \textbf{388} (2025).
\par\noindent\textbf{Relevance.} The direct kinematic frontier: tritium endpoint spectroscopy at $m_\beta < 0.45$~eV, model-independent and laboratory-owned. It brackets the neutrino mass scale from above while cosmology presses from the sum (\S\ref{sec:exp-now}); the derived triplet sits far below its reach, so the bound is consistency, not confrontation --- the confrontations live with DESI and JUNO. First cited in \S\ref{sec:exp-now}.

\bibitem[Kawamura(2000)]{Kawamura2000}
\textbf{Y.~Kawamura}, Prog.\ Theor.\ Phys.\ \textbf{105}, 999
(2001).
\par\noindent\textbf{Relevance.} The orbifold GUT: $\mathrm{diag}(+,+,+,-,-)$ boundary conditions on an extra dimension, centralizer breaking of SU(5), doublet--triplet splitting by parity assignment --- Part~VI's mechanism, item for item. The projection matrix of \S\ref{sec:projection} credits it in full; the descent's addition is that each entry of the matrix is forced rather than chosen, which converts a model-building toolkit into a falsifiable claim. First cited in \S\ref{sec:two-structures}.

\bibitem[Kay(1978)]{Kay1978}
\textbf{B.~S.~Kay}, Commun.\ Math.\ Phys.\ \textbf{62}, 55
(1978).
\par\noindent\textbf{Relevance.} With \citet{AshtekarMagnon1975}, the uniqueness of the positive-energy complex structure: the theorem pair that carries the functional-analytic weight of the clock construction --- domains, closures, and the completion on which $(-A^2)^{-1/2}$ exists. T-P5-B stands on their care (\S\ref{sec:inherited-theorems}); the paper's contribution is the constitutional reading of what their selection clause means. First cited in \S\ref{sec:two-structures}.

\bibitem[Kobayashi and Maskawa(1973)]{KobayashiMaskawa1973}
\textbf{M.~Kobayashi, T.~Maskawa}, Prog.\ Theor.\ Phys.\
\textbf{49}, 652 (1973).
\par\noindent\textbf{Relevance.} One phase at three families: the counting theorem that predicted a third generation from CP violation alone, four years before the $\Upsilon$ --- the boldest and most successful parameter-counting argument in the history of the subject. Its graph form is Eq.~\eqref{eq:phase-count}, its phase is the one the pentagon values, and its counting is drilled from scratch in problem E.1.1 (\S\ref{sec:history-cp}). First cited in \S\ref{sec:history-cp}.

\bibitem[KTeV Collaboration(1999)]{KTeV1999}
\textbf{KTeV Collaboration (A.~Alavi-Harati et al.)}, Phys.\
Rev.\ Lett.\ \textbf{83}, 22 (1999).
\par\noindent\textbf{Relevance.} Direct CP violation established: $\mathrm{Re}(\varepsilon'/\varepsilon)$ measured decisively through the double-ratio technique, closing the thirty-five-year question the superweak hypothesis had kept open --- CP violation lives in decay amplitudes, not only in mixing. It is the author's own decade of shifts and systematics, and the amplitude-level fact any account of CP must carry (\S\ref{sec:history-precision}); the measurement's design logic is derived in Appendix~\ref{app:flavour-kaons}. First cited in \S\ref{sec:history-precision}.

\bibitem[Kuzmin, Rubakov, and
Shaposhnikov(1985)]{KuzminRubakovShaposhnikov1985}
\textbf{V.~A.~Kuzmin, V.~A.~Rubakov, M.~E.~Shaposhnikov},
Phys.\ Lett.\ B \textbf{155}, 36 (1985).
\par\noindent\textbf{Relevance.} The realization that changed the field's target: above the electroweak crossover the hot plasma walks freely over the sphaleron pass, erasing $B{+}L$ and preserving only $B{-}L$ --- so GUT-era asymmetries stored as $B{+}L$ die, and any surviving excess must be written in the sphaleron-proof currency. The 1985 lesson redirected baryogenesis to leptogenesis, and it is the reprocessing channel of Part~VIII (\S\ref{sec:transport}; taught in Appendix~\ref{app:lepto-sakharov}). First cited in \S\ref{sec:history-table}.

\bibitem[Landau(1957)]{Landau1957}
\textbf{L.~D.~Landau}, Nucl.\ Phys.\ \textbf{3}, 127 (1957).
\par\noindent\textbf{Relevance.} CP as the repaired mirror: with parity fallen, Landau proposed that the true mirror composes spatial reflection with charge conjugation --- and paired it with the two-component neutrino. For seven years the repair held, and its elegance is why 1964 was a shock; both the proposal and its fall are load-bearing history in \S\ref{sec:history-parity} and \S\ref{sec:history-cp}. First cited in \S\ref{sec:history-parity}.

\bibitem[Laporte(1924)]{Laporte1924}
\textbf{O.~Laporte}, Z.\ Phys.\ \textbf{23}, 135 (1924).
\par\noindent\textbf{Relevance.} The parity selection rule in atomic spectra: even and odd terms do not combine --- parity as pre-1956 common sense, working silently in every spectroscopic table three decades before anyone thought to ask whether the weak interaction obeyed it. The quiet era it represents is where \S\ref{sec:history-parity}'s story begins. First cited in \S\ref{sec:before-the-fall}.

\bibitem[Lee and Yang(1956)]{LeeYang1956}
\textbf{T.~D.~Lee, C.~N.~Yang}, Phys.\ Rev.\ \textbf{104}, 254
(1956).
\par\noindent\textbf{Relevance.} The audit that found the assumption: parity conservation, tested exquisitely in the strong and electromagnetic interactions, had never been tested in the weak --- and the paper proposed the experiments that would do it. The template of this paper's own register --- find the untested assumption, state the test, accept the verdict --- descends from this audit (\S\ref{sec:history-parity}). First cited in \S\ref{sec:history-parity}.

\bibitem[Lee and Yang(1957)]{LeeYang1957}
\textbf{T.~D.~Lee, C.~N.~Yang}, Phys.\ Rev.\ \textbf{105}, 1671
(1957).
\par\noindent\textbf{Relevance.} The two-component neutrino (with \citet{Landau1957} and \citet{Salam1957}): maximal parity violation not as a small dirt effect but as structure --- the neutrino exists in one handedness only. It is the first reading of the weak interaction as intrinsically chiral, the fact Part~VI's one-gate descent exists to derive (\S\ref{sec:history-parity}). First cited in \S\ref{sec:history-parity}.

\bibitem[Lawson and Michelsohn(1989)]{LawsonMichelsohn}
\textbf{H.~B.~Lawson, M.-L.~Michelsohn}, \emph{Spin Geometry}
(Princeton, 1989).
\par\noindent\textbf{Relevance.} The reference treatment of spin geometry: Clifford algebras, spinor bundles, and the reality classification by signature --- the eight-hour clock in its rigorous form. Lemma~\ref{lem:no-majorana} is a line read off their tables ($B\bar B = -\mathbf 1$ at $(t{-}s) \equiv 4$), and Appendix~\ref{app:anti-algebra} teaches the same material at drill depth (\S\ref{sec:spinor-algebra}). First cited in \S\ref{sec:spinor-algebra}.

\bibitem[LEGEND Collaboration(2025)]{LEGEND2025}
\textbf{LEGEND Collaboration}, arXiv:2505.10440 (2025).
\par\noindent\textbf{Relevance.} LEGEND-200 first result, combined with the GERDA and MAJORANA heritage: $T_{1/2} > 1.9\times10^{26}$~yr in Ge-76, the background-free technique scaled up and running. The bridge between the present generation and the ton scale (\S\ref{sec:exp-now}) --- the staircase on which this paper's registered null will be tested step by step. First cited in \S\ref{sec:exp-now}.

\bibitem[LEGEND-1000 Collaboration(2021)]{LEGEND1000}
\textbf{LEGEND Collaboration}, arXiv:2107.11462 (2021), LEGEND-1000 preconceptual design.
\par\noindent\textbf{Relevance.} The preferred ton-scale Ge-76 experiment: a tonne of enriched germanium, discovery sensitivity $1.3\times10^{28}$~yr, $\mbb$ reach 9--21~meV --- decisive for the inverted ordering, and the sharpest instrument of the one-sided Majorana test of \S\ref{sec:exp-future}. For this paper its verdict is asymmetric by design: silence is consistency, a line is a kill. First cited in \S\ref{sec:exp-future}.

\bibitem[LHCb Collaboration(2019)]{LHCbCharm2019}
\textbf{R.~Aaij et al.\ (LHCb)}, Phys.\ Rev.\ Lett.\ \textbf{122}, 211803 (2019).
\par\noindent\textbf{Relevance.} Discovery of CP violation in charm: $\Delta A_{CP} = (-15.4\pm2.9)\times10^{-4}$, the first CP asymmetry in up-type quarks --- the broken mirror reaching a new quark sector after fifty-five years. Its interpretation strains SM amplitude estimates without requiring new phases; this paper's custody statement (one valued phase, hadronic envelope delegated) is Objection~6 (\S\ref{ssec:exp-b-era}). First cited in \S\ref{ssec:exp-b-era}.

\bibitem[LHCb Collaboration(2025)]{LHCbBaryon2025}
\textbf{R.~Aaij et al.\ (LHCb)}, Nature \textbf{643} (2025), arXiv:2503.16954.
\par\noindent\textbf{Relevance.} First observation of CP violation in baryon decays ($\Lambda_b^0$, $5.2\sigma$): the broken mirror reaches the matter category we are made of --- baryons themselves now carry the asymmetry the mesons discovered. A completion of the experimental record of \S\ref{ssec:exp-b-era}, and a reminder that the one-phase paradigm keeps passing tests in new sectors. First cited in \S\ref{ssec:exp-b-era}.

\bibitem[L\"uders(1954)]{Luders1954}
\textbf{G.~L\"uders}, Dan.\ Mat.\ Fys.\ Medd.\ \textbf{28}, 5
(1954).
\par\noindent\textbf{Relevance.} With \citet{Pauli1955}, the first proofs of the CPT theorem: from Lorentz invariance, locality, and hermiticity, the product of the three reversals is a symmetry of any lawful field theory --- the theorem that survived the decade in which its factors fell one by one. The Lagrangian road of the theorem is drilled in Appendix~\ref{app:d-cpt}; its anatomy is this paper's Part~V (\S\ref{sec:history-cpt}). First cited in \S\ref{sec:history-cpt}.

\bibitem[Minkowski(1977)]{Minkowski1977}
\textbf{P.~Minkowski}, Phys.\ Lett.\ B \textbf{67}, 421 (1977).
\par\noindent\textbf{Relevance.} The seesaw's first paper, a decade ahead of its adoption: heavy neutral leptons make light neutrinos light, with the scale of the partners setting the smallness. Independently found three more times (\citealp{GellMannRamondSlansky}; \citealp{YanagidaSeesaw}; \citealp{MohapatraSenjanovic1980}), it is the mass mechanism of Part~VIII's chain, derived by block diagonalization in Appendix~\ref{app:nu-mass} (\S\ref{sec:history-neutrinos}). First cited in \S\ref{sec:history-neutrinos}.

\bibitem[Minkowski(1908)]{Minkowski1908}
\textbf{H.~Minkowski}, Phys.\ Z.\ \textbf{10}, 104 (1909).
\par\noindent\textbf{Relevance.} Time as the first non-spatial geometric dimension: the union of space and time into one real geometry, with a direction that is genuinely geometric yet not a place one walks. It is the precedent that makes ``a real dimension that is not a place'' respectable physics rather than paradox --- the exact rhetorical ancestor of the scale and phase directions of AX-2 (\S\ref{sec:clauses}). First cited in \S\ref{sec:clauses}.

\bibitem[Moe, Elliott, and Hahn(1987)]{MoeElliottHahn1987}
\textbf{S.~R.~Elliott, A.~A.~Hahn, M.~K.~Moe}, Phys.\ Rev.\ Lett.\ \textbf{59}, 2020 (1987).
\par\noindent\textbf{Relevance.} First direct observation of two-neutrino double beta decay (Se-82, in a time-projection chamber): the allowed second-order weak process made measurable --- the experimental foundation on which every $0\nu\beta\beta$ search stands, since the neutrinoless mode must be found against this irreducible cousin. The opening of the laboratory story told in \S\ref{sec:exp-now}. First cited in \S\ref{sec:exp-now}.

\bibitem[Mohapatra and Pati(1975)]{MohapatraPati1975}
\textbf{R.~N.~Mohapatra, J.~C.~Pati}, Phys.\ Rev.\ D \textbf{11},
566 (1975).
\par\noindent\textbf{Relevance.} With \citet{PatiSalam1974}, left--right symmetry: parity restored at high scales through $SU(2)_R$ and a $W_R$, breaking spontaneously on the way down --- the great structural alternative to intrinsic chirality. It is the alternative F-P5-1 is built to distinguish: the descent forbids the restoration sector at any scale, so a single $W_R$ kills Part~VI outright (\S\ref{sec:wr-falsifier}). First cited in \S\ref{sec:wr-falsifier}.

\bibitem[nEDM Collaboration(2020)]{nEDM2020}
\textbf{C.~Abel et al.}, Phys.\ Rev.\ Lett.\ \textbf{124},
081803 (2020).
\par\noindent\textbf{Relevance.} The neutron EDM bound at $|d_n| < 1.8\times10^{-26}\,e\,$cm: the current terminus of the seventy-year program Purcell and Ramsey opened, and the source of the $|\bar\theta| \lesssim 10^{-10}$ constraint that defines the strong-CP problem. In this paper's two-fate architecture it also bounds the leakage floor from above --- the EDM window between $e^{-K}$ and $10^{-10}$ is a named kill band (\S\ref{sec:history-cpt}). First cited in \S\ref{sec:history-cpt}.

\bibitem[Mohapatra and Senjanovi\'c(1980)]{MohapatraSenjanovic1980}
\textbf{R.~N.~Mohapatra, G.~Senjanovi\'c}, Phys.\ Rev.\ Lett.\
\textbf{44}, 912 (1980).
\par\noindent\textbf{Relevance.} The seesaw's fourth independent finding, in the left--right context: the same light-mass mechanism arrived at through parity restoration, tying neutrino mass to the scale of $SU(2)_R$ breaking. Cited with its three siblings at the mechanism's source (\S\ref{sec:seesaw}); the ledger's use of the mechanism runs through the dial-free door instead of the left--right sector F-P5-1 forbids. First cited in \S\ref{sec:seesaw}.

\bibitem[NA62 Collaboration(2025, 2026)]{NA622025}
\textbf{E.~Cortina Gil et al.\ (NA62)}, JHEP \textbf{02} (2025) 191; update arXiv:2604.12649 (2026).
\par\noindent\textbf{Relevance.} Observation of $K^+ \to \pi^+\nu\bar\nu$: the charged golden mode measured at $\mathrm{BR} = (9.6^{+1.9}_{-1.8})\times10^{-11}$, on the Standard-Model value --- the cleanest loop in flavor physics closing on prediction. For this paper the channel is a consistency check at fixed ledger inputs (Objection~7), and a standing spear against any framework needing new flavored phases (\S\ref{ssec:exp-kaon-era}). First cited in \S\ref{ssec:exp-kaon-era}.

\bibitem[nEXO Collaboration(2022)]{nEXO2022}
\textbf{G.~Adhikari et al.\ (nEXO)}, J.\ Phys.\ G \textbf{49}, 015104 (2022).
\par\noindent\textbf{Relevance.} Five tonnes of enriched Xe-136 in a single monolithic detector: projected sensitivity $1.35\times10^{28}$~yr, $\mbb$ reach 6--27~meV --- the liquid-xenon pillar of the ton-scale triad (\S\ref{sec:exp-future}). One of the three instruments whose collective silence this paper's $1.72$~meV predicts, and whose line would kill the door. First cited in \S\ref{sec:exp-future}.

\bibitem[Noether(1918)]{Noether1918}
\textbf{E.~Noether}, Nachr.\ Ges.\ Wiss.\ G\"ottingen, 235
(1918).
\par\noindent\textbf{Relevance.} Invariance and conservation united: every continuous symmetry a conserved current --- the theorem that taught physics what symmetry is worth, and gave the discrete inversions a prestige they had never earned by test. The unexamined confidence that broke in 1956 traces to the halo of this (correct, continuous) theorem (\S\ref{sec:before-the-fall}); the discrete lesson --- symmetries are empirical --- is this paper's opening move. First cited in \S\ref{sec:before-the-fall}.

\bibitem[NA48 Collaboration(2002)]{NA48_2002}
\textbf{NA48 Collaboration (J.~R.~Batley et al.)}, Phys.\ Lett.\
B \textbf{544}, 97 (2002).
\par\noindent\textbf{Relevance.} The concurrent establishment of direct CP violation at CERN: the European double-ratio program, agreeing with KTeV on $\mathrm{Re}(\varepsilon'/\varepsilon)$ --- two beams, two continents, one amplitude-level fact. The two-experiment closure of the superweak era (\S\ref{sec:history-precision}), and half of the world value the flavor chapter derives the machinery for (Appendix~\ref{app:flavour-kaons}). First cited in \S\ref{sec:history-precision}.

\bibitem[DONUT Collaboration(2001)]{DONUT2001}
\textbf{K.~Kodama et al.\ (DONUT)}, Phys.\ Lett.\ B \textbf{504}, 218 (2001).
\par\noindent\textbf{Relevance.} Direct observation of the tau neutrino: the third row of the lepton table completed by seeing $\nu_\tau$ interactions in emulsion --- the last Standard-Model fermion observed directly. It closes the roster whose three-fold structure carries the one surviving phase (\S\ref{sec:exp-nu-history}). First cited in \S\ref{sec:exp-nu-history}.

\bibitem[DUNE Collaboration(2020)]{DUNE2020}
\textbf{B.~Abi et al.\ (DUNE)}, Eur.\ Phys.\ J.\ C \textbf{80}, 978 (2020).
\par\noindent\textbf{Relevance.} The far-detector technical design report: seventy kilotons of liquid argon a mile underground, ordering at $5\sigma$, $\delta$ resolution of $10$--$20^{\circ}$ depending on the value, and the complementary $p \to \bar\nu K^+$ channel --- three of this Part's confrontation lines in one instrument (\S\ref{sec:exp-future}). Its matter-thick baseline is the consistency check that fake and genuine CP violation cannot both be faked (problem H.3.2). First cited in \S\ref{sec:exp-future}.

\bibitem[NuFit Collaboration(2024)]{NuFit2024}
\textbf{I.~Esteban et al.}, NuFit~6.0 global analysis (2024).
\par\noindent\textbf{Relevance.} The global oscillation fit, NuFit~6.0: the likelihood over all oscillation data whose angles, splittings, and phase band this paper confronts --- the lattice value sits $0.2\sigma$ inside its normal-ordering $\dPMNS$ band (\S\ref{ssec:pmns}). The refresh rules of Part~XI are written against exactly this fit's releases: pulls with dates, rerun each cycle. First cited in \S\ref{sec:history-neutrinos}.

\bibitem[O'Keefe and Nadel(1978)]{OKeefeNadel1978}
\textbf{J.~O'Keefe, L.~Nadel}, \emph{The Hippocampus as a
Cognitive Map} (Oxford, 1978).
\par\noindent\textbf{Relevance.} Place cells and the cognitive map: the founding demonstration that the mammalian brain builds an allocentric model of the world rather than storing egocentric snapshots --- space represented as space, in dedicated hardware. With grid cells it anchors the aside of \S\ref{ssec:real-bulk-mind}: model-building is what at least one physical system demonstrably does; the aside claims analogy, not mechanism. First cited in \S\ref{ssec:real-bulk-mind}.

\bibitem[OPERA Collaboration(2018)]{OPERA2018}
\textbf{N.~Agafonova et al.\ (OPERA)}, Phys.\ Rev.\ Lett.\ \textbf{120}, 211801 (2018).
\par\noindent\textbf{Relevance.} $\nu_\tau$ appearance in a $\nu_\mu$ beam at $6.1\sigma$: oscillations closed in appearance rather than disappearance, with the third-row transition seen event by event in emulsion. It completes the appearance record of the oscillation matrix (\S\ref{sec:exp-nu-history}) whose entries this paper's derived triplet feeds. First cited in \S\ref{sec:exp-nu-history}.

\bibitem[Osterwalder and Schrader(1973, 1975)]{OsterwalderSchrader}
\textbf{K.~Osterwalder, R.~Schrader}, Commun.\ Math.\ Phys.\
\textbf{31}, 83 (1973); \textbf{42}, 281 (1975).
\par\noindent\textbf{Relevance.} Reflection positivity and the
Euclidean reconstruction: the rigorous content of ``Wick
rotation,'' equivalent by T-P5-B1 to the clock's polar
decomposition --- clause (3) of the quartet, and the license
behind every Euclidean computation the boundary performs
(\S\ref{sec:wick-quartet}). First cited in \S\ref{sec:wick-quartet}.

\bibitem[Mohapatra and Pal(2004)]{MohapatraPal2004}
\textbf{R.~N.~Mohapatra, P.~B.~Pal}, \emph{Massive Neutrinos in
Physics and Astrophysics}, 3rd ed.\ (World Scientific, 2004).
\par\noindent\textbf{Relevance.} The seesaw's model space at book length: every variant, every texture, every phenomenological consequence of massive neutrinos, in the field's standard monograph. \S\ref{app:nu-mass}'s block diagonalization is their Chapter~6 done slowly, and its Majorana-versus-Dirac bookkeeping follows their conventions; the model space they map is exactly the freedom the dial-free door closes. First cited in Appendix~\ref{app:neutrino}.

\bibitem[Page and Wootters(1983)]{PageWootters1983}
\textbf{D.~N.~Page, W.~K.~Wootters}, Phys.\ Rev.\ D
\textbf{27}, 2885 (1983).
\par\noindent\textbf{Relevance.} Relational quantum time:
evolution read from entanglement with a clock subsystem, so
that a globally stationary state contains internal change ---
the honest quantum-mechanical attempt to bring time inside the
Hilbert space. Engaged in \S\ref{sec:time-comparison} as a
cousin whose circularity (a clock must be chosen) the
constitutional clock dissolves by fiat of the arena. First
cited in \S\ref{sec:time-standard}.

\bibitem[Pati and Salam(1974)]{PatiSalam1974}
\textbf{J.~C.~Pati, A.~Salam}, Phys.\ Rev.\ D \textbf{10}, 275
(1974).
\par\noindent\textbf{Relevance.} With \citet{MohapatraPati1975}, the left--right program's foundation: quark--lepton unification with parity restored at high energy --- the alternative reading of the weak interaction's handedness in which chirality is environmental, not intrinsic. The distinction is precisely what F-P5-1 makes experimental: any $W_R$ at any scale decides for restoration and against the descent (\S\ref{sec:wr-falsifier}). First cited in \S\ref{sec:wr-falsifier}.

\bibitem[Pauli(1940)]{Pauli1940}
\textbf{W.~Pauli}, Phys.\ Rev.\ \textbf{58}, 716 (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:quartet-second-child} as the first child's sibling ---
the theorem whose analytic machinery T-P5-B1 geometrizes and
whose \GG{} restatement is the quartet's second child. First cited in \S\ref{sec:quartet-second-child}.

\bibitem[Pauli(1955)]{Pauli1955}
\textbf{W.~Pauli}, in \emph{Niels Bohr and the Development of
Physics} (Pergamon, 1955).
\par\noindent\textbf{Relevance.} With \citet{Luders1954}, the CPT theorem's first proofs --- Pauli's contribution framing the reflection as the surviving discrete symmetry of any Lorentz-invariant local theory. The theorem the whole subject leans on, drilled by both roads (Lagrangian and axiomatic) in Appendix~\ref{app:d-cpt}, and given its anatomy in Part~V (\S\ref{sec:history-cpt}). First cited in \S\ref{sec:history-cpt}.

\bibitem[Particle Data Group(2024)]{PDG2024}
\textbf{Particle Data Group (S.~Navas et al.)}, Phys.\ Rev.\ D
\textbf{110}, 030001 (2024).
\par\noindent\textbf{Relevance.} The living compilation: world averages for $\dCKM$, $\JCKM$, the CPT identities, the $\bar\theta$ bounds, and the $W_R$ limits --- the numbers this paper's tables confront, in the edition dated at each use. The refresh discipline of \S\ref{sec:exp-refresh} instructs the reader to recompute every pull against each new edition: the PDG is this paper's standing counterparty. First cited in \S\ref{sec:history-cpt}.

\bibitem[Peccei and Quinn(1977)]{PecceiQuinn1977}
\textbf{R.~D.~Peccei, H.~R.~Quinn}, Phys.\ Rev.\ Lett.\
\textbf{38}, 1440 (1977).
\par\noindent\textbf{Relevance.} The relaxation template: promote $\bar\theta$ to a field and let dynamics roll it to the CP-conserving minimum --- the mechanism class of every axion solution since. The Kaxion executes the template without a PQ scalar, inheriting the roll from the compact direction (\S\ref{sec:strongcp}; \citealp{ArisakaC1Kaxion}), with the leakage floor as the falsifiable residue the original mechanism does not carry. First cited in \S\ref{sec:strongcp}.

\bibitem[Purcell and Ramsey(1950)]{PurcellRamsey1950}
\textbf{E.~M.~Purcell, N.~F.~Ramsey}, Phys.\ Rev.\ \textbf{78},
807 (1950).
\par\noindent\textbf{Relevance.} The lone early doubt, six years before Lee and Yang: parity conservation is ``a purely experimental matter,'' testable through the neutron EDM --- the founding document of the EDM program and of the experimental attitude this paper's register institutionalizes. Seventy years and twelve orders of magnitude later, the program they opened is still the sharpest static probe of T violation (\S\ref{sec:before-the-fall}, \S\ref{sec:history-cpt}). First cited in \S\ref{sec:before-the-fall}.

\bibitem[Peskin and Schroeder(1995)]{PeskinSchroeder1995}
\textbf{M.~E.~Peskin, D.~V.~Schroeder}, \emph{An Introduction to
Quantum Field Theory} (Westview, 1995).
\par\noindent\textbf{Relevance.} The standard graduate QFT text: the propagator machinery, the loop toolbox, and the discrete-symmetry treatment with its bilinear table --- the assigned check of Appendix~\ref{app:discrete-qft} \S\ref{app:d-dirac}, and the source for the GIM loop analysis of Appendix~\ref{app:flavour} and the contour drills of Appendix~\ref{app:anti-contour}. First cited in \S\ref{app:discrete-qft}.

\bibitem[Pilaftsis(1997)]{Pilaftsis1997}
\textbf{A.~Pilaftsis}, Phys.\ Rev.\ D \textbf{56}, 5431 (1997).
\par\noindent\textbf{Relevance.} Resonant CP asymmetries from quasi-degenerate heavy neutrinos: when the mass splitting approaches the width, the self-energy contribution is resonantly enhanced and the Yukawa suppression cancels --- the enhancement Part~VIII's step~4 invokes, credited at source (\S\ref{sec:transport}). The Breit--Wigner structure is derived and drilled in Appendix~\ref{app:lepto-epsilon} (problem F.3.2), including the twelve-order off-peak cliff the chain must face. First cited in \S\ref{sec:transport}.

\bibitem[Pilaftsis and Underwood(2004)]{PilaftsisUnderwood2004}
\textbf{A.~Pilaftsis, T.~E.~J.~Underwood}, Nucl.\ Phys.\ B
\textbf{692}, 303 (2004).
\par\noindent\textbf{Relevance.} Resonant leptogenesis systematized at the TeV scale: the regime of this paper's entire chain, worked out in the literature that engineered the degeneracy by hand --- the honest contrast being that the ledger's $O(1/K)$ splittings must ultimately do what their model-building chose to do. That confrontation is faced in public in \S\ref{sec:resonance-faced}, where C-P5-C's discharge criteria are stated. First cited in \S\ref{sec:history-table}.

\bibitem[Planck Collaboration(2020)]{Planck2018}
\textbf{Planck Collaboration (N.~Aghanim et al.)}, Astron.\
Astrophys.\ \textbf{641}, A6 (2020).
\par\noindent\textbf{Relevance.} The six-parameter fit of the microwave sky: $\Omega_b h^2$ to $0.3\%$, hence the photon-baryon ratio at the percent level --- the CMB leg of the two-epoch $\etaB$ concordance and the target number \eqref{eq:etaB-obs} of Part~VIII's chain. Also the source of the mass-sum pressure (with DESI) that Part~XV registers as a new kill channel. First cited in \S\ref{sec:history-stake}.

\bibitem[Polchinski(1998)]{Polchinski1998}
\textbf{J.~Polchinski}, \emph{String Theory}, Vol.~I (Cambridge,
1998).
\par\noindent\textbf{Relevance.} The standard text of string theory, cited here for one structural point: Chan--Paton factors and worldsheet orientation reversal realize charge conjugation as geometry --- the rival program's own version of ``conjugation is an orientation,'' conceded before the valuation contrast is drawn (\S\ref{sec:road-string}). The kinship is real; the difference, as everywhere in Part~X, is who prices the values. First cited in \S\ref{ssec:string-achieved}.

\bibitem[Reines and Cowan(1956)]{ReinesCowan1956}
\textbf{C.~L.~Cowan, F.~Reines, F.~B.~Harrison, H.~W.~Kruse, A.~D.~McGuire}, Science \textbf{124}, 103 (1956).
\par\noindent\textbf{Relevance.} First detection of the (anti)neutrino: reactor antineutrinos captured on protons, the particle Pauli apologized for made real after twenty-six years --- the founding of experimental neutrino physics, and the start of the lineage (\S\ref{sec:exp-nu-history}) that runs through Homestake and Super-Kamiokande to the instruments that will test this paper. First cited in \S\ref{sec:exp-nu-history}.

\bibitem[Renou et al.(2021)]{RenouEtAl2021}
\textbf{M.-O.~Renou et al.}, Nature \textbf{600}, 625 (2021).
\par\noindent\textbf{Relevance.} The real-Hilbert-space exclusion: network Bell experiments whose correlations no real-amplitude quantum theory can reproduce --- complex quantum mechanics confirmed as such, experimentally. For this paper the result is load-bearing in both directions: it is the configuration the shared $J$ of T-P4-4/T-P5-C1 supplies (\S\ref{sec:inherited-theorems}), and it is the boundary fact that disciplines the real-bulk custody (Objection~80). First cited in \S\ref{sec:two-structures}.

\bibitem[Roussy et al.(2023)]{Roussy2023}
\textbf{T.~S.~Roussy et al.}, Science \textbf{381}, 46 (2023).
\par\noindent\textbf{Relevance.} The electron EDM below $4.1\times10^{-30}\,e\,$cm, from trapped molecular ions: the deepest static T-odd null in any system, and the template for the coming decade's ion-trap program. In this paper's ledger it disciplines every proposed CP phase and pins the upper edge of the strong-CP kill window (\S\ref{sec:exp-now}). First cited in \S\ref{ssec:exp-b-era}.

\bibitem[Rauch \emph{et~al.}(1975)]{Rauch1975}
\textbf{H.~Rauch, A.~Zeilinger, G.~Badurek, A.~Wilfing,
W.~Bauspiess, U.~Bonse}, Phys.\ Lett.\ A \textbf{54}, 425 (1975).
\par\noindent\textbf{Relevance.} The neutron-interferometric
observation that a $2\pi$ spin rotation reverses the sign of the
amplitude and $4\pi$ restores it --- the measured floor under the
projective phases of Appendix~\ref{app:discrete-qft} \S\ref{app:d-wigner} and
the laboratory face of the frame-path memory of
\S\ref{sec:quartet-second-child}. First cited in
\S\ref{app:d-wigner}.

\bibitem[Sakurai and Napolitano(2017)]{Sakurai2017}
\textbf{J.~J.~Sakurai, J.~Napolitano}, \emph{Modern Quantum
Mechanics}, 2nd ed.\ (Cambridge University Press, 2017).
\par\noindent\textbf{Relevance.} The standard first-year graduate quantum text: its Chapter~4 treatment of symmetry --- parity, lattice translation, time reversal, Kramers --- is the floor on which \S\ref{app:d-wigner} builds, and the assigned parallel reading of Appendix~\ref{app:discrete-qft}. The antiunitarity discussion this paper geometrizes is taught there in its standard form. First cited in \S\ref{app:discrete-qft}.

\bibitem[Sakharov(1967)]{Sakharov1967}
\textbf{A.~D.~Sakharov}, JETP Lett.\ \textbf{5}, 24 (1967).
\par\noindent\textbf{Relevance.} The three conditions, stated in four pages a year after the CMB's discovery: baryon violation, C and CP violation, departure from equilibrium --- the scoreboard of every baryogenesis mechanism since, and of Part~VIII, where each condition is mapped to a constitutional pillar (\S\ref{sec:sakharov-custody}). Each condition is derived as a theorem in Appendix~\ref{app:lepto-sakharov}; his companion twin-universe speculation is engaged in Part~XV. First cited in \S\ref{sec:history-cp}.

\bibitem[Salam(1957)]{Salam1957}
\textbf{A.~Salam}, Nuovo Cim.\ \textbf{5}, 299 (1957).
\par\noindent\textbf{Relevance.} With \citet{LeeYang1957} and \citet{Landau1957}, the two-component neutrino: maximal parity violation as structure rather than contamination --- Salam's version arriving through the gauge-theoretic instinct that would later build the electroweak theory. The chiral neutrino the trio proposed is the boundary fact Part~VI's one-gate descent derives (\S\ref{sec:history-parity}). First cited in \S\ref{sec:history-parity}.

\bibitem[SNEWS Collaboration(2021)]{SNEWS2021}
\textbf{S.~Al~Kharusi et al.\ (SNEWS 2.0)}, New J.\ Phys.\ \textbf{23}, 031201 (2021).
\par\noindent\textbf{Relevance.} The supernova early-warning network, version 2.0: the world's neutrino detectors wired into one alarm, so the next galactic collapse is caught from the first second. It is the standing watch for the sector-wide calibration event (\S\ref{sec:exp-future}) --- the event whose burst, per Objection~15's response, would read the mass ordering upstream of every laboratory program. First cited in \S\ref{sec:exp-future}.

\bibitem[SNO Collaboration(2002)]{SNO2002}
\textbf{SNO Collaboration (Q.~R.~Ahmad et al.)}, Phys.\ Rev.\
Lett.\ \textbf{89}, 011301 (2002).
\par\noindent\textbf{Relevance.} The solar case closed: heavy water separating charged-current from neutral-current rates, showing the total solar flux intact while the electron flavor is depleted --- the neutrinos change flavor rather than vanish. With Super-Kamiokande it earned the oscillation Nobel and opened the era in which the mixing matrix became precision physics (\S\ref{sec:history-neutrinos}). First cited in \S\ref{sec:history-neutrinos}.

\bibitem[Sonoda(1989)]{Sonoda1989}
\textbf{H.~Sonoda}, Nucl.\ Phys.\ B \textbf{326}, 135 (1989).
\par\noindent\textbf{Relevance.} Hermiticity and CPT of string amplitudes: the worldsheet reflection transferring to the spacetime S-matrix --- string theory's CPT inheritance made explicit at the amplitude level. Cited in \S\ref{sec:road-string} as the rival program's own proof that the unbroken theorem survives its framework; the contrast, as always, is anatomy and valuation, not validity. First cited in \S\ref{ssec:string-achieved}.

\bibitem[St\"uckelberg(1941)]{Stueckelberg1941}
\textbf{E.~C.~G.~St\"uckelberg}, Helv.\ Phys.\ Acta
\textbf{14}, 588 (1941).
\par\noindent\textbf{Relevance.} The reinterpretation, first: antiparticles as reversed-flow solutions, eight years before Feynman's contours gave the same reading its computational form --- the pairing that carries both names. What neither author could say is why that assignment and no other is consistent; that provenance is exactly what T-P5-C supplies (\S\ref{ssec:qft-antiparticles}), with the contour version derived in Appendix~\ref{app:anti-contour}. First cited in \S\ref{ssec:qft-antiparticles}.

\bibitem[Strominger and Witten(1985)]{StromingerWitten1985}
\textbf{A.~Strominger, E.~Witten}, Commun.\ Math.\ Phys.\
\textbf{101}, 341 (1985).
\par\noindent\textbf{Relevance.} CP in Calabi--Yau vacua: the opening of the line that made CP a geometric, spontaneously broken symmetry of string theory --- the intellectual ancestry of every geometric-CP position, this paper's included. Credited at the head of the lineage in \S\ref{sec:road-string} before the valuation contrast: geometry breaking CP is their result; geometry pricing the breakage is the claim under test here. First cited in \S\ref{sec:pentagon-claims}.

\bibitem[Sozzi(2008)]{Sozzi2008}
\textbf{M.~S.~Sozzi}, \emph{Discrete Symmetries and CP Violation:
From Experiment to Theory} (Oxford University Press, 2008).
\par\noindent\textbf{Relevance.} The operation-centric companion
to \citet{BigiSanda2009}: C, P, and T defined, measured, and
tested system by system, with the experimental techniques given
their full weight --- the closest book-length analogue of this
paper's Parts~V and~X. Shelf text of
Appendix~\ref{app:neutrino}. First cited in
Appendix~\ref{app:neutrino}.

\bibitem[Streater and Wightman(1964)]{StreaterWightman}
\textbf{R.~F.~Streater, A.~S.~Wightman}, \emph{PCT, Spin and
Statistics, and All That} (Benjamin, 1964).
\par\noindent\textbf{Relevance.} The axiomatic frame: Wightman fields, the spin-statistics and PCT theorems from first principles --- the rigorous scaffolding whose per-field phase conventions T-P5-1 replaces with anatomy (\S\ref{sec:table}, Appendix~\ref{app:d-cpt}). This paper's concessions to the axiomatic program --- interacting reach, above all --- are stated wherever its theorems are engaged. First cited in \S\ref{sec:history-cpt}.

\bibitem[Eddington(1928)]{Eddington1928}
\textbf{A.~S.~Eddington}, \emph{The Nature of the Physical
World} (Cambridge, 1928).
\par\noindent\textbf{Relevance.} ``Time's arrow'' coined: the
one physical asymmetry pointing along time located in entropy
increase, with full clarity that the microscopic laws know
nothing of it. The phrase and the problem it names are the
subject of \S\ref{sec:time-arrow}; the \OGGEP{} reading gives
his arrow the law-level home he correctly said the dynamics of
his day could not provide. First cited in
\S\ref{sec:time-standard}.

\bibitem[Sudarshan and Marshak(1958)]{SudarshanMarshak1958}
\textbf{E.~C.~G.~Sudarshan, R.~E.~Marshak}, Phys.\ Rev.\
\textbf{109}, 1860 (1958).
\par\noindent\textbf{Relevance.} $V{-}A$, independently and arguably first: the chiral structure of the weak current identified from the data's contradictions --- the companion discovery to \citet{FeynmanGellMann1958}, credited jointly as the codification of maximal parity violation (\S\ref{sec:history-parity}). The structure both papers wrote down is Part~VI's explanandum. First cited in \S\ref{sec:history-parity}.

\bibitem[Super-Kamiokande Collaboration(1998)]{SuperK1998}
\textbf{Super-Kamiokande Collaboration (Y.~Fukuda et al.)},
Phys.\ Rev.\ Lett.\ \textbf{81}, 1562 (1998).
\par\noindent\textbf{Relevance.} Atmospheric neutrino oscillations: the zenith-angle deficit that established neutrino mass and mixing --- the third arena opened decisively, by the lineage of water-Cherenkov detectors the author helped found. The 1998 discovery is both history (\S\ref{sec:history-neutrinos}) and biography (\S\ref{sec:personal}) in this paper: the instrument line runs from Kamiokande through Super-K to the Hyper-K that will test the lattice point. First cited in \S\ref{sec:history-neutrinos}.

\bibitem[Super-Kamiokande Collaboration(2020)]{SuperKPdecay2020}
\textbf{A.~Takenaka et al.\ (Super-Kamiokande)}, Phys.\ Rev.\ D \textbf{102}, 112011 (2020); $\eta$ channels: Phys.\ Rev.\ D \textbf{110}, 112011 (2024); $\pi^0\pi^0$: arXiv:2604.10975 (2026).
\par\noindent\textbf{Relevance.} The proton-decay ledger: $\tau/B(p \to e^+\pi^0) > 2.4\times10^{34}$~yr and channel-by-channel silence across every mode --- the standing annual audit of the exact $\Uone{B}$ claim. Every year of silence is a passed test of the ledger's most absolute prediction; a single candidate event surviving scrutiny would kill Part~VIII's logic outright (\S\ref{sec:exp-now}). First cited in \S\ref{sec:exp-now}.

\bibitem[Super-Kamiokande Collaboration(2021)]{SuperKnnbar2021}
\textbf{K.~Abe et al.\ (Super-Kamiokande)}, Phys.\ Rev.\ D \textbf{103}, 012008 (2021).
\par\noindent\textbf{Relevance.} The bound-neutron $n\bar n$ oscillation search: free oscillation time above $4.7\times10^{8}$~s --- the $\Delta B = 2$ channel closed to current sensitivity. It disciplines the same ledger from a second direction: baryon number is not merely conserved in decays but protected against coherent oscillation (\S\ref{sec:exp-now}). First cited in \S\ref{sec:exp-now}.

\bibitem[T2K Collaboration(2020)]{T2K2020}
\textbf{T2K Collaboration (K.~Abe et al.)}, Nature \textbf{580},
339 (2020).
\par\noindent\textbf{Relevance.} The first constraint disfavoring leptonic CP conservation: T2K's appearance asymmetry excluding $\delta = 0, \pi$ at $2\sigma$-class --- the early shadow of the measurement that will adjudicate \eqref{eq:deltapmns}. The instrument sits at its oscillation maximum by design (the arithmetic is problem H.1.1), and its successor owns the quiet signature's verdict (\S\ref{sec:history-neutrinos}). First cited in \S\ref{sec:history-neutrinos}.

\bibitem['t Hooft(1979)]{tHooft1979}
\textbf{G.~'t~Hooft}, in \emph{Recent Developments in Gauge
Theories} (Plenum, 1980).
\par\noindent\textbf{Relevance.} Naturalness and chiral protection: fermion masses are protected by chiral symmetry, so light fermions are natural where light scalars are not --- QFT's partial answer to ``why chiral,'' credited beside the basin/gate architecture it parallels (\S\ref{sec:parity-question}). The survival argument explains why chirality persists; Part~VI's claim is about why it was there to survive. First cited in \S\ref{sec:parity-question}.

\bibitem['t Hooft(1976)]{tHooft1976}
\textbf{G.~'t~Hooft}, Phys.\ Rev.\ Lett.\ \textbf{37}, 8
(1976).
\par\noindent\textbf{Relevance.} Baryon-plus-lepton number anomalously violated in the Standard Model itself: the vacuum's winding structure ties fermion number to topology, at a rate absurdly small cold and --- after the sphaleron insight --- unsuppressed hot. The foundation of the reprocessing channel of \S\ref{sec:transport}, and the reason the matter excess must be written in $B{-}L$ (Appendix~\ref{app:lepto-sakharov}). First cited in \S\ref{sec:transport}.

\bibitem[T2K and NO$\nu$A Collaborations(2025)]{T2KNOvA2025}
\textbf{T2K and NO$\nu$A Collaborations}, Nature \textbf{646}, 818 (2025).
\par\noindent\textbf{Relevance.} The first joint long-baseline fit: correlated systematics unified across the two beams, $|\Delta m^2_{32}|$ below two percent, CP conservation excluded at $3\sigma$ under the inverted ordering --- the freshest constraint the lattice value faces, and a rehearsal of the joint-fit methodology the Hyper-K/DUNE era will need (\S\ref{sec:exp-now}). The refresh rules print its pulls with dates. First cited in \S\ref{sec:exp-now}.

\bibitem[Vafa and Witten(1984)]{VafaWitten1984}
\textbf{C.~Vafa, E.~Witten}, Phys.\ Rev.\ Lett.\ \textbf{53}, 535
(1984).
\par\noindent\textbf{Relevance.} Two theorems in one citation: no spontaneous parity breaking in vector-like theories, and the vacuum energy minimized at $\bar\theta_{\mathrm{eff}} = 0$ --- QCD's parity protection and the certified minimum the Kaxion rolls toward. Both are load-bearing: the first defines the boundary between QCD's safety and the chiral electroweak sector's fall (\S\ref{sec:parity-question}); the second certifies the target of the relaxation (\S\ref{sec:strongcp}; Objection~30's exchange). First cited in \S\ref{sec:history-parity}.

\bibitem[Weinberg(1979)]{Weinberg1979}
\textbf{S.~Weinberg}, Phys.\ Rev.\ Lett.\ \textbf{42}, 850
(1979).
\par\noindent\textbf{Relevance.} With \citet{Yoshimura1978}, GUT baryogenesis: the matter excess from heavy boson decays at the unification scale --- the first complete implementations of Sakharov's program, and the route the sphaleron era later erased and the exact $\Uone{B}$ ledger closes twice (\S\ref{sec:why-leptogenesis}). The toolbox entry whose fate taught the field that mechanisms must survive reprocessing. First cited in \S\ref{sec:why-leptogenesis}.

\bibitem[Weyl(1918)]{Weyl1918}
\textbf{H.~Weyl}, Sitzungsber.\ Preuss.\ Akad.\ Wiss.\
Berlin, 465 (1918).
\par\noindent\textbf{Relevance.} Local scale as geometry: the first gauge theory, wrong about electromagnetism and right about everything methodological --- symmetry made local, geometry made dynamical. It is the ur-precedent of the \GG{} scale dimension (\S\ref{sec:clauses}): the descent's radial direction is Weyl's instinct, executed on the dimension he originally intended. First cited in \S\ref{sec:clauses}.

\bibitem[Weyl(1929)]{Weyl1929}
\textbf{H.~Weyl}, Z.\ Phys.\ \textbf{56}, 330 (1929).
\par\noindent\textbf{Relevance.} Local phase as geometry: the corrected gauge principle, now on the quantum phase --- the birth of gauge theory as physics uses it, and the second Weyl insight AX-2 elevates from principle to dimension. The compact phase circle of T-P5-A is this paper's reading of what the 1929 symmetry was always about (\S\ref{sec:clauses}). First cited in \S\ref{sec:clauses}.

\bibitem[Wigner(1927)]{Wigner1927}
\textbf{E.~P.~Wigner}, Z.\ Phys.\ \textbf{43}, 624 (1927).
\par\noindent\textbf{Relevance.} Parity as a quantum number: the mirror promoted from geometric curiosity to formal law of atomic physics, with selection rules attached --- the moment reflection entered the bookkeeping of quantum mechanics. Three decades of silent service followed, and then the fall this paper's history opens with (\S\ref{sec:before-the-fall}). First cited in \S\ref{sec:before-the-fall}.

\bibitem[Wolfenstein(1964)]{Wolfenstein1964}
\textbf{L.~Wolfenstein}, Phys.\ Rev.\ Lett.\ \textbf{13}, 562
(1964).
\par\noindent\textbf{Relevance.} The superweak hypothesis: all CP violation confined to a new $\Delta S = 2$ interaction, so $\varepsilon' = 0$ identically --- the sharp, falsifiable alternative that gave the $\varepsilon'/\varepsilon$ program its purpose and its kill criterion. Thirty-five years later the double-ratio experiments delivered the verdict (\S\ref{sec:history-cp}); the episode is this paper's model of a clean two-framework discrimination (problem E.3.3). First cited in \S\ref{sec:history-cp}.

\bibitem[Wolfenstein(1983)]{Wolfenstein1983}
\textbf{L.~Wolfenstein}, Phys.\ Rev.\ Lett.\ \textbf{51}, 1945
(1983).
\par\noindent\textbf{Relevance.} The hierarchical expansion of the
CKM matrix in powers of $\lambda = \sin\theta_C$, derived as
problem E.1.2 of Appendix~\ref{app:flavour}; the display of the
$1:\lambda:\lambda^2:\lambda^3$ pattern that Step~2 lists among
the quantities the Standard Model cannot price. First cited in
\S\ref{app:flavour-ckm}.

\bibitem[Weinberg(1995)]{WeinbergQTF}
\textbf{S.~Weinberg}, \emph{The Quantum Theory of Fields},
Vol.~I, ch.~5 (Cambridge, 1995).
\par\noindent\textbf{Relevance.} The causality argument ---
antiparticles forced by SR $+$ QM $+$ microcausality $+$ bounded
energy --- stated in this paper as
Theorem~\ref{thm:weinberg}: the orthodoxy's deepest Depth-2
answer, whose two undischarged axioms the handshake discharges
(\S\ref{ssec:qft-antiparticles}, Appendix~\ref{app:anti-weinberg}). First cited in \S\ref{sec:orientation-reversals}.

\bibitem[Zuber(2020)]{Zuber2020}
\textbf{K.~Zuber}, \emph{Neutrino Physics}, 3rd ed.\ (CRC Press,
2020).
\par\noindent\textbf{Relevance.} The experimentalist's companion
volume: sources, detectors, backgrounds, and the double-beta
chapter (Ch.~7) whose nuclear-matrix-element discussion
\S\ref{app:nu-mass} drills; the natural first book for the
incoming collaboration student Appendix~\ref{app:neutrino}
addresses. First cited in Appendix~\ref{app:neutrino}.

\bibitem[Wigner(1931)]{Wigner1931}
\textbf{E.~P.~Wigner}, \emph{Gruppentheorie und ihre Anwendung}
(Vieweg, 1931).
\par\noindent\textbf{Relevance.} The (anti)unitary fork: every quantum symmetry is implemented unitarily or antiunitarily --- the theorem that gives time reversal its strange character by assignment. T-P5-1 converts the fork from case analysis to computation: which operations get which character is read off the orientation content of the texture (\S\ref{sec:orientation-reversals}, Appendix~\ref{app:d-wigner}). First cited in \S\ref{sec:orientation-reversals}.

\bibitem[Wu et al.(1957)]{Wu1957}
\textbf{C.~S.~Wu, E.~Ambler, R.~W.~Hayward, D.~D.~Hoppes,
R.~P.~Hudson}, Phys.\ Rev.\ \textbf{105}, 1413 (1957).
\par\noindent\textbf{Relevance.} The fall of parity: polarized cobalt-60, an asymmetry of order unity, and a symmetry of nature gone in one cryostat --- the experiment this paper's history opens with, and the template of every orientation measurement since. Appendix~\ref{app:d-lab} computes it end to end, and Part~VI's worked example re-reads it as the first direct observation of the descent's handedness. First cited in \S\ref{sec:history-parity}.

\bibitem[McTaggart(1908)]{McTaggart1908}
\textbf{J.~M.~E.~McTaggart}, Mind \textbf{17}, 457 (1908).
\par\noindent\textbf{Relevance.} The unreality of time: the
tensed, flowing A-series is contradictory; only the ordered
B-series survives --- analytic philosophy's sharpest result on
time, and an uncanny anticipation of the block universe.
Part~\ref{part:time} agrees with the theorem and relocates the
flow: what McTaggart proved cannot belong to time's order
belongs instead to the dictionary's one-way writing
(\S\ref{sec:time-for-everyone}). First cited in
\S\ref{sec:time-standard}.

\bibitem[Yanagida(1979)]{YanagidaSeesaw}
\textbf{T.~Yanagida}, in \emph{Proc.\ Workshop on Unified Theory
and Baryon Number in the Universe} (KEK, 1979).
\par\noindent\textbf{Relevance.} With \citet{GellMannRamondSlansky} and \citet{Minkowski1977}, the seesaw's source papers --- Yanagida's version presented at the 1979 workshop that gave the mechanism its community. The mechanism is taught by hand in Appendix~\ref{app:nu-mass}; the ledger fixes the one number all the source papers left free, the scale (\S\ref{sec:history-neutrinos}). First cited in \S\ref{sec:history-neutrinos}.

\bibitem[Yoshimura(1978)]{Yoshimura1978}
\textbf{M.~Yoshimura}, Phys.\ Rev.\ Lett.\ \textbf{41}, 281
(1978).
\par\noindent\textbf{Relevance.} With \citet{Weinberg1979}, the first concrete baryogenesis calculations: the observed asymmetry from GUT-scale boson decays --- Sakharov's program made quantitative for the first time. The route's later erasure by sphaleron reprocessing (\S\ref{sec:why-leptogenesis}) is one of the field's formative lessons, inherited here as the reason the excess must be minted in $B{-}L$.

\end{thebibliography}

\end{document} First cited in \S\ref{sec:why-leptogenesis}.
